Finite Product Research Articles

Complete Hausdorffness and Complete Regularity on Supra Topological Spaces

The supra topological topic is of great importance in preserving some topological properties under conditions weaker than topology and constructing a suitable framework to describe many real-life problems. Herein, we introduce the version of complete Hausdorffness and complete regularity on supra topological spaces and discuss their fundamental properties. We show the relationships between them with the help of examples. In general, we study them in terms of hereditary and topological properties and prove that they are closed under the finite product space. One of the issues we are interested in is showing the easiness and diversity of constructing examples that satisfy supra T i spaces compared with their counterparts on general topology.

Finding best approximation pairs for two intersections of closed convex sets

The problem of finding a best approximation pair of two sets, which in turn generalizes the well known convex feasibility problem, has a long history that dates back to work by Cheney and Goldstein in 1959. In 2018, Aharoni, Censor, and Jiang revisited this problem and proposed an algorithm that can be used when the two sets are finite intersections of halfspaces. Motivated by their work, we present alternative algorithms that utilize projection and proximity operators. Our modeling framework is able to accommodate even convex sets. Numerical experiments indicate that these methods are competitive and sometimes superior to the one proposed by Aharoni et al.

Geometry, Coordinatization and Cardinality of the Rational Numbers from Physical Perspective

In this article, we will first discuss the completeness of real numbers in the context of an alternate definition of the straight line as a geometric continuum. According to this definition, points are not regarded as the basic constituents of a line segment and a line segment is considered to be a fundamental geometric object. This definition is in particular suitable to coordinatize different points on the straight line preserving the order properties of real numbers. Geometrically fundamental nature of line segments are required in physical theories like the string theory. We will construct a new topology suitable for this alternate definition of the straight line as a geometric continuum. We will discuss the cardinality of rational numbers in the later half of the article. We will first discuss what we do in an actual process of counting and define functions well-defined on the set of all positive integers. We will follow an alternate approach that depends on the Hausdorff topology of real numbers to demonstrate that the set of positive rationals can have a greater cardinality than the set of positive integers. This approach is more consistent with an actual act of counting. We will illustrate this aspect further using well-behaved functionals of convergent functions defined on the finite dimensional Cartezian products of the set of positive integers and non-negative integers. These are similar to the partition functions in statistical physics. This article indicates that the axiom of choice can be a better technique to prove theorems that use second-countability. This is important for the metrization theorems and physics of spacetime.

Clark measures and de Branges–Rovnyak spaces in several variables

Let denote the unit ball of , , and let denote a finite product of , . Given a non-constant holomorphic function , we study the corresponding family , , of Clark measures on the distinguished boundary . We construct a natural unitary operator from the de Branges–Rovnyak space onto the Hardy space . As an application, for and an inner function , we show that the property is directly related to the membership of an appropriate explicit function in .

An Inventory Production Model for Deteriorating Items Allowing Price Discount with Permissible Delay in Payments

The present Paper deals with an inventory production policy for a ramp type deteriorating item over a finite planning horizon with constant demand, finite production rate and shortages are not allowed. The optimal number of production cycles that minimizes the average system cost is determined. The model permits inventory shortage in each cycle, which is completely backlogged within the cycle itself. Every cycle starts with zero stock and production. As production continues, the inventory begins to accumulate after meeting current demands. The excess inventory accumulated during the production period is used to account for demand and deterioration in the no-production period. Numerical examples, tables, and final concluding remarks are discussed in the subsequent sections. Numerical example solved by using Mathematica software.

A Totally Relaxed, Self-Adaptive Subgradient Extragradient Method for Variational Inequality and Fixed Point Problems in a Banach Space

Abstract In this paper, we introduce a Totally Relaxed Self-adaptive Subgradient Extragradient Method (TRSSEM) with Halpern iterative scheme for finding a common solution of a Variational Inequality Problem (VIP) and the fixed point of quasi-nonexpansive mapping in a 2-uniformly convex and uniformly smooth Banach space. The TRSSEM does not require the computation of projection onto the feasible set of the VIP; instead, it uses a projection onto a finite intersection of sub-level sets of convex functions. The advantage of this is that any general convex feasible set can be involved in the VIP. We also introduce a modified TRSSEM which involves the projection onto the set of a convex combination of some convex functions. Under some mild conditions, we prove a strong convergence theorem for our algorithm and also present an application of our theorem to the approximation of a solution of nonlinear integral equations of Hammerstein’s type. Some numerical examples are presented to illustrate the performance of our method as well as comparing it with some related methods in the literature. Our algorithm is simple and easy to implement for computation.

