154 Sentences With "non singular" | Random Sentence Generator

The language has four numbers: singular, dual, paucal, and plural. Nouns show all four while pronouns are either singular and non-singular. First-person non-singular shows a distinction in inclusive and exclusive.

Over a field, unimodular has the same meaning as non-singular. Unimodular here refers to matrices with coefficients in some ring (often the integers) which are invertible over that ring, and one uses non-singular to mean matrices that are invertible over the field.

Any surface is birational to a non-singular surface, so for most purposes it is enough to classify the non-singular surfaces. Given any point on a surface, we can form a new surface by blowing up this point, which means roughly that we replace it by a copy of the projective line. For the purpose of this article, a non- singular surface X is called minimal if it cannot be obtained from another non-singular surface by blowing up a point. By Castelnuovo's contraction theorem, this is equivalent to saying that X has no (−1)-curves (smooth rational curves with self-intersection number −1).

In algebraic geometry, a cubic threefold is a hypersurface of degree 3 in 4-dimensional projective space. Cubic threefolds are all unirational, but used intermediate Jacobians to show that non-singular cubic threefolds are not rational. The space of lines on a non-singular cubic 3-fold is a Fano surface.

So to classify all compact complex surfaces up to birational isomorphism it is (more or less) enough to classify the minimal non-singular ones.

Also, for a non-singular M-matrix, the diagonal elements of A must be positive. Here we will further characterize only the class of non-singular M-matrices. Many statements that are equivalent to this definition of non-singular M-matrices are known, and any one of these statements can serve as a starting definition of a non-singular M-matrix.. For example, Plemmons lists 40 such equivalences.. These characterizations has been categorized by Plemmons in terms of their relations to the properties of: (1) positivity of principal minors, (2) inverse-positivity and splittings, (3) stability, and (4) semipositivity and diagonal dominance. It makes sense to categorize the properties in this way because the statements within a particular group are related to each other even when matrix is an arbitrary matrix, and not necessarily a Z-matrix.

The inverse of any non-singular M-matrix is a non-negative matrix. If the non-singular M-matrix is also symmetric then it is called a Stieltjes matrix. The inverse of a non-negative matrix is usually not non-negative. The exception is the non-negative monomial matrices: a non-negative matrix has non-negative inverse if and only if it is a (non-negative) monomial matrix.

The plane curves of degree 3 are called cubic plane curves and, if they are non-singular, elliptic curves. Those of degree 4 are called quartic plane curves.

In mathematics, the Atiyah conjecture on configurations is a conjecture introduced by stating that a certain n by n matrix depending on n points in R3 is always non-singular.

The gradient of is zero at a singular point of the hypersurface (this is the definition of a singular point). At a non- singular point, it is a nonzero normal vector.

The group law of an abelian variety is necessarily commutative and the variety is non- singular. An elliptic curve is an abelian variety of dimension 1. Abelian varieties have Kodaira dimension 0.

For a non-singular projective variety, a result of Kodaira states that this is equivalent to a divisor in the class being the sum of an ample divisor and an effective divisor.

In linear algebra and statistics, the pseudo-determinant PDF is the product of all non-zero eigenvalues of a square matrix. It coincides with the regular determinant when the matrix is non-singular.

If V is finite-dimensional then, relative to some basis for V, a bilinear form is degenerate if and only if the determinant of the associated matrix is zero – if and only if the matrix is singular, and accordingly degenerate forms are also called singular forms. Likewise, a nondegenerate form is one for which the associated matrix is non-singular, and accordingly nondegenerate forms are also referred to as non-singular forms. These statements are independent of the chosen basis.

The Hayward metric is the simplest description of a black hole which is non- singular. The metric was written down by Sean Hayward as the minimal model which is regular, static, spherically symmetric and asymptotically flat. The metric is not derived from any particular alternative theory of gravity, but provides a framework to test the formation and evaporation of non-singular black holes both within general relativity and beyond. Hayward first published his metric in 2005 and numerous papers have studied it since.

In algebraic geometry, a quartic threefold is a degree 4 hypersurface of dimension 3 in 4-dimensional projective space. showed that all non-singular quartic threefolds are irrational, though some of them are unirational.

Consani and Scholten define their hypersurface from the (projectivized) set of solutions to the equation :P(x,y)=P(z,w) in four complex variables, where :P(x,y)=x^5+y^5-(5xy-5)(x^2+y^2-x-y). In this form the resulting hypersurface is singular: it has 120 double points. Its Hodge diamond is The Consani–Scholton quintic itself is the non-singular hypersurface obtained by blowing up these singularities. As a non-singular quintic threefold, it is a Calabi–Yau manifold.

The Hayward metric is the simplest description of a black hole which is non-singular. The metric was written down by Sean Hayward as the minimal model which is regular, static, spherically symmetric and asymptotically flat.

While this model is the simplest way to describe hyperelliptic curves, such an equation will have a singular point at infinity in the projective plane. This feature is specific to the case n > 3. Therefore, in giving such an equation to specify a non- singular curve, it is almost always assumed that a non-singular model (also called a smooth completion), equivalent in the sense of birational geometry, is meant. To be more precise, the equation defines a quadratic extension of C(x), and it is that function field that is meant.

The nine inflection points of a non-singular cubic have the property that every line passing through two of them contains exactly three inflection points. The real points of cubic curves were studied by Isaac Newton. The real points of a non-singular projective cubic fall into one or two 'ovals'. One of these ovals crosses every real projective line, and thus is never bounded when the cubic is drawn in the Euclidean plane; it appears as one or three infinite branches, containing the three real inflection points.

There is also a quick proof, by Morris Hirsch, based on the impossibility of a differentiable retraction. The indirect proof starts by noting that the map f can be approximated by a smooth map retaining the property of not fixing a point; this can be done by using the Weierstrass approximation theorem, for example. One then defines a retraction as above which must now be differentiable. Such a retraction must have a non-singular value, by Sard's theorem, which is also non-singular for the restriction to the boundary (which is just the identity).

A square matrix A is called invertible or non-singular if there exists a matrix B such that :AB=BA=I_n. If B exists, it is unique and is called the inverse matrix of A, denoted A^{-1}.

A general algebraic variety being defined as the common zeros of several polynomials, the condition on a point of to be singular point is that the Jacobian matrix of the first order partial derivatives of the polynomials has a rank at that is lower than the rank at other points of the variety. Points of V that are not singular are called non-singular or regular. It is always true that almost all points are non-singular, in the sense that the non-singular points form a set that is both open and dense in the variety (for the Zariski topology, as well as for the usual topology, in the case of varieties defined over the complexes). In case of a real variety (that is the set of the points with real coordinates of a variety defined by polynomials with real coefficients), the variety is a manifold near every regular point.

In the case of Araki, it is more appropriate to discuss ‘personal markers’ (rather than ‘pronouns’). There are seven morphosyntactic person markings: first, second, third, and in the case of non-singular first person, there is an inclusive/exclusive distinction.

Since the linear equations require O(n^3) operations to solve, high-order quadrature rules perform better because low-order quadrature rules require large n for a given accuracy. Gaussian quadrature is normally a good choice for smooth, non- singular problems.

The BKM is basically a combination of the distance function, non-singular general solution, and dual reciprocity method (DRM). The distance function is employed in the BKM to approximate the inhomogeneous terms via the DRM, whereas the non-singular general solution of the partial differential equation leads to a boundary-only formulation for the homogeneous solution. Without the singular fundamental solution, the BKM removes the controversial artificial boundary in the method of fundamental solutions. Some preliminary numerical experiments show that the BKM can produce excellent results with relatively a small number of nodes for various linear and nonlinear problems.

In algebraic geometry, the Zeuthen–Segre invariant I is an invariant of a projective surface found in a complex projective space which was introduced by and rediscovered by . The invariant I is defined to be d – 4g – b if the surface has a pencil of curves, non-singular of genus g except for d curves with 1 ordinary node, and with b base points where the curves are non-singular and transverse. showed that the Zeuthen–Segre invariant I is χ–4, where χ is the topological Euler–Poincaré characteristic introduced by , which is equal to the Chern number c2 of the surface.

