Russell Markert's original choreography makes abstraction mesmerizingly beautiful as it puts the Euclidean space in motion.
|
|
It remains to be seen whether their methods can be applied to ordinary Euclidean space, like the space we live in.
|
|
The LP's cerebral title references a technical term used to describe a standard way of "assigning a measure to subsets of Euclidean space" (thanks Wikipedia!).
|
|
And it is why Marine Serre's multidimensional Divine Comedy, a narrative in fashion form that imagined a "a futuristic wormhole … way beyond Euclidean space" (excerpts from the short story that was the show notes) and then provided the clothes for it, was so compelling.
|
|
Step forward Fabula, which has patented what it dubs a "new class" of machine learning algorithms to detect "fake news" — in the emergent field of "Geometric Deep Learning"; where the datasets to be studied are so large and complex that traditional machine learning techniques struggle to find purchase on this 'non-Euclidean' space.
|
|
Similar ideas occur in category theory: the category of Euclidean spaces is a concrete category over the category of topological spaces; the forgetful (or "stripping") functor maps the former category to the latter category. A three-dimensional Euclidean space is a special case of a Euclidean space. In Bourbaki's terms, the species of three- dimensional Euclidean space is richer than the species of Euclidean space. Likewise, the species of compact topological space is richer than the species of topological space. Fig.
|
|
The Hausdorff property is not a local one; so even though Euclidean space is Hausdorff, a locally Euclidean space need not be. It is true, however, that every locally Euclidean space is T1. An example of a non-Hausdorff locally Euclidean space is the line with two origins. This space is created by replacing the origin of the real line with two points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero.
|
|
Beyond Euclidean space, there is an infinite set of regular hyperbolic tilings.
|
|
Both transitions are not surjective, that is, not every B-space results from some A-space. First, a 3-dim Euclidean space is a special (not general) case of a Euclidean space. Second, a topology of a Euclidean space is a special case of topology (for instance, it must be non-compact, and connected, etc). We denote surjective transitions by a two-headed arrow, "↠" rather than "→".
|
|
Topological notions (continuity, convergence, open sets, closed sets etc.) are defined naturally in every Euclidean space. In other words, every Euclidean space is also a topological space. Every isomorphism between two Euclidean spaces is also an isomorphism between the corresponding topological spaces (called "homeomorphism"), but the converse is wrong: a homeomorphism may distort distances. In Bourbaki's terms, "topological space" is an underlying structure of the "Euclidean space" structure.
|
|
Another familiar example might be the compact 2-torus or Euclidean space under addition.
|
|
In general, the Euclidean group can be studied by conjugation of isometries in Euclidean space.
|
|
For example, the three- dimensional Euclidean space is not a countable union of its affine planes.
|
|
A ' is a finite-dimensional inner product space over the real numbers. A Euclidean space is an affine space over the reals such that the associated vector space is a Euclidean vector space. Euclidean spaces are sometimes called Euclidean affine spaces for distinguishing them from Euclidean vector spaces. If is a Euclidean space, its associated vector space is often denoted \overrightarrow E. The dimension of a Euclidean space is the dimension of its associated vector space.
|
|
The hyperbolic plane is a plane where every point is a saddle point. There exist various pseudospheres in Euclidean space that have a finite area of constant negative Gaussian curvature. By Hilbert's theorem, it is not possible to isometrically immerse a complete hyperbolic plane (a complete regular surface of constant negative Gaussian curvature) in a three-dimensional Euclidean space. Other useful models of hyperbolic geometry exist in Euclidean space, in which the metric is not preserved.
|
|
One may say that the Euclidean space is the real inner product space that forgot its origin.
|
|
A Euclidean graph (a graph embedded in some Euclidean space) is periodic if there exists a basis of that Euclidean space whose corresponding translations induce symmetries of that graph (i.e., application of any such translation to the graph embedded in the Euclidean space leaves the graph unchanged). Equivalently, a periodic Euclidean graph is a periodic realization of an abelian covering graph over a finite graph. A Euclidean graph is uniformly discrete if there is a minimal distance between any two vertices.
|
|
An important question in metric space geometry is to characterize those metric spaces that can be embedded in some Euclidean space by a bi-Lipschitz function. This means that one can essentially think of the metric space as a subset of Euclidean space. Not all metric spaces may be embedded in Euclidean space. Doubling metric spaces, on the other hand, would seem like they have more of a chance, since the doubling condition says, in a way, that the metric space is not infinite dimensional.
|
|
Every separoid can be represented with a family of convex sets in some Euclidean space and their separations by hyperplanes.
|
|
In a pseudo-Euclidean space, the magnitude of a vector is the value of the quadratic form for that vector.
|
|
For example, a surface in Euclidean space is umbilic if and only if it is a piece of a sphere.
|
|
An isometry from a Euclidean space onto itself is called Euclidean isometry, Euclidean transformation or rigid transformation. The rigid transformations of a Euclidean space form a group (under composition), called the Euclidean group and often denoted of . The simplest Euclidean transformations are translations :P \to P+v. They are in bijective correspondence with vectors.
|
|
Let G be a connected finite graph embedded in Euclidean space of dimension n. Let V be a closed regular neighborhood of G in the Euclidean space. Then V is an n-dimensional handlebody. The graph G is called a spine of V. Any genus zero handlebody is homeomorphic to the three-ball B3.
|
|
An isomorphism to itself is called an automorphism. Automorphisms of a Euclidean space are shifts, rotations, reflections and compositions of these. Euclidean space is homogeneous in the sense that every point can be transformed into every other point by some automorphism. Euclidean axioms leave no freedom; they determine uniquely all geometric properties of the space.
|
|
The rotation group generalizes quite naturally to n-dimensional Euclidean space, \R^n with its standard Euclidean structure. The group of all proper and improper rotations in n dimensions is called the orthogonal group O(n), and the subgroup of proper rotations is called the special orthogonal group SO(n), which is a Lie group of dimension . In special relativity, one works in a 4-dimensional vector space, known as Minkowski space rather than 3-dimensional Euclidean space. Unlike Euclidean space, Minkowski space has an inner product with an indefinite signature.
|
|
The index set of this stochastic process is the non-negative numbers, while its state space is three-dimensional Euclidean space.
|
|
A d-dimensional Euclidean space has isoperimetric dimension d. This is the well known isoperimetric problem -- as discussed above, for the Euclidean space the constant C is known precisely since the minimum is achieved for the ball. An infinite cylinder (i.e. a product of the circle and the line) has topological dimension 2 but isoperimetric dimension 1.
|
|
Topologically, an -sphere can be constructed as a one-point compactification of -dimensional Euclidean space. Briefly, the -sphere can be described as , which is -dimensional Euclidean space plus a single point representing infinity in all directions. In particular, if a single point is removed from an -sphere, it becomes homeomorphic to . This forms the basis for stereographic projection.
|
|
In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°. The universal cover of a complete flat manifold is Euclidean space.
|
|
A common interpretation is that the concept of a point is meant to capture the notion of a unique location in Euclidean space.
|
|
Projective geometry is not necessarily concerned with curvature and the real projective plane may be twisted up and placed in the Euclidean plane or 3-space in many different ways.Apéry, F.; Models of the real projective plane, Vieweg (1987) Some of the more important examples are described below. The projective plane cannot be embedded (that is without intersection) in three-dimensional Euclidean space. The proof that the projective plane does not embed in three-dimensional Euclidean space goes like this: Assuming that it does embed, it would bound a compact region in three- dimensional Euclidean space by the generalized Jordan curve theorem.
|
|
In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal- length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called "the" inner product (or rarely projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more).
|
|
The second- level classification distinguishes, for example, between Euclidean and non- Euclidean spaces; between finite-dimensional and infinite-dimensional spaces; between compact and non-compact spaces, etc. In Bourbaki's terms, the second- level classification is the classification by "species". Unlike biological taxonomy, a space may belong to several species. The third-level classification distinguishes, for example, between spaces of different dimension, but does not distinguish between a plane of a three-dimensional Euclidean space, treated as a two-dimensional Euclidean space, and the set of all pairs of real numbers, also treated as a two-dimensional Euclidean space.
|
|
A manifold is a generalization of the concepts of curve and surface. In topology, a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space. In differential geometry, a differentiable manifold is a space where each neighborhood is diffeomorphic to Euclidean space. Manifolds are used extensively in physics, including in general relativity and string theory.
|
|
In the usual Euclidean space, we can rewrite the formal definition in usual terms. Each Voronoi polygon R_k is associated with a generator point P_k. Let X be the set of all points in the Euclidean space. Let P_1 be a point that generates its Voronoi region R_1, P_2 that generates R_2, and P_3 that generates R_3, and so on.
|
|
A surface is a two- dimensional, topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3—for example, the surface of a ball. On the other hand, there are surfaces, such as the Klein bottle, that cannot be embedded in three-dimensional Euclidean space without introducing singularities or self- intersections.
|
|
Most non-Euclidean geometries can be modeled by a manifold, and embedded in a Euclidean space of higher dimension. For example, an elliptic space can be modeled by an ellipsoid. It is common to represent in a Euclidean space mathematics objects that are a priori not of a geometrical nature. An example among many is the usual representation of graphs.
|
|
A line representation is minimal if it uses four parameters, which is the minimum needed to represent all possible lines in the Euclidean Space (E³).
|
|
1\. Embed M in some high-dimensional Euclidean space. (Use the Whitney embedding theorem.) 2\. Take a small neighborhood of M in that Euclidean space, Nε. Extend the vector field to this neighborhood so that it still has the same zeroes and the zeroes have the same indices. In addition, make sure that the extended vector field at the boundary of Nε is directed outwards. 3\.
|
|
Flexible 4-polytopes in 4-dimensional Euclidean space and 3-dimensional hyperbolic space were studied by . In dimensions n\geq 5, flexible polytopes were constructed by .
|
|
There are several equivalent ways to formally define the integral of a differential form, all of which depend on reducing to the case of Euclidean space.
|
|
In mathematics, the reach of a subset of Euclidean space Rn is a real number that roughly describes how curved the boundary of the set is.
|
|
See rotations in 4-dimensional Euclidean space for some information. In differential geometry, is the only case where admits a non- standard differential structure: see exotic R4.
|
|
In physics, the standard basis vectors for a given Euclidean space are sometimes referred to as the versors of the axes of the corresponding Cartesian coordinate system.
|
|
3d Euclidean space, where λ is a parameter of the space curve (light green). Following are examples of how the line elements are found from the metric.
|
|
In physics and mathematics, a sequence of numbers can be understood as a location in -dimensional space. When , the set of all such locations is called three- dimensional Euclidean space (or simply Euclidean space when the context is clear). It is commonly represented by the symbol . This serves as a three- parameter model of the physical universe (that is, the spatial part, without considering time), in which all known matter exists.
|
|
In quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to the ordered set of points in Euclidean space with no coinciding points. These functions are called the Schwinger functions (named after Julian Schwinger) and they are analytic, symmetric under the permutation of arguments (antisymmetric for fermionic fields), Euclidean covariant and satisfy a property known as reflection positivity.
|
|
While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. A manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an -dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension . Lines and circles, but not figure eights, are one-dimensional manifolds.
|
|
A subset U of a metric space is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with y also belongs to U. Equivalently, U is open if every point in U has a neighborhood contained in U. This generalizes the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.
|
|
The first part of the problem asks whether there are only finitely many essentially different space groups in n-dimensional Euclidean space. This was answered affirmatively by Bieberbach.
|
|
A knot is an embedding of the circle () into three-dimensional Euclidean space (). or the 3-sphere, , since the 3-sphere is compact., p. 33; , pp. 246–250.
|
|
In this language, an affine connection is simply a covariant derivative or (linear) connection on the tangent bundle. However, this approach does not explain the geometry behind affine connections nor how they acquired their name. The term really has its origins in the identification of tangent spaces in Euclidean space by translation: this property means that Euclidean -space is an affine space. (Alternatively, Euclidean space is a principal homogeneous space or torsor under the group of translations, which is a subgroup of the affine group.) As mentioned in the introduction, there are several ways to make this precise: one uses the fact that an affine connection defines a notion of parallel transport of vector fields along a curve.
|
|
For any subset of Euclidean space ℝn, is compact if and only if it is closed and bounded; this is the Heine–Borel theorem. As a Euclidean space is a metric space, the conditions in the next subsection also apply to all of its subsets. Of all of the equivalent conditions, it is in practice easiest to verify that a subset is closed and bounded, for example, for a closed interval or closed -ball.
|
|
The Arzelà–Ascoli theorem holds, more generally, if the functions take values in -dimensional Euclidean space , and the proof is very simple: just apply the -valued version of the Arzelà–Ascoli theorem times to extract a subsequence that converges uniformly in the first coordinate, then a sub-subsequence that converges uniformly in the first two coordinates, and so on. The above examples generalize easily to the case of functions with values in Euclidean space.
|
|
In Euclidean -space, an (open) -ball of radius and center is the set of all points of distance less than from . A closed -ball of radius is the set of all points of distance less than or equal to away from . In Euclidean -space, every ball is bounded by a hypersphere. The ball is a bounded interval when , is a disk bounded by a circle when , and is bounded by a sphere when .
|
|
In some mathematical frameworks, a given point process may be considered as a sequence of points with each point randomly positioned in d-dimensional Euclidean space Rd as well as some other more abstract mathematical spaces. In general, whether or not a random sequence is equivalent to the other interpretations of a point process depends on the underlying mathematical space, but this holds true for the setting of finite-dimensional Euclidean space Rd.
|
|
In the context of a function on a Euclidean space, some texts restrict the vector v to being a unit vector. With this restriction, both the above definitions are equivalent.
|
|
The Levi-Civita symbol allows the determinant of a square matrix, and the cross product of two vectors in three-dimensional Euclidean space, to be expressed in Einstein index notation.
|
|
There is a version of Rademacher's theorem that holds for Lipschitz functions from a Euclidean space into an arbitrary metric space in terms of metric differentials instead of the usual derivative.
|
|
In Euclidean geometry, of special interest are involutions which are linear or affine transformations over the Euclidean space Rn. Such involutions are easy to characterize and they can be described geometrically.
|
|
In higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space Rn + 1.
|
|
In mathematics, the isoperimetric dimension of a manifold is a notion of dimension that tries to capture how the large-scale behavior of the manifold resembles that of a Euclidean space (unlike the topological dimension or the Hausdorff dimension which compare different local behaviors against those of the Euclidean space). In the Euclidean space, the isoperimetric inequality says that of all bodies with the same volume, the ball has the smallest surface area. In other manifolds it is usually very difficult to find the precise body minimizing the surface area, and this is not what the isoperimetric dimension is about. The question we will ask is, what is approximately the minimal surface area, whatever the body realizing it might be.
|
|
In mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an n-dimensional symmetric bilinear space can be described as the composition of at most n reflections. The notion of a symmetric bilinear space is a generalization of Euclidean space whose structure is defined by a symmetric bilinear form (which need not be positive definite, so is not necessarily an inner product – for instance, a pseudo-Euclidean space is also a symmetric bilinear space). The orthogonal transformations in the space are those automorphisms which preserve the value of the bilinear form between every pair of vectors; in Euclidean space, this corresponds to preserving distances and angles. These orthogonal transformations form a group under composition, the orthogonal group.
|
|
To describe the complex projective space in an analogous manner requires a generalization of the idea of vector, line, and direction. Imagine that instead of standing in a real Euclidean space, the artist is standing in a complex Euclidean space Cn+1 (which has real dimension 2n+2) and the landscape is a complex hyperplane (of real dimension 2n). Unlike the case of real Euclidean space, in the complex case there are directions in which the artist can look which do not see the landscape (because it does not have high enough dimension). However, in a complex space, there is an additional "phase" associated with the directions through a point, and by adjusting this phase the artist can guarantee that he typically sees the landscape.
|
|
Reciprocally, according to the Nash-Kuiper theorem, any Riemannian surface with boundary can be embedded in Euclidean space preserving the lengths and area specified by the Riemannian metric. Thus the filling problem can be stated equivalently as a question about Riemannian surfaces, that are not placed in Euclidean space in any particular way. :Conjecture (Gromov's filling area conjecture, 1983): The hemisphere has minimum area among the orientable compact Riemannian surfaces that fill isometrically their boundary curve, of given length.
|
|
The spherical coordinate system is commonly used in physics. It assigns three numbers (known as coordinates) to every point in Euclidean space: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). The symbol ρ (rho) is often used instead of r. In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.
|
|
In geometry, a flat or Euclidean subspace is a subset of a Euclidean space that is itself a Euclidean space (of lower dimension). The flats in two- dimensional space are points and lines, and the flats in three-dimensional space are points, lines, and planes. In a -dimensional space, there are flats of every dimension from 0 to .In addition, a whole -dimensional space, being a subset of itself, may also be considered as an -dimensional flat.
|
|
A manifold is a space that in the neighborhood of each point resembles a Euclidean space. In technical terms, a manifold is a topological space, such that each point has a neighborhood that is homeomorphic to an open subset of a Euclidean space. Manifold can be classified by increasing degree of this "resemblance" into topological manifolds, differentiable manifolds, smooth manifolds, and analytic manifolds. However, none of these types of "resemblance" respect distances and angles, even approximately.
|
|
Volume 10, Number 2 (1975), 277-288. are Riemannian manifolds whose Ricci curvature tensor vanishes. Ricci-flat manifolds are special cases of Einstein manifolds, where the cosmological constant need not vanish. Since Ricci curvature measures the amount by which the volume of a small geodesic ball deviates from the volume of a ball in Euclidean space, small geodesic balls will have no volume deviation, but their "shape" may vary from the shape of the standard ball in Euclidean space.
|
|
The Riemann sphere, the one-dimensional complex projective space, i.e. the complex projective line. In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines through the origin of a complex Euclidean space (see below for an intuitive account).
|
|
Another example. The question "what is the sum of the three angles of a triangle" makes sense in a Euclidean space but not in a projective space. In a non- Euclidean space the question makes sense but is answered differently, which is not an upper-level distinction. Also, the distinction between a Euclidean plane and a Euclidean 3-dimensional space is not an upper-level distinction; the question "what is the dimension" makes sense in both cases.
|
|
Kindle Edition. Absolute space being three-dimensional Euclidean space, infinite and without a center. Being "at rest" means being at the same place in absolute space over time.Tim Maudlin (2012-07-22).
|
|
For example, separability is assumed when constructing and studying random fields, where the collection of random variables is now indexed by sets other than the real line such as n-dimensional Euclidean space.
|
|
The reverse is true for subsets of Euclidean space (with the subspace topology), but not in general. For example, an infinite set equipped with the discrete metric is bounded but not totally bounded.
|
|
As usual for Sobolev inequalities, Hoffman and Spruck were also able to derive new isoperimetric inequalities for submanifolds of Riemannian manifolds. It is well known that there is a wide variety of minimal surfaces in the three-dimensional Euclidean space. Hoffman and William Meeks proved that any minimal surface which is contained in a half-space must fail to be properly immersed. That is, there must exist a compact set in Euclidean space which contains a noncompact region of the minimal surface.
|
|
Euclidean space was introduced by ancient Greeks as an abstraction of our physical space. Their great innovation, appearing in Euclid's Elements was to build and prove all geometry by starting from a few very basic properties, which are abstracted from the physical world, and cannot be mathematically proved because of the lack of more basic tools. These properties are called postulates, or axioms in modern language. This way of defining Euclidean space is still in use under the name of synthetic geometry.
|
|
Various formulas are known for computing distances between different types of objects, such as the distance from a point to a line. In more advanced areas of mathematics, Euclidean space and its distance provides a standard example of a metric space, called the Euclidean metric. When viewing Euclidean space as a vector space, its distance is associated with a norm called the Euclidean norm. It can be extended to more general vector spaces as the L2 norm or L2 distance.
|
|
This generalized definition implies that the above-mentioned geometric entities are a special kind of vectors, as they are elements of a special kind of vector space called Euclidean space. This article is about vectors strictly defined as arrows in Euclidean space. When it becomes necessary to distinguish these special vectors from vectors as defined in pure mathematics, they are sometimes referred to as geometric, spatial, or Euclidean vectors. Being an arrow, a Euclidean vector possesses a definite initial point and terminal point.
|
|
The concept of angles between lines in the plane and between pairs of two lines, two planes or a line and a plane in space can be generalized to arbitrary dimension. This generalization was first discussed by Jordan. For any pair of flats in a Euclidean space of arbitrary dimension one can define a set of mutual angles which are invariant under isometric transformation of the Euclidean space. If the flats do not intersect, their shortest distance is one more invariant.
|
|
More precisely, isomorphic spaces are considered identical, where an isomorphism between two spaces is a one-to-one correspondence between their points that preserves the relationships. For example, the relationships between the points of a three-dimensional Euclidean space are uniquely determined by Euclid's axioms,Reformed by Hilbert, Tarski and Birkhoff in order to avoid hidden assumptions found in Euclid's Elements. and all three- dimensional Euclidean spaces are considered identical. Topological notions such as continuity have natural definitions in every Euclidean space.
|
|
In probability theory, an isotropic measure is any mathematical measure that is invariant under linear isometries. It is a standard simplification and assumption used in probability theory. Generally, it is used in the context of measure theory on n-dimensional Euclidean space, for which it can be intuitive to study measures that are unchanged by rotations and translations. An obvious example of such a measure is the standard way of assigning a measure to subsets of n-dimensional Euclidean space: Lebesgue measure.
|
|
A smooth manifold is a mathematical object which looks locally like a smooth deformation of Euclidean space : for example a smooth curve or surface looks locally like a smooth deformation of a line or a plane. Smooth functions and vector fields can be defined on manifolds, just as they can on Euclidean space, and scalar functions on manifolds can be differentiated in a natural way. However, differentiation of vector fields is less straightforward: this is a simple matter in Euclidean space, because the tangent space of based vectors at a point can be identified naturally (by translation) with the tangent space at a nearby point . On a general manifold, there is no such natural identification between nearby tangent spaces, and so tangent vectors at nearby points cannot be compared in a well-defined way.
|
|
In mathematics, Vitale's random Brunn–Minkowski inequality is a theorem due to Richard Vitale that generalizes the classical Brunn–Minkowski inequality for compact subsets of n-dimensional Euclidean space Rn to random compact sets.
|
|
Two-dimensional space can be seen as a projection of the physical universe onto a plane. Usually, it is thought of as a Euclidean space and the two dimensions are called length and width.
|
|
In mathematics, the intrinsic flat distance is a notion for distance between two Riemannian manifolds which is a generalization of Federer and Fleming's flat distance between submanifolds and integral currents lying in Euclidean space.
|
|
The theorem is valid word by word also for stochastic processes taking values in the -dimensional Euclidean space or the complex vector space . This follows from the one-dimensional version by considering the components individually.
|
|
However S2 × R cannot be, Euclidean space corresponds to two different Bianchi groups, and there are an uncountable number of solvable non-unimodular Bianchi groups, most of which give model geometries with no compact representatives.
|
|
The analogue of a cube in four-dimensional Euclidean space has a special name—a tesseract or hypercube. More properly, a hypercube (or n-dimensional cube or simply n-cube) is the analogue of the cube in n-dimensional Euclidean space and a tesseract is the order-4 hypercube. A hypercube is also called a measure polytope. There are analogues of the cube in lower dimensions too: a point in dimension 0, a line segment in one dimension and a square in two dimensions.
|
|
This concept of time and simultaneity was later generalized by Einstein in his special theory of relativity (1905) where he developed transformations between inertial frames of reference based upon the universal nature of physical laws and their economy of expression (Lorentz transformations). The definition of inertial reference frame can also be extended beyond three-dimensional Euclidean space. Newton's assumed a Euclidean space, but general relativity uses a more general geometry. As an example of why this is important, consider the geometry of an ellipsoid.
|
|
The Johnson-Lindenstrauss lemma states that large sets of vectors in a high-dimensional space can be linearly mapped in a space of much lower (but still high) dimension n with approximate preservation of distances. One of the explanations of this effect is the exponentially high quasiorthogonal dimension of n-dimensional Euclidean space. There are exponentially large (in dimension n) sets of almost orthogonal vectors (with small value of inner products) in n–dimensional Euclidean space. This observation is useful in indexing of high-dimensional data.
|
|
Gersho's conjecture, proven for one and two dimensions, says that "asymptotically speaking, all cells of the optimal CVT, while forming a tessellation, are congruent to a basic cell which depends on the dimension." In two dimensions, the basic cell for the optimal CVT is a regular hexagon as it is proven to be the most dense packing of circles in 2D Euclidean space. Its three dimensional equivalent is the rhombic dodecahedral honeycomb, derived from the most dense packing of spheres in 3D Euclidean space.
|
|
Clifford algebras can be used to represent any classical Lie algebra. In general it is possible to identify Lie algebras of compact groups by using anti-Hermitian elements, which can be extended to non-compact groups by adding Hermitian elements. The bivectors of an n-dimensional Euclidean space are Hermitian elements and can be used to represent the spin(n) Lie algebra. The bivectors of the three-dimensional Euclidean space form the spin(3) Lie algebra, which is isomorphic to the su(2) Lie algebra.
|
|
Topological spaces arose as generalization of the open sets of spaces studied in elementary mathematics, such as open disks in the Euclidean plane, open balls in the Euclidean space, and open intervals of the real line.
|
|
Hilbert's eighteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by mathematician David Hilbert. It asks three separate questions about lattices and sphere packing in Euclidean space.
|
|
Shi is well-known for his foundational work with Luen-Fai Tam on compact and smooth Riemannian manifolds-with-boundary whose scalar curvature is nonnegative and whose boundary is mean-convex. In particular, if the manifold has a spin structure, and if each connected component of the boundary can be isometrically embedded as a strictly convex hypersurface in Euclidean space, then the average value of the mean curvature of each boundary component is less than or equal to the average value of the mean curvature of the corresponding hypersurface in Euclidean space. This is particularly simple in three dimensions, where every manifold has a spin structure and a result of Louis Nirenberg shows that any positively-curved Riemannian metric on the two-dimensional sphere can be isometrically embedded in three-dimensional Euclidean space in a geometrically unique way.Louis Nirenberg.
|
|
The notion of convexity in the Euclidean space may be generalized by modifying the definition in some or other aspects. The common name "generalized convexity" is used, because the resulting objects retain certain properties of convex sets.
|
|
Implicit surface torus (R=40, a=15). Implicit surface of genus 2. Implicit non-algebraic surface (wineglass). In mathematics, an implicit surface is a surface in Euclidean space defined by an equation : F(x,y,z)=0.
|
|
Solid geometry was the traditional name for the geometry of three-dimensional Euclidean space. Stereometry deals with the measurements of volumes of various solid figures (three-dimensional figures) including pyramids, cylinders, cones, truncated cones, spheres, and prisms.
|
|
Is this the minimum possible area? The surface can be imagined as made of a flexible but non-stretchable material, that allows it to be moved around and bended in Euclidean space. None of these transformations modifies the area of the surface nor the length of the curves drawn on it, which are the magnitudes relevant to the problem. The surface can be removed from Euclidean space altogether, obtaining a Riemannian surface, which is an abstract smooth surface with a Riemannian metric that encodes the lengths and area.
|
|
In mathematics, the Hermite constant, named after Charles Hermite, determines how short an element of a lattice in Euclidean space can be. The constant γn for integers n > 0 is defined as follows. For a lattice L in Euclidean space Rn unit covolume, i.e. vol(Rn/L) = 1, let λ1(L) denote the least length of a nonzero element of L. Then is the maximum of λ1(L) over all such lattices L. The square root in the definition of the Hermite constant is a matter of historical convention.
|
|
The Banach–Tarski paradox states that a ball in the ordinary Euclidean space can be doubled using only the operations of partitioning into subsets, replacing a set with a congruent set, and reassembling. Its mathematical structure is greatly elucidated by emphasizing the role played by the group of Euclidean motions and introducing the notions of equidecomposable sets and a paradoxical set. Suppose that is a group acting on a set . In the most important special case, is an -dimensional Euclidean space (for integral n), and consists of all isometries of , i.e.
|
|
Consider a smooth surface in 3-dimensional Euclidean space. Near to any point, can be approximated by its tangent plane at that point, which is an affine subspace of Euclidean space. Differential geometers in the 19th century were interested in the notion of development in which one surface was rolled along another, without slipping or twisting. In particular, the tangent plane to a point of can be rolled on : this should be easy to imagine when is a surface like the 2-sphere, which is the smooth boundary of a convex region.
|
|
It represents in most cases a possible set of symmetries of objects/figures in the space, or functions defined on the space. See symmetry group. A discrete isometry group is an isometry group such that for every point of the space the set of images of the point under the isometries is a discrete set. In pseudo-Euclidean space the metric is replaced with an isotropic quadratic form; transformations preserving this form are sometimes called "isometries", and the collection of them is then said to form an isometry group of the pseudo-Euclidean space.
|
|
The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending without stretching or tearing a page of paper gives an isometric embedding of the page into Euclidean space because curves drawn on the page retain the same arclength however the page is bent. The first theorem is for continuously differentiable (C1) embeddings and the second for analytic embeddings or embeddings that are smooth of class Ck, 3 ≤ k ≤ ∞.
|
|
In Euclidean space, a ball is the volume bounded by a sphere In mathematics, a ball is the volume space bounded by a sphere; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball or hyperball in dimensions is called an -ball and is bounded by an ()-sphere.
|
|
The dot product may be defined algebraically or geometrically. The geometric definition is based on the notions of angle and distance (magnitude of vectors). The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space. In modern presentations of Euclidean geometry, the points of space are defined in terms of their Cartesian coordinates, and Euclidean space itself is commonly identified with the real coordinate space Rn. In such a presentation, the notions of length and angles are defined by means of the dot product.
|
|
In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include Weyl groups and crystallographic Coxeter groups. While the orthogonal group is generated by reflections (by the Cartan–Dieudonné theorem), it is a continuous group (indeed, Lie group), not a discrete group, and is generally considered separately.
|
|
These concepts are named after Eduard Helly (1884-1943); Helly's theorem on convex sets, which gave rise to this notion, states that convex sets in Euclidean space of dimension n are a Helly family of order n + 1.
|
|
D4, Dynkin diagram of SO(8) In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space. It could be either a real or complex simple Lie group of rank 4 and dimension 28.
|
|
Later, von Neumann carried out the mathematical formulation of quantum mechanics, by further developing functional analytic notions such as linear operators on Hilbert spaces, which, very roughly speaking, correspond to Euclidean space, but with an infinity of independent directions.
|
|
Some basic properties of Euclidean spaces depend only of the fact that a Euclidean space is an affine space. They are called affine properties and include the concepts of lines, subspaces, and parallelism. which are detailed in next subsections.
|
|
If a finite weighted graph is geometrically embedded in a Euclidean space, i.e., the graph vertices represent points of this space, then it can be interpreted as a discrete approximation of a related nonlocal operator in the continuum setting.
|
|
Typical choices for the grains include disks, random polygon and segments of random length. Boolean models are also examples of stochastic processes known as coverage processes. The above models can be extended from the plane to general Euclidean space .
|
|
Cartesian products were first developed by René Descartes in the context of analytic geometry. If R denotes the set of all real numbers, then R2 := R × R represents the Euclidean plane and R3 := R × R × R represents three-dimensional Euclidean space.
|
|
In three-dimensional Euclidean space, these three planes represent solutions of linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations.
|
|
In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non- Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff.
|
|
In mathematics, the Menger curvature of a triple of points in n-dimensional Euclidean space Rn is the reciprocal of the radius of the circle that passes through the three points. It is named after the Austrian-American mathematician Karl Menger.
|
|
More exactly: all three-dimensional Euclidean spaces are mutually isomorphic. In this sense we have "the" three- dimensional Euclidean space. In Bourbaki's terms, the corresponding theory is univalent. In contrast, topological spaces are generally non-isomorphic; their theory is multivalent.
|
|
Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector). More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field.
|
|
Openness of V in the subspace topology is automatic.) Both of these statements are not at all obvious and are not generally true if one leaves Euclidean space. A map which is not a homeomorphism onto its image: with g(t) = (t2 − 1, t3 − t) It is of crucial importance that both domain and range of f are contained in Euclidean space of the same dimension. Consider for instance the map f : (0,1) → ℝ2 defined by . This map is injective and continuous, the domain is an open subset of , but the image is not open in .
|
|
For example, any rectangle can be glued left-edge to right-edge with a reversal of orientation. Some, but not all, of these can be smoothly modeled as surfaces in Euclidean space. A closely related, but not homeomorphic, surface is the complete open Möbius band, a boundaryless surface in which the width of the strip is extended infinitely to become a Euclidean line. A half-twist clockwise gives an embedding of the Möbius strip which cannot be moved or stretched to give the half-twist counterclockwise; thus, a Möbius strip embedded in Euclidean space is a chiral object with right- or left-handedness.
|
|
Shiing-Shen Chern published his proof of the theorem in 1944 while at the Institute for Advanced Study. This was historically the first time that the formula was proven without assuming the manifold to be embedded in a Euclidean space, which is what it means by "intrinsic". The special case for a hypersurface (an n-1-dimensional submanifolds in an n-dimensional Euclidean space) was proved by H. Hopf in which the integrand is the Gauss-Kronecker curvature (the product of all principal curvatures at a point of the hypersurface). This was generalized independently by Allendoerfer in 1939 and Fenchel in 1940 to a Riemannian submanifold of a Euclidean space of any codimension, for which they used the Lipschitz-Killing curvature (the average of the Gauss-Kronecker curvature along each unit normal vector over the unit sphere in the normal space; for an even dimensional submanifold, this is an invariant only depending on the Riemann metric of the submanifold).
|
|
If a family of sets is a Helly family of order k, that family is said to have Helly number k. The Helly dimension of a metric space is one less than the Helly number of the family of metric balls in that space; Helly's theorem implies that the Helly dimension of a Euclidean space equals its dimension as a real vector space.. The Helly dimension of a subset S of a Euclidean space, such as a polyhedron, is one less than the Helly number of the family of translates of S.. For instance, the Helly dimension of any hypercube is 1, even though such a shape may belong to a Euclidean space of much higher dimension.. Helly dimension has also been applied to other mathematical objects. For instance defines the Helly dimension of a group (an algebraic structure formed by an invertible and associative binary operation) to be one less than the Helly number of the family of left cosets of the group..
|
|
In geometry, the moment curve is an algebraic curve in d-dimensional Euclidean space given by the set of points with Cartesian coordinates of the form :\left( x, x^2, x^3, \dots, x^d \right)., Definition 5.4.1, p. 97; , Definition 1.6.
|
|
In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of numbers, integral geometry, linear programming, probability theory, game theory, etc.
|
|
Every compact manifold is its own soul. Indeed, the theorem is often stated only for non-compact manifolds. As a very simple example, take to be Euclidean space . The sectional curvature is everywhere, and any point of can serve as a soul of .
|
|
The converse is not true; e.g., a countable discrete space satisfies these three conditions, but is not compact. # is closed and bounded (as a subset of any metric space whose restricted metric is ). The converse may fail for a non-Euclidean space; e.g.
|
|
In mathematics, the Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov, and later independently by Christer Borell, states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian boundary measure.
|
|
Specifically, in a Euclidean space, one considers spatial vectors, also called Euclidean vectors which are used to represent quantities that have both magnitude and direction, and may be added, subtracted and scaled (i.e. multiplied by a real number) for forming a vector space.
|
|
The positive definite case is called Euclidean space, while the case of a single minus, is called Lorentzian space. If , then Lorentzian space is also called Minkowski space or Minkowski spacetime. The special case will be referred to as the split-case.
|
|
Formally the rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. Rotations are direct isometries, i.e., isometries preserving orientation. Therefore, a symmetry group of rotational symmetry is a subgroup of E+(m) (see Euclidean group).
|
|
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfaces that are not necessarily smooth.
|
|
Real Analysis begins with measure theory, Lebesgue integration, and differentiation in Euclidean space. It then covers Hilbert spaces before returning to measure and integration in the context of abstract measure spaces. It concludes with a chapter on Hausdorff measure and fractals.Stein & Shakarchi, Real Analysis.
|
|
To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other three-dimensional spaces and objects other than circles can be used; see knot (mathematics). Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space.
|
|
The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as n-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.
|
|
In mathematics, a set function is a function whose input is a set. The output is usually a number. Often the input is a set of real numbers, a set of points in Euclidean space, or a set of points in some measure space.
|
|
Various equivalent formulations of the problem had been given, such as the d-step conjecture, which states that the diameter of any 2d-facet polytope in d-dimensional Euclidean space is no more than d; Santos Leal's counterexample also disproves this conjecture., p. 84..
|
|
In Riemannian geometry, an isoparametric manifold is a type of (immersed) submanifold of Euclidean space whose normal bundle is flat and whose principal curvatures are constant along any parallel normal vector field. The set of isoparametric manifolds is stable under the mean curvature flow.
|
|
In mathematics, the quadruple product is a product of four vectors in three- dimensional Euclidean space. The name "quadruple product" is used for two different products, the scalar-valued scalar quadruple product and the vector- valued vector quadruple product or vector product of four vectors .
|
|
More in general, the concept can be applied to representing positions on the boundary of a strictly convex bounded subset of k-dimensional Euclidean space, provided that that boundary is a differentiable manifold. In this general case, the n-vector consists of k parameters.
|
|
A metric space is separable space if it has a countable dense subset. Typical examples are the real numbers or any Euclidean space. For metric spaces (but not for general topological spaces) separability is equivalent to second-countability and also to the Lindelöf property.
|
|
For example, in a Ricci-flat manifold, a circle in Euclidean space may be deformed into an ellipse with equal area. This is due to Weyl curvature. Ricci-flat manifolds often have restricted holonomy groups. Important cases include Calabi–Yau manifolds and hyperkähler manifolds.
|
|
The philosopher Roger Scruton described Philosophical Problems of Space and Time as the most comprehensive discussion of non-Euclidean space, though he added that the work was "far from inviting". The philosopher Philip L. Quinn called Grünbaum's thesis about physical geometry and chronometry "striking".
|
|
Isomap defines the geodesic distance to be the sum of edge weights along the shortest path between two nodes (computed using Dijkstra's algorithm, for example). The top n eigenvectors of the geodesic distance matrix, represent the coordinates in the new n-dimensional Euclidean space.
|
|
Paracompact manifolds have all the topological properties of metric spaces. In particular, they are perfectly normal Hausdorff spaces. Manifolds are also commonly required to be second-countable. This is precisely the condition required to ensure that the manifold embeds in some finite-dimensional Euclidean space.
|
|
As this algebraic structure lends itself directly to effective computation, it facilitates exploration of the classical methods of projective geometry and inversive geometry in a concrete, easy-to-manipulate setting. It has also been used as an efficient structure to represent and facilitate calculations in screw theory. CGA has particularly been applied in connection with the projective mapping of the everyday Euclidean space into a five-dimensional vector space , which has been investigated for applications in robotics and computer vision. It can be applied generally to any pseudo-Euclidean space, and the mapping of Minkowski space to the space is being investigated for applications to relativistic physics.
|
|
The book has seven chapters. The first two are introductory, providing material about manifolds in general, the Hauptvermutung proving the existence and equivalence of triangulations for low-dimensional manifolds, the classification of two-dimensional surfaces, covering spaces, and the mapping class group. The third chapter begins the book's material on 3-manifolds, and on the decomposition of manifolds into smaller spaces by cutting them along surfaces. For instance, the three- dimensional Schoenflies theorem states that cutting Euclidean space by a sphere can only produce two topological balls; an analogous theorem of J. W. Alexander states that at least one side of any torus in Euclidean space must be a solid torus.
|
|
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. The rigid transformations include rotations, translations, reflections, or their combination. Sometimes reflections are excluded from the definition of a rigid transformation by imposing that the transformation also preserve the handedness of figures in the Euclidean space (a reflection would not preserve handedness; for instance, it would transform a left hand into a right hand). To avoid ambiguity, this smaller class of transformations is known as rigid motions or proper rigid transformations (informally, also known as roto- translations) .
|
|
The curvature of a curve at a point is normally a scalar quantity, that is, it is expressed by a single real number. For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or manifold. This leads to the concepts of maximal curvature, minimal curvature, and mean curvature. For Riemannian manifolds (of dimension at least two) that are not necessarily embedded in a Euclidean space, one can define the curvature intrinsically, that is without referring to an external space.
|
|
A hermitian matrix of unit determinant and having positive eigenvalues can be uniquely expressed as the exponential of a traceless hermitian matrix, and therefore the topology of this is that of -dimensional Euclidean space. Section 2.5 Since SU(n) is simply connected, Proposition 13.11 we conclude that is also simply connected, for all n. The topology of is the product of the topology of SO(n) and the topology of the group of symmetric matrices with positive eigenvalues and unit determinant. Since the latter matrices can be uniquely expressed as the exponential of symmetric traceless matrices, then this latter topology is that of -dimensional Euclidean space.
|
|
In 1987, Huisken adapted his methods to consider an alternative "mean curvature"-driven flow for closed hypersurfaces in Euclidean space, in which the volume enclosed by the surface is kept constant; the result is directly analogous. Similarly to Hamilton's result, Huisken's results can be viewed as providing proofs that any smooth closed convex hypersurface of Euclidean space is diffeomorphic to a sphere, and is the boundary of a region which is diffeomorphic to a ball. However, both of these results can be proved by more elementary means using the Gauss map. In 1986, Huisken extended the calculations in his proof to consider hypersurfaces in general Riemannian manifolds.
|
|
Hyperbolic trees employ hyperbolic space, which intrinsically has "more room" than Euclidean space. For instance, linearly increasing the radius of a circle in Euclidean space increases its circumference linearly, while the same circle in hyperbolic space would have its circumference increase exponentially. Exploiting this property allows laying out the tree in hyperbolic space in an uncluttered manner: placing a node far enough from its parent gives the node almost the same amount of space as its parent for laying out its own children. Displaying a hyperbolic tree commonly utilizes the Poincaré disk model of hyperbolic geometry, though the Klein-Beltrami model can also be used.
