787 results
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2. A Note on the Paper 'A Common Fixed Point Theorem with Applications'
- Author
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Jiangqian Zou and Hui Huang
- Subjects
Discrete mathematics ,Kantorovich inequality ,021103 operations research ,Control and Optimization ,Applied Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,02 engineering and technology ,Management Science and Operations Research ,01 natural sciences ,Theory of computation ,Common fixed point ,Log sum inequality ,Rearrangement inequality ,0101 mathematics ,Mathematical economics ,Common fixed point theorem ,Mathematics - Abstract
In this note, we give affirmative answers to two open problems posed by Agarwal et al. in the paper (J Optim Theory Appl 163(2):482---490, 2014).
- Published
- 2015
3. Some comments on the paper: Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959
- Author
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Michelle Pierri and Donal O'Regan
- Subjects
Discrete mathematics ,Class (set theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,Differential systems ,01 natural sciences ,010101 applied mathematics ,Controllability ,Algebra ,0101 mathematics ,Differential (mathematics) ,Mathematics - Abstract
The abstract results and applications presented in “Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959, are not correct. Moreover, the class of differential control problems studied in [1] is not H-controllable.
- Published
- 2016
4. A Note on the Paper 'The Asymptotic Behavior of the Composition of Firmly Nonexpansive Mappings'
- Author
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David Ariza-Ruiz, Adriana Nicolae, and Genaro López-Acedo
- Subjects
Discrete mathematics ,021103 operations research ,Control and Optimization ,Applied Mathematics ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,Management Science and Operations Research ,Composition (combinatorics) ,01 natural sciences ,Theory of computation ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this note we correct an error in the paper (Ariza-Ruiz et al. in J Optim Theory Appl 167:409–429, 2015).
- Published
- 2017
5. Formal self duality
- Author
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Robert Schüler and Lukas Kölsch
- Subjects
Discrete mathematics ,20C15, 20K01 ,Computer Networks and Communications ,Applied Mathematics ,010102 general mathematics ,Duality (mathematics) ,0102 computer and information sciences ,01 natural sciences ,Dual (category theory) ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Abelian group ,Connection (algebraic framework) ,Boolean function ,Mathematics - Abstract
We study the notion of formal self duality in finite abelian groups. Formal duality in finite abelian groups has been proposed by Cohn, Kumar, Reiher and Schürmann. In this paper we give a precise definition of formally self dual sets and discuss results from the literature in this perspective. Also, we discuss the connection to formally dual codes. We prove that formally self dual sets can be reduced to primitive formally self dual sets similar to a previously known result on general formally dual sets. Furthermore, we describe several properties of formally self dual sets. Also, some new examples of formally self dual sets are presented within this paper. Lastly, we study formally self dual sets of the form $\{(x,F(x)) \ : \ x\in {\mathbb {F}}_{2^{n}}\}$ { ( x , F ( x ) ) : x ∈ F 2 n } where F is a vectorial Boolean function mapping ${\mathbb {F}}_{2^{n}}$ F 2 n to ${\mathbb {F}}_{2^{n}}$ F 2 n .
- Published
- 2021
6. Positive-definite modification of a covariance matrix by minimizing the matrix $$\ell_{\infty}$$ norm with applications to portfolio optimization
- Author
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Johan Lim, Shota Katayama, Young-Geun Choi, and Seonghun Cho
- Subjects
0106 biological sciences ,Statistics and Probability ,Discrete mathematics ,Economics and Econometrics ,Covariance matrix ,Applied Mathematics ,Estimator ,Positive-definite matrix ,010603 evolutionary biology ,01 natural sciences ,010104 statistics & probability ,Matrix (mathematics) ,Positive definiteness ,Modeling and Simulation ,Norm (mathematics) ,Diagonal matrix ,Convex combination ,0101 mathematics ,Social Sciences (miscellaneous) ,Analysis ,Mathematics - Abstract
The covariance matrix, which should be estimated from the data, plays an important role in many multivariate procedures, and its positive definiteness (PDness) is essential for the validity of the procedures. Recently, many regularized estimators have been proposed and shown to be consistent in estimating the true matrix and its support under various structural assumptions on the true covariance matrix. However, they are often not PD. In this paper, we propose a simple modification to make a regularized covariance matrix be PD while preserving its support and the convergence rate. We focus on the matrix $$\ell_{\infty }$$ norm error in covariance matrix estimation because it could allow us to bound the error in the downstream multivariate procedure relying on it. Our proposal in this paper is an extension of the fixed support positive-definite (FSPD) modification by Choi et al. (2019) from spectral and Frobenius norms to the matrix $$\ell_{\infty }$$ norm. Like the original FSPD, we consider a convex combination between the initial estimator (the regularized covariance matrix without PDness) and a given form of the diagonal matrix minimize the $$\ell_{\infty }$$ distance between the initial estimator and the convex combination, and find a closed-form expression for the modification. We apply the procedure to the minimum variance portfolio (MVP) optimization problem and show that the vector $$\ell_{\infty }$$ error in the estimation of the optimal portfolio weight is bounded by the matrix $$\ell _{\infty }$$ error of the plug-in covariance matrix estimator. We illustrate the MVP results with S&P 500 daily returns data from January 1978 to December 2014.
- Published
- 2021
7. On the turnpike property with interior decay for optimal control problems
- Author
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Martin Gugat
- Subjects
Discrete mathematics ,0209 industrial biotechnology ,Control and Optimization ,Partial differential equation ,Applied Mathematics ,010102 general mathematics ,Order (ring theory) ,Natural number ,02 engineering and technology ,State (functional analysis) ,Optimal control ,01 natural sciences ,020901 industrial engineering & automation ,Control and Systems Engineering ,Ordinary differential equation ,Signal Processing ,Integral element ,ddc:510 ,0101 mathematics ,Stationary state ,Mathematics - Abstract
In this paper the turnpike phenomenon is studied for problems of optimal control where both pointwise-in-time state and control constraints can appear. We assume that in the objective function, a tracking term appears that is given as an integral over the time-interval $$[0,\, T]$$ [ 0 , T ] and measures the distance to a desired stationary state. In the optimal control problem, both the initial and the desired terminal state are prescribed. We assume that the system is exactly controllable in an abstract sense if the time horizon is long enough. We show that that the corresponding optimal control problems on the time intervals $$[0, \, T]$$ [ 0 , T ] give rise to a turnpike structure in the sense that for natural numbers n if T is sufficiently large, the contribution of the objective function from subintervals of [0, T] of the form $$\begin{aligned} {[}t - t/2^n,\; t + (T-t)/2^n] \end{aligned}$$ [ t - t / 2 n , t + ( T - t ) / 2 n ] is of the order $$1/\min \{t^n, (T-t)^n\}$$ 1 / min { t n , ( T - t ) n } . We also show that a similar result holds for $$\epsilon $$ ϵ -optimal solutions of the optimal control problems if $$\epsilon >0$$ ϵ > 0 is chosen sufficiently small. At the end of the paper we present both systems that are governed by ordinary differential equations and systems governed by partial differential equations where the results can be applied.
- Published
- 2021
8. Delone sets in ℝ3: Regularity Conditions
- Author
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N. P. Dolbilin
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Statistics and Probability ,Discrete mathematics ,Euclidean space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Delone set ,01 natural sciences ,Identity (music) ,010305 fluids & plasmas ,Set (abstract data type) ,0103 physical sciences ,Homogeneous space ,Mathematics::Metric Geometry ,0101 mathematics ,Symmetry (geometry) ,Orbit (control theory) ,Link (knot theory) ,Mathematics - Abstract
A regular system is a Delone set in Euclidean space with a transitive group of symmetries or, in other words, the orbit of a crystallographic group. The local theory for regular systems, created by the geometric school of B. N. Delone, was aimed, in particular, to rigorously establish the “local-global-order” link, i.e., the link between the arrangement of a set around each of its points and symmetry/regularity of the set as a whole. The main result of this paper is a proof of the so-called 10R-theorem. This theorem asserts that identity of neighborhoods within a radius 10R of all points of a Delone set (in other words, an (r, R)-system) in 3D Euclidean space implies regularity of this set. The result was obtained and announced long ago independently by M. Shtogrin and the author of this paper. However, a detailed proof remains unpublished for many years. In this paper, we give a proof of the 10R-theorem. In the proof, we use some recent results of the author, which simplify the proof.