The Cartan matrix is non-singular; in fact, its determinant is a power of the characteristic of K. Since a projective indecomposable module in a given block has all its composition factors in that same block, each block has its own Cartan matrix.

In mathematics, the Jacobian variety J(C) of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of C, hence an abelian variety.

In classical algebraic geometry, certain special singular points were also called nodes. A node is a singular point where the Hessian matrix is non-singular; this implies that the singular point has multiplicity two and the tangent cone is not singular outside its vertex.

From 1960 to 1961 and from 1967 to 1968 he was at the Institute for Advanced Study. In 1968, together with Paul Monsky, he introduced the Monsky–Washnitzer cohomology, which is a p-adic cohomology theory for non- singular algebraic varieties. Among his students was William Fulton.

Then Hrushovski and Zilber prove that under these conditions there is an algebraically closed field K, and a non-singular algebraic curve C, such that its Zariski geometry of powers and their Zariski topology is isomorphic to the given one. In short, the geometry can be algebraized.

Suppose that the equation :y^2 = x^3 + ax^2 + bx + c defines a non-singular cubic curve with integer coefficients a, b, c, and let D be the discriminant of the cubic polynomial on the right side: :D = -4a^3c + a^2b^2 + 18abc - 4b^3 - 27c^2.

It is known that Heath-Brown's result is best possible in the sense that there exist non- singular cubic forms over the rationals in 9 variables that don't represent zero. However, Hooley showed that the Hasse principle holds for the representation of 0 by non-singular cubic forms over the rational numbers in at least nine variables. Davenport, Heath-Brown and Hooley all used the Hardy–Littlewood circle method in their proofs. According to an idea of Manin, the obstructions to the Hasse principle holding for cubic forms can be tied into the theory of the Brauer group; this is the Brauer–Manin obstruction, which accounts completely for the failure of the Hasse principle for some classes of variety.

In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that the number of such automorphisms cannot exceed 84(g − 1). A group for which the maximum is achieved is called a Hurwitz group, and the corresponding Riemann surface a Hurwitz surface. Because compact Riemann surfaces are synonymous with non-singular complex projective algebraic curves, a Hurwitz surface can also be called a Hurwitz curve.Technically speaking, there is an equivalence of categories between the category of compact Riemann surfaces with the orientation-preserving conformal maps and the category of non-singular complex projective algebraic curves with the algebraic morphisms.

Geometrically, this means that the graph has no cusps, self-intersections, or isolated points. Algebraically, this holds if and only if the discriminant : \Delta = -16(4a^3 + 27b^2) is not equal to zero. (Although the factor −16 is irrelevant to whether or not the curve is non- singular, this definition of the discriminant is useful in a more advanced study of elliptic curves.) The (real) graph of a non-singular curve has two components if its discriminant is positive, and one component if it is negative. For example, in the graphs shown in figure to the right, the discriminant in the first case is 64, and in the second case is −368.

In mathematics, Weber's theorem, named after Heinrich Martin Weber, is a result on algebraic curves. It states the following. : Consider two non- singular curves C and C′ having the same genus g > 1. If there is a rational correspondence φ between C and C′, then φ is a birational transformation.

Valency-decreasing operations The reciprocal marker t-/to-/tin-/te- indicates that the Actor and Undergoer within a clause are the same referent. Thus, it makes a divalent verb monovalent. The reciprocal marker can only occur with non-singular Actors. :Gi- man ong kantor mi kreyang, :3POSS2\- father this office be.

Wiley, New York. Whether or not the surface is singular: a singular surface is one in which the surface tension as a function of orientation has a pointed minimum. Growth of singular surfaces is known to requires steps, whereas it is generally held that non-singular surfaces can continuously advance normal to themselves.

In differential geometry, an almost symplectic structure on a differentiable manifold M is a two-form ω on M that is everywhere non-singular.. If, in addition, ω is closed, then it is a symplectic form. An almost symplectic manifold is an Sp-structure; requiring ω to be closed is an integrability condition.

For n=1 the intermediate Jacobian is the Picard variety, and for n=2 \dim (M)-1 it is the Albanese variety. In these two extreme cases the constructions of Weil and Griffiths are equivalent. used intermediate Jacobians to show that non-singular cubic threefolds are not rational, even though they are unirational.

The singularities of a projective variety V are canonical if the variety is normal, some power of the canonical line bundle of the non-singular part of V extends to a line bundle on V, and V has the same plurigenera as any resolution of its singularities. V has canonical singularities if and only if it is a relative canonical model. The singularities of a projective variety V are terminal if the variety is normal, some power of the canonical line bundle of the non-singular part of V extends to a line bundle on V, and V the pullback of any section of Vm vanishes along any codimension 1 component of the exceptional locus of a resolution of its singularities.

A square matrix A is called invertible or non-singular if there exists a matrix B such that :AB = BA = I , where I is the n×n identity matrix with 1s on the main diagonal and 0s elsewhere. If B exists, it is unique and is called the inverse matrix of A, denoted A.

Stability conditions are known for RBMs in 1, 2, and 3 dimensions. "The problem of recurrence classification for SRBMs in four and higher dimensions remains open." In the special case where R is an M-matrix then necessary and sufficient conditions for stability are # R is a non- singular matrix and # R−1μ < 0\.

In birational geometry, a complex rational surface is any algebraic surface birationally equivalent to the complex projective plane. It is known that any non-singular rational variety is obtained from the plane by a sequence of blowing up transformations and their inverses ('blowing down') of curves, which must be of a very particular type. As a special case, a non-singular complex quadric in P3 is obtained from the plane by blowing up two points to curves, and then blowing down the line through these two points; the inverse of this transformation can be seen by taking a point P on the quadric Q, blowing it up, and projecting onto a general plane in P3 by drawing lines through P. The group of birational automorphisms of the complex projective plane is the Cremona group.

The only non-singular idempotent matrix is the identity matrix; that is, if a non-identity matrix is idempotent, its number of independent rows (and columns) is less than its number of rows (and columns). This can be seen from writing A^2 = A, assuming that has full rank (is non-singular), and pre- multiplying by A^{-1} to obtain A = IA = A^{-1}A^2 = A^{-1}A = I. When an idempotent matrix is subtracted from the identity matrix, the result is also idempotent. This holds since :(I-A)(I-A) = I-A-A+A^2 = I-A-A+A = I-A. A matrix is idempotent if and only if for all positive integers n, A^n = A. The 'if' direction trivially follows by taking n=2.

The reference system makes a 10-way distinction. Gender is distinguished in second and third person singular pronouns. The first person non-singular pronouns include a dual inclusive form, a plural inclusive form, and a plural exclusive form. The plural inclusive form is a bimorphemic pronoun which combines the first person dual inclusive form with the second person plural form.

The non-singular points of an Alexander horned sphere form a Cantor tree surface In dynamical systems, the Cantor tree is an infinite-genus surface homeomorphic to a sphere with a Cantor set removed. The blooming Cantor tree is a Cantor tree with an infinite number of handles added in such a way that every end is a limit of handles.

If is a homogeneous polynomial in three variables, the equation is the implicit equation of a plane projective curve. The inflection points of the curve are exactly the non-singular points where the Hessian determinant is zero. It follows by Bézout's theorem that a cubic plane curve has at most 9 inflection points, since the Hessian determinant is a polynomial of degree 3.

The Fermat curve is non-singular and has genus :(n - 1)(n - 2)/2.\ This means genus 0 for the case n = 2 (a conic) and genus 1 only for n = 3 (an elliptic curve). The Jacobian variety of the Fermat curve has been studied in depth. It is isogenous to a product of simple abelian varieties with complex multiplication.

In terms of the affine subspace , an isotropic line through the origin is :x_2 = \pm i x_1 . In projective geometry, the isotropic lines are the ones passing through the circular points at infinity. In the real orthogonal geometry of Emil Artin, isotropic lines occur in pairs: :A non-singular plane which contains an isotropic vector shall be called a hyperbolic plane.

