2.1 (Inductive Reasoning and Conjecture).notebook
... Conjecture: DE + EF = DF Answer: Sometimes if E is not between D and F, then it is not true, but it E does fall between D and F, then it is true. SO SOMETIMES IT's TRUE. ...
... Conjecture: DE + EF = DF Answer: Sometimes if E is not between D and F, then it is not true, but it E does fall between D and F, then it is true. SO SOMETIMES IT's TRUE. ...
ADEQUATE SUBCATEGORIES
... This paper introduces and studies the notion of a left adequate subcategory of an arbitrary category (and the dual notion). Definition follows. Let a be a full subcategory of e. For any object X of (, let Map ((, X) denote the contravariant functor on a into the category of all sets and all function ...
... This paper introduces and studies the notion of a left adequate subcategory of an arbitrary category (and the dual notion). Definition follows. Let a be a full subcategory of e. For any object X of (, let Map ((, X) denote the contravariant functor on a into the category of all sets and all function ...
The Kazhdan-Lusztig polynomial of a matroid
... braid matroid. Polo [Pol99] has shown that any polynomial with non-negative coefficients and constant term 1 appears as a Kazhdan-Lusztig polynomial associated to some symmetric group, while Kazhdan-Lusztig polynomials of matroids are far more restrictive (see Proposition 2.14). The original work of ...
... braid matroid. Polo [Pol99] has shown that any polynomial with non-negative coefficients and constant term 1 appears as a Kazhdan-Lusztig polynomial associated to some symmetric group, while Kazhdan-Lusztig polynomials of matroids are far more restrictive (see Proposition 2.14). The original work of ...
Topology in the 20th century
... itself has a fixed point. In the early 20th century Dehn extended the ideas of 3-dimensional topology a long way. Starting from the fundamental group, he created the combinatorial theory of finitely presented groups, stating its main algorithmic problems: the problem of identity of words, conjugacy an ...
... itself has a fixed point. In the early 20th century Dehn extended the ideas of 3-dimensional topology a long way. Starting from the fundamental group, he created the combinatorial theory of finitely presented groups, stating its main algorithmic problems: the problem of identity of words, conjugacy an ...
Projective p-adic representations of the K-rational geometric fundamental group (with G. Frey).
... and put P = (a, 0) ∈ C(K). Then the K-rational geometric fundamental group π1 (C, P ) is infinite; more precisely, for every prime p ≡ 5 (mod 12) (with p 6= char(K)), the group PSL3 (Zp ) is a factor of π1 (C, P ), i.e. there is a surjection ρ̃ : π1 (C, P ) → PSL3 (Zp ). Corollary 1.2 In the above s ...
... and put P = (a, 0) ∈ C(K). Then the K-rational geometric fundamental group π1 (C, P ) is infinite; more precisely, for every prime p ≡ 5 (mod 12) (with p 6= char(K)), the group PSL3 (Zp ) is a factor of π1 (C, P ), i.e. there is a surjection ρ̃ : π1 (C, P ) → PSL3 (Zp ). Corollary 1.2 In the above s ...
de Rham cohomology
... Let X = j Xj . For each j, let ij : Xj −→ X be the canonical injection defined by ij (x) = (x, j). The disjoint union topology on X is defined by the condition U is open in X if and only if i−1 j (U) is open in Xj , for every j. Given two continuous maps F, G : X −→ Y , a homotopy between F and G is ...
... Let X = j Xj . For each j, let ij : Xj −→ X be the canonical injection defined by ij (x) = (x, j). The disjoint union topology on X is defined by the condition U is open in X if and only if i−1 j (U) is open in Xj , for every j. Given two continuous maps F, G : X −→ Y , a homotopy between F and G is ...
Lecture 4 Supergroups
... is G-stable if ∆V (W ) ⊂ k[G] ⊗ W . Equivalently W is G-stable if ρA (g)(A ⊗ W) ⊂ A ⊗ W. Definition 3.7. The right regular representation of the affine algebraic group G is the representation of G in the (infinite dimensional) super vector space k[G] corresponding to the comodule map: ∆ : k[G] −→ k[ ...
... is G-stable if ∆V (W ) ⊂ k[G] ⊗ W . Equivalently W is G-stable if ρA (g)(A ⊗ W) ⊂ A ⊗ W. Definition 3.7. The right regular representation of the affine algebraic group G is the representation of G in the (infinite dimensional) super vector space k[G] corresponding to the comodule map: ∆ : k[G] −→ k[ ...
Cohomology of cyro-electron microscopy
... earlier — we will call this the intrinsic tribar to distinguish it from the nonexistent 3D object. In fact, the intrinsic tribar can be embedded in a three-dimensional manifold R3 /Z, a quotient space of R3 under a certain action of the discrete group Z related to Figure 2 (see [9] for details). We ...