- Published
- 2020
9. Orbits of bounded bijective operators and Gabor frames
- Author
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Rosario Corso and Corso R.
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Context (language use) ,01 natural sciences ,symbols.namesake ,Operator (computer programming) ,Wavelet ,Operator representation of frames ,Settore MAT/05 - Analisi Matematica ,0103 physical sciences ,FOS: Mathematics ,Orthonormal basis ,0101 mathematics ,Representation (mathematics) ,Mathematics ,Discrete mathematics ,Bounded bijective operators ,Applied Mathematics ,010102 general mathematics ,Hilbert space ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Bounded function ,symbols ,Bijection ,010307 mathematical physics ,42C15, 94A20 ,Gabor frames - Abstract
This paper is a contribution to frame theory. Frames in a Hilbert space are generalizations of orthonormal bases. In particular, Gabor frames of $L^2(\mathbb{R})$, which are made of translations and modulations of one or more windows, are often used in applications. More precisely, the paper deals with a question posed in the last years by Christensen and Hasannasab about the existence of overcomplete Gabor frames, with some ordering over $\mathbb{Z}$, which are orbits of bounded operators on $L^2(\mathbb{R})$. Two classes of overcomplete Gabor frames which cannot be ordered over $\mathbb{Z}$ and represented by orbits of operators in $GL(L^2(\mathbb{R}))$ are given. Some results about operator representation are stated in a general context for arbitrary frames, covering also certain wavelet frames., Comment: 12 pages
- Published
- 2020
10. A Proof via Finite Elements for Schiffer’s Conjecture on a Regular Pentagon
- Author
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Bartłomiej Siudeja, Benjamin Young, and Nilima Nigam
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Discrete mathematics ,Sequence ,Conjecture ,Applied Mathematics ,Regular polygon ,Boundary (topology) ,010103 numerical & computational mathematics ,Mathematics::Spectral Theory ,Eigenfunction ,Equilateral triangle ,01 natural sciences ,Computational Mathematics ,Computational Theory and Mathematics ,0101 mathematics ,Laplace operator ,Analysis ,Eigenvalues and eigenvectors ,Mathematics - Abstract
A modified version of Schiffer’s conjecture on a regular pentagon states that Neumann eigenfunctions of the Laplacian do not change sign on the boundary. In a companion paper by Bartlomiej Siudeja it was shown that eigenfunctions which are strictly positive on the boundary exist on regular polygons with at least 6 sides, while on equilateral triangles and cubes it is not even possible to find an eigenfunction which is nonnegative on the boundary. The case for the regular pentagon is more challenging, and has resisted a completely analytic attack. In this paper, we present a validated numerical method to prove this case, which involves iteratively bounding eigenvalues for a sequence of subdomains of the triangle. We use a learning algorithm to find and optimize this sequence of subdomains, making it straightforward to check our computations with standard software. Our proof has a short proof certificate, is checkable without specialized software and is adaptable to other situations.
- Published
- 2020
11. Global in Time and Bounded Solutions to a Parabolic–Elliptic Chemotaxis System with Nonlinear Diffusion and Signal-Dependent Sensitivity
- Author
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Giuseppe Viglialoro
- Subjects
Discrete mathematics ,0209 industrial biotechnology ,Control and Optimization ,Partial differential equation ,Applied Mathematics ,010102 general mathematics ,02 engineering and technology ,01 natural sciences ,Domain (mathematical analysis) ,Nonlinear system ,020901 industrial engineering & automation ,Bounded function ,Uniform boundedness ,Nonlinear diffusion ,Sensitivity (control systems) ,0101 mathematics ,Mathematics - Abstract
This paper deals with a system of two coupled partial differential equations arising in chemotaxis, involving nonlinear diffusion and nonlinear and signal-dependent sensitivity. Depending on the interplay between such nonlinearities, we establish the existence of global classical solutions which are uniformly bounded in time. Precisely, we study the zero-flux chemotaxis-system $$\varOmega $$ being a bounded and smooth domain of $$\mathbb {R}^n$$ , $$n\ge 1$$ , and where $$m,\alpha \in \mathbb {R}$$ , with $$\alpha \le \max \{m,\frac{m+1}{2}\}$$ . Additionally, $$00, k\ge 1$$ and all $$s>0$$ . We prove that for any nonnegative and properly regular initial data u(x, 0), the initial-boundary value problem associated to ( $$\Diamond $$ ) admits a unique globally bounded classical solution, provided some smallness assumptions on $$\chi _0$$ are satisfied. In addition, in this article we compare our results with those achieved in the recent paper (Wang et al. in J Differ Equ 263(5):2851–2873, 2017); we will emphasize how the employment of independent techniques used to solve problem ( $$\Diamond $$ ) may lead to complementary conclusions.
- Published
- 2019
12. An extension of the Hermite–Hadamard inequality for convex and s-convex functions
- Author
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Péter Kórus
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Regular polygon ,010103 numerical & computational mathematics ,Extension (predicate logic) ,01 natural sciences ,Iterated integrals ,Hermite–Hadamard inequality ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Convex function ,Mathematics - Abstract
The Hermite–Hadamard inequality was extended using iterated integrals by Retkes [Acta Sci Math (Szeged) 74:95–106, 2008]. In this paper we further extend the main results of the above paper for convex and also for s-convex functions in the second sense.
- Published
- 2019
13. Approximation by modified $$U^{\rho }_n$$ U n ρ operators
- Author
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Ana Maria Acu, Voichiţa Adriana Radu, and Tuncer Acar
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Discrete mathematics ,Algebra and Number Theory ,Generalization ,Applied Mathematics ,010102 general mathematics ,Lambda ,01 natural sciences ,Shape parameter ,Moduli ,010101 applied mathematics ,Computational Mathematics ,Convergence (routing) ,Geometry and Topology ,0101 mathematics ,Analysis ,Mathematics - Abstract
The purpose of present paper is to extend the study of $$\lambda $$ -Bernstein operators introduce by Cai et al. (J Inequal Appl 12:1–11, 2018). In our paper we consider a generalization of the $$U^{\rho }_n$$ operators introduced in 2007 by Radu Paltanea, using the new Bernstein–Bezier bases $$\{\tilde{b}_{n,k}\}$$ with shape parameter $$\lambda $$ . Some approximation properties are given, including local approximation, error estimation in terms of moduli of continuity and Voronovskaja-type asymptotic formulas. Finally, we give some numerical examples and graphs to put in evidence the convergence of $$U^{\rho }_n(f;x)$$ to f(x).
- Published
- 2019
14. Generating functions of binary products of k-Fibonacci and orthogonal polynomials
- Author
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Mehmet Acikgoz, Serkan Araci, Mohamed Kerada, Ali Boussayoud, Souhila Boughaba, and HKÜ, İktisadi, İdari ve Sosyal Bilimler Fakültesi, İktisat Bölümü
- Subjects
Discrete mathematics ,Algebra and Number Theory ,Fibonacci number ,Applied Mathematics ,010102 general mathematics ,Symmetric functions ,Generating functions ,k-Fibonacci numbers ,k-Pell numbers ,Binary number ,Order (ring theory) ,01 natural sciences ,010101 applied mathematics ,Symmetric function ,Computational Mathematics ,Operator (computer programming) ,Fibonacci polynomials ,Orthogonal polynomials ,Geometry and Topology ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this paper, we introduce a new operator in order to derive some new symmetric properties of k-Fibonacci and k-Pell numbers and Tchebychev polynomials of first and second kind. By making use of the new operator defined in this paper, we give some new generating functions for k-Fibonacci and k-Pell numbers and Fibonacci polynomials.
