Nonempty Family Research Articles

On [formula omitted]-lineability

In the paper we study c-lineability and c-spaceability of some families F of real functions defined on an interval I. The main goal is to formulate general conditions under which any non-empty family F⊂RI of functions is c-spaceable or c-lineable. Generally, we consider the families of function of the form F=F1∖F2. In most cases, families of functions for which lineability and spaceability are studied have such a form. Most often, family F2 is seemingly “very close” to F1 or consists of “almost all” functions. The results obtained in this paper are a generalization of previous ideas. The main idea of our constructions is to “reproduce” one function to obtain c-dimensional (closed) linear space. For this “reproduction” we use the Fichtenholz-Kantorovich Theorem, applied to a countable family of pairwise disjoint intervals contained in the domain of functions from the considered class. The initial function is “squashed” and “pasted” into disjoint intervals included in the domain of constructed function.

The Maximum Sum of Sizes of Non-Empty Cross t-Intersecting Families

The Maximum Sum of Sizes of Non-Empty Cross t-Intersecting Families

Planar Turán Number of the Θ6

Let F be a nonempty family of graphs. A graph 𝐺 is called F -free if it contains no graph from F as a subgraph. For a positive integer 𝑛, the planar Turán number of F, denoted by exp (𝑛, F), is the maximum number of edges in an 𝑛-vertex F -free planar graph.Let Θ𝑘 be the family of Theta graphs on 𝑘 ≥ 4 vertices, that is, graphs obtained by joining a pair of non-consecutive of a 𝑘-cycle with an edge. Lan, Shi and Song determined an upper bound exp (𝑛, Θ6) ≤ 18𝑛/7−36𝑛/7, but for large 𝑛, they did not verify that the bound is sharp. In this paper, we improve their bound by proving exp (𝑛, Θ6) ≤ 18𝑛/−48𝑛/7 and then we demonstrate the existence of infinitely many positive integer 𝑛 and an 𝑛-vertex Θ6-free planar graph attaining the bound.

Exploring well-gradedness in polytomous knowledge structures

Enhancing learning effectiveness and comprehension, well-gradedness plays a crucial role in knowledge structure theory by establishing a systematic and progressive knowledge system. Extensive research has been conducted in this domain, resulting in significant findings. This paper explores the properties of well-gradedness in polytomous knowledge structures, shedding light on both classical confirmations and exceptional cases. A key characteristic of well-gradedness is the presence of adjacent elements within a non-empty family that exhibit a distance of 1. The study investigates various manifestations of well-gradedness, including its discriminative properties and its manifestation in discriminative factorial polytomous structures. Furthermore, intriguing deviations from classical standards in minimal polytomous states are uncovered, revealing unexpected behaviors.

Some inequalities concerning cross-intersecting families of integer sequences

Let [n]={1,2,…,n} and let [n]k be the family of integer sequences (x1,x2,…,xk) with 1≤xi≤n, i=1,2,…,k. Two families F,G⊂[n]k are called cross-intersecting if F and G agree in at least one position for every F∈F and G∈G. A family F⊂[n]k is called non-trivial if for every i∈[k] there exist (x1,x2,…,xk), (y1,y2,…,yk)∈F such that xi≠yi. In the present paper, we show that if F,G⊂[n]k are non-empty cross-intersecting and n≥2, then|F|+|G|≤1+nk−(n−1)k. If F,G⊂[n]k are both non-trivial, cross-intersecting and n≥2, then|F|+|G|≤nk−2(n−1)k+(n−2)k+2. We also establish a similar inequality for non-empty cross-intersecting families of generalized integer sequences.