|
|
In mathematics, a dilation is a function f from a metric space into itself that satisfies the identity :d(f(x),f(y))=rd(x,y) for all points (x, y), where d(x, y) is the distance from x to y and r is some positive real number.. In Euclidean space, such a dilation is a similarity of the space.. See in particular p. 110. Dilations change the size but not the shape of an object or figure. Every dilation of a Euclidean space that is not a congruence has a unique fixed point. that is called the center of dilation.. Some congruences have fixed points and others do not..
|
|
Nirenberg's Ph.D. thesis provided a resolution of the Weyl problem and Minkowski problem of differential geometry. The former asks for the existence of isometric embeddings of positively curved Riemannian metrics on the two-dimensional sphere into three-dimensional Euclidean space, while the latter asks for closed surfaces in three- dimensional Euclidean space of prescribed Gaussian curvature. The now-standard approach to these problems is through the theory of the Monge-Ampère equation, which is a fully nonlinear elliptic partial differential equation. Nirenberg made novel contributions to the theory of such equations in the setting of two-dimensional domains, building on the earlier 1938 work of Charles Morrey.
|
|
Suppose that for some population of 1000 people, each person is asked to provide their age, height, weight, and number of nose hairs. Thus to each member of the population there is associated an ordered quadruple of numbers. Since n-dimensional Euclidean space is defined as all ordered n-tuples of numbers, this means that the data on 1000 people maybe be thought of as 1000 points in 4-dimensional Euclidean space. The grand tour converts the spatial complexity of the multivariate data set into temporal complexity by using the relatively simple 2-dimensional views of the projected data as the individual frames of the movie.
|
|
A symmetry of a Euclidean graph is an isometry of the underlying Euclidean space whose restriction to the graph is an automorphism; the symmetry group of the Euclidean graph is the group of its symmetries. A Euclidean graph in three-dimensional Euclidean space is periodic if there exist three linearly independent translations whose restrictions to the net are symmetries of the net. Often (and always, if one is dealing with a crystal net), the periodic net has finitely many orbits, and is thus uniformly discrete in that there exists a minimum distance between any two vertices. The result is a three- dimensional periodic graph as a geometric object.
|
|
The reason is that the real plane is mapped to the w = 1 plane in real projective space, and so translation in real Euclidean space can be represented as a shear in real projective space. Although a translation is a non-linear transformation in a 2-D or 3-D Euclidean space described by Cartesian coordinates (i.e. it can't be combined with other transformations while preserving commutativity and other properties), it becomes, in a 3-D or 4-D projective space described by homogeneous coordinates, a simple linear transformation (a shear). More affine transformations can be obtained by composition of two or more affine transformations.
|
|
Theorem. Let (M,g) be a Riemannian manifold and ƒ: Mm → Rn a short C∞-embedding (or immersion) into Euclidean space Rn, where n ≥ m+1. Then for arbitrary ε > 0 there is an embedding (or immersion) ƒε: Mm → Rn which is In particular, as follows from the Whitney embedding theorem, any m-dimensional Riemannian manifold admits an isometric C1-embedding into an arbitrarily small neighborhood in 2m-dimensional Euclidean space. The theorem was originally proved by John Nash with the condition n ≥ m+2 instead of n ≥ m+1 and generalized by Nicolaas Kuiper, by a relatively easy trick. The theorem has many counterintuitive implications.
|
|
More recently, it refers largely to the use of nonlinear partial differential equations to study geometric and topological properties of spaces, such as submanifolds of Euclidean space, Riemannian manifolds, and symplectic manifolds. This approach dates back to the work by Tibor Radó and Jesse Douglas on minimal surfaces, John Forbes Nash Jr. on isometric embeddings of Riemannian manifolds into Euclidean space, work by Louis Nirenberg on the Minkowski problem and the Weyl problem, and work by Aleksandr Danilovich Aleksandrov and Aleksei Pogorelov on convex hypersurfaces. In the 1980s fundamental contributions by Karen Uhlenbeck,Jackson, Allyn. (2019). Founder of geometric analysis honored with Abel Prize Retrieved 20 March 2019.
|
|
Wayne has developed wave functions that represent the paths of the semiphotons in Euclidean space and Newtonian time. The transverse wave functions are solutions to the Schrödinger equation that has been modified to directly operate on bosons as opposed to fermions, and the longitudinal wavefunctions are solutions to the classical equations of mechanics. These fields, unlike the fields described by Maxwell’s equations, are consistent with the assumptions of the Kirchhoff's diffraction equation. Wayne’s wave mechanical approach shows that the binary photon can be visualized as an oscillating particulate rotor propagating electromagnetic waves through Euclidean space and Newtonian time at the invariant vacuum speed of light.
|
|
This can be generalized to create n coordinates for any point in n-dimensional Euclidean space. Depending on the direction and order of the coordinate axes, the three-dimensional system may be a right-handed or a left- handed system. This is one of many coordinate systems.
|
|
In mathematics Antoine's necklace is a topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected. It also serves as a counterexample to the claim that all Cantor spaces are ambiently homeomorphic to each other. It was discovered by .
|
|
In knot theory, a virtual knot is a generalization of knots in 3-dimensional Euclidean space, , to knots in thickened surfaces \Sigma \times [0,1] modulo an equivalence relation called stabilization/destabilization. Here \Sigma is required to be closed and oriented. Virtual knots were first introduced by .
|
|
Affine connections may also be used to define (affine) geodesics on a manifold, generalizing the straight lines of Euclidean space, although the geometry of those straight lines can be very different from usual Euclidean geometry; the main differences are encapsulated in the curvature of the connection.
|
|
Three-dimensional Euclidean space \R^3 is irreducible: all smooth 2-spheres in it bound balls. On the other hand, Alexander's horned sphere is a non-smooth sphere in \R^3 that does not bound a ball. Thus the stipulation that the sphere be smooth is necessary.
|
|
In mathematical analysis, a metric differential is a generalization of a derivative for a Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metric space. With this definition of a derivative, one can generalize Rademacher's theorem to metric space-valued Lipschitz functions.
|
|
In convex geometry, two disjoint convex sets in n-dimensional Euclidean space are separated by a hyperplane, a result called the hyperplane separation theorem. In machine learning, hyperplanes are a key tool to create support vector machines for such tasks as computer vision and natural language processing.
|
|
Adam G. Weyhaupt. New families of embedded triply periodic minimal surfaces of genus three in euclidean space. PhD thesis, Indiana University, 2006 Another approach to classifying TPMS is to examine their space groups. For surfaces containing lines the possible boundary polygons can be enumerated, providing a classification.
|
|
If the axiom of choice is assumed to be true, it can be proved that not all subsets of Euclidean space are Lebesgue measurable; examples of such sets include the Vitali set, and the non- measurable sets postulated by the Hausdorff paradox and the Banach–Tarski paradox.
|
|
Suppose that f is a function of more than one variable. For instance, :z = f(x,y) = x^2 + xy + y^2. The graph of this function defines a surface in Euclidean space. To every point on this surface, there are an infinite number of tangent lines.
|
|
Orthogonality and rotation of coordinate systems compared between left: Euclidean space through circular angle ϕ, right: in Minkowski spacetime through hyperbolic angle ϕ (red lines labelled c denote the worldlines of a light signal, a vector is orthogonal to itself if it lies on this line).
|
|
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by Ernst Witt in 1940.
|
|
An extension of metric multidimensional scaling, in which the target space is an arbitrary smooth non-Euclidean space. In cases where the dissimilarities are distances on a surface and the target space is another surface, GMDS allows finding the minimum-distortion embedding of one surface into another.
|
|
In mathematics, the annulus theorem (formerly called the annulus conjecture) states roughly that the region between two well-behaved spheres is an annulus. It is closely related to the stable homeomorphism conjecture (now proved) which states that every orientation-preserving homeomorphism of Euclidean space is stable.
|
|
Differential geometry can either be intrinsic (meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric, which determines how distances are measured near each point) or extrinsic (where the object under study is a part of some ambient flat Euclidean space).
|
|
Lindenstrauss, Varju, Random walks in the group of Euclidean isometries and self-similar measures, Duke Math. J., Band 165, 2016, pp. 1061–1127Varju, Random walks in euclidean space, Annals of Mathematics, Band 181, 2015, pp. 243–301 He received the 2016 EMS Prize and the 2018 Whitehead Prize.
|
|
In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Augustin Louis Cauchy (1789–1857), Bernhard Riemann (1826–1866), and Karl Weierstrass (1815–1897). It was also during this period that the differentiation was generalized to Euclidean space and the complex plane.
|
|
A Heronian tetrahedron (also called a Heron tetrahedron or perfect pyramid) is a tetrahedron whose edge lengths, face areas and volume are all integers. The faces must therefore all be Heronian triangles. Every Heronian tetrahedron can be arranged in Euclidean space so that its vertex coordinates are also integers.
|
|
Every differentiable surface is a topological surface, but the converse is false. For simplicity, unless otherwise stated, "surface" will mean a surface in the Euclidean space of dimension 3 or in . A surface that is not supposed to be included in another space is called an abstract surface.
|
|
Hellmuth Stachel wrote 3 books (in cooperation with other scholars) and approximately 120 scientific articles on classical and descriptive geometry, kinematics and the theory of mechanisms, as well as on computer aided design. He studied flexible polyhedra in the 4-dimensional Euclidean space and 3-dimensional Lobachevsky space.
|
|
Frenet–Serret frame, and the osculating plane (spanned by T and N). In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order of contact at the point. The word osculate is from the Latin osculatus which is a past participle of osculari, meaning to kiss. An osculating plane is thus a plane which "kisses" a submanifold. The osculating plane in the geometry of Euclidean space curves can be described in terms of the Frenet-Serret formulas as the linear span of the tangent and normal vectors.
|
|
Referring to the quadratic form applied to the displacement vector of a point on the quasi-sphere from the centre (i.e. ) as the radial scalar square, in any pseudo-Euclidean space the quasi-spheres may be separated into three disjoint sets: those with positive radial scalar square, those with negative radial scalar square, those with zero radial scalar square. In a space with a positive-definite quadratic form (i.e. a Euclidean space), a quasi-sphere with negative radial scalar square is the empty set, one with zero radial scalar square consists of a single point, one with positive radial scalar square is a standard -sphere, and one with zero curvature is a hyperplane that is partitioned with the -spheres.
|
|
The same set of points would not accumulate to any point of the open unit interval ; so the open unit interval is not compact. Euclidean space itself is not compact since it is not bounded. In particular, the sequence of points , which is not bounded, has no subsequence that converges to any real number. Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces are encountered in mathematical analysis, where the property of compactness of some topological spaces arises in the hypotheses or in the conclusions of many fundamental theorems, such as the Bolzano–Weierstrass theorem, the extreme value theorem, the Arzelà–Ascoli theorem, and the Peano existence theorem.
|
|
The discipline of stochastic geometry entails the mathematical study of random objects defined on some (often Euclidean) space. In the context of wireless networks, the random objects are usually simple points (which may represent the locations of network nodes such as receivers and transmitters) or shapes (for example, the coverage area of a transmitter) and the Euclidean space is either 3-dimensional, or more often, the (2-dimensional) plane, which represents a geographical region. In wireless networks (for example, cellular networks) the underlying geometry (the relative locations of nodes) plays a fundamental role due to the interference of other transmitters, whereas in wired networks (for example, the Internet) the underlying geometry is less important.
|
|
In n dimensions, an affine space group, or Bieberbach group, is a discrete subgroup of isometries of n-dimensional Euclidean space with a compact fundamental domain. proved that the subgroup of translations of any such group contains n linearly independent translations, and is a free abelian subgroup of finite index, and is also the unique maximal normal abelian subgroup. He also showed that in any dimension n there are only a finite number of possibilities for the isomorphism class of the underlying group of a space group, and moreover the action of the group on Euclidean space is unique up to conjugation by affine transformations. This answers part of Hilbert's eighteenth problem.
|
|
Every compact metric space is complete; the real line is non- compact but complete; the open interval (0,1) is incomplete. Every Euclidean space is also a complete metric space. Moreover, all geometric notions immanent to a Euclidean space can be characterized in terms of its metric. For example, the straight segment connecting two given points A and C consists of all points B such that the distance between A and C is equal to the sum of two distances, between A and B and between B and C. The Hausdorff dimension (related to the number of small balls that cover the given set) applies to metric spaces, and can be non-integer (especially for fractals).
|
|
When the two curves are embedded in a metric space other than Euclidean space, such as a polyhedral terrain or some Euclidean space with obstacles, the distance between two points on the curves is most naturally defined as the length of the shortest path between them. The leash is required to be a geodesic joining its endpoints. The resulting metric between curves is called the geodesic Fréchet distance... Cook and Wenk describe a polynomial-time algorithm to compute the geodesic Fréchet distance between two polygonal curves in a simple polygon. If we further require that the leash must move continuously in the ambient metric space, then we obtain the notion of the homotopic Fréchet distance.
|
|
Suppose M and N are two differentiable manifolds with dimensions m and n, respectively, and f is a function from M to N. Since differentiable manifolds are topological spaces we know what it means for f to be continuous. But what does "f is " mean for ? We know what that means when f is a function between Euclidean spaces, so if we compose f with a chart of M and a chart of N such that we get a map that goes from Euclidean space to M to N to Euclidean space we know what it means for that map to be . We define "f is " to mean that all such compositions of f with charts are .
|
|
Page 15 of: D. Leborgne Calcul différentiel et géométrie Puf (1982) . A slightly more general version is as follows:This version follows directly from the previous one because every convex compact subset of a Euclidean space is homeomorphic to a closed ball of the same dimension as the subset; see :;Convex compact set:Every continuous function from a convex compact subset K of a Euclidean space to K itself has a fixed point.V. & F. Bayart Point fixe, et théorèmes du point fixe on Bibmath.net. An even more general form is better known under a different name: :;Schauder fixed point theorem:Every continuous function from a convex compact subset K of a Banach space to K itself has a fixed point.
|
|
However, integer-distance straight line embeddings are known to exist for cubic graphs.. raised the question of whether every graph with a linkless embedding in three-dimensional Euclidean space has a linkless embedding in which all edges are represented by straight line segments, analogously to Fáry's theorem for two-dimensional embeddings.
|
|
In mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lipschitz continuous function. The term is named after the German mathematician Rudolf Lipschitz.
|
|
In a space of dimension three, it is a surface. For example, the equation :x_1^2+x_2^2+\cdots+x_n^2-1=0 defines an algebraic hypersurface of dimension in the Euclidean space of dimension . This hypersurface is also a smooth manifold, and is called a hypersphere or an -sphere.
|
|
Quaternions, one of the ways to describe rotations in three dimensions, consist of a four- dimensional space. Rotations between quaternions, for interpolation, for example, take place in four dimensions. Spacetime, which has three space dimensions and one time dimension is also four-dimensional, though with a different structure to Euclidean space.
|
|
In mathematics, the cone condition is a property which may be satisfied by a subset of a Euclidean space. Informally, it requires that for each point in the subset a cone with vertex in that point must be contained in the subset itself, and so the subset is "non-flat".
|
|
Many Dirac type operators have a covariance under conformal change in metric. This is true for the Dirac operator in euclidean space, and the Dirac operator on the sphere under Möbius transformations. Consequently this holds true for Dirac operators on conformally flat manifolds and conformal manifolds which are simultaneously spin manifolds.
|
|
There are objects called complex polyhedra, for which the underlying space is a complex Hilbert space rather than real Euclidean space. Precise definitions exist only for the regular complex polyhedra, whose symmetry groups are complex reflection groups. The complex polyhedra are mathematically more closely related to configurations than to real polyhedra..
|
|
The real numbers with the usual ordering form a totally ordered vector space. For all integers n ≥ 0, the Euclidean space ℝn considered as a vector space over the reals with the lexicographic ordering forms a preordered vector space whose order is Archimedean if and only if n = 0 or 1.
|
|
In a Euclidean space, two directed line segments, often called vectors in applied mathematics, are antiparallel, if they are supported by parallel lines and have opposite directions., Chapter 6, p. 332 In that case, one of the associated Euclidean vectors is the product of the other by a negative number.
|
|
When embedded to a Euclidean space (of a higher dimension), every point of a hyperbolic space is a saddle point. Another distinctive property is the amount of space covered by the 3-ball in hyperbolic 3-space: it increases exponentially with respect to the radius of the ball, rather than polynomially.
|
|
In geometry, the Killing–Hopf theorem states that complete connected Riemannian manifolds of constant curvature are isometric to a quotient of a sphere, Euclidean space, or hyperbolic space by a group acting freely and properly discontinuously. These manifolds are called space forms. The Killing–Hopf theorem was proved by and .
|
|
Suppose the n-dimensional Euclidean space is partitioned by r hyperplanes that are (n-1)-dimensional. What is the number of components in the partition? The largest number of components is attained when the hyperplanes are in general position, i.e, no two are parallel and no three have the same intersection.
|
|
The coordinate-wise mean of a point set is the centroid, which solves the same variational problem in the plane (or higher-dimensional Euclidean space) that the familiar average solves on the real line -- that is, the centroid has the smallest possible average squared distance to all points in the set.
|
|
The Fréchet distance and its variants find application in several problems, from morphing. and handwriting recognition. to protein structure alignment.. Alt and Godau. were the first to describe a polynomial-time algorithm to compute the Fréchet distance between two polygonal curves in Euclidean space, based on the principle of parametric search.
|
|
The Riemann integral can only integrate functions on a bounded interval. It can however be extended to unbounded intervals by taking limits, so long as this doesn't yield an answer such as . Integrating on structures other than Euclidean space. The Riemann integral is inextricably linked to the order structure of the real line.
|
|
In geometry, Jung's theorem is an inequality between the diameter of a set of points in any Euclidean space and the radius of the minimum enclosing ball of that set. It is named after Heinrich Jung, who first studied this inequality in 1901. Algorithms also exist to solve the smallest-circle problem explicitly.
|
|
An example of negatively curved space is hyperbolic geometry. A space or space-time with zero curvature is called flat. For example, Euclidean space is an example of a flat space, and Minkowski space is an example of a flat spacetime. There are other examples of flat geometries in both settings, though.
|
|
A meron or half-instanton is a Euclidean space-time solution of the Yang–Mills field equations. It is a singular non-self-dual solution of topological charge 1/2. The instanton is believed to be composed of two merons. A meron can be viewed as a tunneling event between two Gribov vacua.
|
|
In classical differential geometry, development refers to the simple idea of rolling one smooth surface over another in Euclidean space. For example, the tangent plane to a surface (such as the sphere or the cylinder) at a point can be rolled around the surface to obtain the tangent plane at other points.
|
|
In mathematics, Tarski's plank problem is a question about coverings of convex regions in n-dimensional Euclidean space by "planks": regions between two hyperplanes. Tarski asked if the sum of the widths of the planks must be at least the minimum width of the convex region. The question was answered affirmatively by .
|
|
Other authors consider a point process as a stochastic process, where the process is indexed by sets of the underlying space on which it is defined, such as the real line or n-dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes.
|
|
If the state space is the real line, then the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space. If the state space is n-dimensional Euclidean space, then the stochastic process is called a n-dimensional vector process or n-vector process.
|
|
This is achieved in three steps. First an ideal point at infinity is added to Euclidean space so that lines (or planes) can be regarded as circles (or spheres) passing through the point at infinity (i.e., having infinite radius). This extension is known as inversive geometry with automorphisms known as "Mobius transformations".
|
|
By definition, all Euclidean vectors have a magnitude (see above). However, the notion of magnitude cannot be applied to all kinds of vectors. A function that maps objects to their magnitudes is called a norm. A vector space endowed with a norm, such as the Euclidean space, is called a normed vector space.
|
|
In information theory, information dimension is an information measure for random vectors in Euclidean space, based on the normalized entropy of finely quantized versions of the random vectors. This concept was first introduced by Alfréd Rényi in 1959.See . Simply speaking, it is a measure of the fractal dimension of a probability distribution.
|
|
A classical theorem of Joseph Liouville shows that there are much fewer conformal maps in higher dimensions than in two dimensions. Any conformal map on a portion of Euclidean space of dimension three or greater can be composed from three types of transformations: a homothety, an isometry, and a special conformal transformation.
|
|
In mathematics, Fubini's theorem on differentiation, named after Guido Fubini, is a result in real analysis concerning the differentiation of series of monotonic functions. It can be proven by using Fatou's lemma and the properties of null sets.Jones, Frank (2001), Lebesgue Integration on Euclidean Space, Jones and Bartlett publishers, pp. 527–529.
|
|
In addition to general purpose calculators, there are those designed for specific markets. For example, there are scientific calculators which include trigonometric and statistical calculations. Some calculators even have the ability to do computer algebra. Graphing calculators can be used to graph functions defined on the real line, or higher-dimensional Euclidean space.
|
|
In Euclidean space, the hypervolume of an -facet of an -simplex is less than or equal to the sum of the hypervolumes of the other facets. In particular, the area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three sides.
|
|
An N×N Euclidean random matrix  is defined with the help of an arbitrary deterministic function f(r, r′) and of N points {ri} randomly distributed in a region V of d-dimensional Euclidean space. The element Aij of the matrix is equal to f(ri, rj): Aij = f(ri, rj).
|
|
Examples of topologies include the Zariski topology in algebraic geometry that reflects the algebraic nature of varieties, and the topology on a differential manifold in differential topology where each point within the space is contained in an open set that is homeomorphic to an open ball in a finite-dimensional Euclidean space.
|
|
The Socolar-Taylor tile forms two-dimensional aperiodic tilings, but is defined by combinatorial matching conditions rather than purely by its shape. In higher dimensions, the problem is solved: the Schmitt-Conway-Danzer tile is the prototile of a monohedral aperiodic tiling of three-dimensional Euclidean space, and cannot tile space periodically.
|
|
Thus, in three dimensions, it is impossible to make the left hand of a human figure into the right hand of the figure by applying a rotation alone, but it is possible to do so by reflecting the figure in a mirror. As a result, in the three-dimensional Euclidean space, the two possible basis orientations are called right-handed and left-handed (or right-chiral and left-chiral). The orientation on a real vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, right- handed bases are typically declared to be positively oriented, but the choice is arbitrary, as they may also be assigned a negative orientation.
|
|
Some authors restrict the phrase "toroidal polyhedra" to mean more specifically polyhedra topologically equivalent to the (genus 1) torus.. In this area, it is important to distinguish embedded toroidal polyhedra, whose faces are flat polygons in three-dimensional Euclidean space that do not cross themselves or each other, from abstract polyhedra, topological surfaces without any specified geometric realization.. Intermediate between these two extremes are polyhedra formed by geometric polygons or star polygons in Euclidean space that are allowed to cross each other. In all of these cases the toroidal nature of a polyhedron can be verified by its orientability and by its Euler characteristic being non-positive. The Euler characteristic generalizes to V − E + F = 2 − 2N, where N is the number of holes.
|
|
Since the introduction, at the end of 19th century, of Non-Euclidean geometries, many sorts of spaces have been considered, about which one can do geometric reasoning in the same way as with Euclidean spaces. In general, they share some properties with Euclidean spaces, but may also have properties that could appear as rather strange. Some of these spaces use Euclidean geometry for their definition, or can be modeled as subspaces of a Euclidean space of higher dimension. When such a space is defined by geometrical axioms, embedding the space in a Euclidean space is a standard way for proving consistency of its definition, or, more precisely for proving that its theory is consistent, if Euclidean geometry is consistent (which cannot be proved).
|
|
Carl Friedrich Gauss in 1828 In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves.
|
|
The crystallographic restriction theorem can be formulated in terms of isometries of Euclidean space. A set of isometries can form a group. By a discrete isometry group we will mean an isometry group that maps each point to a discrete subset of RN, i.e. the orbit of any point is a set of isolated points.
|
|
In mathematics, a time dependent vector field is a construction in vector calculus which generalizes the concept of vector fields. It can be thought of as a vector field which moves as time passes. For every instant of time, it associates a vector to every point in a Euclidean space or in a manifold.
|
|
Lie groups (in honor of Sophus Lie) are groups which also have a manifold structure, i.e., they are spaces looking locally like some Euclidean space of the appropriate dimension. Again, the additional structure, here the manifold structure, has to be compatible, i.e., the maps corresponding to multiplication and the inverse have to be smooth.
|
|
The notion of an "apparent horizon" begins with the notion of a trapped null surface. A (compact, orientable, spacelike) surface always has two independent forward-in-time pointing, lightlike, normal directions. For example, a (spacelike) sphere in Minkowski space has lightlike vectors pointing inward and outward along the radial direction. In Euclidean space (i.e.
|
|
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis and discrete mathematics. It has close connections to convex analysis, optimization and functional analysis and important applications in number theory. Convex geometry dates back to antiquity. Archimedes gave the first known precise definition of convexity.
|
|
The set of proper rigid transformation is called special Euclidean group, denoted SE(n). In kinematics, proper rigid transformations in a 3-dimensional Euclidean space, denoted SE(3), are used to represent the linear and angular displacement of rigid bodies. According to Chasles' theorem, every rigid transformation can be expressed as a screw displacement.
|
|
An example of generalized convexity is orthogonal convexity.Rawlins G.J.E. and Wood D, "Ortho-convexity and its generalizations", in: Computational Morphology, 137-152. Elsevier, 1988. A set in the Euclidean space is called orthogonally convex or ortho-convex, if any segment parallel to any of the coordinate axes connecting two points of lies totally within .
|
|
Other structures considered on include the one of a pseudo-Euclidean space, symplectic structure (even ), and contact structure (odd ). All these structures, although can be defined in a coordinate-free manner, admit standard (and reasonably simple) forms in coordinates. is also a real vector subspace of which is invariant to complex conjugation; see also complexification.
|
|
David Bagsby (born September 4, 1960) is an American musician, film maker, photographer and composer born in Tulsa, Oklahoma.bagsby.com Bagsby has studied under Jazz guitarist Stanley Jordan in St. Louis, Missouri. His music encompasses progressive/symphonic rock, humanly impossible rhythmic hierarchies, translated nature recordings, acoustic guitar works and pieces utilizing perfect tuning systems & Euclidean Space.
|
|
In geometry the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition.Jacobson (2009), p. 34, Ex. 14. By definition, a rotation about the origin is a linear transformation that preserves length of vectors (it is an isometry) and preserves orientation (i.e.
|
|
In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1\. For a lattice in n-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamental domain for the lattice be 1\. The E8 lattice and the Leech lattice are two famous examples.
|
|
The dihedral angle between two non-parallel hyperplanes of a Euclidean space is the angle between the corresponding normal vectors. The product of the transformations in the two hyperplanes is a rotation whose axis is the subspace of codimension 2 obtained by intersecting the hyperplanes, and whose angle is twice the angle between the hyperplanes.
|
|
Cartesian coordinates Analytic geometry is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes in three dimensions. Geometrically, one studies the Euclidean plane (2 dimensions) and Euclidean space (3 dimensions).
|
|
There is an analogous problem in geometric measure theory which asks the following: under what conditions may a subset E of Euclidean space be contained in a rectifiable curve (that is, when is there a curve with finite length that visits every point in E)? This problem is known as the analyst's travelling salesman problem.
|
|
Once again, the chain rule guarantees that the idea of differentiability does not depend on which charts of the atlases on M and N are selected. However, defining the derivative itself is more subtle. If M or N is itself already a Euclidean space, then we don't need a chart to map it to one.
|
|
Campbell's theorem states that any n-dimensional Riemannian manifold can be embedded locally in an (n + 1)-manifold with a Ricci curvature of R'a b = 0\. The theorem also states, in similar form, that an n-dimensional pseudo-Riemannian manifold can be both locally and isometrically embedded in an n(n + 1)/2-pseudo-Euclidean space.
|
|
In 1973, James Michael and Leon Simon established a Sobolev inequality for functions on submanifolds of Euclidean space, in a form which is adapted to the mean curvature of the submanifold and takes on a special form for minimal submanifolds.J.H. Michael and L.M. Simon. Sobolev and mean-value inequalities on generalized submanifolds of . Comm. Pure Appl. Math.
|
|
Like the Möbius strip, the Klein bottle is a two-dimensional manifold which is not orientable. Unlike the Möbius strip, the Klein bottle is a closed manifold, meaning it is a compact manifold without boundary. While the Möbius strip can be embedded in three-dimensional Euclidean space R3, the Klein bottle cannot. It can be embedded in R4, however.
|
|
It is said that the group acts on the space or structure. If a group acts on a structure, it also acts on everything that is built on the structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. In particular, it acts on the set of all triangles.
|
|
Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. It is the most familiar of the theories of physics. The concepts it covers, such as mass, acceleration, and force, are commonly used and known. The subject is based upon a three-dimensional Euclidean space with fixed axes, called a frame of reference.
|
|
The fine topology on the Euclidean space \R^n is defined to be the coarsest topology making all subharmonic functions (equivalently all superharmonic functions) continuous. Concepts in the fine topology are normally prefixed with the word 'fine' to distinguish them from the corresponding concepts in the usual topology, as for example 'fine neighbourhood' or 'fine continuous'.
|
|
In mathematics, the Brunn–Minkowski theorem (or Brunn–Minkowski inequality) is an inequality relating the volumes (or more generally Lebesgue measures) of compact subsets of Euclidean space. The original version of the Brunn–Minkowski theorem (Hermann Brunn 1887; Hermann Minkowski 1896) applied to convex sets; the generalization to compact nonconvex sets stated here is due to Lazar Lyusternik (1935).
|
|
That is all for lattices in the plane. In n-dimensional Euclidean space the same questions can be posed, for example about SO(n)\SLn(R)/SLn(Z). This is harder to compactify. There are a variety of compactifications, such as the Borel–Serre compactification, the reductive Borel-Serre compactification, and the Satake compactifications, that can be formed.
|
|
In mathematics, particularly in the subfield of real analytic geometry, a subanalytic set is a set of points (for example in Euclidean space) defined in a way broader than for semianalytic sets (roughly speaking, those satisfying conditions requiring certain real power series to be positive there). Subanalytic sets still have a reasonable local description in terms of submanifolds.
|
|
The triskelion has 3-fold rotational symmetry. Rotational symmetry is symmetry with respect to some or all rotations in -dimensional Euclidean space. Rotations are direct isometries, which are isometries that preserve orientation.Vladimir G. Ivancevic, Tijana T. Ivancevic (2005) Natural Biodynamics World Scientific Therefore, a symmetry group of rotational symmetry is a subgroup of the special Euclidean group E+().
|
|
A few examples are the following. The rotation group is a symmetry of Euclidean space and all fundamental laws of physics, the Lorentz group is a symmetry group of spacetime of special relativity. The special unitary group is the symmetry group of quantum chromodynamics and the symplectic group finds application in Hamiltonian mechanics and quantum mechanical versions of it.
|
|
The hypothesis also says that we can have equality only when are all equal. In this case, their geometric mean has the same value, Hence, unless are all equal, we have . This completes the proof. This technique can be used in the same manner to prove the generalized AM–GM inequality and Cauchy–Schwarz inequality in Euclidean space .
|
|
In statistics, Wombling is any of a number of techniques used for identifying zones of rapid change, typically in some quantity as it varies across some geographical or Euclidean space. It is named for statistician William H. Womble. The technique may be applied to gene frequency in a population of organisms, and to evolution of language.
|
|
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space Rn, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the German mathematician Carl Friedrich Gauss. One reason why Gaussian measures are so ubiquitous in probability theory is the central limit theorem.
|
|
In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n = 8, the set of all such locations is called 8-dimensional space. Often such spaces are studied as vector spaces, without any notion of distance. Eight-dimensional Euclidean space is eight-dimensional space equipped with the Euclidean metric.
|
|
For example, in geometry, Poincaré believed that the structure of non-Euclidean space can be known analytically. Poincaré held that convention plays an important role in physics. His view (and some later, more extreme versions of it) came to be known as "conventionalism".Yemima Ben-Menahem, Conventionalism: From Poincare to Quine, Cambridge University Press, 2006, p. 39.
|
|
The hemicube should not be confused with the demicube – the hemicube is a projective polyhedron, while the demicube is an ordinary polyhedron (in Euclidean space). While they both have half the vertices of a cube, the hemicube is a quotient of the cube, while the vertices of the demicube are a subset of the vertices of the cube.
|
|
Hadwiger proved a theorem characterizing eutactic stars, systems of points in Euclidean space formed by orthogonal projection of higher-dimensional cross polytopes. He found a higher-dimensional generalization of the space-filling Hill tetrahedra.. And his 1957 book Vorlesungen über Inhalt, Oberfläche und Isoperimetrie was foundational for the theory of Minkowski functionals, used in mathematical morphology.
|
|
By definition, differentiable manifolds of a fixed dimension are all locally diffeomorphic to Euclidean space, so aside from dimension, there are no local invariants. Thus, differentiable structures on a manifold are topological in nature. By contrast, the curvature of a Riemannian manifold is a local (indeed, infinitesimal) invariant (and is the only local invariant under isometry).
|
|
In mathematics, Alexander duality refers to a duality theory presaged by a result of 1915 by J. W. Alexander, and subsequently further developed, particularly by Pavel Alexandrov and Lev Pontryagin. It applies to the homology theory properties of the complement of a subspace X in Euclidean space, a sphere, or other manifold. It is generalized by Spanier–Whitehead duality.
|
|
These can be generalized to tessellations of other spaces, especially uniform tessellations, notably tilings of Euclidean space (honeycombs), which have exceptional objects, and tilings of hyperbolic space. There are various exceptional objects in dimension below 6, but in dimension 6 and above, the only regular polyhedra/tilings/hyperbolic tilings are the simplex, hypercube, cross- polytope, and hypercube lattice.
|
|
For tetrahedra in hyperbolic space or in three- dimensional elliptic geometry, the dihedral angles of the tetrahedron determine its shape and hence its volume. In these cases, the volume is given by the Murakami–Yano formula. However, in Euclidean space, scaling a tetrahedron changes its volume but not its dihedral angles, so no such formula can exist.
|
|
In geometry, space partitioning is the process of dividing a space (usually a Euclidean space) into two or more disjoint subsets (see also partition of a set). In other words, space partitioning divides a space into non-overlapping regions. Any point in the space can then be identified to lie in exactly one of the regions.
|
|
The tesseract, like all hypercubes, tessellates Euclidean space. The self-dual tesseractic honeycomb consisting of 4 tesseracts around each face has Schläfli symbol {4,3,3,4}. Hence, the tesseract has a dihedral angle of 90°. The tesseract's radial equilateral symmetry makes its tessellation the unique regular body-centered cubic lattice of equal-sized spheres, in any number of dimensions.
|
|
The Universal Book of Mathematics provides the following information about pleated surfaces: It is a surface in Euclidean space or hyperbolic space that resembles a polyhedron in the sense that it has flat faces that meet along edges. Unlike a polyhedron, a pleated surface has no corners, but it may have infinitely many edges that form a lamination.
|
|
In mathematics, flat convergence is a notion for convergence of submanifolds of Euclidean space. It was first introduced by Hassler Whitney in 1957, and then extended to integral currents by Federer and Fleming in 1960. It forms a fundamental part of the field of geometric measure theory. The notion was applied to find solutions to Plateau's problem.
|
|
The property of being locally Euclidean is preserved by local homeomorphisms. That is, if X is locally Euclidean of dimension n and f : Y → X is a local homeomorphism, then Y is locally Euclidean of dimension n. In particular, being locally Euclidean is a topological property. Manifolds inherit many of the local properties of Euclidean space.
|
|
The roots of integrative neuroscience originated from the Rashevsky-Rosen school of relational biology that characterizes functional organization mathematically by abstracting away the structure (i.e., physics and chemistry). It was further expanded by Chauvet who introduced hierarchical and functional integration. Hierarchical integration is structural involving spatiotemporal dynamic continuity in Euclidean space to bring about functional organization, viz.
|
|
Subsequently, he worked on multiple Fourier series, posing the question of the Bochner–Riesz means. This led to results on how the Fourier transform on Euclidean space behaves under rotations. In differential geometry, Bochner's formula on curvature from 1946 was published. Joint work with Kentaro Yano (1912–1993) led to the 1953 book Curvature and Betti Numbers.
|
|
A subset of the Euclidean -space is open if, for every point in , there exists a positive real number (depending on ) such that a point in belongs to as soon as its Euclidean distance from is smaller than . Equivalently, a subset of is open if every point in is the center of an open ball contained in .
|
|
Norbert Wiener In mathematics, classical Wiener space is the collection of all continuous functions on a given domain (usually a sub-interval of the real line), taking values in a metric space (usually n-dimensional Euclidean space). Classical Wiener space is useful in the study of stochastic processes whose sample paths are continuous functions. It is named after the American mathematician Norbert Wiener.
|
|
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.Jacobson (2009), p. 34, Ex. 14. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e.
|
|
However, the real plane has exactly the same homotopy groups as a solitary point (as does a Euclidean space of any dimension), and the real plane with a point removed has the same groups as a circle, so groups alone are not enough to distinguish spaces. Although the loss of discrimination power is unfortunate, it can also make certain computations easier.
|
|
Bour's surface. Bour's surface, leaving out the points with r < 0.5 to show the self-crossings more clearly. In mathematics, Bour's minimal surface is a two-dimensional minimal surface, embedded with self-crossings into three- dimensional Euclidean space. It is named after Edmond Bour, whose work on minimal surfaces won him the 1861 mathematics prize of the French Academy of Sciences..
|
|
Levi graphs of configurations are biregular, and every biregular graph with girth at least six can be viewed as the Levi graph of an abstract configuration.. Levi graphs may also be defined for other types of incidence structure, such as the incidences between points and planes in Euclidean space. For every Levi graph, there is an equivalent hypergraph, and vice versa.
|
|
This iterative partitioning process is similar to that of a -d tree, but uses circular (or spherical, hyperspherical, etc.) rather than rectilinear partitions. In two-dimensional Euclidean space, this can be visualized as a series of circles segregating the data. The vantage-point tree is particularly useful in dividing data in a non- standard metric space into a metric tree.
|
|
The definition of an atlas depends on the notion of a chart. A chart for a topological space M (also called a coordinate chart, coordinate patch, coordinate map, or local frame) is a homeomorphism \varphi from an open subset U of M to an open subset of a Euclidean space. The chart is traditionally recorded as the ordered pair (U, \varphi).
|
|
A user of a LBS must be able to seamlessly convert from a Euclidean space (Cartesian Reference Space), to a Linear Reference Space (LRS), to indoor space (to include perhaps the floor, wing, hallway, and room number).Hong, Sang-Ki, 2008. Ubiquitous Geographic Information (UBGI) and address standards. ISO Workshop on address standards: Considering the issues related to an international address standard.
|
|
In mathematics, an exotic \R^4 is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space \R^4. The first examples were found in 1982 by Michael Freedman and others, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds.Kirby (1989), p. 95Freedman and Quinn (1990), p.
|
|
In mathematics, particularly, in asymptotic convex geometry, Milman's reverse Brunn–Minkowski inequality is a result due to Vitali Milman that provides a reverse inequality to the famous Brunn-Minkowski inequality for convex bodies in n-dimensional Euclidean space Rn. Namely, it bounds the volume of the Minkowski sum of two bodies from above in terms of the volumes of the bodies.
|
|
It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space. Just as an ordinary sphere (or 2-sphere) is a two-dimensional surface that forms the boundary of a ball in three dimensions, a 3-sphere is an object with three dimensions that forms the boundary of a ball in four dimensions.
|
|
A line in 3-dimensional space is not a hyperplane, and does not separate the space into two parts (the complement of such a line is connected). Any hyperplane of a Euclidean space has exactly two unit normal vectors. Affine hyperplanes are used to define decision boundaries in many machine learning algorithms such as linear-combination (oblique) decision trees, and perceptrons.
|
|
In a universe with zero curvature, the local geometry is flat. The most obvious global structure is that of Euclidean space, which is infinite in extent. Flat universes that are finite in extent include the torus and Klein bottle. Moreover, in three dimensions, there are 10 finite closed flat 3-manifolds, of which 6 are orientable and 4 are non-orientable.
|
|
A table of all prime knots with seven crossings or fewer (not including mirror images). Overhand knot becomes a trefoil knot by joining the ends. The triangle is associated with the trefoil knot. pretzel link knot In mathematics, a knot is an embedding of a topological circle in 3-dimensional Euclidean space, (also known as ), considered up to continuous deformations (isotopies).
|
|
A Euclidean space is an affine space equipped with a metric. Affine spaces have many other uses in mathematics. In particular, as they are defined over any field, they allow doing geometry in other contexts. As soon as non-linear questions are considered, it is generally useful to consider affine spaces over the complex numbers as an extension of Euclidean spaces.
|
|
This expression also shows that the computation of partial derivatives reduces to the computation of one-variable derivatives. An important example of a function of several variables is the case of a scalar-valued function on a domain in Euclidean space Rn (e.g., on R2 or R3). In this case f has a partial derivative ∂f/∂xj with respect to each variable xj.
|
|
This approach is particularly simple for an embedded surface. Thanks to a result of , the connection 1-form on a surface embedded in Euclidean space is just the pullback under the Gauss map of the connection 1-form on . Using the identification of with the homogeneous space , the connection 1-form is just a component of the Maurer–Cartan 1-form on ..
|
|
In the mathematical field of differential geometry, more precisely, the theory of surfaces in Euclidean space, the Bonnet theorem states that the first and second fundamental forms determine a surface in R3 uniquely up to a rigid motion.. It was proven by Pierre Ossian Bonnet in about 1860. This is not to be confused with the Bonnet–Myers theorem or Gauss–Bonnet theorem.
|
|
A topological space X is said to be equidimensional if for all points p in X the dimension at p that is, dim p(X) is constant. The Euclidean space is an example of an equidimensional space. The disjoint union of two spaces X and Y (as topological spaces) of different dimension is an example of a non- equidimensional space.
