Finite Family Research Articles

External Direct Product JU-Algebras

Lingcong and Endam introduced the idea of the direct product of a finite family of B-algebras. In this paper, we introduce the concept of the direct product of an infinite family of JU-algebras, which we call the external direct product. This is a general concept of the direct product in the sense of Lingcong and Endam. The result is based on different types of subsets of JU-algebras. These include JU-subalgebras, JU-ideals, p-ideals, t-ideals, strong JU-ideals, JU-filters, comparative JU-filters, and implicative JU-filters. Also, we introduce the concept of the weak direct product of JU-algebras. Finally, we provide several fundamental theorems of (anti-)JU-homomorphisms in view of the external direct product JU-algebras.

An iterative approach for addressing monotone inclusion and fixed point problems with generalized demimetric mappings

Throughout this study, we present a new algorithm for finding the common solution of a finite family of monotone inclusion and the fixed point problem of a finite family of generalized demimetric operators in the context of a real Hilbert space and show its strong convergence. Moreover, we utilize our result to solve the minimization problem.

Explicit families of congruences for the overpartition function

AbstractIn this article we exhibit new explicit families of congruences for the overpartition function, making effective the existence results given previously by Treneer. We give infinite families of congruences modulo m for $$m = 3, 5, 7, 11$$ m = 3 , 5 , 7 , 11 , and finite families for $$m = 13, 17, 19$$ m = 13 , 17 , 19 .

Monomial asymptotic polynomials and applications to polynomial optimization problems

The problem of polynomial optimization plays an important role in many fields such as physics, chemistry, and economics. This problem has received research attention from many mathematicians recently. In this paper, we study the polynomial optimization problem over a non-compact semi-algebraic set, for which its constraint set of polynomials G is asymptotic with a finite family of monomials. By changing variables via a suitable monomial mapping, we transform the problem under consideration into the polynomial optimization problem over a compact semi-algebraic feasible set. We then apply the well-known result that the optimal value of a polynomial over a compact semi-algebraic set can be approximated as closely as desired by solving a hierarchy of semi-definite programs and the convergence is finite generically, to obtain results in the general case when the cone C(G) is unimodular. In particular, in the case of polynomials in two variables, we solve the problem quite completely without requiring C(G) to be unimodular

Variants of the Gyárfás–Sumner conjecture: Oriented trees and rainbow paths

AbstractGiven a finite family of graphs, we say that a graph is “‐free” if does not contain any graph in as a subgraph. We abbreviate ‐free to just “‐free” when . A vertex‐colored graph is called “rainbow” if no two vertices of have the same color. Given an integer and a finite family of graphs , let denote the smallest integer such that any properly vertex‐colored ‐free graph having contains an induced rainbow path on vertices. Scott and Seymour showed that exists for every complete graph . A conjecture of N. R. Aravind states that . The upper bound on that can be obtained using the methods of Scott and Seymour setting are, however, super‐exponential. Gyárfás and Sárközy showed that . For , we show that and therefore, . This significantly improves Gyárfás and Sárközy's bound and also covers a bigger class of graphs. We adapt our proof to achieve much stronger upper bounds for graphs of higher girth: we prove that , where . Moreover, in each case, our results imply the existence of at least distinct induced rainbow paths on vertices. Along the way, we obtain some new results on an oriented variant of the Gyárfás–Sumner conjecture. For , let denote the orientations of in which one vertex has out‐degree or in‐degree . We show that every ‐free oriented graph having a chromatic number at least and every bikernel‐perfect oriented graph with girth having a chromatic number at least contains every oriented tree on at most vertices as an induced subgraph.