A totally unimodular matrix The term was coined by Claude Berge, see (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. Equivalently, every square submatrix has determinant 0, +1 or −1\. A totally unimodular matrix need not be square itself. From the definition it follows that any submatrix of a totally unimodular matrix is itself totally unimodular (TU).

In algebraic geometry, Sumihiro's theorem, introduced by , states that a normal algebraic variety with an action of a torus can be covered by torus- invariant affine open subsets. The "normality" in the hypothesis cannot be relaxed.Toric Varieties The hypothesis that the group acting on the variety is a torus can also not be relaxed.Bialynicki-Birula decomposition of a non- singular quasi-projective scheme.

The projective line is a fundamental example of an algebraic curve. From the point of view of algebraic geometry, P1(K) is a non-singular curve of genus 0. If K is algebraically closed, it is the unique such curve over K, up to rational equivalence. In general a (non-singular) curve of genus 0 is rationally equivalent over K to a conic C, which is itself birationally equivalent to projective line if and only if C has a point defined over K; geometrically such a point P can be used as origin to make explicit the birational equivalence.. The function field of the projective line is the field K(T) of rational functions over K, in a single indeterminate T. The field automorphisms of K(T) over K are precisely the group PGL2(K) discussed above.

The index of the vector field as a whole is defined when it has just a finite number of zeroes. In this case, all zeroes are isolated, and the index of the vector field is defined to be the sum of the indices at all zeroes. The index is not defined at any non-singular point (i.e., a point where the vector is non-zero).

The joint distribution of the real eigenvalues of the inverse complex (and real) Wishart are found in Edelman's paper who refers back to an earlier paper by James. In the non-singular case, the eigenvalues of the inverse Wishart are simply the inverted values for the Wishart. Edelman also characterises the marginal distributions of the smallest and largest eigenvalues of complex and real Wishart matrices.

The Clebsch cubic in a local chart Model of the surface In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by and , all of whose 27 exceptional lines can be defined over the real numbers. The term Klein's icosahedral surface can refer to either this surface or its blowup at the 10 Eckardt points.

Non-singular linear transformations of the vector space provide motions of PG(n,q). The first book (1979) covered PG(1,q) and PG(2,q). The second book addressed PG(3,q) and the third PG(n,q). Chapters are numbered sequentially through the trilogy: 14 in the first book, 15 to 21 in the second, and 22 to 27 in the third.

In linguistics, the term "universal grinder" refers to an idea that in some languages most count nouns can be used as if they were mass nouns, which causes a slight change in their meaning. The term "universal grinder" was first used in print by F. Jeffry Pelletier in 1975, after a personal suggestion by David Lewis.Pelletier, F. Jeffry: Non-singular reference: some preliminaries. In Philosophia Vol.

Alexander Grothendieck gave a complete description of the relation of q to h^{0,1}in all characteristics. The dimension of the tangent space to the Picard scheme (at any point) is equal to h^{0,1}. In characteristic 0 a result of Pierre Cartier showed that all groups schemes of finite type are non-singular, so the dimension of their tangent space is their dimension.

In mathematics, Noether's theorem on rationality for surfaces is a classical result of Max Noether on complex algebraic surfaces, giving a criterion for a rational surface. Let S be an algebraic surface that is non-singular and projective. Suppose there is a morphism φ from S to the projective line, with general fibre also a projective line. Then the theorem states that S is rational.

William Fulton in Intersection Theory (1984) writes > ... if and are subvarieties of a non-singular variety , the intersection > product should be an equivalence class of algebraic cycles closely related > to the geometry of how , and are situated in . Two extreme cases have been > most familiar. If the intersection is proper, i.e. , then is a linear > combination of the irreducible components of , with coefficients the > intersection multiplicities.

A line on a (which can also be interpreted as the non-singular quadric in ) has self-intersection , since a line can be moved off itself. (It is a ruled surface.) In terms of intersection forms, we say has one of type – there are two basic classes of lines, which intersect each other in one point (), but have zero self- intersection (no or terms).

An ontological commitment refers to a relation between a language and certain objects postulated to be extant by that language. The 'existence' referred to need not be 'real', but exist only in a universe of discourse. As an example, legal systems use vocabulary referring to 'legal persons' that are collective entities that have rights. One says the legal doctrine has an ontological commitment to non-singular individuals.

In 1960 Donald Samuel Ornstein constructed an example of a non-singular transformation on the Lebesgue space of the unit interval, which does not admit a \sigma–finite invariant measure equivalent to Lebesgue measure, thus solving a long-standing problem in ergodic theory. A few years later, Rafael V. Chacón gave an example of a positive (linear) isometry of L_1 for which the individual ergodic theorem fails in L_1.

In algebraic geometry, the Klein cubic threefold is the non-singular cubic threefold in 4-dimensional projective space given by the equation :V^2W+W^2X+X^2Y+Y^2Z+Z^2V =0 \, studied by . Its automorphism group is the group PSL2(11) of order 660 . It is unirational but not a rational variety. showed that it is birational to the moduli space of (1,11)-polarized abelian surfaces.

Every non-singular rational surface can be obtained by repeatedly blowing up a minimal rational surface. The minimal rational surfaces are the projective plane and the Hirzebruch surfaces Σr for r = 0 or r ≥ 2. Invariants: The plurigenera are all 0 and the fundamental group is trivial. Hodge diamond: where n is 0 for the projective plane, and 1 for Hirzebruch surfaces and greater than 1 for other rational surfaces.

A counterexample by Ernst S. Selmer shows that the Hasse–Minkowski theorem cannot be extended to forms of degree 3: The cubic equation 3x3 + 4y3 + 5z3 = 0 has a solution in real numbers, and in all p-adic fields, but it has no nontrivial solution in which x, y, and z are all rational numbers. Roger Heath-Brown showed that every cubic form over the integers in at least 14 variables represents 0, improving on earlier results of Davenport. Since every cubic form over the p-adic numbers with at least ten variables represents 0, the local–global principle holds trivially for cubic forms over the rationals in at least 14 variables. Restricting to non-singular forms, one can do better than this: Heath-Brown proved that every non-singular cubic form over the rational numbers in at least 10 variables represents 0, thus trivially establishing the Hasse principle for this class of forms.

A local field is sometimes called a one-dimensional local field. A non-Archimedean local field can be viewed as the field of fractions of the completion of the local ring of a one-dimensional arithmetic scheme of rank 1 at its non- singular point. For a non-negative integer n, an n-dimensional local field is a complete discrete valuation field whose residue field is an (n − 1)-dimensional local field.Definition 1.4.

In algebraic geometry, Monsky–Washnitzer cohomology is a p-adic cohomology theory defined for non-singular affine varieties over fields of positive characteristic p introduced by and , who were motivated by the work of . The idea is to lift the variety to characteristic 0, and then take a suitable subalgebra of the algebraic de Rham cohomology of . The construction was simplified by . Its extension to more general varieties is called rigid cohomology.

Ferdinand Georg Frobenius (1849–1917), German mathematician. His main interests were elliptic functions differential equations, and later group theory. The Frobenius covariants of a matrix can be obtained from any eigendecomposition , where is non-singular and is diagonal with . If has no multiple eigenvalues, then let ci be the th right eigenvector of , that is, the th column of ; and let ri be the th left eigenvector of , namely the th row of −1.

A topological version of Zariski's main theorem says that if x is a (closed) point of a normal complex variety it is unibranch; in other words there are arbitrarily small neighborhoods U of x such that the set of non-singular points of U is connected . The property of being normal is stronger than the property of being unibranch: for example, a cusp of a plane curve is unibranch but not normal.

The automorphisms of the real projective line are constructed with 2 × 2 real matrices. A matrix is required to be non-singular, and following the identification of proportional projective coordinates, proportional matrices (having identical actions on the real projective line) determine the same automorphism of P(R). Such an automorphism is sometimes called a homography of the projective line. With due regard for the point at infinity, an automorphism may be called a linear fractional transformation.