|
|
Ioannis "John" N. Hazzidakis (Ιωάννης Χατζιδάκις, or Hatzidakis or Chatzidakis, 1844 – 1921) was a Greek mathematician, known for the Hazzidakis transform in differential geometry. Erratum: Trans. Amer. Math. Soc. 8 (1907), 535 The Hazzidakis formula for the Hazzidakis transform can be applied in proving Hilbert's theorem on negative curvature, stating that hyperbolic geometry does not have a model in 3-dimensional Euclidean space.
|
|
The localization (or global positioning) problem emerges naturally in IoT sensor networks. The problem is to recover the sensor map in Euclidean space from a local or partial set of pairwise distances. Thus it is a matrix completion problem with rank two if the sensors are located in a 2-D plane and three if they are in a 3-D space.
|
|
His first work on this subject, in his thesis, concerned the compactness degree of a space: this is a number, defined to be −1 for a compact space, and 1 + x if every point in the space has a neighbourhood the boundary of which has compactness degree x. He made an important conjecture, only solved much later in 1982 by Pol and 1988 by Kimura, that the compactness degree was the same as the minimum dimension of a set that could be adjoined to the space to compactify it. Thus, for instance the familiar Euclidean space has compactness degree zero; it is not compact itself, but every point has a neighborhood bounded by a compact sphere. This compactness degree, zero, equals the dimension of the single point that may be added to Euclidean space to form its one-point compactification.
|
|
In the following year, Hadamard generalised the theorem under discussion to an arbitrary finite dimension, but he employed different methods. Hans Freudenthal comments on the respective roles as follows: "Compared to Brouwer's revolutionary methods, those of Hadamard were very traditional, but Hadamard's participation in the birth of Brouwer's ideas resembles that of a midwife more than that of a mere spectator." Brouwer's approach yielded its fruits, and in 1910 he also found a proof that was valid for any finite dimension, as well as other key theorems such as the invariance of dimension.If an open subset of a manifold is homeomorphic to an open subset of a Euclidean space of dimension n, and if p is a positive integer other than n, then the open set is never homeomorphic to an open subset of a Euclidean space of dimension p.
|
|
In mathematics and theoretical physics, a quasi-sphere is a generalization of the hypersphere and the hyperplane to the context of a pseudo-Euclidean space. It may be described as the set of points for which the quadratic form for the space applied to the displacement vector from a centre point is a constant value, with the inclusion of hyperplanes as a limiting case.
|
|
The basic object used in mesh modeling is a vertex, a point in three-dimensional space. Two vertices connected by a straight line become an edge. Three vertices, connected to each other by three edges, define a triangle, which is the simplest polygon in Euclidean space. More complex polygons can be created out of multiple triangles, or as a single object with more than 3 vertices.
|
|
In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set- valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing it. The Kakutani fixed point theorem is a generalization of Brouwer fixed point theorem.
|
|
Parallel transport Polyhedral space a simplicial complex with a metric such that each simplex with induced metric is isometric to a simplex in Euclidean space. Principal curvature is the maximum and minimum normal curvatures at a point on a surface. Principal direction is the direction of the principal curvatures. Path isometry Proper metric space is a metric space in which every closed ball is compact.
|
|
In geometry a parallelohedron is a polyhedron that can be translated without rotations in 3-dimensional Euclidean space to fill space with a honeycomb in which all copies of the polyhedron meet face-to-face. There are five types of parallelohedron, first identified by Evgraf Fedorov in 1885 in his studies of crystallographic systems: the cube, hexagonal prism, rhombic dodecahedron, elongated dodecahedron, and truncated octahedron.
|
|
One can tessellate 4-dimensional Euclidean space by regular 16-cells. This is called the 16-cell honeycomb and has Schläfli symbol {3,3,4,3}. Hence, the 16-cell has a dihedral angle of 120°. Each 16-cell has 16 neighbors with which it shares a tetrahedron, 24 neighbors with which it shares only an edge, and 72 neighbors with which it shares only a single point.
|
|
In mathematics, the Minkowski–Steiner formula is a formula relating the surface area and volume of compact subsets of Euclidean space. More precisely, it defines the surface area as the "derivative" of enclosed volume in an appropriate sense. The Minkowski–Steiner formula is used, together with the Brunn–Minkowski theorem, to prove the isoperimetric inequality. It is named after Hermann Minkowski and Jakob Steiner.
|
|
If n is some positive integer, we can consider the set of all invertible n by n matrices over the reals, say. This is a group with matrix multiplication as operation. It is called the general linear group, GL(n). Geometrically, it contains all combinations of rotations, reflections, dilations and skew transformations of n-dimensional Euclidean space that fix a given point (the origin).
|
|
Nb3Sn is used for some high field applications, for example high-end MRI scanners and NMR spectrometers. A relaxed form of the Voronoi diagram of the A15 phase seems to have the least surface area among all the possible partitions of three-dimensional Euclidean space in regions of equal volume. This partition, also known as the Weaire–Phelan structure, is often present in clathrate hydrates.
|
|
Stress majorization is an optimization strategy used in multidimensional scaling (MDS) where, for a set of n m-dimensional data items, a configuration X of n points in r(<\sigma(X). Usually r is 2 or 3, i.e. the (n x r) matrix X lists points in 2- or 3-dimensional Euclidean space so that the result may be visualised (i.e. an MDS plot).
|
|
Typically, the Cartesian coordinates of the elements of a Euclidean space form a real coordinate spaces. This explains the name of coordinate space and the fact that geometric terms are often used when working with coordinate spaces. For example, is a plane. Coordinate spaces are widely used in geometry and physics, as their elements allow locating points in Euclidean spaces, and computing with them.
|
|
Gerald Budge Folland is an American mathematician and a professor of mathematics at the University of Washington. His areas of interest are harmonic analysis (on both Euclidean space and Lie groups), differential equations, and mathematical physics. The title of his doctoral dissertation at Princeton University (1971) is "The Tangential Cauchy-Riemann Complex on Spheres". He is the author of several textbooks in mathematical analysis.
|
|
In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4. In this article rotation means rotational displacement. For the sake of uniqueness rotation angles are assumed to be in the segment except where mentioned or clearly implied by the context otherwise.
|
|
Peter's studies specialise in areas such as multiple points of immersions of manifolds in Euclidean space. He has also taught the history of mathematics and probability theory. In 1997 Cambridge University Press published his book 'Introduction to mathematical reasoning: numbers, sets and functions’. As a research mathematician, Eccles specialised in topology and homotopy theory, publishing numerous journal papers in this area of study[4].
|
|
Each point in this polytope is a fractional matching. For example, in the triangle graph there are 3 edges, and the corresponding linear program has the following 6 constraints: > Maximize x1+x2+x3 Subject to: x1≥0, x2≥0, x3≥0. __________ x1+x2≤1, x2+x3≤1, > x3+x1≤1. This set of inequalities represents a polytope in R3 \- the 3-dimensional Euclidean space.
|
|
In mathematics, a sequence of n real numbers can be understood as a location in n-dimensional space. When n = 7, the set of all such locations is called 7-dimensional space. Often such a space is studied as a vector space, without any notion of distance. Seven-dimensional Euclidean space is seven-dimensional space equipped with a Euclidean metric, which is defined by the dot product.
|
|
In three-dimensional Euclidean space, the densest packing of equal spheres is achieved by a family of structures called close-packed structures. One method for generating such a structure is as follows. Consider a plane with a compact arrangement of spheres on it. Call it A. For any three neighbouring spheres, a fourth sphere can be placed on top in the hollow between the three bottom spheres.
|
|
Parallel transport of polarization vectors along such sphere gives rise to Thomas precession, which is analogous to the rotation of the swing plane of Foucault pendulum due to parallel transport along a sphere S2 in 3-dimensional Euclidean space. In physics, the evolution of such systems is determined by geometric phases."Geometric Phases in Physics", eds. Frank Wilczek and Alfred Shapere (World Scientific, Singapore, 1989).
|
|
The Alexander horned sphere is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting with a standard torus:. #Remove a radial slice of the torus. #Connect a standard punctured torus to each side of the cut, interlinked with the torus on the other side. #Repeat steps 1–2 on the two tori just added ad infinitum.
|
|
In special relativity the orthogonal complement is used to determine the simultaneous hyperplane at a point of a world line. The bilinear form η used in Minkowski space determines a pseudo-Euclidean space of events. The origin and all events on the light cone are self-orthogonal. When a time event and a space event evaluate to zero under the bilinear form, then they are hyperbolic-orthogonal.
|
|
The volume of a metric ball of radius r in a complete n-dimensional Riemannian manifold with positive Ricci curvature has volume at most that of the volume of a ball of the same radius r in Euclidean space. #Gromov's compactness theorem. The set of all Riemannian manifolds with positive Ricci curvature and diameter at most D is pre-compact in the Gromov-Hausdorff metric.
|
|
Matrices allow arbitrary linear transformations to be displayed in a consistent format, suitable for computation. This also allows transformations to be concatenated easily (by multiplying their matrices). Linear transformations are not the only ones that can be represented by matrices. Some transformations that are non-linear on an n-dimensional Euclidean space Rn can be represented as linear transformations on the n+1-dimensional space Rn+1.
|
|
A conservative force can be expressed in the language of differential geometry as a closed form. As Euclidean space is contractible, its de Rham cohomology vanishes, so every closed form is also an exact form, and can be expressed as the gradient of a scalar field. This gives a mathematical justification of the fact that all conservative forces are gradients of a potential field.
|
|
The elements of are called points and are commonly denoted by capital letters. The elements of \overrightarrow E are called Euclidean vectors or free vectors. They are also called translations, although, properly speaking, a translation is the geometric transformation resulting of the action of a Euclidean vector on the Euclidean space. The action of a translation on a point provides a point that is denoted .
|
|
Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple. In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. Similarly, Euclidean space is given coordinates where every point has three coordinates.
|
|
Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric. This is a concept of distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point. Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, although they still resemble the Euclidean space at each point infinitesimally, i.e. in the first order of approximation.
|
|
The problem of finding the local maxima and minima subject to constraints can be generalized to finding local maxima and minima on a differentiable manifold M. In what follows, it is not necessary that M be a Euclidean space, or even a Riemannian manifold. All appearances of the gradient abla (which depends on a choice of Riemannian metric) can be replaced with the exterior derivative d.
|
|
In Riemannian geometry, a Bryant surface is a 2-dimensional surface embedded in 3-dimensional hyperbolic space with constant mean curvature equal to 1... These surfaces take their name from the geometer Robert Bryant, who proved that every simply-connected minimal surface in 3-dimensional Euclidean space is isometric to a Bryant surface by a holomorphic parameterization analogous to the (Euclidean) Weierstrass–Enneper parameterization..
|
|
The neighbourhood swept out has similar properties to balls in Euclidean space, namely any two points in it are joined by a unique geodesic. This property is called "geodesic convexity" and the coordinates are called "normal coordinates". The explicit calculation of normal coordinates can be accomplished by considering the differential equation satisfied by geodesics. The convexity properties are consequences of Gauss's lemma and its generalisations.
|
|
Then () and () hold for every vector in -dimensional Euclidean space, where the exponent in the definition of denotes the transpose. #Equations () and () hold for an arbitrary linear operator as long as is the difference of and the identity operator . #The probabilistic versions (), () and () can be generalized to every finite measure space. For textbook presentations of the probabilistic Schuette–Nesbitt formula () and their applications to actuarial science, cf. .
|
|
If X is a compact subspace of Euclidean space, the cone on X is homeomorphic to the union of segments from X to any fixed point v ot\in X such that these segments intersect only by v itself. That is, the topological cone agrees with the geometric cone for compact spaces when the latter is defined. However, the topological cone construction is more general.
|
|
Bi-dimensional Cartesian coordinate system Two-dimensional space (also known as bi-dimensional space) is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point). The set of pairs of real numbers with appropriate structure often serves as the canonical example of a two-dimensional Euclidean space. For a generalization of the concept, see dimension.
|
|
The sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees. This fact is equivalent to Euclid's parallel postulate. This allows determination of the measure of the third angle of any triangle, given the measure of two angles. An exterior angle of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle.
|
|
The transition from "Euclidean" to "topological" is forgetful. Topology distinguishes continuous from discontinuous, but does not distinguish rectilinear from curvilinear. Intuition tells us that the Euclidean structure cannot be restored from the topology. A proof uses an automorphism of the topological space (that is, self-homeomorphism) that is not an automorphism of the Euclidean space (that is, not a composition of shifts, rotations and reflections).
|
|
For a Euclidean space, the Hausdorff dimension is equal to n. Uniform spaces do not introduce distances, but still allow one to use uniform continuity, Cauchy sequences (or filters or nets), completeness and completion. Every uniform space is also a topological space. Every linear topological space (metrizable or not) is also a uniform space, and is complete in finite dimension but generally incomplete in infinite dimension.
|
|
On each open subset of the topological space, the sheaf specifies a collection of functions, called "regular functions". The topological space and the structure sheaf together are required to satisfy conditions that mean the functions come from algebraic operations. Like manifolds, schemes are defined as spaces that are locally modeled on a familiar space. In the case of manifolds, the familiar space is Euclidean space.
|
|
More generally, if a given manifold is thought of as an embedded submanifold of Euclidean space, then one can picture a tangent space in this literal fashion. This was the traditional approach toward defining parallel transport. Many authors in differential geometry and general relativity use it. : More strictly, this defines an affine tangent space, which is distinct from the space of tangent vectors described by modern terminology.
|
|
Vector optimization is a subarea of mathematical optimization where optimization problems with a vector-valued objective functions are optimized with respect to a given partial ordering and subject to certain constraints. A multi-objective optimization problem is a special case of a vector optimization problem: The objective space is the finite dimensional Euclidean space partially ordered by the component-wise "less than or equal to" ordering.
|
|
In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem. Roughly speaking, the theorem asserts that if A is a closed subset of a Euclidean space, then it is possible to extend a given function of A in such a way as to have prescribed derivatives at the points of A. It is a result of Hassler Whitney.
|
|
For any d-dimensional Riemannian manifold the equilateral dimension is at least . For a d-dimensional sphere, the equilateral dimension is , the same as for a Euclidean space of one higher dimension into which the sphere can be embedded. At the same time as he posed Kusner's conjecture, Kusner asked whether there exist Riemannian metrics with bounded dimension as a manifold but arbitrarily high equilateral dimension.
|
|
An ideal, unfrustrated, structure is defined in this curved space. Then, specific distortions are applied to this ideal template in order to embed it into three dimensional Euclidean space. The final structure is a mixture of ordered regions, where the local order is similar to that of the template, and defects arising from the embedding. Among the possible defects, disclinations play an important role.
|
|
The frustration has a topological character: it is impossible to fill Euclidean space with tetrahedra, even severely distorted, if we impose that a constant number of tetrahedra (here five) share a common edge. The next step is crucial: the search for an unfrustrated structure by allowing for curvature in the space, in order for the local configurations to propagate identically and without defects throughout the whole space.
|
|
He is best known for his discovery of the Möbius strip, a non-orientable two- dimensional surface with only one side when embedded in three-dimensional Euclidean space. It was independently discovered by Johann Benedict Listing a few months earlier. The Möbius configuration, formed by two mutually inscribed tetrahedra, is also named after him. Möbius was the first to introduce homogeneous coordinates into projective geometry.
|
|
Their result would be valid for the general case if the Nash embedding theorem can be assumed. However, this theorem was not available then, as John Nash published his famous embedding theorem for Riemannian manifolds in 1956. In 1943 Allendoerfer and Weil published their proof for the general case, in which they first used an approximation theorem of H. Whitney to reduce the case to analytic Riemannian manifolds, then they embedded "small" neighborhoods of the manifold isometrically into a Euclidean space with the help of the Cartan-Janet local embedding theorem, so that they can patch these embedded neighborhoods together and apply the above theorem of Allendoerfer and Fenchel to establish the global result. This is, of course, unsatisfactory for the reason that the theorem only involves intrinsic invariants of the manifold, then the validity of the theorem should not rely on its embedding into a Euclidean space.
|
|
This often involves finding a subtle correspondence between solutions of two seemingly quite different equations. An early example of this which Atiyah used repeatedly is the Penrose transform, which can sometimes convert solutions of a non-linear equation over some real manifold into solutions of some linear holomorphic equations over a different complex manifold. In a series of papers with several authors, Atiyah classified all instantons on 4-dimensional Euclidean space. It is more convenient to classify instantons on a sphere as this is compact, and this is essentially equivalent to classifying instantons on Euclidean space as this is conformally equivalent to a sphere and the equations for instantons are conformally invariant. With Hitchin and Singer he calculated the dimension of the moduli space of irreducible self-dual connections (instantons) for any principal bundle over a compact 4-dimensional Riemannian manifold (the Atiyah–Hitchin–Singer theorem).
|
|
It includes the theorem of Reidemeister and Singer on common refinements ("stabilizations") of Heegaard splittings, the reducibility of splittings, the uniqueness of splittings of a given genus for Euclidean space, and the Rubinstein–Scharlemann graphic, a tool for studying Heegaard splittings. A final chapter surveys more advanced topics including the geometrization conjecture, Dehn surgery, foliations, laminations, and curve complexes. There are two appendices, on general position and Morse theory.
|
|
Formally, the result is as follows. Let be a function or multivalued function from a -dimensional Euclidean space to itself, and suppose that, for every pair of points and that are at unit distance from each other, every pair of images and are also at unit distance from each other. Then must be an isometry: it is a one-to-one function that preserves distances between all pairs of points.
|
|
The circular sections of a quadric may be computed from the implicit equation of the quadric, as it is done in the following sections. They may also be characterised and studied by using synthetic projective geometry. Let be the intersection of a quadric surface and a plane . In this section, and are surfaces in the three-dimensional Euclidean space, which are extended to the projective space over the complex numbers.
|
|
The collection of convex subsets of a vector space, an affine space, or a Euclidean space has the following properties:Soltan, Valeriu, Introduction to the Axiomatic Theory of Convexity, Ştiinţa, Chişinău, 1984 (in Russian). #The empty set and the whole space are convex. #The intersection of any collection of convex sets is convex. #The union of a sequence of convex sets is convex, if they form a non-decreasing chain for inclusion.
|
|
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. The surface of a sphere can be represented by a collection of two dimensional maps.
|
|
With Caffarelli, they studied the Yamabe equation on Euclidean space, proving a positive mass-style theorem on the asymptotic behavior of isolated singularities. In 1974, Spruck and David Hoffman extended a mean curvature-based Sobolev inequality of James H. Michael and Leon Simon to the setting of submanifolds of Riemannian manifolds.Michael, J.H.; Simon, L.M. Sobolev and mean-value inequalities on generalized submanifolds of . Comm. Pure Appl. Math. 26 (1973), 361–379.
|
|
Orthogonal transformations in two- or three-dimensional Euclidean space are stiff rotations, reflections, or combinations of a rotation and a reflection (also known as improper rotations). Reflections are transformations that reverse the direction front to back, orthogonal to the mirror plane, like (real-world) mirrors do. The matrices corresponding to proper rotations (without reflection) have a determinant of +1. Transformations with reflection are represented by matrices with a determinant of −1.
|
|
Yet another way to describe the torus is to say that it is a computer screen with wrap, both left-to-right and up-to-down. The torus is a classical example of what is known in mathematics as a manifold. This is a space that looks somewhat like ordinary Euclidean space at each point, but somehow is hooked together differently. A sphere is another example of a manifold.
|
|
The BFR algorithm, named after its inventors Bradley, Fayyad and Reina, is a variant of k-means algorithm that is designed to cluster data in a high- dimensional Euclidean space. It makes a very strong assumption about the shape of clusters: they must be normally distributed about a centroid. The mean and standard deviation for a cluster may differ for different dimensions, but the dimensions must be independent.
|
|
Consider the first- order differential operators Di to be infinitesimal operators on Euclidean space. That is, Di in a sense generates the one-parameter group of translations parallel to the xi-axis. These groups commute with each other, and therefore the infinitesimal generators do also; the Lie bracket : [Di, Dj] = 0 is this property's reflection. In other words, the Lie derivative of one coordinate with respect to another is zero.
|
|
Historically, people used covariant derivative (or Levi-Civita connection given by metric) to describe the variation rate of a vector along the direction of another vector. Here on the punctured 2-dimensional Euclidean space, the blue vector field sends the one-form to 0.07 everywhere. The red vector field sends the one-form to everywhere. Endorsed by the metric , the Levi-Civita connection is 0 everywhere, indicating has no change along .
|
|
The Minkowski space of special relativity (SR) and general relativity (GR) is a 4-dimensional "pseudo-Euclidean space" vector space. The spacetime underlying Albert Einstein's field equations, which mathematically describe gravitation, is a real 4-dimensional "Pseudo-Riemannian manifold". In QM, wave functions describing particles are complex-valued functions of real space and time variables. The set of all wavefunctions for a given system is an infinite- dimensional complex Hilbert space.
|
|
Example of geometric median (in yellow) of a series of points. In blue the Center of mass. The geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances for one-dimensional data, and provides a central tendency in higher dimensions.
|
|
In mathematics, a ternary equivalence relation is a kind of ternary relation analogous to a binary equivalence relation. A ternary equivalence relation is symmetric, reflexive, and transitive. The classic example is the relation of collinearity among three points in Euclidean space. In an abstract set, a ternary equivalence relation determines a collection of equivalence classes or pencils that form a linear space in the sense of incidence geometry.
|
|
Every (bounded) convex polytope is the image of a simplex, as every point is a convex combination of the (finitely many) vertices. However, polytopes are not in general isomorphic to simplices. This is in contrast to the case of vector spaces and linear combinations, every finite-dimensional vector space being not only an image of, but in fact isomorphic to, Euclidean space of some dimension (or analog over other fields).
|
|
A penetrable space can be perceived in two different ways: navigable and navigated. The navigated perception is realized by the hodological space while the former by ambient space. According to this conceptualization, we do not move within a hodological space but that it is created once we move through an ambient space. The space is interpreted as "lived" and is distinguished from the Euclidean space, which is considered "represented".
|
|
In 1906, L. E. J. Brouwer was the first mathematician to consider the parallel transport of a vector for the case of a space of constant curvature. In 1917, Levi-Civita pointed out its importance for the case of a hypersurface immersed in a Euclidean space, i.e., for the case of a Riemannian manifold immersed in a "larger" ambient space. In 1918, independently of Levi-Civita, Schouten obtained analogous results.
|
|
David Preiss FRS (born 1947) is a professor of mathematics at the University of WarwickMathematics Staff and Postgraduates, University of Warwick Institute of Mathematics. Revised 2 March 2012. Retrieved 26 October 2015. and the winner of the 2008 London Mathematical Society Pólya Prize for his 1987 result on Geometry of Measures, where he solved the remaining problem in the geometric theoretic structure of sets and measures in Euclidean space.
|
|
In mathematics, the Pompeiu problem is a conjecture in integral geometry, named for Dimitrie Pompeiu, who posed the problem in 1929, as follows. Suppose f is a nonzero continuous function defined on a Euclidean space, and K is a simply connected Lipschitz domain, so that the integral of f vanishes on every congruent copy of K. Then the domain is a ball. A special case is Schiffer's conjecture.
|
|
In mathematics and statistics, random projection is a technique used to reduce the dimensionality of a set of points which lie in Euclidean space. Random projection methods are known for their power, simplicity, and low error rates when compared to other methods. According to experimental results, random projection preserves distances well, but empirical results are sparse. They have been applied to many natural language tasks under the name random indexing.
|
|
The most common coordinate system to use is the Cartesian coordinate system, where each point has an x-coordinate representing its horizontal position, and a y-coordinate representing its vertical position. These are typically written as an ordered pair (x, y). This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates (x, y, z).
|
|
Illustration of the hyperplane separation theorem. In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap.
|
|
On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°. Two dimensional Euclidean geometry is modelled by our notion of a "flat plane".
|
|
In physics and engineering, a vector is typically regarded as a geometric entity characterized by a magnitude and a direction. It is formally defined as a directed line segment, or arrow, in a Euclidean space. In pure mathematics, a vector is defined more generally as any element of a vector space. In this context, vectors are abstract entities which may or may not be characterized by a magnitude and a direction.
|
|
In signal processing, multidimensional discrete convolution refers to the mathematical operation between two functions f and g on an n-dimensional lattice that produces a third function, also of n-dimensions. Multidimensional discrete convolution is the discrete analog of the multidimensional convolution of functions on Euclidean space. It is also a special case of convolution on groups when the group is the group of n-tuples of integers.
|
|
Fig. 7: Relations between mathematical spaces: normed, Banach etc Vectors in a Euclidean space form a linear space, but each vector x has also a length, in other words, norm, \lVert x\rVert. A real or complex linear space endowed with a norm is a normed space. Every normed space is both a linear topological space and a metric space. A Banach space is a complete normed space.
|
|
The Lie group E8 has dimension 248. Its rank, which is the dimension of its maximal torus, is eight (8). Therefore, the vectors of the root system are in eight-dimensional Euclidean space: they are described explicitly later in this article. The Weyl group of E8, which is the group of symmetries of the maximal torus which are induced by conjugations in the whole group, has order 2357 = .
|
|
The affine root system of type G2. In mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space. They are used in the classification of affine Lie algebras and superalgebras, and semisimple p-adic algebraic groups, and correspond to families of Macdonald polynomials. The reduced affine root systems were used by Kac and Moody in their work on Kac–Moody algebras.
|
|
Discrete isometry groups of more general Riemannian manifolds generated by reflections have also been considered. The most important class arises from Riemannian symmetric spaces of rank 1: the n-sphere Sn, corresponding to finite reflection groups, the Euclidean space Rn, corresponding to affine reflection groups, and the hyperbolic space Hn, where the corresponding groups are called hyperbolic reflection groups. In two dimensions, triangle groups include reflection groups of all three kinds.
|
|
The edge, or boundary, of a Möbius strip is homeomorphic (topologically equivalent) to a circle. Under the usual embeddings of the strip in Euclidean space, as above, the boundary is not a true circle. However, it is possible to embed a Möbius strip in three dimensions so that the boundary is a perfect circle lying in some plane. For example, see Figures 307, 308, and 309 of "Geometry and the imagination".
|
|
A closely related 'strange' geometrical object is the Klein bottle. A Klein bottle could in theory be produced by gluing two Möbius strips together along their edges; however this cannot be done in ordinary three-dimensional Euclidean space without creating self-intersections. Another closely related manifold is the real projective plane. If a circular disk is cut out of the real projective plane, what is left is a Möbius strip.
|
|
A manifold is a topological space that near each point resembles Euclidean space. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot be realized in three dimensions, but can be realized in four dimensions. Typically, results in algebraic topology focus on global, non- differentiable aspects of manifolds; for example Poincaré duality.
|
|
A subset V of a given Euclidean space E is semianalytic if each point has a neighbourhood U in E such that the intersection of V and U lies in the Boolean algebra of sets generated by subsets defined by inequalities f > 0, where f is a real analytic function. There is no Tarski–Seidenberg theorem for semianalytic sets, and projections of semianalytic sets are in general not semianalytic. A subset V of E is a subanalytic set if for each point there exists a relatively compact semianalytic set X in a Euclidean space F of dimension at least as great as E, and a neighbourhood U in E, such that the intersection of V and U is a linear projection of X into E from F. In particular all semianalytic sets are subanalytic. On an open dense subset, subanalytic sets are submanifolds and so they have a definite dimension "at most points".
|
|
Calculus on Manifolds is a brief monograph on the theory of vector-valued functions of several real variables (f : Rn→Rm) and differentiable manifolds in Euclidean space. In addition to extending the concepts of differentiation (including the inverse and implicit function theorems) and Riemann integration (including Fubini's theorem) to functions of several variables, the book treats the classical theorems of vector calculus, including those of Cauchy–Green, Ostrogradsky–Gauss (divergence theorem), and Kelvin–Stokes, in the language of differential forms on differentiable manifolds embedded in Euclidean space, and as corollaries of the generalized Stokes' theorem on manifolds-with-boundary. The book culminates with the statement and proof of this vast and abstract modern generalization of several classical results: The cover of Calculus on Manifolds features snippets of a July 2, 1850 letter from Lord Kelvin to Sir George Stokes containing the first disclosure of the classical Stokes' theorem (i.e., the Kelvin–Stokes theorem).
|
|
In geometry, a hyperplane of an n-dimensional space V is a subspace of dimension n − 1, or equivalently, of codimension 1 in V. The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings; in all cases however, any hyperplane can be given in coordinates as the solution of a single (due to the "codimension 1" constraint) algebraic equation of degree 1\. If V is a vector space, one distinguishes "vector hyperplanes" (which are linear subspaces, and therefore must pass through the origin) and "affine hyperplanes" (which need not pass through the origin; they can be obtained by translation of a vector hyperplane). A hyperplane in a Euclidean space separates that space into two half spaces, and defines a reflection that fixes the hyperplane and interchanges those two half spaces.
|
|
In geometry, two subsets of a Euclidean space have the same shape if one can be transformed to the other by a combination of translations, rotations (together also called rigid transformations), and uniform scalings. In other words, the shape of a set of points is all the geometrical information that is invariant to translations, rotations, and size changes. Having the same shape is an equivalence relation, and accordingly a precise mathematical definition of the notion of shape can be given as being an equivalence class of subsets of a Euclidean space having the same shape. Mathematician and statistician David George Kendall writes: > In this paper ‘shape’ is used in the vulgar sense, and means what one would > normally expect it to mean. [...] We here define ‘shape’ informally as ‘all > the geometrical information that remains when location, scaleHere, scale > means only uniform scaling, as non-uniform scaling would change the shape of > the object (e.g.
|
|
Relativistic speed refers to speed at which relativistic effects become significant to the desired accuracy of measurement of the phenomenon being observed. Relativistic effects are those discrepancies between values calculated by models considering and not considering relativity. Related words are velocity, rapidity, and celerity which is proper velocity. Speed is a scalar, being the magnitude of the velocity vector which in relativity is the four-velocity and in three-dimension Euclidean space a three-velocity.
|
|
The Hounds of Tindalos are Long's most famous fictional creation. The Hounds were a pack of foul and incomprehensibly alien beasts "emerging from strange angles in dim recesses of non-Euclidean space before the dawn of time" (Long) to pursue travelers down the corridors of time. They could only enter our reality via angles, where they would mangle and exsanguinate their victims, leaving behind only a "peculiar bluish pus or ichor" (Long).
|
|
In geometry, the Beckman-Quarles theorem, named after F. S. Beckman and D. A. Quarles, Jr., states that if a transformation of the Euclidean plane or a higher-dimensional Euclidean space preserves unit distances, then it preserves all distances. Equivalently, every automorphism of the unit distance graph of the plane must be an isometry of the plane. Beckman and Quarles published this result in 1953;. it was later rediscovered by other authors...
|
|
A Klein bottle is homeomorphic to the connected sum of two projective planes. It is also homeomorphic to a sphere plus two cross caps. When embedded in Euclidean space, the Klein bottle is one-sided. However, there are other topological 3-spaces, and in some of the non-orientable examples a Klein bottle can be embedded such that it is two-sided, though due to the nature of the space it remains non-orientable.
|
|
The viewer would be inside one of the cubes, and would be able to see cubes in front of, behind, above, below, to the left and right of himself. If one could travel in these directions, one could explore the array of cubes, and gain an understanding of its geometrical structure. An infinite array of cubes is not a polytope in the traditional sense. In fact, it is a tessellation of 3-dimensional (Euclidean) space.
|
|
Various definitions of compactness may apply, depending on the level of generality. A subset of Euclidean space in particular is called compact if it is closed and bounded. This implies, by the Bolzano–Weierstrass theorem, that any infinite sequence from the set has a subsequence that converges to a point in the set. Various equivalent notions of compactness, such as sequential compactness and limit point compactness, can be developed in general metric spaces.
|
|
The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set.
|
|
If we restrict ourselves to matrices with determinant 1, then we get another group, the special linear group, SL(n). Geometrically, this consists of all the elements of GL(n) that preserve both orientation and volume of the various geometric solids in Euclidean space. If instead we restrict ourselves to orthogonal matrices, then we get the orthogonal group O(n). Geometrically, this consists of all combinations of rotations and reflections that fix the origin.
|
|
Digital geometry deals with discrete sets (usually discrete point sets) considered to be digitized models or images of objects of the 2D or 3D Euclidean space. Simply put, digitizing is replacing an object by a discrete set of its points. The images we see on the TV screen, the raster display of a computer, or in newspapers are in fact digital images. Its main application areas are computer graphics and image analysis.
|
|
Huisken is widely known for his foundational work on the mean curvature flow of hypersurfaces. In 1984, he adapted Richard Hamilton's work on the Ricci flow to the setting of mean curvature flow, proving that a normalization of the flow which preserves surface area will deform any smooth closed convex hypersurface of Euclidean space into a round sphere.Richard S. Hamilton. Three-manifolds with positive Ricci curvature. J. Differential Geometry 17 (1982), no.
|
|
In 2013, Bishop became one of the inaugural fellows of the American Mathematical Society.List of Fellows of the American Mathematical Society, retrieved 2014-06-16. The Bishop–Gromov inequality in Riemannian geometry, one form of which appeared in his book with Crittenden, is named after him and Mikhail Gromov, who gave an improved formulation of Bishop's result. He introduced the "Bishop frame" of curves in Euclidean space, an alternative to the better- known Frenet frame.
|
|
The human visual system perceives visual information as a pattern on the retina, which is 2-dimensional. Thus walking around the sculpture to understand it better creates a temporal sequence of 2-dimensional images in the brain. The multivariate data that is the original input for any grand tour visualization is a (finite) set of points in some high-dimensional Euclidean space. This kind of set arises naturally when data is collected.
|
|
A checkered sphere, without (left) and with (right) UV mapping (3D checkered or 2D checkered). In the example to the right, a sphere is given a checkered texture in two ways. On the left, without UV mapping, the sphere is carved out of three-dimensional checkers tiling Euclidean space. With UV mapping, the checkers tile the two-dimensional UV space, and points on the sphere map to this space according to their latitude and longitude.
|
|
A Lévy process can be defined such that its state space is some abstract mathematical space, such as a Banach space, but the processes are often defined so that they take values in Euclidean space. The index set is the non-negative numbers, so I= [0,\infty) , which gives the interpretation of time. Important stochastic processes such as the Wiener process, the homogeneous Poisson process (in one dimension), and subordinators are all Lévy processes.
|
|
A rotating penrose triangle model to show illusion The tribar appears to be a solid object, made of three straight beams of square cross- section which meet pairwise at right angles at the vertices of the triangle they form. The beams may be broken, forming cubes or cuboids. This combination of properties cannot be realized by any three-dimensional object in ordinary Euclidean space. Such an object can exist in certain Euclidean 3-manifolds.
|
|
In geometry, a ball is a region in space comprising all points within a fixed distance from a given point; that is, it is the region enclosed by a sphere or hypersphere. An -ball is a ball in -dimensional Euclidean space. The volume of a unit -ball is an important expression that occurs in formulas throughout mathematics; it generalizes the notion of the volume enclosed by a sphere in 3-dimensional space.
|
|
In an arbitrary topological space, the class of closed sets with empty interior consists precisely of the boundaries of dense open sets. These sets are, in a certain sense, "negligible". Some examples are finite sets in , smooth curves in the plane, and proper affine subspaces in a Euclidean space. If a topological space is a Baire space then it is "large", meaning that it is not a countable union of negligible subsets.
|
|
Since ancient Greeks, Euclidean space is used for modeling shapes in the physical world. It is thus used in many sciences such as physics, mechanics, and astronomy. It is also widely used in all technical areas that are concerned with shapes, figure, location and position, such as architecture, geodesy, topography, navigation, industrial design, or technical drawing. Space of dimensions higher than three occurs in several modern theories of physics; see Higher dimension.
|
|
In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that take the cone onto itself. By the Koecher–Vinberg theorem these correspond to the cone of squares in finite- dimensional real Euclidean Jordan algebras, originally studied and classified by . The tube domain associated with a symmetric cone is a noncompact Hermitian symmetric space of tube type.
|
|
Since Cartesian coordinates are unique and non-ambiguous, the points of a Cartesian plane can be identified with pairs of real numbers; that is with the Cartesian product \R^2 = \R\times\R, where \R is the set of all real numbers. In the same way, the points in any Euclidean space of dimension n be identified with the tuples (lists) of n real numbers, that is, with the Cartesian product \R^n.
|
|
In Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of 90° (π/2 radians), or one of the vectors is zero. Hence orthogonality of vectors is an extension of the concept of perpendicular vectors to spaces of any dimension. The orthogonal complement of a subspace is the space of all vectors that are orthogonal to every vector in the subspace.
|
|
An important example is the Borel algebra over any topological space: the σ-algebra generated by the open sets (or, equivalently, by the closed sets). Note that this σ-algebra is not, in general, the whole power set. For a non-trivial example that is not a Borel set, see the Vitali set or Non-Borel sets. On the Euclidean space Rn, another σ-algebra is of importance: that of all Lebesgue measurable sets.
|
|
All manufactured components have finite size and well behaved boundaries, so initially the focus was on mathematically modeling rigid parts made of homogeneous isotropic material that could be added or removed. These postulated properties can be translated into properties of subsets of three- dimensional Euclidean space. The two common approaches to define solidity rely on point-set topology and algebraic topology respectively. Both models specify how solids can be built from simple pieces or cells.
|
|
In 2004 he was elected to the Brazilian Academy of Sciences since 2004. His students include Norbert A'Campo, Christian Bonatti, and Michael Herman. In 1993, he studied the hypersurfaces in Euclidean space with a given constant value of an elementary symmetric polynomial of the shape operator, known as a "higher-order mean curvature". His primary result was to obtain some control of the height of such a surface over a plane containing its boundary.
|
|
For distinction, the latter is then called a Hamel basis. Not only does the theorem exhibit suitable basis functions as sufficient for approximation purposes, but also together with the Gram–Schmidt process, it enables one to construct a basis of orthogonal vectors. Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional Euclidean space. The solutions to various differential equations can be interpreted in terms of Hilbert spaces.
|
|
In 1964, Keldysh was made a full professor at Moscow State University and in 1966 she published the book Topological embeddings into Euclidean space to help her students understand her lectures. She lectured until 1974, when she resigned in protest of the expulsion of one of her students. At the time, Novikov was ill and he died in January 1975. She died one year and one month later, on 16 February 1976 in Moscow.
|
|
In modern treatments, Dedekind's assertion is often taken to be the definition of a line, thereby reducing geometry to arithmetic. Three- dimensional Euclidean space is defined to be an affine space whose associated vector space of differences of its elements is equipped with an inner product. See also OpenCourseWare. A definition "from scratch", as in Euclid, is now not often used, since it does not reveal the relation of this space to other spaces.
|
|
Likewise it does not distinguish between different Euclidean models of the same non-Euclidean space. More formally, the third level classifies spaces up to isomorphism. An isomorphism between two spaces is defined as a one-to-one correspondence between the points of the first space and the points of the second space, that preserves all relations stipulated according to the first level. Mutually isomorphic spaces are thought of as copies of a single space.
|
|
In the case of a Euclidean space, both topological dimensions are equal to n. Every subset of a topological space is itself a topological space (in contrast, only linear subsets of a linear space are linear spaces). Arbitrary topological spaces, investigated by general topology (called also point-set topology) are too diverse for a complete classification up to homeomorphism. Compact topological spaces are an important class of topological spaces ("species" of this "type").
|
|
In geometry, a plesiohedron is a special kind of space-filling polyhedron, defined as the Voronoi cell of a symmetric Delone set. Three-dimensional Euclidean space can be completely filled by copies of any one of these shapes, with no overlaps. The resulting honeycomb will have symmetries that take any copy of the plesiohedron to any other copy. The plesiohedra include such well- known shapes as the cube, hexagonal prism, rhombic dodecahedron, and truncated octahedron.
|
|
In a Euclidean space, the sum of angles of a triangle equals the straight angle (180 degrees, radians, two right angles, or a half-turn). A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides. It was unknown for a long time whether other geometries exist, for which this sum is different. The influence of this problem on mathematics was particularly strong during the 19th century.
|
|
In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space Rn. The theorem states that each bounded sequence in Rn has a convergent subsequence.Bartle and Sherbert 2000, p. 78 (for R). An equivalent formulation is that a subset of Rn is sequentially compact if and only if it is closed and bounded.Fitzpatrick 2006, p.
|
|
In 1906, L. E. J. Brouwer was the first mathematician to consider the parallel transport of a vector for the case of a space of constant curvature. In 1917, Levi-Civita pointed out its importance for the case of a hypersurface immersed in a Euclidean space, i.e., for the case of a Riemannian manifold embedded in a "larger" ambient space. In 1918, independently of Levi-Civita, Jan Arnoldus Schouten obtained analogous results.
|
|
Vectors in Three- Dimensional Space has six chapters, each divided into five or more subsections. The first on linear spaces and displacements including these sections: Introduction, Scalar multiplication of vectors, Addition and subtraction of vectors, Displacements in Euclidean space, Geometrical applications. The second on Scalar products and components including these sections: Scalar products, Linear dependence and dimension, Components of a vector, Geometrical applications, Coordinate systems. The third on Other products of vectors.
|
|
Quadrance and distance (as its square root) both measure separation of points in Euclidean space. Following Pythagoras's theorem, the quadrance of two points and in a plane is therefore defined as the sum of squares of differences in the x and y coordinates: : Q(A_1, A_2) = (x_2 - x_1)^2 + (y_2 - y_1)^2. The triangle inequality d_3 \leq d_1 + d_2 is expressed under rational trigonometry as (Q_3 - Q_1 - Q_2)^2 \leq 4 Q_1 Q_2 .
|
|
Besides the fixed-point theorems for more or less contracting functions, there are many that have emerged directly or indirectly from the result under discussion. A continuous map from a closed ball of Euclidean space to its boundary cannot be the identity on the boundary. Similarly, the Borsuk–Ulam theorem says that a continuous map from the n-dimensional sphere to Rn has a pair of antipodal points that are mapped to the same point.
