694 results
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2. Sampling Discretization of Integral Norms
- Author
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Alexei Shadrin, Feng Dai, Andriy Prymak, Sergey Tikhonov, and Vladimir Temlyakov
- Subjects
Discretization ,General Mathematics ,Numerical analysis ,010102 general mathematics ,Probabilistic logic ,010103 numerical & computational mathematics ,Extension (predicate logic) ,01 natural sciences ,Computational Mathematics ,Uniform norm ,Entropy (information theory) ,Applied mathematics ,0101 mathematics ,Trigonometry ,Analysis ,Subspace topology ,Mathematics - Abstract
The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. Even though this problem is extremely important in applications, its systematic study has begun only recently. In this paper we obtain a conditional theorem for all integral norms $$L_q$$ , $$1\le q
- Published
- 2021
3. On non-monotonicity height of piecewise monotone functions
- Author
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Yingying Zeng and Lin Li
- Subjects
Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Monotonic function ,010103 numerical & computational mathematics ,Interval (mathematics) ,Variance (accounting) ,Composition (combinatorics) ,Infinity ,01 natural sciences ,Instability ,Discrete Mathematics and Combinatorics ,Applied mathematics ,0101 mathematics ,Piecewise monotone ,Mathematics ,media_common - Abstract
Non-monotonicity height is an important index to describe the complexity of dynamics for piecewise monotone functions. Although it is used extensively in the theory of iterative roots, its calculation is still difficult especially in the infinite case. In this paper, by introducing the concept of spanning interval, we first present a sufficient condition for piecewise monotone functions to have height infinity and then an algorithm for finding the spanning intervals is given. We further investigate the density of all piecewise monotone functions with infinite and finite height, respectively, and the results indicate the instability of height. At the end of this paper, the variance of height under composition, especially for functions of height 1 and infinity, are also discussed.
- Published
- 2021
4. On a new class of functional equations satisfied by polynomial functions
- Author
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Chisom Prince Okeke, Timothy Nadhomi, Maciej Sablik, and Tomasz Szostok
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Polynomial functions ,Polynomial ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Fr'echet operator ,Functional equations ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Continuity of monomial functions ,Monomial functions ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Linear combination ,Linear equation ,Mathematics - Abstract
The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi’s result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation$$\begin{aligned} F(x + y) - F(x) - F(y) = yf(x) + xf(y) \end{aligned}$$F(x+y)-F(x)-F(y)=yf(x)+xf(y)considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation.
- Published
- 2021
5. On graphs with equal total domination and Grundy total domination numbers
- Author
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Tilen Marc, Tim Kos, Tanja Dravec, and Marko Jakovac
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Sequence ,Domination analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Characterization (mathematics) ,01 natural sciences ,Vertex (geometry) ,Combinatorics ,Dominating set ,Chordal graph ,Bipartite graph ,Discrete Mathematics and Combinatorics ,Projective plane ,0101 mathematics ,Mathematics - Abstract
A sequence $$(v_1,\ldots ,v_k)$$ of vertices in a graph G without isolated vertices is called a total dominating sequence if every vertex $$v_i$$ in the sequence totally dominates at least one vertex that was not totally dominated by $$\{v_1,\ldots , v_{i-1}\}$$ and $$\{v_1,\ldots ,v_k\}$$ is a total dominating set of G. The length of a shortest such sequence is the total domination number of G ( $$\gamma _{t}(G)$$ ), while the length of a longest such sequence is the Grundy total domination number of G ( $$\gamma _{gr}^t(G)$$ ). In this paper we study graphs with equal total and Grundy total domination numbers. We characterize bipartite graphs with both total and Grundy total dominations number equal to 4, and show that there is no connected chordal graph G with $$\gamma _{t}(G)=\gamma _{gr}^t(G)=4$$ . The main result of the paper is a characterization of bipartite graphs with $$\gamma _{t}(G)=\gamma _{gr}^t(G)=6$$ proved by establishing a surprising correspondence between the existence of such graphs and a classical but still open problem of the existence of certain finite projective planes.
- Published
- 2021
6. Multivariate approximation of functions on irregular domains by weighted least-squares methods
- Author
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Giovanni Migliorati, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)
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Christoffel symbols ,Computational complexity theory ,Basis (linear algebra) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Estimator ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Domain (mathematical analysis) ,Computational Mathematics ,Bounded function ,FOS: Mathematics ,Applied mathematics ,Orthonormal basis ,Mathematics - Numerical Analysis ,[MATH]Mathematics [math] ,0101 mathematics ,Mathematics - Abstract
We propose and analyse numerical algorithms based on weighted least squares for the approximation of a real-valued function on a general bounded domain $\Omega \subset \mathbb{R}^d$. Given any $n$-dimensional approximation space $V_n \subset L^2(\Omega)$, the analysis in [6] shows the existence of stable and optimally converging weighted least-squares estimators, using a number of function evaluations $m$ of the order $n \log n$. When an $L^2(\Omega)$-orthonormal basis of $V_n$ is available in analytic form, such estimators can be constructed using the algorithms described in [6,Section 5]. If the basis also has product form, then these algorithms have computational complexity linear in $d$ and $m$. In this paper we show that, when $\Omega$ is an irregular domain such that the analytic form of an $L^2(\Omega)$-orthonormal basis is not available, stable and quasi-optimally weighted least-squares estimators can still be constructed from $V_n$, again with $m$ of the order $n \log n$, but using a suitable surrogate basis of $V_n$ orthonormal in a discrete sense. The computational cost for the calculation of the surrogate basis depends on the Christoffel function of $\Omega$ and $V_n$. Numerical results validating our analysis are presented., Comment: Version of the paper accepted for publication
- Published
- 2020
7. Packing colorings of subcubic outerplanar graphs
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Nicolas Gastineau, Olivier Togni, Boštjan Brešar, Faculty of Natural Sciences and Mathematics [Maribor], University of Maribor, Laboratoire d'Informatique de Bourgogne [Dijon] (LIB), Université de Bourgogne (UB), and Togni, Olivier
- Subjects
05C15, 05C12, 05C70 ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,01 natural sciences ,Graph ,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO] ,Combinatorics ,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM] ,Integer ,Outerplanar graph ,Bounded function ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Bipartite graph ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Invariant (mathematics) ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
Given a graph $G$ and a nondecreasing sequence $S=(s_1,\ldots,s_k)$ of positive integers, the mapping $c:V(G)\longrightarrow \{1,\ldots,k\}$ is called an $S$-packing coloring of $G$ if for any two distinct vertices $x$ and $y$ in $c^{-1}(i)$, the distance between $x$ and $y$ is greater than $s_i$. The smallest integer $k$ such that there exists a $(1,2,\ldots,k)$-packing coloring of a graph $G$ is called the packing chromatic number of $G$, denoted $\chi_{\rho}(G)$. The question of boundedness of the packing chromatic number in the class of subcubic (planar) graphs was investigated in several earlier papers; recently it was established that the invariant is unbounded in the class of all subcubic graphs. In this paper, we prove that the packing chromatic number of any 2-connected bipartite subcubic outerplanar graph is bounded by $7$. Furthermore, we prove that every subcubic triangle-free outerplanar graph has a $(1,2,2,2)$-packing coloring, and that there exists a subcubic outerplanar graph with a triangle that does not admit a $(1,2,2,2)$-packing coloring. In addition, there exists a subcubic triangle-free outerplanar graph that does not admit a $(1,2,2,3)$-packing coloring. A similar dichotomy is shown for bipartite outerplanar graphs: every such graph admits an $S$-packing coloring for $S=(1,3,\ldots,3)$, where $3$ appears $\Delta$ times ($\Delta$ being the maximum degree of vertices), and this property does not hold if one of the integers $3$ is replaced by $4$ in the sequence $S$., Comment: 24 pages
- Published
- 2020
8. Extensions of linear operators from hyperplanes and strong uniqueness of best approximation in L(X,W)
- Author
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Paweł Wójcik
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Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Banach space ,010103 numerical & computational mathematics ,Codimension ,Extension (predicate logic) ,01 natural sciences ,Projection (linear algebra) ,Operator (computer programming) ,Hyperplane ,Uniqueness ,0101 mathematics ,Analysis ,Subspace topology ,Mathematics - Abstract
The aim of this paper is to present some results concerning the problem of minimal projections and extensions. Let X be a reflexive Banach space and let Y be a closed subspace of X of codimension one. Let W be a finite-dimensional Banach space. We present a new sufficient condition under which any minimal extension of an operator A ∈ L ( Y , W ) is strongly unique. In this paper we show (in some circumstances) that if 1 λ ( Y , X ) , then a minimal projection from X onto Y is a strongly unique minimal projection. Moreover, we introduce and study a new geometric property of normed spaces. In this paper we also present a result concerning the strong unicity of best approximation.