- Published
- 2019
15. Equiangular tight frames from group divisible designs
- Author
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John Jasper and Matthew Fickus
- Subjects
Open problem ,0211 other engineering and technologies ,Equiangular polygon ,02 engineering and technology ,Type (model theory) ,01 natural sciences ,symbols.namesake ,FOS: Mathematics ,Mathematics - Combinatorics ,0101 mathematics ,Algebraic number ,Mathematics ,Discrete mathematics ,Strongly regular graph ,42C15 ,Group (mathematics) ,Applied Mathematics ,010102 general mathematics ,Hilbert space ,021107 urban & regional planning ,Graph theory ,Functional Analysis (math.FA) ,Computer Science Applications ,Mathematics - Functional Analysis ,symbols ,Combinatorics (math.CO) - Abstract
An equiangular tight frame (ETF) is a type of optimal packing of lines in a real or complex Hilbert space. In the complex case, the existence of an ETF of a given size remains an open problem in many cases. In this paper, we observe that many of the known constructions of ETFs are of one of two types. We further provide a new method for combining a given ETF of one of these two types with an appropriate group divisible design (GDD) in order to produce a larger ETF of the same type. By applying this method to known families of ETFs and GDDs, we obtain several new infinite families of ETFs. The real instances of these ETFs correspond to several new infinite families of strongly regular graphs. Our approach was inspired by a seminal paper of Davis and Jedwab which both unified and generalized McFarland and Spence difference sets. We provide combinatorial analogs of their algebraic results, unifying Steiner ETFs with hyperoval ETFs and Tremain ETFs.
- Published
- 2018
16. Piecewise Convex Deterministic Dynamical Systems and Weakly Convex Random Dynamical Systems and Their Invariant Measures
- Author
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Shafiqul Islam
- Subjects
Discrete mathematics ,Random map ,Dynamical systems theory ,Applied Mathematics ,010102 general mathematics ,Convex position ,Absolute continuity ,01 natural sciences ,010305 fluids & plasmas ,Computational Mathematics ,Spline (mathematics) ,0103 physical sciences ,Piecewise ,Invariant measure ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
Absolutely continuous invariant measures of deterministic dynamical systems and random dynamical systems respectively are studied via a general spline maximum entropy optimization method. In the first part of this paper, we consider piecewise convex deterministic dynamical systems (maps) $$\tau : [0, 1]\rightarrow [0, 1]$$ and we study their absolutely continuous invariant measures. We assume that the deterministic piecewise convex transformation $$\tau $$ has a unique absolutely continuous invariant measure (acim) $$\mu ^*$$ with density $$f^*.$$ We present a general spline maximum entropy optimization method for the approximation of $$f^*.$$ The proof of convergence of our general spline maximum entropy optimization method is presented. A numerical example is presented for the general spline (linear, quadratic and cubic respectively) maximum entropy numerical scheme for the approximation of $$f^*.$$ In the second part of this paper, we generalize above results for weakly convex position dependent random map $$T=\{\tau _1(x),\tau _2(x),\ldots , \tau _K(x); p_1(x),p_2(x),\ldots ,p_K(x)\}$$ on $$I=[0, 1],$$ where $$\tau _k: [0, 1] \rightarrow [0, 1]), k=1, 2, \dots , K$$ is a piecewise convex map and $$\{p_1(x), p_2(x),\ldots ,p_K(x) \}$$ is a set of position dependent probabilities on [0, 1]. We assume that T has a unique acim $$\nu ^*$$ with density $$h^*.$$ We present a general spline maximum entropy optimization method for the approximation of $$h^*.$$ The proof of convergence of our numerical schemes is presented. Also, we present a numerical example of the general spline maximum entropy method for the approximation of $$h^*.$$
- Published
- 2021
17. Approximation of function using generalized Zygmund class
- Author
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H. K. Nigam, Mohammad Mursaleen, and Supriya Rani
- Subjects
Discrete mathematics ,Class (set theory) ,Matrix-Cesàro (δ order) ( T C δ ) $(TC^{\delta })$ ,Algebra and Number Theory ,lcsh:Mathematics ,Applied Mathematics ,010102 general mathematics ,Function (mathematics) ,lcsh:QA1-939 ,Space (mathematics) ,01 natural sciences ,Prime (order theory) ,Generalized Minkowski inequality (GMI) ,010101 applied mathematics ,Matrix ( T ) $(T)$ means ,C δ $C^{\delta }$ means ,Generalized Zygmund class ,Best approximation ,0101 mathematics ,Fourier series ,Analysis ,Mathematics - Abstract
In this paper we review some of the previous work done by the earlier authors (Singh et al. in J. Inequal. Appl. 2017:101, 2017; Lal and Shireen in Bull. Math. Anal. Appl. 5(4):1–13, 2013), etc., on error approximation of a functiongin the generalized Zygmund space and resolve the issue of these works. We also determine the best error approximation of the functionsgand$g^{\prime }$g′, where$g^{\prime }$g′is a derived function of a 2π-periodic functiong, in the generalized Zygmund class$X_{z}^{(\eta )}$Xz(η),$z\geq 1$z≥1, using matrix-Cesàro$(TC^{\delta })$(TCδ)means of its Fourier series and its derived Fourier series, respectively. Theorem 2.1 of the present paper generalizes eight earlier results, which become its particular cases. Thus, the results of (Dhakal in Int. Math. Forum 5(35):1729–1735, 2010; Dhakal in Int. J. Eng. Technol. 2(3):1–15, 2013; Nigam in Surv. Math. Appl. 5:113–122, 2010; Nigam in Commun. Appl. Anal. 14(4):607–614, 2010; Nigam and Sharma in Kyungpook Math. J. 50:545–556, 2010; Nigam and Sharma in Int. J. Pure Appl. Math. 70(6):775–784, 2011; Kushwaha and Dhakal in Nepal J. Sci. Technol. 14(2):117–122, 2013; Shrivastava et al. in IOSR J. Math. 10(1 Ver. I):39–41, 2014) become particular cases of our Theorem 2.1. Several corollaries are also deduced from our Theorem 2.1.
- Published
- 2021
18. An Operator-Valued Kantorovich Metric on Complete Metric Spaces
- Author
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Trubee Davison
- Subjects
Discrete mathematics ,Applied Mathematics ,010102 general mathematics ,Operator theory ,Space (mathematics) ,01 natural sciences ,Complete metric space ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,010101 applied mathematics ,Metric space ,Iterated function system ,Operator algebra ,Metric (mathematics) ,FOS: Mathematics ,0101 mathematics ,Probability measure ,Mathematics - Abstract
The Kantorovich metric provides a way of measuring the distance between two Borel probability measures on a metric space. This metric has a broad range of applications from bioinformatics to image processing, and is commonly linked to the optimal transport problem in computer science. Noteworthy to this paper will be the role of the Kantorovich metric in the study of iterated function systems, which are families of contractive mappings on a complete metric space. When the underlying metric space is compact, it is well known that the space of Borel probability measures on this metric space, equipped with the Kantorovich metric, constitutes a compact, and thus complete metric space. In previous work, we generalized the Kantorovich metric to operator-valued measures for a compact underlying metric space, and applied this generalized metric to the setting of iterated function systems. We note that the work of P. Jorgensen, K. Shuman, and K. Kornelson provided the framework for our application to this setting. The situation when the underlying metric space is complete, but not necessarily compact, has been studied by A. Kravchenko. In this paper, we extend the results of Kravchenko to the generalized Kantorovich metric on operator-valued measures., Comment: The current version has been updated with small changes to match the final publication, Acta Applicandae Mathematicae (2018)
- Published
- 2018
19. Redheffer type bounds for Bessel and modified Bessel functions of the first kind
- Author
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Árpád Baricz and Khaled Mehrez
- Subjects
Discrete mathematics ,Pure mathematics ,Hankel transform ,Cylindrical harmonics ,Bessel process ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dirichlet eta function ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Bessel polynomials ,Struve function ,symbols ,Discrete Mathematics and Combinatorics ,Bessel's inequality ,0101 mathematics ,Bessel function ,Mathematics - Abstract
In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research.