Disjoint union of fuzzy soft topological spaces

<abstract><p>In this work, sums of fuzzy soft topological spaces are defined with free union of a pairwise disjoint non-empty family of fuzzy soft topological spaces. Firstly, we give general information of fuzzy soft topological spaces. Then, we define free union of fuzzy soft topological spaces and disjoint union topology of fuzzy soft topological spaces. We call the free union of a pairwise disjoint non-empty family of fuzzy soft topological spaces the sum of fuzzy soft topological spaces. We show what are the interchangeable properties between fuzzy soft topological spaces and the sum of fuzzy soft topological spaces. For example, there are fuzzy soft interior, fuzzy soft closure, fuzzy soft limit points. Also, we provide some properties showing the relationships between fuzzy soft topological spaces and their sums. Some of these are fuzzy soft base, fuzzy soft sequences, fuzzy soft connected-disconnected, fuzzy soft compact spaces. Also, part of the research for this article is work on fuzzy soft convergence on fuzzy soft topological sum. With this paper, some results, theorems and definitions for fuzzy soft topological sum have been acquired with the help of results, theorems and definitions given in previous studies about fuzzy soft topological spaces.</p></abstract>

Non-split supermanifolds associated with the cotangent bundle

Here, I study the problem of classification of non-split supermanifolds having as retract the split supermanifold $(M,\Omega)$, where $\Omega$ is the sheaf of holomorphic forms on a given complex manifold $M$ of dimension $> 1$. I propose a general construction associating with any $d$-closed $(1,1)$-form $\omega$ on $M$ a supermanifold with retract $(M,\Omega)$ which is non-split whenever the Dolbeault class of $\omega$ is non-zero. In particular, this gives a non-empty family of non-split supermanifolds for any flag manifold $M\ne \mathbb{CP}^1$. In the case where $M$ is an irreducible compact Hermitian symmetric space, I get a complete classification of non-split supermanifolds with retract $(M,\Omega)$. For each of these supermanifolds, the 0- and 1-cohomology with values in the tangent sheaf are calculated. As an example, I study the $\Pi$-symmetric super-Grassmannians introduced by Yu. Manin.

Families of finite sets in which no set is covered by the union of the others

Let ℱ be a finite nonempty family of finite nonempty sets. We prove the following: (1) ℱ satisfies the condition of the title if and only if for every pair of distinct subfamilies {A1,…,Ar}, {B1,…,Bs} of ℱ, ⋃i=1rAi≠⋃i=1sBi. (2) If ℱ satisfies the condition of the title, then the number of subsets of ⋃A∈ℱA containing at least one set of ℱ is odd. We give two applications of these results, one to number theory and one to commutative algebra.

γ– algebra of Sets and Some of its Properties

The main objective of this work is to generalize the concept of fuzzy algebra by introducing the notion of fuzzy algebra. Characterization and examples of the proposed generalization are presented, as well as several different properties of fuzzy algebra are proven. Furthermore, the relationship between fuzzy algebra and fuzzy algebra is studied, where it is shown that the fuzzy algebra is a generalization of fuzzy algebra too. In addition, the notion of restriction, as an important property in the study of measure theory, is studied as well. Many properties of restriction of a nonempty family of fuzzy subsets of fuzzy power set are investigated and it is shown that the restriction of fuzzy algebra is fuzzy algebra too.

