Morphisms of Algebraic Stacks
... (1) f is quasi-separated, (2) ∆f is quasi-compact, or (3) ∆f is of finite type. Proof. The statements “f is quasi-separated”, “∆f is quasi-compact”, and “∆f is of finite type” refer to the notions defined in Properties of Stacks, Section 3. Note that (2) and (3) are equivalent in view of the fact th ...
... (1) f is quasi-separated, (2) ∆f is quasi-compact, or (3) ∆f is of finite type. Proof. The statements “f is quasi-separated”, “∆f is quasi-compact”, and “∆f is of finite type” refer to the notions defined in Properties of Stacks, Section 3. Note that (2) and (3) are equivalent in view of the fact th ...
Vector Bundles And F Theory
... general polynomial in x and y with at most an nth order pole at infinity, and (modulo the Weierstrass equation) at most a linear dependence on y. To allow for a completely general set of Qi , one restricts the ak only by requiring that they are not all identically zero. (For example, an vanishes if ...
... general polynomial in x and y with at most an nth order pole at infinity, and (modulo the Weierstrass equation) at most a linear dependence on y. To allow for a completely general set of Qi , one restricts the ak only by requiring that they are not all identically zero. (For example, an vanishes if ...
Symplectic structures -- a new approach to geometry.
... where the flux group Γω is a subgroup of H 1 (M, R). Example. In the case of the torus T 2 with a symplectic form dx ∧ dy of total area 1, the group Γω is H 1 (M, Z). The family of rotations Rt : (x, y) 7→ (x+t, y) of the torus T 2 consists of symplectomorphisms that are not Hamiltonian. Its image u ...
... where the flux group Γω is a subgroup of H 1 (M, R). Example. In the case of the torus T 2 with a symplectic form dx ∧ dy of total area 1, the group Γω is H 1 (M, Z). The family of rotations Rt : (x, y) 7→ (x+t, y) of the torus T 2 consists of symplectomorphisms that are not Hamiltonian. Its image u ...
ISOSPECTRAL AND ISOSCATTERING MANIFOLDS: A SURVEY
... If (M, g) is a compact manifold with boundary and Dirichlet or Neumann conditions are imposed, the spectrum is again an infinite sequence of eigenvalues. Find manifolds (M1 , g1 ) and (M2 , g2 ) with the same spectrum. We will call closed manifolds with the same spectrum (including multiplicities) i ...
... If (M, g) is a compact manifold with boundary and Dirichlet or Neumann conditions are imposed, the spectrum is again an infinite sequence of eigenvalues. Find manifolds (M1 , g1 ) and (M2 , g2 ) with the same spectrum. We will call closed manifolds with the same spectrum (including multiplicities) i ...
HEIGHTS OF VARIETIES IN MULTIPROJECTIVE SPACES AND
... height of polynomials satisfying such an identity became a widely considered question in connection with problems in computer algebra and Diophantine approximation. The results in this direction are generically known as arithmetic Nullstellensätze and they play an important role in number theory an ...
... height of polynomials satisfying such an identity became a widely considered question in connection with problems in computer algebra and Diophantine approximation. The results in this direction are generically known as arithmetic Nullstellensätze and they play an important role in number theory an ...
Algebraic D-groups and differential Galois theory
... In particular, the above map defines a K-rational isomorphism between the vector groups τ (G)e and T (G)e = L(G). Note that we have again an exact sequence 0 → τ (G)e → τ (G) → G → e of algebraic groups over K, which by virtue of the (canonical) isomorphism between τ (G)e and L(G) given by Lemma 2.3 ...
... In particular, the above map defines a K-rational isomorphism between the vector groups τ (G)e and T (G)e = L(G). Note that we have again an exact sequence 0 → τ (G)e → τ (G) → G → e of algebraic groups over K, which by virtue of the (canonical) isomorphism between τ (G)e and L(G) given by Lemma 2.3 ...
Noncommutative geometry @n
... Ever since the dawn of noncommutative algebraic geometry in the midseventies, see for example the work of P. Cohn [21], J. Golan [38], C. Procesi [86], F. Van Oystaeyen and A. Verschoren [103],[105], it has been ring theorists’ hope that this theory might one day be relevant to commutative geometry, ...
... Ever since the dawn of noncommutative algebraic geometry in the midseventies, see for example the work of P. Cohn [21], J. Golan [38], C. Procesi [86], F. Van Oystaeyen and A. Verschoren [103],[105], it has been ring theorists’ hope that this theory might one day be relevant to commutative geometry, ...