|
|
Although Gauss was the first to study the differential geometry of surfaces in Euclidean space E3, it was not until Riemann's Habilitationsschrift of 1854 that the notion of a Riemannian space was introduced. Christoffel introduced his eponymous symbols in 1869. Tensor calculus was developed by Ricci, who published a systematic treatment with Levi-Civita in 1901. Covariant differentiation of tensors was given a geometric interpretation by who introduced the notion of parallel transport on surfaces.
|
|
Homogeneous distributions on the Euclidean space } with the origin deleted are always of the form where ƒ is a distribution on the unit sphere Sn−1. The number λ, which is the degree of the homogeneous distribution S, may be real or complex. Any homogeneous distribution of the form () on } extends uniquely to a homogeneous distribution on Rn provided . In fact, an analytic continuation argument similar to the one-dimensional case extends this for all .
|
|
As a map , a similarity of ratio takes the form :f(x) = rAx + t, where is an orthogonal matrix and is a translation vector. Similarities preserve planes, lines, perpendicularity, parallelism, midpoints, inequalities between distances and line segments. Similarities preserve angles but do not necessarily preserve orientation, direct similitudes preserve orientation and opposite similitudes change it. The similarities of Euclidean space form a group under the operation of composition called the similarities group .
|
|
In Euclidean geometry, AAA (Angle-Angle-Angle) (or just AA, since in Euclidean geometry the angles of a triangle add up to 180°) does not provide information regarding the size of the two triangles and hence proves only similarity and not congruence in Euclidean space. However, in spherical geometry and hyperbolic geometry (where the sum of the angles of a triangle varies with size) AAA is sufficient for congruence on a given curvature of surface.
|
|
In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non- positive sectional curvature. The theorem states that the universal cover of such a manifold is diffeomorphic to a Euclidean space via the exponential map at any point. It was first proved by Hans Carl Friedrich von Mangoldt for surfaces in 1881, and independently by Jacques Hadamard in 1898. Élie Cartan generalized the theorem to Riemannian manifolds in 1928 (; ; ).
|
|
Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space (R3) are cylindrical and spherical polar coordinates. A Cartesian coordinate surface in this space is a coordinate plane; for example z = 0 defines the x-y plane. In the same space, the coordinate surface r = 1 in spherical polar coordinates is the surface of a unit sphere, which is curved. The formalism of curvilinear coordinates provides a unified and general description of the standard coordinate systems.
|
|
This principle (or, theorem) can be generalized. Thus, consider a system S in the form of a closed domain of finite volume in the Euclidean space. And let us further consider the situation where there is a stream of ”equivalent” particles into S (number of particles per time unit) where each particle retains its identity while being in S and eventually - after a finite time - leaves the system irreversibly (i.e. for these particles the system is ”open”).
|
|
In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of faces at each vertex. Its dimension can be clarified as n-honeycomb for an n-dimensional honeycomb. An n-dimensional uniform honeycomb can be constructed on the surface of n-spheres, in n-dimensional Euclidean space, and n-dimensional hyperbolic space.
|
|
A regular polytope has a regular vertex figure. The vertex figure of a regular polytope {p,q,r,...,y,z} is {q,r,...,y,z}. Regular polytopes can have star polygon elements, like the pentagram, with symbol {}, represented by the vertices of a pentagon but connected alternately. The Schläfli symbol can represent a finite convex polyhedron, an infinite tessellation of Euclidean space, or an infinite tessellation of hyperbolic space, depending on the angle defect of the construction.
|
|
In geometry, a glide plane operation is a type of isometry of the Euclidean space: the combination of a reflection in a plane and a translation in that plane. Reversing the order of combining gives the same result. Depending on context, we may consider a reflection a special case, where the translation vector is the zero vector. The combination of a reflection in a plane and a translation in a perpendicular direction is a reflection in a parallel plane.
|
|
In mathematics, equivalent definitions are used in two somewhat different ways. First, within a particular mathematical theory (for example, Euclidean geometry), a notion (for example, ellipse or minimal surface) may have more than one definition. These definitions are equivalent in the context of a given mathematical structure (Euclidean space, in this case). Second, a mathematical structure may have more than one definition (for example, topological space has at least seven definitions; ordered field has at least two definitions).
|
|
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols , (where is the nabla operator) or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form.
|
|
In collaboration with his former student H. Blaine Lawson, Osserman studied the minimal surface problem in the case that the codimension is larger than one. They considered the case of a graphical minimal submanifold of euclidean space. Their conclusion was that most of the analytical properties which hold in the codimension-one case fail to extend. Solutions to the boundary value problem may exist and fail to be unique, or in other situations may simply fail to exist.
|
|
Point processes are mathematical objects that are defined on some underlying mathematical space. Since these processes are often used to represent collections of points randomly scattered in space, time or both, the underlying space is usually d-dimensional Euclidean space denoted here by Rd, but they can be defined on more abstract mathematical spaces. Point processes have a number of interpretations, which is reflected by the various types of point process notation.F. Baccelli and B. Błaszczyszyn.
|
|
In statistics, multiple correspondence analysis (MCA) is a data analysis technique for nominal categorical data, used to detect and represent underlying structures in a data set. It does this by representing data as points in a low-dimensional Euclidean space. The procedure thus appears to be the counterpart of principal component analysis for categorical data. MCA can be viewed as an extension of simple correspondence analysis (CA) in that it is applicable to a large set of categorical variables.
|
|
This prototype example can be suitably generalized to Fourier integral expansions in higher dimensions, both in Euclidean space and other non-compact rank-one symmetric spaces. Also related are eigenfunction expansions on a geodesic ball in a rank-one symmetric space, but one must consider boundary conditions. Pinsky and others also represent some results on the asymptotic behavior of the Fejer approximation in one dimension, inspired by work of Bump, Persi Diaconis, and J. B. Keller.
|
|
These two operations (referred to as meet and join) make the set of all flats in the Euclidean -space a lattice and can build systematic coordinates for flats in any dimension, leading to Grassmann coordinates or dual Grassmann coordinates. For example, a line in three-dimensional space is determined by two distinct points or by two distinct planes. However, the lattice of all flats is not a distributive lattice. If two lines and intersect, then is a point.
|
|
However, this is still not the case in general. The Heisenberg group with its Carnot metric is an example of a doubling metric space which cannot be embedded in any Euclidean space. Assouad's Theorem states that, for a M-doubling metric space X, if we give it the metric d(x, y)ε for some 0 < ε < 1, then there is a L-bi-Lipschitz map f:X → ℝd, where d and L depend on M and ε.
|
|
In mathematics, the Hilbert metric, also known as the Hilbert projective metric, is an explicitly defined distance function on a bounded convex subset of the n-dimensional Euclidean space Rn. It was introduced by as a generalization of the Cayley's formula for the distance in the Cayley–Klein model of hyperbolic geometry, where the convex set is the n-dimensional open unit ball. Hilbert's metric has been applied to Perron–Frobenius theory and to constructing Gromov hyperbolic spaces.
|
|
From a local point of view one can take X to be Euclidean space. On the other hand, it is useful to be more general, in that (for example) it is possible to define the tangent vectors to X by means of this notion of curve. If X is a smooth manifold, a smooth curve in X is a smooth map :\gamma \colon I \rightarrow X. This is a basic notion. There are less and more restricted ideas, too.
|
|
If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the Kolmogorov continuity theorem says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version. The theorem can also be generalized to random fields so the index set is n-dimensional Euclidean space as well as to stochastic processes with metric spaces as their state spaces.
|
|
The definition of separability can also be stated for other index sets and state spaces,, p. 22 such as in the case of random fields, where the index set as well as the state space can be n-dimensional Euclidean space. The concept of separability of a stochastic process was introduced by Joseph Doob,. The underlying idea of separability is to make a countable set of points of the index set determine the properties of the stochastic process.
|
|
In Euclidean space of n dimensions, the -dimensional unit sphere is the set of all points (x_1, \ldots, x_n) which satisfy the equation : x_1^2 + x_2^2 + \cdots + x_n ^2 = 1. The n-dimensional open unit ball is the set of all points satisfying the inequality : x_1^2 + x_2^2 + \cdots + x_n ^2 < 1, and the n-dimensional closed unit ball is the set of all points satisfying the inequality : x_1^2 + x_2^2 + \cdots + x_n ^2 \le 1.
|
|
A pre- distance matrix that can be embedded in a euclidean space is called a Euclidean distance matrix. Another common example of a metric distance matrix arises in coding theory when in a block code the elements are strings of fixed length over an alphabet and the distance between them is given by the Hamming distance metric. The smallest non-zero entry in the distance matrix measures the error correcting and error detecting capability of the code.
|
|
Eric Weisstein Rectangular hyperbola from Wolfram Mathworld The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry. A prominent instance is the depiction of spacetime as a pseudo-Euclidean space. There the asymptotes of the unit hyperbola form a light cone. Further, the attention to areas of hyperbolic sectors by Gregoire de Saint-Vincent led to the logarithm function and the modern parametrization of the hyperbola by sector areas.
|
|
By contrast, a highly nonspherical solid, the hexahedron (cube) represents "earth". These clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cube's being the only regular solid that tessellates Euclidean space was believed to cause the solidity of the Earth. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarked, "...the god used [it] for arranging the constellations on the whole heaven".
|
|
Basic examples of Riemannian symmetric spaces are Euclidean space, spheres, projective spaces, and hyperbolic spaces, each with their standard Riemannian metrics. More examples are provided by compact, semi- simple Lie groups equipped with a bi-invariant Riemannian metric. Every compact Riemann surface of genus greater than 1 (with its usual metric of constant curvature −1) is a locally symmetric space but not a symmetric space. An example of a non-Riemannian symmetric space is anti-de Sitter space.
|
|
Despite the wide use of Descartes' approach, which was called analytic geometry, the definition of Euclidean space remained unchanged until the end of 19th century. The introduction of abstract vector spaces allowed their use in defining Euclidean spaces with a purely algebraic definition. This new definition has been shown to be equivalent to the classical definition in terms of geometric axioms. It is this algebraic definition that is now most often used for introducing Euclidean spaces.
|
|
Later G. D. Birkhoff and Alfred Tarski proposed simpler sets of axioms, which use real numbers (see Birkhoff's axioms and Tarski's axioms). In Geometric Algebra, Emil Artin has proved that all these definitions of a Euclidean space are equivalent. It is rather easy to prove that all definitions of Euclidean spaces satisfy Hilbert's axioms, and that those involving real numbers (including the above given definition) are equivalent. The difficult part of Artin's proof is the following.
|
|
He also introduced the Wick rotation, in which computations are analytically continued from Minkowski space to four-dimensional Euclidean space using a coordinate change to imaginary timeThe Wick rotation, D. M. O'Brien, Australian Journal of Physics 28 (February 1975), pp. 7–13, . He developed the helicity formulation for collisions between particles with arbitrary spin, worked with Geoffrey Chew on the impulse approximation, and worked on meson theory, symmetry principles in physics, and the vacuum structure of quantum field theory.
|
|
This paper was sufficiently well-regarded that it was translated into German and published in the German scientific journal Physikalische Zeitschrift in 1922. That year, Fermi submitted his article "On the phenomena occurring near a world line" (') to the Italian journal '. In this article he examined the Principle of Equivalence, and introduced the so-called "Fermi coordinates". He proved that on a world line close to the time line, space behaves as if it were a Euclidean space.
|
|
There are infinitely many honeycombs, which have only been partially classified. The more regular ones have attracted the most interest, while a rich and varied assortment of others continue to be discovered. The simplest honeycombs to build are formed from stacked layers or slabs of prisms based on some tessellations of the plane. In particular, for every parallelepiped, copies can fill space, with the cubic honeycomb being special because it is the only regular honeycomb in ordinary (Euclidean) space.
|
|
The graphs that do not have knotless embeddings (that is, they are intrinsically knotted) include K7 and K3,3,1,1.; . However, there also exist minimal forbidden minors for knotless embedding that are not formed (as these two graphs are) by adding one vertex to an intrinsically linked graph.. One may also define graph families by the presence or absence of more complex knots and links in their embeddings,; . or by linkless embedding in three-dimensional manifolds other than Euclidean space.
|
|
The FPP is also preserved by any retraction. According to Brouwer fixed point theorem every compact and convex subset of a Euclidean space has the FPP. More generally, according to the Schauder-Tychonoff fixed point theorem every compact and convex subset of a locally convex topological vector space has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP.
|
|
Pseudodifferential operators can be explained easily in the case of constant coefficient operators on Euclidean space. In this case, constant coefficient differential operators are just the Fourier transforms of multiplication by polynomials, and constant coefficient pseudodifferential operators are just the Fourier transforms of multiplication by more general functions. Many proofs of the index theorem use pseudodifferential operators rather than differential operators. The reason for this is that for many purposes there are not enough differential operators.
|
|
A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat faces can be thought of as a lens, an example of which is the fundamental domain of a lens space L(p,q). Dihedra have also been called bihedra,. flat polyhedra, or doubly covered polygons.
|
|
To what extent do they correspond to an experimental reality? This important physical problem no longer has anything to do with mathematics. Even if a "geometry" does not correspond to an experimental reality, its theorems remain no less "mathematical truths". A Euclidean model of a non-Euclidean geometry is a choice of some objects existing in Euclidean space and some relations between these objects that satisfy all axioms (and therefore, all theorems) of the non-Euclidean geometry.
|
|
In differential geometry, the Angenent torus is a smooth embedding of the torus into three-dimensional Euclidean space, with the property that it remains self-similar as it evolves under the mean curvature flow. Its existence shows that, unlike the one-dimensional curve-shortening flow (for which every embedded closed curve converges to a circle as it shrinks to a point), the two-dimensional mean-curvature flow has embedded surfaces that form more complex singularities as they collapse.
|
|
Suppose X, Y, and Z are Banach spaces (which includes Euclidean space) and B : X × Y → Z is a continuous bilinear operator. Then B is differentiable, and its derivative at the point (x,y) in X × Y is the linear map D(x,y)B : X × Y → Z given by : (D_\left( x,y \right)\,B)\left( u,v \right) = B\left( u,y \right) + B\left( x,v \right)\qquad\forall (u,v)\in X \times Y.
|
|
In mathematics, a Möbius strip, band, or loop ( , ; ), also spelled Mobius or Moebius, is a surface with only one side (when embedded in three-dimensional Euclidean space) and only one boundary curve. The Möbius strip is the simplest non-orientable surface. It can be realized as a ruled surface. Its discovery is attributed independently to the German mathematicians Johann Benedict Listing and August Ferdinand Möbius in 1858,August Ferdinand Möbius, The MacTutor History of Mathematics archive. History.mcs.st-andrews.ac.uk.
|
|
Finite groups often occur when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of Lie groups, which may be viewed as dealing with "continuous symmetry", is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finite- dimensional Euclidean space. The properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry.
|
|
However, it is often assumed to have a structure of -vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real numbers of a given size, or an -algebra, such as the complex numbers or the quaternions. The structure -vector space of the codomain induces a structure of -vector space on the functions. If the codomain has a structure of -algebra, the same is true for the functions.
|
|
Extract of page 204 The dark blue vertical line represents an inertial observer measuring a coordinate time interval t between events E1 and E2. The red curve represents a clock measuring its proper time interval τ between the same two events. In terms of four-dimensional spacetime, proper time is analogous to arc length in three-dimensional (Euclidean) space. By convention, proper time is usually represented by the Greek letter τ (tau) to distinguish it from coordinate time represented by t.
|
|
A star domain is simply connected since any loop can be contracted to the center of the domain, denoted x_0. This section lists some basic examples of fundamental groups. To begin with, in Euclidean space (\R^n) or any convex subset of \R^n, there is only one homotopy class of loops, and the fundamental group is therefore the trivial group with one element. More generally, any star domain and, yet more generally any contractible space has a trivial fundamental group.
|
|
Geometry is the discipline devoted to the study of space and the rules relating the elements to each other. For example, in Euclidean space the Pythagorean theorem provides a rule to compute distances from Cartesian coordinates. In a two- dimensional space of constant curvature, like the surface of a sphere, the rule is somewhat more complex but applies everywhere. On the two-dimensional surface of a football, the rule is more complex still and has different values depending on location.
|
|
3D representation of the human color space. One can picture this space as a region in three-dimensional Euclidean space if one identifies the x, y, and z axes with the stimuli for the long-wavelength (L), medium-wavelength (M), and short-wavelength (S) light receptors. The origin, (S,M,L) = (0,0,0), corresponds to black. White has no definite position in this diagram; rather it is defined according to the color temperature or white balance as desired or as available from ambient lighting.
|
|
A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set of fixed points is either empty or an affine space. For an object, any unique centre and, more generally, any point with unique properties with respect to the object is a fixed point of its symmetry group. In particular this applies for the centroid of a figure, if it exists.
|
|
A rigid graph is an embedding of a graph in a Euclidean space which is structurally rigid. That is, a graph is rigid if the structure formed by replacing the edges by rigid rods and the vertices by flexible hinges is rigid. A graph that is not rigid is called flexible. More formally, a graph embedding is flexible if the vertices can be moved continuously, preserving the distances between adjacent vertices, with the result that the distances between some nonadjacent vertices are altered.
|
|
In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space. At every point in the field, the curl of that point is represented by a vector. The attributes of this vector (length and direction) characterize the rotation at that point. The direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl is the magnitude of rotation.
|
|
'In premodern societies, space and place largely coincided....Modernity increasingly tears space away from place'.Anthony Giddens, quoted in Kellerman, p. 46 Whereas in the premodern 'every thing has its assigned place in social space',Emile Durkheim, The Elementary Forms of the Religious Life (London 1971) p. 442 postmodernists would proudly proclaim that 'we need to substitute for the magisterial space of the past...a less upright, less Euclidean space where no one would ever be in his final place '.
|
|
Such submanifolds (given as graphs) might not even solve the Plateau problem, as they automatically must in the case of graphical hypersurfaces of Euclidean space. Their results pointed to the deep analytical difficulty of general elliptic systems and of the minimal submanifold problem in particular. Many of these issues have still failed to be fully understood, despite their great significance in the theory of calibrated geometry and the Strominger–Yau–Zaslow conjecture.Reese Harvey and H. Blaine Lawson, Jr. Calibrated geometries.
|
|
The hypermeridians generates a set of concentric spheres. In geometry, a hypercone (or spherical cone) is the figure in the 4-dimensional Euclidean space represented by the equation :x^2 + y^2 + z^2 - w^2 = 0. It is a quadric surface, and is one of the possible 3-manifolds which are 4-dimensional equivalents of the conical surface in 3 dimensions. It is also named spherical cone because its intersections with hyperplanes perpendicular to the w-axis are spheres.
|
|
The Conjecture claims that any convex, closed and sufficiently smooth surface in three dimensional Euclidean space needs to admit at least two umbilic points. In the sense of the Conjecture, the spheroid with only two umbilic points and the sphere, all points of which are umbilic, are examples of surfaces with minimal and maximal numbers of the umbilicus. For the conjecture to be well posed, or the umbilic points to be well-defined, the surface needs to be at least twice differentiable.
|
|
Formally, six-dimensional Euclidean space, ℝ6, is generated by considering all real 6-tuples as 6-vectors in this space. As such it has the properties of all Euclidean spaces, so it is linear, has a metric and a full set of vector operations. In particular the dot product between two 6-vectors is readily defined and can be used to calculate the metric. 6 × 6 matrices can be used to describe transformations such as rotations that keep the origin fixed.
|
|
In classical continuum mechanics, the space of interest is usually 3-dimensional Euclidean space, as is the tangent space at each point. If we restrict the local coordinates to be Cartesian coordinates with the same scale centered at the point of interest, the metric tensor is the Kronecker delta. This means that there is no need to distinguish covariant and contravariant components, and furthermore there is no need to distinguish tensors and tensor densities. All Cartesian-tensor indices are written as subscripts.
|
|
The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is around 74.05%. In 1998 Thomas Hales, following an approach suggested by , announced that he had a proof of the Kepler conjecture.
|
|
Alexander also proved that the theorem does hold in three dimensions for piecewise linear/smooth embeddings. This is one of the earliest examples where the need for distinction between the categories of topological manifolds, differentiable manifolds, and piecewise linear manifolds became apparent. Now consider Alexander's horned sphere as an embedding into the 3-sphere, considered as the one-point compactification of the 3-dimensional Euclidean space R3. The closure of the non-simply connected domain is called the solid Alexander horned sphere.
|
|
In three dimensions, there are 219 distinct types, or 230 if chiral copies are considered distinct. Space groups are also studied in dimensions other than 3 where they are sometimes called Bieberbach groups, and are discrete cocompact groups of isometries of an oriented Euclidean space. In crystallography, space groups are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallography ().
|
|
The paper is a joint work by Martin Dyer, Alan Frieze and Ravindran Kannan. The main result of the paper is a randomised algorithm for finding an \epsilon approximation to the volume of a convex body K in n-dimensional Euclidean space by assume the existence of a membership oracle. The algorithm takes time bounded by a polynomial in n, the dimension of K and 1/\epsilon. The algorithm is a sophisticated usage of the so-called Markov chain Monte Carlo (MCMC) method.
|
|
In Euclidean geometry one can expect the standard circular angle to be characteristic, but in pseudo-Euclidean space there is also the hyperbolic angle. In the study of special relativity the various frames of reference, for varying velocity with respect to a rest frame, are related by rapidity, a hyperbolic angle. One way to describe a Lorentz boost is as a hyperbolic rotation which preserves the differential angle between rapidities. Thus they are conformal transformations with respect to the hyperbolic angle.
|
|
Frequently D is required to be a complete set of coset representatives with some repetitions, but the repeated part has measure zero. This is a typical situation in ergodic theory. If a fundamental domain is used to calculate an integral on X/G, sets of measure zero do not matter. For example, when X is Euclidean space Rn of dimension n, and G is the lattice Zn acting on it by translations, the quotient X/G is the n-dimensional torus.
|
|
The determinant can be thought of as assigning a number to every sequence of n vectors in Rn, by using the square matrix whose columns are the given vectors. For instance, an orthogonal matrix with entries in Rn represents an orthonormal basis in Euclidean space. The determinant of such a matrix determines whether the orientation of the basis is consistent with or opposite to the orientation of the standard basis. If the determinant is +1, the basis has the same orientation.
|
|
In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space. More generally, a half-space is either of the two parts into which a hyperplane divides an affine space. That is, the points that are not incident to the hyperplane are partitioned into two convex sets (i.e., half-spaces), such that any subspace connecting a point in one set to a point in the other must intersect the hyperplane.
|
|
In this general case, necessary and sufficient conditions for boundedness have not been established, even for Euclidean space or the unit circle. However, several necessary conditions and several sufficient conditions are known. For instance it is known that in order for a multiplier operator to be bounded on even a single Lp space, the multiplier must be bounded and measurable (this follows from the characterisation of L2 multipliers above and the inclusion property). However, this is not sufficient except when p = 2.
|
|
A gallery room in Antichamber demonstrating the use of the impossible object geometries; looking face on to each wall of these individual cubes will show a different scene. The player's manipulation "gun" is shown in the bottom right. In Antichamber, the player controls the unnamed protagonist from a first-person perspective as they wander through levels. Regarding typical notions of Euclidean space, Bruce has stated that "breaking down all those expectations and then remaking them is essentially the core mechanic of the game".
|
|
The facility of versors illustrate elliptic geometry, in particular elliptic space, a three-dimensional realm of rotations. The versors are the points of this elliptic space, though they refer to rotations in 4-dimensional Euclidean space. Given two fixed versors u and v, the mapping q \mapsto u q v is an elliptic motion. If one of the fixed versors is 1, then the motion is a Clifford translation of the elliptic space, named after William Kingdon Clifford who was a proponent of the space.
|
|
Inversion of a cylinder passing through the sphere. In geometry, inversion in a sphere is a transformation of Euclidean space that fixes the points of a sphere while sending the points inside of the sphere to the outside of the sphere, and vice versa. Intuitively, it "swaps the inside and outside" of the sphere while leaving the points on the sphere unchanged. Inversion is a conformal transformation, and is the basic operation of inversive geometry; it is a generalization of inversion in a circle.
|
|
The theory of curves is much simpler and narrower in scope than the theory of surfaces and its higher- dimensional generalizations because a regular curve in a Euclidean space has no intrinsic geometry. Any regular curve may be parametrized by the arc length (the natural parametrization). From the point of view of a theoretical point particle on the curve that does not know anything about the ambient space, all curves would appear the same. Different space curves are only distinguished by how they bend and twist.
|
|
The origin of a Cartesian coordinate system In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry.
|
|
A point reflection in 2 dimensions is the same as a 180° rotation. In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invariant under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric. Point reflection can be classified as an affine transformation.
|
|
In this tiling of the plane by congruent squares, the green and violet squares meet edge-to-edge as do the blue and orange squares. In geometry, Keller's conjecture is the conjecture that in any tiling of Euclidean space by identical hypercubes there are two cubes that meet face to face. For instance, as shown in the illustration, in any tiling of the plane by identical squares, some two squares must meet edge to edge. This conjecture was introduced by , after whom it is named.
|
|
An affine quadric is the set of zeros of a polynomial of degree two. When not specified otherwise, the polynomial is supposed to have real coefficients, and the zeros are points in a Euclidean space. However, most properties remain true when the coefficients belong to any field and the points belong in an affine space. As usually in algebraic geometry, it is often useful to consider points over an algebraically closed field containing the polynomial coefficients, generally the complex numbers, when the coefficients are real.
|
|
In three-dimensional Euclidean space, these three planes represent solutions of linear equations, and their intersection represents the set of common solutions: in this case, a unique point. The blue line is the common solution to two of these equations. Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\ldots +a_nx_n, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics.
|
|
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space . It states: :If is an open subset of and is an injective continuous map, then is open in and is a homeomorphism between and . The theorem and its proof are due to L. E. J. Brouwer, published in 1912. Beweis der Invarianz des n-dimensionalen Gebiets, Mathematische Annalen 71 (1912), pages 305–315; see also 72 (1912), pages 55–56 The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem.
|
|
In topological data analysis, data sets are regarded as a point cloud sampling of a manifold or algebraic variety embedded in Euclidean space. By linking nearest neighbor points in the cloud into a triangulation, a simplicial approximation of the manifold is created and its simplicial homology may be calculated. Finding techniques to robustly calculate homology using various triangulation strategies over multiple length scales is the topic of persistent homology. In sensor networks, sensors may communicate information via an ad-hoc network that dynamically changes in time.
|
|
See also the discussion at Polytope of simplicial complexes as subspaces of Euclidean space made up of subsets, each of which is a simplex. That somewhat more concrete concept is there attributed to Alexandrov. Any finite simplicial complex in the sense talked about here can be embedded as a polytope in that sense, in some large number of dimensions. In algebraic topology, a compact topological space which is homeomorphic to the geometric realization of a finite simplicial complex is usually called a polyhedron (see , , ).
|
|
Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called "infinitesimal calculus". Bernhard Riemann used these ideas to give a precise definition of the integral. It was also during this period that the ideas of calculus were generalized to Euclidean space and the complex plane. In modern mathematics, the foundations of calculus are included in the field of real analysis, which contains full definitions and proofs of the theorems of calculus.
|
|
The dihedral angle of a Euclidean regular dodecahedron is ~116.6°, so no more than three of them can fit around an edge in Euclidean 3-space. In hyperbolic space, however, the dihedral angle is smaller than it is in Euclidean space, and depends on the size of the figure; the smallest possible dihedral angle is 60°, for an ideal hyperbolic regular dodecahedron with infinitely long edges. The dodecahedra in this dodecahedral honeycomb are sized so that all of their dihedral angles are exactly 72°.
|
|
In general, the kissing number problem seeks the maximum possible kissing number for n-dimensional spheres in (n + 1)-dimensional Euclidean space. Ordinary spheres correspond to two-dimensional closed surfaces in three-dimensional space. Finding the kissing number when centers of spheres are confined to a line (the one-dimensional case) or a plane (two-dimensional case) is trivial. Proving a solution to the three-dimensional case, despite being easy to conceptualise and model in the physical world, eluded mathematicians until the mid-20th century.
|
|
Six-dimensional space is any space that has six dimensions, six degrees of freedom, and that needs six pieces of data, or coordinates, to specify a location in this space. There are an infinite number of these, but those of most interest are simpler ones that model some aspect of the environment. Of particular interest is six-dimensional Euclidean space, in which 6-polytopes and the 5-sphere are constructed. Six-dimensional elliptical space and hyperbolic spaces are also studied, with constant positive and negative curvature.
|
|
In algebraic geometry, a Newton–Okounkov body, also called an Okounkov body, is a convex body in Euclidean space associated to a divisor (or more generally a linear system) on a variety. The convex geometry of a Newton–Okounkov body encodes (asymptotic) information about the geometry of the variety and the divisor. It is a large generalization of the notion of the Newton polytope of a projective toric variety. It was introduced (in passing) by Andrei Okounkov in his papers in the late 1990s and early 2000s.
|
|
The occasion of the proof by Hassler Whitney of the embedding theorem for smooth manifolds is said (rather surprisingly) to have been the first complete exposition of the manifold concept precisely because it brought together and unified the differing concepts of manifolds at the time: no longer was there any confusion as to whether abstract manifolds, intrinsically defined via charts, were any more or less general than manifolds extrinsically defined as submanifolds of Euclidean space. See also the history of manifolds and varieties for context.
|
|
In mathematics, cylinder set measure (or promeasure, or premeasure, or quasi- measure, or CSM) is a kind of prototype for a measure on an infinite- dimensional vector space. An example is the Gaussian cylinder set measure on Hilbert space. Cylinder set measures are in general not measures (and in particular need not be countably additive but only finitely additive), but can be used to define measures, such as classical Wiener measure on the set of continuous paths starting at the origin in Euclidean space.
|
|
This uniqueness is a mathematical result that relies on the Pontryagin duality theorem, the Tannaka–Krein duality theorem, and related results of Iwahori-Sugiura, and Tatsuuma. Algorithms exist for recovering bandlimited functions from their triple correlation on euclidean space, as well as rotation groups in two and three dimensions. There is also an interesting link with Wiener's tauberian theorem: any function whose translates are dense in L_1(G), where G is a locally compact abelian group, is also uniquely identified by its triple correlation.
|
|
The graph of an Icosidodecahedron, an example for which the conjecture is true. In mathematical programming and polyhedral combinatorics, the Hirsch conjecture is the statement that the edge-vertex graph of an n-facet polytope in d-dimensional Euclidean space has diameter no more than n − d. That is, any two vertices of the polytope must be connected to each other by a path of length at most n − d. The conjecture was first put forth in a letter by to George B. Dantzig in 1957, pp.
|
|
Because vectors are matrices with only one column, the simplest matrix derivatives are vector derivatives. The notations developed here can accommodate the usual operations of vector calculus by identifying the space M(n,1) of n-vectors with the Euclidean space Rn, and the scalar M(1,1) is identified with R. The corresponding concept from vector calculus is indicated at the end of each subsection. NOTE: The discussion in this section assumes the numerator layout convention for pedagogical purposes. Some authors use different conventions.
|
|
Let K be a convex body in n-dimensional Euclidean space Rn containing the origin in its interior. Let S be an (n − 2)-dimensional linear subspace of Rn. For each unit vector θ in S⊥, the orthogonal complement of S, let Sθ denote the (n − 1)-dimensional hyperplane containing θ and S. Define r(θ) to be the (n − 1)-dimensional volume of K ∩ Sθ. Let C be the curve {θr(θ)} in S⊥. Then C forms the boundary of a convex body in S⊥.
|
|
In Euclidean geometry, oblique reflections generalize ordinary reflections by not requiring that reflection be done using perpendiculars. If two points are oblique reflections of each other, they will still stay so under affine transformations. Consider a plane P in the three-dimensional Euclidean space. The usual reflection of a point A in space in respect to the plane P is another point B in space, such that the midpoint of the segment AB is in the plane, and AB is perpendicular to the plane.
|
|
The Feynman path integral in Euclidean space generalizes this to other problems of interest to statistical mechanics. Any probability distribution which obeys a condition on correlation functions called reflection positivity leads to a local quantum field theory after Wick rotation to Minkowski spacetime ( see Osterwalder- Schrader axioms ). The operation of renormalization is a specified set of mappings from the space of probability distributions to itself. A quantum field theory is called renormalizable if this mapping has a fixed point which gives a quantum field theory.
|
|
In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space. It assigns to any two vectors a, b in a vector also in . Like the cross product in three dimensions, the seven- dimensional product is anticommutative and is orthogonal both to a and to b. Unlike in three dimensions, it does not satisfy the Jacobi identity, and while the three-dimensional cross product is unique up to a sign, there are many seven-dimensional cross products.
|
|
Let f:E\mapsto R be a grayscale image, mapping points from a Euclidean space or discrete grid E (such as R2 or Z2) into the real line. Let b(x) be a structuring element of grayscale. Then, the white top-hat transform of f is given by: :T_w(f)=f-f \circ b, where \circ denotes the opening operation. The black top-hat transform of f (sometimes called the bottom-hat transform ) is given by: :T_b(f)=f\bullet b-f, where \bullet is the closing operation.
|
|
Line art drawing of parallel lines and curves. In geometry, parallel lines are lines in a plane which do not meet; that is, two straight lines in a plane that do not intersect at any point are said to be parallel. Colloquially, curves that do not touch each other or intersect and keep a fixed minimum distance are said to be parallel. A line and a plane, or two planes, in three- dimensional Euclidean space that do not share a point are also said to be parallel.
|
|
Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the N-dimensional Euclidean space, so any point in phase space can be represented by a vector with N numbers. The analysis of linear systems is possible because they satisfy a superposition principle: if u(t) and w(t) satisfy the differential equation for the vector field (but not necessarily the initial condition), then so will u(t) + w(t).
|
|
Many textbooks erroneously state that a flat universe implies an infinite universe; however, the correct statement is that a flat universe that is also simply connected implies an infinite universe. For example, Euclidean space is flat, simply connected, and infinite, but the torus is flat, multiply connected, finite, and compact. In general, local to global theorems in Riemannian geometry relate the local geometry to the global geometry. If the local geometry has constant curvature, the global geometry is very constrained, as described in Thurston geometries.
|
|
A eutactic star consisting of 5 pairs of vectors in three-dimensional space (n = 3, s = 5) In Euclidean geometry, a eutactic star is a geometrical figure in a Euclidean space. A star is a figure consisting of any number of opposing pairs of vectors (or arms) issuing from a central origin. A star is eutactic if it is the orthogonal projection of plus and minus the set of standard basis vectors (i.e., the vertices of a cross-polytope) from a higher-dimensional space onto a subspace.
|
|
The open interval , again with the absolute value metric, is not complete either. The sequence defined by = is Cauchy, but does not have a limit in the given space. However the closed interval is complete; for example the given sequence does have a limit in this interval and the limit is zero. The space R of real numbers and the space C of complex numbers (with the metric given by the absolute value) are complete, and so is Euclidean space Rn, with the usual distance metric.
|
|
Around 1909, Hilbert dedicated himself to the study of differential and integral equations; his work had direct consequences for important parts of modern functional analysis. In order to carry out these studies, Hilbert introduced the concept of an infinite dimensional Euclidean space, later called Hilbert space. His work in this part of analysis provided the basis for important contributions to the mathematics of physics in the next two decades, though from an unanticipated direction. Later on, Stefan Banach amplified the concept, defining Banach spaces.
|
|
If is a subset of a Euclidean space, then is an interior point of if there exists an open ball centered at which is completely contained in . (This is illustrated in the introductory section to this article.) This definition generalizes to any subset of a metric space with metric : is an interior point of if there exists , such that is in whenever the distance . This definition generalises to topological spaces by replacing "open ball" with "open set". Let be a subset of a topological space .
|
|
By the same construction, projective spaces can be considered in higher dimensions. For instance, the real projective 3-space is a Euclidean space together with a plane at infinity that represents the horizon that an artist (who must, necessarily, live in four dimensions) would see. These real projective spaces can be constructed in a slightly more rigorous way as follows. Here, let Rn+1 denote the real coordinate space of n+1 dimensions, and regard the landscape to be painted as a hyperplane in this space.
|
|
The dimension of the Petersen graph is 2. In mathematics, and particularly in graph theory, the dimension of a graph is the least integer such that there exists a "classical representation" of the graph in the Euclidean space of dimension with all the edges having unit length. In a classical representation, the vertices must be distinct points, but the edges may cross one another.Some mathematicians regard this strictly as an "immersion", but many graph theorists, including Erdős, Harary and Tutte, use the term "embedding".
|
|
Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space; these are called hyperbolic geometry and elliptic geometry. In the modern theory of manifolds, these notions correspond to manifolds with constant, negative and positive curvature, respectively. Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right. His theorema egregium gives a method for computing the curvature of a surface without considering the ambient space in which the surface lies.
|
|
In mathematical measure theory, for every positive integer the ham sandwich theorem states that given measurable "objects" in -dimensional Euclidean space, it is possible to divide all of them in half (with respect to their measure, e.g. volume) with a single -dimensional hyperplane. It was proposed by Hugo Steinhaus and proved by Stefan Banach (explicitly in dimension 3, without bothering to automatically state the theorem in the n-dimensional case), and also years later called the Stone–Tukey theorem after Arthur H. Stone and John Tukey.
|
|
Points A, B, C, D and A′, B′, C′, D′ are related by a perspectivity, which is a projective transformation. Historically, the concept of homography had been introduced to understand, explain and study visual perspective, and, specifically, the difference in appearance of two plane objects viewed from different points of view. In three-dimensional Euclidean space, a central projection from a point O (the center) onto a plane P that does not contain O is the mapping that sends a point A to the intersection (if it exists) of the line OA and the plane P. The projection is not defined if the point A belongs to the plane passing through O and parallel to P. The notion of projective space was originally introduced by extending the Euclidean space, that is, by adding points at infinity to it, in order to define the projection for every point except O. Given another plane Q, which does not contain O, the restriction to Q of the above projection is called a perspectivity. With these definitions, a perspectivity is only a partial function, but it becomes a bijection if extended to projective spaces.
|
|
In mathematics, the Cayley plane (or octonionic projective plane) P2(O) is a projective plane over the octonions.Baez (2002). It was discovered in 1933 by Ruth Moufang, and is named after Arthur Cayley (for his 1845 paper describing the octonions). More precisely, there are two objects called Cayley planes, namely the real and the complex Cayley plane. The real Cayley plane is the symmetric space F4/Spin(9), where F4 is a compact form of an exceptional Lie group and Spin(9) is the spin group of nine-dimensional Euclidean space (realized in F4).
|
|
Example of a regular grid A regular grid is a tessellation of n-dimensional Euclidean space by congruent parallelotopes (e.g. bricks). Grids of this type appear on graph paper and may be used in finite element analysis, finite volume methods, finite difference methods, and in general for discretization of parameter spaces. Since the derivatives of field variables can be conveniently expressed as finite differences, structured grids mainly appear in finite difference methods. Unstructured grids offer more flexibility than structured grids and hence are very useful in finite element and finite volume methods.
|
|
However, for more complicated manifolds, cutting along incompressible surfaces can be used to construct the JSJ decomposition of a manifold. This chapter also includes material on Seifert fiber spaces. Chapter four concerns knot theory, knot invariants, thin position, and the relation between knots and their invariants to manifolds via knot complements, the subspaces of Euclidean space on the other sides of tori. Reviewer Bruno Zimmermann calls chapters 5 and 6 "the heart of the book", although reviewer Michael Berg disagrees, viewing chapter 4 on knot theory as more central.
|
|
A quasi-sphere is a submanifold of a pseudo- Euclidean space consisting of the points for which the displacement vector from a reference point satisfies the equation :, where and . Since in permitted, this definition includes hyperplanes; it is thus a generalization of generalized circles and their analogues in any number of dimensions. This inclusion provides a more regular structure under conformal transformations than if they are omitted. This definition has been generalized to affine spaces over complex numbers and quaternions by replacing the quadratic form with a Hermitian form.
|
|
When M is a Euclidean space and all the fields are taken to be invariant by translations by the vectors of M, we get back to a situation where a tensor field is synonymous with a tensor 'sitting at the origin'. This does no great harm, and is often used in applications. As applied to tensor densities, it does make a difference. The bundle of densities cannot seriously be defined 'at a point'; and therefore a limitation of the contemporary mathematical treatment of tensors is that tensor densities are defined in a roundabout fashion.
|
|
In mathematics, the capacity of a set in Euclidean space is a measure of that set's "size". Unlike, say, Lebesgue measure, which measures a set's volume or physical extent, capacity is a mathematical analogue of a set's ability to hold electrical charge. More precisely, it is the capacitance of the set: the total charge a set can hold while maintaining a given potential energy. The potential energy is computed with respect to an idealized ground at infinity for the harmonic or Newtonian capacity, and with respect to a surface for the condenser capacity.
|
|
200x200px 200x200px In mathematics, a fractal is a self-similar subset of Euclidean space whose fractal dimension strictly exceeds its topological dimension. Fractals appear the same at different levels, as illustrated in successive magnifications of the Mandelbrot set. Fractals exhibit similar patterns at increasingly small scales called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, it is called affine self- similar. Fractal geometry lies within the mathematical branch of measure theory.
|
|
Underactuated mechanisms have a lower number of actuators than the number of degrees of freedom (DOF). In a two-dimensional plane, a mechanism can have up to three DOF (two translations, one rotation), and in three-dimensional Euclidean space, up to six (three translations, three rotations). In the case of self-adaptive mechanisms, the lack of actuators is compensated by passive elements that constrain the motion of the system. Springs are a good example of such elements, but other can be used depending on the type of mechanisms.
|
|
Usually, the world is perceived as a 3D Euclidean space. In some cases, it is not possible to use the full Euclidean structure of 3D space. The simplest being projective, then the affine geometry which forms the intermediate layers and finally Euclidean geometry. The concept of stratification is closely related to the series of transformations on geometric entities: in the projective stratum is a series of projective transformations (a homography), in the affine stratum is a series of affine transformations, and in Euclidean stratum is a series of Euclidean transformations.