Parallel inertial forward–backward splitting methods for solving variational inequality problems with variational inclusion constraints

The inertial forward–backward splitting algorithm can be considered as a modified form of the forward–backward algorithm for variational inequality problems with monotone and Lipschitz continuous cost mappings. By using parallel and inertial techniques and the forward–backward splitting algorithm, in this paper, we propose a new parallel inertial forward–backward splitting algorithm for solving variational inequality problems, where the constraints are the intersection of common solution sets of a finite family of variational inclusion problems. Then, strong convergence of proposed iteration sequences is showed under standard assumptions imposed on cost mappings in a real Hilbert space. Finally, some numerical experiments demonstrate the reliability and benefits of the proposed algorithm.

Numerical Algorithms for Divergence-Free Velocity Applications

This work focuses on the well-known issue of mass conservation in the context of the finite element technique for computational fluid dynamic simulations. Specifically, non-conventional finite element families for solving Navier–Stokes equations are investigated to address the mathematical constraint of incompressible flows. Raviart–Thomas finite elements are employed for the achievement of a discrete free-divergence velocity. In particular, the proposed algorithm projects the velocity field into the discrete free-divergence space by using the lowest-order Raviart–Thomas element. This decomposition is applied in the context of the projection method, a numerical algorithm employed for solving Navier–Stokes equations. Numerical examples validate the approach’s effectiveness, considering different types of computational grids. Additionally, the presented paper considers an interface advection problem using marker approximation in the context of multiphase flow simulations. Numerical tests, equipped with an analytical velocity field for the surface advection, are presented to compare exact and non-exact divergence-free velocity interpolation.

Majorization requires infinitely many second laws

Majorization is a fundamental model of uncertainty with several applications in areas ranging from thermodynamics to entanglement theory, and constitutes one of the pillars of the resource-theoretic approach to physics. Here, we improve on its relation to measurement apparatuses. In particular, after discussing what the proper notion of second law in this scenario is, we show that, for a sufficiently large state space, any family of entropy-like functions constituting a second law must be countably infinite. Moreover, we provide an analogous result for a variation of majorization known as thermo-majorization which, in fact, does not require any constraint on the state space provided the equilibrium distribution is not uniform. Finally, we discuss the applicability of our results to molecular diffusion and catalytic majorization. In this regard, we consider a variation of majorization used in plasma physics as a model of molecular diffusion and show that no finite family of entropy-like functions constituting a second law of molecular diffusion exists. Moreover, we show how our results are useful when dealing with a conjecture regarding catalytic majorization (i.e. trumping). In particular, we show that the sort of characterizations of trumping that have been considered before require an infinite family of real-valued functions.

On the functional regression model and its finite-dimensional approximations

AbstractThe problem of linearly predicting a scalar response Y from a functional (random) explanatory variable $$X=X(t),\ t\in I$$ X = X ( t ) , t ∈ I is considered. It is argued that the term “linearly” can be interpreted in several meaningful ways. Thus, one could interpret that (up to a random noise) Y could be expressed as a linear combination of a finite family of marginals $$X(t_i)$$ X ( t i ) of the process X, or a limit of a sequence of such linear combinations. This simple point of view (which has some precedents in the literature) leads to a formulation of the linear model in terms of the RKHS space generated by the covariance function of the process X(t). It turns out that such RKHS-based formulation includes the standard functional linear model, based on the inner product in the space $$L^2[0,1]$$ L 2 [ 0 , 1 ] , as a particular case. It includes as well all models in which Y is assumed to be (up to an additive noise) a linear combination of a finite number of linear projections of X. Some consistency results are proved which, in particular, lead to an asymptotic approximation of the predictions derived from the general (functional) linear model in terms of finite-dimensional models based on a finite family of marginals $$X(t_i)$$ X ( t i ) , for an increasing grid of points $$t_j$$ t j in I. We also include a discussion on the crucial notion of coefficient of determination (aimed at assessing the fit of the model) in this setting. A few experimental results are given.

