On effective σ‐boundedness and σ‐compactness

We prove several dichotomy theorems which extend some known results on σ‐bounded and σ‐compact pointsets. In particular we show that, given a finite number of $\Delta ^{1}_{1}$ equivalence relations $\mathrel {\mathsf {F}}_1,\dots ,\mathrel {\mathsf {F}}_n$, any $\Sigma ^{1}_{1}$ set A of the Baire space either is covered by compact $\Delta ^{1}_{1}$ sets and lightface $\Delta ^{1}_{1}$ equivalence classes of the relations $\mathrel {\mathsf {F}}_i$, or A contains a superperfect subset which is pairwise $\mathrel {\mathsf {F}}_i$‐inequivalent for all i = 1, …, n. Further generalizations to $\Sigma ^{1}_{2}$ sets A are obtained.     

Family of Pearson discrete distributions generated by the univariate hypergeometric function 3F2(α1, α2, α3; γ1, γ2; λ)

In this work we present a study of the Pearson discrete distributions generated by the hypergeometric function ₃F₂(α₁, α₂, α₃;γ₁, γ₂; λ), a univariate extension of the Gaussian hypergeometric function, through a constructive methodology. We start from the polynomial coefficients of the difference equation that lead to such a function as a solution. Immediately after, we obtain the generating probability function and the differential equation that it satisfies, valid for any admissible values of the parameters. We also obtain the differential equations that satisfy the cumulants generating function, moments generating function and characteristic function, From this point on, we obtain a relation in recurrences between the moments about the origin, allowing us to create an equation system for estimating the parameters by the moment method. We also establish a classification of all possible distributions of such type and conclude with a summation theorem that allows us study some distributions belonging to this family. © 1997 by John Wiley & Sons, Ltd.     

Maximal σ‐polynomials of connected 3‐chromatic graphs

In the set of graphs of order n and chromatic number k the following partial order relation is defined. One says that a graph G is less than a graph H if c_i(G) ≤ c_i(H) holds for every i, k ≤ i ≤ n and at least one inequality is strict, where c_i(G) denotes the number of i‐color partitions of G. In this paper the first ? n/2 ? levels of the diagram of the partially ordered set of connected 3‐chromatic graphs of order n are described. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 210–222, 2003     

On the Loebl–Komlós–Sós conjecture

The Loebl–Komlós–Sós conjecture says that any graph G on n vertices with at least half of vertices of degree at least k contains each tree of size k. We prove that the conjecture is true for paths as well as for large values of k(k ≥ n − 3). © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 269–276, 2000     

On Schur σ—Groups

In this paper, we consider some conditions of finiteness related to the p-class field tower problem over an imaginary quadratic field, where p is an odd prime.     

σ‐short Boolean algebras

We introduce properties of Boolean algebras which are closely related to the existence of winning strategies in the Banach‐Mazur Boolean game. A σ‐short Boolean algebra is a Boolean algebra that has a dense subset in which every strictly descending sequence of length ω does not have a nonzero lower bound. We give a characterization of σ‐short Boolean algebras and study properties of σ‐short Boolean algebras. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Hankel Convolution and the Zemanian Spaces βμ and β'μ

In this paper we give structure theorems for the elements of the Zemanian spaces β'_μ and β'_μ,a' Also, bounded sets and convergent sequences in β'_μ,a' are characterized through representations as derivatives of measurable functions. Finally, we analyze the Hankel convolution on the above spaces.     

The denormalized 3×3 lemma

The classical 3×3 lemma holds in any regular protomodular category with a zero object. It is investigated here whether there is a “denormalized” version when the category no longer has a zero object, as, for instance, any slice category of the category , or any slice category of an abelian category. The answer is actually positive in the weaker context of regular Mal'cev categories.     

α-Stable characterization of Banach spaces (1 < α < 2)

This work characterizes some subclasses of α-stable (0 < α < 1) Banach spaces in terms of the extendibility to Radon laws of certain α-stable cylinder measures. These result extend the work of S. Chobanian and V. Tarieladze (J. Multivar. Anal.7, 183–203 (1977)). For these spaces it is shown that every Radon stable measure is the continuous image of a stable measure on a suitable L_β space with β = α(1 − α)⁻¹. The latter result extends some work of Garling (Ann. Probab.4, 600–611 (1976)) and Jain (Proceedings, Symposia in Pure Math. XXXI, p. 55–65, Amer. Math. Soc., Providence, R.I.).     

Nonexistence results for classes of 3 × 3 elliptic systems

We consider the system

     

Judiciously 3‐partitioning 3‐uniform hypergraphs

Bollobás, Reed, and Thomason proved every 3‐uniform hypergraph ? with m edges has a vertex‐partition V()=V₁?V₂?V₃ such that each part meets at least edges, later improved to 0.6m by Halsegrave and improved asymptotically to 0.65m+o(m) by Ma and Yu. We improve this asymptotic bound to , which is best possible up to the error term, resolving a special case of a conjecture of Bollobás and Scott.     