- Published
- 2019
9. On the Generalized Laplace Transform
- Author
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Paul Bosch, Héctor José Carmenate García, José M. Rodríguez, José M. Sigarreta, Comunidad de Madrid, and Ministerio de Ciencia, Innovación y Universidades (España)
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Work (thermodynamics) ,Physics and Astronomy (miscellaneous) ,Matemáticas ,General Mathematics ,Inverse ,010103 numerical & computational mathematics ,01 natural sciences ,Convolution ,Computer Science (miscellaneous) ,Applied mathematics ,convolution ,0101 mathematics ,Harmonic oscillator ,Mathematics ,Laplace transform ,lcsh:Mathematics ,010102 general mathematics ,Order (ring theory) ,fractional derivative ,Fractional derivative ,lcsh:QA1-939 ,Generalized Laplace transform ,Fractional calculus ,generalized Laplace transform ,Chemistry (miscellaneous) ,Fractional differential - Abstract
This article belongs to the Special Issue Discrete and Fractional Mathematics: Symmetry and Applications. In this paper we introduce a generalized Laplace transform in order to work with a very general fractional derivative, and we obtain the properties of this new transform. We also include the corresponding convolution and inverse formula. In particular, the definition of convolution for this generalized Laplace transform improves previous results. Additionally, we deal with the generalized harmonic oscillator equation, showing that this transform and its properties allow one to solve fractional differential equations. We would like to thank the referees for their comments, which have improved the paper. The research of José M. Rodríguez and José M. Sigarreta was supported by a grant from Agencia Estatal de Investigación (PID2019-106433GB-I00/AEI/10.13039/501100011033), Spain. The research of José M. Rodríguez is supported by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation).
- Published
- 2021
10. Existence of meromorphic solutions of some generalized Fermat functional equations
- Author
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Linlin Wu, Weiran Lü, Chun He, and Feng Lü
- Subjects
Fermat's Last Theorem ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,Quadratic equation ,Functional equation ,Discrete Mathematics and Combinatorics ,Order (group theory) ,0101 mathematics ,Meromorphic function ,Mathematics - Abstract
The aim of this paper is twofold. Firstly, we study the non-existence of finite order meromorphic solutions to the Cubic type of Fermat functional equation $$f(z)^3-3\tau f(z)f(z+c)+f(z+c)^3=1$$. In addition, the paper is concerned with the description of finite order entire solutions of the Quadratic type of Fermat functional equation $$f(z)^2-2\mu f(z)f(z+c)+f(z+c)^2=1$$.
- Published
- 2019
11. Exceptional sets for sums of almost equal prime cubes
- Author
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Mengdi Wang
- Subjects
Combinatorics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Waring–Goldbach problem ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Prime (order theory) ,Mathematics - Abstract
In this paper, we continue to investigate the exceptional sets for sums of five and six almost equal cubes of primes. We would also like to establish that almost all natural numbers n, subjected to certain congruence conditions, can be written as n = p 1 3 + ⋯ + p s 3 {n=p_{1}^{3}+\cdots+p_{s}^{3}} ( s = 5 , 6 {s=5,6} ) with | p j - ( n / s ) 1 / 3 | ≤ n θ s / 3 + ε {|p_{j}-(n/s)^{1/3}|\leq n^{\theta_{s}/3+\varepsilon}} ( 1 ≤ j ≤ s {1\leq j\leq s} ), where θ s {\theta_{s}} is as small as possible. The main result of this paper is to improve θ 6 = 5 / 6 + ε {\theta_{6}=5/6+\varepsilon} , which is proven in [M. Wang, Exceptional sets for sums of five and six almost equal prime cubes, Acta Math. Hungar. 156 2018, 2, 424–434], to θ 6 = 9 / 11 + ε {\theta_{6}=9/11+\varepsilon} , as well as prove θ 5 = 8 / 9 + ε {\theta_{5}=8/9+\varepsilon} in another way.
- Published
- 2019
12. Reproducing kernel orthogonal polynomials on the multinomial distribution
- Author
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Robert C. Griffiths and Persi Diaconis
- Subjects
Numerical Analysis ,Stationary distribution ,Markov chain ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Poisson kernel ,010103 numerical & computational mathematics ,Kravchuk polynomials ,01 natural sciences ,Combinatorics ,symbols.namesake ,Kernel (statistics) ,Orthogonal polynomials ,symbols ,Test statistic ,Multinomial distribution ,0101 mathematics ,Analysis ,Mathematics - Abstract
Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q n ( x , y ; N , p ) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n = 0 , 1 , … , N . The Poisson kernel ∑ n = 0 N ρ n Q n ( x , y ; N , p ) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new duplication formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The duplication formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The χ 2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.
- Published
- 2019
13. 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
14. Comparison of probabilistic and deterministic point sets on the sphere
- Author
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Peter J. Grabner and T. A. Stepanyuk
- Subjects
Unit sphere ,Numerical Analysis ,Sequence ,Applied Mathematics ,General Mathematics ,Existential quantification ,010102 general mathematics ,Probabilistic logic ,Sampling (statistics) ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Point (geometry) ,0101 mathematics ,Constant (mathematics) ,Analysis ,Mathematics - Abstract
In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (especially spherical t -designs) are better or as good as probabilistic ones like the jittered sampling model. We find asymptotic equalities for the discrete Riesz s -energy of sequences of well separated t -designs on the unit sphere S d ⊂ R d + 1 , d ≥ 2 . The case d = 2 was studied in Hesse (2009) and Hesse and Leopardi (2008). In Bondarenko et al., (2015) it was established that for d ≥ 2 , there exists a constant c d , such that for every N > c d t d there exists a well-separated spherical t -design on S d with N points. This paper gives results, based on recent developments that there exists a sequence of well separated spherical t -designs such that t and N are related by N ≍ t d .
