IDEAL BICOMBINGS FOR HYPERBOLIC GROUPS
... Let (X, d) be a metric space. Denote by B(x, r), respectively B(x, r), the open, respectively closed, r-ball with centre x, and by N (S, r) the closed r-neighbourhood of the subset S in X. One calls X proper if all closed balls are compact. A background reference for hyperbolic metric spaces is [Gdl ...
... Let (X, d) be a metric space. Denote by B(x, r), respectively B(x, r), the open, respectively closed, r-ball with centre x, and by N (S, r) the closed r-neighbourhood of the subset S in X. One calls X proper if all closed balls are compact. A background reference for hyperbolic metric spaces is [Gdl ...
Atom structures
... AC atom-canonical if (At A)+ is in V for every atomic A in V, or equivalently, if At V ⊆ Str V.4 AX atom-complex if every atomic algebra in V is isomorphic to a complex algebra over its atom structure. AO atom-corresponding if there is a set ∆ of first order sentences in the frame language such that ...
... AC atom-canonical if (At A)+ is in V for every atomic A in V, or equivalently, if At V ⊆ Str V.4 AX atom-complex if every atomic algebra in V is isomorphic to a complex algebra over its atom structure. AO atom-corresponding if there is a set ∆ of first order sentences in the frame language such that ...
HOW TO PROVE THAT A NON-REPRESENTABLE FUNCTOR IS
... which implies that g = g 0 by the universal property of idA ∈ hom(A, A). X We have shown that u = a2 has the universal property that for every square b2 in any ring B, there is a unique homomorphism A → B sending u to b2 . Now we have transferred the problem to ring theory, where we will finish it o ...
... which implies that g = g 0 by the universal property of idA ∈ hom(A, A). X We have shown that u = a2 has the universal property that for every square b2 in any ring B, there is a unique homomorphism A → B sending u to b2 . Now we have transferred the problem to ring theory, where we will finish it o ...
ALGEBRAIC FORMULAS FOR THE COEFFICIENTS OF HALF
... binary quadratic forms Q(x, y) = ax2 + bxy + cy 2 with the property that 6 | a. The group Γ0 (6) acts on such forms, and we let Qn be any set of representatives of those equivalence classes with a > 0 and b ≡ 1 (mod 12). For each Q(x, y), we let αQ be the CM point in H, the upper half of the complex ...
... binary quadratic forms Q(x, y) = ax2 + bxy + cy 2 with the property that 6 | a. The group Γ0 (6) acts on such forms, and we let Qn be any set of representatives of those equivalence classes with a > 0 and b ≡ 1 (mod 12). For each Q(x, y), we let αQ be the CM point in H, the upper half of the complex ...
Quaternion Algebras and Quadratic Forms - UWSpace
... Corollary 1.2.6 If (V, B) is a quadratic space and S is a regular subspace, then: 1. V = S⊥S ⊥ 2. If T is a subspace of V such that V = S⊥T , then T = S ⊥ . Proof: (1) Since S is regular, S ∩ S ⊥ = 0. Since we already have the dimension theorem for regular subspaces, it suffices to show that V is s ...
... Corollary 1.2.6 If (V, B) is a quadratic space and S is a regular subspace, then: 1. V = S⊥S ⊥ 2. If T is a subspace of V such that V = S⊥T , then T = S ⊥ . Proof: (1) Since S is regular, S ∩ S ⊥ = 0. Since we already have the dimension theorem for regular subspaces, it suffices to show that V is s ...
DILATION OF THE WEYL SYMBOL AND BALIAN
... Theorem 1.5 implies a new version for Gabor systems with window in the standard class M 1 , of a recent non-existence result for time-frequency localized Riesz bases, found in [32]. In that paper, Gröchenig and Malinnikova prove the nonexistence of Riesz bases when the functions used are well local ...
... Theorem 1.5 implies a new version for Gabor systems with window in the standard class M 1 , of a recent non-existence result for time-frequency localized Riesz bases, found in [32]. In that paper, Gröchenig and Malinnikova prove the nonexistence of Riesz bases when the functions used are well local ...
Simple Proof of the Prime Number Theorem
... Prime Number Theorem from the non-vanishing of ζ(s) on Re (s) = 1. [Erdos 1950] and [Selberg 1950] gave proofs of the Prime Number Theorem elementary in the sense of using no complex analysis or other limiting procedure devices. At the time, it was hoped that this might shed light on the behavior of ...
... Prime Number Theorem from the non-vanishing of ζ(s) on Re (s) = 1. [Erdos 1950] and [Selberg 1950] gave proofs of the Prime Number Theorem elementary in the sense of using no complex analysis or other limiting procedure devices. At the time, it was hoped that this might shed light on the behavior of ...
THE DYNAMICAL MORDELL-LANG PROBLEM FOR NOETHERIAN SPACES
... Petsche uses methods from topological dynamics and ergodic theory; in particular, he uses Berkovich spaces and a strong topological version of Furstenberg’s Poincaré Recurrence Theorem. William Gignac indicated to us that Theorem 1.4 can also be derived using arguments that come from a deep result ...
... Petsche uses methods from topological dynamics and ergodic theory; in particular, he uses Berkovich spaces and a strong topological version of Furstenberg’s Poincaré Recurrence Theorem. William Gignac indicated to us that Theorem 1.4 can also be derived using arguments that come from a deep result ...
Cyclic groups and elementary number theory
... Proof. Let H ≤ Z. If H = {0}, then H = h0i and hence H is cyclic. Thus we may assume that there exists an a ∈ H, a 6= 0. Then −a ∈ H as well, and either a > 0 or −a > 0. In particular, the set H ∩ N is nonempty. Let d be the smallest element of H ∩N, which exists by the well-ordering principle. To p ...
... Proof. Let H ≤ Z. If H = {0}, then H = h0i and hence H is cyclic. Thus we may assume that there exists an a ∈ H, a 6= 0. Then −a ∈ H as well, and either a > 0 or −a > 0. In particular, the set H ∩ N is nonempty. Let d be the smallest element of H ∩N, which exists by the well-ordering principle. To p ...
Revised version
... where 3 denotesQthe usual Legendre symbol. Thus, the degree of any solution has the form m2 j pj where the pj are primes ≡ 0, 1 (mod 3). Here is the organization of the paper. In Section 2, we review the addition law for points on elliptic curves. This endows V with the structure of an abelian group ...
... where 3 denotesQthe usual Legendre symbol. Thus, the degree of any solution has the form m2 j pj where the pj are primes ≡ 0, 1 (mod 3). Here is the organization of the paper. In Section 2, we review the addition law for points on elliptic curves. This endows V with the structure of an abelian group ...