- Published
- 2018
20. On solving generalized convex MINLP problems using supporting hyperplane techniques
- Author
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Ville-Pekka Eronen, Marko M. Mäkelä, and Tapio Westerlund
- Subjects
Discrete mathematics ,021103 operations research ,Control and Optimization ,Applied Mathematics ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,Subderivative ,Management Science and Operations Research ,Lipschitz continuity ,01 natural sciences ,Computer Science Applications ,Quasiconvex function ,Convex optimization ,Supporting hyperplane ,Differentiable function ,0101 mathematics ,Convex function ,Pseudoconvex function ,Mathematics - Abstract
Solution methods for convex mixed integer nonlinear programming (MINLP) problems have, usually, proven convergence properties if the functions involved are differentiable and convex. For other classes of convex MINLP problems fewer results have been given. Classical differential calculus can, though, be generalized to more general classes of functions than differentiable, via subdifferentials and subgradients. In addition, more general than convex functions can be included in a convex problem if the functions involved are defined from convex level sets, instead of being defined as convex functions only. The notion generalized convex, used in the heading of this paper, refers to such additional properties. The generalization for the differentiability is made by using subgradients of Clarke’s subdifferential. Thus, all the functions in the problem are assumed to be locally Lipschitz continuous. The generalization of the functions is done by considering quasiconvex functions. Thus, instead of differentiable convex functions, nondifferentiable $$f^{\circ }$$ -quasiconvex functions can be included in the actual problem formulation and a supporting hyperplane approach is given for the solution of the considered MINLP problem. Convergence to a global minimum is proved for the algorithm, when minimizing an $$f^{\circ }$$ -pseudoconvex function, subject to $$f^{\circ }$$ -pseudoconvex constraints. With some additional conditions, the proof is also valid for $$f^{\circ }$$ -quasiconvex functions, which sums up the properties of the method, treated in the paper. The main contribution in this paper is the generalization of the Extended Supporting Hyperplane method in Eronen et al. (J Glob Optim 69(2):443–459, 2017) to also solve problems with $$f^{\circ }$$ -pseudoconvex objective function.
- Published
- 2018
21. Strong Duality and Dual Pricing Properties in Semi-Infinite Linear Programming: A non-Fourier–Motzkin Elimination Approach
- Author
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Qinghong Zhang
- Subjects
Discrete mathematics ,021103 operations research ,Control and Optimization ,Duality gap ,Linear programming ,Applied Mathematics ,0211 other engineering and technologies ,Duality (optimization) ,Perturbation function ,010103 numerical & computational mathematics ,02 engineering and technology ,Management Science and Operations Research ,01 natural sciences ,Weak duality ,Fourier–Motzkin elimination ,Applied mathematics ,Strong duality ,Wolfe duality ,0101 mathematics ,Mathematics - Abstract
Following the idea of the conjecture for semi-infinite programming in a paper by Kortanek and Zhang, recently published in Optimization, in this paper we show that the Fourier–Motzkin elimination is not needed in the study of the strong duality and dual pricing properties for semi-infinite programming. We also prove several new results on the strong duality and dual pricing properties. Specifically, we propose a new subspace, under which the strong duality property holds. We give a necessary and sufficient condition for the dual pricing property to hold under this subspace, which is further used to examine the examples presented in the Basu–Martin–Ryan paper.
- Published
- 2017
22. On p-convergent Operators on Banach Lattices
- Author
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Elroy D. Zeekoei and Jan Fourie
- Subjects
Unbounded operator ,Discrete mathematics ,Mathematics::Functional Analysis ,Approximation property ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Spectrum (functional analysis) ,Finite-rank operator ,Compact operator ,01 natural sciences ,Strictly singular operator ,010101 applied mathematics ,Pseudo-monotone operator ,0101 mathematics ,C0-semigroup ,Mathematics - Abstract
The notion of a p-convergent operator on a Banach space was originally introduced in 1993 by Castillo and Sanchez in the paper entitled “Dunford–Pettis-like properties of continuous vector function spaces”. In the present paper we consider the p-convergent operators on Banach lattices, prove some domination properties of the same and consider their applications (together with the notion of a weak p-convergent operator, which we introduce in the present paper) to a study of the Schur property of order p. Also, the notion of a disjoint p-convergent operator on Banach lattices is introduced, studied and its applications to a study of the positive Schur property of order p are considered.
- Published
- 2017
23. On the Enumeration of Hypermaps Which are Self-Equivalent with Respect to Reversing the Colors of Vertices
- Author
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M. A. Deryagina
- Subjects
Statistics and Probability ,Connected component ,Discrete mathematics ,Mathematics::Combinatorics ,Applied Mathematics ,General Mathematics ,Riemann surface ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Graph ,symbols.namesake ,Colored ,010201 computation theory & mathematics ,Enumeration ,symbols ,Bipartite graph ,Bibliography ,Reversing ,0101 mathematics ,Mathematics - Abstract
A map (S,G) is a closed Riemann surface S with embedded graph G such that S \G is the disjoint union of connected components, called faces, each of which is homeomorphic to an open disk. Tutte began a systematic study of maps in the 1960s and contemporary authors are actively developing it. In the present paper, after recalling the concept of a circular map introduced by the author and Mednykh, a relationship between bipartite maps and circular maps is demonstrated via the concept of the duality of maps. In this way an enumeration formula for the number of bipartite maps with a given number of edges is obtained. A hypermap is a map whose vertices are colored black and white in such a way that every edge connects vertices of different colors. The hypermaps are also known as dessins d’enfants (or Grothendieck’s dessins). A hypermap is self-equivalent with respect to reversing the colors of vertices if it is equivalent to the hypermap obtained by reversing the colors of its vertices. The main result of the present paper is an enumeration formula for the number of unrooted hypermaps, regardless of genus, which have n edges and are self-equivalent with respect to reversing the colors of vertices. Bibliography: 13 titles.
- Published
- 2017
24. Some weak specification properties and strongly mixing
- Author
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Jiandong Yin, Tao Wang, and Qi Yan
- Subjects
010101 applied mathematics ,Discrete mathematics ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Equivalence (formal languages) ,01 natural sciences ,Mathematics - Abstract
In this paper, the authors first construct a dynamical system which is strongly mixing but has no weak specification property. Then the authors introduce two new concepts which are called the quasi-weak specification property and the semi-weak specification property in this paper, respectively, and the authors prove the equivalence of quasi-weak specification property, semi-weak specification property and strongly mixing.
- Published
- 2017
25. Quasidense Monotone Multifunctions
- Author
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Stephen Simons
- Subjects
Statistics and Probability ,Mathematics::Optimization and Control ,0211 other engineering and technologies ,Banach space ,02 engineering and technology ,Subderivative ,Type (model theory) ,01 natural sciences ,Fuzzy logic ,FOS: Mathematics ,0101 mathematics ,Mathematics ,Discrete mathematics ,Mathematics::Functional Analysis ,Numerical Analysis ,021103 operations research ,Applied Mathematics ,010102 general mathematics ,47H05, 47N10, 52A41, 46A20 ,Extension (predicate logic) ,Strongly monotone ,Functional Analysis (math.FA) ,Computer Science::Other ,Dual (category theory) ,Mathematics - Functional Analysis ,Monotone polygon ,Geometry and Topology ,Analysis - Abstract
In this paper, we discuss quasidense multifunctions from a Banach space into its dual, and use the two sum theorems proved in a previous paper to give various characterizations of quasidensity. We investigate the Fitzpatrick extension of such a multifunction. We prove that, for closed monotone multifunctions, quasidensity implies type (FPV) and strong maximality, and that quasidensity is equivalent to type (FP). This version differs from Version 3 in that a few minor errors have been corrected., 24 pages
- Published
- 2017
26. Structure Graphs of Rings: Definitions and First Results
- Author
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Aleksandar Lipkovski
- Subjects
Statistics and Probability ,Discrete mathematics ,Cayley graph ,Algebraic structure ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Directed graph ,01 natural sciences ,010101 applied mathematics ,Quadratic equation ,Vieta's formulas ,Branched covering ,0101 mathematics ,Commutative property ,Complex number ,Mathematics - Abstract
The quadratic Vieta formulas (x, y) ↦ (u, v) = (x + y, xy) in the complex geometry define a two-fold branched covering ℂ2 → ℂ2 ramified over the parabola u 2 = 4v. Thinking about topics considered in Arnold’s paper Topological content of the Maxwell theorem on multipole representation of spherical functions, I came to a very simple idea: in fact, these formulas describe the algebraic structure, i.e., addition and multiplication, of complex numbers. What if, instead of the field of complex numbers, we consider an arbitrary ring? Namely for an arbitrary ring A (commutative, with unity) consider the mapping Φ: A 2 → A 2 defined by the Vieta formulas (x, y) ↦ (u, v) = (x + y, xy). What kind of algebraic properties of the ring itself does this map reflect? At first, it is interesting to investigate the simplest finite rings A = ℤ m and A = ℤ k ×ℤ m . Recently, it has been very popular to consider graphs associated to rings (the zero-divisor graph, the Cayley graph, etc.). In the present paper, we study the directed graph defined by the Vieta mapping Φ.