Two elements v and w of V are called orthogonal if . The kernel of a bilinear form B consists of the elements that are orthogonal to every element of V. Q is non-singular if the kernel of its associated bilinear form is {0}. If there exists a non-zero v in V such that , the quadratic form Q is isotropic, otherwise it is anisotropic. This terminology also applies to vectors and subspaces of a quadratic space.

In mathematics, Belyi's theorem on algebraic curves states that any non- singular algebraic curve C, defined by algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only. This is a result of G. V. Belyi from 1979. At the time it was considered surprising, and it spurred Grothendieck to develop his theory of dessins d'enfant, which describes nonsingular algebraic curves over the algebraic numbers using combinatorial data.

The exact inner product used does not matter, because a different inner product will give an equivalent norm on , and so give an equivalent metric. If the ground field is arbitrary and is considered as an algebraic group, then this construction shows that the Grassmannian is a non- singular algebraic variety. It follows from the existence of the Plücker embedding that the Grassmannian is complete as an algebraic variety. In particular, is a parabolic subgroup of .

The topological plane, or its equivalent the open disc, is the basic topological neighborhood used to construct surfaces (or 2-manifolds) classified in low-dimensional topology. Isomorphisms of the topological plane are all continuous bijections. The topological plane is the natural context for the branch of graph theory that deals with planar graphs, and results such as the four color theorem. The plane may also be viewed as an affine space, whose isomorphisms are combinations of translations and non-singular linear maps.

The Serre duality theorem is also true in complex geometry more generally, for compact complex manifolds that are not necessarily projective complex algebraic varieties. In this setting, the Serre duality theorem is an application of Hodge theory for Dolbeault cohomology, and may be seen as a result in the theory of elliptic operators. These two different interpretations of Serre duality coincide for non-singular projective complex algebraic varieties, by an application of Dolbeault's theorem relating sheaf cohomology to Dolbeault cohomology.

The necessity for Faddeev–Popov ghosts follows from the requirement that quantum field theories yield unambiguous, non-singular solutions. This is not possible in the path integral formulation when a gauge symmetry is present since there is no procedure for selecting among physically equivalent solutions related by gauge transformation. The path integrals overcount field configurations corresponding to the same physical state; the measure of the path integrals contains a factor which does not allow obtaining various results directly from the action.

In mathematics, the Segre class is a characteristic class used in the study of cones, a generalization of vector bundles. For vector bundles the total Segre class is inverse to the total Chern class, and thus provides equivalent information; the advantage of the Segre class is that it generalizes to more general cones, while the Chern class does not. The Segre class was introduced in the non-singular case by . In the modern treatment of intersection theory in algebraic geometry, as developed e.g.

In mathematics, the Hasse–Witt matrix H of a non-singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping (p-th power mapping where F has q elements, q a power of the prime number p) with respect to a basis for the differentials of the first kind. It is a g × g matrix where C has genus g. The rank of the Hasse–Witt matrix is the Hasse or Hasse–Witt invariant.

If the restriction of Q to a subspace U of V is identically zero, U is totally singular. The orthogonal group of a non-singular quadratic form Q is the group of the linear automorphisms of V that preserve Q, that is, the group of isometries of into itself. If a quadratic space has a product so that A is an algebra over a field, and satisfies :\forall x, y \isin A \quad Q(x y) = Q(x) Q(y) , then it is a composition algebra.

Churchill has debated historian Deborah Lipstadt over the singularity of the Holocaust. In a 1996 review of Lipstadt’s book Denying the Holocaust, Churchill defended the German philosopher Ernst Nolte, whom Lipstadt criticized for asserting that the Holocaust was a non-singular event. Churchill argues that the Holocaust was one of many genocides, as opposed to Lipstadt’s view of the Holocaust as a singular event. In A Little Matter of Genocide, Churchill accuses Lipstadt of denying the genocide of Native Americans, notwithstanding his respect of her work.

Nominal morphology is very simple. The only affix which can apply to a noun is the accusative suffix -ra, which is essentially extinct in Jarawara (although still common in Jamamadí and Banawá); a remnant of this can be seen, however, in the non-singular object pronouns era, otara, tera, mera. There is also a grammatically bound but phonologically independent marker taa, which signifies contrast with a previous noun (not X "but Y"). This inflection follows the noun, but precedes the accusative -ra in Jamamadí and Banawá.

When successive powers of a matrix T become small (that is, when all of the entries of T approach zero, upon raising T to successive powers), the matrix T converges to the zero matrix. A regular splitting of a non-singular matrix A results in a convergent matrix T. A semi-convergent splitting of a matrix A results in a semi-convergent matrix T. A general iterative method converges for every initial vector if T is convergent, and under certain conditions if T is semi- convergent.

Mary Elizabeth Kramer was born on August 6, 1948 in Cleveland, Ohio, United States. From 1965 to 1969, she was educated at Swarthmore College in Pennsylvania, majoring in mathematics, philosophy and history, and graduating summa cum laude in 1969. She then attended Harvard University as a mathematics graduate student: she completed her Master of Arts (MA) degree in 1971 and her Doctor of Philosophy (PhD) degree in 1972. Her doctoral thesis was titled "Non-singular deformations of space curves, using determinantal schemes": her advisors were David Mumford and Heisuke Hironaka.

A weak del Pezzo surface is a complete non-singular surface with anticanonical bundle that is nef and big. The blowdown of any (−1)-curve on a weak del Pezzo surface is a weak del Pezzo surface of degree 1 more. The blowup of any point on a weak del Pezzo surface is a weak del Pezzo surface of degree 1 less, provided that the point does not lie on a −2-curve and the degree is greater than 1. Any curve on a weak del Pezzo surface has self intersection number at least −2\.

In this section we assume that V is finite-dimensional and its bilinear form is non-singular. (If K has characteristic 2 this implies that the dimension of V is even.) The Pin group PinV(K) is the subgroup of the Lipschitz group Γ of elements of spinor norm 1, and similarly the Spin group SpinV(K) is the subgroup of elements of Dickson invariant 0 in PinV(K). When the characteristic is not 2, these are the elements of determinant 1. The Spin group usually has index 2 in the Pin group.

Some proofs produced by the school are not considered satisfactory because of foundational difficulties. These included frequent use of birational models in dimension three of surfaces that can have non-singular models only when embedded in higher-dimensional projective space. In order to avoid these issues, a sophisticated theory of handling a linear system of divisors was developed (in effect, a line bundle theory for hyperplane sections of putative embeddings in projective space). Many modern techniques were found, in embryonic form, and in some cases the articulation of these ideas exceeded the available technical language.

In mathematics, especially linear algebra, an M-matrix is a Z-matrix with eigenvalues whose real parts are nonnegative. The set of non-singular M-matrices are a subset of the class of P-matrices, and also of the class of inverse-positive matrices (i.e. matrices with inverses belonging to the class of positive matrices).. The name M-matrix was seemingly originally chosen by Alexander Ostrowski in reference to Hermann Minkowski, who proved that if a Z-matrix has all of its row sums positive, then the determinant of that matrix is positive..

Dissipative systems can also be used as a tool to study economic systems and complex systems. For example, a dissipative system involving self-assembly of nanowires has been used as a model to understand the relationship between entropy generation and the robustness of biological systems. The Hopf decomposition states that dynamical systems can be decomposed into a conservative and a dissipative part; more precisely, it states that every measure space with a non-singular transformation can be decomposed into an invariant conservative set and an invariant dissipative set.

At the other extreme, if is a non-singular > subvariety, the self-intersection formula says that is represented by the > top Chern class of the normal bundle of in . To give a definition, in the general case, of the intersection multiplicity was the major concern of André Weil's 1946 book Foundations of Algebraic Geometry. Work in the 1920s of B. L. van der Waerden had already addressed the question; in the Italian school of algebraic geometry the ideas were well known, but foundational questions were not addressed in the same spirit.