On Non-Empty Cross-Intersecting Families

Let 2[n] and ( $$\matrix{{\left[ n \right]} \cr i \cr } $$ ) be the power set and the collection of all i-subsets of {1, 2, …, n}, respectively. We call t (t ≥ 2) families $${{\cal A}_1},{{\cal A}_2}, \ldots ,{{\cal A}_t} \subseteq {2^{\left[ n \right]}}$$ cross-intersecting if Ai ∩ Aj ≠ ∅ for any $${A_i} \in {{\cal A}_i}$$ and $${A_j} \in {{\cal A}_j}$$ with i ≠ j. We show that, for n ≥ k +l, l ≥ r ≥ 1, c > 0 and $${\cal A} \subseteq \left( {\matrix{{\left[ n \right]} \cr k \cr } } \right),{\cal B} \subseteq \left( {\matrix{{\left[ n \right]} \cr l \cr } } \right)$$ , if $${\cal A}$$ and $${\cal B}$$ are cross-intersecting and $$\left( {\matrix{{n - r} \cr {l - r} \cr } } \right) \le \left| {\cal B} \right| \le \left( {\matrix{{n - 1} \cr {l - 1} \cr } } \right)$$ , then $$\left| {\cal A} \right| + c\left| {\cal B} \right| \le \max \left\{ {\left( {\matrix{n \cr k \cr } } \right) - \left( {\matrix{{n - r} \cr k \cr } } \right) + c\left( {\matrix{{n - r} \cr {l - r} \cr } } \right),\left( {\matrix{{n - 1} \cr {k - 1} \cr } } \right) + c\left( {\matrix{{n - 1} \cr {l - 1} \cr } } \right)} \right\}.$$ This implies a result of Tokushige and the second author (Theorem 3.1) and also yields that, for n ≥ 2k, if $${{\cal A}_1},{{\cal A}_2}, \ldots ,{{\cal A}_t} \subseteq \left( {\matrix{{\left[ n \right]} \cr k \cr } } \right)$$ are non-empty cross-intersecting, then $$\sum\limits_{i = 1}^t {\left| {{{\cal A}_i}} \right| \le \max \left\{ {\left( {\matrix{n \cr k \cr } } \right) - \left( {\matrix{{n - k} \cr k \cr } } \right) + t - 1,\,\,t\left( {\matrix{{n - 1} \cr {k - 1} \cr } } \right)} \right\},} $$ which generalizes the corresponding result of Hilton and Milner for t = 2. Moreover, the extremal families attaining the two upper bounds above are also characterized.

Relaxed Lagrangian duality in convex infinite optimization: reducibility and strong duality

We associate with each convex optimization problem, posed on some locally convex space, with infinitely many constraints indexed by the set T, and a given non-empty family of finite subsets of T, a suitable Lagrangian-Haar dual problem. We obtain necessary and sufficient conditions for -reducibility, that is, equivalence to some subproblem obtained by replacing the whole index set T by some element of . Special attention is addressed to linear optimization, infinite and semi-infinite, and to convex problems with a countable family of constraints. Results on zero -duality gap and on -(stable) strong duality are provided. Examples are given along the paper to illustrate the meaning of the results.

On relation between statistical ideal and ideal generated by a modulus function

Ideal on an arbitrary non-empty set $\Omega$ it's a non-empty family of subset $\mathfrak{I}$ of the set $\Omega$ which satisfies the following axioms: $\Omega \notin \mathfrak{I}$, if $A, B \in \mathfrak{I}$, then $A \cup B \in \mathfrak{I}$, if $A \in \mathfrak{I}$ and $D \subset A$, then $D \in \mathfrak{I}$. The ideal theory is a very popular branch of modern mathematical research. In our paper we study some classes of ideals on the set of all positive integers $\mathbb{N}$, namely the ideal of statistical convergence $\mathfrak{I}_s$ and the ideal $\mathfrak{I}_f$ generated by a modular function $f$. Statistical ideal it's a family of subsets of $\mathbb{N}$ whose natural density is equal to 0, i.e. $A \in \mathfrak{I}_s$ if and only if $\displaystyle\lim\limits_{n \rightarrow \infty}\frac{\#\{k \leq n: k \in A\}}{n} = 0$. A function $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ is called a modular function, if $f(x) = 0$ only if $x = 0$, $f(x + y) \leq f(x) + f(y)$ for all $x, y \in\mathbb{R}^+$, $f(x) \le f(y)$ whenever $x \le y$, $f$ is continuous from the right 0, and finally $\lim\limits_{n \rightarrow \infty} f(n) = \infty$. Ideal, generated by the modular function $f$ it's a family of subsets of $\mathbb{N}$ with zero $f$-density, in other words, $A \in \mathfrak{I}_f$ if and only if $\displaystyle\lim\limits_{n \rightarrow \infty}\frac{f(\#\{k \leq n: k \in A\})}{f(n)} = 0$. It is known that for an arbitrary modular function $f$ the following is true: $\mathfrak{I}_f \subset \mathfrak{I}_s$. In our research we give the complete description of those modular functions $f$ for which $\mathfrak{I}_f = \mathfrak{I}_s$. Then we analyse obtained result, give some partial cases of it and prove one simple sufficient condition for the equality $\mathfrak{I}_f = \mathfrak{I}_s$. The last section of this article is devoted to examples of some modulus functions $f, g$ for which $\mathfrak{I}_f = \mathfrak{I}_s$ and $\mathfrak{I}_g \neq \mathfrak{I}_s$. Namely, if $f(x) = x^p$ where $p \in (0, 1]$ we have $\mathfrak{I}_f = \mathfrak{I}_s$; for $g(x) = \log(1 + x)$, we obtain $\mathfrak{I}_g \neq \mathfrak{I}_s$. Then we consider more complicated function $f$ which is given recursively to demonstrate that the conditions of the main theorem of our paper can't be reduced to the sufficient condition mentioned above.

