ABSTRACT APPROACH TO FINITE RAMSEY
... More examples can be found in papers [17] and [18]. (Note that the terminology in [17] differs somewhat from the one in the present paper.) Section 8: This section contains applications of the abstract Ramsey approach to concrete situations. As a consequence of the general theory we obtain a new self ...
... More examples can be found in papers [17] and [18]. (Note that the terminology in [17] differs somewhat from the one in the present paper.) Section 8: This section contains applications of the abstract Ramsey approach to concrete situations. As a consequence of the general theory we obtain a new self ...
LOCALIZATION OF ALGEBRAS OVER COLOURED OPERADS
... We prove the following. Let C be any set and P a cofibrant C-coloured operad in the category of simplicial sets (or compactly generated spaces) acting on a simplicial (or topological) monoidal model category M. Let L be a homotopical localization functor on M whose class of equivalences is closed un ...
... We prove the following. Let C be any set and P a cofibrant C-coloured operad in the category of simplicial sets (or compactly generated spaces) acting on a simplicial (or topological) monoidal model category M. Let L be a homotopical localization functor on M whose class of equivalences is closed un ...
Limiting Absorption Principle for Schrödinger Operators with
... and Theorem 2.14 in [GM]). “Strongly singular” terms (more singular than our Vc ) are also considered in Section 3 in [GM]. Remark 1.4. When w = 0, H has a good enough regularity w.r.t. A (see Section 3 and Appendix B for details) thus the Mourre theory based on A can be applied to get Theorem 1.2. ...
... and Theorem 2.14 in [GM]). “Strongly singular” terms (more singular than our Vc ) are also considered in Section 3 in [GM]. Remark 1.4. When w = 0, H has a good enough regularity w.r.t. A (see Section 3 and Appendix B for details) thus the Mourre theory based on A can be applied to get Theorem 1.2. ...
NSE CHARACTERIZATION OF THE SIMPLE GROUP L2(3n) Hosein
... power. Is G isomorphic to L2 (q)? Our main purpose is to show that the problem has an affirmative answer for q = 3n and |π(L2 (q))| = 4. In fact, we have the following main theorem. Main Theorem. Let G be a group such that nse(G) = nse(L2 (3n )), where n, (3 − 1)/2 and (3n + 1)/4 are odd primes. The ...
... power. Is G isomorphic to L2 (q)? Our main purpose is to show that the problem has an affirmative answer for q = 3n and |π(L2 (q))| = 4. In fact, we have the following main theorem. Main Theorem. Let G be a group such that nse(G) = nse(L2 (3n )), where n, (3 − 1)/2 and (3n + 1)/4 are odd primes. The ...
Representation schemes and rigid maximal Cohen
... So assume that f annihilates the ideal (Wi : i = 1, . . . , s) ⊂ T . Then f ∗ (φ) factors through an R-module map f ∗ (φ) : A → End(V• ) ⊗ R ⊗ T 0 . For this to be an R-algebra map, it must satisfy two conditions. First, f ∗ (φ)(1A ) = id ⊗1 ⊗ 1. This means that φ(1A ) − id ⊗1 ⊗ 1 must map to zero u ...
... So assume that f annihilates the ideal (Wi : i = 1, . . . , s) ⊂ T . Then f ∗ (φ) factors through an R-module map f ∗ (φ) : A → End(V• ) ⊗ R ⊗ T 0 . For this to be an R-algebra map, it must satisfy two conditions. First, f ∗ (φ)(1A ) = id ⊗1 ⊗ 1. This means that φ(1A ) − id ⊗1 ⊗ 1 must map to zero u ...
Weights for Objects of Monoids
... (i.e. a strict symmetric monoidal 2-category whose 0-cells are natural numbers such that I = 0 and n ⊗ m = n + m) but the Eilenberg-Moore object has sufficiently simple structure that we are able to get it as a weighted limit from the functor with domain s∆. Thus in this case no coalgebra is needed. ...