Viro developed a "patchworking" technique in algebraic geometry, which allows real algebraic varieties to be constructed by a sort of "cut and paste" method. Using this technique, Viro was able to complete the isotopy classification of non- singular plane projective curves of degree 7. The patchworking technique was one of the fundamental ideas which motivated the development of tropical geometry. In topology, Viro is most known for his joint work with Vladimir Turaev, in which the Turaev-Viro invariants and related topological quantum field theory notions were introduced.

In mechanics, a stress-induced transformation is called a deformation and may be described by a diffeomorphism. A diffeomorphism f : U → V between two surfaces U and V has a Jacobian matrix Df that is an invertible matrix. In fact, it is required that for p in U, there is a neighborhood of p in which the Jacobian Df stays non-singular. Since the Jacobian is a 2 × 2 real matrix, Df can be read as one of three types of complex number: ordinary complex, split complex number, or dual number.

Their work used an intermediate Jacobian. showed that all non-singular quartic threefolds are irrational, though some of them are unirational. found some unirational 3-folds with non-trivial torsion in their third cohomology group, which implies that they are not rational. For any field K, János Kollár proved in 2000 that a smooth cubic hypersurface of dimension at least 2 is unirational if it has a point defined over K. This is an improvement of many classical results, beginning with the case of cubic surfaces (which are rational varieties over an algebraic closure).

The other oval, if it exists, does not contain any real inflection point and appears either as an oval or as two infinite branches. Like for conic sections, a line cuts this oval at, at most, two points. A non-singular plane cubic defines an elliptic curve, over any field K for which it has a point defined. Elliptic curves are now normally studied in some variant of Weierstrass's elliptic functions, defining a quadratic extension of the field of rational functions made by extracting the square root of a cubic.

In algebraic geometry, in contrast, there is an intrinsic definition of the tangent space at a point of an algebraic variety V that gives a vector space with dimension at least that of V itself. The points p at which the dimension of the tangent space is exactly that of V are called non-singular points; the others are called singular points. For example, a curve that crosses itself does not have a unique tangent line at that point. The singular points of V are those where the ‘test to be a manifold’ fails.

In algebraic geometry, a Fano surface is a surface of general type (in particular, not a Fano variety) whose points index the lines on a non-singular cubic threefold. They were first studied by . Hodge diamond: Fano surfaces are perhaps the simplest and most studied examples of irregular surfaces of general type that are not related to a product of two curves and are not a complete intersection of divisors in an Abelian variety. The Fano surface S of a smooth cubic threefold F into P4 carries many remarkable geometric properties.

In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non- singular) and so of dimension four as a smooth manifold. The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of (real) dimension two). Many results were obtained, however, in the Italian school of algebraic geometry, and are up to 100 years old.

For r>R>0 (where R is the radius of some mass shell), mass acts as a delta function at the origin. For r, shells of mass may exist externally, but for the metric to be non-singular at the origin, M must be zero in the metric. This reduces the metric to flat Minkowski space; thus external shells have no gravitational effect. This result illuminates the gravitational collapse leading to a black hole and its effect on the motion of light-rays and particles outside and inside the event horizon (Hartle 2003, chapter 12).

In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In case of varieties defined over the reals, this notion generalizes the notion of local non-flatness. A point of an algebraic variety which is not singular is said to be regular. An algebraic variety which has no singular point is said to be non-singular or smooth.

Every homogeneous system has at least one solution, known as the zero (or trivial) solution, which is obtained by assigning the value of zero to each of the variables. If the system has a non-singular matrix () then it is also the only solution. If the system has a singular matrix then there is a solution set with an infinite number of solutions. This solution set has the following additional properties: # If u and v are two vectors representing solutions to a homogeneous system, then the vector sum is also a solution to the system.

Elliptic curves can be defined over any field K; the formal definition of an elliptic curve is a non-singular projective algebraic curve over K with genus 1 and endowed with a distinguished point defined over K. If the characteristic of K is neither 2 nor 3, then every elliptic curve over K can be written in the form :y^2 = x^3 - px - q where p and q are elements of K such that the right hand side polynomial x3 − px − q does not have any double roots. If the characteristic is 2 or 3, then more terms need to be kept: in characteristic 3, the most general equation is of the form :y^2 = 4x^3 + b_2 x^2 + 2b_4 x + b_6 for arbitrary constants b2, b4, b6 such that the polynomial on the right-hand side has distinct roots (the notation is chosen for historical reasons). In characteristic 2, even this much is not possible, and the most general equation is :y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6 provided that the variety it defines is non-singular. If characteristic were not an obstruction, each equation would reduce to the previous ones by a suitable change of variables.

In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general type, which have ample canonical class. They are named for Pasquale del Pezzo who studied the surfaces with the more restrictive condition that they have a very ample anticanonical divisor class, or in his language the surfaces with a degree n embedding in n-dimensional projective space , which are the del Pezzo surfaces of degree at least 3.

From 1979 to 1991 he was a professor at the College of Electrical Engineering in Leningrad. In 1985 he received his Russian doctorate (higher doctoral degree) with thesis Fourth-degree areas in three-dimensional space (Russian). He has been a professor at the University of Strasbourg since 1991, where he is a permanent member of the team at the Institut de Recherche Mathématique Avancée, UMRI 7501, CNRS. From 1972 he succeeded in solving a part of the Hilbert's sixteenth problem, concerning the number of components and the topology of non-singular fourth-order algebraic surfaces in three dimensions.

In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed field such as the complex numbers, these fibers are elliptic curves, perhaps without a chosen origin.) This is equivalent to the generic fiber being a smooth curve of genus one. This follows from proper base change. The surface and the base curve are assumed to be non-singular (complex manifolds or regular schemes, depending on the context).

The Hodge index theorem was a result on the intersection number theory for curves on an algebraic surface: it determines the signature of the corresponding quadratic form. This result was sought by the Italian school of algebraic geometry, but was proved by the topological methods of Lefschetz. The Theory and Applications of Harmonic Integrals summed up Hodge's development during the 1930s of his general theory. This starts with the existence for any Kähler metric of a theory of Laplacians – it applies to an algebraic variety V (assumed complex, projective and non-singular) because projective space itself carries such a metric.

The n-vector representation (also called geodetic normal or ellipsoid normal vector) is a three-parameter non-singular representation well-suited for replacing latitude and longitude as horizontal position representation in mathematical calculations and computer algorithms. Geometrically, the n-vector for a given position on an ellipsoid is the outward-pointing unit vector that is normal in that position to the ellipsoid. For representing horizontal positions on Earth, the ellipsoid is a reference ellipsoid and the vector is decomposed in an Earth-centered Earth-fixed coordinate system. It behaves smoothly at all Earth positions, and it holds the mathematical one-to-one property.

Special affine curvature, also known as the equiaffine curvature or affine curvature, is a particular type of curvature that is defined on a plane curve that remains unchanged under a special affine transformation (an affine transformation that preserves area). The curves of constant equiaffine curvature are precisely all non-singular plane conics. Those with are ellipses, those with are parabolae, and those with are hyperbolae. The usual Euclidean curvature of a curve at a point is the curvature of its osculating circle, the unique circle making second order contact (having three point contact) with the curve at the point.

This is the only known derivation of this formula from a fundamental theory, for the case of generic non singular black holes. Older attempts at this calculation had difficulties. The problem was that although Loop quantum gravity predicted that the entropy of a black hole is proportional to the area of the event horizon, the result depended on a crucial free parameter in the theory, the above-mentioned Immirzi parameter. However, there is no known computation of the Immirzi parameter, so it had to be fixed by demanding agreement with Bekenstein and Hawking's calculation of the black hole entropy.

Graphs of curves y2 = x3 − x and y2 = x3 − x + 1 Although the formal definition of an elliptic curve requires some background in algebraic geometry, it is possible to describe some features of elliptic curves over the real numbers using only introductory algebra and geometry. In this context, an elliptic curve is a plane curve defined by an equation of the form :y^2 = x^3 + ax + b where a and b are real numbers. This type of equation is called a Weierstrass equation. The definition of elliptic curve also requires that the curve be non-singular.

