Algorithmic Information Distortions in Node-Aligned and Node-Unaligned Multidimensional Networks

In this article, we investigate limitations of importing methods based on algorithmic information theory from monoplex networks into multidimensional networks (such as multilayer networks) that have a large number of extra dimensions (i.e., aspects). In the worst-case scenario, it has been previously shown that node-aligned multidimensional networks with non-uniform multidimensional spaces can display exponentially larger algorithmic information (or lossless compressibility) distortions with respect to their isomorphic monoplex networks, so that these distortions grow at least linearly with the number of extra dimensions. In the present article, we demonstrate that node-unaligned multidimensional networks, either with uniform or non-uniform multidimensional spaces, can also display exponentially larger algorithmic information distortions with respect to their isomorphic monoplex networks. However, unlike the node-aligned non-uniform case studied in previous work, these distortions in the node-unaligned case grow at least exponentially with the number of extra dimensions. On the other hand, for node-aligned multidimensional networks with uniform multidimensional spaces, we demonstrate that any distortion can only grow up to a logarithmic order of the number of extra dimensions. Thus, these results establish that isomorphisms between finite multidimensional networks and finite monoplex networks do not preserve algorithmic information in general and highlight that the algorithmic information of the multidimensional space itself needs to be taken into account in multidimensional network complexity analysis.     

Two graph isomorphism polytopes

The convex hull ψ_n,n of certain (n!)² tensors was considered recently in connection with graph isomorphism. We consider the convex hull ψ_n of the n! diagonals among these tensors. We show: 1. The polytope ψ_n is a face of ψ_n,n. 2. Deciding if a graph G has a subgraph isomorphic to H reduces to optimization over ψ_n. 3. Optimization over ψ_n reduces to optimization over ψ_n,n. In particular, this implies that the subgraph isomorphism problem reduces to optimization over ψ_n,n.     

Simple 7-designs with small parameters

We describe a computer search for the construction of simple designs with prescribed automorphism groups. Using our program package DISCRETA this search yields designs with parameter sets 7-(33, 8, 10), 7-(27, 9, 60), 7-(26, 9, λ) for λ = 54, 63, 81, 7-(26, 8, 6), 7-(25, 9, λ) for λ = 45, 54, 72, 7-(24, 9, λ) for λ = 40, 48, 64, 7-(24, 8, λ) for λ = 4, 5, 6, 7, 8, 6-(25, 8, λ) for λ = 36, 45, 54, 63, 72, 81, 6-(24, 8, λ) for λ = 36, 45, 54, 63, 72, 5-(19, 6, 4), and 5-(19, 6, 6). In several of these cases we are able to determine the exact number of isomorphism types of designs with that prescribed automorphism group. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 79–94, 1999     

FR^A模的Fuzzy同态的几个重要结果

证明了ＦＲ＾Ａ模的第一同构定理、第二同构定理、同构扩展定理以及其他一些重要性质。     

On finite groups with the cayley isomorphism property

Let G be a finite group, and let Cay(G, S) be a Cayley digraph of G. If, for all T ⊂ G, Cay(G, S) ≅ Cay(G, T) implies S^α = T for some α ∈ Aut(G), then Cay(G, S) is called a CI-graph of G. For a group G, if all Cayley digraphs of valency m are CI-graphs, then G is said to have the m-DCI property; if all Cayley graphs of valency m are CI-graphs, then G is said to have the m-CI property. It is shown that every finite group of order greater than 2 has a nontrivial CI-graph, and all finite groups with the m-CI property and with the m-DCI property are characterized for small values of m. A general investigation is made of the structure of Sylow subgroups of finite groups with the m-DCI property and with the m-CI property for large values of m. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 21–31, 1998     

基于同构理论的三支概念格的构造方法与算法研究

三支概念分析理论目前已经发展成为数据分析与知识发现的有效工具。主要通过形式背景特征的研究, 讨论三支概念格与概念格的同构关系, 进一步研究了三支概念格的构造方法。首先给出了对偶属性、属性对偶背景的定义, 并证明了在对偶背景下三支概念格与概念格是同构的。其次, 推广了对偶属性、属性诱导的对偶背景, 给出对偶可交属性及属性对偶可交背景的定义, 同时, 证明了在对偶可交背景下,三支概念格与概念格是同构的。最后基于上述理论, 给出了判定属性对偶背景与属性对偶可交背景的2种算法以及三支概念格的构造方法。     

Isomorphism of inclusion compounds built with composite host lattices of thiadiazole and V-shaped molecules with the existence of tetrabutylammonium guest template

By using 4,4′-dihydroxydiphenylsulfone (BPS) and 4,4′-dihydroxydiphenylsulfide (BTS) react with amidinethiourea and tetrabutylammonium respectively, two isomorphs of inclusion compounds of (n-C₄H₉)₄N⁺·C₁₂H₉O₄S^?·C₂H₄N₄S (1) and (n-C₄H₉)₄N⁺·C₁₂H₉O₂S^?·C₂H₄N₄S (2) were prepared and characterized by Single crystal X-ray diffraction. Although two varied types of V-shaped molecules were selected to react in the actual experimental process, two isomorphs with similar packing modes were finally obtained. Interestingly, amidinothiourea molecule occurred in situ oxidation reaction to generate a new compound of 3,5-dimido-1,2,4,-thiadiazole during the experiments, and these two inclusion compounds here are the pioneer examples of thiadiazole and tetraalkylammonium and can be regarded as two isomorphs due to their similar crystallization modes. In these two isomorphs, thiadiazole molecules derived from amidinothiourea link with various V-type molecules to develop the similar dumbbell-type hydrogen-bonded ribbons by the alike hydrogen bonding modes, then the ribbons interpenetrate with tetrabutylammonium to result in two stable isomorphism structures.     

Complete fuzzy topological hyperfields and fuzzy multiplication in the fuzzy real lines

Recently E. Sanchez has given a necessary and sufficient condition for the existence of a solution of a fuzzy $\text{A 1 X = C}$, with 1 the extension of a given binary operation in the universe of discourse.Using many valued logic we generalize this result and get a characterization also of the cases if $\text{A 1 X = C}$ has only nearly a solution.Additionally, we consider fuzzy relation equations and get similar results.     

The modular group algebra problem for metacyclic -groups

It is shown that the isomorphism type of a metacyclic -group is determined by its group algebra over the field of elements. This completes work of Baginski. It is also shown that, if a -group has a cyclic commutator subgroup , then the order of the largest cyclic subgroup containing is determined by .