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"let be" Antonyms

324 Sentences With "let be"

How to use let be in a sentence? Find typical usage patterns (collocations)/phrases/context for "let be" and check conjugation/comparative form for "let be". Mastering all the usages of "let be" from sentence examples published by news publications.

It's up to you which incidents you confront and which you let be.
But let be real — if you can mooch of someone else's account, you totally should.
We will call for separation even from this country and let be what will be.
And let be honest, seeing me in tight white shorts is not appropriate for 5 p.m.
They are very guarded by what they let be used of his stuff, especially after his death.
We are working very hard for deal and let be clear that is by far our preferred option.
"It's something we need to remember and not let be drift off into the annals of history where it's something that we don't really contribute to the person who's now running our country," she said.
According to the Post, investigators seem "particularly focused on what data Facebook let be collected from its platform and under what conditions" — and what Facebook said about it, not only when it was happening but also at the congressional hearings.
Nor is it any help that the production from the director-designer Stewart Laing (a 1997 Tony Award winner for his sets for the Broadway musical "Titanic") backs leading player Fiona Glascott into defiantly strident mode: this is one quilting bee that I was happy, sorry to say, to let be.
But really, in the end, today's harsh realities are not all that surprising for some of us — for people of color, or for people from marginalized communities — who have long since given up on being shocked or dismayed by the news, by what this or that administration will allow, what this or that police department will excuse, who will be exonerated, what this or that fellow American is willing to let be, either by contribution or complicity.
Let be any linear functional on (not necessarily continuous). Fix . Let be the set :} and let be the Minkowski functional of . Then : for all .
Pompeiu's construction is described here. Let denote the real cube root of the real number . Let be an enumeration of the rational numbers in the unit interval . Let be positive real numbers with .
Let be in with . Then is in ; the -span of .
The characters discussed in this section are assumed to be complex-valued. Let be a subgroup of the finite group . Given a character of , let denote its restriction to . Let be a character of .
Let be the number of processing elements (PE), numbered from to .
Given a right triangle with hypotenuse , construct a circle Ω whose diameter is . Let be the center of Ω. Let be the intersection of Ω and the ray . By Thales's theorem, ∠ is right. But then must equal .
Let be a topological vector space (TVS) (we do not assume that is Hausdorff or locally convex) and let be a linear functional on . The following are equivalent: 1. is continuous. 2. is continuous at the origin. 3.
Let A be a finite-dimensional complex unital Jordan algebra which is semisimple, i.e. the trace form Tr L(ab) is non-degenerate. Let be the quotient of by the equivalence relation. Let be the subset of X of classes .
In this section it is assumed that every set can be endowed with a group structure . Let be a set. Let be the Hartogs number of . This is the least cardinal number such that there is no injection from into .
Let be a weighted directed graph with vertex set and edge set (figure A); let be a designated source vertex in , and let be a designated destination vertex. Let each edge in , from vertex to vertex , have a non-negative cost . Define to be the cost of the shortest path to vertex from vertex in the shortest path tree rooted at (figure C). Note: Node and Vertex are often used interchangeably.
The infinite- dimensional version of the Frobenius theorem also holds on Banach manifolds. The statement is essentially the same as the finite-dimensional version. Let be a Banach manifold of class at least C2. Let be a subbundle of the tangent bundle of .
Let be a subspace of that generates as an algebra and which is minimal with respect to this property. Let be an orthonormal basis of with respect to . Then orthonormality implies that: :e_i^2 =-1, \quad e_i e_j = - e_j e_i. If , then is isomorphic to .
Let be a continuous real-valued function defined on a closed interval . Let be the function defined, for all in , by :F(x) = \int_a^x f(t)\, dt. Then, is continuous on , differentiable on the open interval , and :F'(x) = f(x) for all in .
Let be the collection of all sets in together with their complements (taken in ). 3. Let be the collection of all possible finite intersections of sets in .Since , there is some such that its complement also belongs to . The intersection of these two sets implies that .
Let be a closed interval in , and let be a collection of open sets that covers . Then the Heine–Borel theorem states that some finite subcollection of covers as well. This statement can be proved by considering the set :. This set must have a least upper bound .
Pick some mathematical object that has an underlying set, for instance a group, ring, vector space, etc. For any subset of , let be the smallest subobject of that contains , i.e. the subgroup, subring or subspace generated by . For any subobject of , let be the underlying set of .
Paul Erdős gave a proof that also relies on the fundamental theorem of arithmetic. Every positive integer has a unique factorization into a square-free number and a square number . For example, . Let be a positive integer, and let be the number of primes less than or equal to .
To convert quote notation into standard notation, the following algorithm can be used. :Let and be sequences of digits, as in x'y. :Let be the digit 1 followed by a sequence of zeros of the same length as . :Let be the largest valued digit (one less than the base).
Formally, let be a set and let be a family of subsets of . Then is called a topology on if: # Both the empty set and are elements of . # Any union of elements of is an element of . # Any intersection of finitely many elements of is an element of .
Let be a fibered manifold. A generalized connection on is a section , where is the jet manifold of .
Let be the set of all minimal Cauchy filters on and let be the map defined by sending to the neighborhood filter of in . We endow with the following vector space structure. Given and a scalar , let (resp. ) denote the unique minimal Cauchy filter contained in the filter generated by } (resp. }).
The motivating example comes from Galois theory: suppose is a field extension. Let be the set of all subfields of that contain , ordered by inclusion ⊆. If is such a subfield, write for the group of field automorphisms of that hold fixed. Let be the set of subgroups of , ordered by inclusion ⊆.
Let be a sequence of independent, symmetric, and }-valued random variables. For every let be the σ-algebra generated by and define when is the first random variable taking the value . Note that , hence by the ratio test. The assumptions (), () and (), hence () and () with , (), (), and () hold, hence also (), and () and Wald's equation applies.
Moreover, since . So commutes with and hence . Thus has inverse . Now let be the mutation of A defined by a.
Suppose and are integers. Let be a ratio given in its lowest terms. Draw the arcs and with centre . Join .
Linearisation of the expression. Each letter of the alphabet appearing in the expression is renamed, so that each letter occurs at most once in the new expression e'. Glushkov's construction essentially relies on the fact that e' represents a local language L(e'). Let be the old alphabet and let be the new one.
Let be a measure- preserving transformation on a measure space , with . Then is ergodic if for every in with , either or .
Let be a binary relation on set . The transitive extension of , denoted , is the smallest binary relation on such that contains , and if and then . For example, suppose is a set of towns, some of which are connected by roads. Let be the relation on towns where if there is a road directly linking town and town .
Furthermore, is complementary to in . Let be given by and by . It is clear that . Thus is a split extension of by .
Throughout, let be a partially ordered set and let . By using instead of in the above definition, one defines the least element of .
One may encounter, in many textbooks, sentences such as "Let be a function ...". This is an abuse of notation, as the name of the function is , and usually denotes the value of the function for the element of its domain. The correct phrase would be "Let be a function of the variable ..." or "Let be a function ..." This abuse of notation is widely used, as it simplifies the formulation, and the systematic use of a correct notation quickly becomes pedantic. A similar abuse of notation occurs in sentences such as "Let us consider the function ...", when in fact is not a function.
Let be a covering map where both X and C are path-connected. Let be a basepoint of X and let be one of its pre-images in C, that is . There is an induced homomorphism of fundamental groups which is injective by the lifting property of coverings. Specifically if γ is a closed loop at c such that , that is is null-homotopic in X, then consider a null-homotopy of as a map from the 2-disc D2 to X such that the restriction of f to the boundary S1 of D2 is equal to .
Let be a ring and let be a simple right -module. If is a non-zero element of , (where is the cyclic submodule of generated by ). Therefore, if are non-zero elements of , there is an element of that induces an endomorphism of transforming to . The natural question now is whether this can be generalized to arbitrary (finite) tuples of elements.
Substituting for itself results in :P = P(X), explaining why the sentences "Let be a polynomial" and "Let be a polynomial" are equivalent. The polynomial function defined by a polynomial is the function from into that is defined by x\mapsto P(x). If is an infinite field, two different polynomials define different polynomial functions, but this property is false for finite fields.
The simplest root-finding algorithm is the bisection method. Let be a continuous function, for which one knows an interval such that and have opposite signs (a bracket). Let be the middle of the interval (the midpoint or the point that bisects the interval). Then either and , or and have opposite signs, and one has divided by two the size of the interval.
Affine normal line for the curve at The affine normal vector field for a curve in the plane has a nice geometrical interpretation. Let be an open interval and let be a smooth parametrisation of a plane curve. We assume that γ(I) is a non-degenerate curve (in the sense of Nomizu and Sasaki), i.e. is without inflexion points.
Let be an integrable function. The span of translations = is dense in if and only if the Fourier transform of has no real zeros.
Let be a scheme with structure sheaf If: :(1) is quasi-compact, and :(2) for every quasi-coherent ideal sheaf of -modules, , then is affine..
In affine d-space, two flats of any dimension may be parallel. However, in projective space, parallelism does not exist; two flats must either intersect or be skew. Let be the set of points on an i-flat, and let be the set of points on a j-flat. In projective d-space, if then the intersection of and must contain a (i+j−d)-flat.
Diagonal triangle of quadrangle on conic. Polars of diagonal points are colored the same as the points. The theory of poles and polars of a conic in a projective plane can be developed without the use of coordinates and other metric concepts. Let be a conic in where is a field not of characteristic two, and let be a point of this plane not on .
Let be a differentiable function, and let be its derivative. The derivative of (if it has one) is written and is called the second derivative of . Similarly, the derivative of the second derivative, if it exists, is written and is called the third derivative of . Continuing this process, one can define, if it exists, the th derivative as the derivative of the th derivative.
Let be an unknown, deterministic parameter, and let be a random variable, interpreted as a measurement of θ. Suppose the probability density function of X is given by p(x; θ). It is assumed that p(x; θ) is well-defined and that for all values of x and θ. Suppose δ(X) is an unbiased estimate of an arbitrary scalar function of θ, i.e.
Specifically, let be an arbitrary cyclic quadrilateral and let , , , be the incenters of the triangles , , , . Then the quadrilateral formed by , , , is a rectangle. Note that this theorem is easily extended to prove the Japanese theorem for cyclic polygons. To prove the quadrilateral case, simply construct the parallelogram tangent to the corners of the constructed rectangle, with sides parallel to the diagonals of the quadrilateral.
Let be a subset of . Define by :\mu_B(A)=\mu(A\cap B). One can check directly from the definitions that is another outer measure on .
Let be an -dimensional vector space over a field . The Grassmannian is the set of all -dimensional linear subspaces of . The Grassmannian is also denoted or .
A proof runs as follows: let be the convex hull of }. Note that is an absorbing disk in , and call its Minkowski functional . Then on and on .
To distinguish the degenerate cases from the non-degenerate cases (including the empty set with the latter) using matrix notation, let be the determinant of the 3×3 matrix of the conic section—that is, ; and let be the discriminant. Then the conic section is non-degenerate if and only if . If we have a point when , two parallel lines (possibly coinciding) when , or two intersecting lines when .
Let be a non-trivial (i.e. }) real or complex vector space and let be the translation-invariant trivial metric on defined by and for all such that . The topology that induces on is the discrete topology, which makes into a commutative topological group under addition but does not form a vector topology on because is disconnected but every vector topology is connected. What fails is that scalar multiplication isn't continuous on .
De Witte, 1959, p.294, col.2. Now let be a point on close to , and let be a point on close to . Then, by the construction, :(i) the time taken for a secondary wavefront from to reach has at most a second-order dependence on the displacement , and :(ii) the time taken for a secondary wavefront to reach from has at most a second-order dependence on the displacement .
We will first prove: :Proposition. Let be a compact Lie group. There exists a contractible space on which acts freely. The projection is a -principal fibre bundle. Proof.
Let be 4, then must be 124, which is impossible since by hypothesis is not amongst the numbers 1, 2, 4, 8, 16, 31, 62, 124 or 248.
There is a general procedure, the substitution rule, that allows additional tautologies to be constructed from a given tautology (Kleene 1967 sec. 3). Suppose that is a tautology and for each propositional variable in a fixed sentence is chosen. Then the sentence obtained by replacing each variable in with the corresponding sentence is also a tautology. For example, let be the tautology :(A \land B) \lor \lnot A \lor \lnot B. Let be C \lor D and let be C \to E. It follows from the substitution rule that the sentence :((C \lor D) \land (C \to E)) \lor \lnot (C \lor D) \lor \lnot (C \to E) is a tautology, too.