Graphs with Few 3‐Cliques and 3‐Anticliques are 3‐Universal

For given integers we ask whether every large graph with a sufficiently small number of k‐cliques and k‐anticliques must contain an induced copy of every l‐vertex graph. Here we prove this claim for with a sharp bound. A similar phenomenon is established as well for tournaments with .     

σ‐algebras for quasirandom hypergraphs

We examine the correspondence between the various notions of quasirandomness for k‐uniform hypergraphs and σ‐algebras related to measurable hypergraphs. This gives a uniform formulation of most of the notions of quasirandomness for dense hypergraphs which have been studied, with each notion of quasirandomness corresponding to a σ‐algebra defined by a collection of subsets of . We associate each notion of quasirandomness with a collection of hypergraphs, the ‐adapted hypergraphs, so that G is quasirandom exactly when it contains roughly the correct number of copies of each ‐adapted hypergraph. We then identify, for each , a particular ‐adapted hypergraph with the property that if G contains roughly the correct number of copies of then G is quasirandom in the sense of . This generalizes recent results of Kohayakawa, Nagle, Rödl, and Schacht; Conlon, Hàn, Person, and Schacht; and Lenz and Mubayi giving this result for some particular notions of quasirandomness. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 114–139, 2017     

On σ-equivalence and χ-equivalence of graphs

We define a partial ordering on the set of σ-polynomials as well as a vertex splitting operation on the set of graphs, and introduce the notions of σ-equivalence and σ-uniqueness of graphs. Let σ(G) be the σ-polynomial of a graph G and (OVERBAR)σ(G) = σ(G^c). Let H = (G, v, A, B) be a vertex splitting graph of G. We prove that (OVERBAR)σ(G) ≤ (OVERBAR)σ(H) and the equality holds if and only if every vertex of A is adjacent to every vertex of B. This gives us an effective means to find σ-equivalent and χ-equivalent graphs. A necessary and sufficient condition for a graph to be χ-unique but not σ-unique is also obtained. © 1996 John Wiley & Sons, Inc.     

Unsolvability in 3 × 3 Matrices

A set of 3 × 3 matrices over the integers will be said to be mortal if the zero matrix can be expressed as a finite product of members of the set. It is shown in this paper that the problem of deciding whether a given finite set is mortal is recursively unsolvable.     

Difference between irregular chaotic patterns of second-order double-loop ΣΔ modulators and second-order interpolative bandpass ΣΔ modulators

In this paper, we find that, by computing the difference between two consecutive state vectors of second-order double-loop sigma-delta modulators (SDMs) and plotting one component of the subtracted vectors against the other component, irregular chaotic patterns will become two vertical lines. By multiplying a matrix on the subtracted vectors, it can be further transformed to two fixed points. However, second-order interpolative bandpass SDMs still exhibit chaotic behaviors after applying the same transformations. Moreover, it is found that the Lyapunov exponent of state vectors of second-order double-loop SDMs is higher than that of second-order interpolative bandpass SDMs, whereas the Lyapunov exponent of transformed vectors becomes negative infinity for second-order double-loop SDMs and increases for second-order interpolative bandpass SDMs. Hence, by examining the occurrence of chaotic behaviors of the transformed vectors of these two SDMs, these two SDMs can be distinguished from their state vectors and their transformed vectors without solving the state equations and knowing the information of input signals.     

On the Roots of σ‐Polynomials

Given a graph G of order n, the σ‐polynomial of G is the generating function where is the number of partitions of the vertex set of G into i nonempty independent sets. Such polynomials arise in a natural way from chromatic polynomials. Brenti (Trans Am Math Soc 332 (1992), 729–756) proved that σ‐polynomials of graphs with chromatic number at least had all real roots, and conjectured the same held for chromatic number . We affirm this conjecture.     

Destructibility of stationary subsets of Pκλ

For a regular cardinal κ with κ ^<κ = κ and κ ≤ λ , we construct generically (forcing by a < κ‐closed κ ⁺‐c. c. p. o.‐set ℙ₀) a subset S of {x ∈ P _κ λ : x ∩ κ is a singular ordinal} such that S is stationary in a strong sense (F ^IA_κ λ ‐stationary in our terminology) but the stationarity of S can be destroyed by a κ ⁺‐c. c. forcing ℙ* (in V ^ℙ) which does not add any new element of P _κ λ . Actually ℙ* can be chosen so that ℙ* is κ‐strategically closed. However we show that such ℙ* itself cannot be κ‐strategically closed or even <κ‐strategically closed if κ is inaccessible. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Σ‐algebraically compact modules and ‐compact cardinals

We prove that the property characterizes Σ‐algebraically compact modules if is not ω‐measurable. Moreover, under a large cardinal assumption, we show that over any ring R where is not ω‐measurable, any free module M of ω‐measurable rank satisfies , hence the assumption on cannot be dropped in general (e.g., over small non‐right perfect rings). In this way, we extend results from a recent paper by Simion Breaz 4 .