- Published
- 2019
15. Approximation by modified Jain–Baskakov operators
- Author
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Marius Mihai Birou, Vishnu Narayan Mishra, and Preeti Sharma
- Subjects
Baskakov operator ,General Mathematics ,010102 general mathematics ,Applied mathematics ,Basis function ,Asymptotic formula ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Modulus of continuity ,Mathematics - Abstract
In the present paper, we discuss the approximation properties of Jain–Baskakov operators with parameter c. The paper deals with the modified forms of the Baskakov basis functions. Some direct results are established, which include the asymptotic formula, error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct the King modification of these operators, which preserves the test functions e 0 {e_{0}} and e 1 {e_{1}} . It is shown that these King type operators provide a better approximation order than some Baskakov–Durrmeyer operators for continuous functions defined on some closed intervals.
- Published
- 2019
16. On the existence of optimal meshes in every convex domain on the plane
- Author
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András Kroó
- Subjects
Numerical Analysis ,Polynomial ,Conjecture ,Degree (graph theory) ,Plane (geometry) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Polytope ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Cardinality ,Polygon mesh ,0101 mathematics ,Constant (mathematics) ,Analysis ,Mathematics - Abstract
In this paper we study the so called optimal polynomial meshes for domains in K ⊂ R d , d ≥ 2 . These meshes are discrete point sets Y n of cardinality c n d which have the property that ‖ p ‖ K ≤ A ‖ p ‖ Y n for every polynomial p of degree at most n with a constant A > 1 independent of n . It was conjectured earlier that optimal polynomial meshes exist in every convex domain. This statement was previously shown to hold for polytopes and C 2 like domains. In this paper we give a complete affirmative answer to the above conjecture when d = 2 .
- Published
- 2019
17. Geometric properties of F-normed Orlicz spaces
- Author
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Yunan Cui, Paweł Kolwicz, Radosław Kaczmarek, and Henryk Hudzik
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Function space ,Applied Mathematics ,General Mathematics ,Uniform convergence ,010102 general mathematics ,Monotonic function ,010103 numerical & computational mathematics ,01 natural sciences ,Linear subspace ,Monotone polygon ,Norm (mathematics) ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Mathematics - Abstract
The paper deals with F-normed functions and sequence spaces. First, some general results on such spaces are presented. But most of the results in this paper concern various monotonicity properties and various Kadec–Klee properties of F-normed Orlicz functions and sequence spaces and their subspaces of elements with order continuous norm, when they are generated by monotone Orlicz functions on $${\mathbb {R}}_{+}$$ and equipped with the classical Mazur–Orlicz F-norm. Strict monotonicity, lower (and upper) local uniform monotonicity and uniform monotonicity in the classical sense as well as their orthogonal counterparts are considered. It follows from the criteria that are presented for these properties that all the above classical monotonicity properties except for uniform monotonicity differ from their orthogonal counterparts [in contrast to Kothe spaces (see Hudzik et al. in Rocky Mt J Math 30(3):933–950, 2000)]. The Kadec–Klee properties that are considered in this paper correspond to various kinds of convergence: convergence locally in measure and convergence globally in measure for function spaces, uniform convergence and coordinatewise convergence in the case of sequence spaces.
- Published
- 2018
18. On symmetries of roots of rational functions and the classification of rational function solutions of functional equations arising from multiplication of quantum integers with prime semigroup supports
- Author
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Lan Nguyen
- Subjects
Polynomial ,Pure mathematics ,Semigroup ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,010103 numerical & computational mathematics ,Rational function ,01 natural sciences ,Prime (order theory) ,Additive number theory ,Discrete Mathematics and Combinatorics ,Grothendieck group ,Multiplication ,0101 mathematics ,Mathematics - Abstract
The study of quantum integers and their operations is closely related to the studies of symmetries of roots of polynomials and of fundamental questions of decompositions in Additive Number Theory. In his papers on quantum arithmetics, Melvyn Nathanson raises the question of classifying solutions of functional equations arising from the multiplication of quantum integers, starting with polynomial solutions and then generalizing to rational function solutions. The classification of polynomial solutions with fields of coefficients of characteristic zero and support base P has been completed. In a paper concerning the Grothendieck group associated to the collection of polynomial solutions, Nathanson poses a problem which asks whether the set of rational function solutions strictly contains the set of ratios of polynomial solutions. It is now known that there are infinitely many rational function solutions $$\Gamma $$ with fields of coefficients of characteristic zero not constructible as ratios of polynomial solutions, even in the purely cyclotomic case, which is the case most similar to the polynomial solution case. The classification of polynomial solutions is thus not sufficient, in essential ways, to resolve the classification problem of all rational function solutions with fields of coefficients of characteristic zero. In this paper we study symmetries of roots of rational functions and the classification of the important class-the last and main obstruction to the classification problem-of rational function solutions, the purely cyclotomic, purely nonrational primitive solutions with fields of coefficients of characteristic zero and support base P, which allows us to complete the classification problem raised by Nathanson.
- Published
- 2018
19. The weak core inverse
- Author
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Néstor Thome, D. E. Ferreyra, Fabián Eduardo Levis, and A. N. Priori
- Subjects
Pure mathematics ,Multilinear algebra ,Class (set theory) ,Generalized inverse ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Inverse ,Generalized inverses ,010103 numerical & computational mathematics ,Weak group inverse ,01 natural sciences ,Square matrix ,Core EP decomposition ,Core (graph theory) ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Core inverse ,Mathematics - Abstract
[EN] In this paper, we introduce a new generalized inverse, called weak core inverse (or, in short, WC inverse) of a complex square matrix. This new inverse extends the notion of the core inverse defined by Baksalary and Trenkler (Linear Multilinear Algebra 58(6):681-697, 2010). We investigate characterizations, representations, and properties for this generalized inverse. In addition, we introduce weak core matrices (or, in short, WC matrices) and we show that these matrices form a more general class than that given by the known weak group matrices, recently investigated by H. Wang and X. Liu., In what follows, we detail the acknowledgements. D.E. Ferreyra, F.E. Levis, A.N. Priori - Partially supported by Universidad Nacional de Rio Cuarto (Grant PPI 18/C559) and CONICET (Grant PIP 112-201501-00433CO). D.E. Ferreyra F.E. Levis - Partially supported by ANPCyT (Grant PICT 201803492). D.E. Ferreyra, N. Thome -Partially supported by Universidad Nacional de La Pampa, Facultad de Ingenieria (Grant Resol. Nro. 135/19). N. Thome -Partially supported by Ministerio de Economia, Industria y Competitividad of Spain (Grant Red de Excelencia MTM2017-90682-REDT) and by Universidad Nacional del Sur of Argentina (Grant 24/L108). We would like to thank the Referees for their valuable comments and suggestions which helped us to considerably improve the presentation of the paper
- Published
- 2021
20. Convergence of a multidimensional Glimm-like scheme for the transport of fronts
- Author
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Olivier Hurisse, Thierry Gallouët, Aix Marseille Université (AMU), Mécanique des Fluides, Energies et Environnement (EDF R&D MFEE), EDF R&D (EDF R&D), and EDF (EDF)-EDF (EDF)
- Subjects
Scheme (programming language) ,Convection ,Class (set theory) ,Advection ,Applied Mathematics ,General Mathematics ,Numerical analysis ,010102 general mathematics ,010103 numerical & computational mathematics ,Glimm’s scheme ,01 natural sciences ,Projection (linear algebra) ,multidimensional problem ,Computational Mathematics ,Convergence (routing) ,Applied mathematics ,random choice ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,front propagation ,Preprint ,0101 mathematics ,computer ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,Mathematics ,computer.programming_language - Abstract
International audience; This paper is devoted to the numerical analysis of a numerical scheme dedicated to the simulation of front advection (see https://hal.archives-ouvertes.fr/hal-02940407v1 for a preprint version presenting this scheme and some numerical results). The latter has been recently proposed and it is based on the ideas used for the Glimm's scheme. It relies on a two-step approach: a convection step is followed by a projection step which is based on a random choice. The main advantage of this scheme is that it applies to multi-dimensional problems. In the present paper a convergence result for this scheme is provided for a particular class of multi-dimensional problems. This work has been accepted for publication in IMA Journal of Numerical Analysis: https://doi.org/10.1093/imanum/drab053.