The proof also relies on the following theorem proven in p. 185: :Theorem. Let be a simple right -module, , and a finite set. Write for the annihilator of in .
Let be a Lie group acting smoothly on a manifold . If the action is a locally free action or free action, then the orbits of define a foliation of .
This proof is inspired by . Let be a simple undirected graph. We proceed by induction on , the number of edges. If the graph is empty, the theorem trivially holds.
Let be algebraic varieties. We say and are isomorphic, and write , if there are regular maps and such that the compositions and are the identity maps on and respectively.
Let be a random variable; we assume for the sake of presentation that is finite, that is, takes on only finitely many values . Let be an event, then the conditional probability of given is defined as the random variable, written , that takes on the value : P(A\mid X=x) whenever :X=x. More formally, :P(A\mid X)(\omega)=P(A\mid X=X(\omega)) . The conditional probability is a function of .
Given a prime number and prime power with positive integers and such that , a primitive narrow-sense BCH code over the finite field (or Galois field) with code length and distance at least is constructed by the following method. Let be a primitive element of . For any positive integer , let be the minimal polynomial with coefficients in of . The generator polynomial of the BCH code is defined as the least common multiple .
Suppose is provable. Let be the Gödel number of a proof of . Then, as seen earlier, the formula is provable. Proving both and violates the consistency of the formal theory.
Let be a simple graph and let consist of all 2-element subsets of . Then is the complement of ,. Electronic edition, page 4. where is the relative complement of in .
Let be a continuously differentiable function. Write for the solid of revolution of the graph about the -axis. If the surface area of is finite, then so is the volume.
Ellipsoidal coordinates The parametric latitude can also be extended to a three-dimensional coordinate system. For a point not on the reference ellipsoid (semi-axes and ) construct an auxiliary ellipsoid which is confocal (same foci , ) with the reference ellipsoid: the necessary condition is that the product of semi-major axis and eccentricity is the same for both ellipsoids. Let be the semi-minor axis () of the auxiliary ellipsoid. Further let be the parametric latitude of on the auxiliary ellipsoid.
The Moschovakis coding lemma is a lemma from descriptive set theory involving sets of real numbers under the axiom of determinacy (the principle — incompatible with choice — that every two-player integer game is determined). The lemma was developed and named after the mathematician Yiannis N. Moschovakis. The lemma may be expressed generally as follows: :Let be a non- selfdual pointclass closed under real quantification and , and a -well-founded relation on of rank . Let be such that .
There are general conjectures due to Shouwu Zhang and others concerning subvarieties that contain infinitely many periodic points or that intersect an orbit in infinitely many points. These are dynamical analogues of, respectively, the Manin-Mumford conjecture, proven by Raynaud, and the Mordell–Lang conjecture, proven by Faltings. The following conjectures illustrate the general theory in the case that the subvariety is a curve. :Conjecture. Let be a morphism and let be an irreducible algebraic curve.
This allows us to not even have to assign a symbol to a monoid's multiplication operation. Moreover, when we use this juxtaposition notation then we will automatically assume that the monoid's identity element is denoted by . :Definition: Let be a monoid with identity element whose operation is denoted by juxtaposition and let be a set. A monoid action of on is a map , which we will also denote by juxtaposition, such that and for all and all .
Formally, let be any graph, and let be any subset of vertices of . Then the induced subgraph is the graph whose vertex set is and whose edge set consists of all of the edges in that have both endpoints in .. The same definition works for undirected graphs, directed graphs, and even multigraphs. The induced subgraph may also be called the subgraph induced in by , or (if context makes the choice of unambiguous) the induced subgraph of .
If M admits an almost complex structure, it must be even-dimensional. This can be seen as follows. Suppose M is n-dimensional, and let be an almost complex structure. If then .
This scheme also uses special sequences of integers. Let be integers. We consider a sequence of pairwise coprime positive integers m_0 < ... < m_n such that m_0.m_{n-k+2}...m_n < m_1...m_k.
Let be a convex balanced neighborhood of 0 in a locally convex topological vector space and suppose is not an element of . Then there exists a continuous linear functional on such that : .
The tetrahedron is a 2-complex. The link of a vertex of a tetrahedron is the triangle. Let be a simplicial complex. The link of a vertex is the graph constructed as follows.
Let be an n-dimensional, non-degenerate symmetric bilinear space over a field with characteristic not equal to 2. Then, every element of the orthogonal group is a composition of at most n reflections.
More precisely, let be a point and a line. If is the point of intersection of and the unique line through that is perpendicular to , then is called the foot of this perpendicular through .
Extending this to get the anchor day, the procedure is often described as accumulating a running total in six steps, as follows: # Let be the year's last two digits. #If is odd, add 11.
Define an incidence structure with "points," the points of not on and "lines" the planes of meeting in a line of . Then is a translation affine plane of order . Let be the projective completion of .
Let be a (finite and simple) graph with vertices. We denote by the degree of a vertex in , i.e. the number of incident edges in to . Then, Ore's theorem states that if then is Hamiltonian.
The free category on a quiver can be described up to isomorphism by a universal property. Let : Quiv → Cat be the functor that takes a quiver to the free category on that quiver (as described above), let be the forgetful functor defined above, and let be any quiver. Then there is a graph homomorphism : → (()) and given any category D and any graph homomorphism : → , there is a unique functor : () → D such that ()∘=, i.e. the following diagram commutes: 300px The functor is left adjoint to the forgetful functor .
Here we consider the simplest case, i.e. manifolds of codimension one. Let be an n-dimensional manifold, and let ξ be a vector field on transverse to such that for all where ⊕ denotes the direct sum and Span the linear span. For a smooth manifold, say N, let Ψ(N) denote the module of smooth vector fields over N. Let be the standard covariant derivative on Rn+1 where We can decompose DXY into a component tangent to M and a transverse component, parallel to ξ.
An algebraic representation of (affine) translation planes can be obtained as follows: Let be a -dimensional vector space over a field . A spread of is a set of -dimensional subspaces of that partition the non-zero vectors of . The members of are called the components of the spread and if and are distinct components then . Let be the incidence structure whose points are the vectors of and whose lines are the cosets of components, that is, sets of the form where is a vector of and is a component of the spread .
Accordingly, let be a graph on vertices that is not Hamiltonian, and let be formed from by adding edges one at a time that do not create a Hamiltonian cycle, until no more edges can be added. Let and be any two non-adjacent vertices in . Then adding edge to would create at least one new Hamiltonian cycle, and the edges other than in such a cycle must form a Hamiltonian path in with and . For each index in the range , consider the two possible edges in from to and from to .
Qvist's theorem: to the proof in case of n odd Qvist's theorem: to the proof in case of n even ;Proof: (a) Let be the tangent to at point and let be the remaining points of this line. For each , the lines through partition into sets of cardinality 2 or 1 or 0. Since the number is even, for any point , there must exist at least one more tangent through that point. The total number of tangents is , hence, there are exactly two tangents through each , and one other.
There are several techniques for constructing configurations, generally starting from known configurations. Some of the simplest of these techniques construct symmetric () configurations. Any finite projective plane of order is an (( configuration. Let be a projective plane of order .
Trisection of the angle using marked ruler Let be the horizontal line in the adjacent diagram. Angle (left of point ) is the subject of trisection. First, a point is drawn at an angle's ray, one unit apart from .
Bezhanishvili et al. (2010) Finally, let be set-theoretic inclusion on the set of prime filters of and let . Then is a Priestley space. Moreover, is a lattice isomorphism from onto the lattice of all clopen up-sets of .
Let be the non-cyclic subgroup of called the Klein four-group. :. Let . Since both and are subgroups of , is also a subgroup of . From Lagrange's theorem, the order of must divide both and , the orders of and respectively.
To be concrete, let be a function, and fix a point in the domain of . is a point on the graph of the function. If is a number close to zero, then is a number close to . Therefore, is close to .
Prove that is a perfect square. # Fix some value that is a non-square positive integer. Assume there exist positive integers for which . # Let be positive integers for which and such that is minimized, and without loss of generality assume .
An endofunction is a function whose domain is equal to its codomain. A homomorphic endofunction is an endomorphism. Let be an arbitrary set. Among endofunctions on one finds permutations of and constant functions associating to every in the same element in .
In decimal, we have . :Let be a sequence of s of the same length as . Then the number represented by x'y is given by y-xz/w. As an example, we will take 12'345 and convert it to a standard notation.
Let be a topological field, namely a field with a topology such that addition, multiplication, and division are continuous. In most applications will be either the field of complex numbers or the field of real numbers with the familiar topologies.
Define a circulant numbering of a circulant graph to be a labeling of the vertices of the graph by the numbers from 0 to in such a way that, if some two vertices numbered and are adjacent, then every two vertices numbered and are adjacent. Equivalently, a circulant numbering is a numbering of the vertices for which the adjacency matrix of the graph is a circulant matrix. Let be an integer that is relatively prime to , and let be any integer. Then the linear function that takes a number to transforms a circulant numbering to another circulant numbering.
A very general comment of William LawvereWilliam Lawvere, Adjointness in foundations, Dialectica, 1969, available here. The notation is different nowadays; an easier introduction by Peter Smith in these lecture notes, which also attribute the concept to the article cited. is that syntax and semantics are adjoint: take to be the set of all logical theories (axiomatizations), and the power set of the set of all mathematical structures. For a theory , let be the set of all structures that satisfy the axioms ; for a set of mathematical structures , let be the minimum of the axiomatizations which approximate .
Let be a TVS (not necessarily Hausdorff or locally convex). :Definition: For any , the convex (resp. balanced, disked, closed convex, closed balanced, closed disked) hull of is the smallest subset of that has this property and contains . We denote the closure (resp.
From the maximality of , there exists such that . From the definition of this holds: : Let be the -path from with respect to . From (1), has to end in . But is missing in , so it has to end with an edge of color .
Thus, for any point not in oval , if is on any tangent to it is on exactly two tangents. (b) Let be a secant, } and }. Because is odd, through any , there passes at least one tangent . The total number of tangents is .
In mathematics, an H-matrix is a matrix whose comparison matrix is an M-matrix. It is useful in iterative methods. Definition: Let be a complex matrix. Then comparison matrix M(A) of complex matrix A is defined as where for all and for all .
A nonempty compact subset of the real numbers has a greatest element and a least element. Let be a simply ordered set endowed with the order topology. Then is compact if and only if is a complete lattice (i.e. all subsets have suprema and infima).
Let be the complex manifold constructed for . Because is a direct sum of copies of C, Z is just a product of Riemann spheres, one for each . In particular it is compact. There is a natural map of Z into X which is continuous.
This also explains the definition of the multiplication rule in . The direct product is a special case of the semidirect product. To see this, let be the trivial homomorphism (i.e., sending every element of to the identity automorphism of ) then is the direct product .
Let be the additive group of the integers, and the subgroup . Then the cosets of in are the three sets , , and , where }. These three sets partition the set ℤ, so there are no other right cosets of . Due to the commutivity of addition and .
Finally we may define the homotopy category. :Definition. Let be a finite-dimensional Noetherian scheme, and let denote the category of smooth schemes over . Equip with the Nisnevich topology to get the site . We let the affine line play the role of the interval.
Let be an integrable, -valued random variable, which is independent of the integrable, real-valued random variable with . Define for all . Then assumptions (), (), (), and () with are satisfied, hence also () and (), and Wald's equation applies. If the distribution of is not symmetric, then () does not hold.
Let be a real-valued function defined on a closed interval [] that admits an antiderivative on . That is, and are functions such that for all in , :f(x) = F'(x). If is integrable on then :\int_a^b f(x)\,dx = F(b) - F(a).
There is an analogue of the prime number theorem that describes the "distribution" of irreducible polynomials over a finite field; the form it takes is strikingly similar to the case of the classical prime number theorem. To state it precisely, let be the finite field with elements, for some fixed , and let be the number of monic irreducible polynomials over whose degree is equal to . That is, we are looking at polynomials with coefficients chosen from , which cannot be written as products of polynomials of smaller degree. In this setting, these polynomials play the role of the prime numbers, since all other monic polynomials are built up of products of them.