Generic object of dark energy (also known as GEODE and GEODEs) refers to a class of non-singular theoretical objects that mimic black holes, but with dark energy interiors instead. They have been hypothesized to result from the collapse of very large stars by Leningrad physicist Erast Gliner at the Ioffe Physico-Technical Institute in 1966. Such GEODEs appear to be black holes when viewed from afar but, different from black holes, these objects contain dark energy instead of a gravitational singularity. Contrary to classical black holes, GEODEs may intrinsically gain mass via the same relativistic effect responsible for the photon redshift.

The geometric genus can be defined for non-singular complex projective varieties and more generally for complex manifolds as the Hodge number (equal to by Serre duality), that is, the dimension of the canonical linear system plus one. In other words for a variety of complex dimension it is the number of linearly independent holomorphic -forms to be found on .Danilov & Shokurov (1998), [ p. 53] This definition, as the dimension of : then carries over to any base field, when is taken to be the sheaf of Kähler differentials and the power is the (top) exterior power, the canonical line bundle.

A matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. In the system of linear equations () above, with A non-singular, the matrix A can be split, that is, written as a difference so that () can be re-written as () above. The expression () is a regular splitting of A if and only if B−1 ≥ 0 and C ≥ 0, that is, and C have only nonnegative entries. If the splitting () is a regular splitting of the matrix A and A−1 ≥ 0, then ρ(T) < 1 and T is a convergent matrix.

Alternatively one can think of the dual of the tautological line bundle as the Serre twist sheaf O(1) on projective space, and use it to twist the structure sheaf OV any number of times, say k times, obtaining a sheaf OV(k). Then V is called k-normal if the global sections of O(k) map surjectively to those of OV(k), for a given k, and if V is 1-normal it is called linearly normal. A non-singular variety is projectively normal if and only if it is k-normal for all k ≥ 1.

A cubic curve may have a singular point, in which case it has a parametrization in terms of a projective line. Otherwise a non-singular cubic curve is known to have nine points of inflection, over an algebraically closed field such as the complex numbers. This can be shown by taking the homogeneous version of the Hessian matrix, which defines again a cubic, and intersecting it with C; the intersections are then counted by Bézout's theorem. However, only three of these points may be real, so that the others cannot be seen in the real projective plane by drawing the curve.

Using the intrinsic concept of tangent space, points P on an algebraic curve C are classified as smooth (synonymous: non- singular), or else singular. Given n−1 homogeneous polynomials in n+1 variables, we may find the Jacobian matrix as the (n−1)×(n+1) matrix of the partial derivatives. If the rank of this matrix is n−1, then the polynomials define an algebraic curve (otherwise they define an algebraic variety of higher dimension). If the rank remains n−1 when the Jacobian matrix is evaluated at a point P on the curve, then the point is a smooth or regular point; otherwise it is a singular point.

But this convergence may only be guaranteed within infinite time. In TSM, a nonlinear term is introduced in the sliding surface design so that the manifold is formulated as an attractor. After the sliding surface is intercepted, the trajectory is attracted within the manifold and converges to the origin following a power rule. There are some variations of the TSM including: Non-singular TSM, Fast TSM, Terminal sliding mode also has been widely applied to nonlinear process control, for example, rigid robot control etc.. Several open questions still remain on the mathematical treatment of the system's behavior at the origin since it is non- Lipschitz.

It is possible to derive, from the covariant formulation of full quantum theory (Spinfoam) the correct relation between energy and area (1st law), the Unruh temperature and the distribution that yields Hawking entropy. The calculation makes use of the notion of dynamical horizon and is done for non-extremal black holes. A recent success of the theory in this direction is the computation of the entropy of all non singular black holes directly from theory and independent of Immirzi parameter. The result is the expected formula S=A/4, where S is the entropy and A the area of the black hole, derived by Bekenstein and Hawking on heuristic grounds.

In algebraic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic p > 0 with unusually large endomorphism rings. Elliptic curves over such fields which are not supersingular are called ordinary and these two classes of elliptic curves behave fundamentally differently in many aspects. discovered supersingular elliptic curves during his work on the Riemann hypothesis for elliptic curves by observing that in positive characteristic elliptic curves could have endomorphism rings of unusually large rank 4, and developed their basic theory. The term "supersingular" has nothing to do with singular points of curves, and all supersingular elliptic curves are non-singular.

Mbonye held a postdoctoral position at the University of Michigan and was a professor at the Rochester Institute of Technology until 2011. He also carried out research at the NASA Goddard Space Flight Center in Greenbelt, Maryland as a National Research Council Senior Research Associate. Mbonye has made important contributions to theoretical physics, including the Mbonye-Kazanas model of non-singular black holes, cosmology with interacting dark energy, and models for the origin of the M-sigma relation. Mbonye is currently Vice-Rector in Charge of Academic Affairs at the National University of Rwanda, and RIT-NUR Research Professor at the Rochester Institute of Technology.

The largest class, in some sense, was that of surfaces of general type: those for which the consideration of differential forms provides linear systems that are large enough to make all the geometry visible. The work of the Italian school had provided enough insight to recognise the other main birational classes. Rational surfaces and more generally ruled surfaces (these include quadrics and cubic surfaces in projective 3-space) have the simplest geometry. Quartic surfaces in 3-spaces are now classified (when non-singular) as cases of K3 surfaces; the classical approach was to look at the Kummer surfaces, which are singular at 16 points.

Kaluza's approach to unification was to embed space-time into a five- dimensional cylindrical world, consisting of four space dimensions and one time dimension. Unlike Weyl's approach, Riemannian geometry was maintained, and the extra dimension allowed for the incorporation of the electromagnetic field vector into the geometry. Despite the relative mathematical elegance of this approach, in collaboration with Einstein and Einstein's aide Grommer it was determined that this theory did not admit a non-singular, static, spherically symmetric solution. This theory did have some influence on Einstein's later work and was further developed later by Klein in an attempt to incorporate relativity into quantum theory, in what is now known as Kaluza–Klein theory.

Structural stability of non-singular smooth vector fields on the torus can be investigated using the theory developed by Poincaré and Arnaud Denjoy. Using the Poincaré recurrence map, the question is reduced to determining structural stability of diffeomorphisms of the circle. As a consequence of the Denjoy theorem, an orientation preserving C2 diffeomorphism ƒ of the circle is structurally stable if and only if its rotation number is rational, ρ(ƒ) = p/q, and the periodic trajectories, which all have period q, are non-degenerate: the Jacobian of ƒq at the periodic points is different from 1, see circle map. Dmitri Anosov discovered that hyperbolic automorphisms of the torus, such as the Arnold's cat map, are structurally stable.

Specifically, it comprises two masses (the pendulum, mass m and counterweight, mass M) connected by an inextensible, massless string suspended on two frictionless pulleys of zero radius such that the pendulum can swing freely around its pulley without colliding with the counterweight. The conventional Atwood's machine allows only "runaway" solutions (i.e. either the pendulum or counterweight eventually collides with its pulley), except for M=m. However, the swinging Atwood's machine with M>m has a large parameter space of conditions that lead to a variety of motions that can be classified as terminating or non-terminating, periodic, quasiperiodic or chaotic, bounded or unbounded, singular or non-singular due to the pendulum's reactive centrifugal force counteracting the counterweight's weight.

If V is finite-dimensional then, relative to some basis for V, a bilinear form is degenerate if and only if the determinant of the associated matrix is zero. Likewise, a nondegenerate form is one for which the determinant of the associated matrix is non-zero (the matrix is non-singular). These statements are independent of the chosen basis. For a module over a commutative ring, a unimodular form is one for which the determinant of the associate matrix is a unit (for example 1), hence the term; note that a form whose matrix is non- zero but not a unit will be nondegenerate but not unimodular, for example over the integers.

In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1-forms. Given a complex manifold M, a differential of the first kind ω is therefore the same thing as a 1-form that is everywhere holomorphic; on an algebraic variety V that is non-singular it would be a global section of the coherent sheaf Ω1 of Kähler differentials. In either case the definition has its origins in the theory of abelian integrals. The dimension of the space of differentials of the first kind, by means of this identification, is the Hodge number :h1,0.