- Published
- 2020
21. On the Cauchy Problem of Vectorial Thermostatted Kinetic Frameworks
- Author
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Marco Menale, Carlo Bianca, Bruno Carbonaro, Bianca, Carlo, Carbonaro, Bruno, and Menale, Marco
- Subjects
State variable ,Physics and Astronomy (miscellaneous) ,integro-differential equation ,General Mathematics ,Complex system ,010103 numerical & computational mathematics ,complexity ,kinetic theory ,Cauchy problem ,nonlinearity ,Mathematical models, Boltzmann equation, Vlasov equation, Kinetic Theory for Active Particles, well-posedness problems ,01 natural sciences ,Quadratic equation ,Computer Science (miscellaneous) ,Applied mathematics ,Initial value problem ,Uniqueness ,0101 mathematics ,Mathematics ,Variable (mathematics) ,lcsh:Mathematics ,010102 general mathematics ,lcsh:QA1-939 ,Nonlinear system ,Chemistry (miscellaneous) - Abstract
This paper is devoted to the derivation and mathematical analysis of new thermostatted kinetic theory frameworks for the modeling of nonequilibrium complex systems composed by particles whose microscopic state includes a vectorial state variable. The mathematical analysis refers to the global existence and uniqueness of the solution of the related Cauchy problem. Specifically, the paper is divided in two parts. In the first part the thermostatted framework with a continuous vectorial variable is proposed and analyzed. The framework consists of a system of partial integro-differential equations with quadratic type nonlinearities. In the second part the thermostatted framework with a discrete vectorial variable is investigated. Real world applications, such as social systems and crowd dynamics, and future research directions are outlined in the paper.
- Published
- 2020
22. A note on Korobov lattice rules for integration of analytic functions
- Author
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Friedrich Pillichshammer
- Subjects
Statistics and Probability ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Numerical Analysis (math.NA) ,11K45, 65D30 ,01 natural sciences ,Numerical integration ,Periodic function ,Lattice (order) ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Fourier series ,SIMPLE algorithm ,Analytic function ,Mathematics - Abstract
We study numerical integration for a weighted Korobov space of analytic periodic functions for which the Fourier coefficients decay exponentially fast. In particular, we are interested in how the error depends on the dimension d . Many recent papers deal with this problem or similar problems and provide matching necessary and sufficient conditions for various notions of tractability. In most cases even simple algorithms are known which allow to achieve these notions of tractability. However, there is a gap in the literature: while for the notion of exponential-weak tractability one knows matching necessary and sufficient conditions, so far no explicit algorithm has been known which yields the desired result. In this paper we close this gap and prove that Korobov lattice rules are suitable algorithms in order to achieve exponential-weak tractability for integration in weighted Korobov spaces of analytic periodic functions.
- Published
- 2020
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23. Derivations and Leibniz differences on rings
- Author
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Bruce Ebanks
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Discrete Mathematics and Combinatorics ,Order (ring theory) ,010103 numerical & computational mathematics ,Commutative ring ,0101 mathematics ,Composition (combinatorics) ,01 natural sciences ,Mathematics ,Integral domain - Abstract
In an earlier paper we discussed the composition of derivations of order 1 on a commutative ring R, showing that (i) the composition of n derivations of order 1 yields a derivation of order at most n, and (ii) under additional conditions on R the composition of n derivations of order exactly 1 forms a derivation of order exactly n. In the present paper we consider the composition of derivations of any orders on rings. We show that on any commutative ring R the composition of a derivation of order at most n with a derivation of order at most m results in a derivation of order at most $$n+m$$. If R is an integral domain of sufficiently large characteristic, then the composition of a derivation of order exactly n with a derivation of order exactly m results in a derivation of order exactly $$n+m$$. As in the previous paper, the results are proved using Leibniz difference operators.
- Published
- 2018
24. Two refinements of Frink’s metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces
- Author
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Filip Turoboś, Jacek Jachymski, and Katarzyna Chrząszcz
- Subjects
Intersection theorem ,Pure mathematics ,Sequence ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Fixed point ,01 natural sciences ,Metric space ,Iterated function ,Metrization theorem ,Metric (mathematics) ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Contraction principle ,Mathematics - Abstract
Quasimetric spaces have been an object of thorough investigation since Frink’s paper appeared in 1937 and various generalisations of the axioms of metric spaces are now experiencing their well-deserved renaissance. The aim of this paper is to present two improvements of Frink’s metrization theorem along with some fixed point results for single-valued mappings on quasimetric spaces. Moreover, Cantor’s intersection theorem for sequences of sets which are not necessarily closed is established in a quasimetric setting. This enables us to give a new proof of a quasimetric version of the Banach Contraction Principle obtained by Bakhtin. We also provide error estimates for a sequence of iterates of a mapping, which seem to be new even in a metric setting.
- Published
- 2018
25. The CMV Matrix and the Generalized Lanczos Process
- Author
-
Kh. D. Ikramov
- Subjects
Statistics and Probability ,Pure mathematics ,Basis (linear algebra) ,Applied Mathematics ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Block matrix ,010103 numerical & computational mathematics ,Mathematics::Spectral Theory ,01 natural sciences ,Matrix (mathematics) ,Unit circle ,Orthogonal polynomials ,Multiplication ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The CMV matrix is the five-diagonal matrix that represents the operator of multiplication by the independent variable in a special basis formed of Laurent polynomials orthogonal on the unit circle C. The article by Cantero, Moral, and Velazquez, published in 2003 and describing this matrix, has attracted much attention because it implies that the conventional orthogonal polynomials on C can be interpreted as the characteristic polynomials of the leading principal submatrices of a certain five-diagonal matrix. The present paper recalls that finite-dimensional sections of the CMV matrix appeared in papers on the unitary eigenvalue problem long before the article by Cantero et al. was published. Moreover, band forms were also found for a number of other situations in the normal eigenvalue problem.
- Published
- 2018
26. On the invariance equation for two-variable weighted nonsymmetric Bajraktarević means
- Author
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Zsolt Páles and Amr Zakaria
- Subjects
Weight function ,39B12, 39.35, 26E60 ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Hyperbolic function ,Zero (complex analysis) ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Monotone polygon ,Természettudományok ,Mathematics - Classical Analysis and ODEs ,Functional equation ,Discrete Mathematics and Combinatorics ,Matematika- és számítástudományok ,0101 mathematics ,Mathematics ,Variable (mathematics) - Abstract
The purpose of this paper is to investigate the invariance of the arithmetic mean with respect to two weighted Bajraktarevic means, i.e., to solve the functional equation $$\begin{aligned} \left( \frac{f}{g}\right) ^{\!\!-1}\!\!\left( \frac{tf(x)+sf(y)}{tg(x)+sg(y)}\right) +\left( \frac{h}{k}\right) ^{\!\!-1}\!\!\left( \frac{sh(x)+th(y)}{sk(x)+tk(y)}\right) =x+y \qquad (x,y\in I), \end{aligned}$$ where $$f,g,h,k:I\rightarrow \mathbb {R}$$ are unknown continuous functions such that g, k are nowhere zero on I, the ratio functions f / g, h / k are strictly monotone on I, and $$t,s\in \mathbb {R}_+$$ are constants different from each other. By the main result of this paper, the solutions of the above invariance equation can be expressed either in terms of hyperbolic functions or in terms of trigonometric functions and an additional weight function. For the necessity part of this result, we will assume that $$f,g,h,k:I\rightarrow \mathbb {R}$$ are four times continuously differentiable.
