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On -adic Saito-Kurokawa lifting and its application
On -adic Saito-Kurokawa lifting and its application

Subgroups of Finite Index in Profinite Groups
Subgroups of Finite Index in Profinite Groups

... Theorem 1.1. Suppose that G is a topologically finitely generated profinite group. Then every subgroup of G of finite index is open. One way to view Theorem 1.1 is as a statement that the algebraic structure of a finitely generated profinite group somehow also encodes the topological structure. That ...
CLASS NOTES ON LINEAR ALGEBRA 1. Matrices Suppose that F is
CLASS NOTES ON LINEAR ALGEBRA 1. Matrices Suppose that F is

... Theorem 4.13. (Universal Property of Vector Spaces) Suppose that V and W are finite dimensional vector spaces, β = {v1 , . . . , vn } is a basis of V , and w1 , . . . , wn ∈ W . Then there exists a unique linear map L : V → W such that L(vi ) = wi for 1 ≤ i ≤ n. Proof. We first prove existence. For ...
Chapter I, Section 6
Chapter I, Section 6

... Definition (6.6.1). — [Liu, Ex. 2.3.17] A morphism f : X → Y is quasi-compact if f −1 (V ) is quasi-compact for every quasi-compact open V ⊆ Y . Suppose B is a base of the topology on Y which consists of quasi-compact open sets (affines, for example). For f to be quasi-compact, it suffices that f −1 ...
A cursory introduction to spin structure
A cursory introduction to spin structure

... by the units of norm one in V. Spin(V ) := Pin(V ) ∩ Cl0 (V ). The generators of Pin(V ) are in Cl1 (V ) so Spin(V ) is the subgroup of index two consisting of all elements in Pin(V ) expressible as a product of an even number of generators in Pin(V ). There is a natural action of SO(n) on V , SO(V ...
PDF
PDF

Continuous Logic and Probability Algebras THESIS Presented in
Continuous Logic and Probability Algebras THESIS Presented in

... Multi-valued logics are formal systems that deviate from classical logic by allowing for more than two truth values. An early version of multi-valued logic appeared in 1920 with the development of a three-valued logic by Łukasiewicz. This was extended to an infinitevalued propositional logic for whi ...
On bimeasurings
On bimeasurings

... in general only if A is commutative. Proposition 2.7. The universal bimeasuring bialgebra B(k[x], A) exists if and only if the algebra A is commutative. Proof. It is sufficient to see that every element of A is in the image of some bimeasuring N ⊗ k[x] → A. This is observed by noting that  : k[x] ⊗ ...
Cyclic A structures and Deligne`s conjecture
Cyclic A structures and Deligne`s conjecture

... by Kn the associahedron of dimension n 2. Recall the associahedron Kn is an abstract polytope whose vertices correspond to full bracketings of n letters and whose codimension m faces correspond to partial bracketings with m brackets. Hence K2 is a point, K3 is an interval, K4 is a pentagon. For more ...
Universiteit Leiden Super-multiplicativity of ideal norms in number
Universiteit Leiden Super-multiplicativity of ideal norms in number

Coarse structures on groups - St. John`s University Unofficial faculty
Coarse structures on groups - St. John`s University Unofficial faculty

... The notion of asymptotic dimension was introduced by Gromov as a tool for studying the large scale geometry of groups. Yu stimulated widespread interest in this concept when he proved that the Baum-Connes assembly map in topological K-theory is a split injection for torsion-free groups with finite a ...
pdf-file. - Fakultät für Mathematik
pdf-file. - Fakultät für Mathematik

The Z-densities of the Fibonacci sequence
The Z-densities of the Fibonacci sequence

... if the order of α is n then F2n ≡ 0 (mod p) and L2n ≡ 1 (mod p). Thereby we can recover the divisibility properties of the Fibonacci sequence by considering the order α in G(Fp ). Hence we can relate Z(p) to the order of α = (3/2, 1/2) in G(Fp ), as is shown in Theorem 3.5. We define a n-th preimag ...
The Knot Quandle
The Knot Quandle