Let be any finite set, be any function from to itself, and be any element of . For any , let . Let be the smallest index such that the value reappears infinitely often within the sequence of values , and let (the loop length) be the smallest positive integer such that . The cycle detection problem is the task of finding and .. One can view the same problem graph-theoretically, by constructing a functional graph (that is, a directed graph in which each vertex has a single outgoing edge) the vertices of which are the elements of and the edges of which map an element to the corresponding function value, as shown in the figure.
The survival probability is the probability that no jump has occurred in the interval . The change in the survival probability is :d p_s(t) = -p_s(t) h(t) \, dt. So :p_s(t) = \exp \left(-\int_0^t h(u) \, du \right). Let be a discontinuous stochastic process.
A regular spread may be constructed in the following way. Let be a field and an -dimensional extension field of . Let considered as a -dimensional vector space over . The set of all 1-dimensional subspaces of over (and hence, -dimensional over ) is a regular spread of .
Induction step for proof of Fáry's theorem. One way of proving Fáry's theorem is to use mathematical induction.The proof that follows can be found in . Let be a simple plane graph with vertices; we may add edges if necessary so that is a maximally plane graph.
Let be a Riemannian manifold of dimension . One says that a coordinate chart , defined on an open subset of , is harmonic if each individual coordinate function is a harmonic function on . That is, one requires that :\Delta^g x^i = 0.\, where is the Laplace–Beltrami operator.
Define then :, which easily works. On the other hand, suppose is a winning strategy for II. From the s-m-n theorem, let be continuous such that for all , , , and , :. By the recursion theorem, there exists such that . A straightforward induction on for shows that :, and :.
Let be an infinite set and let denote the set of all finite subsets of . There is a natural multiplication on . For , let , where denotes the symmetric difference. This turns into a group with the empty set, , being the identity and every element being its own inverse; .
Let be a continuous function, and suppose that and . In this case, the intermediate value theorem states that must have a root in the interval . This theorem can be proved by considering the set :. That is, is the initial segment of that takes negative values under .
Let be a continuous function and let , where if has no upper bound. The extreme value theorem states that is finite and for some . This can be proved by considering the set :. If is the least upper bound of this set, then it follows from continuity that .
The two key ideas are the following. Let be the polynomial obtained from by taking the coefficients . Now: # is divisible by if and only if ; and # has no more than roots. More rigorously, start by noting that if and only if each coefficient of is divisible by .
Therefore, the last edge of is . Now, let be the -path from with respect to . Since is uniquely determined and the inner edges of are not changed in , the path uses the same edges as in reverse order and visits . The edge leading to clearly has color .
We may also define functions on discontinuous stochastic processes. Let be the jump intensity. The Poisson process model for jumps is that the probability of one jump in the interval is plus higher order terms. could be a constant, a deterministic function of time, or a stochastic process.
Let and be partially ordered sets and let be an order-preserving map. The map is a bounded morphism (also known as p-morphism) if for each and , if , then there exists such that and . Theorem:Esakia (1974), Esakia (1985). The following conditions are equivalent: :(1) is a bounded morphism.
Let be a group. Two elements and of are conjugate, if there exists an element in such that . One says also that is a conjugate of and that is a conjugate of . In the case of the group of invertible matrices, the conjugacy relation is called matrix similarity.
The following is a standard proof that a complete pseudometric space \scriptstyle X is a Baire space. Let be a countable collection of open dense subsets. We want to show that the intersection is dense. A subset is dense if and only if every nonempty open subset intersects it.
Let be a morphism of a category containing two objects A and B. Associated with these objects are the identity morphisms and . By composing these with f, we construct two morphisms: :, and :. Both are morphisms between the same objects as f. We have, accordingly, the following coherence statement: :.
Liu Hui's algorithm Liu Hui began with an inscribed hexagon. Let be the length of one side of hexagon, is the radius of circle. Bisect with line , becomes one side of dodecagon (12-gon), let its length be . Let the length of be and the length of be .
In general, given some linear map (where is a finite-dimensional vector space), we can define the trace of this map by considering the trace of a matrix representation of , that is, choosing a basis for and describing as a matrix relative to this basis, and taking the trace of this square matrix. The result will not depend on the basis chosen, since different bases will give rise to similar matrices, allowing for the possibility of a basis-independent definition for the trace of a linear map. Such a definition can be given using the canonical isomorphism between the space of linear maps on and , where is the dual space of . Let be in and let be in .
We use induction on . If is empty, then the theorem is vacuously true and the base case for induction is verified. Assume is non-empty, let be an element of and write If is any -linear transformation on , by the induction hypothesis there exists such that for all in . Write .
The evolute of a circle is therefore a single point at its center. Equivalently, an evolute is the envelope of the normals to a curve. The evolute of a curve, a surface, or more generally a submanifold, is the caustic of the normal map. Let be a smooth, regular submanifold in .
Let be a root of this polynomial (in the polynomial representation this would be ), that is, . Now , so is not a primitive element of GF(28) and generates a multiplicative subgroup of order 51. However, is a primitive polynomial. Consider the field element (in the polynomial representation this would be + 1).
Let be an open subset of . Give its standard orientation and the restriction of that orientation. Every smooth -form on has the form :\omega = f(x)\,dx^1 \wedge \cdots \wedge dx^n for some smooth function . Such a function has an integral in the usual Riemann or Lebesgue sense.
Let be any integer. A lattice in the complex plane with period ratio has a sublattice with period ratio . The latter lattice is one of a finite set of sublattices permuted by the modular group , which is based on changes of basis for . Let denote the elliptic modular function of Felix Klein.
Let be a root of the primitive polynomial . The traditional representation of elements of this field is as polynomials in α of degree 2 or less. A table of Zech logarithms for this field are , , , , , , , and . The multiplicative order of α is 7, so the exponential representation works with integers modulo 7.
Let be a set in a real or complex vector space. is star convex (star- shaped) if there exists an in such that the line segment from to any point in is contained in . Hence a non-empty convex set is always star-convex but a star-convex set is not always convex.
Let , , be three quadratic spaces over a field k. Assume that : (V_1,q_1)\oplus(V,q) \simeq (V_2,q_2)\oplus(V,q). Then the quadratic spaces and are isometric: : (V_1,q_1)\simeq (V_2,q_2). In other words, the direct summand appearing in both sides of an isomorphism between quadratic spaces may be "cancelled".
The basic scheme that protects against abuses is as follows: Let be sender, be recipient, and be an e-mail. If has agreed beforehand to receive e-mail from , then is transmitted in the usual way. Otherwise, computes some function and sends to . checks if what it receives from is of the form .
Although simpler, a direct-mapped cache needs to be much larger than an associative one to give comparable performance, and it is more unpredictable. Let be block number in cache, be block number of memory, and be number of blocks in cache, then mapping is done with the help of the equation .
A 2D construction of perspective viewing, showing the formation of a vanishing point The vanishing point may also be referred to as the "direction point", as lines having the same directional vector, say D, will have the same vanishing point. Mathematically, let be a point lying on the image plane, where is the focal length (of the camera associated with the image), and let be the unit vector associated with , where . If we consider a straight line in space with the unit vector and its vanishing point , the unit vector associated with is equal to , assuming both point towards the image plane.B. Caprile, V. Torre "Using Vanishing Points for Camera Calibration", International Journal of Computer Vision, Volume 4, Issue 2, pp.
Suppose that is a weakly closed equicontinuous disk in (this implies that is weakly compact) and let } be the polar of . Since by the bipolar theorem, it follows that a continuous linear functional belongs to if and only if belongs to the continuous dual space of , where is the Minkowski functional of defined by .
Let be a probability space, } with or a finite or an infinite index set, a filtration of , and an adapted stochastic process with for all . Then there exists a martingale and an integrable predictable process starting with such that for every . Here predictable means that is -measurable for every }. This decomposition is almost surely unique.
For this reason, the derivative is sometimes called the slope of the function . Here is a particular example, the derivative of the squaring function at the input 3. Let be the squaring function. The derivative of a curve at a point is the slope of the line tangent to that curve at that point.
A simple proof using only elliptic operators discovered in 1988 can be found in . Let be the Green's function on satisfying , where is the point measure at a fixed point of . The equation , has a smooth solution , because the right hand side has integral 0 by the Gauss–Bonnet theorem. Thus satisfies away from .
To see that this implies the standard pigeonhole principle, take any fixed arrangement of pigeons into holes and let be the number of pigeons in a hole chosen uniformly at random. The mean of is , so if there are more pigeons than holes the mean is greater than one. Therefore, is sometimes at least 2.
All Pythagorean quadruples (including non-primitives, and with repetition, though , and do not appear in all possible orders) can be generated from two positive integers and as follows: If and have different parity, let be any factor of such that . Then and . Note that . A similar method existsSierpiński, Wacław, Pythagorean Triangles, Dover, 2003 (orig.
Let be a commutative ring. The tensor product of -modules applies, in particular, if and are -algebras. In this case, the tensor product is an -algebra itself by putting :(a_1 \otimes b_1) \cdot (a_2 \otimes b_2) = (a_1 \cdot a_2) \otimes (b_1 \cdot b_2). For example, :R[x] \otimes_R R[y] \cong R[x, y].
Let be an instance of the travelling salesman problem. That is, is a complete graph on the set of vertices, and the function assigns a nonnegative real weight to every edge of . According to the triangle inequality, for every three vertices , , and , it should be the case that . Then the algorithm can be described in pseudocode as follows.
It turns out that this is the only way that an orbit can contain infinitely many integers. :Theorem. Let be a rational function of degree at least two, and assume that no iterateAn elementary theorem says that if and if some iterate of is a polynomial, then already the second iterate is a polynomial. of is a polynomial. Let .
Asymptotic stability of fixed points of a non-linear system can often be established using the Hartman–Grobman theorem. Suppose that is a -vector field in which vanishes at a point , . Then the corresponding autonomous system :x'=v(x) has a constant solution : x(t)=p. Let be the Jacobian matrix of the vector field at the point .
It makes use of classic compass and straightedge construction, proving the theorem by a method similar to that employed by ancient Greek geometers. It is essentially the algebraic proof of the previous section viewed geometrically in yet another way. Let be a right isosceles triangle with hypotenuse length and legs as shown in Figure 2. By the Pythagorean theorem, .
While defined on all schemes, the h and qfh topology are only ever used on Noetherian schemes. The h topology has various non-equivalent extensions to non-Noetherian schemes including the ph topologyA cohomological bound for the h-topology and the v topology. The proper cdh topology is defined as follows. Let be a proper morphism.
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group. Let be an real or complex matrix.
Let be the distance of the object A to the k-th nearest neighbor. Note that the set of the k nearest neighbors includes all objects at this distance, which can in the case of a "tie" be more than k objects. We denote the set of k nearest neighbors as . Illustration of the reachability distance.
Let be the height of the communication structure with the root at processor and the two trees below it. After steps, the first data block has reached every node in both trees. Afterwards, each processor receives one block in every step until it received all blocks. The total number of steps is resulting in a total communication time of .
Let be the fundamental group corresponding to the spanning tree . For every vertex and edge , and can be identified with their images in . It is possible to define a graph with vertices and edges the disjoint union of all coset spaces and respectively. This graph is a tree, called the universal covering tree, on which acts.
The discovery of his children's talent is considered to have been a life-transforming event for Mozart. He once referred to his son as the "miracle which God let be born in Salzburg". Of Leopold's attitude, the Grove Dictionary says: By "missionary", the Grove Dictionary refers to the family's concert tours. Scholars differ on whether the tours made substantial profits.
A vector space is an abelian group with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as multiplication by −1). Hence, every topological vector space is an abelian topological group. Every TVS is completely regular but a TVS need not be normal. Let be a topological vector space.
The theorem can be stated either for a complex semisimple Lie group or for its compact form . Let be a connected complex semisimple Lie group, a Borel subgroup of , and the flag variety. In this scenario, is a complex manifold and a nonsingular algebraic . The flag variety can also be described as a compact homogeneous space , where is a (compact) Cartan subgroup of .
Let be a TVS. Say that a vector subspace of has the extension property if any continuous linear functional on can be extended to a continuous linear functional on . Say that has the Hahn-Banach extension property (HBEP) if every vector subspace of has the extension property. The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP.