Inertial iterative method for solving generalized equilibrium, variational inequality, and fixed point problems of multivalued mappings in Banach space

We devise an iterative algorithm incorporating inertial techniques to approximate the shared solution of a generalized equilibrium problem, a fixed point problem for a finite family of relatively nonexpansive multivalued mappings, and a variational inequality problem. Our discussion encompasses the strong convergence of the proposed algorithm and highlights specific outcomes derived from our theorem. Additionally, we provide a computational analysis to underscore the significance of our findings and draw comparisons. The results presented in this paper serve to extend and unify numerous previously established outcomes in this particular research domain.

A novel family of strain-based finite elements for the analysis of the material softening of planar frames

The article presents a novel strain-based finite element family for the analysis of material softening of planar frame structures. In our case, the softening zone is described by a discrete crack, which is considered as an ‘excluded’ finite element point, i.e., the deformation quantities in the crack are considered separately from the deformation quantities of the element. They are connected to the element only through kinematic quantities, used to describe the crack opening. The criterion for crack initiation is defined as the limit axial-bending resistance of the cross-section. The advantage of the presented model is that it is not necessary to define cracks or softening zones in advance and that the solution is mesh-independent in the sense that no further densification of the mesh is needed purely on account of capturing material softening. The accuracy and efficiency of the presented finite element family is illustrated by the example of a clamped–simply supported concrete beam and a portal concrete frame. The examples demonstrate that even with a minimum number of finite elements of suitable accuracy, sufficiently accurate results are obtained for normal engineering practice.

Split common null point problem for generalized monotone operators

ABSTRACT In this article, we present some properties of the resolvent of maximally comonotone (generalized monotone) operators in Hilbert spaces. Also, we study the split common null point problem for a finite family of maximally comonotone operators in Hilbert spaces. For solving this split common null point problem, we propose an inertial algorithm with the step size independent on the prior estimate of the norm of the bounded linear operators and prove strong convergence theorem for this inertial algorithm. The obtained result of this article complements many recent results in this direction.

Online class cover problem

In this paper, we study the online class cover problem where a (finite or infinite) family F of geometric objects and a set Pr of red points in Rd are given a prior, and blue points from Rd arrives one after another. Upon the arrival of a blue point, the online algorithm must make an irreversible decision to cover it with objects from F that do not cover any points of Pr. The objective of the problem is to place a minimum number of objects. When F consists of axis-parallel unit squares in R2, we prove that the competitive ratio of any deterministic online algorithm is Ω(log⁡|Pr|), and also propose an O(log⁡|Pr|)-competitive deterministic algorithm for the problem.

Halpern inertial subgradient extragradient algorithm for solving equilibrium problems in Banach spaces

ABSTRACT In this work, we study an equilibrium problem involving a pseudomonotone bifunction in the context of uniformly smooth and 2-uniformly convex real Banach spaces. For the purpose of solving the fixed point problem of a finite family of multi-valued relatively nonexpansive mappings and the pseudomonotone equilibrium problem, we introduce an inertial subgradient extragradient method and establish its strong convergence. To increase the step size and enhance the performance of our iterative method, we use a new parameter that is independent of the inertial extrapolation method. We point out that our step size is chosen in a self-adaptive manner, making it simple to calculate our iterative algorithm without having to know the Lipschitz constants beforehand. Finally, we give some numerical results for the proposed algorithm and comparison with some other known algorithms.

Modified subgradient extragradient method for solution of split variational inequality problem with application to generalized Nash equilibrium problem

In this research, we propose modified subgradient extragradient method to find the common solution of split variational inequality problem and common fixed point problem of a finite family of demicontractive mappings without knowing norm of bounded linear operator and Lipschitz constants of the associated pseudomonotone operators. We obtain the strong convergence result for the proposed method under certain mild conditions. This result extends and generalizes many recent results in this direction. Furthermore, we demonstrate how our findings may be used to solve a generalized Nash equilibrium problem and LASSO problem. Finally, we give numerical experiments to validate the performance of our algorithm and compare it with other existing methods.