Closed sets of finitary functions between products of finite fields of coprime order

We investigate the finitary functions from a finite product of finite fields prod _{j =1}^mmathbb {F}_{q_j} = {mathbb K} to a finite product of finite fields prod _{i =1}^nmathbb {F}_{p_i} = {mathbb {F}}, where |{mathbb K}| and |{mathbb {F}}| are coprime. An ({mathbb {F}},{mathbb K})-linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these subsets of functions through the {mathbb {F}}_p[{mathbb K}^{times }]-submodules of mathbb {F}_p^{{mathbb K}}, where {mathbb K}^{times } is the multiplicative monoid of {mathbb K}= prod _{i=1}^m {mathbb {F}}_{q_i}. Furthermore we prove that each of these subsets of functions is generated by a set of unary functions and we provide an upper bound for the number of distinct ({mathbb {F}},{mathbb K})-linearly closed clonoids.

Certain Inequalities Pertaining to Some New Generalized Fractional Integral Operators

In this paper, we introduce the generalized left-side and right-side fractional integral operators with a certain modified ML kernel. We investigate the Chebyshev inequality via this general family of fractional integral operators. Moreover, we derive new results of this type of inequalities for finite products of functions. In addition, we establish an estimate for the Chebyshev functional by using the new fractional integral operators. From our above-mentioned results, we find similar inequalities for some specialized fractional integrals keeping some of the earlier results in view. Furthermore, two important results and some interesting consequences for convex functions in the framework of the defined class of generalized fractional integral operators are established. Finally, two basic examples demonstrated the significance of our results.

Angle Sums of Schläfli Orthoschemes

We consider the simplices KnA={x∈Rn+1:x1≥x2≥⋯≥xn+1,x1-xn+1≤1,x1+⋯+xn+1=0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} K_n^A=\\{x\\in {\\mathbb {R}}^{n+1}:x_1\\ge x_2\\ge \\cdots \\ge x_{n+1},x_1-x_{n+1}\\le 1,\\,x_1+\\cdots +x_{n+1}=0\\} \\end{aligned}$$\\end{document}and KnB={x∈Rn:1≥x1≥x2≥⋯≥xn≥0},\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} K_n^B=\\{x\\in {\\mathbb {R}}^n:1\\ge x_1\\ge x_2\\ge \\cdots \\ge x_n\\ge 0\\}, \\end{aligned}$$\\end{document}which are called the Schläfli orthoschemes of types A and B, respectively. We describe the tangent cones at their j-faces and compute explicitly the sums of the conic intrinsic volumes of these tangent cones at all j-faces of K_n^A and K_n^B. This setting contains sums of external and internal angles of K_n^A and K_n^B as special cases. The sums are evaluated in terms of Stirling numbers of both kinds. We generalize these results to finite products of Schläfli orthoschemes of type A and B and, as a probabilistic consequence, derive formulas for the expected number of j-faces of the Minkowski sums of the convex hulls of a finite number of Gaussian random walks and random bridges. Furthermore, we evaluate the analogous angle sums for the tangent cones of Weyl chambers of types A and B and finite products thereof.

Intrinsic characterizations of C-realcompact spaces

<pre>c-realcompact spaces are introduced by Karamzadeh and Keshtkar in Quaest. Math. 41, no. 8 (2018), 1135-1167. We offer a characterization of these spaces X via c-stable family of closed sets in X by showing that X is c-realcompact if and only if each c-stable family of closed sets in X with finite intersection property has nonempty intersection. This last condition which makes sense for an arbitrary topological space can be taken as an alternative definition of a c-realcompact space. We show that each topological space can be extended as a dense subspace to a c-realcompact space with some desired extension properties. An allied class of spaces viz CP-compact spaces akin to that of c-realcompact spaces are introduced. The paper ends after examining how far a known class of c-realcompact spaces could be realized as CP-compact for appropriately chosen ideal P of closed sets in X.</pre>

Dynamical IP⋆-sets in weak rings

V. Bergelson and N. Hindman in [1], proved that IP⋆- sets contain all possible finite sums and products of a sum subsystem of any sequence in N. In a recent work [2], the second author of this article has proved that a stronger result holds for dynamical IP⋆- sets. In this article we will establish a non-commutative version of this result. We will prove that a richer configuration is contained in dynamical IP⋆- sets in weak rings.

Some explicit formulas for partial sums of Möbius functions

The purpose of this paper is to give some explicit formulas involving Möbius functions. Such explicit formulas may be known under the generalized Riemann Hypothesis, but unconditional in this paper. Concretely, we prove explicit formulas of partial sums of the Möbius function in arithmetic progressions and partial sums of the Möbius functions on an Abelian number field. In addition, to obtain these explicit formulas, we study a certain finite Euler product appearing from a relation between primitive characters and imprimitive characters.