Barbara Liskov and Jeannette Wing formulated the principle succinctly in a 1994 paper as follows: > Subtype Requirement: Let be a property provable about objects of type . Then > should be true for objects of type where is a subtype of . Thus, normally one must distinguish subtyping and subclassing. Most current object-oriented languages distinguish subtyping and subclassing, however some approaches to design do not.
It will be shown that , and hence a one-to-one correspondence between and the group exists. For , let be the subset of consisting of all subsets of cardinality exactly . Then is the disjoint union of the . The number of subsets of of cardinality is at most because every subset with elements is an element of the -fold cartesian product of .
Although , the convex balanced hull of is not necessarily equal to the balanced hull of the convex hull of . For an example where , let be the real vector space and let }. Then is a strict subset of cobal(S) that is not even convex. In particular, this example also shows that the balanced hull of a convex set is not necessarily convex.
Under certain additional conditions we can simplify the analysis of () theories by introducing an auxiliary field . Assuming f(R) eq 0 for all , let () be the Legendre transformation of () so that \Phi = f'(R) and R=V'(\Phi). Then, one obtains the O'Hanlon (1972) action: \right].}} We have the Euler–Lagrange equations Eliminating , we obtain exactly the same equations as before.
Let be a rational function of degree at least two with coefficients in . A theorem of Northcott says that has only finitely many -rational preperiodic points, i.e., has only finitely many preperiodic points in . The Uniform Boundedness Conjecture of Morton and Silverman says that the number of preperiodic points of in is bounded by a constant that depends only on the degree of .
The line through which is perpendicular to the line is called the polarAlthough no duality has yet been defined these terms are being used in anticipation of the existence of one. of the point with respect to circle . Let be a line not passing through . Drop a perpendicular from to , meeting at the point (this is the point of that is closest to ).
Proof of Apollonius's theorem The theorem can be proved as a special case of Stewart's theorem, or can be proved using vectors (see parallelogram law). The following is an independent proof using the law of cosines. Let the triangle have sides with a median drawn to side . Let be the length of the segments of formed by the median, so is half of .
One can check that is indeed a measure. It is not -finite, as not every Borel set is at most a countable union of finite sets. Let be the usual Lebesgue measure on this Borel algebra. Then, is absolutely continuous with respect to , since for a set one has only if is the empty set, and then is also zero.
Unlike the ratio, the difference between and increases without bound as increases. On the other hand, switches sign infinitely many times. Let be the prime-counting function that gives the number of primes less than or equal to , for any real number . For example, because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10.
Let be a Hilbert space and the bounded operators on . Consider a self-adjoint unital subalgebra of (this means that contains the adjoints of its members, and the identity operator on ). The theorem is equivalent to the combination of the following three statements: :(i) :(ii) :(iii) where the and subscripts stand for closures in the weak and strong operator topologies, respectively.
That is, every left coset of is also a right coset, so is a normal subgroup. (The same argument shows that every subgroup of an Abelian group is normal.) This example may be generalized. Again let be the additive group of the integers, , and now let the subgroup , where is a positive integer. Then the cosets of in are the sets , , ..., , where }.
Formally, let be a set S with a closed binary operation • on it (known as a magma). A zero element is an element z such that for all s in S, . A refinement are the notions of left zero, where one requires only that , and right zero, where . Absorbing elements are particularly interesting for semigroups, especially the multiplicative semigroup of a semiring.
For an order theoretic example, let be some set, and let and both be the power set of , ordered by inclusion. Pick a fixed subset of . Then the maps and , where , and , form a monotone Galois connection, with being the lower adjoint. A similar Galois connection whose lower adjoint is given by the meet (infimum) operation can be found in any Heyting algebra.
Also, let be the hyperplane that splits the longest side of in two. Here is the algorithm in pseudo-code: if // is a hyperrectangle which each side has a length of zero. Store in the only point in . else repeat Compute Let the i-th dimension be the one where while and } } Let and be respectively, the left and right children of .
The theorem relates two quantities: the maximum flow through a network, and the minimum weight of a cut of the network. To state the theorem, each of these quantities must first be defined. Let be a directed graph, where V denotes the set of vertices and E is the set of edges. Let and be the source and the sink of , respectively.
Importantly, may fail to be a filter on even if every is a filter on . However, if every is a prefilter on then is a prefilter on ; moreover, this prefilter is equal to the coarsest prefilter on such that for every . This filter has as a filter subbasis. ;Kowalsky's filter Let and be sets and for every let be a dual ideal on .
If is any dual ideal on then is a dual ideal on called _Kowalsky's filter_. ;Using duality of ideals and dual ideals Let be a map and suppose that . Define : which contains the empty set if and only if does. It's possible for to be an ultrafilter and for to be empty or not closed under finite intersections (see footnote for example).
Therefore each vertex in has deficiency at most three, so there are at least four vertices with positive deficiency. In particular we can choose a vertex with at most five neighbors that is different from , and . Let be formed by removing from and retriangulating the face formed by removing . By induction, has a combinatorially isomorphic straight line re-embedding in which is the outer face.
This step was taken by Sommerfeld, who proposed the general quantization rule for an integrable system, : J_k = h n_k. \, Each action variable is a separate integer, a separate quantum number. This condition reproduces the circular orbit condition for two dimensional motion: let be polar coordinates for a central potential. Then is already an angle variable, and the canonical momentum conjugate is , the angular momentum.
Parallel transport of a tangent vector along a curve in the sphere. Comparison of tangent vectors at different points on a manifold is generally not a well-defined process. An affine connection provides one way to remedy this using the notion of parallel transport, and indeed this can be used to give a definition of an affine connection. Let be a manifold with an affine connection .
Let be a topological space. Consider the equivalence relation on continuous paths in in which two continuous paths are equivalent if they are homotopic with fixed endpoints. The fundamental groupoid assigns to each ordered pair of points in the collection of equivalence classes of continuous paths from to . As suggested by its name, the fundamental groupoid of naturally has the structure of a groupoid.
Let be the finite field with 211 elements. Its group of units has order − 1 = 2047 = 23 · 89, so it has a cyclic subgroup of order 23. The Mathieu group M23 can be identified with the group of -linear automorphisms of that stabilize . More precisely, the action of this automorphism group on can be identified with the 4-fold transitive action of M23 on 23 objects.
It is possible to prove the least-upper-bound property using the assumption that every Cauchy sequence of real numbers converges. Let be a nonempty set of real numbers, and suppose that has an upper bound . Since is nonempty, there exists a real number that is not an upper bound for . Define sequences and recursively as follows: # Check whether is an upper bound for .
Let K be a quadratic extension of Q, and let be its ring of integers. By extending to a Z-basis, we see that every order O in K has the form for some positive integer c. The conductor of this order equals the ideal cOK. Indeed, it is clear that cOK is an ideal of OK contained in O, so it is contained in the conductor.
This is the only legal cover they have obtained till date. Other carnal activities, red-light districts and brothels remain illegal business and operate as an open secret let be by offering huge sums in bribe to the police. Many areas in Pakistan like Heera Mandi in Lahore and 12 No Chungi in Sargodha are govt licensed areas and police protect them due to some political reasons.
The choice of a total order on the monomials allows sorting the terms of a polynomial. The leading term of a polynomial is thus the term of the largest monomial (for the chosen monomial ordering). Concretely, let be any ring of polynomials. Then the set of the (monic) monomials in is a basis of , considered as a vector space over the field of the coefficients.
Suppose that (X,Y,b) is a pairing of vector spaces over . :Notation: For all , let denote the linear functional on defined by and let . Similarly, for all , let be defined by and let . The weak topology on induced by Y (and ) is the weakest TVS topology on , denoted by \sigma(X,Y,b) or simply \sigma(X,Y), making all maps continuous, as y ranges over .
Let be a volume form defined on Rn+1. We can induce a volume form on M given by given by This is a natural definition: in Euclidean differential geometry where ξ is the Euclidean unit normal then the standard Euclidean volume spanned by X1,...,Xn is always equal to ω(X1,...,Xn). Notice that ω depends on the choice of transverse vector field ξ.
Then is a proper -edge-coloring of due to definition of . Also, note that the missing colors in are the same with respect to for all . Let be the color missing in with respect to , then is also missing in with respect to for all . Note that cannot be missing in , otherwise we could easily extend , therefore an edge with color is incident to for all .
The previous example can be generalized to Dedekind domains. Let be a Dedekind domain, its field of fractions, and let P be a non- zero prime ideal of . Then, the localization of at P, denoted RP, is a principal ideal domain whose field of fractions is . The construction of the previous section applied to the prime ideal PRP of RP yields the -adic valuation of .
Let be a compact Hausdorff space and k= \R or \Complex. Then K_k(X) is defined to be the Grothendieck group of the commutative monoid of isomorphism classes of finite-dimensional -vector bundles over under Whitney sum. Tensor product of bundles gives -theory a commutative ring structure. Without subscripts, K(X) usually denotes complex -theory whereas real -theory is sometimes written as KO(X).
Let be a topological space with topology . A subbase of is usually defined as a subcollection of satisfying one of the two following equivalent conditions: #The subcollection generates the topology . This means that is the smallest topology containing : any topology on containing must also contain . #The collection of open sets consisting of all finite intersections of elements of , together with the set , forms a basis for .
If maps to , then maps back to . Let be a function whose domain is the set , and whose codomain is the set . Then is invertible if there exists a function with domain and image (range) , with the property: : f(x) = y\,\,\Leftrightarrow\,\,g(y) = x. If is invertible, then the function is unique, which means that there is exactly one function satisfying this property.
By taking and to be the identity map on in the last condition, it follows that the class of sequential spaces consists precisely of those spaces whose topological structure is determined by convergent sequences. (1) ⇔ (2): Assume that any sequentially open subsets is open and let be sequentially closed. It is proved above that the complement is sequentially open and thus open so that is closed. The converse is similar.
It is possible, using classical results, to construct explicitly a polynomial whose Galois group over is the cyclic group for any positive integer . To do this, choose a prime such that ; this is possible by Dirichlet's theorem. Let be the cyclotomic extension of generated by , where is a primitive root of unity; the Galois group of is cyclic of order . Since divides , the Galois group has a cyclic subgroup of order .
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.
In mathematics, duality theory for distributive lattices provides three different (but closely related) representations of bounded distributive lattices via Priestley spaces, spectral spaces, and pairwise Stone spaces. This duality, which is originally also due to Marshall H. Stone Stone (1938), generalizes the well-known Stone duality between Stone spaces and Boolean algebras. Let be a bounded distributive lattice, and let denote the set of prime filters of . For each , let }.
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 concept of zeros and poles extends naturally to functions on a complex curve, that is complex analytic manifold of dimension one (over the complex numbers). The simplest examples of such curves are the complex plane and the Riemann surface. This extension is done by transferring structures and properties through charts, which are analytic isomorphisms. More precisely, let be a function from a complex curve to the complex numbers.
The second stage in the proof is to use the Gödel numbering, described above, to show that the notion of provability can be expressed within the formal language of the theory. Suppose the theory has deduction rules: . Let be their corresponding relations, as described above. Every provable statement is either an axiom itself, or it can be deduced from the axioms by a finite number of applications of the deduction rules.
The Bernoulli numbers are related to Fermat's Last Theorem (FLT) by Kummer's theorem , which says: :If the odd prime does not divide any of the numerators of the Bernoulli numbers then has no solutions in nonzero integers. Prime numbers with this property are called regular primes. Another classical result of Kummer are the following congruences. :Let be an odd prime and an even number such that does not divide .
Preimages of sets under functions can be described as pullbacks as follows: Suppose , . Let be the inclusion map . Then a pullback of and (in ) is given by the preimage together with the inclusion of the preimage in : and the restriction of to :. Because of this example, in a general category the pullback of a morphism and a monomorphism can be thought of as the "preimage" under of the subobject specified by .
Let (, ) be a pair of amicable numbers with , and write and where is the greatest common divisor of and . If and are both coprime to and square free then the pair (, ) is said to be regular , otherwise it is called irregular or exotic. If (, ) is regular and and have and prime factors respectively, then is said to be of type . For example, with , the greatest common divisor is and so and .
The definitions of strict positive homogeneity that was given for -valued functions on immediately extends, without change, to functions that are valued in other codomains. :Definition: Let be a function on valued in (or even in ). We say that is strictly positively homogeneous if for all and all positive real . If never takes the value then we say that is non-negative homogeneous if for all and all non-negative real .