|
|
The proof is a simple application of the maximum principle and unique continuation for minimal surfaces, based on comparison with a family of catenoids. This enhances a result of Meeks, Leon Simon, and Shing-Tung Yau, which states that any two complete and properly immersed minimal surfaces in three-dimensional Euclidean space, if both are nonplanar, either have a point of intersection or are separated from each other by a plane.William Meeks III, Leon Simon, and Shing Tung Yau. Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature. Ann.
|
|
Using ideas from Whitney's embedding theorem, A can be embedded in k-dimensional Euclidean space with :k > 2 d_A. That is, there is a diffeomorphism \phi that maps A into \R^k such that the derivative of \phi has full rank. A delay embedding theorem uses an observation function to construct the embedding function. An observation function \alpha must be twice-differentiable and associate a real number to any point of the attractor A. It must also be typical, so its derivative is of full rank and has no special symmetries in its components.
|
|
An ideal polyhedron has ideal polygons as its faces, meeting along lines of the hyperbolic space. The Platonic solids and Archimedean solids have ideal versions, with the same combinatorial structure as their more familiar Euclidean versions. Several uniform hyperbolic honeycombs divide hyperbolic space into cells of these shapes, much like the familiar division of Euclidean space into cubes. However, not all polyhedra can be represented as ideal polyhedra – a polyhedron can be ideal only when it can be represented in Euclidean geometry with all its vertices on a circumscribed sphere.
|
|
For mathematical models the Poisson point process is often defined in Euclidean space, but has been generalized to more abstract spaces and plays a fundamental role in the study of random measures, which requires an understanding of mathematical fields such as probability theory, measure theory and topology. In general, the concept of distance is of practical interest for applications, while topological structure is needed for Palm distributions, meaning that point processes are usually defined on mathematical spaces with metrics.A. E. Gelfand, P. Diggle, P. Guttorp, and M. Fuentes. Handbook of spatial statistics, Chapter 9.
|
|
Given an oriented matroid M = (E,T), given in terms of its topes T, we can define a separoid on E by saying that two subsets are separated if they are contained in opposite signs of a tope. In other words, the topes of an oriented matroid are the maximal separations of a separoid. This example includes, of course, all directed graphs. 3\. Given a family of objects in an Euclidean space, we can define a separoid in it by saying that two subsets are separated if there exists a hyperplane that separates them; i.e.
|
|
During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Gauss' Theorema Egregium (remarkable theorem) that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in an Euclidean space. This implies that surfaces can be studied intrinsically, that is as stand alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any contradiction.
|
|
They demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his 1867 inauguration lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based), published only after his death. Riemann's new idea of space proved crucial in Albert Einstein's general relativity theory. Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.
|
|
Since the solid tori are chosen to become arbitrarily small as the iteration number increases, the connected components of A must be single points. It is then easy to verify that A is closed, dense-in-itself, and totally disconnected, having the cardinality of the continuum. This is sufficient to conclude that as an abstract metric space A is homeomorphic to the Cantor set. However, as a subset of Euclidean space A is not ambiently homeomorphic to the standard Cantor set C, embedded in R3 on a line segment.
|
|
For example, in the two dimensional Euclidean plane, every orthogonal transformation is either a reflection across a line through the origin or a rotation about the origin (which can be written as the composition of two reflections). Any arbitrary composition of such rotations and reflections can be rewritten as a composition of no more than 2 reflections. Similarly, in three-dimensional Euclidean space, every orthogonal transformation can be described as a single reflection, a rotation (2 reflections), or an improper rotation (3 reflections). In four-dimensions, double rotations are added that represent 4 reflections.
|
|
The cube is the cell of the only regular tiling of three-dimensional Euclidean space. It is also unique among the Platonic solids in having faces with an even number of sides and, consequently, it is the only member of that group that is a zonohedron (every face has point symmetry). The cube can be cut into six identical square pyramids. If these square pyramids are then attached to the faces of a second cube, a rhombic dodecahedron is obtained (with pairs of coplanar triangles combined into rhombic faces).
|
|
Instead of learning the output vectors directly, the semidefinite programming aims to find an inner product matrix that maximizes the pairwise distances between any two inputs that are not connected in the neighbourhood graph while preserving the nearest neighbors distances. # The low-dimensional embedding is finally obtained by application of multidimensional scaling on the learned inner product matrix. The steps of applying semidefinite programming followed by a linear dimensionality reduction step to recover a low-dimensional embedding into a Euclidean space were first proposed by Linial, London, and Rabinovich.
|
|
The observable universe is thus a sphere with a diameter of about 28.5 gigaparsecs (93 billion light-years or ). Assuming that space is roughly flat (in the sense of being a Euclidean space), this size corresponds to a comoving volume of about ( or ). The figures quoted above are distances now (in cosmological time), not distances at the time the light was emitted. For example, the cosmic microwave background radiation that we see right now was emitted at the time of photon decoupling, estimated to have occurred about years after the Big Bang, (see p.
|
|
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n-volume, or simply volume.The term volume is also used, more strictly, as a synonym of 3-dimensional volume It is used throughout real analysis, in particular to define Lebesgue integration.
|
|
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally. A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve.
|
|
Visibility in geometry is a mathematical abstraction of the real-life notion of visibility. Given a set of obstacles in the Euclidean space, two points in the space are said to be visible to each other, if the line segment that joins them does not intersect any obstacles. (In the Earth's atmosphere light follows a slightly curved path that is not perfectly predictable, complicating the calculation of actual visibility.) Computation of visibility is among the basic problems in computational geometry and has applications in computer graphics, motion planning, and other areas.
|
|
Others consider a point process as a stochastic process, where the process is indexed by sets of the underlying space on which it is defined, such as the real line or n-dimensional Euclidean space. Other stochastic processes such as renewal and counting processes are studied in the theory of point processes. Sometimes the term "point process" is not preferred, as historically the word "process" denoted an evolution of some system in time, so point process is also called a random point field. Point processes are well studied objects in probability theoryKallenberg, O. (1986).
|
|
The state of a vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space. The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions.
|
|
Huisken and Hamilton's ideas were later adapted by Grigori Perelman to the setting of the "backwards" heat equation for volume forms along the Ricci flow.Grisha Perelman. The entropy formula for the Ricci flow and its geometric applications. Huisken and Klaus Ecker made repeated use of the monotonicity result to show that, for a certain class of noncompact graphical hypersurfaces in Euclidean space, the mean curvature flow exists for all positive time and deforms any surface in the class to a self- expanding solution of the mean curvature flow.
|
|
The Minkowski content (named after Hermann Minkowski), or the boundary measure, of a set is a basic concept that uses concepts from geometry and measure theory to generalize the notions of length of a smooth curve in the plane, and area of a smooth surface in space, to arbitrary measurable sets. It is typically applied to fractal boundaries of domains in the Euclidean space, but it can also be used in the context of general metric measure spaces. It is related to, although different from, the Hausdorff measure.
|
|
The author(s) will usually make it clear whether a subscript is intended as an index or as a label. For example, in 3-D Euclidean space and using Cartesian coordinates; the coordinate vector shows a direct correspondence between the subscripts 1, 2, 3 and the labels . In the expression , is interpreted as an index ranging over the values 1, 2, 3, while the subscripts are not variable indices, more like "names" for the components. In the context of spacetime, the index value 0 conventionally corresponds to the label .
|
|
The second part of the problem asks whether there exists a polyhedron which tiles 3-dimensional Euclidean space but is not the fundamental region of any space group; that is, which tiles but does not admit an isohedral (tile-transitive) tiling. Such tiles are now known as anisohedral. In asking the problem in three dimensions, Hilbert was probably assuming that no such tile exists in two dimensions; this assumption later turned out to be incorrect. The first such tile in three dimensions was found by Karl Reinhardt in 1928.
|
|
Each "view" (i.e., frame) of the animation is an orthogonal projection of the data set onto a 2-dimensional subspace of the Euclidean space Rp where the data resides. The subspaces are selected by taking small steps along a continuous curve, parametrized by time, in the space of all 2-dimensional subspaces of Rp, known as the Grassmannian G(2,p). To display these views on a computer screen, it is necessary to pick one particular rotated position of each view (in the plane of the computer screen) for display.
|
|
One disadvantage to dyadic cubes in Euclidean space is that they rely too much on the specific position of the cubes. For example, for the dyadic cubes Δ described above, it is not possible to contain an arbitrary ball inside some Q in Δ (consider, for example, the unit ball centered at zero). Alternatively, there may be such a cube that contains the ball, but the sizes of the ball and cube are very different. Because of this caveat, it is sometimes useful to work with two or more collections of dyadic cubes simultaneously.
|
|
In spherical geometry, a direct motion of the -sphere (an example of the elliptic geometry) is the same as a rotation of -dimensional Euclidean space about the origin (). For odd , most of these motions do not have fixed points on the -sphere and, strictly speaking, are not rotations of the sphere; such motions are sometimes referred to as Clifford translations. Rotations about a fixed point in elliptic and hyperbolic geometries are not different from Euclidean ones. Affine geometry and projective geometry have not a distinct notion of rotation.
|
|
Real-world objects that approximate a solid torus include O-rings, non- inflatable lifebuoys, ring doughnuts, and bagels. In topology, a ring torus is homeomorphic to the Cartesian product of two circles: S1 × S1, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into Euclidean space, but another way to do this is the Cartesian product of the embedding of S1 in the plane with itself.
|
|
According to the Brouwer fixed-point theorem, every compact and convex subset of a Euclidean space has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP. In 1932 Borsuk asked whether compactness together with contractibility could be a necessary and sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by Kinoshita who found an example of a compact contractible space without the FPP.
|
|
The "latitude" on the hypersphere will be half of the corresponding angle of rotation, and the neighborhood of any point will become "flatter" (i.e. be represented by a 3D Euclidean space of points) as the neighborhood shrinks. This behavior is matched by the set of unit quaternions: A general quaternion represents a point in a four-dimensional space, but constraining it to have unit magnitude yields a three-dimensional space equivalent to the surface of a hypersphere. The magnitude of the unit quaternion will be unity, corresponding to a hypersphere of unit radius.
|
|
This state space can be, for example, the integers, the real line or n-dimensional Euclidean space. An increment is the amount that a stochastic process changes between two index values, often interpreted as two points in time. A stochastic process can have many outcomes, due to its randomness, and a single outcome of a stochastic process is called, among other names, a sample function or realization. A single computer-simulated sample function or realization, among other terms, of a three-dimensional Wiener or Brownian motion process for time 0 ≤ t ≤ 2.
|
|
A lattice arrangement (commonly called a regular arrangement) is one in which the centers of the spheres form a very symmetric pattern which needs only n vectors to be uniquely defined (in n-dimensional Euclidean space). Lattice arrangements are periodic. Arrangements in which the spheres do not form a lattice (often referred to as irregular) can still be periodic, but also aperiodic (properly speaking non-periodic) or random. Lattice arrangements are easier to handle than irregular ones—their high degree of symmetry makes it easier to classify them and to measure their densities.
|
|
In mathematics, a Meyer set or almost lattice is a set relatively dense X of points in the Euclidean plane or a higher-dimensional Euclidean space such that its Minkowski difference with itself is uniformly discrete. Meyer sets have several equivalent characterizations; they are named after Yves Meyer, who introduced and studied them in the context of diophantine approximation. Nowadays Meyer sets are best known as mathematical model for quasicrystals. However, Meyer's work precedes the discovery of quasicrystals by more than a decade and was entirely motivated by number theoretic questions. ..
|
|
In mathematics, specifically the field of algebraic number theory, a Minkowski space is a Euclidean space associated with an algebraic number field. If K is a number field of degree d then there are d distinct embeddings of K into C. We let KC be the image of K in the product Cd, considered as equipped with the usual Hermitian inner product. If c denotes complex conjugation, let KR denote the subspace of KC fixed by c, equipped with a scalar product. This is the Minkowski space of K.
|
|
The 2n-vertex crown graph may be embedded into four-dimensional Euclidean space in such a way that all of its edges have unit length. However, this embedding may also place some non-adjacent vertices a unit distance apart. An embedding in which edges are at unit distance and non- edges are not at unit distance requires at least n − 2 dimensions. This example shows that a graph may require very different dimensions to be represented as a unit distance graphs and as a strict unit distance graph.
|
|
In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations or difference equations. State variables are variables whose values evolve over time in a way that depends on the values they have at any given time and on the externally imposed values of input variables. Output variables’ values depend on the values of the state variables. The "state space" is the Euclidean space in which the variables on the axes are the state variables.
|
|
In the mathematical fields of geometry and linear algebra, a principal axis is a certain line in a Euclidean space associated with an ellipsoid or hyperboloid, generalizing the major and minor axes of an ellipse or hyperbola. The principal axis theorem states that the principal axes are perpendicular, and gives a constructive procedure for finding them. Mathematically, the principal axis theorem is a generalization of the method of completing the square from elementary algebra. In linear algebra and functional analysis, the principal axis theorem is a geometrical counterpart of the spectral theorem.
|
|
Several mathematicians have demonstrated workable procedures based on a variety of approximations. Bending the rules by introducing a supplemental tool, allowing an infinite number of compass-and- straightedge operations or by performing the operations in certain non- Euclidean geometries also makes squaring the circle possible in some sense. For example, the quadratrix of Hippias provides the means to square the circle and also to trisect an arbitrary angle, as does the Archimedean spiral. Although the circle cannot be squared in Euclidean space, it sometimes can be in hyperbolic geometry under suitable interpretations of the terms.
|
|
Two subspaces and of the same dimension in a Euclidean space are parallel if they have the same direction. Equivalently, they are parallel, if there is a translation vector that maps one to the other: :T= S+v. Given a point and a subspace , there exists exactly one subspace that contains and is parallel to , which is P + \overrightarrow S. In the case where is a line (subspace of dimension one), this property is Playfair's axiom. It follows that in a Euclidean plane, two lines either meet in one point or are parallel.
|
|
The inner product that is defined to define Euclidean spaces is a positive definite bilinear form. If it is replaced by an indefinite quadratic form which is non-degenerate, one gets a pseudo-Euclidean space. A fundamental example of such a space is the Minkowski space, which is the space-time of Einstein's special relativity. It is a four-dimensional space, where the metric is defined by the quadratic form :x^2+y^2+z^2-t^2, where the last coordinate (t) is temporal, and the other three (x, y, z) are spatial.
|
|
Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other self- consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only over short distances (relative to the strength of the gravitational field).Misner, Thorne, and Wheeler (1973), p.
|
|
Given a point A0 in a Euclidean space and a translation S, define the point Ai to be the point obtained from i applications of the translation S to A0, so Ai = Si A0. The set of vertices Ai with i any integer, together with edges connecting adjacent vertices, is a sequence of equal segments of a line. The constructed sequence Ai is called the regular apeirogon as defined by H. S. M. Coxeter. An example of apeirogonal tiling in the hyperbolic plane, visualised using the Poincaré disk model.
|
|
This part of the book also describes the regular tessellations of the Euclidean plane and the sphere, and the regular honeycombs of Euclidean space. Chapter 6 discusses the star polyhedra including the Kepler–Poinsot polyhedra. The remaining chapters cover higher-dimensional generalizations of these topics, including two chapters on the enumeration and construction of the regular polytopes, two chapters on higher-dimensional Euler characteristics and background on quadratic forms, two chapters on higher-dimensional Coxeter groups, a chapter on cross-sections and projections of polytopes, and a chapter on star polytopes and polytope compounds.
|
|
Three linked golden rectangles in a regular icosahedron A realization of the Borromean rings by three mutually perpendicular golden rectangles can be found within a regular icosahedron by connecting three opposite pairs of its edges. Every three unknotted polygons in Euclidean space may be combined, after a suitable scaling transformation, to form the Borromean rings. If all three polygons are planar, then scaling is not needed. More generally, Matthew Cook has conjectured that any three unknotted simple closed curves in space, not all circles, can be combined without scaling to form the Borromean rings.
|
|
The uniquely defined dimension of every connected topological manifold can be calculated. A connected topological manifold is locally homeomorphic to Euclidean -space, in which the number is the manifold's dimension. For connected differentiable manifolds, the dimension is also the dimension of the tangent vector space at any point. In geometric topology, the theory of manifolds is characterized by the way dimensions 1 and 2 are relatively elementary, the high-dimensional cases are simplified by having extra space in which to "work"; and the cases and are in some senses the most difficult.
|
|
This problem was resolved by defining measure only on a sub-collection of all subsets; the so- called measurable subsets, which are required to form a -algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement.Halmos, Paul (1950), Measure theory, Van Nostrand and Co. Indeed, their existence is a non- trivial consequence of the axiom of choice.
|
|
It is closely related to quantum field theory, which describes the quantum mechanics of fields, and shares with it many techniques, such as the path integral formulation and renormalization. If the system involves polymers, it is also known as polymer field theory. In fact, by performing a Wick rotation from Minkowski space to Euclidean space, many results of statistical field theory can be applied directly to its quantum equivalent. The correlation functions of a statistical field theory are called Schwinger functions, and their properties are described by the Osterwalder–Schrader axioms.
|
|
In this context, a simplex in d-dimensional Euclidean space is the convex hull of d+1 points that do not all lie in a common hyperplane. For example, a 2-dimensional simplex is just a triangle (the convex hull of three points in the plane) and a 3-dimensional simplex is a tetrahedron (the convex of four points in three-dimensional space). The points that form the simplex in this way are called its vertices. An orthoscheme, also called a path simplex, is a special kind of simplex.
|
|
In lattice gauge theory, the spacetime is Wick rotated into Euclidean space and discretized into a lattice with sites separated by distance a and connected by links. In the most commonly considered cases, such as lattice QCD, fermion fields are defined at lattice sites (which leads to fermion doubling), while the gauge fields are defined on the links. That is, an element U of the compact Lie group G (not algebra) is assigned to each link. Hence, to simulate QCD with Lie group SU(3), a 3×3 unitary matrix is defined on each link.
|
|
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field), that measures the extent to which the metric tensor is not locally isometric to that of Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.
|
|
It is linear in u and v, and so defines a tensor. Occasionally, the curvature tensor is defined with the opposite sign. If u = \partial/\partial x^i and v = \partial/\partial x^j are coordinate vector fields then [u, v] = 0 and therefore the formula simplifies to :R(u, v)w = abla_u abla_v w - abla_v abla_u w . The curvature tensor measures noncommutativity of the covariant derivative, and as such is the integrability obstruction for the existence of an isometry with Euclidean space (called, in this context, flat space).
|
|
In a three-dimensional Euclidean vector space, the orthogonal complement of a line through the origin is the plane through the origin perpendicular to it, and vice versa. Note that the geometric concept of two planes being perpendicular does not correspond to the orthogonal complement, since in three dimensions a pair of vectors, one from each of a pair of perpendicular planes, might meet at any angle. In four- dimensional Euclidean space, the orthogonal complement of a line is a hyperplane and vice versa, and that of a plane is a plane.
|
|
In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras. Specifically, the question is how many linearly independent smooth nowhere-zero vector fields can be constructed on a sphere in N-dimensional Euclidean space. A definitive answer was provided in 1962 by Frank Adams. It was already known, by direct construction using Clifford algebras, that there were at least ρ(N)-1 such fields (see definition below).
|
|
81–85 In a three- dimensional Euclidean space, lines with true length are parallel to the projection plane. For example, in a top view of a pyramid, which is an orthographic projection, the base edges (which are parallel to the projection plane) have true length, whereas the remaining edges in this view are not true lengths. The same is true with an orthographic side view of a pyramid. If any face of a pyramid was parallel to the projection plane (for a particular view), all edges would demonstrate true length.
|
|
Many of these problems, when the container size is increased in all directions, become equivalent to the problem of packing objects as densely as possible in infinite Euclidean space. This problem is relevant to a number of scientific disciplines, and has received significant attention. The Kepler conjecture postulated an optimal solution for packing spheres hundreds of years before it was proven correct by Thomas Callister Hales. Many other shapes have received attention, including ellipsoids, Platonic and Archimedean solids including tetrahedra, tripods (unions of cubes along three positive axis-parallel rays),.
|
|
Minkowski in his earlier works in 1907 and 1908 followed Poincaré in representing space and time together in complex form (x,y,z,ict) emphasizing the formal similarity with Euclidean space. He noted that space-time is in a certain sense a four- dimensional non-Euclidean manifold.Goettingen lecture 1907, see comments in Walter 1999 Sommerfeld (1910) used Minkowski's complex representation to combine non-collinear velocities by spherical geometry and so derive Einstein's addition formula. Subsequent writers,Walter (1999b) principally Varićak, dispensed with the imaginary time coordinate, and wrote in explicitly non-Euclidean (i.e.
|
|
All cones are path-connected since every point can be connected to the vertex point. Furthermore, every cone is contractible to the vertex point by the homotopy :h_t(x,s) = (x, (1-t)s). The cone is used in algebraic topology precisely because it embeds a space as a subspace of a contractible space. When X is compact and Hausdorff (essentially, when X can be embedded in Euclidean space), then the cone CX can be visualized as the collection of lines joining every point of X to a single point.
|
|
On a Riemannian manifold, a normal coordinate system at p facilitates the introduction of a system of spherical coordinates, known as polar coordinates. These are the coordinates on M obtained by introducing the standard spherical coordinate system on the Euclidean space TpM. That is, one introduces on TpM the standard spherical coordinate system (r,φ) where r ≥ 0 is the radial parameter and φ = (φ1,...,φn−1) is a parameterization of the (n−1)-sphere. Composition of (r,φ) with the inverse of the exponential map at p is a polar coordinate system.
|
|
The Kervaire invariant is an invariant of a (4k + 2)-dimensional manifold that measures whether the manifold could be surgically converted into a sphere. This invariant evaluates to 0 if the manifold can be converted to a sphere, and 1 otherwise. More specifically, the Kervaire invariant applies to a framed manifold, that is, to a manifold equipped with an embedding into Euclidean space and a trivialization of the normal bundle. The Kervaire invariant problem is the problem of determining in which dimensions the Kervaire invariant can be nonzero.
|
|
The set of distances between the vertices of a claw provides an example of a finite metric space that cannot be embedded isometrically into a Euclidean space of any dimension. The star network, a computer network modeled after the star graph, is important in distributed computing. A geometric realization of the star graph, formed by identifying the edges with intervals of some fixed length, is used as a local model of curves in tropical geometry. A tropical curve is defined to be a metric space that is locally isomorphic to a star shaped metric graph.
|
|
Based on assumed mathematical properties, any scheme of representing solids is a method for capturing information about the class of semi-analytic subsets of Euclidean space. This means all representations are different ways of organizing the same geometric and topological data in the form of a data structure. All representation schemes are organized in terms of a finite number of operations on a set of primitives. Therefore, the modeling space of any particular representation is finite, and any single representation scheme may not completely suffice to represent all types of solids.
|
|
By giving a collection of subspaces of some vector space a topological structure, it is possible to talk about a continuous choice of subspace or open and closed collections of subspaces; by giving them the structure of a differential manifold one can talk about smooth choices of subspace. A natural example comes from tangent bundles of smooth manifolds embedded in Euclidean space. Suppose we have a manifold of dimension embedded in . At each point in , the tangent space to can be considered as a subspace of the tangent space of , which is just .
|
|
Harmonic functions can be defined on an arbitrary Riemannian manifold, using the Laplace–Beltrami operator Δ. In this context, a function is called harmonic if :\ \Delta f = 0. Many of the properties of harmonic functions on domains in Euclidean space carry over to this more general setting, including the mean value theorem (over geodesic balls), the maximum principle, and the Harnack inequality. With the exception of the mean value theorem, these are easy consequences of the corresponding results for general linear elliptic partial differential equations of the second order.
|
|
The name paravector is used for the sum of a scalar and a vector in any Clifford algebra (Clifford algebra is also known as geometric algebra in the physics community.) This name was given by J. G. Maks, Doctoral Dissertation, Technische Universiteit Delft (Netherlands), 1989. The complete algebra of paravectors along with corresponding higher grade generalizations, all in the context of the Euclidean space of three dimensions, is an alternative approach to the spacetime algebra (STA) introduced by David Hestenes. This alternative algebra is called algebra of physical space (APS).
|
|
In the worst case, every pair of vertices is connected, giving a complete graph. To immerse the complete graph K_n with all the edges having unit length, we need the Euclidean space of dimension n-1. For example, it takes two dimensions to immerse K_3 (an equilateral triangle), and three to immerse K_4 (a regular tetrahedron) as shown to the right. :\dim K_n = n-1 In other words, the dimension of the complete graph is the same as that of the simplex having the same number of vertices.
|
|
Square grid graph Triangular grid graph A lattice graph, mesh graph, or grid graph, is a graph whose drawing, embedded in some Euclidean space Rn, forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a lattice in the group-theoretical sense. Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid.
|
|
The natural topology of Euclidean space 𝔼n implies a topology for the Euclidean group E(n). Namely, a sequence fi of isometries of 𝔼n (i∈ℕ) is defined to converge if and only if, for any point p of 𝔼n, the sequence of points pi converges. From this definition it follows that a function f:[0,1]→E(n) is continuous if and only if, for any point p of 𝔼n, the function fp:[0,1]→𝔼n defined by fp(t) = (f(t))(p) is continuous. Such a function is called a "continuous trajectory" in E(n).
|
|
Such transformation turns the given Euclidean structure into a (isomorphic but) different Euclidean structure; both Euclidean structures correspond to a single topological structure. In contrast, the transition from "3-dim Euclidean" to "Euclidean" is not forgetful; a Euclidean space need not be 3-dimensional, but if it happens to be 3-dimensional, it is full-fledged, no structure is lost. In other words, the latter transition is injective (one-to- one), while the former transition is not injective (many-to-one). We denote injective transitions by an arrow with a barbed tail, "↣" rather than "→".
|
|
The elements of the tangent space at x are called the tangent vectors at x . This is a generalization of the notion of a bound vector in a Euclidean space. The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself. For example, if the given manifold is a 2 -sphere, then one can picture the tangent space at a point as the plane that touches the sphere at that point and is perpendicular to the sphere's radius through the point.
|
|
The first stage of the dogbone space construction. In geometric topology, the dogbone space, constructed by , is a quotient space of three-dimensional Euclidean space \R^3 such that all inverse images of points are points or tame arcs, yet it is not homeomorphic to \R^3. The name "dogbone space" refers to a fanciful resemblance between some of the diagrams of genus 2 surfaces in R.H. Bing's paper and a dog bone. showed that the product of the dogbone space with \R^1 is homeomorphic to \R^4.
|
|
Federer's mathematical work separates thematically into the periods before and after his watershed 1960 paper Normal and integral currents, co- authored with Fleming. That paper provided the first satisfactory general solution to Plateau's problem — the problem of finding a (k+1)-dimensional least-area surface spanning a given k-dimensional boundary cycle in n-dimensional Euclidean space. Their solution inaugurated a new and fruitful period of research on a large class of geometric variational problems — especially minimal surfaces — via what came to be known as Geometric Measure Theory.
|
|
The concept of a secant line can be applied in a more general setting than Euclidean space. Let be a finite set of points in some geometric setting. A line will be called an -secant of if it contains exactly points of . For example, if is a set of 50 points arranged on a circle in the Euclidean plane, a line joining two of them would be a 2-secant (or bisecant) and a line passing through only one of them would be a 1-secant (or unisecant).
|
|
The tangent plane to a surface at a given point p is defined in an analogous way to the tangent line in the case of curves. It is the best approximation of the surface by a plane at p, and can be obtained as the limiting position of the planes passing through 3 distinct points on the surface close to p as these points converge to p. More generally, there is a k-dimensional tangent space at each point of a k-dimensional manifold in the n-dimensional Euclidean space.
|
|
Every two-graph is equivalent to a set of lines in some dimensional euclidean space each pair of which meet in the same angle. The set of lines constructed from a two graph on n vertices is obtained as follows. Let -ρ be the smallest eigenvalue of the Seidel adjacency matrix, A, of the two-graph, and suppose that it has multiplicity n - d. Then the matrix is positive semi-definite of rank d and thus can be represented as the Gram matrix of the inner products of n vectors in euclidean d-space.
|
|
The inradius or filling radius of a given outer figure is the radius of the inscribed circle or sphere, if it exists. The definition given above assumes that the objects concerned are embedded in two- or three-dimensional Euclidean space, but can easily be generalized to higher dimensions and other metric spaces. For an alternative usage of the term "inscribed", see the inscribed square problem, in which a square is considered to be inscribed in another figure (even a non-convex one) if all four of its vertices are on that figure.
|
|
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simplest forms of Brouwer's theorem are for continuous functions f from a closed interval I in the real numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a convex compact subset K of Euclidean space to itself.
|
|
The theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows: :;In the plane: Every continuous function from a closed disk to itself has at least one fixed point.D. Violette Applications du lemme de Sperner pour les triangles Bulletin AMQ, V. XLVI N° 4, (2006) p 17. This can be generalized to an arbitrary finite dimension: :;In Euclidean space:Every continuous function from a closed ball of a Euclidean space into itself has a fixed point.
|
|
Adding a seventh sphere gives a new cluster consisting in two "axial" balls touching each other and five others touching the latter two balls, the outer shape being an almost regular pentagonal bi- pyramid. However, we are facing now a real packing problem, analogous to the one encountered above with the pentagonal tiling in two dimensions. The dihedral angle of a tetrahedron is not commensurable with 2; consequently, a hole remains between two faces of neighboring tetrahedra. As a consequence, a perfect tiling of the Euclidean space R3 is impossible with regular tetrahedra.
|
|
An extreme point of a convex set is a point in the set that does not lie on a line segment between any other two points of the same set. For a convex hull, every extreme point must be part of the given set, because otherwise it cannot be formed as a convex combination of given points. According to the Krein–Milman theorem, every compact convex set in a Euclidean space (or more generally in a locally convex topological vector space) is the convex hull of its extreme points.; , p. 43.
|
|
Inversion with respect to the origin corresponds to additive inversion of the position vector, and also to scalar multiplication by −1\. The operation commutes with every other linear transformation, but not with translation: it is in the center of the general linear group. "Inversion" without indicating "in a point", "in a line" or "in a plane", means this inversion; in physics 3-dimensional reflection through the origin is also called a parity transformation. In mathematics, reflection through the origin refers to the point reflection of Euclidean space Rn across the origin of the Cartesian coordinate system.
|
|
The orthogonal convex hull of a point set In geometry, a set is defined to be orthogonally convex if, for every line that is parallel to one of standard basis vectors, the intersection of with is empty, a point, or a single segment. The term "orthogonal" refers to corresponding Cartesian basis and coordinates in Euclidean space, where different basis vectors are perpendicular, as well as corresponding lines. Unlike ordinary convex sets, an orthogonally convex set is not necessarily connected. The orthogonal convex hull of a set is the intersection of all connected orthogonally convex supersets of .
|
|
The weighted version of the minimum covering circle problem takes as input a set of points in a Euclidean space, each with weights; the goal is to find a single point that minimizes the maximum weighted distance to any point. The original minimum covering circle problem can be recovered by setting all weights to the same number. As with the unweighted problem, the weighted problem may be solved in linear time in any space of bounded dimension, using approaches closely related to bounded dimension linear programming algorithms, although slower algorithms are again frequent in the literature...
|
|
A consequence of the Theorema Egregium is that the Earth cannot be displayed on a map without distortion. The Mercator projection, shown here, preserves angles but fails to preserve area. Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry (proved by Carl Friedrich Gauss in 1827) that concerns the curvature of surfaces. The theorem is that Gaussian curvature can be determined entirely by measuring angles, distances and their rates on a surface, without reference to the particular manner in which the surface is embedded in the ambient 3-dimensional Euclidean space.
|
|
Plane equation in normal form In mathematics, a plane is a flat, two- dimensional surface that extends infinitely far. A plane is the two- dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as subspaces of some higher- dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the plane refers to the whole space.
|
|
In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds. Every Hermitian symmetric space is a homogeneous space for its isometry group and has a unique decomposition as a product of irreducible spaces and a Euclidean space. The irreducible spaces arise in pairs as a non-compact space that, as Borel showed, can be embedded as an open subspace of its compact dual space.
|
|
As proved independently by Leroy and Simpson, the Banach-Tarski paradox does not violate volumes if one works with locales rather than topological spaces. In this abstract setting, it is possible to have subspaces without point but still nonempty. The parts of the paradoxical decomposition do intersect a lot in the sense of locales, so much that some of these intersections should be given a positive mass. Allowing for this hidden mass to be taken into account, the theory of locales permits all subsets (and even all sublocales) of the Euclidean space to be satisfactorily measured.
|
|
As shown in the figure, the singular values can be interpreted as the magnitude of the semiaxes of an ellipse in 2D. This concept can be generalized to -dimensional Euclidean space, with the singular values of any square matrix being viewed as the magnitude of the semiaxis of an -dimensional ellipsoid. Similarly, the singular values of any matrix can be viewed as the magnitude of the semiaxis of an -dimensional ellipsoid in -dimensional space, for example as an ellipse in a (tilted) 2D plane in a 3D space. Singular values encode magnitude of the semiaxis, while singular vectors encode direction.
|
|
Although it may be embedded in two dimensions, the Desargues configuration has a very simple construction in three dimensions: for any configuration of five planes in general position in Euclidean space, the ten points where three planes meet and the ten lines formed by the intersection of two of the planes together form an instance of the configuration . This construction is closely related to the property that every projective plane that can be embedded into a 3-dimensional projective space obeys Desargues' theorem. This three-dimensional realization of the Desargues configuration is also called the complete pentahedron .
|
|
The angle and axis unit vector define a rotation, concisely represented by the rotation vector . In mathematics, the axis–angle representation of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector indicating the direction of an axis of rotation, and an angle describing the magnitude of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector rooted at the origin because the magnitude of is constrained. For example, the elevation and azimuth angles of suffice to locate it in any particular Cartesian coordinate frame.
|
|
The solutions of a linear equation form a line in the Euclidean plane, and, conversely, every line can be viewed as the set of all solutions of a linear equation in two variables. This is the origin of the term linear for describing this type of equations. More generally, the solutions of a linear equation in variables form a hyperplane (a subspace of dimension ) in the Euclidean space of dimension . Linear equations occur frequently in all mathematics and their applications in physics and engineering, partly because non-linear systems are often well approximated by linear equations.
|
|
In the mathematics of structural rigidity, a rigidity matroid is a matroid that describes the number of degrees of freedom of an undirected graph with rigid edges of fixed lengths, embedded into Euclidean space. In a rigidity matroid for a graph with n vertices in d-dimensional space, a set of edges that defines a subgraph with k degrees of freedom has matroid rank dn − k. A set of edges is independent if and only if, for every edge in the set, removing the edge would increase the number of degrees of freedom of the remaining subgraph....
|
|
Also, the space of distributions and all Lp-spaces over open subsets of Euclidean space as well as many other spaces that occur in analysis are Souslin spaces. The Borel graph theorem states: :Let X and Y be locally convex Hausdorff spaces and let u : X \to Y be linear. If X is the inductive limit of an arbitrary family of Banach spaces, if Y is a Souslin space, and if the graph of u is a Borel set in X \times Y, then u is continuous. An improvement upon this theorem, proved by A. Martineau, uses K-analytic spaces.
|
|
In mathematics, a Carleson measure is a type of measure on subsets of n-dimensional Euclidean space Rn. Roughly speaking, a Carleson measure on a domain Ω is a measure that does not vanish at the boundary of Ω when compared to the surface measure on the boundary of Ω. Carleson measures have many applications in harmonic analysis and the theory of partial differential equations, for instance in the solution of Dirichlet problems with "rough" boundary. The Carleson condition is closely related to the boundedness of the Poisson operator. Carleson measures are named after the Swedish mathematician Lennart Carleson.
|
|
A set of points in a -dimensional affine space (-dimensional Euclidean space is a common example) is in general linear position (or just general position) if no of them lie in a -dimensional flat for . These conditions contain considerable redundancy since, if the condition holds for some value then it also must hold for all with . Thus, for a set containing at least points in -dimensional affine space to be in general position, it suffices to know that no hyperplane contains more than points -- i.e. the points do not satisfy any more linear relations than they must.
|
|
Space is one of the few fundamental quantities in physics, meaning that it cannot be defined via other quantities because there is nothing more fundamental known at present. Thus, similar to the definition of other fundamental quantities (like time and mass), space is defined via measurement. Currently, the standard space interval, called a standard metre or simply metre, is defined as the distance traveled by light in a vacuum during a time interval of 1/299792458 of a second (exact). In classical physics, space is a three-dimensional Euclidean space where any position can be described using three coordinates and parameterised by time.
|
|
Spurred by Weyl's brilliant results on compact groups, he developed new methods for the study of global properties of Lie groups; in particular he showed that topologically a connected Lie group is a product of a Euclidean space and a compact group, and for compact Lie groups he discovered that the possible fundamental groups of the underlying manifold can be read from the structure of the Lie algebra of the group. Finally, he outlined a method of determining the Betti numbers of compact Lie groups, again reducing the problem to an algebraic question on their Lie algebras, which has since been completely solved.
|
|
This is achieved via the fundamental assumption that images are projected from a Euclidean space through a linear, 5 degree of freedom (in the simplest case), pinhole camera model with non-linear optical distortion. The linear pinhole parameters are the focal length, the aspect ratio, the skew, and the 2D principal point. With only a set of uncalibrated (or calibrated) images, a scene may be reconstructed up to a six degree of freedom euclidean transform and an isotropic scaling. A mathematical theory for general multi-view camera self-calibration was originally demonstrated in 1992 by Olivier Faugeras, QT Luong, and Stephen J. Maybank.
|
|
In geometric measure theory, Falconer's conjecture, named after Kenneth Falconer, is an unsolved problem concerning the sets of Euclidean distances between points in compact d-dimensional spaces. Intuitively, it states that a set of points that is large in its Hausdorff dimension must determine a set of distances that is large in measure. More precisely, if S is a compact set of points in d-dimensional Euclidean space whose Hausdorff dimension is strictly greater than d/2, then the conjecture states that the set of distances between pairs of points in S must have nonzero Lebesgue measure.
|
|
Nirenberg's work on the Minkowski problem was significantly extended by Aleksei Pogorelov, Shiu-Yuen Cheng, and Shing-Tung Yau, among other authors. In a separate contribution to differential geometry, Nirenberg and Philip Hartman characterized the cylinders in Euclidean space as the only complete hypersurfaces which are intrinsically flat. In the same year as his resolution of the Weyl and Minkowski problems, Nirenberg made a major contribution to the understanding of the maximum principle, proving the strong maximum principle for second-order parabolic partial differential equations. This is now regarded as one of the most fundamental results in this setting.
|
|
In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of surfaces evolves under mean curvature flow if the normal component of the velocity of which a point on the surface moves is given by the mean curvature of the surface. For example, a round sphere evolves under mean curvature flow by shrinking inward uniformly (since the mean curvature vector of a sphere points inward). Except in special cases, the mean curvature flow develops singularities.
|
|
Bieberbach wrote a habilitation thesis in 1911 about groups of Euclidean motions – identifying conditions under which the group must have a translational subgroup whose vectors span the Euclidean space – that helped solve Hilbert's 18th problem. He worked on complex analysis and its applications to other areas in mathematics. He is known for his work on dynamics in several complex variables, where he obtained results similar to Fatou's. In 1916 he formulated the Bieberbach conjecture, stating a necessary condition for a holomorphic function to map the open unit disc injectively into the complex plane in terms of the function's Taylor series.
|
|
The deviation of a Riemannian manifold X from the standard metric on Euclidean space is measured by sectional curvature, which is a real number associated to any real 2-plane in the tangent space of X at a point. For example, the sectional curvature of the standard metric on CPn (for ) varies between 1/4 and 1. For a Hermitian manifold (for example, a Kähler manifold), the holomorphic sectional curvature means the sectional curvature restricted to complex lines in the tangent space. This behaves more simply, in that CPn has holomorphic sectional curvature equal to 1.
|
|
The concept of "curvature of space" is fundamental to cosmology. A space without curvature is called a "flat space" or Euclidean space. Whether the universe is “flat″ could determine its ultimate fate; whether it will expand forever, or ultimately collapse back into itself. The geometry of spacetime has been measured by the Wilkinson Microwave Anisotropy Probe (WMAP) to be nearly flat. According to the WMAP 5-year results and analysis, “WMAP determined that the universe is flat, from which it follows that the mean energy density in the universe is equal to the critical density (within a 1% margin of error).
|
|
As for symmetry in the usual sense of the term, we may start with the theorem that the symmetries of the n-dimensional Laplace equation are exactly the conformal symmetries of the n-dimensional Euclidean space. This fact has several implications. First of all, one can consider harmonic functions which transform under irreducible representations of the conformal group or of its subgroups (such as the group of rotations or translations). Proceeding in this fashion, one systematically obtains the solutions of the Laplace equation which arise from separation of variables such as spherical harmonic solutions and Fourier series.
|
|
So the family of separable states is the closed convex hull of pure product states. We will make use of the following variant of Hahn–Banach theorem: Theorem Let S_1 and S_2 be disjoint convex closed sets in a real Banach space and one of them is compact, then there exists a bounded functional f separating the two sets. This is a generalization of the fact that, in real Euclidean space, given a convex set and a point outside, there always exists an affine subspace separating the two. The affine subspace manifests itself as the functional f.
|
|
For example, the box dimension of a single point is 0, but the box dimension of the collection of rational numbers in the interval [0, 1] has dimension 1. The Hausdorff measure by comparison, is countably additive. An interesting property of the upper box dimension not shared with either the lower box dimension or the Hausdorff dimension is the connection to set addition. If A and B are two sets in a Euclidean space then A + B is formed by taking all the pairs of points a,b where a is from A and b is from B and adding a+b.
|
|
For points in Euclidean space, a set X is a Meyer set if it is relatively dense and its difference set X − X is uniformly discrete. Equivalently, X is a Meyer set if both X and X − X are Delone. Meyer sets are named after Yves Meyer, who introduced them (with a different but equivalent definition based on harmonic analysis) as a mathematical model for quasicrystals. They include the point sets of lattices, Penrose tilings, and the Minkowski sums of these sets with finite sets.. The Voronoi cells of symmetric Delone sets form space-filling polyhedra called plesiohedra..