- Published
- 2017
27. Sequential Analogues of the Lyapunov and Krein–Milman Theorems in Fréchet Spaces
- Author
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F. S. Stonyakin
- Subjects
Statistics and Probability ,Discrete mathematics ,Mathematics::Functional Analysis ,Dual space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Eberlein–Šmulian theorem ,Banach space ,Hahn–Banach theorem ,02 engineering and technology ,01 natural sciences ,Fréchet space ,Locally convex topological vector space ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Closed graph theorem ,0101 mathematics ,Open mapping theorem (functional analysis) ,Mathematics - Abstract
In this paper we develop the theory of anti-compact sets we introduced earlier. We describe the class of Frechet spaces where anti-compact sets exist. They are exactly the spaces that have a countable set of continuous linear functionals. In such spaces we prove an analogue of the Hahn–Banach theorem on extension of a continuous linear functional from the original space to a space generated by some anti-compact set. We obtain an analogue of the Lyapunov theorem on convexity and compactness of the range of vector measures, which establishes convexity and a special kind of relative weak compactness of the range of an atomless vector measure with values in a Frechet space possessing an anti-compact set. Using this analogue of the Lyapunov theorem, we prove the solvability of an infinite-dimensional analogue of the problem of fair division of resources. We also obtain an analogue of the Lyapunov theorem for nonadditive analogues of measures that are vector quasi-measures valued in an infinite-dimensional Frechet space possessing an anti-compact set. In the class of Frechet spaces possessing an anti-compact set, we obtain analogues of the Krein–Milman theorem on extreme points for convex bounded sets that are not necessarily compact. A special place is occupied by analogues of the Krein–Milman theorem in terms of extreme sequences introduced in the paper (the so-called sequential analogues of the Krein–Milman theorem).
- Published
- 2017
28. Weak and strong laws of large numbers for arrays of rowwise END random variables and their applications
- Author
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Aiting Shen and Andrei Volodin
- Subjects
Statistics and Probability ,Discrete mathematics ,Uniform integrability ,010102 general mathematics ,01 natural sciences ,Nonparametric regression ,Orthant ,Moment (mathematics) ,010104 statistics & probability ,Convergence of random variables ,Law of large numbers ,Convergence (routing) ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics - Abstract
In the paper, the Marcinkiewicz–Zygmund type moment inequality for extended negatively dependent (END, in short) random variables is established. Under some suitable conditions of uniform integrability, the $$L_r$$ convergence, weak law of large numbers and strong law of large numbers for usual normed sums and weighted sums of arrays of rowwise END random variables are investigated by using the Marcinkiewicz–Zygmund type moment inequality. In addition, some applications of the $$L_r$$ convergence, weak and strong laws of large numbers to nonparametric regression models based on END errors are provided. The results obtained in the paper generalize or improve some corresponding ones for negatively associated random variables and negatively orthant dependent random variables.
- Published
- 2017
29. On $$\varvec{n}$$ n -norm preservers and the Aleksandrov conservative $$\varvec{n}$$ n -distance problem
- Author
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György Pál Gehér
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Order (ring theory) ,010103 numerical & computational mathematics ,01 natural sciences ,Surjective function ,Nonlinear system ,Transformation (function) ,Norm (mathematics) ,Distance problem ,Discrete Mathematics and Combinatorics ,Affine transformation ,0101 mathematics ,Unit (ring theory) ,Mathematics - Abstract
The goal of this paper is to point out that the results obtained in the recent papers (Chen and Song in Nonlinear Anal 72:1895–1901, 2010; Chu in J Math Anal Appl 327:1041–1045, 2007; Chu et al. in Nonlinear Anal 59:1001–1011, 2004a, J. Math Anal Appl 289:666–672, 2004b) can be seriously strengthened in the sense that we can significantly relax the assumptions of the main results so that we still get the same conclusions. In order to do this first, we prove that for $$n \ge 3$$ any transformation which preserves the n-norm of any n vectors is automatically plus-minus linear. This will give a re-proof of the well-known Mazur–Ulam-type result that every n-isometry is automatically affine ( $$n \ge 2$$ ) which was proven in several papers, e.g. in Chu et al. (Nonlinear Anal 70:1068–1074, 2009). Second, following the work of Rassias and Semrl (Proc Am Math Soc 118:919–925, 1993), we provide the solution of a natural Aleksandrov-type problem in n-normed spaces, namely, we show that every surjective transformation which preserves the unit n-distance in both directions ( $$n\ge 2$$ ) is automatically an n-isometry.
- Published
- 2017
30. Krasnoselski-Mann type iterative method for hierarchical fixed point problem and split mixed equilibrium problem
- Author
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Mohd Furkan, K. R. Kazmi, and Rehan Ali
- Subjects
Discrete mathematics ,021103 operations research ,Weak convergence ,Iterative method ,Applied Mathematics ,Numerical analysis ,0211 other engineering and technologies ,Solution set ,02 engineering and technology ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,Monotone polygon ,Theory of computation ,Applied mathematics ,Equilibrium problem ,0101 mathematics ,Mathematics - Abstract
In this paper, we suggest and analyze a Krasnoselski-Mann type iterative method to approximate a common element of solution sets of a hierarchical fixed point problem for nonexpansive mappings and a split mixed equilibrium problem. We prove that sequences generated by the proposed iterative method converge weakly to a common element of solution sets of these problems. Further, we derive some consequences from our main result. Furthermore, we extend the considered iterative method to a split monotone variational inclusion problem and deduce some consequences. Finally, we give a numerical example to justify the main result. The method and results presented in this paper generalize and unify the corresponding known results in this area.
- Published
- 2017
31. Fast multipole methods for approximating a function from sampling values
- Author
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Guidong Liu and Shuhuang Xiang
- Subjects
Discrete mathematics ,Applied Mathematics ,Fast multipole method ,Numerical analysis ,Lagrange polynomial ,Sampling (statistics) ,010103 numerical & computational mathematics ,Function (mathematics) ,Barycentric coordinate system ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,symbols.namesake ,symbols ,0101 mathematics ,Multipole expansion ,Mathematics ,Interpolation - Abstract
Both barycentric Lagrange interpolation and barycentric rational interpolation are thought to be stable and effective methods for approximating a given function on some special point sets. A direct evaluation of these interpolants due to N interpolation points at M sampling points requires \(\mathcal {O}(NM)\) arithmetic operations. In this paper, we introduce two fast multipole methods to reduce the complexity to \(\mathcal {O}(\max \left \{N,M\right \})\). The convergence analysis is also presented in this paper.
