The Coarse Baum-Connes Conjecuture for Relatively Hyperbolic
... Objects: Coarse equivalence classes of metric spaces. Morphisms: Hom(X, Y) = {f : X → Ycoarse map}/close. Definition The coarse K-homology KX∗ (−) is a coarse version of the K-homology which is a covariant functor from the category of coarse spaces to the category of abelian groups. Let X be a metri ...
... Objects: Coarse equivalence classes of metric spaces. Morphisms: Hom(X, Y) = {f : X → Ycoarse map}/close. Definition The coarse K-homology KX∗ (−) is a coarse version of the K-homology which is a covariant functor from the category of coarse spaces to the category of abelian groups. Let X be a metri ...
On linearly ordered H-closed topological semilattices
... VI.1.14 of [6] imply that ↑e and ↓e are closed subsets in S for any e ∈ S. Let E be a linearly ordered topological semilattice. Since ↓e and ↑e are closed for each e ∈ E, it follows that the topology of E refines the order topology. Thus the principal objects whick we shall consider can be alternati ...
... VI.1.14 of [6] imply that ↑e and ↓e are closed subsets in S for any e ∈ S. Let E be a linearly ordered topological semilattice. Since ↓e and ↑e are closed for each e ∈ E, it follows that the topology of E refines the order topology. Thus the principal objects whick we shall consider can be alternati ...
finitely generated powerful pro-p groups
... To avoid confusion most of the notation in this thesis follows [DSMS91]. One distinct difference in the presentation of the material is that any homomorphism, say ι, will act on an element a by ι(a). This is consistent with the entirety of undergraduate lecturing at UNSW though not the aι notation u ...
... To avoid confusion most of the notation in this thesis follows [DSMS91]. One distinct difference in the presentation of the material is that any homomorphism, say ι, will act on an element a by ι(a). This is consistent with the entirety of undergraduate lecturing at UNSW though not the aι notation u ...
1 Weakly Perfect Generalized Ordered Spaces by Harold R Bennett
... base of open, convex subsets, where we say that a set C is convex (in X) provided for any a < b < c in X, if {a, c} ⊂ C then b ∈ C. It is known that the class of GO-spaces coincides with the class of subspaces of linearly ordered topological spaces. It will be important to distinguish between subset ...
... base of open, convex subsets, where we say that a set C is convex (in X) provided for any a < b < c in X, if {a, c} ⊂ C then b ∈ C. It is known that the class of GO-spaces coincides with the class of subspaces of linearly ordered topological spaces. It will be important to distinguish between subset ...
1. Postulate 11 Through any two points there is exactly one line 2
... Postulate 11 Through any two points there is exactly one line Postulate 12 If two lines intersect, then they intersect in exactly one point Postulate 13 If two planes intersect, then they intersect in exactly one line Postulate 14 Through any three noncollinear points there is exactly one pla ...
... Postulate 11 Through any two points there is exactly one line Postulate 12 If two lines intersect, then they intersect in exactly one point Postulate 13 If two planes intersect, then they intersect in exactly one line Postulate 14 Through any three noncollinear points there is exactly one pla ...
On some problems in computable topology
... [31] and think of the basic open sets as easy to encode observations that can be made about the computational process determining the elements. Therefore, we let the topological basis be indexed in a total way. By doing better and better observation we want finally be able to determine every element ...
... [31] and think of the basic open sets as easy to encode observations that can be made about the computational process determining the elements. Therefore, we let the topological basis be indexed in a total way. By doing better and better observation we want finally be able to determine every element ...
Topological types of Algebraic stacks - IBS-CGP
... 1.4.1. In this paper, an algebraic space X over a scheme S is a functor X : (Sch/S)op → Set such that the following holds: (i) X is a sheaf with respect to the big étale topology. (ii) The diagonal ∆ : X → X ×S X is representable by schemes. (iii) There exists a S-scheme U and an étale surjection U ...
