1 Graph Homomorphisms 2 1.1 Graphs and Digraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Graph Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The H-coloring Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Polymorphism...

Booleanization of frames or uniform frames, which is not functorial under the basic choice of morphisms, becomes functorial in the categories with weakly open homomorphisms or weakly open uniform homomorphisms. Then, the construction becomes a reflection. In the uniform case, moreover, it also has a left adjoint. In connection with this, certain dual equivalences concerning uniform spaces and u...

Let A be a separable C∗-algebra and let B be a stable C∗-algebra with a strictly positive element. We consider the (semi)group Ext(A,B) (resp. Ext(A,B)) of homotopy classes of asymptotic (resp. of genuine) homomorphisms from A to the corona algebra M(B)/B and the natural map i : Ext(A,B) −→ Ext(A,B). We show that if A is a suspension then Ext(A,B) coincides with E-theory of Connes and Higson an...

This note consists of three unrelated remarks. First, we demonstrate how roughly speaking $*$-homomorphisms between matrix stable $C^*$-algebras are exactly the uniformly continuous $*$-preserving group homomorphisms their genral linear groups. Second, using Cuntz picture in $KK$-theory bring morphisms represented by generators and relations to a particular simple form. Third, show that for an ...

11 More work should be done on learning homomorphisms into larger alphabets. More importantly, it would be interesting to nd natural linguistic limitations of the type of morphological transformations such that the resultant learning problem would become tractable. References 1] Angluin, D. and C. Smith, \Inductive inference: theory and methods". Comput. QUESTION: Is there a K-state determinist...

Suppose that is an innnite set and k is a natural number. Let ] k denote the set of all k-subsets of and let F be a eld. In this paper we study the FSym(()-submodule structure of the permutation module F] k. Using the representation theory of nite symmetric groups, we show that every submodule of F] k can be written as an intersection of kernels of certain FSym(()-homomorphisms F] k ?! F] l for...

Let S(r) denote the w-sphere with a linear involution having a fixed point set of codimension r, where O ^ r ^ w . We pick some fixed point as a base point and consider the set [S(r); S(t)] of base point preserving equivariant homotopy classes of maps from S(r) to S(t). This has a natural group structure for n—r^l and is abelian if n—r^2. There is a suspension functor S without action and one 2...

Let C(X,G) denote the group of continuous functions from a topological space X into a topological group G with the pointwise multiplication and the compact-open topology. We show that there is a natural topology on the collection of normal subgroups A(X) of C(X,G) of the M {f C(X,G): f(p) e} which is P analogous to the hull-kernel topology on the commutative Banach algegra C(X) of all continuou...

We study, from a set-theoretic point of view, those subgroups of the infinite direct product Z0 for which all homomorphisms to Z annihilate all but finitely many of the standard unit vectors. Specifically, we relate the smallest possible size of such a subgroup to several of the standard cardinal characteristics of the continuum. We also study some related properties and cardinals, both group-t...