- Published
- 2017
32. Approximation by (p, q)-Baskakov–Durrmeyer–Stancu Operators
- Author
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Tuncer Acar, Syed Abdul Mohiuddine, Mohammad Mursaleen, and Kırıkkale Üniversitesi
- Subjects
Discrete mathematics ,Generalization ,Applied Mathematics ,010102 general mathematics ,(p,q)-Integers ,Weighted approximation ,Operator theory ,01 natural sciences ,Modulus of continuity ,010101 applied mathematics ,Computational Mathematics ,Baskakov operator ,(p,q)-Baskakov-Durrmeyer operators ,Computational Theory and Mathematics ,Rate of convergence ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
Acar, Tuncer/0000-0003-0982-9459; Mursaleen, M./0000-0003-4128-0427; Mohiuddine, S. A./0000-0002-9050-9104 WOS: 000439367300005 The present paper deals with the Stancu-type generalization of (p, q)-Baskakov-Durrmeyer operators. We investigate local approximation, weighted approximation properties of new operators and present the rate of convergence by means of suitable modulus of continuity. At the end of the paper, we introduce a new modification of (p, q)-Baskakov-Durrmeyer-Stancu operators with King approach. King Abdulaziz University, Jeddah, Saudi Arabia The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
- Published
- 2017
33. A study on distributive and modular lattice ordered fuzzy soft group and its duality
- Author
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J. Arockia Reeta and J. Vimala
- Subjects
Discrete mathematics ,Modular lattice ,Mathematics::General Mathematics ,Group (mathematics) ,Astrophysics::High Energy Astrophysical Phenomena ,Applied Mathematics ,Duality (optimization) ,010103 numerical & computational mathematics ,02 engineering and technology ,Fuzzy subalgebra ,01 natural sciences ,Fuzzy logic ,Algebra ,Distributive property ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,0101 mathematics ,Parametric family ,Soft set ,Mathematics - Abstract
Soft set theory has a rich potential application in several fields. A soft group is a parameterized family of subgroups and a fuzzy soft group is a parameterized family of fuzzy subgroups. The concept of fuzzy soft group is the generalization of soft group. Abdulkadir Aygunoglu and Halis Aygun introduced the notion of fuzzy soft groups in 2009[1]. In this paper, the concept of lattice ordered fuzzy soft groups and its duality has been introduced. Then distributive and modular lattice ordered fuzzy soft groups are analysed. The objective of this paper is to study the lattice theory over the collection of fuzzy soft group in a parametric manner. Some pertinent properties have been analysed and hence established duality principle.
- Published
- 2016
34. Homoclinic Solutions for p-Laplacian Hamiltonian Systems with Combined Nonlinearities
- Author
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Rong Yuan and Ziheng Zhang
- Subjects
Discrete mathematics ,Applied Mathematics ,010102 general mathematics ,01 natural sciences ,Hamiltonian system ,010101 applied mathematics ,Bounded function ,Mountain pass theorem ,p-Laplacian ,Discrete Mathematics and Combinatorics ,Nabla symbol ,Homoclinic orbit ,0101 mathematics ,Mathematics - Abstract
In this paper we investigate the existence of homoclinic solutions for the following p-Laplacian Hamiltonian systems $$\begin{aligned} \frac{d}{dt}(|{\dot{u}}(t)|^{p-2} {\dot{u}}(t))-a(t)|u(t)|^{p-2}u(t)+\nabla W(t,u(t))=0, \end{aligned}$$ where $$t\in {\mathbb R}$$ , $$u\in {\mathbb R}^n$$ , $$p>1$$ , $$a\in C({\mathbb R},{\mathbb R})$$ , $$W\in C^1({\mathbb R}\times {\mathbb R}^n,{\mathbb R})$$ and $$\nabla W(t,u)$$ is the gradient of W(t, u) in u. The novelty of this paper is that, assuming that a(t) is bounded in the sense that there are two constants $$0
- Published
- 2016
35. On a new property of n-poised and G C n sets
- Author
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Vahagn Bayramyan and Hakop Hakopian
- Subjects
Discrete mathematics ,Polynomial (hyperelastic model) ,Conjecture ,Applied Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Polynomial interpolation ,Combinatorics ,Set (abstract data type) ,Computational Mathematics ,Product (mathematics) ,Line (geometry) ,Node (circuits) ,Algebraic curve ,0101 mathematics ,Mathematics - Abstract
In this paper we consider n-poised planar node sets, as well as more special ones, called GCn sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line l is called k-node line for a node set ??$\mathcal X$ if it passes through exactly k nodes. An (n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every GCn set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of ??$\mathcal X$ is used by each node in ???M,$\mathcal X\setminus M, $meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of GCn set ??$\mathcal {X}$ is used either by exactly n2$\binom {n}{2}$ nodes or by exactly n?12$\binom {n-1}{2}$ nodes. We prove also similar statements concerning n-node or (n ? 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and GCn sets. At the end we present a conjecture concerning any k-node line.
- Published
- 2016
36. Fixed point theorems via generalized $$\varvec{F}$$ F -contractions with applications to functional equations occurring in dynamic programming
- Author
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Vishal Joshi, Mohammad Imdad, Poom Kumam, and Deepak Singh
- Subjects
Discrete mathematics ,021103 operations research ,Applied Mathematics ,0211 other engineering and technologies ,Structure (category theory) ,Fixed-point theorem ,Context (language use) ,02 engineering and technology ,Fixed point ,Fixed-point property ,01 natural sciences ,Cauchy sequence ,010101 applied mathematics ,Metric space ,Modeling and Simulation ,Geometry and Topology ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
The purpose of this paper is threefold. Firstly, recognizing the concept of Piri and Kumam (Fixed Point Theor Appl 210, 2014), we define generalized F-contractive mappings in the framework of G-metric spaces and by employing this, some fixed point theorems in the structure of G-metric spaces are established that can not be obtained from the existing results in the context of allied metric spaces and do not meet the remarks of Samet et al. (Int J Anal. Article ID 917158, 2013) and Jleli et al. (Fixed Point Theor Appl 210, 2012). Infact, we utilize the pattern, mentioned in Karapinar and Agrawal (Fixed Point Theor Appl 154, 2013), a counter paper to remarks of Samet et al. (Int J Anal. Article ID 917158, 2013) and Jleli et al. (Fixed Point Theor Appl 210, 2012). Secondly, in the setting of G-metric spaces, certain fixed point results for integral inequalities under generalized F-contraction are presented. Finally, as an application, our results are utilized to establish the existence and uniqueness of solution the equations arising in Oscillation of a spring. In the sequel, another application is given to set-up the existence and uniqueness of solution of functional equations occurring in dynamic programming. Our investigations are also authenticated with the aid of some appropriate and innovative examples.
- Published
- 2016
37. The adjacency matrix of a graph as a data table: a geometric perspective
- Author
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Paolo A. Oliverio, Tommaso Gentile, Giampiero Chiaselotti, and Federico Infusino
- Subjects
Discrete mathematics ,Applied Mathematics ,010102 general mathematics ,Neighbourhood (graph theory) ,0102 computer and information sciences ,01 natural sciences ,Semi-symmetric graph ,law.invention ,Combinatorics ,Circulant graph ,010201 computation theory & mathematics ,Graph power ,law ,Line graph ,Regular graph ,Adjacency matrix ,0101 mathematics ,Null graph ,Mathematics - Abstract
In this paper we continue a research project concerning the study of a graph from the perspective of granular computation. To be more specific, we interpret the adjacency matrix of any simple undirected graph G in terms of data information table, which is one of the most studied structures in database theory. Granular computing (abbreviated GrC) is a well-developed research field in applied and theoretical information sciences; nevertheless, in this paper we address our efforts toward a purely mathematical development of the link between GrC and graph theory. From this perspective, the well-studied notion of indiscernibility relation in GrC becomes a symmetry relation with respect to a given vertex subset in graph theory; therefore, the investigation of this symmetry relation turns out to be the main object of study in this paper. In detail, we study a simple undirected graph G by assuming a generic vertex subset W as reference system with respect to which examine the symmetry of all vertex subsets of G. The change of perspective from G without reference system to the pair (G, W) is similar to what occurs in the transition from an affine space to a vector space. We interpret the symmetry blocks in the reference system (G, W) as particular equivalence classes of vertices in G, and we study the geometric properties of all reference systems (G, W), when W runs over all vertex subsets of G. We also introduce three hypergraph models and a vertex set partition lattice associated to G, by taking as general models of reference several classical notions of GrC. For all these constructions, we provide a geometric characterization and we determine their structure for basic graph families. Finally, we apply a wide part of our work to study the important case of the Petersen graph.