|
|
When one considers motions of the Euclidean space that preserve the origin, the distinction between points and vectors, important in pure mathematics, can be erased because there is a canonical one- to-one correspondence between points and position vectors. The same is true for geometries other than Euclidean, but whose space is an affine space with a supplementary structure; see an example below. Alternatively, the vector description of rotations can be understood as a parametrization of geometric rotations up to their composition with translations. In other words, one vector rotation presents many equivalent rotations about all points in the space.
|
|
In virtual reality (VR), positional tracking detects the precise position of the head-mounted displays, controllers, other objects or body parts within Euclidean space. Positional tracking registers the exact position due to recognition of the rotation (pitch, yaw and roll) and recording of the translational movements. Since VR is about emulating and altering reality it's important that we can track accurately how objects (like the head or the hands) move in real life in order to represent them. Defining the position and orientation of a real object in space is determined with the help of special sensors or markers.
|
|
But then it can be defined on the n-dimensional Euclidean space or other mathematical spaces, where it is often interpreted as a random set or a random counting measure, instead of a stochastic process. In this setting, the Poisson process, also called the Poisson point process, is one of the most important objects in probability theory, both for applications and theoretical reasons. But it has been remarked that the Poisson process does not receive as much attention as it should, partly due to it often being considered just on the real line, and not on other mathematical spaces.
|
|
The set T is called the index set or parameter set of the stochastic process. Often this set is some subset of the real line, such as the natural numbers or an interval, giving the set T the interpretation of time. In addition to these sets, the index set T can be other linearly ordered sets or more general mathematical sets, such as the Cartesian plane R^2 or n-dimensional Euclidean space, where an element t\in T can represent a point in space. But in general more results and theorems are possible for stochastic processes when the index set is ordered.
|
|
Convex polyhedra can be defined in three-dimensional hyperbolic space in the same way as in Euclidean space, as the convex hulls of finite sets of points. However, in hyperbolic space, it is also possible to consider ideal points as well as the points that lie within the space. An ideal polyhedron is the convex hull of a finite set of ideal points. Its faces are ideal polygons, but its edges are defined by entire hyperbolic lines rather than line segments, and its vertices (the ideal points of which it is the convex hull) do not lie within the hyperbolic space.
|
|
When the disk undergoes a full clockwise revolution, the bicycle wheel will not return to its original position, but will have undergone a net rotation of . Foucault-like precession is observed in a virtual system wherein a massless particle is constrained to remain on a rotating plane that is inclined with respect to the axis of rotation. Spin of a relativistic particle moving in a circular orbit precesses similar to the swing plane of Foucault pendulum. The relativistic velocity space in Minkowski spacetime can be treated as a sphere S3 in 4-dimensional Euclidean space with imaginary radius and imaginary timelike coordinate.
|
|
In 1990, Claus Gerhardt showed that this situation is characteristic of the more general case of mean-convex star-shaped smooth hypersurfaces of Euclidean space. In particular, for any such initial data, the inverse mean curvature flow exists for all positive time and consists only of mean-convex and star- shaped smooth hypersurfaces. Moreover the surface area grows exponentially, and after a rescaling that fixes the surface area, the surfaces converge smoothly to a round sphere. The geometric estimates in Gerhardt's work follow from the maximum principle; the asymptotic roundness then becomes a consequence of the Krylov-Safonov theorem.
|
|
In 1951, Bing proved that all hereditarily indecomposable arc-like continua are homeomorphic — this implies that Knaster's K, Moise's M, and Bing's B are all homeomorphic. Bing also proved that the pseudo-arc is typical among the continua in a Euclidean space of dimension at least 2 or an infinite-dimensional separable Hilbert space. Bing and F. Burton Jones constructed a decomposable planar continuum that admits an open map onto the circle, with each point preimage homeomorphic to the pseudo-arc, called the circle of pseudo-arcs. Bing and Jones also showed that it is homogeneous.
|
|
In computational geometry, the fixed-radius near neighbor problem is a variant of the nearest neighbor search problem. In the fixed-radius near neighbor problem, one is given as input a set of points in d-dimensional Euclidean space and a fixed distance Δ. One must design a data structure that, given a query point q, efficiently reports the points of the data structure that are within distance Δ of q. The problem has long been studied; cites a 1966 paper by Levinthal that uses this technique as part of a system for visualizing molecular structures, and it has many other applications..
|
|
Thus, in the Seifert–Weber space, each edge is surrounded by five pentagonal faces, and the dihedral angle between these pentagons is 72°. This does not match the 117° dihedral angle of a regular dodecahedron in Euclidean space, but in hyperbolic space there exist regular dodecahedra with any dihedral angle between 60° and 117°, and the hyperbolic dodecahedron with dihedral angle 72° may be used to give the Seifert–Weber space a geometric structure as a hyperbolic manifold. It is a quotient space of the order-5 dodecahedral honeycomb, a regular tessellation of hyperbolic 3-space by dodecahedra with this dihedral angle.
|
|
If one applies Minkowski space-based special relativity to expansion of the universe, without resorting to the concept of a curved spacetime, then one obtains the Milne model. Any spatial section of the universe of a constant age (the proper time elapsed from the Big Bang) will have a negative curvature; this is merely a pseudo-Euclidean geometric fact analogous to one that concentric spheres in the flat Euclidean space are nevertheless curved. Spatial geometry of this model is an unbounded hyperbolic space. The entire universe is contained within a light cone, namely the future cone of the Big Bang.
|
|
He deduced from this that the intersection form must be a sum of one-dimensional ones, which led to several spectacular applications to smooth 4-manifolds, such as the existence of non- equivalent smooth structures on 4-dimensional Euclidean space. Donaldson went on to use the other moduli spaces studied by Atiyah to define Donaldson invariants, which revolutionized the study of smooth 4-manifolds, and showed that they were more subtle than smooth manifolds in any other dimension, and also quite different from topological 4-manifolds. Atiyah described some of these results in a survey talk.
|
|
Because spherical elliptic geometry can be modeled as, for example, a spherical subspace of a Euclidean space, it follows that if Euclidean geometry is self-consistent, so is spherical elliptic geometry. Therefore it is not possible to prove the parallel postulate based on the other four postulates of Euclidean geometry. Tarski proved that elementary Euclidean geometry is complete: there is an algorithm which, for every proposition, can show it to be either true or false.Tarski (1951) (This does not violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.
|
|
As a consequence the incompleteness identified by Aerts is not due to missing hidden variables, but to the impossibility to model separated quantum entities using quantum theory. Elaborating further on this analysis of separated quantum entities, Aerts explored a conceptual view on quantum reality substituting the notion of non locality by that of non spatiality, hence interpreting three-dimensional Euclidean space as a theatre for macroscopical material objects, but 'not' as the space containing all of reality. More concretely, quantum entities are considered to be not inside this three-dimensional space when in non local states.
|
|
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space (three dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom.
|
|
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Informally, it is a line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
|
|
It is further known as a Euclidean 2-torus (the "2" is its topological dimension); figures drawn on it obey Euclidean geometry as if it were flat, whereas the surface of a common "doughnut"-shaped torus is positively curved on the outer rim and negatively curved on the inner. Although having a different geometry than the standard embedding of a torus in three-dimensional Euclidean space, the square torus can also be embedded into three-dimensional space, by the Nash embedding theorem; one possible embedding modifies the standard torus by a fractal set of ripples running in two perpendicular directions along the surface..
|
|
A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines. Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three- dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds.
|
|
An important example is provided by affine connections. For a surface in R3, tangent planes at different points can be identified using a natural path-wise parallelism induced by the ambient Euclidean space, which has a well-known standard definition of metric and parallelism. In Riemannian geometry, the Levi-Civita connection serves a similar purpose. (The Levi- Civita connection defines path-wise parallelism in terms of a given arbitrary Riemannian metric on a manifold.) More generally, differential geometers consider spaces with a vector bundle and an arbitrary affine connection which is not defined in terms of a metric.
|
|
The dimension of a manifold depends on the base field with respect to which Euclidean space is defined. While analysis usually assumes a manifold to be over the real numbers, it is sometimes useful in the study of complex manifolds and algebraic varieties to work over the complex numbers instead. A complex number (x + iy) has a real part x and an imaginary part y, where x and y are both real numbers; hence, the complex dimension is half the real dimension. Conversely, in algebraically unconstrained contexts, a single complex coordinate system may be applied to an object having two real dimensions.
|
|
Curves on a surface which minimize length between the endpoints are called geodesics; they are the shape that an elastic band stretched between the two points would take. Mathematically they are described using ordinary differential equations and the calculus of variations. The differential geometry of surfaces revolves around the study of geodesics. It is still an open question whether every Riemannian metric on a 2-dimensional local chart arises from an embedding in 3-dimensional Euclidean space: the theory of geodesics has been used to show this is true in the important case when the components of the metric are analytic.
|
|
Hyperbolic paraboloid A model of an elliptic hyperboloid of one sheet A 300px A saddle surface is a smooth surface containing one or more saddle points. Classical examples of two-dimensional saddle surfaces in the Euclidean space are second order surfaces, the hyperbolic paraboloid z=x^2-y^2 (which is often referred to as "the saddle surface" or "the standard saddle surface") and the hyperboloid of one sheet. The Pringles potato chip or crisp is an everyday example of a hyperbolic paraboloid shape. Saddle surfaces have negative Gaussian curvature which distinguish them from convex/elliptical surfaces which have positive Gaussian curvature.
|
|
Special relativity places space and time on equal footing, so that one considers the geometry of a unified spacetime instead of considering space and time separately. Minkowski geometry replaces Galilean geometry (which is the three-dimensional Euclidean space with time of Galilean relativity). In relativity, rather than considering Euclidean, elliptic and hyperbolic geometries, the appropriate geometries to consider are Minkowski space, de Sitter space and anti-de Sitter space, corresponding to zero, positive and negative curvature respectively. Hyperbolic geometry enters special relativity through rapidity, which stands in for velocity, and is expressed by a hyperbolic angle.
|
|
The RGB color model mapped to a cube. The horizontal x-axis as red values increasing to the left, y-axis as blue increasing to the lower right and the vertical z-axis as green increasing towards the top. The origin, black is the vertex hidden from view. :See also RGB color space Since colors are usually defined by three components, not only in the RGB model, but also in other color models such as CIELAB and Y'UV, among others, then a three-dimensional volume is described by treating the component values as ordinary Cartesian coordinates in a Euclidean space.
|
|
Illustratiom from De ichnographica campi published in Acta Eruditorum, 1763 La perspective affranchie de l'embarras du plan géometral, French edition, 1759 Lambert was the first to introduce hyperbolic functions into trigonometry. Also, he made conjectures regarding non-Euclidean space. Lambert is credited with the first proof that π is irrational by using a generalized continued fraction for the function tan x. Euler believed the conjecture but could not prove that π was irrational, and it is speculated that Aryabhata also believed this, in 500 CE. Lambert also devised theorems regarding conic sections that made the calculation of the orbits of comets simpler.
|
|
Each of these ideas can be carefully defined, and correspond reasonably well to the equivalent concepts from ordinary mathematics. The analogy does not stop there: one has an entire branch of supermathematics, where the analog of Euclidean space is superspace, the analog of a manifold is a supermanifold, the analog of a Lie algebra is a Lie superalgebra and so on. The Grassmann numbers are the underlying construct that make this all possible. Of course, one could pursue a similar program for any other field, or even ring, and this is indeed widely and commonly done in mathematics.
|
|
In higher-dimensional algebra (HDA), a double groupoid is a generalisation of a one-dimensional groupoid to two dimensions, and the latter groupoid can be considered as a special case of a category with all invertible arrows, or morphisms. Double groupoids are often used to capture information about geometrical objects such as higher-dimensional manifolds (or n-dimensional manifolds). In general, an n-dimensional manifold is a space that locally looks like an n-dimensional Euclidean space, but whose global structure may be non-Euclidean. Double groupoids were first introduced by Ronald Brown in 1976, in ref.
|
|
A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two antipodal points, u and v, are also shown. The great-circle distance, orthodromic distance, or spherical distance is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a straight line through the sphere's interior). The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics.
|
|
The modern, more abstract treatment, first formulated by Giuseppe Peano in 1888, encompasses more general objects than Euclidean space, but much of the theory can be seen as an extension of classical geometric ideas like lines, planes and their higher-dimensional analogs. Today, vector spaces are applied throughout mathematics, science and engineering. They are the appropriate linear-algebraic notion to deal with systems of linear equations. They offer a framework for Fourier expansion, which is employed in image compression routines, and they provide an environment that can be used for solution techniques for partial differential equations.
|
|
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by , is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original space of vectors. The magnitudeStrictly speaking, the magnitude depends on some additional structure, namely that the vectors be in a Euclidean space. We do not generally assume that this structure is available, except where it is helpful to develop intuition on the subject.
|
|
In a Euclidean space, the separation between two points is measured by the distance between the two points. The distance is purely spatial, and is always positive. In spacetime, the separation between two events is measured by the invariant interval between the two events, which takes into account not only the spatial separation between the events, but also their separation in time. The interval, , between two events is defined as: :s^2 = \Delta r^2 - c^2\Delta t^2 \,(spacetime interval), where is the speed of light, and and denote differences of the space and time coordinates, respectively, between the events.
|
|
In mathematics, a Euclidean group is the group of (Euclidean) isometries of an Euclidean space 𝔼n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension n of the space, and is commonly denoted E(n) or ISO(n). The Euclidean group E(n) comprises all translations, rotations, and reflections of 𝔼n; and arbitrary finite combinations of them. The Euclidean group can be seen as the symmetry group of the space itself, and contains the group of symmetries of any figure (subset) of that space.
|
|
Hermann Weyl gave an intrinsic definition for differentiable manifolds in 1912. During the 1930s Hassler Whitney and others clarified the foundational aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through differential geometry and Lie group theory. The Whitney embedding theorem showed that manifolds intrinsically defined by charts could always be embedded in Euclidean space, as in the extrinsic definition, showing that the two concepts of manifold were equivalent. Due to this unification, it is said to be the first complete exposition of the modern concept of manifold.
|
|
The word "geometry" (from Ancient Greek: geo- "earth", -metron "measurement") initially meant a practical way of processing lengths, regions and volumes in the space in which we live, but was then extended widely (as well as the notion of space in question here). According to Bourbaki, the period between 1795 (Géométrie descriptive of Monge) and 1872 (the "Erlangen programme" of Klein) can be called the golden age of geometry. The original space investigated by Euclid is now called three-dimensional Euclidean space. Its axiomatization, started by Euclid 23 centuries ago, was reformed with Hilbert's axioms, Tarski's axioms and Birkhoff's axioms.
|
|
Let (X, d) be a metric space and A be a nonempty subset of X. If x is a point of X, the distance of x from A is defined as d(x, A) = inf{ d(x, a): a in A}. If A and B are both nonempty subsets of X then the equidistant set determined by A and B is defined to be the set {x in X: d(x, A) = d(x, B)}. This equidistant set is denoted by { A = B }. The study of equidistant sets is more interesting in the case when the background metric space is the Euclidean space.
|
|
In mathematics, the Johnson–Lindenstrauss lemma is a result named after William B. Johnson and Joram Lindenstrauss concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space. The lemma states that a set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are nearly preserved. The map used for the embedding is at least Lipschitz, and can even be taken to be an orthogonal projection. The lemma has uses in compressed sensing, manifold learning, dimensionality reduction, and graph embedding.
|
|
Knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
|
|
In mathematics and probability theory, continuum percolation theory is a branch of mathematics that extends discrete percolation theory to continuous space (often Euclidean space ). More specifically, the underlying points of discrete percolation form types of lattices whereas the underlying points of continuum percolation are often randomly positioned in some continuous space and form a type of point process. For each point, a random shape is frequently placed on it and the shapes overlap each with other to form clumps or components. As in discrete percolation, a common research focus of continuum percolation is studying the conditions of occurrence for infinite or giant components.
|
|
Let E be a finite-dimensional Euclidean space. A finite reflection group is a subgroup of the general linear group of E which is generated by a set of orthogonal reflections across hyperplanes passing through the origin. An affine reflection group is a discrete subgroup of the affine group of E that is generated by a set of affine reflections of E (without the requirement that the reflection hyperplanes pass through the origin). The corresponding notions can be defined over other fields, leading to complex reflection groups and analogues of reflection groups over a finite field.
|
|
Since Euclid, geometry had meant the geometry of Euclidean space of two dimensions (plane geometry) or of three dimensions (solid geometry). In the first half of the nineteenth century there had been several developments complicating the picture. Mathematical applications required geometry of four or more dimensions; the close scrutiny of the foundations of the traditional Euclidean geometry had revealed the independence of the parallel postulate from the others, and non-Euclidean geometry had been born. Klein proposed an idea that all these new geometries are just special cases of the projective geometry, as already developed by Poncelet, Möbius, Cayley and others.
|
|
In statistics and computational geometry, the notion of centerpoint is a generalization of the median to data in higher-dimensional Euclidean space. Given a set of points in d-dimensional space, a centerpoint of the set is a point such that any hyperplane that goes through that point divides the set of points in two roughly equal subsets: the smaller part should have at least a 1/(d + 1) fraction of the points. Like the median, a centerpoint need not be one of the data points. Every non-empty set of points (with no duplicates) has at least one centerpoint.
|
|
In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rn and Tn through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in harmonic analysis are the harmonic conjugation operator on the circle, the Hilbert transform on the circle and the real line, the Beurling transform in the complex plane and the Riesz transforms in Euclidean space. The continuity of these operators on L2 is evident because the Fourier transform converts them into multiplication operators. Continuity on Lp spaces was first established by Marcel Riesz.
|
|
In the case of an embedded surface, this lift is very simply described in terms of orthogonal projection. Indeed, the vector bundles associated with the frame bundle are all sub-bundles of trivial bundles that extend to the ambient Euclidean space; a first order differential operator can always be applied to a section of a trivial bundle, in particular to a section of the original sub- bundle, although the resulting section might no longer be a section of the sub-bundle. This can be corrected by projecting orthogonally. The Riemannian connection can also be characterized abstractly, independently of an embedding.
|
|
In mathematics, the equilateral dimension of a metric space is the maximum number of points that are all at equal distances from each other. Equilateral dimension has also been called "metric dimension", but the term "metric dimension" also has many other inequivalent usages. The equilateral dimension of a d-dimensional Euclidean space is , and the equilateral dimension of a d-dimensional vector space with the Chebyshev distance (L∞ norm) is 2d. However, the equilateral dimension of a space with the Manhattan distance (L1 norm) is not known; Kusner's conjecture, named after Robert B. Kusner, states that it is exactly 2d.
|
|
An eigenvector, interpreted in euclidean space, can be seen as a sequence of k euclidean vectors associated to corresponding landmark and designating a compound move for the whole shape. Global nonlinear variation is usually well handled provided nonlinear variation is kept to a reasonable level. Typically, a twisting nematode worm is used as an example in the teaching of kernel PCA-based methods. Due to the PCA properties: eigenvectors are mutually orthogonal, form a basis of the training set cloud in the shape space, and cross at the 0 in this space, which represents the mean shape.
|
|
In mathematics, the Calderón–Zygmund lemma is a fundamental result in Fourier analysis, harmonic analysis, and singular integrals. It is named for the mathematicians Alberto Calderón and Antoni Zygmund. Given an integrable function , where denotes Euclidean space and denotes the complex numbers, the lemma gives a precise way of partitioning into two sets: one where is essentially small; the other a countable collection of cubes where is essentially large, but where some control of the function is retained. This leads to the associated Calderón–Zygmund decomposition of , wherein is written as the sum of "good" and "bad" functions, using the above sets.
|
|
A similarity (also called a similarity transformation or similitude) of a Euclidean space is a bijection from the space onto itself that multiplies all distances by the same positive real number , so that for any two points and we have :d(f(x),f(y)) = r d(x,y), \, where "" is the Euclidean distance from to . The scalar has many names in the literature including; the ratio of similarity, the stretching factor and the similarity coefficient. When = 1 a similarity is called an isometry (rigid transformation). Two sets are called similar if one is the image of the other under a similarity.
|
|
The convex hull of the red set is the blue and red convex set. In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.
|
|
The Hopf fibration is a continuous function from the 3-sphere (a three-dimensional surface in four-dimensional Euclidean space) into the more familiar 2-sphere, with the property that the inverse image of each point on the 2-sphere is a circle. Thus, these images decompose the 3-sphere into a continuous family of circles, and each two distinct circles form a Hopf link. This was Hopf's motivation for studying the Hopf link: because each two fibers are linked, the Hopf fibration is a nontrivial fibration. This example began the study of homotopy groups of spheres..
|
|
Example of unstructured grid for a finite element analysis mesh An unstructured (or irregular) grid is a tessellation of a part of the Euclidean plane or Euclidean space by simple shapes, such as triangles or tetrahedra, in an irregular pattern. Grids of this type may be used in finite element analysis when the input to be analyzed has an irregular shape. Unlike structured grids, unstructured grids require a list of the connectivity which specifies the way a given set of vertices make up individual elements (see graph (data structure)). Ruppert's algorithm is often used to convert an irregularly shaped polygon into an unstructured grid of triangles.
|
|
The case of may be interpreted as the centre being a well-defined point at infinity with either infinite or zero radial scalar square (the latter for the case of a null hyperplane). Knowing (and ) in this case does not determine the hyperplane's position, though, only its orientation in space. The radial scalar square may take on a positive, zero or negative value. When the quadratic form is definite, even though and may be determined from the above expressions, the set of vectors satisfying the defining equation may be empty, as is the case in a Euclidean space for a negative radial scalar square.
|
|
In contrast to other Hough transform-based approaches for analytical shapes, Fernandes' technique does not depend on the shape one wants to detect nor on the input data type. The detection can be driven to a type of analytical shape by changing the assumed model of geometry where data have been encoded (e.g., euclidean space, projective space, conformal geometry, and so on), while the proposed formulation remains unchanged. Also, it guarantees that the intended shapes are represented with the smallest possible number of parameters, and it allows the concurrent detection of different kinds of shapes that best fit an input set of entries with different dimensionalities and different geometric definitions (e.g.
|
|
A visual depiction of a Poisson point process starting from 0, in which increments occur continuously and independently at rate λ. In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space. The Poisson point process is often called simply the Poisson process, but it is also called a Poisson random measure, Poisson random point field or Poisson point field. This point process has convenient mathematical properties, which has led to it being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy,G.
|
|
However, a 4-polytope can be considered a tessellation of a 3-dimensional non- Euclidean space, namely, a tessellation of the surface of a four-dimensional sphere (a 4-dimensional spherical tiling). A regular dodecahedral honeycomb, {5,3,4}, of hyperbolic space projected into 3-space. Locally, this space seems like the one we are familiar with, and therefore, a virtual-reality system could, in principle, be programmed to allow exploration of these "tessellations", that is, of the 4-dimensional regular polytopes. The mathematics department at UIUC has a number of pictures of what one would see if embedded in a tessellation of hyperbolic space with dodecahedra.
|
|
One such generalization is that a topological space is sequentially compact if every infinite sequence of points sampled from the space has an infinite subsequence that converges to some point of the space. The Bolzano–Weierstrass theorem states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses an infinite number of points in the closed unit interval , some of those points will get arbitrarily close to some real number in that space. For instance, some of the numbers in the sequence accumulate to 0 (while others accumulate to 1).
|
|
Since the (red) part of the (black and red) line-segment joining the points x and y lies outside of the (green) set, the set is non-convex. In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points, it contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersect every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex.
|
|
If the configuration space is continuous (something like the real line or Euclidean space, see above), then there are no valid quantum states corresponding to particular , and the probability that the system is "in the state " will always be zero. An archetypical example of this is the space constructed with 1-dimensional Lebesgue measure; it is used to study a motion in one dimension. This presentation of the infinite-dimensional Hilbert space corresponds to the spectral decomposition of the coordinate operator: in this example. Although there are no such vectors as , strictly speaking, the expression can be made meaningful, for instance, with spectral theory.
|
|
These 24 points also form the 24 roots in the root system D_4. They can be grouped into pairs of points opposite each other on a line through the origin. The lines and planes through the origin of four-dimensional Euclidean space have the geometry of the points and lines of three-dimensional projective space, and in this three-dimensional projective space the lines through opposite pairs of these 24 points and the central planes through these points become the points and lines of the Reye configuration . The permutations of (\pm 1, \pm 1, 0, 0) form the homogeneous coordinates of the 12 points in this configuration.
|
|
In addition, infinitely many sites are allowed in the definition (this setting has applications in geometry of numbers and crystallography), but again, in many cases only finitely many sites are considered. In the particular case where the space is a finite-dimensional Euclidean space, each site is a point, there are finitely many points and all of them are different, then the Voronoi cells are convex polytopes and they can be represented in a combinatorial way using their vertices, sides, two- dimensional faces, etc. Sometimes the induced combinatorial structure is referred to as the Voronoi diagram. In general however, the Voronoi cells may not be convex or even connected.
|
|
Grid cells were discovered in 2005 by Edvard Moser, May-Britt Moser, and their students Torkel Hafting, Marianne Fyhn, and Sturla Molden at the Centre for the Biology of Memory (CBM) in Norway. They were awarded the 2014 Nobel Prize in Physiology or Medicine together with John O'Keefe for their discoveries of cells that constitute a positioning system in the brain. The arrangement of spatial firing fields, all at equal distances from their neighbors, led to a hypothesis that these cells encode a neural representation of Euclidean space. The discovery also suggested a mechanism for dynamic computation of self-position based on continuously updated information about position and direction.
|
|
Consider the problem of estimating the probability that a test point in N-dimensional Euclidean space belongs to a set, where we are given sample points that definitely belong to that set. Our first step would be to find the centroid or center of mass of the sample points. Intuitively, the closer the point in question is to this center of mass, the more likely it is to belong to the set. However, we also need to know if the set is spread out over a large range or a small range, so that we can decide whether a given distance from the center is noteworthy or not.
|
|
Given a set of points in Euclidean space, the first principal component corresponds to a line that passes through the multidimensional mean and minimizes the sum of squares of the distances of the points from the line. The second principal component corresponds to the same concept after all correlation with the first principal component has been subtracted from the points. The singular values (in Σ) are the square roots of the eigenvalues of the matrix XTX. Each eigenvalue is proportional to the portion of the "variance" (more correctly of the sum of the squared distances of the points from their multidimensional mean) that is associated with each eigenvector.
|
|
In the point of view of Cartan connections, however, the affine subspaces of Euclidean space are model surfaces -- they are the simplest surfaces in Euclidean 3-space, and are homogeneous under the affine group of the plane -- and every smooth surface has a unique model surface tangent to it at each point. These model surfaces are Klein geometries in the sense of Felix Klein's Erlangen programme. More generally, an -dimensional affine space is a Klein geometry for the affine group , the stabilizer of a point being the general linear group . An affine -manifold is then a manifold which looks infinitesimally like -dimensional affine space.
|
|
The class of n\times n doubly stochastic matrices is a convex polytope known as the Birkhoff polytope B_n. Using the matrix entries as Cartesian coordinates, it lies in an (n-1)^2-dimensional affine subspace of n^2-dimensional Euclidean space defined by 2n-1 independent linear constraints specifying that the row and column sums all equal one. (There are 2n-1 constraints rather than 2n because one of these constraints is dependent, as the sum of the row sums must equal the sum of the column sums.) Moreover, the entries are all constrained to be non-negative and less than or equal to one.
|
|
The Heawood graph is a toroidal graph; that is, it can be embedded without crossings onto a torus. One embedding of this type places its vertices and edges into three-dimensional Euclidean space as the set of vertices and edges of a nonconvex polyhedron with the topology of a torus, the Szilassi polyhedron. The graph is named after Percy John Heawood, who in 1890 proved that in every subdivision of the torus into polygons, the polygonal regions can be colored by at most seven colors. The Heawood graph forms a subdivision of the torus with seven mutually adjacent regions, showing that this bound is tight.
|
|
The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for the parameter space is domain of a function. The ranges of values of the parameters may form the axes of a plot, and particular outcomes of the model may be plotted against these axes to illustrate how different regions of the parameter space produce different types of behavior in the model. In statistics, parameter spaces are particularly useful for describing parametric families of probability distributions.
|
|
It is interesting to consider the space as the three-dimensional sphere S3, the boundary of a disk in 4-dimensional Euclidean space. For doing this, we will have to define how we represent a rotation with this 4D-embedded surface. The way in which the radius can be used to specify the angle of rotation is not straightforward. It can be related to circles of latitude in a sphere with a defined north pole and is explained as follows: Beginning at the north pole of a sphere in three-dimensional space, we specify the point at the north pole to represent the identity rotation.
|
|
One area in which these considerations, in some form, become inevitable, is the kinematics of a rigid body. One can take as definition the idea of a curve in the Euclidean group E(3) of three- dimensional Euclidean space, starting at the identity (initial position). The translation subgroup T of E(3) is a normal subgroup, with quotient SO(3) if we look at the subgroup E+(3) of direct isometries only (which is reasonable in kinematics). The translational part can be decoupled from the rotational part in standard Newtonian kinematics by considering the motion of the center of mass, and rotations of the rigid body about the center of mass.
|
|
A random field is a collection of random variables indexed by a n-dimensional Euclidean space or some manifold. In general, a random field can be considered an example of a stochastic or random process, where the index set is not necessarily a subset of the real line. But there is a convention that an indexed collection of random variables is called a random field when the index has two or more dimensions. If the specific definition of a stochastic process requires the index set to be a subset of the real line, then the random field can be considered as a generalization of stochastic process.
|
|
Its index set and state space are the non-negative numbers and real numbers, respectively, so it has both continuous index set and states space. But the process can be defined more generally so its state space can be n-dimensional Euclidean space. If the mean of any increment is zero, then the resulting Wiener or Brownian motion process is said to have zero drift. If the mean of the increment for any two points in time is equal to the time difference multiplied by some constant \mu, which is a real number, then the resulting stochastic process is said to have drift \mu.
|
|
Poincaré believed that Newton's first law was not empirical but is a conventional framework assumption for mechanics (Gargani, 2012). He also believed that the geometry of physical space is conventional. He considered examples in which either the geometry of the physical fields or gradients of temperature can be changed, either describing a space as non-Euclidean measured by rigid rulers, or as a Euclidean space where the rulers are expanded or shrunk by a variable heat distribution. However, Poincaré thought that we were so accustomed to Euclidean geometry that we would prefer to change the physical laws to save Euclidean geometry rather than shift to a non-Euclidean physical geometry.
|
|
Sphere packing finds practical application in the stacking of cannonballs In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, spaces of other dimensions (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space. A typical sphere packing problem is to find an arrangement in which the spheres fill as much of the space as possible.
|
|
The Borel graph theorem, proved by L. Schwartz, shows that the closed graph theorem is valid for linear maps defined on and valued in most spaces encountered in analysis. Recall that a topological space is called a Polish space if it is a separable complete metrizable space and that a Souslin space is the continuous image of a Polish space. The weak dual of a separable Fréchet space and the strong dual of a separable Fréchet-Montel space are Souslin spaces. Also, the space of distributions and all Lp-spaces over open subsets of Euclidean space as well as many other spaces that occur in analysis are Souslin spaces.
|
|
The quadrilaterals formed by the edges between the points in any two symmetric pairs of points can be thought of as equators of the octahedron. These equators have the property (by their symmetry) that opposite pairs of quadrilateral sides have equal length. Every quadrilateral with opposite pairs of equal sides, embedded in Euclidean space, has axial symmetry, and some (such as the rectangle) have other symmetries besides. If one cuts the Bricard octahedron into two open-bottomed pyramids by slicing it along one of its equators, both of these open pyramids can flex, and the flexing motion can be made to preserve the axis of symmetry of the whole shape.
|
|
A piecewise linear function over two dimensions (top) and the polygonal areas on which it is linear (bottom) In computational geometry, a polyhedral terrain in three-dimensional Euclidean space is a polyhedral surface that intersects every line parallel to some particular line in a connected set (i.e., a point or a line segment) or the empty set.Richard Cole, Micha Sharir, "Visibility problems for polyhedral terrains" 1989, Without loss of generality, we may assume that the line in question is the z-axis of the Cartesian coordinate system. Then a polyhedral terrain is the image of a piecewise-linear function in x and y variables.
|
|
The Banach fixed-point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. By contrast, the Brouwer fixed-point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point,Eberhard Zeidler, Applied Functional Analysis: main principles and their applications, Springer, 1995. but it doesn't describe how to find the fixed point (See also Sperner's lemma). For example, the cosine function is continuous in [−1,1] and maps it into [−1, 1], and thus must have a fixed point.
|
|
As shown by the tick marks, lines a and b are parallel. This can be proved because the transversal t produces congruent corresponding angles \theta, shown here both to the right of the transversal, one above and adjacent to line a and the other above and adjacent to line b. Given parallel straight lines l and m in Euclidean space, the following properties are equivalent: #Every point on line m is located at exactly the same (minimum) distance from line l (equidistant lines). #Line m is in the same plane as line l but does not intersect l (recall that lines extend to infinity in either direction).
|
|
Most books generally define a manifold as a space that is, locally, diffeomorphic to Euclidean space, thus by this definition, every manifold does not include its boundary. However, this definition is too specific as it doesn’t cover even basic objects such as a closed disk, so authors usually define a manifold with boundary and abusively say manifold without reference to the boundary. Due to this, a compact manifold (compact with respect to its underlying topology) can synonymously be used for closed manifolds if the definition is taken to be original definition. The notion of a closed manifold is unrelated with that of a closed set.
|
|
In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors {v1, ... , vk} in an inner product space (most commonly the Euclidean space Rn), orthogonalization results in a set of orthogonal vectors {u1, ... , uk} that generate the same subspace as the vectors v1, ... , vk. Every vector in the new set is orthogonal to every other vector in the new set; and the new set and the old set have the same linear span. In addition, if we want the resulting vectors to all be unit vectors, then the procedure is called orthonormalization.
|
|
A metric space M is said to be complete if every Cauchy sequence converges in M. That is to say: if d(x_n, x_m) \to 0 as both n and m independently go to infinity, then there is some y\in M with d(x_n, y) \to 0. Every Euclidean space is complete, as is every closed subset of a complete space. The rational numbers, using the absolute value metric d(x,y) = \vert x - y \vert, are not complete. Every metric space has a unique (up to isometry) completion, which is a complete space that contains the given space as a dense subset.
|
|
One may define Fréchet manifolds as spaces that "locally look like" Fréchet spaces (just like ordinary manifolds are defined as spaces that locally look like Euclidean space ), and one can then extend the concept of Lie group to these manifolds. This is useful because for a given (ordinary) compact manifold , the set of all diffeomorphisms forms a generalized Lie group in this sense, and this Lie group captures the symmetries of . Some of the relations between Lie algebras and Lie groups remain valid in this setting. Another important example of a Fréchet Lie group is the loop group of a compact Lie group , the smooth () mappings , multiplied pointwise by .
|
|
During his time at the University of Bonn, Delboeuf published Prolégomènes philosophiques à la géométrie (1860), disputing his mentor Ueberweg’s concept of Euclidean space and earning the praise of Bertrand Russell. He argued that, in order to use geometry to find the fundamental qualities of “determinations of space,” we must first understand the concepts of both “determination” and “space.” In this paper, he independently discovered Euclidean postulate 5. Postulate 5 states that, if a line intersects two straight lines that together form two interior angles on the same side that sum to less than 180 degrees, then the two straight lines must meet on that side.
|
|
The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.
|
|
From the point of view of the Erlangen program, one may understand that "all points are the same", in the geometry of X. This was true of essentially all geometries proposed before Riemannian geometry, in the middle of the nineteenth century. Thus, for example, Euclidean space, affine space and projective space are all in natural ways homogeneous spaces for their respective symmetry groups. The same is true of the models found of non- Euclidean geometry of constant curvature, such as hyperbolic space. A further classical example is the space of lines in projective space of three dimensions (equivalently, the space of two-dimensional subspaces of a four- dimensional vector space).
|
|
The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 'position space' to a function of momentum (or a function of space and time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex- valued, and possibly vector-valued.In relativistic quantum mechanics one encounters vector-valued Fourier transforms of multi-component wave functions.
|
|
Using the Pythagorean theorem to compute two-dimensional Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is a number, the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem. It is occasionally called the Pythagorean distance, especially as a way to contrast three-dimensional Euclidean distance with geodesic distance, the length of a shortest curve along some surface such as that of the earth. The distance between two objects that are not points is usually defined to be the smallest distance between any two points from the two objects.
|
|
Signed sets are fundamental to the definition of oriented matroids. They may also be used to define the faces of a hypercube. If the hypercube consists of all points in Euclidean space of a given dimension whose Cartesian coordinates are in the interval [-1,+1], then a signed subset of the coordinate axes can be used to specify the points whose coordinates within the subset are -1 or +1 (according to the sign in the signed subset) and whose other coordinates may be anywhere in the interval [-1,+1]. This subset of points forms a face, whose codimension is the cardinality of the signed subset.
|
|
A Tverberg partition of the vertices of a regular heptagon into three subsets with intersecting convex hulls. In discrete geometry, Tverberg's theorem, first stated by , is the result that sufficiently many points in d-dimensional Euclidean space can be partitioned into subsets with intersecting convex hulls. Specifically, for any set of :(d + 1)(r - 1) + 1\ points there exists a point x (not necessarily one of the given points) and a partition of the given points into r subsets, such that x belongs to the convex hull of all of the subsets. The partition resulting from this theorem is known as a Tverberg partition.
|
|
It is sometimes called the nucleus of the integral, whence the term nuclear operator arises. In the general theory, and may be points on any manifold; the real number line or -dimensional Euclidean space in the simplest cases. The general theory also often requires that the functions belong to some given function space: often, the space of square-integrable functions is studied, and Sobolev spaces appear often. The actual function space used is often determined by the solutions of the eigenvalue problem of the differential operator; that is, by the solutions to :L\psi_n(x)=\omega_n \psi_n(x) where the are the eigenvalues, and the are the eigenvectors.
|
|
In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into Euclidean space in such a way that no two cycles of the graph are linked. A flat embedding is an embedding with the property that every cycle is the boundary of a topological disk whose interior is disjoint from the graph. A linklessly embeddable graph is a graph that has a linkless or flat embedding; these graphs form a three- dimensional analogue of the planar graphs.. Complementarily, an intrinsically linked graph is a graph that does not have a linkless embedding. Flat embeddings are automatically linkless, but not vice versa.
|
|
A portion of the curve rotated around the -axis A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. A circle that is rotated around any diameter generates a sphere of which it is then a great circle, and if the circle is rotated around an axis that does not intersect the interior of a circle, then it generates a torus which does not intersect itself (a ring torus).
|
|
It may also easily be seen that many different elements in the "physical" space Hcolor can all result in the same single perceived color in R3color, so a perceived color is not unique to one physical color. Thus human color perception is determined by a specific, non-unique linear mapping from the infinite-dimensional Hilbert space Hcolor to the 3-dimensional Euclidean space R3color. Technically, the image of the (mathematical) cone over the simplex whose vertices are the spectral colors, by this linear mapping, is also a (mathematical) cone in R3color. Moving directly away from the vertex of this cone represents maintaining the same chromaticity while increasing its intensity.
|
|
In the active transformation (left), a point moves from position P to P' by rotating clockwise by an angle θ about the origin of the coordinate system. In the passive transformation (right), point P does not move, while the coordinate system rotates counterclockwise by an angle θ about its origin. The coordinates of P' in the active case (that is, relative to the original coordinate system) are the same as the coordinates of P relative to the rotated coordinate system. In analytic geometry, spatial transformations in the 3-dimensional Euclidean space \R^3 are distinguished into active or alibi transformations, and passive or alias transformations.
|
|
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advantage of the notation is that it describes these groups in a way which indicates many of the groups' properties: in particular, it follows William Thurston in describing the orbifold obtained by taking the quotient of Euclidean space by the group under consideration. Groups representable in this notation include the point groups on the sphere (S^2), the frieze groups and wallpaper groups of the Euclidean plane (E^2), and their analogues on the hyperbolic plane (H^2).
|
|
In mathematics the Markov theorem gives necessary and sufficient conditions for two braids to have closures that are equivalent links. In algebraic topology, Alexander's theorem states that every knot or link in three- dimensional Euclidean space is the closure of a braid. The Markov theorem, proved by Russian mathematician Andrei Andreevich Markov Jr.A. A. Markov Jr., Über die freie Äquivalenz der geschlossenen Zöpfe states that three conditions are necessary and sufficient for two braids to have equivalent closures: # They are equivalent braids # They are conjugate braids # Appending or removing on the right of the braid a strand that crosses the strand to its left exactly once.
|
|
Even in a three-dimensional Euclidean space, there is typically no natural way to prescribe a basis of the tangent plane, and so it is conceived of as an abstract vector space rather than a real coordinate space. The tangent space is the generalization to higher- dimensional differentiable manifolds. Riemannian manifolds are manifolds whose tangent spaces are endowed with a suitable inner product.. See also Lorentzian manifold. Derived therefrom, the Riemann curvature tensor encodes all curvatures of a manifold in one object, which finds applications in general relativity, for example, where the Einstein curvature tensor describes the matter and energy content of space-time.
|
|
From Peano's example, it was easy to deduce continuous curves whose ranges contained the n-dimensional hypercube (for any positive integer n). It was also easy to extend Peano's example to continuous curves without endpoints, which filled the entire n-dimensional Euclidean space (where n is 2, 3, or any other positive integer). Most well-known space- filling curves are constructed iteratively as the limit of a sequence of piecewise linear continuous curves, each one more closely approximating the space-filling limit. Peano's ground-breaking article contained no illustrations of his construction, which is defined in terms of ternary expansions and a mirroring operator.
