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 ...
... 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
... 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 ...
... 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
... 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 ...
... 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
... 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 ...
... 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 ...
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 ...
... 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
... 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] ⊗ ...
... 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
... 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 ...
... 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 ...
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 ...
... 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 ...
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 ...
... 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
... 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 ...
... 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
... with integer coefficients H (CP ) ; Z . In [9], Cohen gives a com1 ...
... with integer coefficients H (CP ) ; Z . In [9], Cohen gives a com1 ...
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 ...
... 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 ...
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 ...
... 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
... 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 ...
... 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
... 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 ...
... 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
... 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 ...
... 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
... 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 ...
... 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
... 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 ...
... 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 ...