Let be positive integers. If :q_1 + q_2 + \cdots + q_n - n + 1 objects are distributed into boxes, then either the first box contains at least objects, or the second box contains at least objects, ..., or the th box contains at least objects. The simple form is obtained from this by taking , which gives objects. Taking gives the more quantified version of the principle, namely: Let and be positive integers.
The symmetric algebra can also be built from polynomial rings. If is a -vector space or a free -module, with a basis , let be the polynomial ring that has the elements of as indeterminates. The homogeneous polynomials of degree one form a vector space or a free module that can be identified with . It is straightforward to verify that this makes a solution to the universal problem stated in the introduction.
For an arbitrary field , let be a set of -dimensional subspaces of the vector space , any two of which intersect only in {0} (called a partial spread). The members of , and their cosets in , form the lines of a translation net on the points of . If this is a -net of order . Starting with an affine translation plane, any subset of the parallel classes will form a translation net.
For univariate polynomials over a field, this results from Bézout's identity, which itself results from Euclidean algorithm. So, let be a unique factorization domain, which is not a field, and the univariate polynomial ring over . An irreducible element in is either an irreducible element in or an irreducible primitive polynomial. If is in and divides a product P_1P_2 of two polynomials, then it divides the content c(P_1P_2) = c(P_1)c(P_2).
The number of squares in a square grid is similarly counted by the square pyramidal numbers. The identity also admits a natural probabilistic interpretation as follows. Let be four integer numbers independently and uniformly chosen at random between and . Then, the probability that be the largest of the four numbers is equal to the probability that both is at least as large as and is at least as large as , that is, .
In 1935, the Base Area was the object of two Nationalist Government Encirclement Campaigns in 1935, both of which failed. Uniquely in the history of these campaigns, the target Soviet was then let be: its forces and apparatus were neither destroyed nor driven out by subsequent strategy. This may be due to the Long March then concluding, which made much of Shaanxi into a fairly secure, Soviet Union- backed territory for the Chinese Communists.
An affine connection defines a notion of development of curves. Intuitively, development captures the notion that if is a curve in , then the affine tangent space at may be rolled along the curve. As it does so, the marked point of contact between the tangent space and the manifold traces out a curve in this affine space: the development of . In formal terms, let be the linear parallel transport map associated to the affine connection.
Let be a duality of for . If is composed with the natural isomorphism between and , the composition is an incidence preserving bijection between and . By the Fundamental theorem of projective geometry is induced by a semilinear map with associated isomorphism , which can be viewed as an antiautomorphism of . In the classical literature, would be called a reciprocity in general, and if it would be called a correlation (and would necessarily be a field).
Let be an individual's utility function, where is the person's wealth and is a dummy variable that takes the value 1 in the presence of an undesired feature and takes the value 0 in the absence of that feature. The utility function is assumed to be increasing in wealth and decreasing in . Also, define as the person's initial wealth. Then the willingness to accept is defined by :u(w_0 + WTA , 1) = u(w_0 , 0).
Minimal Bottleneck Spanning Tree In an undirected graph and a function , let be the set of all spanning trees Ti. Let B(Ti) be the maximum weight edge for any spanning tree Ti. We define subset of minimum bottleneck spanning trees S′ such that for every and we have for all i and k. The graph on the right is an example of MBST, the red edges in the graph form a MBST of .
Let be a compatible couple, and assume that is dense in X0 and in X1. In this case, the restriction map from the (continuous) dual X'_j of , to the dual of is one-to-one. It follows that the pair of duals \left (X'_0, X'_1 \right ) is a compatible couple continuously embedded in the dual . For the complex interpolation method, the following duality result holds: :Theorem.see 12.1 and 12.2, p. 121 in .
Now, for , let be a root of the th derivative f=p^{(i)} of , which is not a root of p^{(i-1)}. There are two cases to be considered. If the multiplicity of the root is even, then f=p^{(i)} and p^{(i-1)} keep a constant sign when pass through . This implies that the number of sign of variation in the sequence of derivatives decrease by the even number .
In the complex numbers, , there are just two numbers, i and −i, whose square is −1 . In there are infinitely many square roots of minus one: the quaternion solution for the square root of −1 is the unit sphere in . To see this, let be a quaternion, and assume that its square is −1. In terms of , and , this means :a^2 - b^2 - c^2 - d^2 = -1, :2ab = 0, :2ac = 0, :2ad = 0.
Let be a Banach space. The tensor product X' \widehat \otimes_\varepsilon X is identified isometrically with the closure in of the set of finite rank operators. When has the approximation property, this closure coincides with the space of compact operators on . For every Banach space , there is a natural norm linear map : Y \widehat\otimes_\pi X \to Y \widehat\otimes_\varepsilon X obtained by extending the identity map of the algebraic tensor product.
Let be a subgroup of the group whose operation is written multiplicatively (juxtaposition means apply the group operation). Given an element of , the left cosets of in are the sets obtained by multiplying each element of by a fixed element of (where is the left factor). In symbols these are, :} for each in . The right cosets are defined similarly, except that the element is now a right factor, that is, :} for in .
This function does not have a derivative at the marked point, as the function is not continuous there (specifically, it has a jump discontinuity). If is differentiable at , then must also be continuous at . As an example, choose a point and let be the step function that returns the value 1 for all less than , and returns a different value 10 for all greater than or equal to . cannot have a derivative at .
In fact, this is necessary and sufficient, because if is any nilpotent, then one of its powers will be nilpotent of order at most . In particular, if is a field then the Frobenius endomorphism is injective. The Frobenius morphism is not necessarily surjective, even when is a field. For example, let be the finite field of elements together with a single transcendental element; equivalently, is the field of rational functions with coefficients in .
For tangent vectors X1,...,Xn let be the given by We define a second volume form on M given by where Again, this is a natural definition to make. If M = Rn and h is the Euclidean scalar product then ν(X1,...,Xn) is always the standard Euclidean volume spanned by the vectors X1,...,Xn. Since h depends on the choice of transverse vector field ξ it follows that ν does too.
The typical use of discriminants in algebraic geometry is for studying algebraic curve and, more generally algebraic hypersurfaces. Let be such a curve or hypersurface; is defined as the zero set of a multivariate polynomial. This polynomial may be considered as a univariate polynomial in one of the indeterminates, with polynomials in the other indeterminates as coefficients. The discriminant with respect to the selected indeterminate defines a hypersurface in the space of the other indeterminates.
Let be a vector space over the field . Informally, multiplication in is performed by manipulating symbols and imposing a distributive law, an associative law, and using the identity for . Formally, is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associative -algebra containing with alternating multiplication on must contain a homomorphic image of . In other words, the exterior algebra has the following universal property:See , and .
There is one significant result concerning subbases, due to James Waddell Alexander II. Note that the corresponding result for basic open covers is much easier to prove. :Alexander Subbase Theorem: Let be a topological space. If has a subbasis such that every cover of by elements from has a finite subcover, then is compact. The converse to this theorem also holds and it is proven by using (since every topology is a subbasis for itself).
In this article we distinguish between Huygens' principle, which states that every point crossed by a traveling wave becomes the source of a secondary wave, and Huygens' construction, which is described below. Let the surface be a wavefront at time , and let the surface be the same wavefront at the later time (Fig.4). Let be a general point on . Then, according to Huygens' construction,Huygens, 1690, tr. Thompson, pp.19,50–51,63–65,68,75.
A linear function of a matrix is a linear combination of its elements (with given coefficients), where is the matrix of the coefficients; see Trace (linear algebra)#Inner product. A random orthogonal matrix is said to be distributed uniformly, if its distribution is the normalized Haar measure on the orthogonal group ; see Rotation matrix#Uniform random rotation matrices. > Theorem. Let be a random orthogonal matrix distributed uniformly, and a > fixed matrix such that , and let .
Proving existence is relatively straightforward: let be the set of all normal subgroups that can not be written as a product of indecomposable subgroups. Moreover, any indecomposable subgroup is (trivially) the one-term direct product of itself, hence decomposable. If Krull-Schmidt fails, then contains ; so we may iteratively construct a descending series of direct factors; this contradicts the DCC. One can then invert the construction to show that all direct factors of appear in this way.
Let be a manifold, and a principal -bundle over . Then an affine connection is a 1-form on with values in satisfying the following properties # is equivariant with respect to the action of on and ; # for all in the Lie algebra of all matrices; # is a linear isomorphism of each tangent space of with . The last condition means that is an absolute parallelism on , i.e., it identifies the tangent bundle of with a trivial bundle (in this case ).
Let be a topological vector space (TVS). :Definition: A convex and balanced subset of a real or complex vector space is called a disk and it is said to be disked, absolutely convex, or convex balanced. :Definition: A barrel or a barrelled set in a TVS is a subset that is a closed absorbing disk. Note that the only topological requirement on a barrel is that it be a closed subset of the TVS; all other requirements (i.e.
Closed linear operators are a class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the spectrum and (with certain assumptions) functional calculus for such operators. Many important linear operators which fail to be bounded turn out to be closed, such as the derivative and a large class of differential operators. Let be two Banach spaces.
Let be an abelian group and we consider the group homomorphisms from A into A. Then addition of two such homomorphisms may be defined pointwise to produce another group homomorphism. Explicitly, given two such homomorphisms f and g, the sum of f and g is the homomorphism [f + g](x) := f(x) + g(x). Under this operation End(A) is an abelian group. With the additional operation of composition of homomorphisms, End(A) is a ring with multiplicative identity.
An M-matrix is commonly defined as follows: Definition: Let be a real Z-matrix. That is, where for all . Then matrix A is also an M-matrix if it can be expressed in the form , where with , for all , where is at least as large as the maximum of the moduli of the eigenvalues of , and is an identity matrix. For the non-singularity of , according to the Perron-Frobenius theorem, it must be the case that .
The derivative at different points of a differentiable function. In this case, the derivative is equal to:\sin \left(x^2\right) + 2x^2 \cos\left(x^2\right) Let be a function that has a derivative at every point in its domain. We can then define a function that maps every point x to the value of the derivative of f at x. This function is written and is called the derivative function or the derivative of .
Let be a commutative ring with prime characteristic (an integral domain of positive characteristic always has prime characteristic, for example). The Frobenius endomorphism F is defined by :F(r) = r^p for all r in R. It respects the multiplication of R: :F(rs) = (rs)^p = r^ps^p = F(r)F(s), and is clearly 1 also. What is interesting, however, is that it also respects the addition of . The expression can be expanded using the binomial theorem.
We give pseudocode for the Split tree computation below. Let be the node for if // is a hyperrectangle which each side has a length of zero. Store in the only point in S. else Compute Let the i-th dimension be the one where Split along the i-th dimension in two same-size hyperrectangles and take the points contained in these hyperrectangles to form the two sets and . Store and as, respectively, the left and right children of .
Let be any non-empty open subset of (e.g. could be a non-empty bounded open interval in ) and let denote the subspace topology on that inherits from (so ). Then the topology generated by on is equal to the union (see this footnote for an explanation),Since is a topology on and is an open subset of , it is easy to verify that is a topology on . Since isn't a topology on , is clearly the smallest topology on containing ).
Let be a finite-dimensional real or complex vector space with a nondegenerate quadratic form . The (real or complex) linear maps preserving form the orthogonal group . The identity component of the group is called the special orthogonal group . (For real with an indefinite quadratic form, this terminology is not standard: the special orthogonal group is usually defined to be a subgroup with two components in this case.) Up to group isomorphism, has a unique connected double cover, the spin group .
Let be a quadratic space over a field k. Then it admits a Witt decomposition: : (V,q)\simeq (V_0,0)\oplus(V_a, q_a)\oplus (V_h,q_h), where is the radical of q, is an anisotropic quadratic space and is a split quadratic space. Moreover, the anisotropic summand, termed the core form, and the hyperbolic summand in a Witt decomposition of are determined uniquely up to isomorphism. Quadratic forms with the same core form are said to be similar or Witt equivalent.
Barbara Liskov and Jeannette Wing described the principle succinctly in a 1994 paper as follows: > Subtype Requirement: Let be a property provable about objects of type . Then > should be true for objects of type where is a subtype of . In the same paper, Liskov and Wing detailed their notion of behavioral subtyping in an extension of Hoare logic, which bears a certain resemblance to Bertrand Meyer's design by contract in that it considers the interaction of subtyping with preconditions, postconditions and invariants.