In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\\!\Omega^n = \omega, which is the nth exterior power of the cotangent bundle Ω on V. Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V. This is the dualising object for Serre duality on V. It may equally well be considered as an invertible sheaf. The canonical class is the divisor class of a Cartier divisor K on V giving rise to the canonical bundle -- it is an equivalence class for linear equivalence on V, and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor −K with K canonical.

An algebraic plane curve is a curve in an affine or projective plane given by one polynomial equation (or , where F is a homogeneous polynomial, in the projective case.) Algebraic curves have been studied extensively since the 18th century. Every algebraic plane curve has a degree, the degree of the defining equation, which is equal, in case of an algebraically closed field, to the number of intersections of the curve with a line in general position. For example, the circle given by the equation has degree 2. The non-singular plane algebraic curves of degree 2 are called conic sections, and their projective completion are all isomorphic to the projective completion of the circle (that is the projective curve of equation ).

For example, ramification is a phenomenon of codimension 1 (in the geometry of complex manifolds, reflecting as for Riemann surfaces that ramify at single points that it happens in real codimension two). A classical result, Zariski-Nagata purity of Masayoshi Nagata and Oscar Zariski, called also purity of the branch locus, proves that on a non-singular algebraic variety a branch locus, namely the set of points at which a morphism ramifies, must be made up purely of codimension 1 subvarieties (a Weil divisor). There have been numerous extensions of this result into theorems of commutative algebra and scheme theory, establishing purity of the branch locus in the sense of description of the restrictions on the possible "open subsets of failure" to be an étale morphism.

For an example more geometrical in nature, take the ring R = {f/g : f, g polynomials in R[X] and g(0) ≠ 0}, considered as a subring of the field of rational functions R(X) in the variable X. R can be identified with the ring of all real-valued rational functions defined (i.e. finite) in a neighborhood of 0 on the real axis (with the neighborhood depending on the function). It is a discrete valuation ring; the "unique" irreducible element is X and the valuation assigns to each function f the order (possibly 0) of the zero of f at 0. This example provides the template for studying general algebraic curves near non-singular points, the algebraic curve in this case being the real line.

If all extrema of are isolated, then an inflection point is a point on the graph of at which the tangent crosses the curve. A falling point of inflection is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A rising point of inflection is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing. For an algebraic curve, a non singular point is an inflection point if and only if the intersection number of the tangent line and the curve (at the point of tangency) is greater than 2.

The variety V in its projective embedding is projectively normal if R is integrally closed. This condition implies that V is a normal variety, but not conversely: the property of projective normality is not independent of the projective embedding, as is shown by the example of a rational quartic curve in three dimensions.Robin Hartshorne, Algebraic Geometry (1977), p. 23. Another equivalent condition is in terms of the linear system of divisors on V cut out by the dual of the tautological line bundle on projective space, and its d-th powers for d = 1, 2, 3, ... ; when V is non- singular, it is projectively normal if and only if each such linear system is a complete linear system.Hartshorne, p. 159.

The rational maps play a role analogous to the meromorphic functions of complex analysis, and indeed in the case of compact Riemann surfaces the two concepts coincide. If V is now taken to be of dimension 1, we get a picture of a typical algebraic curve C presented 'over' P1(K). Assuming C is non-singular (which is no loss of generality starting with K(C)), it can be shown that such a rational map from C to P1(K) will in fact be everywhere defined. (That is not the case if there are singularities, since for example a double point where a curve crosses itself may give an indeterminate result after a rational map.) This gives a picture in which the main geometric feature is ramification.

In the fall of 1988 and in the academic year 1992–1993 he was a visiting scholar at the Institute for Advanced Study. In 1986 he was awarded the Fields Medal at the ICM at Berkeley for proving the Tate conjecture for abelian varieties over number fields, the Shafarevich conjecture for abelian varieties over number fields and the Mordell conjecture, which states that any non-singular projective curve of genus g > 1 defined over a number field K contains only finitely many K-rational points. As a Fields Medalist he gave an ICM plenary talk Recent progress in arithmetic algebraic geometry. In 1994 as an ICM invited speaker in Zurich he gave a talk Mumford-Stabilität in der algebraischen Geometrie.

For a real-valued smooth function f : M → R on a differentiable manifold M, the points where the differential of f vanishes are called critical points of f and their images under f are called critical values. If at a critical point b, the matrix of second partial derivatives (the Hessian matrix) is non-singular, then b is called a non-degenerate critical point; if the Hessian is singular then b is a degenerate critical point. For the functions :f(x)=a + b x+ c x^2+d x^3+\cdots from R to R, f has a critical point at the origin if b = 0, which is non-degenerate if c ≠ 0 (i.e. f is of the form a + cx2 + ...) and degenerate if c = 0 (i.e.

In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. More specifically, the conjecture states that certain de Rham cohomology classes are algebraic; that is, they are sums of Poincaré duals of the homology classes of subvarieties. It was formulated by the Scottish mathematician William Vallance Douglas Hodge as a result of a work in between 1930 and 1940 to enrich the description of de Rham cohomology to include extra structure that is present in the case of complex algebraic varieties. It received little attention before Hodge presented it in an address during the 1950 International Congress of Mathematicians, held in Cambridge, Massachusetts.

He proved that every non-singular cubic form over the rational numbers in at least ten variables represents 0.D. R. Heath-Brown, Cubic forms in ten variables, Proceedings of the London Mathematical Society, 47(3), pages 225–257 (1983) Heath-Brown also showed that Linnik's constant is less than or equal to 5.5.D. R. Heath-Brown, Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression, Proceedings of the London Mathematical Society, 64(3), pages 265–338 (1992) More recently, Heath-Brown is known for his pioneering work on the so-called determinant method. Using this method he was able to prove a conjecture of Serre in the four variable case in 2002.

In coherent sheaf theory, pushing to the limit of what could be done with Serre duality without the assumption of a non-singular scheme, the need to take a whole complex of sheaves in place of a single dualizing sheaf became apparent. In fact the Cohen–Macaulay ring condition, a weakening of non-singularity, corresponds to the existence of a single dualizing sheaf; and this is far from the general case. From the top-down intellectual position, always assumed by Grothendieck, this signified a need to reformulate. With it came the idea that the 'real' tensor product and Hom functors would be those existing on the derived level; with respect to those, Tor and Ext become more like computational devices.

American Physical Society which can potentially resolve the Schwarzschild singularity for mini black holes, yielding a non-singular compact object without event horizon, and cosmological singularities. He has also conjectured with Koshelev that astrophysical black hole has no curvature singularity and devoid of an event horizon,Alexey Koshelev and Anupam Mazumdar (31 October 2017)massive compact objects without event horizon exist in infinite derivative gravity? American Physical Society in infinite derivative theories of gravity, because the scale of non-locality in gravitational interaction can engulf the gravitational radius of the compact object. At time scales and at distances below the effective scale of non-locality the gravitational interaction weakens sufficiently enough that a finite pressure from normal matter satisfying null, strong and weak energy conditions can avoid forming blackhole with event horizon and cosmological singularities.

In differential geometry, a Hodge cycle or Hodge class is a particular kind of homology class defined on a complex algebraic variety V, or more generally on a Kähler manifold. A homology class x in a homology group :H_k(V, \Complex) = H where V is a non-singular complex algebraic variety or Kähler manifold is a Hodge cycle, provided it satisfies two conditions. Firstly, k is an even integer 2p, and in the direct sum decomposition of H shown to exist in Hodge theory, x is purely of type (p,p). Secondly, x is a rational class, in the sense that it lies in the image of the abelian group homomorphism :H_k(V, \Q) \to H defined in algebraic topology (as a special case of the universal coefficient theorem).

In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory. The historical roots of the theory lie in the idea of the adjoint linear system of a linear system of divisors in classical algebraic geometry. This was re-expressed, with the advent of sheaf theory, in a way that made an analogy with Poincaré duality more apparent. Then according to a general principle, Grothendieck's relative point of view, the theory of Jean-Pierre Serre was extended to a proper morphism; Serre duality was recovered as the case of the morphism of a non-singular projective variety (or complete variety) to a point.

