Closure systems over effect algebras

The present paper is an attempt to introduce the closure systems over effect algebras. At first, we will define closure systems over effect algebras, and for arbitrary set $ U $ and arbitrary subset S of all functions from U to an effect algebra L we will obtain the closure system containing S. Then, we will define the base of this closure system, and for arbitrary subset S of all functions from U to an effect algebra L we will obtain the base of this closure system.

On certain compactification of an arbitrary subset of $\mathbb{R}^m$ and its applications to DiPerna-Majda measures theory

We present a constructive proof of the fact, that for any subset $A \subseteq \R^m$ and a countable family $F$ of bounded functions $f: A \to R$ there exists a compactification $A' \subset \ell^2$ of $A$ such that every function $f \in F$ possesses a continuous extension to a function $\bar{f}: A' \to \R$. However related to some classical theorems, our result is direct and hence applicable in Calculus of Variations. Our construction is then used to represent limits of weakly convergent sequences $\{f(u^\nu)\}$ via methods related to DiPerna-Majda measures. In particular, as our main application, we generalise the Representation Theorem from the Calculus of Variations due to Kałamajska.

An Econometric Perspective on Algorithmic Subsampling

Data sets that are terabytes in size are increasingly common, but computer bottlenecks often frustrate a complete analysis of the data, and diminishing returns suggest that we may not need terabytes of data to estimate a parameter or test a hypothesis. But which rows of data should we analyze, and might an arbitrary subset preserve the features of the original data? We review a line of work grounded in theoretical computer science and numerical linear algebra that finds that an algorithmically desirable sketch, which is a randomly chosen subset of the data, must preserve the eigenstructure of the data, a property known as subspace embedding. Building on this work, we study how prediction and inference can be affected by data sketching within a linear regression setup. We use statistical arguments to provide “inference-conscious” guides to the sketch size and show that an estimator that pools over different sketches can be nearly as efficient as the infeasible one using the full sample.

On the distribution of an arbitrary subset of the eigenvalues for some finite dimensional random matrices

We present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of complex Wishart, double Wishart, and Gaussian hermitian random matrices of finite dimensions, using a tensor pseudo-determinant operator. Specifically, we derive compact expressions for the joint probability distribution function of the eigenvalues and the expectation of functions of the eigenvalues, including joint moments, for the case of both ordered and unordered eigenvalues.

The k-Rainbow Domination Number of Cn□Cm

Given a graph G and a set of k colors, assign an arbitrary subset of these colors to each vertex of G. If each vertex to which the empty set is assigned has all k colors in its neighborhood, then the assignment is called a k-rainbow dominating function (kRDF) of G. The minimum sum of numbers of assigned colors over all vertices of G is called the k-rainbow domination number of graph G, denoted by γ r k ( G ) . In this paper, we focus on the study of the k-rainbow domination number of the Cartesian product of cycles, C n □ C m . For k ≥ 8 , based on the results of J. Amjadi et al. (2017), γ r k ( C n □ C m ) = m n . For ( 4 ≤ k ≤ 7 ) , we give a proof for the new lower bound of γ r 4 ( C n □ C 3 ) . We construct some novel and recursive kRDFs which are good enough and upon these functions we get sharp upper bounds of γ r k ( C n □ C m ) . Therefore, we obtain the following results: (1) γ r 4 ( C n □ C 3 ) = 2 n ; (2) γ r k ( C n □ C m ) = k m n 8 for n ≡ 0 ( mod 4 ) , m ≡ 0 ( mod 4 ) ( 4 ≤ k ≤ 7 ) ; (3) for n ≢ 0 ( mod 4 ) or m ≢ 0 ( mod 4 ) , m n 2 ≤ γ r 4 ( C n □ C m ) ≤ m n 2 + m + n 2 − 1 and k m n 8 ≤ γ r k ( C n □ C m ) ≤ k m n + n 8 + m for 5 ≤ k ≤ 7 . We also discuss Vizing’s conjecture on the k-rainbow domination number of C n □ C m .

Infinite-dimensional triangularizable algebras

Let {\mathrm{End}_{k}(V)} denote the ring of all linear transformations of an arbitrary k-vector space V over a field k. We define {X\subseteq\mathrm{End}_{k}(V)} to be triangularizable if V has a well-ordered basis such that X sends each vector in that basis to the subspace spanned by basis vectors no greater than it. We then show that an arbitrary subset of {\mathrm{End}_{k}(V)} is strictly triangularizable (defined in the obvious way) if and only if it is topologically nilpotent. This generalizes the theorem of Levitzki that every nilpotent semigroup of matrices is triangularizable. We also give a description of the triangularizable subalgebras of {\mathrm{End}_{k}(V)} , which generalizes a theorem of McCoy classifying triangularizable algebras of matrices over algebraically closed fields.

Achieving Optimal Degrees of Freedom for an Interference Network with General Message Demand

The concept of degrees of freedom (DoF) has been adopted to resolve the difficulty of studying the multi-user wireless network capacity regions. Interference alignment (IA) is an important technique developed recently for quantifying the DoF of such networks. In the present study, a single-hop interference network with K transmitters and N receivers is taken into account. Each transmitter emits an independent message and each receiver requests an arbitrary subset of the messages. Using the linear IA techniques, the optimal DoF assignment has been analyzed. Assuming generic channel coefficients, it has been shown that the perfect IA cannot be achieved for a broad class of interference networks. Analytical evaluation of DoF feasibility for general interference channels (IFCs) is complicated and not available yet. Iterative algorithm designed to minimize the leakage interference at each receiver is extended to work with general IFCs. This algorithm provides numerical insights into the feasibility of IA, which is not yet available in theory.

Exhaustion 2-subsets in dihedral groups of order 2p

Let [Formula: see text] be the dihedral group of order [Formula: see text], where [Formula: see text] is an odd prime. A nonempty subset [Formula: see text] of [Formula: see text] is said to be exhaustive if there exists a positive integer [Formula: see text] such that [Formula: see text] covers all elements in [Formula: see text]. The smallest such [Formula: see text] is called the exhaustion number of [Formula: see text]. In general, finding [Formula: see text] for an arbitrary subset [Formula: see text] of [Formula: see text] is an interesting but difficult task. In this paper, we classify all possible exhaustion 2-subsets in [Formula: see text] by considering a 2-subset [Formula: see text] of [Formula: see text] with either [Formula: see text] or [Formula: see text] and [Formula: see text]. Some explicit formulas for [Formula: see text] are identified and hence some bounds are derived to prove the existence for certain families of exhaustive sets.

Edge stability in secure graph domination

Graph Theory International audience A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X-v)∪u is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. A graph G is p-stable if the largest arbitrary subset of edges whose removal from G does not increase the secure domination number of the resulting graph, has cardinality p. In this paper we study the problem of computing p-stable graphs for all admissible values of p and determine the exact values of p for which members of various infinite classes of graphs are p-stable. We also consider the problem of determining analytically the largest value ωn of p for which a graph of order n can be p-stable. We conjecture that ωn=n-2 and motivate this conjecture.

Tietze Extension Theorem for n-dimensional Spaces

Summary In this article we prove the Tietze extension theorem for an arbitrary convex compact subset of εn with a non-empty interior. This theorem states that, if T is a normal topological space, X is a closed subset of T, and A is a convex compact subset of εn with a non-empty interior, then a continuous function f : X → A can be extended to a continuous function g : T → εn. Additionally we show that a subset A is replaceable by an arbitrary subset of a topological space that is homeomorphic with a convex compact subset of En with a non-empty interior. This article is based on [20]; [23] and [22] can also serve as reference books.