Retracts of structures

It is proved that the category of all structures does not contain nontrivial injective objects. Necessary and sufficient conditions are derived under which any substructure (ideal, filter) of a structure L is a retract of L.Translated from Matematicheskie Zametki, Vol. 7, No. 6, pp. 687–692, June, 1970.     

Counting hypercubes in hypercubes

Subcubes of a hypercube are counted in three different ways, yielding a new graph theory interpretation of a known combinatorial identity. From this and the binomial inversion some additional combinatorial identities related to hypercubes are obtained.     

Retracts of Products of Chordal Graphs

In this article, we characterize the graphs G that are the retracts of Cartesian products of chordal graphs. We show that they are exactly the weakly modular graphs that do not contain K_{2, 3}, the 4‐wheel minus one spoke , and the k‐wheels (for as induced subgraphs. We also show that these graphs G are exactly the cage‐amalgamation graphs as introduced by Bre?ar and Tepeh Horvat (Cage‐amalgamation graphs, a common generalization of chordal and median graphs, Eur J Combin 30 (2009), 1071–1081); this solves the open question raised by these authors. Finally, we prove that replacing all products of cliques of G by products of Euclidean simplices, we obtain a polyhedral cell complex which, endowed with an intrinsic Euclidean metric, is a CAT(0) space. This generalizes similar results about median graphs as retracts of hypercubes (products of edges) and median graphs as 1‐skeletons of CAT(0) cubical complexes. © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 161–180, 2013     

拓扑空间的收缩核与半收缩核(英)

本文的主要目的是建立收缩核与近似连续之间的联系，基于半连续函数概念，我们构造了一类收缩核，得到了关于这一概念的许多的性质与例子，证明并解释了半收缩核与拓扑折送之间的关系。     

Polynomial Retracts and the Jacobian Conjecture

Let be the polynomial algebra in two variables over a field of characteristic . A subalgebra of is called a retract if there is an idempotent homomorphism (a retraction, or projection) such that The presence of other, equivalent, definitions of retracts provides several different methods of studying and applying them, and brings together ideas from combinatorial algebra, homological algebra, and algebraic geometry. In this paper, we characterize all the retracts of up to an automorphism, and give several applications of this characterization, in particular, to the well-known Jacobian conjecture.

     

Retracts and extensors of compact transformation groups

In this paper, we study G-spaces, i.e., topological spaces with actions of a compact group G, from the point of view of the theory of retracts. We recall definitions and properties of absolute retracts, absolute neighborhood retracts, absolute extensors, and absolute neighborhood extensors with respect to the equivariant weakly hereditary class $ \mathfrak{G} $ of G-spaces. Also, we give basic notions of the theory of topological transformation groups.     

Groups of Measure-Preserving Homeomorphisms as Deformation Retracts

It is known that the group of homeomorphisms of a non-compact,connected, metrizable manifold acts transitively on certainspaces of measures defined, among other properties, by theirbehaviour at the manifold's set of ‘ends’. As ageneralization of previous work of Fathi in the realm of compactmanifolds, it is shown that the given action admits a continuoussection when restricted to ‘biregular’ homeomorphismsand measures. This result exhibits the isotropy groups of measure-preservinghomeomorphisms as deformation retracts of the correspondingacting groups.     

Maxima in hypercubes

We derive a Berry‐Esseen bound, essentially of the order of the square of the standard deviation, for the number of maxima in random samples from (0,1)^d. The bound is, although not optimal, the first of its kind for the number of maxima in dimensions higher than two. The proof uses Poisson processes and Stein's method. We also propose a new method for computing the variance and derive an asymptotic expansion. The methods of proof we propose are of some generality and applicable to other regions such as d‐dimensional simplex. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Distance-preserving subgraphs of hypercubes

We give a characterization of connected subgraphs G of hypercubes H such that the distance in G between any two vertices a, b?G is the same as their distance in H. The hypercubes are graphs which generalize the ordinary cube graph.     

Nonfactorizable nonsingular hypercubes

We investigate nonsingular hypercubes, and prove several results, stating a condition for a hypercube to be the product of hypercubes of smaller dimensions. There is a shortage of higher dimensional nonsingular hypercubes in the literature. However, we show that the product of two nonsingular hypercubes is always nonsingular. Then we show how to construct four-dimensional nonsingular hypercubes that are not the products of two three-dimensional hypercubes. It is noted that higher dimensional nonsingular hypercubes, that are not products of smaller ones, correspond to many semifields.     

The carvingwidth of hypercubes

The notion of the carvingwidth of a graph was introduced by Seymour and Thomas [Call routing and the ratcatcher, Combinatorica 14 (1994) 217-241]. In this note, we show that the carvingwidth of a d-dimensional hypercube equals 2^d-1.     

Induced subgraphs of hypercubes

Let _Qk$Q_k$ denote the k$k$-dimensional hypercube on 2^k$2^k$ vertices. A vertex in a subgraph of _Qk$Q_k$ is full   if its degree is k$k$. We apply the Kruskal–Katona Theorem to compute the maximum number of full vertices an induced subgraph on n≤2^k$n\le 2^k$ vertices of _Qk$Q_k$ can have, as a function of k$k$ and n$n$. This is then used to determine min(max(|V(_H1)|,|V(_H2)|))$min(max(\left|V\left(H_1\right)\right|,\left|V\left(H_2\right)\right|))$ where (i) _H1$H_1$ and _H2$H_2$ are induced subgraphs of _Qk$Q_k$, and (ii) together they cover all the edges of _Qk$Q_k$, that is E(_H1)∪E(_H2)=E(_Qk)$E\left(H_1\right)\cup E\left(H_2\right)=E\left(Q_k\right)$.     

Hexagon-free subgraphs of hypercubes

It is shown (for all n ≥ 3) that the edges of the n-cube can be 3-colored in such a way that there is no monochromatic 4-cycle or 6-cycle. © 1993 John Wiley & Sons, Inc.     

Symmetric edge-decompositions of hypercubes

We call a set of edgesE of the n-cubeQ _n a fundamental set for Q _n if for some subgroupG of the automorphism group ofQ _n , theG-translates ofE partition the edge set ofQ _n .Q _n possesses an abundance of fundamental sets. For example, a corollary of one of our main results is that if |E| =n and the subgraph induced byE is connected, then if no three edges ofE are mutually parallel,E is a fundamental set forQ _n . The subgroupG is constructed explicitly. A connected graph onn edges can be embedded intoQ _n so that the image of its edges forms such a fundamental set if and only if each of its edges belongs to at most one cycle.We also establish a necessary condition forE to be a fundamental set. This involves a number-theoretic condition on the integersa _j (E), where for 1 j n, a _j (E) is the number of edges ofE in thej ^th direction (i.e. parallel to thej ^th coordinate axis).