- Published
- 2018
27. Continued fraction expansions for the Lambert $$\varvec{W}$$ W function
- Author
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Cristina B. Corcino, István Mező, and Roberto B. Corcino
- Subjects
Pure mathematics ,Integral representation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,symbols.namesake ,Lambert W function ,symbols ,Discrete Mathematics and Combinatorics ,Fraction (mathematics) ,0101 mathematics ,Principal branch ,Mathematics - Abstract
In the first part of the paper we give a new integral representation for the principal branch of the Lambert W function. Then we deduce two continued fraction expansions for this branch. At the end of the paper we study the numerical behavior of the approximants of these expansions.
- Published
- 2018
28. Gamma and Factorial in the Monthly
- Author
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Robert M. Corless and Jonathan M. Borwein
- Subjects
Factorial ,Stirling engine ,Mathematics - History and Overview ,History and Overview (math.HO) ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,law.invention ,law ,FOS: Mathematics ,Applied mathematics ,Asymptotic formula ,0101 mathematics ,Algorithm ,Mathematics - Abstract
The Monthly has published roughly fifty papers on the $\Gamma$ function or Stirling's formula. We survey those papers (discussing only our favourites in any detail) and place them in the context of the larger mathematical literature on $\Gamma$., Comment: 25 page
- Published
- 2018
29. ASYMPTOTIC AND CONVERGENT EXPANSIONS FOR SOLUTIONS OF THIRD-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH A LARGE PARAMETER
- Author
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Chelo Ferreira, Ester Pérez Sinusía, José L. López, Universidad Pública de Navarra. Departamento de Ingeniería Matemática e Informática, and Nafarroako Unibertsitate Publikoa. Matematika eta Informatika Ingeniaritza Saila
- Subjects
Green’s functions ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Pearcey integral ,Third order ,Asymptotic expansions ,Linear differential equation ,Third-order differential equations ,Banach’s fixed point theorem ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In previous papers [6–8,10], we derived convergent and asymptotic expansions of solutions of second order linear differential equations with a large parameter. In those papers we generalized and developed special cases not considered in Olver’s theory [Olver, 1974]. In this paper we go one step forward and consider linear differential equations of the third order: y ′′′ +aΛ2y′ +bΛ3y = f(x)y′ +g(x)y, with a, b ∈ C fixed, f′ and g continuous, and Λ a large positive parameter. We propose two different techniques to handle the problem: (i) a generalization of Olver’s method and (ii) the transformation of the differential problem into a fixed point problem from which we construct an asymptotic sequence of functions that converges to the unique solution of the problem. Moreover, we show that this second technique may also be applied to nonlinear differential equations with a large parameter. As an application of the theory, we obtain new convergent and asymptotic expansions of the Pearcey integral P(x, y) for large |x|. The Ministerio de Econom´ıa y Competitividad (REF. MTM2014-52859-P) is acknowledged by its financial support.
- Published
- 2018
30. On the relation of the spectral test to isotropic discrepancy and L-approximation in Sobolev spaces
- Author
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Mathias Sonnleitner and Friedrich Pillichshammer
- Subjects
Statistics and Probability ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Isotropy ,Mathematical analysis ,Convex set ,Boundary (topology) ,010103 numerical & computational mathematics ,01 natural sciences ,Upper and lower bounds ,Spectral test ,Sobolev space ,Dimension (vector space) ,Unit cube ,0101 mathematics ,Mathematics - Abstract
This paper is a follow-up to the recent paper of Pillichshammer and Sonnleitner (2020) [12] . We show that the isotropic discrepancy of a lattice point set is at most d 2 2 ( d + 1 ) times its spectral test, thereby correcting the dependence on the dimension d and an inaccuracy in the proof of the upper bound in Theorem 2 of the mentioned paper. The major task is to bound the volume of the neighbourhood of the boundary of a convex set contained in the unit cube. Further, we characterize averages of the distance to a lattice point set in terms of the spectral test. As an application, we infer that the spectral test – and with it the isotropic discrepancy – is crucial for the suitability of the lattice point set for the approximation of Sobolev functions.
- Published
- 2021
31. Comparing Two Numerical Methods for Approximating a New Giving Up Smoking Model Involving Fractional Order Derivatives
- Author
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Baha Alzalg, Vedat Suat Erturk, Anwar Zeb, Gul Zaman, Shaher Momani, and Ondokuz Mayıs Üniversitesi
- Subjects
Caputo fractional derivative ,Numerical solution ,General Mathematics ,Numerical analysis ,010102 general mathematics ,General Physics and Astronomy ,010103 numerical & computational mathematics ,General Chemistry ,Differential transform method ,01 natural sciences ,Smoking dynamics ,Generalized Euler method ,Euler method ,symbols.namesake ,symbols ,General Earth and Planetary Sciences ,Applied mathematics ,Order (group theory) ,0101 mathematics ,General Agricultural and Biological Sciences ,Giving Up Smoking ,Mathematics - Abstract
Alzalg, Baha/0000-0002-1839-8083; Momani, Shaher M./0000-0002-6326-8456; WOS: 000413785300005 In a recent paper (Zeb et al. in Appl Math Model 37(7):5326-5334, 2013), the authors presented a new model of giving up smoking model. In the present paper, the dynamics of this new model involving the Caputo derivative was studied numerically. For this purpose, generalized Euler method and the multistep generalized differential transform method are employed to compute accurate approximate solutions to this new giving up smoking model of fractional order. The unique positive solution for the fractional order model is presented. A comparative study between these two methods and the well-known Runge-Kutta method is presented in the case of integer-order derivatives. The solutions obtained are also presented graphically.
- Published
- 2017
32. A novel simulation methodology of fractional order nuclear science model
- Author
-
Ali Akgül and Yasir Khan
- Subjects
Scheme (programming language) ,Mathematical optimization ,Computer simulation ,Process (engineering) ,General Mathematics ,010102 general mathematics ,General Engineering ,Order (ring theory) ,010103 numerical & computational mathematics ,Nuclear reactor ,01 natural sciences ,law.invention ,Fractional calculus ,law ,Simple (abstract algebra) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Applied mathematics ,0101 mathematics ,Nuclear science ,computer ,Mathematics ,computer.programming_language - Abstract
In this paper, a novel simulation methodology based on the reproducing kernels is proposed for solving the fractional order integro-differential transport model for a nuclear reactor. The analysis carried out in this paper thus forms a crucial step in the process of development of fractional calculus as well as nuclear science models. The fractional derivative is described in the Captuo Riemann–Liouville sense. Results are presented graphically and in tabulated forms to study the efficiency and accuracy of method. The present scheme is very simple, effective, and appropriate for obtaining numerical simulation of nuclear science models. Copyright © 2017 John Wiley & Sons, Ltd.