|
|
During the 15 years or so years prior to that paper, Federer worked at the technical interface of geometry and measure theory. He focused particularly on surface area, rectifiability of sets, and the extent to which one could substitute rectifiability for smoothness in the analysis of surfaces. His 1947 paper on the rectifiable subsets of n-space characterized purely unrectifiable sets by their "invisibility" under almost all projections. A. S. Besicovitch had proven this for 1-dimensional sets in the plane, but Federer's generalization, valid for subsets of arbitrary dimension in any Euclidean space, was a major technical accomplishment, and later played a key role in Normal and Integral Currents.
|
|
In an affine or Euclidean space of higher dimension, the points at infinity are the points which are added to the space to get the projective completion. The set of the points at infinity is called, depending on the dimension of the space, the line at infinity, the plane at infinity or the hyperplane at infinity, in all cases a projective space of one less dimension. As a projective space over a field is a smooth algebraic variety, the same is true for the set of points at infinity. Similarly, if the ground field is the real or the complex field, the set of points at infinity is a manifold.
|
|
Leibniz, G., "Nova Methodus pro Maximis et Minimis", Acta Eruditorum, Oct. 1684. More precisely, a straight line is said to be a tangent of a curve at a point if the line passes through the point on the curve and has slope , where f is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space. As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.
|
|
If S and T are topological spheres in Euclidean space, with S contained in T, then it is not true in general that the region between them is an annulus, because of the existence of wild spheres in dimension at least 3. So the annulus theorem has to be stated to exclude these examples, by adding some condition to ensure that S and T are well behaved. There are several ways to do this. The annulus theorem states that if any homeomorphism h of Rn to itself maps the unit ball B into its interior, then B − h(interior(B)) is homeomorphic to the annulus Sn−1×[0,1].
|
|
The approach of Cartan, using connection 1-forms on the frame bundle of M, gives a third way to understand the Riemannian connection, which is particularly easy to describe for an embedded surface. Thanks to a result of , later generalized by , the Riemannian connection on a surface embedded in Euclidean space E3 is just the pullback under the Gauss map of the Riemannian connection on S2. Using the identification of S2 with the homogeneous space SO(3)/SO(2), the connection 1-form is just a component of the Maurer–Cartan 1-form on SO(3). In other words, everything reduces to understanding the 2-sphere properly.
|
|
Direct projection of 3-sphere into 3D space and covered with surface grid, showing structure as stack of 3D spheres (2-spheres) In mathematics, a 3-sphere, or glome, is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere (or 2-sphere, a two-dimensional surface), the boundary of a ball in four dimensions is a 3-sphere (an object with three dimensions). A 3-sphere is an example of a 3-manifold and an n-sphere.
|
|
In probability theory and statistical mechanics, the Gaussian free field (GFF) is a Gaussian random field, a central model of random surfaces (random height functions). gives a mathematical survey of the Gaussian free field. The discrete version can be defined on any graph, usually a lattice in d-dimensional Euclidean space. The continuum version is defined on Rd or on a bounded subdomain of Rd. It can be thought of as a natural generalization of one-dimensional Brownian motion to d time (but still one space) dimensions; in particular, the one-dimensional continuum GFF is just the standard one- dimensional Brownian motion or Brownian bridge on an interval.
|
|
In a Euclidean system, congruence is fundamental; it is the counterpart of equality for numbers. In analytic geometry, congruence may be defined intuitively thus: two mappings of figures onto one Cartesian coordinate system are congruent if and only if, for any two points in the first mapping, the Euclidean distance between them is equal to the Euclidean distance between the corresponding points in the second mapping. A more formal definition states that two subsets A and B of Euclidean space Rn are called congruent if there exists an isometry f : Rn → Rn (an element of the Euclidean group E(n)) with f(A) = B. Congruence is an equivalence relation.
|
|
Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus. Many specific curves have been thoroughly investigated using the synthetic approach. Differential geometry takes another path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using vector calculus. One of the most important tools used to analyze a curve is the Frenet frame, a moving frame that provides a coordinate system at each point of the curve that is "best adapted" to the curve near that point.
|
|
A manifold is a space whose topology, near any of its points, is the same as the topology near a point of a Euclidean space; however, its global structure may be non-Euclidean. Familiar examples of two-dimensional manifolds include the sphere, torus, and Klein bottle; this book concentrates on three-dimensional manifolds, and on two-dimensional surfaces within them. A particular focus is a Heegaard splitting, a two- dimensional surface that partitions a 3-manifold into two handlebodies. It aims to present the main ideas of this area, but does not include detailed proofs for many of the results that it states, in many cases because these proofs are long and technical.
|
|
Every smooth surface or curve in Euclidean space is a metric space, in which the (intrinsic) distance between two points of is defined as the infimum of the lengths of the curves that go from to along . For example, on a closed curve C of length , for each point of the curve there is a unique other point of the curve (called the antipodal of ) at distance from . A compact surface fills a closed curve if its border (also called boundary, denoted ) is the curve . The filling is said isometric if for any two points of the boundary curve , the distance between them along is the same (not less) than the distance along the boundary.
|
|
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a pure number plus a direction, like velocity), a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space. Many mathematical structures called "tensors" are tensor fields.
|
|
Beckman and Quarles observe that the theorem is not true for the real line (one- dimensional Euclidean space). For, the function that returns if is an integer and returns otherwise obeys the preconditions of the theorem (it preserves unit distances) but is not an isometry. Beckman and Quarles also provide a counterexample for Hilbert space, the space of square-summable sequences of real numbers. This example involves the composition of two discontinuous functions: one that maps every point of the Hilbert space onto a nearby point in a countable dense subspace, and a second that maps this dense set into a countable unit simplex (an infinite set of points all at unit distance from each other).
|
|
A family of closed sets called tiles forms a tessellation or tiling of a Euclidean space if their union is the whole space and every two distinct sets in the family have disjoint interiors. A tiling is said to be monohedral if all of the tiles are congruent to each other. Keller's conjecture concerns monohedral tilings in which all of the tiles are hypercubes of the same dimension as the space. As formulates the problem, a cube tiling is a tiling by congruent hypercubes in which the tiles are additionally required to all be translations of each other, without any rotation, or equivalently to have all of their sides parallel to the coordinate axes of the space.
|
|
Although it lives in a 4-dimensional space, it forms a line segment (a 1-dimensional polytope) within that space. Similarly, the associahedron K4 can be realized in this way as a regular pentagon in five-dimensional Euclidean space, whose vertex coordinates are the cyclic permutations of the vector (1, 2 + φ, 1, 1 + φ, 1 + φ) where φ denotes the golden ratio. Because the possible triangles within a regular hexagon have areas that are integer multiples of each other, this construction can be used to give integer coordinates (in six dimensions) to the three-dimensional associahedron K5; however (as the example of K4 already shows) this construction in general leads to irrational numbers as coordinates.
|
|
The Pythagorean tiling may be generalized to a three- dimensional tiling of Euclidean space by cubes of two different sizes, which also is unilateral and equitransitive. Attila Bölcskei calls this three- dimensional tiling the Rogers filling. He conjectures that, in any dimension greater than three, there is again a unique unilateral and equitransitive way of tiling space by hypercubes of two different sizes.. See also , which includes an illustration of the three-dimensional tiling, credited to "Rogers" but cited to a 1960 paper by Richard K. Guy: . Burns and Rigby found several prototiles, including the Koch snowflake, that may be used to tile the plane only by using copies of the prototile in two or more different sizes.. .
|
|
Real projective space RPn is a compactification of Euclidean space Rn. For each possible "direction" in which points in Rn can "escape", one new point at infinity is added (but each direction is identified with its opposite). The Alexandroff one-point compactification of R we constructed in the example above is in fact homeomorphic to RP1. Note however that the projective plane RP2 is not the one-point compactification of the plane R2 since more than one point is added. Complex projective space CPn is also a compactification of Cn; the Alexandroff one-point compactification of the plane C is (homeomorphic to) the complex projective line CP1, which in turn can be identified with a sphere, the Riemann sphere.
|
|
A solution of such an equation is a -tuples such that substituting each element of the tuple for the corresponding variable transforms the equation into a true equality. For an equation to be meaningful, the coefficient of at least one variable must be non-zero. In fact, if every variable has a zero coefficient, then, as mentioned for one variable, the equation is either inconsistent (for ) as having no solution, or all are solutions. The -tuples that are solutions of a linear equation in are the Cartesian coordinates of the points of an -dimensional hyperplane in an Euclidean space (or affine space if the coefficients are complex numbers or belong to any field).
|
|
In the context of abstract polytopes, one instead refers to "locally projective polytopes" – see Abstract polytope: Local topology. For example, the 11-cell is a "locally projective polytope", but is not a globally projective polyhedron, nor indeed tessellates any manifold, as it not locally Euclidean, but rather locally projective, as the name indicates. Projective polytopes can be defined in higher dimension as tessellations of projective space in one less dimension. Defining k-dimensional projective polytopes in n-dimensional projective space is somewhat trickier, because the usual definition of polytopes in Euclidean space requires taking convex combinations of points, which is not a projective concept, and is infrequently addressed in the literature, but has been defined, such as in .
|
|
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component (dependent on the embedding) and the intrinsic covariant derivative component.
|
|
A framework is an undirected graph, embedded into d-dimensional Euclidean space by providing a d-tuple of Cartesian coordinates for each vertex of the graph. From a framework with n vertices and m edges, one can define a matrix with m rows and nd columns, an expanded version of the incidence matrix of the graph called the rigidity matrix. In this matrix, the entry in row e and column (v,i) is zero if v is not an endpoint of edge e. If, on the other hand, edge e has vertices u and v as endpoints, then the value of the entry is the difference between the ith coordinates of v and u.
|
|
The conjugate of a split-octonion x is given by :\bar x = x_0 - x_1\,i - x_2\,j - x_3\,k - x_4\,\ell - x_5\,\ell i - x_6\,\ell j - x_7\,\ell k , just as for the octonions. The quadratic form on x is given by :N(x) = \bar x x = (x_0^2 + x_1^2 + x_2^2 + x_3^2) - (x_4^2 + x_5^2 + x_6^2 + x_7^2) . This quadratic form N(x) is an isotropic quadratic form since there are non-zero split-octonions x with N(x) = 0. With N, the split-octonions form a pseudo-Euclidean space of eight dimensions over R, sometimes written R4,4 to denote the signature of the quadratic form.
|
|
Eccles's most significant contributions are concerned with the multiple points of immersions of manifolds in Euclidean space and their relationship with classical problems in the homotopy groups of spheres. His interest in this area began when he clarified the relationship between multiple points and the Hopf invariant (disproving a conjecture by Michael Freedman) and the Kervaire invariant. His teaching ranged over most areas of pure mathematics as well as the history of mathematics, relativity theory and probability theory. He became particularly interested in the transition from school to university mathematics and this led in 1967 to the publication by Cambridge University Press of his book 'Introduction to mathematical reasoning: numbers, sets and functions’.
|
|
A set that is made up only of isolated points is called a discrete set (see also discrete space). Any discrete subset S of Euclidean space must be countable, since the isolation of each of its points together with the fact that rationals are dense in the reals means that the points of S may be mapped into a set of points with rational coordinates, of which there are only countably many. However, not every countable set is discrete, of which the rational numbers under the usual Euclidean metric are the canonical example. A set with no isolated point is said to be dense-in-itself (every neighbourhood of a point contains other points of the set).
|
|
If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead. The values of a stochastic process are not always numbers and can be vectors or other mathematical objects. Based on their mathematical properties, stochastic processes can be grouped into various categories, which include random walks, martingales, Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, and branching processes. The study of stochastic processes uses mathematical knowledge and techniques from probability, calculus, linear algebra, set theory, and topology as well as branches of mathematical analysis such as real analysis, measure theory, Fourier analysis, and functional analysis.
|
|
In contrast, if the system intrinsically cannot be represented by independent coordinates (parameters), then it is truly an anholonomic system. Some authors make much of this by creating a distinction between so-called internal and external states of the system, but in truth, all parameters are necessary to characterize the system, be they representative of "internal" or "external" processes, so the distinction is in fact artificial. However, there is a very real and irreconcilable difference between physical systems that obey conservation principles and those that do not. In the case of parallel transport on a sphere, the distinction is clear: a Riemannian manifold has a metric fundamentally distinct from that of a Euclidean space.
|
|
One of the earliest results of this kind was due to Hermann Weyl who used David Hilbert's theory of integral equation in 1911 to show that the volume of a bounded domain in Euclidean space can be determined from the asymptotic behavior of the eigenvalues for the Dirichlet boundary value problem of the Laplace operator. This question is usually expressed as "Can one hear the shape of a drum?", the popular phrase due to Mark Kac. A refinement of Weyl's asymptotic formula obtained by Pleijel and Minakshisundaram produces a series of local spectral invariants involving covariant differentiations of the curvature tensor, which can be used to establish spectral rigidity for a special class of manifolds.
|
|
Poincaré's new condition—i.e., "trivial fundamental group"—can be restated as "every loop can be shrunk to a point." The original phrasing was as follows: Poincaré never declared whether he believed this additional condition would characterize the 3-sphere, but nonetheless, the statement that it does is known as the Poincaré conjecture. Here is the standard form of the conjecture: Note that "closed" here means, as customary in this area, the condition of being compact in terms of set topology, and also without boundary (3-dimensional Euclidean space is an example of a simply connected 3-manifold not homeomorphic to the 3-sphere; but it is not compact and therefore not a counter-example).
|
|
In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The concept was used by Sophie Germain in her work on elasticity theory.Marie-Louise Dubreil-Jacotin on Sophie Germain Jean Baptiste Marie Meusnier used it in 1776, in his studies of minimal surfaces. It is important in the analysis of minimal surfaces, which have mean curvature zero, and in the analysis of physical interfaces between fluids (such as soap films) which, for example, have constant mean curvature in static flows, by the Young- Laplace equation.
|
|
A realization of an abstract apeirogon is defined as a mapping from its vertices to a finite-dimensional geometric space (typically a Euclidean space) such that every symmetry of the abstract apeirogon corresponds to an isometry of the images of the mapping. Two realizations are called congruent if the natural bijection between their sets of vertices is induced by an isometry of their ambient Euclidean spaces. The classical definition of an apeirogon as an equally-spaced subdivision of the Euclidean line is a realization in this sense, as is the convex subset in the hyperbolic plane formed by the convex hull of equally-spaced points on a horocycle. Other realizations are possible in higher-dimensional spaces.
|
|
From the beginning and through the middle of the 18th century, differential geometry was studied from the extrinsic point of view: curves and surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the differential geometry of curves and differential geometry of surfaces. Starting with the work of Riemann, the intrinsic point of view was developed, in which one cannot speak of moving "outside" the geometric object because it is considered to be given in a free-standing way. The fundamental result here is Gauss's theorema egregium, to the effect that Gaussian curvature is an intrinsic invariant.
|
|
In it, the vertices can be connected by a path, such that every two edges in the path are at right angles to each other. A two-dimensional orthoscheme is a right triangle. A three-dimensional orthoscheme can be constructed from a cube by finding a path of three edges of the cube that do not all lie on the same square face, and forming the convex hull of the four points on this path. Dissection of a cube into six orthoschemes A dissection of a shape S (which may be any closed set in Euclidean space) is a representation of S as a union of other shapes whose interiors are disjoint from each other.
|
|
A formulation that was accepted for a long period was that the question was to characterize Lie groups as the topological groups that were also topological manifolds. In terms closer to those that Hilbert would have used, near the identity element of the group in question, there is an open set in Euclidean space containing , and on some open subset of there is a continuous mapping : that satisfies the group axioms where those are defined. This much is a fragment of a typical locally Euclidean topological group. The problem is then to show that is a smooth function near (since topological groups are homogeneous spaces, they look the same everywhere as they do near ).
|
|
In Riemannian geometry and pseudo-Riemannian geometry, the Gauss-Codazzi equations (also called the Gauss–Codazzi–Mainardi equations) are fundamental formulas which link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold. The equations were originally discovered in the context of surfaces in three-dimensional Euclidean space. In this context, the first equation, often called the Gauss equation (after its discoverer Carl Friedrich Gauss), says that the Gauss curvature of the surface, at any given point, is dictated by the derivatives of the Gauss map at that point, as encoded by the second fundamental form.This equation is the basis for Gauss's theorema egregium. .
|
|
Consider a scalar field φ contained in a large box of volume V in flat spacetime at the temperature T = β−1. The partition function is defined by a path integral over all fields φ on the Euclidean space obtained by putting τ = it which are zero on the walls of the box and which are periodic in τ with period β. In this situation from the partition function he computes energy, entropy and pressure of the radiation of the field φ. In case of flat spaces the eigenvalues appearing in the physical quantities are generally known, while in case of curved space they are not known: in this case asymptotic methods are needed.
|
|
Monge laid down the foundations of their theory in his classical memoir L'application de l'analyse à la géometrie which appeared in 1795. The defining contribution to the theory of surfaces was made by Gauss in two remarkable papers written in 1825 and 1827. This marked a new departure from tradition because for the first time Gauss considered the intrinsic geometry of a surface, the properties which are determined only by the geodesic distances between points on the surface independently of the particular way in which the surface is located in the ambient Euclidean space. The crowning result, the Theorema Egregium of Gauss, established that the Gaussian curvature is an intrinsic invariant, i.e.
|
|
In mathematics, a Swiss cheese is a compact subset of the complex plane obtained by removing from a closed disc some countable union of open discs, usually with some restriction on the centres and radii of the removed discs. Traditionally the deleted discs should have pairwise disjoint closures which are subsets of the interior of the starting disc, the sum of the radii of the deleted discs should be finite, and the Swiss cheese should have empty interior. This is the type of Swiss cheese originally introduced by the Swiss mathematician Alice Roth. More generally, a Swiss cheese may be all or part of Euclidean space Rn - or of an even more complicated manifold - with "holes" in it.
|
|
This theorem has been generalized by to a tight bound on the dimension of the height-three partially ordered sets formed analogously from the vertices, edges and faces of a convex polyhedron, or more generally of an embedded planar graph: in both cases, the order dimension of the poset is at most four. However, this result cannot be generalized to higher-dimensional convex polytopes, as there exist four-dimensional polytopes whose face lattices have unbounded order dimension. Even more generally, for abstract simplicial complexes, the order dimension of the face poset of the complex is at most , where is the minimum dimension of a Euclidean space in which the complex has a geometric realization .
|
|
Two linked curves forming a Hopf link. When the circle is mapped to three- dimensional Euclidean space by an injective function (a continuous function that does not map two different points of the circle to the same point of space), its image is a closed curve. Two disjoint closed curves that both lie on the same plane are unlinked, and more generally a pair of disjoint closed curves is said to be unlinked when there is a continuous deformation of space that moves them both onto the same plane, without either curve passing through the other or through itself. If there is no such continuous motion, the two curves are said to be linked.
|
|
The Gauss map provides a mapping from every point on a curve or a surface to a corresponding point on a unit sphere In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere S2. Namely, given a surface X lying in R3, the Gauss map is a continuous map N: X → S2 such that N(p) is a unit vector orthogonal to X at p, namely the normal vector to X at p. The Gauss map can be defined (globally) if and only if the surface is orientable, in which case its degree is half the Euler characteristic. The Gauss map can always be defined locally (i.e.
|
|
Outer Löwner-John ellipsoid containing a set of a points in R2 In mathematics, the John ellipsoid or Löwner-John ellipsoid E(K) associated to a convex body K in n-dimensional Euclidean space Rn can refer to the n-dimensional ellipsoid of maximal volume contained within K or the ellipsoid of minimal volume that contains K. Often, the minimal volume ellipsoid is called as Löwner ellipsoid, and the maximal volume ellipsoid as the John ellipsoid (although John worked with the minimal volume ellipsoid in its original paper). One also refer to the minimal volume circumscribed ellipsoid as the outer Löwner-John ellipsoid and the maximum volume inscribed ellipsoid as the inner Löwner-John ellipsoid.
|
|
If D is a differential operator on a Euclidean space of order n in k variables x_1, \dots, x_k, then its symbol is the function of 2k variables x_1, \dots, x_k, y_1, \dots, y_k, given by dropping all terms of order less than n and replacing \partial/\partial x_i by y_i. So the symbol is homogeneous in the variables y, of degree n. The symbol is well defined even though \partial/\partial x_i does not commute with x_i because we keep only the highest order terms and differential operators commute "up to lower-order terms". The operator is called elliptic if the symbol is nonzero whenever at least one y is nonzero.
|
|
As an application, he was able to derive some rigidity results for complete surfaces with constant higher-order mean curvature. In 2004, he and Uwe Abresch extended the classical Hopf differential, discovered by Heinz Hopf in the 1950s, from the setting of surfaces in three-dimensional Euclidean space to the setting of surfaces in products of two-dimensional space forms with the real line. They showed that, if the surface has constant mean curvature, then their Hopf differential is holomorphic relative to the natural complex structure on the surface. As an application, they were able to show that any immersed sphere of constant mean curvature must be rotationally symmetric, thereby extending a classical theorem of Alexandrov.
|
|
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a semilinear manner, thus the name; which originates from the Latin numerical prefix sesqui- meaning "one and a half". The basic concept of the dot product – producing a scalar from a pair of vectors – can be generalized by allowing a broader range of scalar values and, perhaps simultaneously, by widening the definition of a vector. A motivating special case is a sesquilinear form on a complex vector space, .
|
|
An implicit surface in a Euclidean space (or, more generally, in an affine space) of dimension 3 is the set of the common zeros of a differentiable function of three variables :f(x, y, z)=0. Implicit means that the equation defines implicitly one of the variables as a function of the other variables. This is made more exact by the implicit function theorem: if , and the partial derivative in of is not zero at , then there exists a differentiable function such that :f(x,y,\varphi(x,y))=0 in a neighbourhood of . In other words, the implicit surface is the graph of a function near a point of the surface where the partial derivative in is nonzero.
|
|
It is never actually used in practice computationally, but is conceptually useful in understanding other concepts in vision and graphics (Wong 2002). Since rays in space can be parameterized by three coordinates, x, y, and z and two angles θ and ϕ, as shown at left, it is a five-dimensional function, that is, a function over a five-dimensional manifold equivalent to the product of 3D Euclidean space and the 2-sphere. Summing the irradiance vectors D1 and D2 arising from two light sources I1 and I2 produces a resultant vector D having the magnitude and direction shown (Gershun, fig 17). Like Adelson, Gershun defined the light field at each point in space as a 5D function.
|
|
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice-versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations).
|
|
In mathematics and physics, a de Sitter space is the analog in Minkowski space, or spacetime, of a sphere in ordinary, Euclidean space. The n-dimensional de Sitter space, denoted dSn, is the Lorentzian manifold analog of an n-sphere (with its canonical Riemannian metric); it is maximally symmetric, has constant positive curvature, and is simply connected for n at least 3. The de Sitter space, as well as the anti-de Sitter space is named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked in the 1920s in Leiden closely together on the spacetime structure of our universe.
|
|
In graph theory, a voltage graph is a directed graph whose edges are labelled invertibly by elements of a group. It is formally identical to a gain graph, but it is generally used in topological graph theory as a concise way to specify another graph called the derived graph of the voltage graph. Typical choices of the groups used for voltage graphs include the two-element group ℤ2 (for defining the bipartite double cover of a graph), free groups (for defining the universal cover of a graph), d-dimensional integer lattices ℤd (viewed as a group under vector addition, for defining periodic structures in d-dimensional Euclidean space),; ; . and finite cyclic groups ℤn for n > 2\.
|
|
In 1986, Hamilton and Michael Gage applied Hamilton's Nash-Moser theorem and well-posedness result for parabolic equations to prove the well-posedness for mean curvature flow; they considered the general case of a one-parameter family of immersions of a closed manifold into a smooth Riemannian manifold. Then, they specialized to the case of immersions of the circle into the two-dimensional Euclidean space , which is the simplest context for curve shortening flow. Using the maximum principle as applied to the distance between two points on a curve, they proved that if the initial immersion is an embedding, then all future immersions in the mean curvature flow are embeddings as well. Furthermore, convexity of the curves is preserved into the future.
|
|
Convex geometry is a relatively young mathematical discipline. Although the first known contributions to convex geometry date back to antiquity and can be traced in the works of Euclid and Archimedes, it became an independent branch of mathematics at the turn of the 20th century, mainly due to the works of Hermann Brunn and Hermann Minkowski in dimensions two and three. A big part of their results was soon generalized to spaces of higher dimensions, and in 1934 T. Bonnesen and W. Fenchel gave a comprehensive survey of convex geometry in Euclidean space Rn. Further development of convex geometry in the 20th century and its relations to numerous mathematical disciplines are summarized in the Handbook of convex geometry edited by P. M. Gruber and J. M. Wills.
|
|
First steps toward determining the computational complexity were undertaken in proving that the problem is in larger complexity classes, which contain the class P. By using normal surfaces to describe the Seifert surfaces of a given knot, showed that the unknotting problem is in the complexity class NP. claimed the weaker result that unknotting is in AM ∩ co- AM; however, later they retracted this claim.Mentioned as a "personal communication" in reference [15] of . In 2011, Greg Kuperberg proved that (assuming the generalized Riemann hypothesis) the unknotting problem is in co- NP, and in 2016, Marc Lackenby provided an unconditional proof of co-NP membership. The unknotting problem has the same computational complexity as testing whether an embedding of an undirected graph in Euclidean space is linkless.
|
|
In three dimensions, a surface normal, or simply normal, to a surface at point P is a vector perpendicular to the tangent plane of the surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality (right angles). The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at point P is the set of vectors which are orthogonal to the tangent space at P. Normal vectors are of special interest in the case of smooth curves and smooth surfaces.
|
|
A case in which there is no minimal sufficient statistic was shown by Bahadur, 1954.Lehmann and Casella (1998), Theory of Point Estimation, 2nd Edition, Springer, p 37 However, under mild conditions, a minimal sufficient statistic does always exist. In particular, in Euclidean space, these conditions always hold if the random variables (associated with P_\theta ) are all discrete or are all continuous. If there exists a minimal sufficient statistic, and this is usually the case, then every complete sufficient statistic is necessarily minimal sufficientLehmann and Casella (1998), Theory of Point Estimation, 2nd Edition, Springer, page 42(note that this statement does not exclude the option of a pathological case in which a complete sufficient exists while there is no minimal sufficient statistic).
|
|
Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold. Lie groups (and their associated Lie algebras) play a major role in modern physics, with the Lie group typically playing the role of a symmetry of a physical system.
|
|
Manhattan metric, has a connected orthogonal convex hull, then that hull coincides with the tight span of the points. In metric geometry, the metric envelope or tight span of a metric space M is an injective metric space into which M can be embedded. In some sense it consists of all points "between" the points of M, analogous to the convex hull of a point set in a Euclidean space. The tight span is also sometimes known as the injective envelope or hyperconvex hull of M. It has also been called the injective hull, but should not be confused with the injective hull of a module in algebra, a concept with a similar description relative to the category of R-modules rather than metric spaces.
|
|
In a four-dimensional spacetime manifold, a hypersurface is a three-dimensional submanifold that can be either timelike, spacelike, or null. A particular hyper-surface \Sigma can be selected either by imposing a constraint on the coordinates :f (x^\alpha) = 0, or by giving parametric equations, :x^\alpha = x^\alpha (y^a), where y^a (a=1,2,3) are coordinates intrinsic to the hyper-surface. For example, a two-sphere in three-dimensional Euclidean space can be described either by :f (x^\alpha) = x^2 + y^2 + z^2 - r^2 = 0, where r is the radius of the sphere, or by :x = r \sin \theta \cos \phi, \quad y = r \sin \theta \sin \phi, \quad z = r \cos \theta, where \theta and \phi are intrinsic coordinates.
|
|
The function that maps every subset of a given field to its algebraic closure is also a finitary closure operator, and in general it is different from the operator mentioned before. Finitary closure operators that generalize these two operators are studied in model theory as dcl (for definable closure) and acl (for algebraic closure). The convex hull in n-dimensional Euclidean space is another example of a finitary closure operator. It satisfies the anti- exchange property: If x is in the closure of the union of {y} and A, but not in the union of {y} and closure of A, then y is not in the closure of the union of {x} and A. Finitary closure operators with this property give rise to antimatroids.
|
|
The seven chapters of the book are largely self-contained, and consider different problems combining tessellations and algebra. Throughout the book, the history of the subject as well as the state of the art is discussed, and there are many illustrations. The first chapter concerns a conjecture of Hermann Minkowski that, in any lattice tiling of a Euclidean space by unit hypercubes (a tiling in which a lattice of translational symmetries takes any hypercube to any other hypercube) some two cubes must meet face-to-face. This result was resolved positively by Hajós's theorem in group theory, but a generalization of this question to non-lattice tilings (Keller's conjecture) was disproved shortly before the publication of the book, in part by using similar group-theoretic methods.
|
|
Estimating the box-counting dimension of the coast of Great Britain In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set S in a Euclidean space Rn, or more generally in a metric space (X, d). It is named after the German mathematician Hermann Minkowski and the French mathematician Georges Bouligand. To calculate this dimension for a fractal S, imagine this fractal lying on an evenly spaced grid, and count how many boxes are required to cover the set. The box-counting dimension is calculated by seeing how this number changes as we make the grid finer by applying a box- counting algorithm.
|
|
A point process is a collection of points randomly located on some mathematical space such as the real line, n-dimensional Euclidean space, or more abstract spaces. Sometimes the term point process is not preferred, as historically the word process denoted an evolution of some system in time, so a point process is also called a random point field. There are different interpretations of a point process, such a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is not clear.
|
|
Knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3 (since we're using topology, a circle isn't bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
|
|
Knot theory is the study of mathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3 (since we're using topology, a circle isn't bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
|
|
In mathematics, the Lévy–Steinitz theorem identifies the set of values to which rearrangements of an infinite series of vectors in Rn can converge. It was proved by Paul Lévy in his first published paper when he was 19 years old.. In 1913 Ernst Steinitz filled in a gap in Lévy's proof and also proved the result by a different method.. In an expository article, Peter Rosenthal stated the theorem in the following way.. : The set of all sums of rearrangements of a given series of vectors in a finite-dimensional real Euclidean space is either the empty set or a translate of a subspace (i.e., a set of the form v + M, where v is a given vector and M is a linear subspace).
|
|
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for Rn arises in this fashion. For a general inner product space V, an orthonormal basis can be used to define normalized orthogonal coordinates on V. Under these coordinates, the inner product becomes a dot product of vectors.
|
|
In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold. A metric tensor is called positive-definite if it assigns a positive value to every nonzero vector .
|
|
This can be expressed as : u(A) = \int_A f \, d\mu, where is the new measure being defined for any measurable subset and the function is the density at a given point. The integral is with respect to an existing measure , which may often be the canonical Lebesgue measure on the Real line or the n-dimensional Euclidean space (corresponding to our standard notions of length, area and volume). For example, if represented mass density and was the Lebesgue measure in three-dimensional space , then would equal the total mass in a spatial region . The Radon–Nikodym theorem essentially states that, under certain conditions, any measure can be expressed in this way with respect to another measure on the same space.
|
|
In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a paravector (3-dimensional vector plus a 1-dimensional scalar). The Clifford algebra Cl3,0(R) has a faithful representation, generated by Pauli matrices, on the spin representation C2; further, Cl3,0(R) is isomorphic to the even subalgebra Cl(R) of the Clifford algebra Cl3,1(R). APS can be used to construct a compact, unified and geometrical formalism for both classical and quantum mechanics. APS should not be confused with spacetime algebra (STA), which concerns the Clifford algebra Cl1,3(R) of the four-dimensional Minkowski spacetime.
|
|
Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral. Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure.
|
|
The Dedekind eta-function is an automorphic form in the complex plane. In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups. Modular forms are automorphic forms defined over the groups SL(2, R) or PSL(2, R) with the discrete subgroup being the modular group, or one of its congruence subgroups; in this sense the theory of automorphic forms is an extension of the theory of modular forms.
|
|
As Hans Freudenthal observed: :To be able to consider von Staudt's approach as a rigorous foundation of projective geometry, one need only add explicitly the topological axioms which are tacitly used by von Staudt. ... how can one formulate the topology of projective space without the support of a metric? Von Staudt was still far from raising this question, which a quarter of a century later would become urgent. ... Felix Klein noticed the gap in von Staudt's approach; he was aware of the need to formulate the topology of projective space independently of Euclidean space.... the Italians were the first to find truly satisfactory solutions for the problem of a purely projective foundation of projective geometry, which von Staudt had tried to solve.
|
|
Certain types of world lines are called geodesics of the spacetime – straight lines in the case of flat Minkowski spacetime and their closest equivalent in the curved spacetime of general relativity. In the case of purely time-like paths, geodesics are (locally) the paths of greatest separation (spacetime interval) as measured along the path between two events, whereas in Euclidean space and Riemannian manifolds, geodesics are paths of shortest distance between two points.This characterization is not universal: both the arcs between two points of a great circle on a sphere are geodesics. The concept of geodesics becomes central in general relativity, since geodesic motion may be thought of as "pure motion" (inertial motion) in spacetime, that is, free from any external influences.
|
|
The Sormani–Wenger intrinsic flat (SWIF) distance is a distance between compact oriented Riemannian manifolds of the same dimension. More generally it defines the distance between two integral current spaces, (X,d,T), of the same dimension (see below). This class of spaces and this distance were first announced by mathematicians Sormani and Wenger at the Geometry Festival in 2009 and the detailed development of these notions appeared in the Journal of Differential Geometry in 2011."Intrinsic Flat Distance between Riemannian Manifolds and other Integral Current Spaces" by Sormani and Wenger, Journal of Differential Geometry, Vol 87, 2011, 117–199 The SWIF distance is an intrinsic notion based upon the (extrinsic) flat distance between submanifolds and integral currents in Euclidean space developed by Federer and Fleming.
|
|
A spherically symmetric spacetime is a spacetime whose isometry group contains a subgroup which is isomorphic to the rotation group SO(3) and the orbits of this group are 2-spheres (ordinary 2-dimensional spheres in 3-dimensional Euclidean space). The isometries are then interpreted as rotations and a spherically symmetric spacetime is often described as one whose metric is "invariant under rotations". The spacetime metric induces a metric on each orbit 2-sphere (and this induced metric must be a multiple of the metric of a 2-sphere). Conventionally, the metric on the 2-sphere is written in polar coordinates as :g_\Omega = d\theta^2 + \sin^2\theta \, d\varphi^2, and so the full metric includes a term proportional to this.
|
|
Initially Jim Stasheff considered these objects as curvilinear polytopes. Subsequently, they were given coordinates as convex polytopes in several different ways; see the introduction of for a survey.. One method of realizing the associahedron is as the secondary polytope of a regular polygon. In this construction, each triangulation of a regular polygon with n + 1 sides corresponds to a point in (n + 1)-dimensional Euclidean space, whose ith coordinate is the total area of the triangles incident to the ith vertex of the polygon. For instance, the two triangulations of the unit square give rise in this way to two four-dimensional points with coordinates (1, 1/2, 1, 1/2) and (1/2, 1, 1/2, 1). The convex hull of these two points is the realization of the associahedron K3.
|
|
Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three- dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry; 3.11 Any two congruent triangles are related by a unique isometry. the isometry that relates them is either a rigid motion (translation or rotation), or a composition of a rigid motion and a reflection. Isometries are often used in constructions where one space is embedded in another space.
|
|
The strong version of the paradox claims: : Any two bounded subsets of 3-dimensional Euclidean space with non- empty interiors are equidecomposable. While apparently more general, this statement is derived in a simple way from the doubling of a ball by using a generalization of the Bernstein–Schroeder theorem due to Banach that implies that if is equidecomposable with a subset of and is equidecomposable with a subset of , then and are equidecomposable. The Banach–Tarski paradox can be put in context by pointing out that for two sets in the strong form of the paradox, there is always a bijective function that can map the points in one shape into the other in a one-to-one fashion. In the language of Georg Cantor's set theory, these two sets have equal cardinality.
|
|
On the other hand, in the Banach–Tarski paradox, the number of pieces is finite and the allowed equivalences are Euclidean congruences, which preserve the volumes. Yet, somehow, they end up doubling the volume of the ball! While this is certainly surprising, some of the pieces used in the paradoxical decomposition are non-measurable sets, so the notion of volume (more precisely, Lebesgue measure) is not defined for them, and the partitioning cannot be accomplished in a practical way. In fact, the Banach–Tarski paradox demonstrates that it is impossible to find a finitely- additive measure (or a Banach measure) defined on all subsets of an Euclidean space of three (and greater) dimensions that is invariant with respect to Euclidean motions and takes the value one on a unit cube.
|
|
For S a subset of a Euclidean space, x is a point of closure of S if every open ball centered at x contains a point of S (this point may be x itself). This definition generalizes to any subset S of a metric space X. Fully expressed, for X a metric space with metric d, x is a point of closure of S if for every r > 0, there is a y in S such that the distance d(x, y) < r. (Again, we may have x = y.) Another way to express this is to say that x is a point of closure of S if the distance d(x, S) := inf{d(x, s) : s in S} = 0. This definition generalizes to topological spaces by replacing "open ball" or "ball" with "neighbourhood".
|
|
The definition of a unit distance graph may naturally be generalized to any higher-dimensional Euclidean space. Any graph may be embedded as a set of points in a sufficiently high dimension; show that the dimension necessary to embed a graph in this way may be bounded by twice its maximum degree. The dimension needed to embed a graph so that all edges have unit distance, and the dimension needed to embed a graph so that the edges are exactly the unit distance pairs, may greatly differ from each other: the 2n-vertex crown graph may be embedded in four dimensions so that all its edges have unit length, but requires at least n − 2 dimensions to be embedded so that the edges are the only unit-distance pairs.
|
|
A set of six points (red), its six 2-sets (the sets of points contained in the blue ovals), and lines separating each k-set from the remaining points (dashed black). In discrete geometry, a k-set of a finite point set S in the Euclidean plane is a subset of k elements of S that can be strictly separated from the remaining points by a line. More generally, in Euclidean space of higher dimensions, a k-set of a finite point set is a subset of k elements that can be separated from the remaining points by a hyperplane. In particular, when k = n/2 (where n is the size of S), the line or hyperplane that separates a k-set from the rest of S is a halving line or halving plane.
|
|
Piccard was an invited speaker at the International Congress of Mathematicians in 1932 and again in 1936.. In 1939 she published the book Sur les ensembles de distances des ensembles de points d'un espace Euclidean (Mémoires de L’Université de Neuchâtel 13, Paris, France: Libraire Gauthier-Villars and Cie., 1939).. Its subject was the sets of distances that a collection of points in a Euclidean space might determine. This book included early research on Golomb rulers, finite sets of integer points in a one-dimensional space with the property that their distances are all distinct. She published a theorem claiming that every two Golomb rulers with the same distance set must be congruent to each other; this turned out to be false for certain sets of six points, but true otherwise....
|
|
"Structures métriques pour les variétés riemanniennes", edited by Lafontaine and Pierre Pansu, 1981.M. Gromov, Groups of Polynomial growth and Expanding Maps, Publications mathematiques I.H.É.S., 53, 1981 This distance measures how far two compact metric spaces are from being isometric. If X and Y are two compact metric spaces, then dGH (X, Y) is defined to be the infimum of all numbers dH(f(X), g(Y)) for all metric spaces M and all isometric embeddings f : X → M and g : Y → M. Here dH denotes Hausdorff distance between subsets in M and the isometric embedding is understood in the global sense, i.e. it must preserve all distances, not only infinitesimally small ones; for example no compact Riemannian manifold admits such an embedding into Euclidean space of the same dimension.
|
|
The dual graph depends on how the primal graph is embedded: different planar embeddings of a single graph may lead to different dual graphs. Matroid duality is an algebraic extension of planar graph duality, in the sense that the dual matroid of the graphic matroid of a planar graph is isomorphic to the graphic matroid of the dual graph. A kind of geometric duality also occurs in optimization theory, but not one that reverses dimensions. A linear program may be specified by a system of real variables (the coordinates for a point in Euclidean space a system of linear constraints (specifying that the point lie in a halfspace; the intersection of these halfspaces is a convex polytope, the feasible region of the program), and a linear function (what to optimize).
|
|
In mathematics and theoretical physics, quasiperiodic motion is in rough terms the type of motion executed by a dynamical system containing a finite number (two or more) of incommensurable frequencies. That is, if we imagine that the phase space is modelled by a torus T (that is, the variables are periodic like angles), the trajectory of the system is modelled by a curve on T that wraps around the torus without ever exactly coming back on itself. A quasiperiodic function on the real line is the type of function (continuous, say) obtained from a function on T, by means of a curve :R -> T which is linear (when lifted from T to its covering Euclidean space), by composition. It is therefore oscillating, with a finite number of underlying frequencies.
|
|
In mathematics, Blanuša became known for discovering the second and third known snarks in 1946 (the Petersen graph was the first), triggering a new area of graph theory. The study of snarks had its origin in the 1880 work of P. G. Tait, who at that time had proved that the four color theorem is equivalent to the statement that no snark is planar. Snarks were so named later by the American mathematician Martin Gardner in 1976, after the mysterious and elusive object of Lewis Carroll's poem The Hunting of the Snark. Blanuša's most important works were related to isometric immersions of two-dimensional Lobachevsky plane into six-dimensional Euclidean space and generalizations, in the theory of the special functions (Bessel functions), in differential geometry, and in graph theory.
|
|
In other words, the box definition is extrinsic -- one assumes the fractal space S is contained in a Euclidean space, and defines boxes according to the external geometry of the containing space. However, the dimension of S should be intrinsic, independent of the environment into which S is placed, and the ball definition can be formulated intrinsically. One defines an internal ball as all points of S within a certain distance of a chosen center, and one counts such balls to get the dimension. (More precisely, the Ncovering definition is extrinsic, but the other two are intrinsic.) The advantage of using boxes is that in many cases N(ε) may be easily calculated explicitly, and that for boxes the covering and packing numbers (defined in an equivalent way) are equal.