Relaxed Lagrangian duality in convex infinite optimization: Reverse strong duality and optimality

We associate with each convex optimization problem posed on some locally convex space with an infinite index set T, and a given non-empty family H formed by finite subsets of T, a suitable Lagrangian-Haar dual problem. We provide reverse H-strong duality theorems, H-Farkas type lemmas and optimality theorems. Special attention is addressed to infinite and semi-infinite linear optimization problems.

Neutrosophic Submodule of Direct Sum M ⊕ N

The paper focuses on neutrosophic algebraic structures and operations applicability to the study of classical al-gebraic structures, particularly the R-module. The definition of neutrosophic submodules P and Q was further developed upon in this work in order to create neutrosophic submodules of P + Q. In this study, the neutrosophic submodule of the direct sum M ⊕ N is constructed, analyzed, and its associated results are examined. Additionally, several algebraic results of the neutrosophic submodule’s direct sum of a non-empty arbitrary family of submodules are examined.

Fuzzy geometric spaces associated to fuzzy hypermodules

In this paper, we concentrate on fuzzy geometric spaces associated to fuzzy hypermodules and the feature of strongly transitive. First, we determine the fuzzy geometric space (Δ, FP * (M)) whose Δ is a nonzero fuzzy subset of M and FP *(M) is a nonempty family of fuzzy subsets of a fuzzy hypermodule of M, such that μ ≤ Δ, for all μ ∈ FP * (M), whose elements we called fuzzy blocks that are the fuzzy hypersums of elements of M. In addition, we will present some important and interesting results in this respect. Ultimately, we will verify for a fuzzy hypermodule under the specific circumstances the fuzzy geometric space is strongly transitive and also, the relation θ is transitive.

Characterizing 3-Sets in Union-Closed Families

Let and let a k-set denote a set of cardinality k. A family of sets is union-closed (UC) if the union of any two sets in the family is also in the family. Frankl’s conjecture states that for any nonempty UC family such that , there exists an element that is contained in at least half the sets of , where denotes the power set on . The 3-sets conjecture of Morris states that the smallest number of distinct 3-sets (whose union is an n-set) that ensure Frankl’s conjecture is satisfied (in an element of the n-set) for any UC family that contains them is for all . For an UC family , Poonen’s theorem characterizes the existence of weights on which ensure all UC families that contain satisfy Frankl’s conjecture, however the determination of such weights for specific is nontrivial even for small n. We classify families of 3-sets on using a polyhedral interpretation of Poonen’s theorem and exact rational integer programming. This yields a proof of the 3-sets conjecture.

Old and new applications of Katona’s circle

The present paper is to honour Gyula Katona, my teacher on the occasion of his 80th birthday. Its main content is threefold, new proofs of old results (e.g. the De Bonis, Katona, Swanepoel Theorem on butterfly-free families), new results obtained via the Katona Circle (e.g. for the sum of the sizes of non-empty cross-intersecting families), the solution for the Katona Circle of some notoriously difficult problems (e.g. Erdős Matching Conjecture).