- Published
- 2017
33. On the entropy numbers of the mixed smoothness function classes
- Author
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Vladimir Temlyakov
- Subjects
Numerical Analysis ,Multivariate statistics ,Nonlinear approximation ,Greedy approximation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Applied mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Analysis ,Mathematics - Abstract
Behavior of the entropy numbers of classes of multivariate functions with mixed smoothness is studied here. This problem has a long history and some fundamental problems in the area are still open. The main goal of this paper is to develop a new method of proving the upper bounds for the entropy numbers. This method is based on recent developments of nonlinear approximation, in particular, on greedy approximation. This method consists of the following two steps strategy. At the first step we obtain bounds of the best m -term approximations with respect to a dictionary. At the second step we use general inequalities relating the entropy numbers to the best m -term approximations. For the lower bounds we use the volume estimates method, which is a well known powerful method for proving the lower bounds for the entropy numbers. It was used in a number of previous papers.
- Published
- 2017
34. Kantorovich variant of a new kind ofq-Bernstein-Schurer operators
- Author
-
Nurhayat Ispir, Ruchi Ruchi, and Purshottam Narain Agrawal
- Subjects
Constant coefficients ,General Mathematics ,010102 general mathematics ,General Engineering ,Microlocal analysis ,010103 numerical & computational mathematics ,Spectral theorem ,Operator theory ,Lipschitz continuity ,01 natural sciences ,Fourier integral operator ,Algebra ,Rate of convergence ,Applied mathematics ,Differentiable function ,0101 mathematics ,Mathematics - Abstract
Ren and Zeng (2013) introduced a new kind of q-Bernstein-Schurer operators and studied some approximation properties. Acu etal. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using q-Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's K-functional. Next, we introduce the bivariate case of q-Bernstein-Schurer-Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's K-functional. Finally, we define the generalized Boolean sum operators of the q-Bernstein-Schurer-Kantorovich type and investigate the approximation of the Bogel continuous and Bogel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright (c) 2017 John Wiley & Sons, Ltd.
- Published
- 2017
35. On $$\varvec{n}$$ n -norm preservers and the Aleksandrov conservative $$\varvec{n}$$ n -distance problem
- Author
-
György Pál Gehér
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Order (ring theory) ,010103 numerical & computational mathematics ,01 natural sciences ,Surjective function ,Nonlinear system ,Transformation (function) ,Norm (mathematics) ,Distance problem ,Discrete Mathematics and Combinatorics ,Affine transformation ,0101 mathematics ,Unit (ring theory) ,Mathematics - Abstract
The goal of this paper is to point out that the results obtained in the recent papers (Chen and Song in Nonlinear Anal 72:1895–1901, 2010; Chu in J Math Anal Appl 327:1041–1045, 2007; Chu et al. in Nonlinear Anal 59:1001–1011, 2004a, J. Math Anal Appl 289:666–672, 2004b) can be seriously strengthened in the sense that we can significantly relax the assumptions of the main results so that we still get the same conclusions. In order to do this first, we prove that for $$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
36. Weyl and Bernstein numbers of embeddings of Sobolev spaces with dominating mixed smoothness
- Author
-
Van Kien Nguyen
- Subjects
Statistics and Probability ,Mathematics::Functional Analysis ,Numerical Analysis ,Pure mathematics ,Control and Optimization ,Algebra and Number Theory ,Smoothness (probability theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Sobolev space ,Continuation ,FOS: Mathematics ,0101 mathematics ,Mathematics::Representation Theory ,Lp space ,Mathematics - Abstract
This paper is a continuation of the papers [21] and [22]. Here we shall investigate the asymptotic behaviour of Weyl and Bernstein numbers of embeddings of Sobolev spaces with dominating mixed smoothness into Lebesgue spaces., 33 pages
- Published
- 2016
37. Polynomial control on weighted stability bounds and inversion norms of localized matrices on simple graphs
- Author
-
Qiquan Fang, Chang Eon Shin, and Qiyu Sun
- Subjects
Polynomial (hyperelastic model) ,Mathematics::Functional Analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics - Operator Algebras ,Inverse ,010103 numerical & computational mathematics ,Muckenhoupt weights ,01 natural sciences ,Functional Analysis (math.FA) ,Combinatorics ,Mathematics - Functional Analysis ,Matrix (mathematics) ,Bounded function ,Banach algebra ,Beurling algebra ,FOS: Mathematics ,0101 mathematics ,Operator Algebras (math.OA) ,Operator norm ,Analysis ,Mathematics - Abstract
The (un)weighted stability for some matrices on a graph is one of essential hypotheses in time-frequency analysis and applied harmonic analysis. In the first part of this paper, we show that for a localized matrix in a Beurling algebra, its weighted stabilities for different exponents and Muckenhoupt weights are equivalent to each other, and reciprocal of its optimal lower stability bound for one exponent and weight is controlled by a polynomial of reciprocal of its optimal lower stability bound for another exponent and weight. Banach algebras of matrices with certain off-diagonal decay is of great importance in many mathematical and engineering fields, and its inverse-closed property can be informally interpreted as localization preservation. Let $${{\mathcal {B}}}(\ell ^p_w)$$ be the Banach algebra of bounded linear operators on the weighted sequence space $$\ell ^p_w$$ on a graph. In the second part of this paper, we prove that Beurling algebras of localized matrices on a connected simple graph are inverse-closed in $${{\mathcal {B}}}(\ell ^p_w)$$ for all $$1\le p
- Published
- 2019
38. Equilibrium problems in weakly admissible external fields created by pointwise charges
- Author
-
J.F. Sánchez Lara, Ramón Orive, Franck Wielonsky, Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Universidad de La Laguna [Tenerife - SP] (ULL), and Universidad de Granada = University of Granada (UGR)
- Subjects
Pointwise ,Numerical Analysis ,Pure mathematics ,Mathematics - Complex Variables ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,010103 numerical & computational mathematics ,Infinity ,01 natural sciences ,Measure (mathematics) ,Potential theory ,Compact space ,Simple (abstract algebra) ,FOS: Mathematics ,Uniqueness ,Complex Variables (math.CV) ,0101 mathematics ,[MATH]Mathematics [math] ,Complex plane ,Analysis ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,media_common - Abstract
The main subject of this paper is equilibrium problems on an unbounded conductor $\Sigma$ of the complex plane in the presence of a weakly admissible external field. An admissible external field $Q$ on $\Sigma$ satisfies, along with other mild conditions, the following growth property at infinity: $$\lim_{|x| \rightarrow \infty}(Q(x) - \log |x|) = +\infty.$$ This condition guarantees the existence and uniqueness of the equilibrium measure in the presence of $Q$, and the compactness of its support. In the last 10-15 years, several papers have dealt with weakly admissible external fields, in the sense that $Q$ satisfies a weaker condition at infinity, namely, $$\exists M\in(-\infty,\infty],\quad\liminf_{|x| \rightarrow \infty}(Q(x) - \log |x|) = M.$$ Under this last assumption, there still exists a unique equilibrium measure in the external field $Q$, but the support need not be a compact subset of $\Sigma$ anymore. In most examples considered in the literature the support is indeed unbounded. Our main goal in this paper is to illustrate this topic by means of a simple class of external fields on the real axis created by a pair of attractive and repellent charges in the complex plane, and to study the dynamics of the associated equilibrium measures as the strength of the charges evolves. As one of our findings, we exhibit configurations where the support of the equilibrium measure in a weakly admissible external field is a compact subset of the real axis. To achieve our goal, we extend some results from potential theory, known for admissible external fields, to the weakly admissible case. These new results may be of independent interest. Finally, the so--called signed equilibrium measure is an important tool in our analysis. Its relationship with the (positive) equilibrium measure is also explored., Comment: To appear in Journal of Approximation Theory