Reiterman’s Theorem on Finite Algebras for a Monad

Profinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman’s theorem states that they precisely specify pseudovarieties, i.e., classes of finite algebras closed under finite products, subalgebras and quotients. In this article, Reiterman’s theorem is generalized to finite Eilenberg-Moore algebras for a monad T on a category D: we prove that a class of finite T -algebras is a pseudovariety iff it is presentable by profinite equations. As a key technical tool, we introduce the concept of a profinite monad T ^ associated to the monad T , which gives a categorical view of the construction of the space of profinite terms.

One-sided Shewhart control charts for monitoring the ratio of two normal variables in short production runs

Monitoring the ratio of two normal random variables plays an important role in several manufacturing environments. In addition, the traditional control charts that have been developed for infinite production horizons can not function effectively to detect anomalies in short production runs. In this paper, we tackle this problem by proposing two one-sided Shewhart-type charts to monitor the ratio of two normal random variables for a finite horizon production. The statistical performance of the proposed charts is investigated using the truncated average run length as a performance measure in short production runs. In order to help the quality practitioner to implement these control charts, we have provided ready-to-use tables of the control limit parameters. An illustrative example from the food industry is given for illustration.

Directed sets and topological spaces definable in o‐minimal structures

We study directed sets definable in o-minimal structures, showing that in expansions of ordered fields these admit cofinal definable curves, as well as a suitable analogue in expansions of ordered groups, and furthermore that no analogue holds in full generality. We use the theory of tame pairs to extend the results in the field case to definable families of sets with the finite intersection property. We then apply our results to the study of definable topologies. We prove that all definable topological spaces display properties akin to first countability, and give several characterizations of a notion of definable compactness due to Peterzil and Steinhorn (J. Lond. Math. Soc. (2) 59 (1999) 769–786) generalized to this setting.

Proper actions on finite products of quasi-trees

We say that a finitely generated group G has property (QT) if it acts isometrically on a finite product of quasi-trees so that orbit maps are quasi-isometric embeddings. A quasi-tree is a connected graph with path metric quasi-isometric to a tree, and product spaces are equipped with the ℓ 1 -metric.

A finite product of ordinals is hereditarily dually discrete

In this article, we mainly show that a finite product of ordinals is hereditarily dually discrete. This gives an affirmative answer to a problem posed by Peng in [16, Problem 12]. By this conclusion and a known conclusion we have that if Y is a subspace of the product of a finitely many ordinals, then Y is hereditarily a Lindelöf D-space if and only if Y has countable spread.

Periodic matrix difference equations and companion matrices in blocks: some applications

This study is devoted to some periodic matrix difference equations, through their associated product of companion matrices in blocks. Linear recursive sequences in the algebra of square matrices in blocks and the generalized Cayley–Hamilton theorem are considered for working out some results about the powers of matrices in blocks. Two algorithms for computing the finite product of periodic companion matrices in blocks are built. Illustrative examples and applications are considered to demonstrate the effectiveness of our approach.

Connectedness and Local Connectedness on Infra Soft Topological Spaces

This year, we introduced the concept of infra soft topology as a new generalization of soft topology. To complete our analysis of this space, we devote this paper to presenting the concepts of infra soft connected and infra soft locally connected spaces. We provide some descriptions for infra soft connectedness and elucidate that there is no relationship between an infra soft topological space and its parametric infra topological spaces with respect to the property of infra soft connectedness. We discuss the behaviors of infra soft connected and infra soft locally connected spaces under infra soft homeomorphism maps and a finite product of soft spaces. We complete this manuscript by defining a component of a soft point and establishing its main properties. We determine the conditions under which the number of components is finite or countable, and we discuss under what conditions the infra soft connected subsets are components.

Measurable functions and covering properties

Let X be a set and A⊆P(X) be a family closed under finite intersections such that ∅,X∈A. If ψ=o,ω,γ, then Ψ(A) is the family of those ψ-covers U for which U⊆A. In [3], properties (Ψ0) of a family F⊆XR of real functions have been introduced. The main result of the paper Theorem 4.1 reads as follows: if Φ=Ω,Γ and Ψ=O,Ω,Γ, then for any pair 〈Φ,Ψ〉 different from 〈Ω,O〉, X has the covering property S1(Φ(A),Ψ(A)) if and only if the family of non-negative upper A-semimeasurable real functions satisfies the selection principle S1(Φ0,Ψ0). Similarly for Sfin and Ufin. Some related results are also presented.