The Lambek–Moser theorem is universal, in the sense that it can explain any partition of the integers into two infinite parts. If and are any two infinite subsets forming a partition of the integers, one may construct a pair of functions and from which this partition may be derived using the Lambek–Moser theorem: define and . For instance, consider the partition of integers into even and odd numbers: let be the even numbers and be the odd numbers. Then , so and similarly .
Let be a line bundle over a Kähler manifold , and fix a hermitian bundle metric whose curvature form is a Kähler form on . Suppose that for sufficiently large , an orthonormal set of holomorphic sections of the line bundle defines a projective embedding of . One can pull back the Fubini-Study metric to define a sequence of metrics on as increases. Tian showed that a certain rescaling of this sequence will necessarily converge in the topology to the original Kähler metric.
Let be a vector space, be a linearly independent set of elements of , and be a generating set. One has to prove that the cardinality of is not larger than that of . If is finite, this results from the Steinitz exchange lemma. (Indeed, the Steinitz exchange lemma implies every finite subset of has cardinality not larger than that of , hence is finite with cardinality not larger than that of .) If is finite, a proof based on matrix theory is also possible.
Let p be the number of pairs of vertices that are not connected by an edge in the given graph , and let be the unique integer for which . Then the intersection number of is at most .. As cited by . Graphs that are the complement of a sparse graph have small intersection numbers: the intersection number of any -vertex graph is at most , where is the base of the natural logarithm and d is the maximum degree of the complement graph of ..
Let be a field containing all x_i, and P_n the vector space of the polynomials of degree less than with coefficients in . Let :\varphi:P_n\to F^n be the linear map defined by :p(x)\mapsto (p(x_1), \ldots, p(x_n)). The Vandermonde matrix is the matrix of \varphi with respect to the canonical bases of P_n and F^n. Changing the basis of P_n amounts to multiplying the Vandermonde matrix by a change-of-basis matrix (from the right).
But the points of a parametric net are not well distributed. So it is better to calculate the design parameters a,b,c,d and to use the parametric representation above: Cyclide (blue) as image by an inversion of a torus (black) at the unit sphere (red) Given: A torus, which is shifted out of the standard position along the x-axis. Let be x_1, x_2,x_3,x_4 the intersections of the torus with the x-axis (see diagram). All not zero.
The original proof of the De Bruijn–Erdős theorem, by De Bruijn, used transfinite induction. provided the following very short proof, based on Tychonoff's compactness theorem in topology. Suppose that, for the given infinite graph , every finite subgraph is -colorable, and let be the space of all assignments of the colors to the vertices of (regardless of whether they form a valid coloring). Then may be given a topology as a product space , where denotes the set of vertices of the graph.
By Tychonoff's theorem this topological space is compact. For each finite subgraph of , let be the subset of consisting of assignments of colors that validly color . Then the system of sets is a family of closed sets with the finite intersection property, so by compactness it has a nonempty intersection. Every member of this intersection is a valid coloring of .. Gottschalk states his proof more generally as a proof of the theorem of that generalizes the De Bruijn–Erdős theorem.
Let be a point on (as before), and a point on . And let and be given, so that the problem is to find . If satisfies Huygens' construction, so that the secondary wavefront from is tangential to at , then is a path of stationary traversal time from to . Adding the fixed time from to , we find that is the path of stationary traversal time from to (possibly with a restricted domain of comparison, as noted above), in accordance with Fermat's principle.
Let be a set equipped with a binary operation ∗. Then an element of is called a left identity if for all in , and a right identity if for all in . If is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity. An identity with respect to addition is called an additive identity (often denoted as 0) and an identity with respect to multiplication is called a multiplicative identity (often denoted as 1).
Let be a real or complex vector space. ;Trivial topology The trivial topology (or the indiscrete topology) } is always a TVS topology on any vector space , so it is obviously the coarsest TVS topology possible. This simple observation allows us to conclude that the intersection of any collection of TVS topologies on always contains a TVS topology. Any vector space (including those that are infinite dimensional) endowed with the trivial topology is a compact (and thus locally compact) complete pseudometrizable seminormable locally convex topological vector space.
Therefore, there is a well-defined function such that all reverses of -state cellular automata with the von Neumann neighborhood use a neighborhood with radius at most : simply let be the maximum, among all of the finitely many reversible -state cellular automata, of the neighborhood size needed to represent the time-reversed dynamics of the automaton. However, because of Kari's undecidability result, there is no algorithm for computing and the values of this function must grow very quickly, more quickly than any computable function..
The next generalization we consider is that of bounded self-adjoint operators on a Hilbert space. Such operators may have no eigenvalues: for instance let be the operator of multiplication by on , that is, Section 6.1 : [A \varphi](t) = t \varphi(t). \; Now, a physicist would say that A does have eigenvectors, namely the \varphi(t)=\delta(t-t_0), where \delta is a Dirac delta-function. A delta-function, however, is not a normalizable function; that is, it is not actually in the Hilbert space .
If is a differentiable function, a critical point of is a point where the rank of the Jacobian matrix is not maximal. This means that the rank at the critical point is lower than the rank at some neighbour point. In other words, let be the maximal dimension of the open balls contained in the image of ; then a point is critical if all minors of rank of are zero. In the case where , a point is critical if the Jacobian determinant is zero.
Using estimates on Kloosterman sums he was able to derive estimates for Fourier coefficients of modular forms in cases where Pierre Deligne's proof of the Weil conjectures was not applicable. It was later translated by Jacquet to a representation theoretic framework. Let be a reductive group over a number field F and H\subset G be a subgroup. While the usual trace formula studies the harmonic analysis on G, the relative trace formula a tool for studying the harmonic analysis on the symmetric space .
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 .
The LCF notation is a notation by Joshua Lederberg, Coxeter and Frucht, for the representation of cubic graphs that are Hamiltonian. The two edges along the cycle adjacent to any of the vertices are not written down. Let be the vertices of the graph and describe the Hamiltonian circle along the vertices by the edge sequence . Halting at a vertex , there is one unique vertex at a distance joined by a chord with , : j=i+d_i\quad (\bmod\, p),\quad 2\le d_i\le p-2.
For a simplified proof of Läuchli's theorem by Mycielski, see . The De Bruijn–Erdős theorem for countable graphs can also be shown to be equivalent in axiomatic power, within a theory of second-order arithmetic, to Kőnig's infinity lemma. For a counterexample to the theorem in models of set theory without choice, let be an infinite graph in which the vertices represent all possible real numbers. In , connect each two real numbers and by an edge whenever one of the values is a rational number.
If the th syzygy module is free for some , then by taking a basis as a generating set, the next syzygy module (and every subsequent one) is the zero module. If one does not take a bases as generating sets, then all subsequent syzygy modules are free. Let be the smallest integer, if any, such that the th syzygy module of a module is free or projective. The above property of invariance, up to the sum direct with free modules, implies that does not depend on the choice of generating sets.
Map graphs can be represented combinatorially as the "half-squares of planar bipartite graphs". That is, let be a planar bipartite graph, with bipartition . The square of is another graph on the same vertex set, in which two vertices are adjacent in the square when they are at most two steps apart in . The half- square or bipartite half is the induced subgraph of one side of the bipartition (say ) in the square graph: its vertex set is and it has an edge between each two vertices in that are two steps apart in .
A diagram showing a representation of the equivalent classes of pairs of integers The rational numbers may be built as equivalence classes of ordered pairs of integers. More precisely, let be the set of the pairs of integers such . An equivalence relation is defined on this set by : if and only if . Addition and multiplication can be defined by the following rules: :\left(m_1, n_1\right) + \left(m_2, n_2\right) \equiv \left(m_1n_2 + n_1m_2, n_1n_2\right), :\left(m_1, n_1\right) \times \left(m_2, n_2\right) \equiv \left(m_1m_2, n_1n_2\right).
Let be a derivation (background) of and denote by the one- dimensional Lie algebra spanned by . Define the Lie bracket on byTo show that the Jacobi identity holds, one writes everything out, uses the fact that the underlying Lie algebras have a Lie product satisfying the Jacobi identity, and that . :[G_1 + H_1, G_2 + H_2] = [\lambda\delta + H_1, \mu\delta + H_2] = [H_1, H_2] + \lambda \delta(H_1) - \mu \delta(H_2). It is obvious from the definition of the bracket that is and ideal in in and that is a subalgebra of .
In mathematics, in particular in topology, a subset of a topological space (X, τ) is saturated if it is an intersection of open subsets of X. In a T1 space every set is saturated. An alternative definition for saturated sets comes from surjections, these definitions are not equivalent: let be a surjection; a subset C of X is called saturated with respect to p if for every p−1(A) that intersects C, p−1(A) is contained in C. This is equivalent to the statement that p−1p(C)=C.
More precisely, find necessary and sufficient conditions on the tuple and separately, so that there is an element of with the property that for all . If is the set of all -module endomorphisms of , then Schur's lemma asserts that is a division ring, and the Jacobson density theorem answers the question on tuples in the affirmative, provided that the are linearly independent over . With the above in mind, the theorem may be stated this way: :The Jacobson Density Theorem. Let be a simple right -module, , and a finite and -linearly independent set.
A 1959 paper of Erdős (see reference cited below) addressed the following problem in graph theory: given positive integers and , does there exist a graph containing only cycles of length at least , such that the chromatic number of is at least ? It can be shown that such a graph exists for any and , and the proof is reasonably simple. Let be very large and consider a random graph on vertices, where every edge in exists with probability . We show that with positive probability, satisfies the following two properties: :Property 1.
The cost of the solution produced by the algorithm is within 3/2 of the optimum. To prove this, let be the optimal traveling salesman tour. Removing an edge from produces a spanning tree, which must have weight at least that of the minimum spanning tree, implying that . Next, number the vertices of in cyclic order around , and partition into two sets of paths: the ones in which the first path vertex in cyclic order has an odd number and the ones in which the first path vertex has an even number.
The weak dual of a plane graph is the subgraph of the dual graph whose vertices correspond to the bounded faces of the primal graph. A plane graph is outerplanar if and only if its weak dual is a forest. For any plane graph , let be the plane multigraph formed by adding a single new vertex in the unbounded face of , and connecting to each vertex of the outer face (multiple times, if a vertex appears multiple times on the boundary of the outer face); then, is the weak dual of the (plane) dual of ..
Letbe a reducibility relation and let A and B be two of its degrees. Then A ≤ B if and only if there is a set A in A and a set B in B such that A ≤ B. This is equivalent to the property that for every set A in A and every set B in B, A ≤ B, because any two sets in A are equivalent and any two sets in B are equivalent. It is common, as shown here, to use boldface notation to denote degrees.
Let and be respectively the -coordinate and the -coordinate of the vertex of the parabola (that is the point with maximal or minimal -coordinate. The quadratic function may be rewritten : y = a(x - h)^2 + k. Let be the distance between the point of -coordinate on the axis of the parabola, and a point on the parabola with the same -coordinate (see the figure; there are two such points, which give the same distance, because of the symmetry of the parabola). Then the real part of the roots is , and their imaginary part are .
The following explains the Weyl transformation on the simplest, two-dimensional Euclidean phase space. Let the coordinates on phase space be , and let be a function defined everywhere on phase space. In what follows, we fix operators P and Q satisfying the canonical commutation relations, such as the usual position and momentum operators in the Schrödinger representation. We assume that the exponentiated operators e^{iaQ} and e^{ibP} constitute an irreducible representation of the Weyl relations, so that the Stone–von Neumann theorem (guaranteeing uniqueness of the canonical commutation relations) holds.
Most of the cases, where such a generalized discriminant is defined, are instances of the following. Let be a homogeneous polynomial in indeterminates over a field of characteristic 0, or of a prime characteristic that does not divides the degree of the polynomial. The polynomial defines a projective hypersurface, which has singular points if and only the partial derivatives of have a nontrivial common zero. This is the case if and only if the multivariate resultant of these partial derivatives is zero, and this resultant may be considered as the discriminant of .
Let be a compact oriented Riemannian manifold. The Hodge decomposition states that any -form on uniquely splits into the sum of three components: :\omega = \alpha + \beta + \gamma , where is exact, is co-exact, and is harmonic. One says that a form is co-closed if and co-exact if for some form , and that is harmonic if . This follows by noting that exact and co- exact forms are orthogonal; the orthogonal complement then consists of forms that are both closed and co-closed: that is, of harmonic forms.