R I biorthogonal polynomials of Hahn type

ABSTRACT A finite family of R I polynomials is introduced and studied. It consists in a set of polynomials of 3 F 2 form whose biorthogonality to an ensemble of rational functions is spelled out. These polynomials are shown to satisfy two generalized eigenvalue problems: in addition to their recurrence relation of R I type, they are also found to obey a difference equation. Underscoring this bispectrality is a triplet of operators with tridiagonal actions. The algebra associated to these operators is provided.

Subgradient Extragradient Method for Finite Lipschitzian Demicontractions and Variational Inequality Problems in a Hilbert Space

In this research, the modified subgradient extragradient method and K-mapping generated by a finite family of finite Lipschitzian demicontractions are introduced. Then, a strong convergence theorem for finding a common element of the common fixed point set of finite Lipschitzian demicontraction mappings and the common solution set of variational inequality problems is established. Furthermore, numerical examples are given to support the main theorem.

Colored Unavoidable Patterns and Balanceable Graphs

We study a Turán-type problem on edge-colored complete graphs. We show that for any $r$ and $t$, any sufficiently large $r$-edge-colored complete graph on $n$ vertices with $\Omega(n^{2-1/tr^r})$ edges in each color contains a member from certain finite family $\mathcal{F}_t^r$ of $r$-edge-colored complete graphs. We conjecture that $\Omega(n^{2-1/t})$ edges in each color are sufficient to find a member from ${\mathcal{F}}_t^r$. A result of Girão and Narayanan confirms this conjecture when $r=2$. Next, we study a related problem where the corresponding Turán threshold is linear. We call an edge-coloring of a path $P_{rk}$ balanced if each color appears $k$ times in the coloring. We show that any $3$-edge-coloring of a large complete graph with $kn+o(n)$ edges in each color contains a balanced $P_{3k}$. This is tight up to a constant factor of $2$. For more colors, the problem becomes surprisingly more delicate. Already for $r=7$, we show that even $n^{2-o(1)}$ edges from each color does not guarantee existence of a balanced $P_{7k}$.

Strong and Weak Convergence Theorems for the Split Feasibility Problem of (β,k)-Enriched Strict Pseudocontractive Mappings with an Application in Hilbert Spaces

The concept of symmetry has played a major role in Hilbert space setting owing to the structure of a complete inner product space. Subsequently, different studies pertaining to symmetry, including symmetric operators, have investigated real Hilbert spaces. In this paper, we study the solutions to multiple-set split feasibility problems for a pair of finite families of β-enriched, strictly pseudocontractive mappings in the setup of a real Hilbert space. In view of this, we constructed an iterative scheme that properly included these two mappings into the formula. Under this iterative scheme, an appropriate condition for the existence of solutions and strong and weak convergent results are presented. No sum condition is imposed on the countably finite family of the iteration parameters in obtaining our results unlike for several other results in this direction. In addition, we prove that a slight modification of our iterative scheme could be applied in studying hierarchical variational inequality problems in a real Hilbert space. Our results improve, extend and generalize several results currently existing in the literature.

Fixed point results involving a finite family of enriched strictly pseudocontractive and pseudononspreading mappings

In this study, we introduce a method for finding common fixed points of a finite family of (ηi,ki)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\eta _{i}, k_{i})$\\end{document}-enriched strictly pseudocontractive (ESPC) maps and (ηi,βi)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\eta _{i}, \\beta _{i})$\\end{document}-enriched strictly pseudononspreading (ESPN) maps in the setting of real Hilbert spaces. Further, we prove the strong convergence theorem of the proposed method under mild conditions on the control parameters. Our main results are also applied in proving strong convergence theorems for ηi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\eta _{i}$\\end{document}-enriched nonexpansive, strongly inverse monotone, and strictly pseudononspreading maps. Some nontrivial examples are given, and the results obtained extend, improve, and generalize several well-known results in the current literature.