ON THE WEAK LEFSCHETZ PROPERTY FOR POWERS OF
... shown the following: Let I = h`t1 , . . . , `tn i ⊂ k[x1 , . . . , x4 ] with `i general linear forms. If n ∈ {5, 6, 7, 8} then the WLP fails, respectively, for t ≥ {3, 27, 140, 704}. A famous conjecture of Fröberg gives the expected Hilbert function for an ideal of s general forms of prescribed deg ...
... shown the following: Let I = h`t1 , . . . , `tn i ⊂ k[x1 , . . . , x4 ] with `i general linear forms. If n ∈ {5, 6, 7, 8} then the WLP fails, respectively, for t ≥ {3, 27, 140, 704}. A famous conjecture of Fröberg gives the expected Hilbert function for an ideal of s general forms of prescribed deg ...
On fusion categories - Annals of Mathematics
... results which are partially or fully due to other authors, and our own results are blended in at appropriate places. Sections 3-7 are mostly devoted to review of the technical tools and to proofs of the results of Section 2. Namely, in Section 3, we prove one of the main theorems of this paper, sayi ...
... results which are partially or fully due to other authors, and our own results are blended in at appropriate places. Sections 3-7 are mostly devoted to review of the technical tools and to proofs of the results of Section 2. Namely, in Section 3, we prove one of the main theorems of this paper, sayi ...
Problems in the classification theory of non-associative
... unital algebra A is associative if and only if every isotope of A that is unital again is isomorphic to A. Over any field k, the only one-dimensional division algebra is the field itself. In dimension two, every division algebra can be constructed by isotopy from a quadratic field extension of k. Th ...
... unital algebra A is associative if and only if every isotope of A that is unital again is isomorphic to A. Over any field k, the only one-dimensional division algebra is the field itself. In dimension two, every division algebra can be constructed by isotopy from a quadratic field extension of k. Th ...
Math 257A: Introduction to Symplectic Topology, Lecture 2
... Moreover, U (n + 1) transitively on S 2n+1 (and, hence, on CP n ). Therefore, ωF S is nondegenerate at every point of CP n (since the action of U (n + 1) defines a family of diffeomorphisms from CP n to itself that send [1 : 0 : · · · : 0] to any given point and preserve ωF S ). Fact. Any complex su ...
... Moreover, U (n + 1) transitively on S 2n+1 (and, hence, on CP n ). Therefore, ωF S is nondegenerate at every point of CP n (since the action of U (n + 1) defines a family of diffeomorphisms from CP n to itself that send [1 : 0 : · · · : 0] to any given point and preserve ωF S ). Fact. Any complex su ...
Real Algebraic Sets
... stratification: this means that for each cell C, the closure clos(C) is the union of C and of cells of smaller dimension. This property of incidence between the cells may not be satisfied by a cad (where the cells have to be arranged in cylinders whose directions are given by the coordinate axes), b ...
... stratification: this means that for each cell C, the closure clos(C) is the union of C and of cells of smaller dimension. This property of incidence between the cells may not be satisfied by a cad (where the cells have to be arranged in cylinders whose directions are given by the coordinate axes), b ...
Publikationen - Mathematisches Institut
... Blanc, Jérémy; Hedén, Isac: The group of Cremona transformations generated by linear maps and the standard involution, in: Annales de l'institut Fourier 65, 2015, H. 6, S. 2641-2680.Blanc, Jérémy; Hedén, Isac: The group of Cremona transformations generated by linear maps and the standard involution, ...
... Blanc, Jérémy; Hedén, Isac: The group of Cremona transformations generated by linear maps and the standard involution, in: Annales de l'institut Fourier 65, 2015, H. 6, S. 2641-2680.Blanc, Jérémy; Hedén, Isac: The group of Cremona transformations generated by linear maps and the standard involution, ...
Topological groups and stabilizers of types
... • In M = (Q, <, +) the cut of π in Q is not a definable type in M. • In Gset every type is (trivially) definable, because every subset is definable. ...
... • In M = (Q, <, +) the cut of π in Q is not a definable type in M. • In Gset every type is (trivially) definable, because every subset is definable. ...
Conjugacy and cocycle conjugacy of automorphisms of O2 are not
... on the Hilbert space shows that the relation of conjugacy of unitary operators is Borel. We will show that Theorem 1.1 holds even if one only considers automorphisms of finite order whose induced finite group action has Rokhlin dimension at most one in the sense of [20]. Moreover, it will follow fro ...