... earlier — we will call this the intrinsic tribar to distinguish it from the nonexistent 3D object. In fact, the intrinsic tribar can be embedded in a three-dimensional manifold R3 /Z, a quotient space of R3 under a certain action of the discrete group Z related to Figure 2 (see [9] for details). We ...
x - TeacherWeb
... 11-2 Simplifying Algebraic Expressions In the expression 7x + 9y, 7x and 9y are called terms. A term can be a number, a variable, or a product of numbers and variables. Terms in an expression are separated by plus or minus signs. ...
... 11-2 Simplifying Algebraic Expressions In the expression 7x + 9y, 7x and 9y are called terms. A term can be a number, a variable, or a product of numbers and variables. Terms in an expression are separated by plus or minus signs. ...
The structure of Coh(P1) 1 Coherent sheaves
... order of f at p. In the case where f has degree 1 (for example, f = x), we call Of a skyscraper sheaf. We define Of to have degree equal to deg(f ). (Intuitively, we should think of torsion sheaves as being very small. For c, where M = k[x0 , x1 ]/(f ). Note example, one can easily show that Of = M ...
... order of f at p. In the case where f has degree 1 (for example, f = x), we call Of a skyscraper sheaf. We define Of to have degree equal to deg(f ). (Intuitively, we should think of torsion sheaves as being very small. For c, where M = k[x0 , x1 ]/(f ). Note example, one can easily show that Of = M ...
Oct 2013
... 2) One strategy is to draw a line through C that is parallel to the other two parallel lines, thus creating pairs of angles that are either congruent or supplementary, as shown in this diagram: ...
... 2) One strategy is to draw a line through C that is parallel to the other two parallel lines, thus creating pairs of angles that are either congruent or supplementary, as shown in this diagram: ...
5.1- Patterns and Conjectures
... Conjecture – An unproven statement that is based on a pattern or observations. Counterexample – An example that disproves a statement or a conjecture. ...
... Conjecture – An unproven statement that is based on a pattern or observations. Counterexample – An example that disproves a statement or a conjecture. ...
Homotopy Theory of Topological Spaces and Simplicial Sets
... character of the subject decided to add this material as an appendix. So the reader might decide for himself whether or not to read this before starting on the article. Furthermore the reader is assumed to know some algebraic topology. In algebraic topology we know the definition of homotopic maps a ...
... character of the subject decided to add this material as an appendix. So the reader might decide for himself whether or not to read this before starting on the article. Furthermore the reader is assumed to know some algebraic topology. In algebraic topology we know the definition of homotopic maps a ...
- Departament de matemàtiques
... monoid as well as the Eckmann-Hilton argument make sense in any monoidal category in place of the category of sets. A double monoid in Cat is the same thing as a category with two compatible strict monoidal structures, and the Eckmann-Hilton argument shows that such are commutative. It is natural to ...
... monoid as well as the Eckmann-Hilton argument make sense in any monoidal category in place of the category of sets. A double monoid in Cat is the same thing as a category with two compatible strict monoidal structures, and the Eckmann-Hilton argument shows that such are commutative. It is natural to ...
The structure of reductive groups - UBC Math
... a change of coordinates identifies the group over C with Gm . A torus defined over F that is isomorphic to some (Gm )n over the field F itself is called an F -split torus. ...
... a change of coordinates identifies the group over C with Gm . A torus defined over F that is isomorphic to some (Gm )n over the field F itself is called an F -split torus. ...
A model structure for quasi-categories
... and target and s0 : X0 → X1 picking out the identities. Composition is freely generated by elements of X1 subject to relations given by elements of X2 . More specifically, if x ∈ X2 , then we impose the relation that d1 x = d0 x ◦ d2 x. This functor is very destructive. In particular, it only depend ...
... and target and s0 : X0 → X1 picking out the identities. Composition is freely generated by elements of X1 subject to relations given by elements of X2 . More specifically, if x ∈ X2 , then we impose the relation that d1 x = d0 x ◦ d2 x. This functor is very destructive. In particular, it only depend ...
4. Morphisms
... So far we have defined and studied regular functions on an affine variety X. They can be thought of as the morphisms (i. e. the “nice” maps) from open subsets of X to the ground field K = A1 . We now want to extend this notion of morphisms to maps to other affine varieties than just A1 (and in fact ...
... So far we have defined and studied regular functions on an affine variety X. They can be thought of as the morphisms (i. e. the “nice” maps) from open subsets of X to the ground field K = A1 . We now want to extend this notion of morphisms to maps to other affine varieties than just A1 (and in fact ...