... 1.4.1. In this paper, an algebraic space X over a scheme S is a functor X : (Sch/S)op → Set such that the following holds: (i) X is a sheaf with respect to the big étale topology. (ii) The diagonal ∆ : X → X ×S X is representable by schemes. (iii) There exists a S-scheme U and an étale surjection U ...
The periodic table of n-categories for low
... objects we “forget” structure in the direction shown, but this does not necessarily give an n-functor. So we examine lower-dimensional “truncations” of the (n + 1)categories in question as follows. For each 1 ≤ j ≤ n we write • nCat(k)j for the j-dimensional “truncation” of nCat(k) • PT(n, k)j to be ...
... objects we “forget” structure in the direction shown, but this does not necessarily give an n-functor. So we examine lower-dimensional “truncations” of the (n + 1)categories in question as follows. For each 1 ≤ j ≤ n we write • nCat(k)j for the j-dimensional “truncation” of nCat(k) • PT(n, k)j to be ...
Boyarchenko on associativity.pdf
... 1.6. Finally, in §6, I will explain one reason why sometimes thinking about associativity constraints is important, from my personal (hence biased) point of view. Namely, in at least one of the approaches to the theory of quantum groups, associativity constraints on certain monoidal categories turn ...
... 1.6. Finally, in §6, I will explain one reason why sometimes thinking about associativity constraints is important, from my personal (hence biased) point of view. Namely, in at least one of the approaches to the theory of quantum groups, associativity constraints on certain monoidal categories turn ...
Chapter 7 Duality
... DM(S) can be constructed from the “naive” version Amot (SmS )0 , i.e., we may replace all the homotopy identities in the construction of the motivic DG tensor category A mot (SmS ) with strict identities. Combining this with (3.2.6), we arrive at a construction of Dbmot (S) as a localization of the ...
... DM(S) can be constructed from the “naive” version Amot (SmS )0 , i.e., we may replace all the homotopy identities in the construction of the motivic DG tensor category A mot (SmS ) with strict identities. Combining this with (3.2.6), we arrive at a construction of Dbmot (S) as a localization of the ...
Automorphism groups of cyclic codes Rolf Bienert · Benjamin Klopsch
... = Sym(a) × Sym(b). Interestingly, some of the codes C0 (a, b) were recently studied by Key and Seneviratne in the context of regular lattice graphs and permutation decoding. In fact, we provide a new, unified treatment of a related family C1 (a, b) of binary linear codes whose study was initiated in ...
... = Sym(a) × Sym(b). Interestingly, some of the codes C0 (a, b) were recently studied by Key and Seneviratne in the context of regular lattice graphs and permutation decoding. In fact, we provide a new, unified treatment of a related family C1 (a, b) of binary linear codes whose study was initiated in ...
Group action
In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object. A group action formalizes of the relationship between the group and the symmetries of the object. It relates each element of the group to a particular transformation of the object.In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group G is a representation of G as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of G by permutation matrices. It is the same as a group action of G on an ordered basis of a vector space.A group action is an extension to the notion of a symmetry group in which every element of the group ""acts"" like a bijective transformation (or ""symmetry"") of some set, without being identified with that transformation. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets of features, such as the set of vertices, the set of edges and the set of faces of the polyhedron.If G is a group and X is a set, then a group action may be defined as a group homomorphism h from G to the symmetric group on X. The action assigns a permutation of X to each element of the group in such a way that the permutation of X assigned to the identity element of G is the identity transformation of X; a product gk of two elements of G is the composition of the permutations assigned to g and k.The abstraction provided by group actions is a powerful one, because it allows geometrical ideas to be applied to more abstract objects. Many objects in mathematics have natural group actions defined on them. In particular, groups can act on other groups, or even on themselves. Despite this generality, the theory of group actions contains wide-reaching theorems, such as the orbit stabilizer theorem, which can be used to prove deep results in several fields.