ON THE FIELDS GENERATED BY THE LENGTHS OF CLOSED
... type, or one of them is of type Bn and the other of type Cn for some n > 3.) Much more precise results are available when the groups Γ1 and Γ2 are arithmetic (cf. [13], §1, and §5 below regarding the notion of arithmeticity). As follows from Theorem 1, we only need to consider the case where w1 = w2 ...
... type, or one of them is of type Bn and the other of type Cn for some n > 3.) Much more precise results are available when the groups Γ1 and Γ2 are arithmetic (cf. [13], §1, and §5 below regarding the notion of arithmeticity). As follows from Theorem 1, we only need to consider the case where w1 = w2 ...
Cellular Resolutions of Monomial Modules
... two comparable faces F 0 ⊂ F of the complex X have distinct degrees aF 6= aF 0 . The simplest example of a cellular resolution is the Taylor resolution for monomial ideals [Tay]. The Taylor resolution is easily generalized to arbitrary monomial modules M as follows. Let { mj | j ∈ I } be the minimal ...
... two comparable faces F 0 ⊂ F of the complex X have distinct degrees aF 6= aF 0 . The simplest example of a cellular resolution is the Taylor resolution for monomial ideals [Tay]. The Taylor resolution is easily generalized to arbitrary monomial modules M as follows. Let { mj | j ∈ I } be the minimal ...
(maximal) ideal in . Theorem
... the coz homeomorphism of the uniform spaces uX and vY with the first countability axiom (Corollaries 2.9, 2.10). For coz fine uniform spaces similar results for the rings of all bounded uniformly continuous functions have been obtained (Corollary 2.11). Maximal ideals of a ring Cu* ( X ) ( Cu ( ...
... the coz homeomorphism of the uniform spaces uX and vY with the first countability axiom (Corollaries 2.9, 2.10). For coz fine uniform spaces similar results for the rings of all bounded uniformly continuous functions have been obtained (Corollary 2.11). Maximal ideals of a ring Cu* ( X ) ( Cu ( ...
Algebra II (MA249) Lecture Notes Contents
... A large part of group theory consists of classifying groups with various properties. This means finding representatives of the isomorphism classes of groups with these properties. As an example of this, let us consider groups of order 4. Let G = {1, a, b, c} be such a group. Let us consider the orde ...
... A large part of group theory consists of classifying groups with various properties. This means finding representatives of the isomorphism classes of groups with these properties. As an example of this, let us consider groups of order 4. Let G = {1, a, b, c} be such a group. Let us consider the orde ...
On embeddings of spheres
... be couples (M, (I)) rather than just topological spaces. In order to justify the definition of 7?/-~ we must prove the following two lemmas. L]~[MA. X is closed under this multiplication. We have to show that [77l-~[ satisfies (i). The proof is however simple and can be omitted. I might also include ...
... be couples (M, (I)) rather than just topological spaces. In order to justify the definition of 7?/-~ we must prove the following two lemmas. L]~[MA. X is closed under this multiplication. We have to show that [77l-~[ satisfies (i). The proof is however simple and can be omitted. I might also include ...
Lecture notes
... Theorem 2.6 (i) The Euclidean Algorithm works: given positive integers a and b with a > b, applying the Euclidean Algorithm calculates gcd(a, b). (ii) Given integers a and b (one of which is non-zero) there exist integers u and v such that gcd(a, b) = ua + vb. Proof of (ii): First note that gcd(a, ...
... Theorem 2.6 (i) The Euclidean Algorithm works: given positive integers a and b with a > b, applying the Euclidean Algorithm calculates gcd(a, b). (ii) Given integers a and b (one of which is non-zero) there exist integers u and v such that gcd(a, b) = ua + vb. Proof of (ii): First note that gcd(a, ...
Proper holomorphic immersions into Stein manifolds with the density
... O(S)-convex subset K of the source Stein manifold S. The paper is organized as follows. In Sec. 2 we collect some preliminaries and develop the notion of a very special Cartan pair which plays an important role in the proof. In Sec. 3 we prove the main technical result, Lemma 3.1. Although it is sim ...
... O(S)-convex subset K of the source Stein manifold S. The paper is organized as follows. In Sec. 2 we collect some preliminaries and develop the notion of a very special Cartan pair which plays an important role in the proof. In Sec. 3 we prove the main technical result, Lemma 3.1. Although it is sim ...
Russell Sets, Topology, and Cardinals
... formulate something close to the Axiom of Choice in 1908. At first, many were reluctant to accept this axiom, especially because it would imply the truth of Georg Cantor’s Well-Ordering Principle which had been debated for almost three decades at the time. But as it was revealed that mathematicians, ...
... formulate something close to the Axiom of Choice in 1908. At first, many were reluctant to accept this axiom, especially because it would imply the truth of Georg Cantor’s Well-Ordering Principle which had been debated for almost three decades at the time. But as it was revealed that mathematicians, ...
filter convergence structures on posets
... We note that A1 ↓ = {x11 , x21 , ..., a1 }, A2 ↓ = {x12 , x22 , ..., a2 }, ..., Ak ↓ = {x1k , x2k , ..., ak } and > ∈ cl( A∈A A↓ ). Therefore, in Pc , the filter A converges to >. On the other hand, we see that the elements in each of the sets A1 ↓ , ..., Ak ↓ are incomparable. Moreover, the only ki ...
... We note that A1 ↓ = {x11 , x21 , ..., a1 }, A2 ↓ = {x12 , x22 , ..., a2 }, ..., Ak ↓ = {x1k , x2k , ..., ak } and > ∈ cl( A∈A A↓ ). Therefore, in Pc , the filter A converges to >. On the other hand, we see that the elements in each of the sets A1 ↓ , ..., Ak ↓ are incomparable. Moreover, the only ki ...
ADEQUATE SUBCATEGORIES
... denote the contravariant functor on a into the category of all sets and all functions which takes each object A of ( to the set Map (A, X), and each A in a to the function from Map (A, X) to Map (A’, X) mapping f:A’ defined by [Map (a, X)(f)](g) gf. Observe that every mapping h X --. Y in e induces ...
... denote the contravariant functor on a into the category of all sets and all functions which takes each object A of ( to the set Map (A, X), and each A in a to the function from Map (A, X) to Map (A’, X) mapping f:A’ defined by [Map (a, X)(f)](g) gf. Observe that every mapping h X --. Y in e induces ...
CONVERGENCE THEOREMS FOR PSEUDO
... yn = xn − x0 . Then lim yn = 0. Let U be a 0-neighborhood in E. There exists n→∞ a balanced 0-neighborhood V in E such that V ⊂ U . Then V ⊂ νV for all ν with |ν| ≥ 1. As yn −→τ 0 as n −→ ∞, there exists n0 ∈ N such that yn ∈ V whenever n ≥ n0 . Hence yn ∈ V ⊂ µV ⊂ µU whenever n ≥ n0 and µ ≥ 1. Let ...
... yn = xn − x0 . Then lim yn = 0. Let U be a 0-neighborhood in E. There exists n→∞ a balanced 0-neighborhood V in E such that V ⊂ U . Then V ⊂ νV for all ν with |ν| ≥ 1. As yn −→τ 0 as n −→ ∞, there exists n0 ∈ N such that yn ∈ V whenever n ≥ n0 . Hence yn ∈ V ⊂ µV ⊂ µU whenever n ≥ n0 and µ ≥ 1. Let ...
Birkhoff's representation theorem
This is about lattice theory. For other similarly named results, see Birkhoff's theorem (disambiguation).In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937.The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and Birkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.