Let A be a symmetric square matrix of order n with real entries. Any non-singular matrix S of the same size is said to transform A into another symmetric matrix , also of order n, where ST is the transpose of S. It is also said that matrices A and B are congruent. If A is the coefficient matrix of some quadratic form of Rn, then B is the matrix for the same form after the change of basis defined by S. A symmetric matrix A can always be transformed in this way into a diagonal matrix D which has only entries 0, +1 and −1 along the diagonal. Sylvester's law of inertia states that the number of diagonal entries of each kind is an invariant of A, i.e.

In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex manifolds are defined by homogeneous polynomials. Kunihiko Kodaira's result is that for a compact Kähler manifold M, with a Hodge metric, meaning that the cohomology class in degree 2 defined by the Kähler form ω is an integral cohomology class, there is a complex-analytic embedding of M into complex projective space of some high enough dimension N. The fact that M embeds as an algebraic variety follows from its compactness by Chow's theorem. A Kähler manifold with a Hodge metric is occasionally called a Hodge manifold (named after W. V. D. Hodge), so Kodaira's results states that Hodge manifolds are projective.

The eigenvectors are used as the basis when representing the linear transformation as Λ. Conversely, suppose a matrix A is diagonalizable. Let P be a non-singular square matrix such that P−1AP is some diagonal matrix D. Left multiplying both by P, AP = PD. Each column of P must therefore be an eigenvector of A whose eigenvalue is the corresponding diagonal element of D. Since the columns of P must be linearly independent for P to be invertible, there exist n linearly independent eigenvectors of A. It then follows that the eigenvectors of A form a basis if and only if A is diagonalizable. A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvectors generalizes to generalized eigenvectors and the diagonal matrix of eigenvalues generalizes to the Jordan normal form.

In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) C is determined by its Jacobian variety J(C), when the latter is given in the form of a principally polarized abelian variety. In other words, the complex torus J(C), with certain 'markings', is enough to recover C. The same statement holds over any algebraically closed field.James S. Milne, Jacobian Varieties, Theorem 12.1 in From more precise information on the constructed isomorphism of the curves it follows that if the canonically principally polarized Jacobian varieties of curves of genus \geq 2 are k-isomorphic for k any perfect field, so are the curves.James S. Milne, Jacobian Varieties, Corollary 12.2 in This result has had many important extensions.

A catalog of elliptic curves. Region shown is [−3,3]2 (For (a, b) = (0, 0) the function is not smooth and therefore not an elliptic curve.) In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. Every elliptic curve over a field of characteristic different from 2 and 3 can be described as a plane algebraic curve given by an equation of the form :y^2 = x^3 + ax + b. The curve is required to be non-singular, which means that the curve has no cusps or self- intersections. (This is equivalent to the condition 4a^3+27b^2 e0.) It is always understood that the curve is really sitting in the projective plane, with the point O being the unique point at infinity.

Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic curves; see below.) An elliptic curve is an abelian variety – that is, it has a group law defined algebraically, with respect to which it is an abelian group – and O serves as the identity element. If y2 = P(x), where P is any polynomial of degree three in x with no repeated roots, the solution set is a nonsingular plane curve of genus one, an elliptic curve. If P has degree four and is square-free this equation again describes a plane curve of genus one; however, it has no natural choice of identity element.

In general the ℓ-adic cohomology groups of a variety tend to have similar properties to the singular cohomology groups of complex varieties, except that they are modules over the ℓ-adic integers (or numbers) rather than the integers (or rationals). They satisfy a form of Poincaré duality on non-singular projective varieties, and the ℓ-adic cohomology groups of a "reduction mod p" of a complex variety tend to have the same rank as the singular cohomology groups. A Künneth formula also holds. For example, the first cohomology group of a complex elliptic curve is a free module of rank 2 over the integers, while the first ℓ-adic cohomology group of an elliptic curve over a finite field is a free module of rank 2 over the ℓ-adic integers, provided ℓ is not the characteristic of the field concerned, and is dual to its Tate module.

In mathematics, the pluricanonical ring of an algebraic variety V (which is non-singular), or of a complex manifold, is the graded ring :R(V,K)=R(V,K_V) \, of sections of powers of the canonical bundle K. Its nth graded component (for n\geq 0) is: :R_n := H^0(V, K^n),\ that is, the space of sections of the n-th tensor product Kn of the canonical bundle K. The 0th graded component R_0 is sections of the trivial bundle, and is one-dimensional as V is projective. The projective variety defined by this graded ring is called the canonical model of V, and the dimension of the canonical model is called the Kodaira dimension of V. One can define an analogous ring for any line bundle L over V; the analogous dimension is called the Iitaka dimension. A line bundle is called big if the Iitaka dimension equals the dimension of the variety.

As Freudenthal notes :...up to Hilbert, there is no other example for such a direct derivation of the algebraic laws from geometric axioms as found in von Staudt's Beiträge. Another affirmation of von Staudt's work with the harmonic conjugates comes in the form of a theorem: :The only one-to-one correspondence between the real points on a line which preserves the harmonic relation between four points is a non-singular projectivity.Dirk Struik (1953) Lectures on Analytic and Projective Geometry, p 22, "theorem of von Staudt" The algebra of throws was described as "projective arithmetic" in The Four Pillars of Geometry (2005).John Stillwell, Sheldon Axler, Ken A. Ribet (2005) The Four Pillars of Geometry, page 128, Springer: Undergraduate Texts in Mathematics In a section called "Projective arithmetic", he says :The real difficulty is that the construction of a + b , for example, is different from the construction of b + a, so it is a "coincidence" if a + b = b + a.

Every algebraic curve C of genus g ≥ 1 is associated with an abelian variety J of dimension g, by means of an analytic map of C into J. As a torus, J carries a commutative group structure, and the image of C generates J as a group. More accurately, J is covered by C: J is covered by Cg: any point in J comes from a g-tuple of points in C. The study of differential forms on C, which give rise to the abelian integrals with which the theory started, can be derived from the simpler, translation- invariant theory of differentials on J. The abelian variety J is called the Jacobian variety of C, for any non-singular curve C over the complex numbers. From the point of view of birational geometry, its function field is the fixed field of the symmetric group on g letters acting on the function field of Cg.

Classical results of Bernhard Riemann describe Θ in another way, in the case that A is the Jacobian variety J of an algebraic curve (compact Riemann surface) C. There is, for a choice of base point P on C, a standard mapping of C to J, by means of the interpretation of J as the linear equivalence classes of divisors on C of degree 0. That is, Q on C maps to the class of Q − P. Then since J is an algebraic group, C may be added to itself k times on J, giving rise to subvarieties Wk. If g is the genus of C, Riemann proved that Θ is a translate on J of Wg − 1. He also described which points on Wg − 1 are non-singular: they correspond to the effective divisors D of degree g − 1 with no associated meromorphic functions other than constants. In more classical language, these D do not move in a linear system of divisors on C, in the sense that they do not dominate the polar divisor of a non constant function.

Her work unifies and extends these two remarkable results. It shows, by methods of Baire category, that the seemingly isolated examples of non-singular transformations first discovered by Ornstein and later by Chacón, were in fact the typical case. Beginning in the early 1980s Bellow began a series of papers that brought about a revival of that area of ergodic theory dealing with limit theorems and the delicate question of pointwise a.e. convergence. This was accomplished by exploiting the interplay with probability and harmonic analysis, in the modern context (the Central limit theorem, transference principles, square functions and other singular integral techniques are now part of the daily arsenal of people working in this area of ergodic theory) and by attracting a number of talented mathematicians who were very active in this area. One of the two problems that she raised at the Oberwolfach meeting on "Measure Theory" in 1981, was the question of the validity, for f in L_1, of the pointwise ergodic theorem along the ‘sequence of squares’, and along the ‘sequence of primes’ (A similar question was raised independently, a year later, by Hillel Furstenberg).