- Published
- 2019
39. Burnside rings for Real $2$-representation theory: The linear theory
- Author
-
Dmitriy Rumynin and Matthew B. Young
- Subjects
General Mathematics ,Character theory ,010103 numerical & computational mathematics ,Algebraic topology ,01 natural sciences ,Representation theory ,Mathematics::Category Theory ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Computer Science::General Literature ,Algebraic Topology (math.AT) ,Quantum Algebra (math.QA) ,Category Theory (math.CT) ,Mathematics - Algebraic Topology ,0101 mathematics ,Representation Theory (math.RT) ,2-group ,Category theory ,Mathematics ,20J99 (Primary), 18D05 (Secondary) ,Applied Mathematics ,010102 general mathematics ,Linear system ,Astrophysics::Instrumentation and Methods for Astrophysics ,Quantum algebra ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,Mathematics - Category Theory ,Algebra ,Mathematics - Representation Theory ,2-category - Abstract
This paper is a fundamental study of the Real $2$-representation theory of $2$-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a $2$-equivariant Morita bicategory, where a novel construction of induction is introduced. We identify the Grothendieck ring of Real $2$-representations as a Real variant of the Burnside ring of the fundamental group of the $2$-group and study the Real categorical character theory. This paper unifies two previous lines of inquiry, the approach to $2$-representation theory via Morita theory and Burnside rings, initiated by the first author and Wendland, and the Real $2$-representation theory of $2$-groups, as studied by the second author., Comment: Version 2: many minor improvements, appears as an MPI preprint. Version 3: the final published version
- Published
- 2019
- Full Text
- View/download PDF
40. A higher order Faber spline basis for sampling discretization of functions
- Author
-
Tino Ullrich and Nadiia Derevianko
- Subjects
Numerical Analysis ,Mathematics::Functional Analysis ,Discretization ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Basis function ,42C40, 46E35, 41A15, 65T60 ,010103 numerical & computational mathematics ,Numerical Analysis (math.NA) ,Lipschitz continuity ,01 natural sciences ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Piecewise linear function ,Spline (mathematics) ,Compact space ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Remainder ,Real line ,Analysis ,Mathematics - Abstract
This paper is devoted to the question of constructing a higher order Faber spline basis for the sampling discretization of functions with higher regularity than Lipschitz. The basis constructed in this paper has similar properties as the piecewise linear classical Faber–Schauder basis (Faber, 1908) except for the compactness of the support. Although the new basis functions are supported on the real line they are very well localized (exponentially decaying) and the main parts are concentrated on a segment. This construction gives a complete answer to Problem 3.13 in Triebel’s monograph (Triebel, 2012) by extending the classical Faber basis to higher orders. Roughly, the crucial idea to obtain a higher order Faber spline basis is to apply Taylor’s remainder formula to the dual Chui–Wang wavelets. As a first step we explicitly determine these dual wavelets which may be of independent interest. Using this new basis we provide sampling characterizations for Besov and Triebel–Lizorkin spaces and overcome the smoothness restriction coming from the classical piecewise linear Faber–Schauder system. This basis is unconditional and coefficient functionals are computed from discrete function values similar as for the Faber–Schauder situation.
- Published
- 2019
- Full Text
- View/download PDF
41. Dynamics of the Almost Periodic Discrete Mackey–Glass Model
- Author
-
Jehad Alzabut, Debaldev Jana, and Zhijian Yao
- Subjects
Class (set theory) ,exponential convergence ,Exponential convergence ,lcsh:Mathematics ,General Mathematics ,010102 general mathematics ,Dynamics (mechanics) ,almost periodic solution ,010103 numerical & computational mathematics ,lcsh:QA1-939 ,01 natural sciences ,contraction mapping principle ,Mackey–Glass model ,Lyapunov functional ,Computer Science (miscellaneous) ,Applied mathematics ,Contraction mapping ,0101 mathematics ,Engineering (miscellaneous) ,Mathematics - Abstract
This paper is concerned with a class of the discrete Mackey–Glass model that describes the process of the production of blood cells. Prior to proceeding to the main results, we prove the boundedness and extinction of its solutions. By means of the contraction mapping principle and under appropriate assumptions, we prove the existence of almost periodic positive solutions. Furthermore and by the implementation of the discrete Lyapunov functional, sufficient conditions are established for the exponential convergence of the almost periodic positive solution. Examples, as well as numerical simulations are illustrated to demonstrate the effectiveness of the theoretical findings of the paper. Our results are new and generalize some previously-reported results in the literature.
- Published
- 2018
- Full Text
- View/download PDF
42. Stability of Fredholm properties on interpolation Banach spaces
- Author
-
Mieczysław Mastyło, Natan Kruglyak, and Irina Asekritova
- Subjects
Mathematics::Functional Analysis ,Numerical Analysis ,Pure mathematics ,Functor ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Banach space ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Surjective function ,Operator (computer programming) ,Interpolation space ,0101 mathematics ,Equivalence (measure theory) ,Analysis ,Interpolation ,Mathematics - Abstract
The main aim of this paper is to prove novel results on stability of the semi-Fredholm property of operators on interpolation spaces generated by interpolation functors. The methods are based on some general ideas we develop in the paper. This allows us to extend some previous work in literature to the abstract setting. We show an application to interpolation methods introduced by Cwikel–Kalton–Milman–Rochberg which includes, as special cases, the real and complex methods up to equivalence of norms and also some other well known methods of interpolation. A by-product of these results get the stability of isomorphisms on Calderon products of Banach function lattices. We also study the important characteristics in operator Banach space theory, the so-called modules of injection and surjection, and we prove interpolation estimates of these modules of operators on scales of the Calderon complex interpolation spaces.
- Published
- 2020
43. Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals
- Author
-
Peter Kritzer, Henryk Woźniakowski, and Friedrich Pillichshammer
- Subjects
Statistics and Probability ,Discrete mathematics ,Numerical Analysis ,Polynomial ,Control and Optimization ,Algebra and Number Theory ,Logarithm ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,010103 numerical & computational mathematics ,01 natural sciences ,Exponential polynomial ,Exponential function ,Singular value ,symbols.namesake ,Tensor product ,Bounded function ,symbols ,0101 mathematics ,Mathematics - Abstract
We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an e -approximation for the d -variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function that depends polynomially on d and logarithmically on e − 1 . The corresponding unweighted problem was studied in Hickernell et al. (2020) with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on d and logarithmic dependence on e − 1 , we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential ( s , t ) -weak tractability with max ( s , t ) ≥ 1 . For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential ( s , t ) -weak tractability with max ( s , t ) 1 is left for future study. The paper uses some general results obtained in Hickernell et al. (2020) and Kritzer and Woźniakowski (2019).