The battle then split into two separate events. #The Polish flagship Sankt Georg (galleon, 400t, 31 guns), supported by the smaller Meerwieb (Panna Wodna -160t, 12 guns) attacked the Swedish flagship Tigern (Tiger, 320t, 22 guns), that was commanded by Admiral Niels Stiernskold. Entangled together, the Polish marines boarded and captured the Tigern. #The Vice Admiral's ship of the Polish Navy, the small galleon Meerman (Wodnik, 200t, 17 guns) attacked the larger Swedish Solen (Sun, 300t, 38 guns), whose captain blew up the ship, rather than let be captured.
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).
Let be a topological vector space. A vector subspace of has the extension property if any continuous linear functional on can be extended to a continuous linear functional on , and we say that has the Hahn–Banach extension property (HBEP) if every vector subspace of has the extension property. The Hahn–Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable topological vector spaces there is a converse, due to Kalton: every complete metrizable TVS with the Hahn–Banach extension property is locally convex.
A pairing or a pair over a field is a triple , which may also be denoted by , consisting of two vector spaces and over (which this article assumes is either the real numbers or the complex numbers ) and a bilinear map , which we call the bilinear map associated with the pairing or simply the pairing's map/bilinear form. :Notation: For all , let denote the linear functional on defined by and let . Similarly, for all , let be defined by and let . :Notation: It is common practice to write instead of , in which case the pair is often denoted by rather than .
Let A be a non-empty set, X a subset of A, F a set of functions in A, and X_+ the inductive closure of X under F. Let be B any non-empty set and let G be the set of functions on B, such that there is a function d:F\to G in G that maps with each function f of arity n in F the following function d(f):B^n\to B in G (G cannot be a bijection). From this lemma we can now build the concept of unique homomorphic extension.
The notion and terminology is generalized to a partial order. Letbe a partial order over a set X and let f: X → X be a function over X. Then a prefixpoint (also spelled pre-fixpoint) of f is any p such that f(p) ≤ p. Analogously a postfixpoint (or post- fixpoint) of f is any p such that p ≤ f(p). One way to express the Knaster–Tarski theorem is to say that a monotone function on a complete lattice has a least fixpoint that coincides with its least prefixpoint (and similarly its greatest fixpoint coincides with its greatest postfixpoint).
However, a translational gauge symmetry may be introduced thus: Instead of seeing tetrads as fundamental, we introduce a fundamental translational gauge symmetry instead (which acts upon the internal Minkowski space fibers affinely so that this fiber is once again made local) with a connection and a "coordinate field" taking on values in the Minkowski space fiber. More precisely, let be the Minkowski fiber bundle over the spacetime manifold . For each point , the fiber is an affine space. In a fiber chart , coordinates are usually denoted by , where are coordinates on spacetime manifold , and are coordinates in the fiber .
He was said to have described Wolfgang as "The miracle which God let be born in Salzburg." Mozart biographer Wolfgang Hildesheimer has suggested that, at least in the case of Wolfgang, this venture was premature: "Too soon, [the] father dragged [the] son all over Western Europe for years. This continual change of scene would have worn out even a robust child..."Hildesheimer, pp. 30-31 However, there is little evidence to suggest that Wolfgang was physically harmed or musically hindered by these childhood exertions; it seems that he felt equal to the challenge from the start.
Let be a real valued function defined in an open neighborhood of a real number . In classical geometry, the tangent line to the graph of the function at was the unique line through the point that did not meet the graph of transversally, meaning that the line did not pass straight through the graph. The derivative of with respect to at is, geometrically, the slope of the tangent line to the graph of at . The slope of the tangent line is very close to the slope of the line through and a nearby point on the graph, for example .
Wolfgang Paalen, Form and Sense, Meanings and Movements in Twentieth-Century Art, New York (Arcade Publishing/Artists and Art) 2013 Rothko described his new method as "unknown adventures in an unknown space", free from "direct association with any particular, and the passion of organism". Breslin described this change of attitude as "both self and painting are now fields of possibilities – an effect conveyed ... by the creation of protean, indeterminate shapes whose multiplicity is let be."Breslin, p. 378 In 1949, Rothko became fascinated by Henri Matisse's Red Studio, acquired by the Museum of Modern Art that year.
The specialization of the above section to skewfields was a consequence of the application to projective geometry, and not intrinsic to the nature of sesquilinear forms. Only the minor modifications needed to take into account the non-commutativity of multiplication are required to generalize the arbitrary field version of the definition to arbitrary rings. Let be a ring, an -module and an antiautomorphism of . A map is -sesquilinear if :\varphi(x + y, z + w) = \varphi(x, z) + \varphi(x, w) + \varphi(y, z) + \varphi(y, w) :\varphi(c x, d y) = c \, \varphi(x,y) \, \sigma(d) for all in and all in .
A spread of is a set of -dimensional subspaces of that partition the non-zero vectors of . The members of are called the components of the spread and if and are distinct components then . Let be the incidence structure whose points are the vectors of and whose lines are the cosets of components, that is, sets of the form where is a vector of and is a component of the spread . Then: : is an affine plane and the group of translations for a vector is an automorphism group acting regularly on the points of this plane.
Let be a prime power, and be the splitting field of the polynomial :P=X^q-X over the prime field . This means that is a finite field of lowest order, in which has distinct roots (the formal derivative of is , implying that , which in general implies that the splitting field is a separable extension of the original). The above identity shows that the sum and the product of two roots of are roots of , as well as the multiplicative inverse of a root of . In other words, the roots of form a field of order , which is equal to by the minimality of the splitting field.
Let be a set with an outer measure . One says that a subset of is -measurable (sometimes "Carathéodory-measurable relative to ") if and only if : \mu(A) = \mu\big(A \cap E\big) + \mu\big(A\smallsetminus E\big) for every subset of . Informally, this says that a -measurable subset is one which may be used as a building block, breaking any other subset apart into pieces (namely, the piece which is inside of the measurable set together with the piece which is outside of the measurable set). In terms of the motivation for measure theory, one would expect that area, for example, should be an outer measure on the plane.
The Kazhdan–Lusztig polynomials arise as transition coefficients between their canonical basis and the natural basis of the Hecke algebra. The Inventiones paper also put forth two equivalent conjectures, known now as Kazhdan–Lusztig conjectures, which related the values of their polynomials at 1 with representations of complex semisimple Lie groups and Lie algebras, addressing a long-standing problem in representation theory. Let W be a finite Weyl group. For each w ∈ W denote by be the Verma module of highest weight where ρ is the half-sum of positive roots (or Weyl vector), and let be its irreducible quotient, the simple highest weight module of highest weight .
Let be a finite-dimensional vector space over a field k of characteristic different from 2 together with a non-degenerate symmetric or skew-symmetric bilinear form. If is an isometry between two subspaces of V then f extends to an isometry of V. Witt's theorem implies that the dimension of a maximal totally isotropic subspace (null space) of V is an invariant, called the index' or ' of b, and moreover, that the isometry group of acts transitively on the set of maximal isotropic subspaces. This fact plays an important role in the structure theory and representation theory of the isometry group and in the theory of reductive dual pairs.
More generally, let be a morphism of degree at least two defined over a number field . Northcott's theorem says that has only finitely many preperiodic points in , and the general Uniform Boundedness Conjecture says that the number of preperiodic points in may be bounded solely in terms of , the degree of , and the degree of over . The Uniform Boundedness Conjecture is not known even for quadratic polynomials over the rational numbers . It is known in this case that cannot have periodic points of period four, five, or six, although the result for period six is contingent on the validity of the conjecture of Birch and Swinnerton-Dyer.
Consider the group (the group of order 12 that is the direct product of the symmetric group of order 6 and a cyclic group of order 2). The center of is its second factor . Note that the first factor, , contains subgroups isomorphic to , for instance ; let be the morphism mapping onto the indicated subgroup. Then the composition of the projection of onto its second factor , followed by , followed by the inclusion of into as its first factor, provides an endomorphism of under which the image of the center, , is not contained in the center, so here the center is not a fully characteristic subgroup of .
It is equivalent to require that around each x, there exists an open affine subset such that , where f is a non-zero divisor in A. The sum of two effective Cartier divisors corresponds to multiplication of ideal sheaves. There is a good theory of families of effective Cartier divisors. Let be a morphism. A relative effective Cartier divisor for X over S is an effective Cartier divisor D on X which is flat over S. Because of the flatness assumption, for every S'\to S, there is a pullback of D to X \times_S S', and this pullback is an effective Cartier divisor.
Throughout this section, we will let be a topological vector space (TVS) with continuous dual space X' and (X, X', \langle \cdot, \cdot \rangle) will be the canonical pairing (where \langle x, X'\rangle := X'(x)). Note that always distinguishes/separates the points of X' but X' may fail to distinguishes the points of (this necessarily happens if, for instance, is not Hausdorff), in which case the pairing (X, X', \langle \cdot, \cdot\rangle) is not a dual pair. By the Hahn-Banach theorem, if is a Hausdorff locally convex space then X' separates points of and thus (X, X',\langle \cdot, \cdot\rangle) forms a dual pair.
The Hand as hieroglyphic also forms the word for 'hand' in the Ancient Egyptian hieroglyphic language: "ţet." In line 13, (R-13), one of ten ways for honoring the Pharaoh Ptolemy V was to: :...."and let be engraved the Rank: "Priest of the god appearing-(epiphanous), lord of benefits -(eucharistos-Greek)", upon the rings worn on their hands-(hieroglyph)."Budge, The Rosetta Stone, p. 167. In the 1st half of the Rosetta Stone, (the Decree of Memphis (Ptolemy V)), supplied by the Nubayrah Stele, line N-22, there is use of the Hand-hieroglyph as part of an important word that implies the use of 'hands', or 'action'.
Let be a function with the assumed properties established above: and is convex, and . From we can establish :\Gamma(x+n)=(x+n-1)(x+n-2)(x+n-3)\cdots(x+1)x\Gamma(x) The purpose of the stipulation that forces the property to duplicate the factorials of the integers so we can conclude now that if and if exists at all. Because of our relation for , if we can fully understand for then we understand for all values of . The slope of a line connecting two points and , call it , is monotonically increasing in each argument with since we have stipulated is convex.
If the valuations of the partners are absolutely continuous with respect to each other, then there exists a WPR division which is also weighted-envy-free (WEF) and Pareto efficient (PE), and the ratio between the values of the partners is exactly w1/w2. Proof. For every angle t, let be the angle in which the ratio The function is a continuous function of t that achieves a maximum for some . Cut the pie with radial cuts at and , giving the piece to partner #1 and the complement to partner #2. The partition is WEF because the value of each partner is exactly his due share.
Suurballe's algorithm performs the following steps: # Find the shortest path tree rooted at node by running Dijkstra's algorithm (figure C). This tree contains for every vertex , a shortest path from to . Let be the shortest cost path from to (figure B). The edges in are called tree edges and the remaining edges (the edges missing from figure C) are called non-tree edges. # Modify the cost of each edge in the graph by replacing the cost of every edge by . According to the resulting modified cost function, all tree edges have a cost of 0, and non-tree edges have a non-negative cost.
Let be a family of sets (allowing sets in to be repeated); then the intersection graph of is an undirected graph that has a vertex for each member of and an edge between each two members that have a nonempty intersection. Every graph can be represented as an intersection graph in this way.. The intersection number of the graph is the smallest number such that there exists a representation of this type for which the union of has elements. The problem of finding an intersection representation of a graph with a given number of elements is known as the intersection graph basis problem., Problem GT59.
If there is a majority element, the algorithm will always find it. For, supposing that the majority element is , let be a number defined at any step of the algorithm to be either the counter, if the stored element is , or the negation of the counter otherwise. Then at each step in which the algorithm encounters a value equal to , the value of will increase by one, and at each step at which it encounters a different value, the value of may either increase or decrease by one. If truly is the majority, there will be more increases than decreases, and will be positive at the end of the algorithm.
Ceccherini-Silberstein and Coornaert provide the following proof of the Curtis–Hedlund–Lyndon theorem.. Suppose is a continuous shift-equivariant function on the shift space. For each configuration , let be the pattern consisting of the single symbol that appears at position zero of . By continuity of , there must exist a finite pattern in such that, if the positions outside are changed arbitrarily but the positions within are fixed to their values in , then the result of applying remains the same at position zero. Equivalently, there must exist a fundamental open set such that belongs to and such that for every configuration in , and have the same value at position zero.
