1,034 results
Search Results
2. A Note on the Paper 'The Algebraic Structure of the Arbitrary-Order Cone'
- Author
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Yen Chi Roger Lin, Xin-He Miao, and Jein Shan Chen
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Pure mathematics ,021103 operations research ,Control and Optimization ,Algebraic structure ,Applied Mathematics ,0211 other engineering and technologies ,Structure (category theory) ,Order (ring theory) ,010103 numerical & computational mathematics ,02 engineering and technology ,Management Science and Operations Research ,01 natural sciences ,Cone (formal languages) ,Combinatorics ,Operator (computer programming) ,Product (mathematics) ,Light cone ,0101 mathematics ,Mathematics ,Counterexample - Abstract
In this short paper, we look into a conclusion drawn by Alzalg (J Optim Theory Appl 169:32---49, 2016). We think the conclusion drawn in the paper is incorrect by pointing out three things. First, we provide a counterexample that the proposed inner product does not satisfy bilinearity. Secondly, we offer an argument why a pth-order cone cannot be self-dual under any reasonable inner product structure on $$\mathbb {R}^n$$Rn. Thirdly, even under the assumption that all elements operator commute, the inner product becomes an official inner product and the arbitrary-order cone can be shown as a symmetric cone, we think this condition is still unreasonable and very stringent so that the result can only be applied to very few cases.
- Published
- 2017
3. Computing isogenies between Jacobians of curves of genus 2 and 3
- Author
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Enea Milio
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Pure mathematics ,Algebra and Number Theory ,Mathematics::Number Theory ,Applied Mathematics ,Isotropy ,Prime number ,Theta function ,010103 numerical & computational mathematics ,Paper computing ,Mathematics::Geometric Topology ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Mathematics::Algebraic Geometry ,Genus (mathematics) ,0101 mathematics ,Algebraic number ,Quotient ,Mathematics - Abstract
We present a quasi-linear algorithm to compute isogenies between Jacobians of curves of genus 2 and 3 starting from the equation of the curve and a maximal isotropic subgroup of the l-torsion, for l an odd prime number, generalizing the Velu's formula of genus 1. This work is based from the paper Computing functions on Jacobians and their quotients of Jean-Marc Couveignes and Tony Ezome. We improve their genus 2 case algorithm, generalize it for genus 3 hyperelliptic curves and introduce a way to deal with the genus 3 non-hyperelliptic case, using algebraic theta functions.
- Published
- 2019
4. Shape preserving rational cubic trigonometric fractal interpolation functions
- Author
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Mohammad Sajid, K. R. Tyada, and A. K. B. Chand
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Numerical Analysis ,Pure mathematics ,General Computer Science ,Dense set ,Applied Mathematics ,Basis function ,010103 numerical & computational mathematics ,02 engineering and technology ,Trigonometric polynomial ,01 natural sciences ,Theoretical Computer Science ,Spline (mathematics) ,Fractal ,Modeling and Simulation ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,0101 mathematics ,Trigonometry ,Spline interpolation ,Mathematics ,Interpolation - Abstract
This paper is devoted to a hierarchical approach of constructing a class of fractal interpolants with trigonometric basis functions and to preserve the geometric behavior of given univariate data set by these fractal interpolants. In this paper, we propose a new family of C 1 -rational cubic trigonometric fractal interpolation functions (RCTFIFs) that are the generalized fractal version of the classical rational cubic trigonometric polynomial spline of the form p i ( θ ) / q i ( θ ) , where p i ( θ ) and q i ( θ ) are cubic trigonometric polynomials with four shape parameters in each sub-interval. The convergence of the RCTFIF towards the original function in C 3 is studied. We deduce the simple data dependent sufficient conditions on the scaling factors and shape parameters associated with the C 1 -RCTFIF so that the proposed RCTFIF lies above a straight line when the interpolation data set is constrained by the same condition. The first derivative of the proposed RCTFIF is irregular in a finite or dense subset of the interpolation interval and matches with the first derivative of the classical rational trigonometric cubic interpolation function whenever all scaling factors are zero. The positive shape preservation is a particular case of the constrained interpolation. We derive sufficient conditions on the trigonometric IFS parameters so that the proposed RCTFIF preserves the monotone or comonotone feature of prescribed data.
- Published
- 2021
5. On a new class of functional equations satisfied by polynomial functions
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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
6. Carr–Nadtochiy’s weak reflection principle for Markov chains on $$\mathbf {Z}^d$$
- Author
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Yuri Imamura
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Pure mathematics ,Carr ,Markov chain ,Applied Mathematics ,Mathematical finance ,General Engineering ,Hitting time ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Joint probability distribution ,Lattice (order) ,0101 mathematics ,Algebraic number ,Brownian motion ,Mathematics - Abstract
The reflection principle for Brownian motion gives a way to calculate the joint distribution of a hitting time and a one dimensional marginal. The Carr–Nadtochiy transform is a formulation that generalizes the reflection principle in this respect. The transform originated from a way to hedge so-called barrier options in the literature of financial mathematics. The existence of the transform has been established only for one dimensional diffusion processes. In the present paper, the existence is proved for a fairly general class of Markov chains in the multi dimensional lattice $$\mathbf {Z}^d$$ . The difficulty is that the reflection boundary is not a one-point set, contrasting the one dimensional cases. It is solved in this paper by looking at the problem in an algebraic way.