- Published
- 2016
38. On Weakly Commuting Set-Valued Mappings on a Domain of Sets Endowed with Directed Graph
- Author
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Stojan Radenović, Talat Nazir, Mujahid Abbas, and Branislav Z. Popović
- Subjects
Discrete mathematics ,Applied Mathematics ,010102 general mathematics ,A domain ,Directed graph ,01 natural sciences ,Coincidence ,010101 applied mathematics ,Mathematics (miscellaneous) ,Common fixed point ,Graph (abstract data type) ,Family of sets ,0101 mathematics ,Coincidence point ,Mathematics - Abstract
The aim of this paper is to present coincidence and common fixed point results of set-valued mappings satisfying certain generalized graphic F-contractive conditions on a family of sets endowed with a graph. It is worth mentioning that these results are obtained without appealing to any form of continuity of mappings involved herein. Some examples are presented to support the results proved in this paper. Our results unify, generalize and extend various comparable results in the existing literature.
- Published
- 2016
39. Orthogonal sets: The axiom of choice and proof of a fixed point theorem
- Author
-
Hamid Baghani, Maryam Ramezani, and Madjid Eshaghi Gordji
- Subjects
Discrete mathematics ,Bourbaki–Witt theorem ,Picard–Lindelöf theorem ,Banach fixed-point theorem ,Applied Mathematics ,010102 general mathematics ,Fixed-point theorem ,Fixed-point property ,01 natural sciences ,010101 applied mathematics ,Modeling and Simulation ,Axiom of choice ,Geometry and Topology ,0101 mathematics ,Kakutani fixed-point theorem ,Brouwer fixed-point theorem ,Mathematics - Abstract
In this paper, we prove some fixed point theorem on orthogonal spaces. Our result improve the main result of the paper by Eshaghi Gordji et al. [On orthogonal sets and Banach fixed point theorem, to appear in Fixed Point Theory]. Also we prove a statement which is equivalent to the axiom of choice. In the last section, as an application, we consider the existence and uniqueness of a solution for a Volterra-type integral equation in L p space.
- Published
- 2016
40. Solutions for a class of fractional Hamiltonian systems with a parameter
- Author
-
César E. Torres Ledesma and Ziheng Zhang
- Subjects
Discrete mathematics ,Applied Mathematics ,Image (category theory) ,010102 general mathematics ,Interval (mathematics) ,Lambda ,01 natural sciences ,Dirichlet distribution ,Hamiltonian system ,010101 applied mathematics ,Combinatorics ,Computational Mathematics ,symbols.namesake ,Critical point (thermodynamics) ,Mountain pass theorem ,symbols ,Nabla symbol ,0101 mathematics ,Mathematics - Abstract
In this paper we are concerned with the existence of solutions for the following fractional Hamiltonian systems with a parameter Open image in new window where \(\alpha \in (1/2,1)\), \(t\in {\mathbb {R}}\), \(u\in {\mathbb {R}}^n\), \(\lambda >0\) is a parameter, \(L\in C({\mathbb {R}},{\mathbb {R}}^{n^2})\) is a symmetric matrix for all \(t\in {\mathbb {R}}\), \(W\in C^1({\mathbb {R}}\times {\mathbb {R}}^n,{\mathbb {R}})\) and \(\nabla W(t,u)\) is the gradient of W(t, u) at u. The novelty of this paper is that, assuming L(t) is a symmetric and positive semi-definite matrix for all \(t\in {\mathbb {R}}\), that is, \(L(t)\equiv 0\) is allowed to occur in some finite interval T of \({\mathbb {R}}\), W(t, u) satisfies Ambrosetti–Rabinowitz condition and some other reasonable hypotheses, we show the existence of nontrivial solution of (FHS)\(_\lambda \), which vanishes on \({\mathbb {R}}\backslash T\) as \(\lambda \rightarrow \infty \), and converges to \(\tilde{u}\in H^\alpha ({\mathbb {R}})\); here \(\tilde{u}\in E_0^\alpha \) is a nontrivial solution of the Dirichlet BVP for fractional systems on the finite interval T. Recent results are generalized and significantly improved.
- Published
- 2016
41. Piecewise smooth dynamical systems modeling based on Putzer and Fibonacci-Horner theorems: DC-DC converters case
- Author
-
Kamel Guesmi, Djillali Mahi, A. Khoudiri, Unité d'Enseignement et de Recherche d'Electrotechnique (UER ELT), Ecole Militaire Polytechnique, Centre de Recherche en Sciences et Technologies de l'Information et de la Communication - EA 3804 (CRESTIC), Université de Reims Champagne-Ardenne (URCA), Laboratoire d’Automatique Appliquée Diagnostic Industriel (LAADI), Université Ziane Achour de Djelfa, and GUESMI, Kamel
- Subjects
Discrete mathematics ,Class (set theory) ,Fibonacci number ,Dynamical systems theory ,Computer Applications ,Applied Mathematics ,020208 electrical & electronic engineering ,010103 numerical & computational mathematics ,02 engineering and technology ,Converters ,01 natural sciences ,[SPI.AUTO]Engineering Sciences [physics]/Automatic ,Computer Science Applications ,Algebra ,[SPI.AUTO] Engineering Sciences [physics]/Automatic ,Control and Systems Engineering ,Modeling and Simulation ,0202 electrical engineering, electronic engineering, information engineering ,Piecewise ,0101 mathematics ,Cayley–Hamilton theorem ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
The paper deals with the problem of switched dynamical systems modeling especially in DC-DC converters case study consideration. It presents two approaches to describe accurately the behavior of this class of systems. To clarify the paper's contribution, the proposed approaches are validated through simulations and experimental results. A comparative study, between the obtained results and those of other techniques from the literature, is given to evaluate the performances of the studied approaches.
- Published
- 2016
42. To the History of the Appearance of the Notion of the ε-Entropy of an Authomorphism of a Lebesgue Space and (ε,T)-Entropy of a Dynamical System with Continuous Time
- Author
-
D. Z. Arov
- Subjects
Statistics and Probability ,Discrete mathematics ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Automorphism ,01 natural sciences ,Separable space ,Compact space ,0103 physical sciences ,Entropy (information theory) ,Standard probability space ,Ergodic theory ,010307 mathematical physics ,Invariant measure ,0101 mathematics ,Mathematics - Abstract
The paper is devoted to the master thesis on “information theory” which was written by the author in 1956–57. The topic was suggested by his advisor A. A. Bobrov (a student of A. Ya. Khinchin and A. N. Kolmogorov), and the thesis was written under the influence of lectures by N. I. Gavrilov (a student of I. G. Petrovskii) on the qualitative theory of differential equations, which included the statement of Birkhoff’s theorem for ergodic dynamical systems. In the thesis, the author used the concept of Shannon entropy in the study of ergodic dynamical systems f(p, t) in a separable compact metric space R with an invariant measure μ (where μ(R) = 1) and introduced the notion of the (ϵ, T)-entropy of a system as a quantitative characteristic of the degree of mixing. In the work, not only partitions of R were considered, but also partitions of the interval (−∞,∞) into subintervals of length T > 0. In particular, f(p, T) was regarded as an automorphism S of X = R, and the (ϵ, T)-entropy is essentially the e-entropy of S. But, despite some “oversights” in the definition of the (ϵ, T)-entropy and many years that have passed, the author decided to publish the corresponding chapter of the thesis in connection with the following: 1) There is a number of papers that refer to this work in the explanation of the history of the concept of Kolmogorov’s entropy. 2) Recently, B. M. Gurevich obtained new results on the ϵ-entropy hϵ(S), which show that for two ergodic automorphisms with equal finite entropies their ϵ-entropies also coincide for all ϵ, but, on the other hand, there are unexpected nonergodic automorphisms with equal finite entropies, but different ϵ-entropies for some ϵ. This shows that the concept of ϵ-entropy is of scientific value.
- Published
- 2016
43. Criteria for strong H-tensors
- Author
-
Kaili Zhang, Yiju Wang, and Hongchun Sun
- Subjects
Discrete mathematics ,Multilinear algebra ,Mathematics (miscellaneous) ,Product (mathematics) ,010102 general mathematics ,Principal (computer security) ,Diagonal ,Applied mathematics ,010103 numerical & computational mathematics ,Tensor ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
H-tensor is a new developed concept which plays an important role in tensor analysis and computing. In this paper, we explore the properties of H-tensors and establish some new criteria for strong H-tensors. In particular, based on the principal subtensor, we provide a new necessary and sufficient condition of strong H-tensors, and based on a type of generalized diagonal product dominance, we establish some new criteria for identifying strong H-tensors. The results obtained in this paper extend the corresponding conclusions for strong H-matrices and improve the existing results for strong H-tensors.