Let X be a set of items, and let < be a binary relation on X. Items x and y are said to be incomparable, written here as x ~ y, if neither x < y nor y < x is true. Then the pair (X,<) is a semiorder if it satisfies the following three axioms: describes an equivalent set of four axioms, the first two of which combine the definition of incomparability and the first axiom listed here. #For all x and y, it is not possible for both x < y and y < x to be true. That is, < must be an asymmetric relation #For all x, y, z, and w, if x < y, y ~ z, and z < w, then x < w.
Specifically, let be a finitely presented morphism of pointed schemes and M be an OX-module of finite type whose fiber at x is non- zero. Set n equal to the dimension of and r to the codepth of M at s, that is, to .EGA 0IV, Définition 16.4.9 Then there exist affine étale neighborhoods X′ of x and S′ of s, together with points x′ and s′ lifting x and s, such that the residue field extensions and are trivial, the map factors through S′, this factorization sends x′ to s′, and that the pullback of M to X′ admits a total S′-dévissage at x′ in dimensions between n and .
The threshold value to determine when a data point fits a model , and the number of close data points required to assert that a model fits well to data are determined based on specific requirements of the application and the dataset, and possibly based on experimental evaluation. The number of iterations , however, can be determined as a function of the desired probability of success using a theoretical result. Let be the desired probability that the RANSAC algorithm provides a useful result after running. RANSAC returns a successful result if in some iteration it selects only inliers from the input data set when it chooses the points from which the model parameters are estimated.
Let M and N be (left or right) modules over the same ring, and let be a module homomorphism. If M is simple, then f is either the zero homomorphism or injective because the kernel of f is a submodule of M. If N is simple, then f is either the zero homomorphism or surjective because the image of f is a submodule of N. If M = N, then f is an endomorphism of M, and if M is simple, then the prior two statements imply that f is either the zero homomorphism or an isomorphism. Consequently, the endomorphism ring of any simple module is a division ring. This result is known as Schur's lemma.
More generally, the fundamental theorem of algebra asserts that the complex solutions of a polynomial equation of degree always form a multiset of cardinality . A special case of the above are the eigenvalues of a matrix, whose multiplicity is usually defined as their multiplicity as roots of the characteristic polynomial. However two other multiplicities are naturally defined for eigenvalues, their multiplicities as roots of the minimal polynomial, and the geometric multiplicity, which is defined as the dimension of the kernel of (where is an eigenvalue of the matrix ). These three multiplicities define three multisets of eigenvalues, which may be all different: Let be a matrix in Jordan normal form that has a single eigenvalue.
A line in a projective plane is a translation line if the group of elations with axis acts transitively on the points of the affine plane obtained by removing from the plane . A projective plane with a translation line is called a translation plane and the affine plane obtained by removing the translation line is called an affine translation plane. While in general it is often easier to work with projective planes, in this context the affine planes are preferred and several authors simply use the term translation plane to mean affine translation plane. An alternate view of affine translation planes can be obtained as follows: Let be a -dimensional vector space over a field .
The curve Γ is called the lift of γ. If x and y are two points in X connected by a path, then that path furnishes a bijection between the fiber over x and the fiber over y via the lifting property. More generally, let be a continuous map to X from a path connected and locally path connected space Z. Fix a base-point , and choose a point "lying over" f(z) (i.e. ). Then there exists a lift of f (that is, a continuous map for which and ) if and only if the induced homomorphisms and at the level of fundamental groups satisfy Moreover, if such a lift g of f exists, it is unique.
In the reweighted graph, all paths between a pair and of nodes have the same quantity added to them. The previous statement can be proven as follows: Let be an path. Its weight W in the reweighted graph is given by the following expression: :\bigl(w(s, p_1) + h(s) - h(p_1)\bigr) + \bigl(w(p_1, p_2) + h(p_1) - h(p_2)\bigr) + ... + \bigl(w(p_n, t) + h(p_n) - h(t)\bigr). Every +h(p_i) is cancelled by -h(p_i) in the previous bracketed expression; therefore, we are left with the following expression for W: :\bigl(w(s, p_1) + w(p_1, p_2) + ... + w(p_n, t)\bigr)+ h(s) - h(t) The bracketed expression is the weight of p in the original weighting.
Let be a smooth map between (smooth) manifolds M and N, and suppose is a smooth function on N. Then the pullback of f by φ is the smooth function φ∗f on M defined by . Similarly, if f is a smooth function on an open set U in N, then the same formula defines a smooth function on the open set φ−1(U) in M. (In the language of sheaves, pullback defines a morphism from the sheaf of smooth functions on N to the direct image by φ of the sheaf of smooth functions on M.) More generally, if is a smooth map from N to any other manifold A, then is a smooth map from M to A.
The simplest version of the minhash scheme uses different hash functions, where is a fixed integer parameter, and represents each set by the values of for these functions. To estimate using this version of the scheme, let be the number of hash functions for which , and use as the estimate. This estimate is the average of different 0-1 random variables, each of which is one when and zero otherwise, and each of which is an unbiased estimator of . Therefore, their average is also an unbiased estimator, and by standard deviation for sums of 0-1 random variables, its expected error is .. Therefore, for any constant there is a constant such that the expected error of the estimate is at most .
We record here definitions from . Let be the Coxeter complex associated to a group W generated by a set of reflections S. The vertices of are the elements of W, and the chambers of the complex are the cosets of S in W. The vertices of each chamber can be colored in a one-to-one manner by the elements of S so that no adjacent vertices of the complex receive the same color. This coloring, although essentially canonical, is not quite unique. The coloring of a given chamber is not uniquely determined by its realization as a coset of S. But once the coloring of a single chamber has been fixed, the rest of the Coxeter complex is uniquely colorable.
Dasmariñas even received word that the Portuguese "will try to harm you [Dasmariñas] as much as possible, and that let be clear that if they could, they would set you on fire." Blockaded in El Piñal, the Spanish persisted with the help from sympathetic mendicant orders in Macau supplying El Piñal in secret. However, this help was limited, and throughout the year 1599 the Spanish were worn down by attrition, leaving the men sick and weak and on the verge of revolt. As it became clear that Manila was not going to send reinforcements, the Spanish got ready to leave El Piñal, but was delayed by Chinese bureaucracy, as various customs dues needed to be paid before the Chinese authorized their departure.
This range tree will be used to identify the points in the "main" wedge for a point . # For every node w in the primary tree, and every node e in T(w), calculate the pair (w, e) = (b, r), where b (or r) is defined to be the point with maximum (or minimum) x-coordinate in the left (or right) subtree of e. Let be the set of (w, e) for all pairs w, e in T. This is a superset of the set of pairs of closest points (within the main wedge). # Build a kinetic priority queue on the pairs in , with priorities determined by the distance (measured in the original co-ordinate system) between the points in the pair.
Therefore, the time to compute the completion of a given partial order is . As observe, the problem of listing all cuts in a partially ordered set can be formulated as a special case of a simpler problem, of listing all maximal antichains in a different partially ordered set. If is any partially ordered set, let be a partial order whose elements contain two copies of : for each element of , contains two elements and , with if and only if and . Then the cuts in correspond one-for-one with the maximal antichains in : the elements in the lower set of a cut correspond to the elements with subscript 0 in an antichain, and the elements in the upper set of a cut correspond to the elements with subscript 1 in an antichain.
The points of are exactly the projection of the points of (including the points at infinity), which either are singular or have a tangent hyperplane that is parallel to the axis of the selected indeterminate. For example, let be a bivariate polynomial in and with real coefficients, such that is the implicit equation of a plane algebraic curve. Viewing as a univariate polynomial in with coefficients depending on , then the discriminant is a polynomial in whose roots are the -coordinates of the singular points, of the points with a tangent parallel to the -axis and of some of the asymptotes parallel to the -axis. In other words, the computation of the roots of the -discriminant and the -discriminant allows one to compute all of the remarkable points of the curve, except the inflection points.
Several authors have provided additional conditions that classify some graphs as being of class one or class two, but do not provide a complete classification. For instance, if the vertices of the maximum degree in a graph form an independent set, or more generally if the induced subgraph for this set of vertices is a forest, then must be of class one.. showed that almost all graphs are of class one. That is, in the Erdős–Rényi model of random graphs, in which all -vertex graphs are equally likely, let be the probability that an -vertex graph drawn from this distribution is of class one; then approaches one in the limit as goes to infinity. For more precise bounds on the rate at which converges to one, see .
The reduction needs to solve twice the similar problem where center of the sought-after enclosing circle is constrained to lie on a given line. The solution of the subproblem is either solution of unconstrained problem or it is used to determine the half-plane where the unconstrained solution center is located. The n/16 points to be discarded are found the following way: Points are arranged to pairs what defines n/2 lines as their bisectors. Median of bisectors in order by their directions (oriented to the same half-plane determined by bisector ) is found and pairs from bisectors are made, such that in each pair one bisector has direction at most and the other at least (direction could be considered as -\infty or +\infty according our needs.) Let be intersection of bisectors of -th pair.
More precisely, suppose that the point is moving on the curve at a constant speed of one unit, that is, the position of the point is a function of the parameter , which may be thought as the time or as the arc length from a given origin. Let be a unit tangent vector of the curve at , which is also the derivative of with respect to . Then, the derivative of with respect to is a vector that is normal to the curve and whose length is the curvature. For being meaningful, the definition of the curvature and its different characterizations require that the curve is continuously differentiable near , for having a tangent that varies continuously; it requires also that the curve is twice differentiable at , for insuring the existence of the involved limits, and of the derivative of .
Let be a table, an element to insert, num(T) the number of elements in , and size(T) the allocated size of . We assume the existence of operations create_table(n), which creates an empty table of size , for now assumed to be free, and elementary_insert(T,E), which inserts element into a table that already has space allocated, with a cost of 1. The following pseudocode illustrates the table insertion procedure: function table_insert(T, E) if num(T) = size(T) U := create_table(2 × size(T)) for each F in T elementary_insert(U, F) T := U elementary_insert(T, E) Without amortized analysis, the best bound we can show for n insert operations is O(n) -- this is due to the loop at line 4 that performs num(T) elementary insertions. For analysis using the accounting method, we assign a payment of 3 to each table insertion.
The cross-ratio is a projective invariant in the sense that it is preserved by the projective transformations of a projective line. In particular, if four points lie on a straight line L in R2 then their cross-ratio is a well-defined quantity, because any choice of the origin and even of the scale on the line will yield the same value of the cross-ratio. Furthermore, let be four distinct lines in the plane passing through the same point Q. Then any line L not passing through Q intersects these lines in four distinct points Pi (if L is parallel to Li then the corresponding intersection point is "at infinity"). It turns out that the cross-ratio of these points (taken in a fixed order) does not depend on the choice of a line L, and hence it is an invariant of the 4-tuple of lines {Li}.
In mathematics, let A be a set and letbe a binary relation on A. Then a subset B of A is said to be cofinal if it satisfies the following condition: :For every a ∈ A, there exists some b ∈ B such that a ≤ b. This definition is most commonly applied when A is a partially ordered set or directed set under the relation ≤. Cofinal subsets are very important in the theory of directed sets and nets, where “cofinal subnet” is the appropriate generalization of “subsequence”. They are also important in order theory, including the theory of cardinal numbers, where the minimum possible cardinality of a cofinal subset of A is referred to as the cofinality of A. A subset B of A is said to be coinitial (or dense in the sense of forcing) if it satisfies the following condition: :For every a ∈ A, there exists some b ∈ B such that b ≤ a.
This follows, for instance, from Maekawa's theorem, which states that the number of mountain folds at a flat-folded vertex differs from the number of valley folds by exactly two folds. Therefore, suppose that a crease pattern consists of an even number of creases, and let be the consecutive angles between the creases around the vertex, in clockwise order, starting at any one of the angles. Then Kawasaki's theorem states that the crease pattern may be folded flat if and only if the alternating sum and difference of the angles adds to zero: : An equivalent way of stating the same condition is that, if the angles are partitioned into two alternating subsets, then the sum of the angles in either of the two subsets is exactly 180 degrees. However, this equivalent form applies only to a crease pattern on a flat piece of paper, whereas the alternating sum form of the condition remains valid for crease patterns on conical sheets of paper with nonzero defect at the vertex.
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 .

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