... on the Hilbert space shows that the relation of conjugacy of unitary operators is Borel. We will show that Theorem 1.1 holds even if one only considers automorphisms of finite order whose induced finite group action has Rokhlin dimension at most one in the sense of [20]. Moreover, it will follow fro ...
algebraic density property of homogeneous spaces
... Proposition 10. Let X be an SL2 -variety with associated locally nilpotent derivations δ1 and δ2 , let Y be a normal affine algebraic variety equipped with a trivial SL2 -action, and let r : X → Y be a surjective SL2 -equivariant morphism. Suppose that for any y ∈ Y there exists an étale neighborho ...
... Proposition 10. Let X be an SL2 -variety with associated locally nilpotent derivations δ1 and δ2 , let Y be a normal affine algebraic variety equipped with a trivial SL2 -action, and let r : X → Y be a surjective SL2 -equivariant morphism. Suppose that for any y ∈ Y there exists an étale neighborho ...
FINITE SIMPLICIAL MULTICOMPLEXES
... Γ is called pure if all the maximal facets have the same dimension, equal to dim(Γ). Remark 1.2. An arbitrary intersection and a finite union of finite multicomplexes are again multicomplexes. Therefore, the set of all finite multicomplexes in Nn is the family of closed sets in a topology on Nn , ca ...
... Γ is called pure if all the maximal facets have the same dimension, equal to dim(Γ). Remark 1.2. An arbitrary intersection and a finite union of finite multicomplexes are again multicomplexes. Therefore, the set of all finite multicomplexes in Nn is the family of closed sets in a topology on Nn , ca ...
cylindric algebras and algebras of substitutions^) 167
... in this paper was done while the author held an NSF Faculty ...
... in this paper was done while the author held an NSF Faculty ...
Geometry
... A mathematically generated pattern that is endlessly complex Fractal patterns often resemble natural phenomena in the way they repeat elements with slight variations each time ...
... A mathematically generated pattern that is endlessly complex Fractal patterns often resemble natural phenomena in the way they repeat elements with slight variations each time ...
Trans Dimensional Unified Field Theory
... but also the same happens similarly in the periodic table of elements though perhaps, as in the periodic table, instead of addition perhaps multiplication was the ladder to complexity, and the component was velocity and time, VT (D). So to get to D^4, conceptually I thought of a vector VT (D) in sp ...
... but also the same happens similarly in the periodic table of elements though perhaps, as in the periodic table, instead of addition perhaps multiplication was the ladder to complexity, and the component was velocity and time, VT (D). So to get to D^4, conceptually I thought of a vector VT (D) in sp ...
(IN)CONSISTENCY: SOME LOW-DIMENSIONAL
... 3. D1 (f, ~γ ) for Boolean constants f in higher dimensions Proposition 2 easily generalizes to the following result: Proposition 7. Assume f : 2n → 2n is a Boolean function such that each component fi of f is a Boolean constant. Assume γi > 0 for all i ∈ [n] and that each Pi satisfies Conditions 1 ...
... 3. D1 (f, ~γ ) for Boolean constants f in higher dimensions Proposition 2 easily generalizes to the following result: Proposition 7. Assume f : 2n → 2n is a Boolean function such that each component fi of f is a Boolean constant. Assume γi > 0 for all i ∈ [n] and that each Pi satisfies Conditions 1 ...
Pacal - PAN
... February which has 28.’ So goes the folk rhyme underscoring the illogical nature of the Gregorian calendar. By contrast, a far easier and more logical way to divide the solar year would be by thirteen 28-day months with one extra free day. The point is this: there is no logical or scientific relatio ...
... February which has 28.’ So goes the folk rhyme underscoring the illogical nature of the Gregorian calendar. By contrast, a far easier and more logical way to divide the solar year would be by thirteen 28-day months with one extra free day. The point is this: there is no logical or scientific relatio ...
Exotic spheres and curvature - American Mathematical Society
... manifold to be smooth. Such an atlas can then be extended to a maximal smooth atlas by including all possible charts which satisfy the compatibility condition with the original maps. It is far from obvious that there are manifolds which (up to diffeomorphism) admit more than one distinct smooth struc ...
... manifold to be smooth. Such an atlas can then be extended to a maximal smooth atlas by including all possible charts which satisfy the compatibility condition with the original maps. It is far from obvious that there are manifolds which (up to diffeomorphism) admit more than one distinct smooth struc ...
Dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it – for example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on it – for example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces.In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, but rather are known relative to the motion of an observer. Minkowski space first approximates the universe without gravity; the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to describe string theory, and the state-space of quantum mechanics is an infinite-dimensional function space.The concept of dimension is not restricted to physical objects. High-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space we live in.