... (i.e. a strict symmetric monoidal 2-category whose 0-cells are natural numbers such that I = 0 and n ⊗ m = n + m) but the Eilenberg-Moore object has sufficiently simple structure that we are able to get it as a weighted limit from the functor with domain s∆. Thus in this case no coalgebra is needed. ...
Abstract Vector Spaces, Linear Transformations, and Their
... Corollary 1.8 If V is a vector space and S, T ⊆ V , then the following hold: (1) S ⊆ T ⊆ V =⇒ span(S) ⊆ span(T ) (2) S ⊆ T ⊆ V and span(S) = V =⇒ span(T ) = V (3) span(S ∪ T ) = span(S) + span(T ) (4) span(S ∩ T ) ⊆ span(S) ∩ span(T ) Proof: (1) and (2) are immediate, so we only need to prove 3 and ...
... Corollary 1.8 If V is a vector space and S, T ⊆ V , then the following hold: (1) S ⊆ T ⊆ V =⇒ span(S) ⊆ span(T ) (2) S ⊆ T ⊆ V and span(S) = V =⇒ span(T ) = V (3) span(S ∪ T ) = span(S) + span(T ) (4) span(S ∩ T ) ⊆ span(S) ∩ span(T ) Proof: (1) and (2) are immediate, so we only need to prove 3 and ...
Weights for Objects of Monoids
... (i.e. a strict symmetric monoidal 2-category whose 0-cells are natural numbers such that I = 0 and n ⊗ m = n + m) but the Eilenberg-Moore object has sufficiently simple structure that we are able to get it as a weighted limit from the functor with domain s∆. Thus in this case no coalgebra is needed. ...
... (i.e. a strict symmetric monoidal 2-category whose 0-cells are natural numbers such that I = 0 and n ⊗ m = n + m) but the Eilenberg-Moore object has sufficiently simple structure that we are able to get it as a weighted limit from the functor with domain s∆. Thus in this case no coalgebra is needed. ...
CENTRALIZERS IN DIFFERENTIAL, PSEUDO
... has finite rank over F[a] is that given any b e Cs(a), there exists a nonconstant polynomial q over F in two commuting indeterminates such that q(a, b) = 0. The pattern of these results was clear starting with the work of Flanders [8], who remarked, in the case when R is the ring of complex-valued C ...
... has finite rank over F[a] is that given any b e Cs(a), there exists a nonconstant polynomial q over F in two commuting indeterminates such that q(a, b) = 0. The pattern of these results was clear starting with the work of Flanders [8], who remarked, in the case when R is the ring of complex-valued C ...
4. Number Theory (Part 2)
... In practice, computing the gcd by prime factorization is too slow, especially when the numbers are large. Luckily, an efficient algorithm was given by Euclid way back in the year 300BC. The key idea to find gcd(a, b) is based on two facts: (i) gcd(a, 0) = a. (ii) gcd(a, b) = gcd(b, r ), where r is t ...
... In practice, computing the gcd by prime factorization is too slow, especially when the numbers are large. Luckily, an efficient algorithm was given by Euclid way back in the year 300BC. The key idea to find gcd(a, b) is based on two facts: (i) gcd(a, 0) = a. (ii) gcd(a, b) = gcd(b, r ), where r is t ...
M13/08
... Sp × Sp of Z p × Z p acts ergodically on AαS for any α ∈ Ξp via the dual action: Theorem 2.11. [6, Theorem 3.8] Let p ∈ N, p > 1 and α ∈ Ξp . There exists at least one tracial state on the noncommutative solenoid AαS . Moreover, this tracial state is unique if, and only if α is not periodic. Moreove ...
... Sp × Sp of Z p × Z p acts ergodically on AαS for any α ∈ Ξp via the dual action: Theorem 2.11. [6, Theorem 3.8] Let p ∈ N, p > 1 and α ∈ Ξp . There exists at least one tracial state on the noncommutative solenoid AαS . Moreover, this tracial state is unique if, and only if α is not periodic. Moreove ...
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.