Максимальные сцепленные системы на произведениях широко понимаемых измеримых пространств

Maximal linked systems (MLS) of sets on widely understood measurable spaces (MS) are considered; in addition, every such MS is realized by equipment of a nonempty set with a π-system of its subsets with «zero» and «unit» (π-system is a nonempty family of sets closed with respect to finite intersections). Constructions of the MS product connected with two variants of measurable (in wide sense) rectangles are investigated. Families of MLS are equipped with topologies of the Stone type. The connection of product of above-mentioned topologies considered for box and Tychonoff variants and the corresponding (to every variant) topology of the Stone type on the MLS set for the MS product is studied. The properties of condensation and homeomorphism for resulting variants of topological equipment are obtained.

To the Question on Some Generalizations of Properties of the Linkedness of Families of Sets and the Supercompactness of Topological Spaces

In this paper, we consider natural generalizations of properties of the linkedness of families (of sets) and the supercompactness of topological spaces. In the first case, we analyze the “multiple” linkedness, assuming the nonemptiness of the intersection of sets from subfamilies, whose cardinality does not exceed some given positive integer $\mathbf{n}$ . In the second case, we study the question of the existence of an (open) prebase such that any its covering has a subcovering, whose cardinality does not exceed $\mathbf{n}$ . We consider maximal $\mathbf{n}$ -linked (in the mentioned sense) subfamilies of a $\pi$ -system with “zero” and “unit” (a $\pi$ -system is a nonempty family closed with respect to finite intersections); these subfamilies are said to be maximal $\mathbf{n}$ -linked systems or (for short) $\mathbf{n}$ -MLS. We are interested in correlations between $\mathbf{n}$ -MLS and ultrafilters (u/f) of a $\pi$ -system, including the “dynamics” in dependence of $\mathbf{n}$ . Moreover, we consider bitopological spaces (BTS), whose elements are $\mathbf{n}$ -MLS and u/f; in both cases, for constructing a BTS (a nonempty set with a pair of comparable topologies) we use topologies of Wallman and Stone types. The Wallman-type topology on the set of $\mathbf{n}$ -MLS realizes an $\mathbf{n}$ -supercompact (in the sense mentioned above) T1-space which represents an abstract analog of a superextension of a T1-space. We prove that the BTS of u/f of the initial $\pi$ -system is a subspace of the BTS whose points are $\mathbf{n}$ -MLS; i. e., the corresponding “Wallman” and “Stone” topologies on the set of u/f are induced by the corresponding topologies on the set of $\mathbf{n}$ -MLS.

Unsymmetric T0-quasi-metrics

H.J.K. Junnila [9] called a neighbournet N on a topological space X unsymmetric provided that for each x,y∈X with y∈(N∩N−1)(x) we have that N(x)=N(y).Motivated by this definition, we shall call a T0-quasi-metric d on a set X unsymmetric provided that for each x,y,z∈X the following variant of the triangle inequality holds: d(x,z)≤d(x,y)∨d(y,x)∨d(y,z).Each T0-ultra-quasi-metric is unsymmetric. We also note that for each unsymmetric T0-quasi-metric d, its symmetrization ds=d∨d−1 is an ultra-metric.Furthermore we observe that unsymmetry of T0-quasi-metrics is preserved by subspaces and suprema of nonempty finite families, but not necessarily under conjugation. In addition we show that the bicompletion of an unsymmetric T0-quasi-metric is unsymmetric.The induced T0-quasi-metric of an asymmetrically normed real vector space X is unsymmetric if and only if X={0}. Our results are illustrated by various examples.We also explain how our investigations relate to the theory of ordered topological spaces and questions about (pairwise) strong zero-dimensionality in bitopological spaces.