- Published
- 2020
44. Projection of Rational Lie Rings
- Author
-
A. Lashkhi
- Subjects
Statistics and Probability ,Pure mathematics ,Mathematics::Commutative Algebra ,Isomorphism extension theorem ,Applied Mathematics ,General Mathematics ,Simple Lie group ,010102 general mathematics ,Adjoint representation ,010103 numerical & computational mathematics ,01 natural sciences ,Graded Lie algebra ,Algebra ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,Lie algebra ,Fundamental representation ,0101 mathematics ,Mathematics - Abstract
This paper is a direct continuation of [26], where we proved that every normal lattice isomorphism of supersolvable Lie ring is induced at the isomorphism. In the present paper,we generalize this theorem for rational Lie rings.
- Published
- 2016
45. Persistence and Global Attractivity for a Discretized Version of a General Model of Glucose-Insulin Interaction
- Author
-
Dinh Cong Huong
- Subjects
Persistence (psychology) ,delay difference equations ,Discretization ,General Mathematics ,Insulin ,medicine.medical_treatment ,full time solution ,lcsh:Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,lcsh:QA1-939 ,01 natural sciences ,non-standard difference ,medicine ,numerical discretized model ,Applied mathematics ,!-limit set of a persistent solution ,0101 mathematics ,Mathematics - Abstract
In this paper, we construct a non-standard finite difference scheme for a general model of glucose-insulin interaction. We establish some new sufficient conditions to ensure that the discretized model preserves the persistence and global attractivity of the continuous model. One of the main findings in this paper is that we derive two important propositions (Proposition 3.1 and Proposition 3.2) which are used to prove the global attractivity of the discretized model. Furthermore, when investigating the persistence and, in some cases, the global attractivity of the discretized model, the nonlinear functions f and h are not required to be differentiable. Hence, our results are more realistic because the statistical data of glucose and insulin are collected and reported in discrete time. We also present some numerical examples and their simulations to illustrate our results.
- Published
- 2016
46. ON SEMIDERIVATIONS IN 3-PRIME NEAR-RINGS
- Author
-
Mohammad Ashraf and Abdelkarim Boua
- Subjects
Pure mathematics ,Near-ring ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Multiplicative function ,Zero (complex analysis) ,Center (category theory) ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Combinatorics ,Product (mathematics) ,Domain (ring theory) ,0101 mathematics ,Commutative property ,Mathematics - Abstract
In the present paper, we expand the domain of work on theconcept of semiderivations in 3-prime near-rings through the study ofstructure and commutativity of near-rings admitting semiderivations sat-isfying certain differential identities. Moreover, several examples havebeen provided at places which show that the assumptions in the hypothe-ses of various theorems are not altogether superfluous. 1. IntroductionThroughout this paper, N is a zero-symmetric left near ring. A near ringN is called zero symmetric if 0x= 0 for all x∈ N (recall that in a left nearring x0 = 0 for all x∈ N). N is called 3-prime if xNy = {0} implies x= 0or y = 0. The symbol Z(N) will represent the multiplicative center of N,that is, Z(N) = {x∈ N | xy= yxfor all y∈ N}.For any x,y∈ N; as usual[x,y] = xy−yxand x◦y= xy+yxwill denote the well-known Lie product andJordan product, respectively. Recall that N is called 2-torsion free if 2x= 0implies x= 0 for all x∈ N. For terminologies concerning near-rings we referto G. Pilz [7].An additive mapping d: N → N is said to be a derivation if d(xy) = xd(y)+d(x)yforall x,y∈ N, orequivalently, asnotedin [8], that d(xy) = d(x)y+xd(y)for all x,y ∈ N. An additive mapping d: N → N is called semiderivation ifthere exists a function g : N → N such that d(xy) = xd(y) + d(x)g(y) =g(x)d(y)+d(x)yand d(g(x)) = g(d(x)) for all x,y∈ N.Obviously, any deriva-tion is a semiderivation, but the converse is not true in general (see [6]). Therehas been a greatdeal of workconcerning derivations in near-rings(see [1, 2, 4, 5]where further references can be found). In this paper, we study the commuta-tivity of addition and multiplication of near-rings. Two well-known results forderivations in near-rings have been generalized for semiderivation. In fact, ourresults generalize some theorems obtained by the authors together with Rajiin [1].
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- 2016
47. Lower Bounds for Haar Projections: Deterministic Examples
- Author
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Tino Ullrich and Andreas Seeger
- Subjects
Mathematics::Functional Analysis ,Basis (linear algebra) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Classical Analysis and ODEs ,Probabilistic logic ,Haar ,010103 numerical & computational mathematics ,Triebel–Lizorkin space ,01 natural sciences ,Sobolev space ,Computational Mathematics ,Range (mathematics) ,Mathematics - Classical Analysis and ODEs ,Simple (abstract algebra) ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,46E35, 46B15, 42C40 ,Besov space ,Applied mathematics ,0101 mathematics ,Analysis ,Mathematics - Abstract
In a previous paper by the authors, the existence of Haar projections with growing norms in Sobolev–Triebel–Lizorkin spaces has been shown via a probabilistic argument. This existence was sufficient to determine the precise range of Triebel–Lizorkin spaces for which the Haar system is an unconditional basis. The aim of the present paper is to give simple deterministic examples of Haar projections that show this growth behavior in the respective range of parameters.
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- 2016
48. A note on tractability of multivariate analytic problems
- Author
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Yongping Liu and Guiqiao Xu
- Subjects
Statistics and Probability ,Discrete mathematics ,Numerical Analysis ,Multivariate statistics ,Matching (statistics) ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Algebra ,Random variate ,Linear problem ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we study d -variate approximation for weighted Korobov spaces in the worst-case setting. The considered algorithms use finitely many evaluations of arbitrary linear functionals. We give matching necessary and sufficient conditions for some notions of tractability in terms of two weight parameters. Our result is an affirmative answer to a problem which is left open in a recent paper of Kritzer, Pillichshammer and Wo?niakowski.
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- 2016
49. Fermat curves and a refinement of the reciprocity law on cyclotomic units
- Author
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Tomokazu Kashio
- Subjects
Fermat's Last Theorem ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Reciprocity law ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
We define a “period-ring-valued beta function” and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark’s conjecture implies that the exponentials of the derivatives at s = 0 s=0 of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler’s formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative proof of a weaker result by using the reciprocity law on the period-ring-valued beta function. In other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units.
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- 2016
50. A convergence analysis of SOR iterative methods for linear systems with weak H-matrices
- Author
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Zichen Xue, Shuanghua Luo, and Cheng-yi Zhang
- Subjects
diagonally equipotent matrices ,15a18 ,Iterative method ,General Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,010103 numerical & computational mathematics ,01 natural sciences ,weak h-matrices ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Convergence (routing) ,diagonally dominant matrices ,QA1-939 ,Applied mathematics ,0101 mathematics ,Geometry and topology ,Mathematics ,15a06 ,convergence ,Computer Science::Information Retrieval ,010102 general mathematics ,Linear system ,15a42 ,Relaxation (iterative method) ,nonstricly diagonally dominant matrices ,sor iterative methods - Abstract
It is well known that SOR iterative methods are convergent for linear systems, whose coefficient matrices are strictly or irreducibly diagonally dominant matrices and strong H-matrices (whose comparison matrices are nonsingular M-matrices). However, the same can not be true in case of those iterative methods for linear systems with weak H-matrices (whose comparison matrices are singular M-matrices). This paper proposes some necessary and sufficient conditions such that SOR iterative methods are convergent for linear systems with weak H-matrices. Furthermore, some numerical examples are given to demonstrate the convergence results obtained in this paper.
- Published
- 2016
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