|
|
The uniform matroid U{}^r_n may be represented as the matroid of affinely independent subsets of n points in general position in r-dimensional Euclidean space, or as the matroid of linearly independent subsets of n vectors in general position in an (r+1)-dimensional real vector space. Every uniform matroid may also be realized in projective spaces and vector spaces over all sufficiently large finite fields. However, the field must be large enough to include enough independent vectors. For instance, the n-point line U{}^2_n can be realized only over finite fields of n-1 or more elements (because otherwise the projective line over that field would have fewer than n points): U{}^2_4 is not a binary matroid, U{}^2_5 is not a ternary matroid, etc.
|
|
In the mathematical field of differential geometry, a smooth map from one Riemannian manifold to another Riemannian manifold is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for a mapping also arises as the Euler-Lagrange equation of a functional generalizing the Dirichlet energy (which is often itself called "Dirichlet energy"). As such, the theory of harmonic maps encompasses both the theory of unit-speed geodesics in Riemannian geometry, and the theory of harmonic functions on open subsets of Euclidean space and on Riemannian manifolds. Informally, the Dirichlet energy of a mapping from a Riemannian manifold to a Riemannian manifold can be thought of as the total amount that "stretches" in allocating each of its elements to a point of .
|
|
For instance, a rubber band which is stretched around a (smooth) stone can be mathematically formalized as a mapping from the points on the unstretched band to the surface of the stone. The unstretched band and stone are given Riemannian metrics as embedded submanifolds of three- dimensional Euclidean space; the Dirichlet energy of such a mapping is then a formalization of the notion of the total tension involved. Harmonicity of such a mapping means that, given any hypothetical way of physically deforming the given stretch, the tension (when considered as a function of time) has first derivative zero when the deformation begins. The theory of harmonic maps was initiated in 1964 by James Eells and Joseph Sampson, who showed that in certain geometric contexts, arbitrary smooth maps could be deformed into harmonic maps.
|
|
In mathematics, the Chabauty topology is a certain topological structure introduced in 1950 by Claude Chabauty, on the set of all closed subgroups of a locally compact group G. The intuitive idea may be seen in the case of the set of all lattices in a Euclidean space E. There these are only certain of the closed subgroups: others can be found by in a sense taking limiting cases or degenerating a certain sequence of lattices. One can find linear subspaces or discrete groups that are lattices in a subspace, depending on how one takes a limit. This phenomenon suggests that the set of all closed subgroups carries a useful topology. This topology can be derived from the Vietoris topology construction, a topological structure on all non-empty subsets of a space.
|
|
The combinatorial characterization of a set X ⊂ ℝ3 as a solid involves representing X as an orientable cell complex so that the cells provide finite spatial addresses for points in an otherwise innumerable continuum. The class of semi-analytic bounded subsets of Euclidean space is closed under Boolean operations (standard and regularized) and exhibits the additional property that every semi-analytic set can be stratified into a collection of disjoint cells of dimensions 0,1,2,3. A triangulation of a semi-analytic set into a collection of points, line segments, triangular faces, and tetrahedral elements is an example of a stratification that is commonly used. The combinatorial model of solidity is then summarized by saying that in addition to being semi-analytic bounded subsets, solids are three-dimensional topological polyhedra, specifically three-dimensional orientable manifolds with boundary.
|
|
An element C of Hcolor is a function from the range of visible wavelengths—considered as an interval of real numbers [Wmin,Wmax]—to the real numbers, assigning to each wavelength w in [Wmin,Wmax] its intensity C(w). A humanly perceived color may be modeled as three numbers: the extents to which each of the 3 types of cones is stimulated. Thus a humanly perceived color may be thought of as a point in 3-dimensional Euclidean space. We call this space R3color. Since each wavelength w stimulates each of the 3 types of cone cells to a known extent, these extents may be represented by 3 functions s(w), m(w), l(w) corresponding to the response of the S, M, and L cone cells, respectively.
|
|
Mathematically, scalar fields on a region U is a real or complex-valued function or distribution on U. The region U may be a set in some Euclidean space, Minkowski space, or more generally a subset of a manifold, and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order. A scalar field is a tensor field of order zero, and the term "scalar field" may be used to distinguish a function of this kind with a more general tensor field, density, or differential form. The scalar field of \sin (2\pi(xy+\sigma)) oscillating as \sigma increases. Red represents positive values, purple represents negative values, and sky blue represents values close to zero.
|
|
"The Dreams in the Witch House" was likely inspired by Willem de Sitter's lecture The Size of the Universe, which Lovecraft attended three months prior to writing the story. De Sitter is mentioned by name in the story, described as a mathematical genius, and listed in a group of other intellectual masterminds, including Albert Einstein. Several prominent motifs—including the geometry and curvature of space and using pure mathematics to gain a deeper understanding the nature of the universe—are covered in both Lovecraft's story and de Sitter's lecture. The idea of using higher dimensions of non-Euclidean space as short cuts through normal space can be traced to A. S. Eddington's The Nature of the Physical World which Lovecraft alludes to having read (SL III p 87).
|
|
In the mathematical field of topology, the inductive dimension of a topological space X is either of two values, the small inductive dimension ind(X) or the large inductive dimension Ind(X). These are based on the observation that, in n-dimensional Euclidean space Rn, (n − 1)-dimensional spheres (that is, the boundaries of n-dimensional balls) have dimension n − 1\. Therefore it should be possible to define the dimension of a space inductively in terms of the dimensions of the boundaries of suitable open sets. The small and large inductive dimensions are two of the three most usual ways of capturing the notion of "dimension" for a topological space, in a way that depends only on the topology (and not, say, on the properties of a metric space).
|
|
The algebraic topology of the Lie groups is also largely carried by a maximal compact subgroup K. To be precise, a connected Lie group is a topological product (though not a group theoretic product) of a maximal compact K and a Euclidean space – G = K × Rd – thus in particular K is a deformation retract of G, and is homotopy equivalent, and thus they have the same homotopy groups. Indeed, the inclusion K \hookrightarrow G and the deformation retraction G \twoheadrightarrow K are homotopy equivalences. For the general linear group, this decomposition is the QR decomposition, and the deformation retraction is the Gram-Schmidt process. For a general semisimple Lie group, the decomposition is the Iwasawa decomposition of G as G = KAN in which K occurs in a product with a contractible subgroup AN.
|
|
Approximation by multiple linear segments A curve in the plane can be approximated by connecting a finite number of points on the curve using line segments to create a polygonal path. Since it is straightforward to calculate the length of each linear segment (using the Pythagorean theorem in Euclidean space, for example), the total length of the approximation can be found by summing the lengths of each linear segment; that approximation is known as the (cumulative) chordal distance. If the curve is not already a polygonal path, using a progressively larger number of segments of smaller lengths will result in better approximations. The lengths of the successive approximations will not decrease and may keep increasing indefinitely, but for smooth curves they will tend to a finite limit as the lengths of the segments get arbitrarily small.
|
|
The term reflection is loose, and considered by some an abuse of language, with inversion preferred; however, point reflection is widely used. Such maps are involutions, meaning that they have order 2 – they are their own inverse: applying them twice yields the identity map – which is also true of other maps called reflections. More narrowly, a reflection refers to a reflection in a hyperplane (n-1 dimensional affine subspace – a point on the line, a line in the plane, a plane in 3-space), with the hyperplane being fixed, but more broadly reflection is applied to any involution of Euclidean space, and the fixed set (an affine space of dimension k, where 1 \leq k \leq n-1) is called the mirror. In dimension 1 these coincide, as a point is a hyperplane in the line.
|
|
In contrast, the different notions of compactness are not equivalent in general topological spaces, and the most useful notion of compactness--originally called bicompactness--is defined using covers consisting of open sets (see Open cover definition below). That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the Heine–Borel theorem. Compactness, when defined in this manner, often allows one to take information that is known locally--in a neighbourhood of each point of the space--and to extend it to information that holds globally throughout the space. An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is uniformly continuous; here, continuity is a local property of the function, and uniform continuity the corresponding global property.
|
|
This revolutionary theoretical framework is based on a probabilistic interpretation of states, and evolution and measurements in terms of self-adjoint operators on an infinite dimensional vector space. That is called Hilbert space(introduced by mathematicians David Hilbert (1862–1943), Erhard Schmidt(1876-1959) and Frigyes Riesz (1880-1956) in search of generalization of Euclidean space and study of intrgral equations), and rigorously defined within the axiomatic modern version by John von Neumann in his celebrated book Mathematical Foundations of Quantum Mechanics, where he built up a relevant part of modern functional analysis on Hilbert spaces, the spectral theory(introduced by David Hilbert who investigated quadratic forms with infinitely many variables. Many years later, it had been revealed that his spectral theory is associated with the spectrum of hydrogen atom. He was surprised by this application.) in particular.
|
|
Any six points in general position in four-dimensional space determine 15 points where a line through two of the points intersects the hyperplane through the other four points; thus, the duads of the six points correspond one-for-one with these 15 derived points. Any three duads that together form a syntheme determine a line, the intersection line of the three hyperplanes containing two of the three duads in the syntheme, and this line contains each of the points derived from its three duads. Thus, the duads and synthemes of the abstract configuration correspond one-for-one, in an incidence-preserving way, with these 15 points and 15 lines derived from the original six points, which form a realization of the configuration. The same realization may be projected into Euclidean space or the Euclidean plane.
|
|
Atiyah and Ward used the Penrose correspondence to reduce the classification of all instantons on the 4-sphere to a problem in algebraic geometry. With Hitchin he used ideas of Horrocks to solve this problem, giving the ADHM construction of all instantons on a sphere; Manin and Drinfeld found the same construction at the same time, leading to a joint paper by all four authors. Atiyah reformulated this construction using quaternions and wrote up a leisurely account of this classification of instantons on Euclidean space as a book. Atiyah's work on instanton moduli spaces was used in Donaldson's work on Donaldson theory. Donaldson showed that the moduli space of (degree 1) instantons over a compact simply connected 4-manifold with positive definite intersection form can be compactified to give a cobordism between the manifold and a sum of copies of complex projective space.
|
|
A triangle can be covered by three smaller copies of itself; a square requires four smaller copies In combinatorial geometry, the Hadwiger conjecture states that any convex body in n-dimensional Euclidean space can be covered by 2n or fewer smaller bodies homothetic with the original body, and that furthermore, the upper bound of 2n is necessary if and only if the body is a parallelepiped. There also exists an equivalent formulation in terms of the number of floodlights needed to illuminate the body. The Hadwiger conjecture is named after Hugo Hadwiger, who included it on a list of unsolved problems in 1957; it was, however, previously studied by and independently, . Additionally, there is a different Hadwiger conjecture concerning graph coloring—and in some sources the geometric Hadwiger conjecture is also called the Levi–Hadwiger conjecture or the Hadwiger–Levi covering problem.
|
|
It thus makes sense to define the hyperbolic angle from P0 to an arbitrary point on the curve as a logarithmic function of the point's value of x.Bjørn Felsager, Through the Looking Glass – A glimpse of Euclid's twin geometry, the Minkowski geometry , ICME-10 Copenhagen 2004; p.14. See also example sheets exploring Minkowskian parallels of some standard Euclidean resultsViktor Prasolov and Yuri Solovyev (1997) Elliptic Functions and Elliptic Integrals, page 1, Translations of Mathematical Monographs volume 170, American Mathematical Society Whereas in Euclidean geometry moving steadily in an orthogonal direction to a ray from the origin traces out a circle, in a pseudo-Euclidean plane steadily moving orthogonally to a ray from the origin traces out a hyperbola. In Euclidean space, the multiple of a given angle traces equal distances around a circle while it traces exponential distances upon the hyperbolic line.
|
|
An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of the Euclidean plane, the sphere and the hyperbolic plane. These Lie groups can be used to describe surfaces of constant Gaussian curvature; they also provide an essential ingredient in the modern approach to intrinsic differential geometry through connections. On the other hand, extrinsic properties relying on an embedding of a surface in Euclidean space have also been extensively studied. This is well illustrated by the non-linear Euler–Lagrange equations in the calculus of variations: although Euler developed the one variable equations to understand geodesics, defined independently of an embedding, one of Lagrange's main applications of the two variable equations was to minimal surfaces, a concept that can only be defined in terms of an embedding.
|
|
For any surface embedded in Euclidean space of dimension 3 or higher, it is possible to measure the length of a curve on the surface, the angle between two curves and the area of a region on the surface. This structure is encoded infinitesimally in a Riemannian metric on the surface through line elements and area elements. Classically in the nineteenth and early twentieth centuries only surfaces embedded in were considered and the metric was given as a 2×2 positive definite matrix varying smoothly from point to point in a local parametrization of the surface. The idea of local parametrization and change of coordinate was later formalized through the current abstract notion of a manifold, a topological space where the smooth structure is given by local charts on the manifold, exactly as the planet Earth is mapped by atlases today.
|
|
In 1985, Cohen proved the Immersion Conjecture, which says that each smooth, compact n-manifold has an immersion in Euclidean space of dimension 2 n - \alpha(n), where \alpha(n) is the number of ones in the binary expansion of n. In 1991, Cohen, together with Frederick Cohen, Benjamin Mann, and R. James Milgram gave a complete description of the algebraic topology of the space of rational functions, and in the following years he made several contributions to the study of related moduli spaces. In 1995 Cohen, John D. S. Jones, and Graeme Segal introduced an approach for understanding the homotopy theory underlying Floer homology theory in Symplectic geometry. Since 2002 Cohen has been one of the leading developers and contributors to the theory of String topology, which was introduced originally by Moira Chas and Dennis Sullivan.
|
|
In geometry, exterior dimension is a type of dimension that can be used to characterize the scaling behavior of "fat fractals". A fat fractal is defined to be a subset of Euclidean space such that, for every point p of the set and every sufficiently small number \epsilon, the ball of radius \epsilon centered at p contains both a nonzero Lebesgue measure of points belonging to the fractal, and a nonzero Lebesgue measure of points that do not belong to the fractal. For such a set, the Hausdorff dimension is the same as that of the ambient space. The Hausdorff dimension of a set S can be computed by "fattening" S (taking its Minkowski sum with a ball of radius \epsilon), and examining how the volume of the resulting fattened set scales with \epsilon, in the limit as \epsilon tends to zero.
|
|
The traditional kernel functions K(x, y) of two variables of the theory of integral operators having been expanded in scope to include their generalized function analogues, which are allowed to be more singular in a serious way, a large class of operators from D to its dual space D′ of distributions can be constructed. The point of the theorem is to assert that the extended class of operators can be characterised abstractly, as containing all operators subject to a minimum continuity condition. A bilinear form on D arises by pairing the image distribution with a test function. A simple example is that the natural embedding [.] of the test function space D into D’ - sending every test function f into the corresponding distribution [f] - corresponds to the delta distribution δ(x − y) concentrated at the diagonal of the underlined Euclidean space, in terms of the Dirac delta function δ.
|
|
The first two steps of the Gram–Schmidt process In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn equipped with the standard inner product. The Gram–Schmidt process takes a finite, linearly independent set S = {v1, ..., vk} for and generates an orthogonal set that spans the same k-dimensional subspace of Rn as S. The method is named after Jørgen Pedersen Gram and Erhard Schmidt, but Pierre-Simon Laplace had been familiar with it before Gram and Schmidt. In the theory of Lie group decompositions it is generalized by the Iwasawa decomposition. The application of the Gram–Schmidt process to the column vectors of a full column rank matrix yields the QR decomposition (it is decomposed into an orthogonal and a triangular matrix).
|
|
In measure theory, the "problem of measure" for an -dimensional Euclidean space may be stated as: "does there exist a positive, normalized, invariant, and additive set function on the class of all subsets of ?" The work of Felix Hausdorff and Stefan Banach had implied that the problem of measure has a positive solution if or and a negative solution (because of the Banach–Tarski paradox) in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character": the existence of a measure could be determined by looking at the properties of the transformation group of the given space. The positive solution for spaces of dimension at most two, and the negative solution for higher dimensions, comes from the fact that the Euclidean group is a solvable group for dimension at most two, and is not solvable for higher dimensions.
|
|
The Euclidean distance is not appropriate when the absolute levels of functionally related genes are highly different. Furthermore, if two genes have consistently low expression levels but are otherwise randomly correlated, they might still appear close in Euclidean space. One advantage to mutual information is that it can detect non-linear relationships; however this can turn into a disadvantage because of detecting sophisticated non-linear relationships which does not look biologically meaningful. In addition, for calculating mutual information one should estimate the distribution of the data which needs a large number of samples for a good estimate. Spearman’s rank correlation coefficient is more robust to outliers, but on the other hand it is less sensitive to expression values and in datasets with small number of samples may detect many false positives. Pearson’s correlation coefficient is the most popular co-expression measure used in constructing gene co-expression networks.
|
|
In mathematics, the dyadic cubes are a collection of cubes in Rn of different sizes or scales such that the set of cubes of each scale partition Rn and each cube in one scale may be written as a union of cubes of a smaller scale. These are frequently used in mathematics (particularly harmonic analysis) as a way of discretizing objects in order to make computations or analysis easier. For example, to study an arbitrary subset of A of Euclidean space, one may instead replace it by a union of dyadic cubes of a particular size that cover the set. One can consider this set as a pixelized version of the original set, and as smaller cubes are used one gets a clearer image of the set A. Most notable appearances of dyadic cubes include the Whitney extension theorem and the Calderón–Zygmund lemma.
|
|
An exotic R4 is a differentiable manifold that is homeomorphic but not diffeomorphic to the Euclidean space R4. The first examples were found in the early 1980s by Michael Freedman, by using the contrast between Freedman's theorems about topological 4-manifolds, and Simon Donaldson's theorems about smooth 4-manifolds.. There is a continuum of non-diffeomorphic differentiable structures of R4, as was shown first by Clifford Taubes.Theorem 1.1 of Prior to this construction, non-diffeomorphic smooth structures on spheres—exotic spheres—were already known to exist, although the question of the existence of such structures for the particular case of the 4-sphere remained open (and still remains open as of 2018). For any positive integer n other than 4, there are no exotic smooth structures on Rn; in other words, if n ≠ 4 then any smooth manifold homeomorphic to Rn is diffeomorphic to Rn.Corollary 5.2 of .
|
|
Random walks are stochastic processes that are usually defined as sums of iid random variables or random vectors in Euclidean space, so they are processes that change in discrete time. But some also use the term to refer to processes that change in continuous time, particularly the Wiener process used in finance, which has led to some confusion, resulting in its criticism. There are other various types of random walks, defined so their state spaces can be other mathematical objects, such as lattices and groups, and in general they are highly studied and have many applications in different disciplines. A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one.
|
|
Triple correlation methods are frequently used in signal processing for treating signals that are corrupted by additive white Gaussian noise; in particular, triple correlation techniques are suitable when multiple observations of the signal are available and the signal may be translating in between the observations, e.g.,a sequence of images of an object translating on a noisy background. What makes the triple correlation particularly useful for such tasks are three properties: (1) it is invariant under translation of the underlying signal; (2) it is unbiased in additive Gaussian noise; and (3) it retains nearly all of the relevant phase information in the underlying signal. Properties (1)-(3) of the triple correlation extend in many cases to functions on an arbitrary locally compact group, in particular to the groups of rotations and rigid motions of euclidean space that arise in computer vision and signal processing.
|
|
While developing her thesis, in 1962 Henney researched and published such projects as "Set-Valued Quadratic Functionals" and "One-Parameter Semigroups". Henney also published eight research papers in journals in Europe, Asia and the United States. She is the author of Properties of Set Valued Additive Functions, which serves to "examine certain properties of set-valued additive functions which are defined on the positive cone in Euclidean space";George Washington University Math Department, Faculty Notes on Dagmar Henney, Archives of Gelman Library, Room 704, Washington, DC Elements of Mathematics, which discusses topology, topological vector spaces, integration, set theory, functions of a real variable, and modern algebra; Bourbaki; and best-selling title, Unsolved Questions in Mathematics. In addition to publishing her own research, Henney has experience in editing, including her work in Open Questions in Mathematics, which explores the work of significant scientists and Nobel Prize winners from around the globe.
|
|
If one has a representation of an antimatroid as the closure of a set of d basic words, then this representation can be used to map the feasible sets of the antimatroid into d-dimensional Euclidean space: assign one coordinate per basic word w, and make the coordinate value of a feasible set S be the length of the longest prefix of w that is a subset of S. With this embedding, S is a subset of T if and only if the coordinates for S are all less than or equal to the corresponding coordinates of T. Therefore, the order dimension of the inclusion ordering of the feasible sets is at most equal to the convex dimension of the antimatroid.Korte et al., Corollary 6.10. However, in general these two dimensions may be very different: there exist antimatroids with order dimension three but with arbitrarily large convex dimension.
|
|
The Sylvester–Gallai theorem also does not directly apply to geometries in which points have coordinates that are pairs of complex numbers or quaternions, but these geometries have more complicated analogues of the theorem. For instance, in the complex projective plane there exists a configuration of nine points, Hesse's configuration (the inflection points of a cubic curve), in which every line is non-ordinary, violating the Sylvester–Gallai theorem. Such a configuration is known as a Sylvester–Gallai configuration, and it cannot be realized by points and lines of the Euclidean plane. Another way of stating the Sylvester–Gallai theorem is that whenever the points of a Sylvester–Gallai configuration are embedded into a Euclidean space, preserving colinearities, the points must all lie on a single line, and the example of the Hesse configuration shows that this is false for the complex projective plane.
|
|
A real quadric surface in the Euclidean space of dimension three is a surface that may be defined as the zeros of a polynomial of degree two in three variables. As for the conic sections there are two discriminants that may be naturally defined. Both are useful for getting information on the nature of a quadric surface. Let P(x,y,z) be a polynomial of degree two in three variables that defines a real quadric surface. The first associated quadratic form, Q_4, depends on four variables, and is obtained by homogenizing ; that is :Q_4(x,y,z,t)=t^2P(x/t,y/t, z/t). Let us denote its discriminant by\Delta_4. The second quadratic form, Q_3, depends on three variables, and consists of the terms of degree two of ; that is :Q_3(x,y,z)=Q_4(x, y,z,0). Let us denote its discriminant by\Delta_3.
|
|
Intuitively, a curve in two or three (or higher) dimensions can be thought of as the path of a continuously moving point. To eliminate the inherent vagueness of this notion, Jordan in 1887 introduced the following rigorous definition, which has since been adopted as the precise description of the notion of a curve: In the most general form, the range of such a function may lie in an arbitrary topological space, but in the most commonly studied cases, the range will lie in a Euclidean space such as the 2-dimensional plane (a planar curve) or the 3-dimensional space (space curve). Sometimes, the curve is identified with the image of the function (the set of all possible values of the function), instead of the function itself. It is also possible to define curves without endpoints to be a continuous function on the real line (or on the open unit interval ).
|
|
A square matrix represents a linear operator on the Euclidean space Rn where n is the dimension of the matrix. Since such a space is finite-dimensional, this operator is actually bounded. Using the tools of holomorphic functional calculus, given a holomorphic function f(z) defined on an open set in the complex plane and a bounded linear operator T, one can calculate f(T) as long as f(z) is defined on the spectrum of T. The function f(z)=log z can be defined on any simply connected open set in the complex plane not containing the origin, and it is holomorphic on such a domain. This implies that one can define ln T as long as the spectrum of T does not contain the origin and there is a path going from the origin to infinity not crossing the spectrum of T (e.g.
|
|
Each convex set containing X must (by the assumption that it is convex) contain all convex combinations of points in X, so the set of all convex combinations is contained in the intersection of all convex sets containing X. Conversely, the set of all convex combinations is itself a convex set containing X, so it also contains the intersection of all convex sets containing X, and therefore the second and third definitions are equivalent., p. 12; , p. 17. In fact, according to Carathéodory's theorem, if X is a subset of a d-dimensional Euclidean space, every convex combination of finitely many points from X is also a convex combination of at most d+1 points in X. The set of convex combinations of a (d+1)-tuple of points is a simplex; in the plane it is a triangle and in three-dimensional space it is a tetrahedron.
|
|
When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow). In coordinates, a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law in passing from one coordinate system to the other.
|
|
If three spheres are given, with their centers non-collinear, then their six centers of similitude form the six points of a complete quadrilateral, the four lines of which are called the axes of similitude. And if four spheres are given, with their centers non- coplanar, then they determine 12 centers of similitude and 16 axes of similitude, which together form an instance of the Reye configuration . The Reye configuration can also be realized by points and lines in the Euclidean plane, by drawing the three-dimensional configuration in three-point perspective. An 83122 configuration of eight points in the real projective plane and 12 lines connecting them, with the connection pattern of a cube, can be extended to form the Reye configuration if and only if the eight points are a perspective projection of a parallelepiped The 24 permutations of the points (\pm 1, \pm 1, 0, 0) form the vertices of a 24-cell centered at the origin of four-dimensional Euclidean space.
|
|
His result says that if the hypersurface is sufficiently convex relative to the geometry of the Riemannian manifold, then the mean curvature flow will contract it to a point, and that a normalization of surface area in geodesic normal coordinates will give a smooth deformation to a sphere in Euclidean space (as represented by the coordinates). This shows that such hypersurfaces are diffeomorphic to the sphere, and that they are the boundary of a region in the Riemannian manifold which is diffeomorphic to a ball. In this generality, there is not a simple proof using the Gauss map. Following work of Yoshikazu Giga and Robert Kohn which made extensive use of the Dirichlet energy as weighted by exponentials, Huisken proved in 1990 an integral identity, known as Huisken's monotonicity formula, which shows that, under the mean curvature flow, the integral of the "backwards" Euclidean heat kernel over the evolving hypersurface is always nonincreasing.
|
|
In the mathematical field of differential geometry, Euler's theorem is a result on the curvature of curves on a surface. The theorem establishes the existence of principal curvatures and associated principal directions which give the directions in which the surface curves the most and the least. The theorem is named for Leonhard Euler who proved the theorem in . More precisely, let M be a surface in three-dimensional Euclidean space, and p a point on M. A normal plane through p is a plane passing through the point p containing the normal vector to M. Through each (unit) tangent vector to M at p, there passes a normal plane PX which cuts out a curve in M. That curve has a certain curvature κX when regarded as a curve inside PX. Provided not all κX are equal, there is some unit vector X1 for which k1 = κX1 is as large as possible, and another unit vector X2 for which k2 = κX2 is as small as possible.
|
|
The first versions of an abstract Plancherel formula for the Fourier transform on a unimodular locally compact group G were due to Segal and Mautner., historical notes on the Plancherel theorem for spherical functions At around the same time, Harish-Chandra and Gelfand & Naimark derived an explicit formula for SL(2,R) and complex semisimple Lie groups, so in particular the Lorentz groups. A simpler abstract formula was derived by Mautner for a "topological" symmetric space G/K corresponding to a maximal compact subgroup K. Godement gave a more concrete and satisfactory form for positive definite spherical functions, a class of special functions on G/K. Since when G is a semisimple Lie group these spherical functions φλ were naturally labelled by a parameter λ in the quotient of a Euclidean space by the action of a finite reflection group, it became a central problem to determine explicitly the Plancherel measure in terms of this parametrization.
|
|
In 1868 Beltrami published two memoirs (written in Italian; French translations by J. Hoüel appeared in 1869) dealing with consistency and interpretations of non-Euclidean geometry of János Bolyai and Nikolai Lobachevsky. In his "Essay on an interpretation of non-Euclidean geometry", Beltrami proposed that this geometry could be realized on a surface of constant negative curvature, a pseudosphere. For Beltrami's concept, lines of the geometry are represented by geodesics on the pseudosphere and theorems of non-Euclidean geometry can be proved within ordinary three-dimensional Euclidean space, and not derived in an axiomatic fashion, as Lobachevsky and Bolyai had done previously. In 1840, Ferdinand Minding already considered geodesic triangles on the pseudosphere and remarked that the corresponding "trigonometric formulas" are obtained from the corresponding formulas of spherical trigonometry by replacing the usual trigonometric functions with hyperbolic functions; this was further developed by Delfino Codazzi in 1857, but apparently neither of them noticed the association with Lobachevsky's work.
|
|
In mathematics, especially potential theory, harmonic measure is a concept related to the theory of harmonic functions that arises from the solution of the classical Dirichlet problem. Harmonic measure is the exit distribution of Brownian motion In probability theory, the harmonic measure of a subset of the boundary of a bounded domain in Euclidean space R^n, n\geq 2 is the probability that a Brownian motion started inside a domain hits that subset of the boundary. More generally, harmonic measure of an Itō diffusion X describes the distribution of X as it hits the boundary of D. In the complex plane, harmonic measure can be used to estimate the modulus of an analytic function inside a domain D given bounds on the modulus on the boundary of the domain; a special case of this principle is Hadamard's three-circle theorem. On simply connected planar domains, there is a close connection between harmonic measure and the theory of conformal maps.
|
|
A set S of points in Euclidean space is a Delone set if there exists a number \varepsilon>0 such that every two points of S are at least at distance \varepsilon apart from each other and such that every point of space is within distance 1/\varepsilon of at least one point in S. So S fills space, but its points never come too close to each other. For this to be true, S must be infinite. Additionally, the set S is symmetric (in the sense needed to define a plesiohedron) if, for every two points p and q of S, there exists a rigid motion of space that takes S to S and p to q. That is, the symmetries of S act transitively on S. The Voronoi diagram of any set S of points partitions space into regions called Voronoi cells that are nearer to one given point of S than to any other.
|
|
An affine representation of a topological (Lie) group G on an affine space A is a continuous (smooth) group homomorphism from G to the automorphism group of A, the affine group Aff(A). Similarly, an affine representation of a Lie algebra g on A is a Lie algebra homomorphism from g to the Lie algebra aff(A) of the affine group of A. An example is the action of the Euclidean group E(n) upon the Euclidean space En. Since the affine group in dimension n is a matrix group in dimension n + 1, an affine representation may be thought of as a particular kind of linear representation. We may ask whether a given affine representation has a fixed point in the given affine space A. If it does, we may take that as origin and regard A as a vector space: in that case, we actually have a linear representation in dimension n. This reduction depends on a group cohomology question, in general.
|
|
The most dramatic change in the topology from the classical picture to the new is that points are no longer necessarily closed; by expanding the definition, Grothendieck introduced generic points, which are the points with maximal closure, that is the minimal prime ideals. The closed points correspond to maximal ideals of A. However, the spectrum and projective spectrum are still T0 spaces: given two points P, Q, which are prime ideals of A, at least one of them, say P, does not contain the other. Then D(Q) contains P but, of course, not Q. Just as in classical algebraic geometry, any spectrum or projective spectrum is (quasi)compact, and if the ring in question is Noetherian then the space is a Noetherian space. However, these facts are counterintuitive: we do not normally expect open sets, other than connected components, to be compact, and for affine varieties (for example, Euclidean space) we do not even expect the space itself to be compact.
|
|
Interestingly, the submanifolds for which the inequality is an equality can be characterized as certain products of minimal surfaces of low dimension with Euclidean spaces. Chen introduced and systematically studied the notion of a finite type submanifold of Euclidean space, which is a submanifold for which the position vector is a finite linear combination of eigenfunctions of the Laplace-Beltrami operator. He also introduced and studied a generalization of the class of totally real submanifolds and of complex submanifolds; a slant submanifold of an almost Hermitian manifold is a submanifold for which there is a number such that the image under the almost complex structure of an arbitrary submanifold tangent vector has an angle of with the submanifold's tangent space. In Riemannian geometry, Chen introduced the δ-invariants (also called Chen invariants), which are certain kinds of partial traces of the sectional curvature; they can be viewed as an interpolation between sectional curvature and scalar curvature.
|
|
Some sources, including Kakutani's original paper, use the concept of upper hemicontinuity while stating the theorem: :Let S be a non-empty, compact and convex subset of some Euclidean space Rn. Let φ: S→2S be an upper hemicontinuous set-valued function on S with the property that φ(x) is non-empty, closed, and convex for all x ∈ S. Then φ has a fixed point. This statement of Kakutani's theorem is completely equivalent to the statement given at the beginning of this article. We can show this by using the closed graph theorem for set-valued functions, which says that for a compact Hausdorff range space Y, a set-valued function φ: X→2Y has a closed graph if and only if it is upper hemicontinuous and φ(x) is a closed set for all x. Since all Euclidean spaces are Hausdorff (being metric spaces) and φ is required to be closed-valued in the alternative statement of the Kakutani theorem, the Closed Graph Theorem implies that the two statements are equivalent.
|
|
Robertson's irreducible apex graph, showing that the YΔY- reducible graphs have additional forbidden minors beyond those in the Petersen family. A minor of a graph G is another graph formed from G by contracting and removing edges. As the Robertson–Seymour theorem shows, many important families of graphs can be characterized by a finite set of forbidden minors: for instance, according to Wagner's theorem, the planar graphs are exactly the graphs that have neither the complete graph K5 nor the complete bipartite graph K3,3 as minors. Neil Robertson, Paul Seymour, and Robin Thomas used the Petersen family as part of a similar characterization of linkless embeddings of graphs, embeddings of a given graph into Euclidean space in such a way that every cycle in the graph is the boundary of a disk that is not crossed by any other part of the graph.. Horst Sachs had previously studied such embeddings, shown that the seven graphs of the Petersen family do not have such embeddings, and posed the question of characterizing the linklessly embeddable graphs by forbidden subgraphs.. Robertson et al.
|
|
Given a transformation between input and output values, described by a mathematical function f, optimization deals with generating and selecting a best solution from some set of available alternatives, by systematically choosing input values from within an allowed set, computing the output of the function, and recording the best output values found during the process. Many real-world problems can be modeled in this way. For example, the inputs can be design parameters of a motor, the output can be the power consumption, or the inputs can be business choices and the output can be the obtained profit. An optimization problem, in this case a minimization problem, can be represented in the following way :Given: a function f : A \to R from some set A to the real numbers :Search for: an element x0 in A such that f(x0) ≤ f(x) for all x in A. In continuous optimization, A is some subset of the Euclidean space Rn, often specified by a set of constraints, equalities or inequalities that the members of A have to satisfy.
|
|
Hajós's theorem is named after Hajós, and concerns factorizations of Abelian groups into Cartesian products of subsets of their elements.. This result in group theory has consequences also in geometry: Hajós used it to prove a conjecture of Hermann Minkowski that, if a Euclidean space of any dimension is tiled by hypercubes whose positions form a lattice, then some pair of hypercubes must meet face-to-face. Hajós used similar group- theoretic methods to attack Keller's conjecture on whether cube tilings (without the lattice constraint) must have pairs of cubes that meet face to face; his work formed an important step in the eventual disproof of this conjecture.. Hajós's conjecture is a conjecture made by Hajós that every graph with chromatic number contains a subdivision of a complete graph . However, it is now known to be false: in 1979, Paul A. Catlin found a counterexample for ,. and Paul Erdős and Siemion Fajtlowicz later observed that it fails badly for random graphs.. The Hajós construction is a general method for constructing graphs with a given chromatic number, also due to Hajós.. As cited by .
|
|
In this case, the embedded metric is a finite metric space, whose distances are shortest path lengths in a graph, and the metric into which is embedded is the Euclidean plane. When the graph and its embedding are fixed, but the graph edge weights can vary, the stretch factor is minimized when the weights are exactly the Euclidean distances between the edge endpoints. Research in this area has focused on finding sparse graphs for a given point set that have low stretch factor.. The Johnson–Lindenstrauss lemma asserts that any finite metric space with points can be embedded into a Euclidean space of dimension with distortion , for any constant , where the constant factor in the -notation depends on the choice of .. This result, and related methods of constructing low-distortion metric embeddings, are important in the theory of approximation algorithms. A major open problem in this area is the GNRS conjecture, which (if true) would characterize the families of graphs that have bounded-stretch embeddings into \ell_1 spaces as being all minor-closed graph families.
|
|
In knot theory, the distortion of a knot is a knot invariant, the minimum stretch factor of any embedding of the knot as a space curve in Euclidean space. Undergraduate researcher John Pardon won the 2012 Morgan Prize for his research showing that there is no upper bound on the distortion of torus knots, solving a problem originally posed by Mikhail Gromov... In the study of the curve-shortening flow, in which each point of a curve in the Euclidean plane moves perpendicularly to the curve, with speed proportional to the local curvature, proved that the stretch factor of any simple closed smooth curve (with intrinsic distances measured by arc length) changes monotonically. More specifically, at each pair that forms a local maximum of the stretch factor, the stretch factor is strictly decreasing, except when the curve is a circle. This property was later used to simplify the proof of the Gage–Hamilton–Grayson theorem, according to which every simple closed smooth curve stays simple and smooth until it collapses to a point, converging in shape to a circle before doing so...
|
|
These four-dimensional coordinates may be transformed into three-dimensional coordinates by the formula :(a, b, c, d) → (a + b − c − d, a − b + c − d, −a + b + c − d).. Because the diamond structure forms a distance-preserving subset of the four-dimensional integer lattice, it is a partial cube. Yet another coordinatization of the diamond cubic involves the removal of some of the edges from a three-dimensional grid graph. In this coordinatization, which has a distorted geometry from the standard diamond cubic structure but has the same topological structure, the vertices of the diamond cubic are represented by all possible 3d grid points and the edges of the diamond cubic are represented by a subset of the 3d grid edges.. The diamond cubic is sometimes called the "diamond lattice" but it is not, mathematically, a lattice: there is no translational symmetry that takes the point (0,0,0) into the point (3,3,3), for instance. However, it is still a highly symmetric structure: any incident pair of a vertex and edge can be transformed into any other incident pair by a congruence of Euclidean space.
|
|
In a paper published in 1924, Stefan Banach and Alfred Tarski gave a construction of such a paradoxical decomposition, based on earlier work by Giuseppe Vitali concerning the unit interval and on the paradoxical decompositions of the sphere by Felix Hausdorff, and discussed a number of related questions concerning decompositions of subsets of Euclidean spaces in various dimensions. They proved the following more general statement, the strong form of the Banach–Tarski paradox: : Given any two bounded subsets and of a Euclidean space in at least three dimensions, both of which have a nonempty interior, there are partitions of and into a finite number of disjoint subsets, A=A_1 \cup \cdots\cup A_k, B=B_1 \cup \cdots\cup B_k (for some integer k), such that for each (integer) between and , the sets and are congruent. Now let be the original ball and be the union of two translated copies of the original ball. Then the proposition means that you can divide the original ball into a certain number of pieces and then rotate and translate these pieces in such a way that the result is the whole set , which contains two copies of .
|
|
Refuting the framework of Newton's theory—absolute space and absolute time—special relativity refers to relative space and relative time, whereby length contracts and time dilates along the travel pathway of an object. In 1908, Einstein's former mathematics professor Hermann Minkowski modeled 3D space together with the 1D axis of time by treating the temporal axis like a fourth spatial dimension—altogether 4D spacetime—and declared the imminent demise of the separation of space and time Minkowski, Hermann (1908–1909), "Raum und Zeit" [Space and Time], Physikalische Zeitschrift, 10: 75–88 . Einstein initially called this "superfluous learnedness", but later used Minkowski spacetime with great elegance in his general theory of relativity,Salmon WC & Wolters G, eds, Logic, Language, and the Structure of Scientific Theories (Pittsburgh: University of Pittsburgh Press, 1994), p 125 extending invariance to all reference frames—whether perceived as inertial or as accelerated—and credited this to Minkowski, by then deceased. General relativity replaces Cartesian coordinates with Gaussian coordinates, and replaces Newton's claimed empty yet Euclidean space traversed instantly by Newton's vector of hypothetical gravitational force—an instant action at a distance—with a gravitational field.
|
|
The contact graph of an arbitrary finite packing of unit balls is the graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. The cardinality of the edge set of the contact graph gives the number of touching pairs, the number of 3-cycles in the contact graph gives the number of touching triplets, and the number of tetrahedrons in the contact graph gives the number of touching quadruples (in general for a contact graph associated with a sphere packing in n dimensions that the cardinality of the set of n-simplices in the contact graph gives the number of touching (n + 1)-tuples in the sphere packing). In the case of 3-dimensional Euclidean space, non-trivial upper bounds on the number of touching pairs, triplets, and quadruples were proved by Karoly Bezdek and Samuel Reid at the University of Calgary. The problem of finding the arrangement of n identical spheres that maximizes the number of contact points between the spheres is known as the "sticky-sphere problem".
|
|
In connection with Tverberg's theorem, conjectured that, for every set of r(d + 1) points in d-dimensional Euclidean space, colored with d + 1 colors in such a way that there are r points of each color, there is a way to partition the points into rainbow simplices (sets of d + 1 points with one point of each color) in such a way that the convex hulls of these sets have a nonempty intersection.. For instance, the two-dimensional case (proven by Bárány and Larman) with r = 3 states that, for every set of nine points in the plane, colored with three colors and three points of each color, it is possible to partition the points into three intersecting rainbow triangles, a statement similar to Rota's basis conjecture which states that it is possible to partition the points into three non-degenerate rainbow triangles. The conjecture of Bárány and Larman allows a collinear triple of points to be considered as a rainbow triangle, whereas Rota's basis conjecture disallows this; on the other hand, Rota's basis conjecture does not require the triangles to have a common intersection. Substantial progress on the conjecture of Bárány and Larman was made by ..
|
|
The "girth" terminology generalizes the use of girth in graph theory, meaning the length of the shortest cycle in a graph: the girth of a graphic matroid is the same as the girth of its underlying graph.. The girth of other classes of matroids also corresponds to important combinatorial problems. For instance, the girth of a co-graphic matroid (or the cogirth of a graphic matroid) equals the edge connectivity of the underlying graph, the number of edges in a minimum cut of the graph. The girth of a transversal matroid gives the cardinality of a minimum Hall set in a bipartite graph: this is a set of vertices on one side of the bipartition that does not form the set of endpoints of a matching in the graph.. Any set of points in Euclidean space gives rise to a real linear matroid by interpreting the Cartesian coordinates of the points as the vectors of a matroid representation. The girth of the resulting matroid equals one plus the dimension of the space when the underlying set of point is in general position, and is smaller otherwise.
|
|