- Published
- 2016
44. Graph-Links: Nonrealizability, Orientation, and Jones Polynomial
- Author
-
V. S. Safina and Denis Petrovich Ilyutko
- Subjects
Statistics and Probability ,Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Jones polynomial ,Bracket polynomial ,01 natural sciences ,Graph ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,MathematicsofComputing_DISCRETEMATHEMATICS ,Writhe ,Mathematics - Abstract
The present paper is devoted to graph-links with many components and consists of two parts. In the first part of the paper we classify vertices of a labeled graph according to the component they belong to. Using this classification, we construct an invariant of graph-links. This invariant shows that the labeled second Bouchet graph generates a nonrealizable graph-link. In the second part of the work we introduce the notion of an oriented graph-link. We define a writhe number for the oriented graph-link and we get an invariant of oriented graph-links, the Jones polynomial, by normalizing the Kauffman bracket with the writhe number.
- Published
- 2016
45. Vector Quasi-Equilibrium Problems for the Sum of Two Multivalued Mappings
- Author
-
Gábor Kassay, Mihaela Miholca, and Nguyen The Vinh
- Subjects
Discrete mathematics ,Pure mathematics ,021103 operations research ,Control and Optimization ,Quantitative Biology::Molecular Networks ,Applied Mathematics ,010102 general mathematics ,Ky Fan inequality ,0211 other engineering and technologies ,02 engineering and technology ,Management Science and Operations Research ,01 natural sciences ,Theory of computation ,0101 mathematics ,Quasistatic process ,Vector space ,Mathematics - Abstract
In this paper, we study vector quasi-equilibrium problems for the sum of two multivalued bifunctions. The assumptions are required separately on each of these bifunctions. Sufficient conditions for the existence of solutions of such problems are shown in the setting of topological vector spaces. The results in this paper unify, improve and extend some well-known existence theorems from the literature.
- Published
- 2016
46. Sudoku-like arrays, codes and orthogonality
- Author
-
Melissa A. Huggan, Gary L. Mullen, David Thomson, and Brett Stevens
- Subjects
Discrete mathematics ,business.industry ,Applied Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Cryptography ,0102 computer and information sciences ,Construct (python library) ,01 natural sciences ,Computer Science Applications ,Algebra ,Orthogonality ,Construction method ,010201 computation theory & mathematics ,Hypercube ,0101 mathematics ,business ,Mathematics of Sudoku ,Mathematics - Abstract
This paper is concerned with constructions and orthogonality of generalized Sudoku arrays of various forms. We characterize these arrays based on their constraints; for example Sudoku squares are characterized by having strip and sub-square constraints. First, we generalize Sudoku squares to be multi-dimensional arrays with strip and sub-cube constraints and construct mutually orthogonal sets of these arrays using linear polynomials. We add additional constraints motivated by elementary intervals for low discrepancy sequences and again give a construction of these arrays using linear polynomials in detail for 3 dimensional and a general construction method for arbitrary dimension. Then we give a different construction of these hypercubes due to MDS codes. We also analyze the orthogonality of all of the Sudoku-like hypercubes we consider in this paper.
- Published
- 2016
47. Generalised and Quotient Models for Random And/Or Trees and Application to Satisfiability
- Author
-
Cécile Mailler, Antoine Genitrini, Algorithmes, Programmes et Résolution (APR), Laboratoire d'Informatique de Paris 6 (LIP6), Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematical Sciences [Bath], and University of Bath [Bath]
- Subjects
General Computer Science ,Catalan trees ,[INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS] ,0102 computer and information sciences ,01 natural sciences ,Boolean domain ,Combinatorics ,FOS: Mathematics ,Saturation (graph theory) ,Mathematics - Combinatorics ,Equivalence relation ,Boolean expression ,0101 mathematics ,Quotient ,Mathematics ,Discrete mathematics ,Sequence ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Boolean formulas/functions ,Computer Science Applications ,010201 computation theory & mathematics ,Maximum satisfiability problem ,Combinatorics (math.CO) ,Boolean data type ,Mathematics - Probability - Abstract
This article is motivated by the following satisfiability question: pick uniformly at random an and/or Boolean expression of length n, built on a set of k_n Boolean variables. What is the probability that this expression is satisfiable? asymptotically when n tends to infinity? The model of random Boolean expressions developed in the present paper is the model of Boolean Catalan trees, already extensively studied in the literature for a constant sequence (k_n)_{n\geq 1}. The fundamental breakthrough of this paper is to generalise the previous results to any (reasonable) sequence of integers (k_n)_{n\geq 1}, which enables us, in particular, to solve the above satisfiability question. We also analyse the effect of introducing a natural equivalence relation on the set of Boolean expressions. This new "quotient" model happens to exhibit a very interesting threshold (or saturation) phenomenon at k_n = n/ln n., Comment: Long version of arXiv:1304.5615
- Published
- 2016
48. Omega-Limit Sets of Generic Points of Partially Hyperbolic Diffeomorphisms
- Author
-
Alexey Okunev and Stanislav Minkov
- Subjects
Discrete mathematics ,0209 industrial biotechnology ,Pure mathematics ,Class (set theory) ,Mathematics::Dynamical Systems ,Conjecture ,Lebesgue measure ,Applied Mathematics ,010102 general mathematics ,02 engineering and technology ,01 natural sciences ,020901 industrial engineering & automation ,Corollary ,Product (mathematics) ,Attractor ,Limit (mathematics) ,Diffeomorphism ,0101 mathematics ,Analysis ,Mathematics - Abstract
We prove that, for any Eu ⊕ Ecs partially hyperbolic C2 diffeomorphism, the ω-limit set of a generic (with respect to the Lebesgue measure) point is a union of unstable leaves. As a corollary, we prove a conjecture made by Ilyashenko in his 2011 paper that the Milnor attractor is a union of unstable leaves. In the paper mentioned above, Ilyashenko reduced the local generecity of the existence of a “thick” Milnor attractor in the class of boundary-preserving diffeomorphisms of the product of the interval and the 2-torus to this conjecture.
- Published
- 2016
49. Parametric Functional Representation of Interval Number with Arithmetic Operations
- Author
-
D. Pal and G. S. Mahapatra
- Subjects
Discrete mathematics ,Applied Mathematics ,010103 numerical & computational mathematics ,02 engineering and technology ,Interval (mathematics) ,01 natural sciences ,Computational Mathematics ,Product (mathematics) ,0202 electrical engineering, electronic engineering, information engineering ,Computational Science and Engineering ,020201 artificial intelligence & image processing ,Convex combination ,0101 mathematics ,Arithmetic ,Parametric equation ,Representation (mathematics) ,Mathematics ,Parametric statistics - Abstract
The objective of this paper is to represent the interval number in different functional forms. We have represented the interval number by parametric product functional form, symmetric functional form, asymmetric functional form and convex combination functional form. We represent positive interval number in parametric product functional form, symmetric and asymmetric functional forms. However any interval number can be represented by a convex combination functional form. We also study the arithmetic operations of interval numbers based on the different forms of functional representation. Numerical examples are given to illustrate our proposed approach for arithmetic operations of the interval number in different functional form. Finally some open problems are mentioned at the end of the paper.
- Published
- 2015
50. On two questions of A. Petruşel and G. Petruşel in b-metric fixed point theory
- Author
-
Nguyen Van Dung and Vo Thi Le Hang
- Subjects
010101 applied mathematics ,Discrete mathematics ,Applied Mathematics ,Modeling and Simulation ,010102 general mathematics ,Metric (mathematics) ,Fixed-point theorem ,Geometry and Topology ,0101 mathematics ,Fixed point ,01 natural sciences ,Mathematics - Abstract
In this paper, we study two questions of A. Petrusel and G. Petrusel in b-metric fixed point theory [12]. The main results of the paper are as follows.
- Published
- 2018
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