... Knots that the quandle does allow us to distinguish are, for example 5-1 and the unknot, and 6-3 and 5-1. We couldn’t distinguish these knots using the 3-coloring invariant. Def. The 3-coloring invariant is the number of ways to color a knot diagram with three colors. To three color a diagram, each ...
THE COHOMOLOGY RING OF FREE LOOP SPACES 1. Introduction
THE COHOMOLOGY RING OF FREE LOOP SPACES 1. Introduction

... with integer coefficients H (CP ) ; Z . In [9], Cohen gives a com1 ...
Contents - Harvard Mathematics Department
Contents - Harvard Mathematics Department

HOMOLOGY OF LIE ALGEBRAS WITH Λ/qΛ COEFFICIENTS AND
HOMOLOGY OF LIE ALGEBRAS WITH Λ/qΛ COEFFICIENTS AND

... Proof. First note that Ln V2 (α, γ) = Ln V2 (h0 , h1 ), n ≥ 0. Then using Proposition 3.1 it is easy to get the following natural long exact sequence (compare [El1, Lemma 31]) · · · → Ln V(P ) → Ln V(M/M ∩ N ) ⊕ Ln V(N/M ∩ N ) → Ln−1 V2 (α, γ) → · · · → L0 V2 (α, γ) → L0 V(P ) → L0 V(M/M ∩ N ) ⊕ L0 ...
Associative Operations - Parallel Programming in Scala
Associative Operations - Parallel Programming in Scala

Soergel diagrammatics for dihedral groups
Soergel diagrammatics for dihedral groups

127 A GENERALIZATION OF BAIRE CATEGORY IN A
127 A GENERALIZATION OF BAIRE CATEGORY IN A

... Now for all U1 , U2 ∈ U (2) if U1 6= U2 then U1 ∩ U2 = ∅. Further, if N is nowhere dense in C, and A is any subset of C, then both N × A and A × N are nowhere dense in C 2 . Therefore if B is a set of first ωα -category in C, and A is any subset of C then both B × A and A × B are sets of first ωα -c ...
Two-Variable Logic over Countable Linear Orderings
Two-Variable Logic over Countable Linear Orderings

... We say that a language L ⊆ A◦ is recognised by the ◦-monoid M, if there is a morphism, γ : A◦ → M and a subset S ⊆ M such that L = γ −1 (S). The syntactic ◦-monoid of a language L is the minimal ◦-monoid M recognising L that has the following universal property: any ◦-monoid recognising L has a morp ...
Quotient Rings of Noncommutative Rings in the First Half of the 20th
Quotient Rings of Noncommutative Rings in the First Half of the 20th

... Germany at that time. This is made most obvious in Vol. 2 of [52], but we also see instances of it in Vol. 1. One of these is a comment van der Waerden added to [52, §12]. After giving the proof that every (commutative) domain can be embedded in a field (its quotient field), he says that it is an op ...
Lattices in Lie groups
Lattices in Lie groups

... of left cosets of H in G is a topological space under the quotient topology, which declares a set in G/H to be open if and only if its preimage under the quotient map G → G/H is open. Then it is easy to see that G/H is a locally compact Hausdorff space. Further, G acts by left translations on G/H an ...
Stable range one for rings with many units
Stable range one for rings with many units

... Except where specifically noted otherwise, all rings in this paper are associative with unit, and all modules are unital. Recall that a ring R satisfies stable range 1 provided that for any a, b E R satisfying aR + bR = R, there exists y E R such that a + by is right invertible. This condition is le ...
On the homology and homotopy of commutative shuffle algebras
On the homology and homotopy of commutative shuffle algebras

... group Σn , k[Σn ]. Usually one replaces the operad Com by an E∞ -operad to make things homotopy invariant. For instance Mike Mandell showed [M03, 1.8, 1.3] that the normalization functor induces an isomorphism between André-Quillen homology for simplicial E∞ -algebras and André-Quillen homology fo ...
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Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). The word homomorphism comes from the ancient Greek language: ὁμός (homos) meaning ""same"" and μορφή (morphe) meaning ""form"" or ""shape"". Isomorphisms, automorphisms, and endomorphisms are special types of homomorphisms.
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