- Published
- 2020
7. Quasi-tight framelets with high vanishing moments derived from arbitrary refinable functions
- Author
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Chenzhe Diao and Bin Han
- Subjects
Pure mathematics ,Hilbert's syzygy theorem ,Applied Mathematics ,Refinable function ,010102 general mathematics ,Explained sum of squares ,010103 numerical & computational mathematics ,Vanishing moments ,16. Peace & justice ,Filter bank ,01 natural sciences ,Factorization ,Real algebraic geometry ,Sum rule in quantum mechanics ,0101 mathematics ,Mathematics - Abstract
Construction of multivariate tight framelets is known to be a challenging problem because it is linked to the difficult problem on sum of squares of multivariate polynomials in real algebraic geometry. Multivariate dual framelets with vanishing moments generalize tight framelets and are not easy to be constructed either, since their construction is related to syzygy modules and factorization of multivariate polynomials. On the other hand, compactly supported multivariate framelets with directionality or high vanishing moments are of interest and importance in both theory and applications. In this paper we introduce the notion of a quasi-tight framelet, which is a dual framelet, but behaves almost like a tight framelet. Let ϕ ∈ L 2 ( R d ) be an arbitrary compactly supported real-valued M -refinable function with a general dilation matrix M and ϕ ˆ ( 0 ) = 1 such that its underlying real-valued low-pass filter satisfies the basic sum rule. We first constructively prove by a step-by-step algorithm that we can always easily derive from the arbitrary M -refinable function ϕ a directional compactly supported real-valued quasi-tight M -framelet in L 2 ( R d ) associated with a directional quasi-tight M -framelet filter bank, each of whose high-pass filters has one vanishing moment and only two nonzero coefficients. If in addition all the coefficients of its low-pass filter are nonnegative, then such a quasi-tight M -framelet becomes a directional tight M -framelet in L 2 ( R d ) . Furthermore, we show by a constructive algorithm that we can always derive from the arbitrary M -refinable function ϕ a compactly supported quasi-tight M -framelet in L 2 ( R d ) with the highest possible order of vanishing moments. We shall also present a result on quasi-tight framelets whose associated high-pass filters are purely differencing filters with the highest order of vanishing moments. Several examples will be provided to illustrate our main theoretical results and algorithms in this paper.
- Published
- 2020
8. Further generalizations on some hardy type RL-integral inequalities
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Amina Khameli, Zoubir Dahmani, and Karima Freha
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Pure mathematics ,Mathematics::Complex Variables ,Applied Mathematics ,010103 numerical & computational mathematics ,02 engineering and technology ,Mathematics::Spectral Theory ,Type (model theory) ,01 natural sciences ,symbols.namesake ,Riemann–Liouville integral ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,020201 artificial intelligence & image processing ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this paper, we use the Riemnn-Liouville fractional integrals to establish new results related to Hardy inequalities. For our results, some result of the paper [ A.Khameli et al : New Riemann-Lio...
- Published
- 2020
9. Generalized matrix spectral factorization and quasi-tight framelets with a minimum number of generators
- Author
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Bin Han and Chenzhe Diao
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010101 applied mathematics ,Computational Mathematics ,Pure mathematics ,Matrix (mathematics) ,Algebra and Number Theory ,Applied Mathematics ,010103 numerical & computational mathematics ,Vanishing moments ,Spectral theorem ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
As a generalization of orthonormal wavelets in L 2 ( R ) L_2({\mathbb {R}}) , tightframelets (also called tight wavelet frames) are of importance in wavelet analysis and applied sciences due to their many desirable properties in applications such as image processing and numerical algorithms. Tight framelets are often derived from particular refinable functions satisfying certain stringent conditions. Consequently, a large family of refinable functions cannot be used to construct tight framelets. This motivates us to introduce the notion of a quasi-tight framelet, which is a dual framelet but behaves almost like a tight framelet. It turns out that the study of quasi-tight framelets is intrinsically linked to the problem of the generalized matrix spectral factorization for matrices of Laurent polynomials. In this paper, we provide a systematic investigation on the generalized matrix spectral factorization problem and compactly supported quasi-tight framelets. As an application of our results on generalized matrix spectral factorization for matrices of Laurent polynomials, we prove in this paper that from any arbitrary compactly supported refinable function in L 2 ( R ) L_2({\mathbb {R}}) , we can always construct a compactly supported one-dimensional quasi-tight framelet having the minimum number of generators and the highest possible order of vanishing moments. Our proofs are constructive and supplemented by step-by-step algorithms. Several examples of quasi-tight framelets will be provided to illustrate the theoretical results and algorithms developed in this paper.
- Published
- 2020
10. Simulation of elliptic and hypo-elliptic conditional diffusions
- Author
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Joris Bierkens, Frank van der Meulen, Moritz Schauer, and Mathematics
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FOS: Computer and information sciences ,Statistics and Probability ,Pure mathematics ,guided proposal ,Monte Carlo method ,Identity matrix ,010103 numerical & computational mathematics ,twice-integrated diffusion ,Statistics - Computation ,01 natural sciences ,partially observed diffusion ,010104 statistics & probability ,Matrix (mathematics) ,FitzHugh-Nagumo model ,FOS: Mathematics ,60J60 (Primary) 65C30, 65C05 (Secondary) ,FitzHugh–Nagumo model ,0101 mathematics ,Diffusion (business) ,Linear combination ,Computation (stat.CO) ,Mathematics ,Applied Mathematics ,Probability (math.PR) ,State (functional analysis) ,Diffusion process ,Langevin sampler ,Diffusion bridge ,Mathematics - Probability - Abstract
Suppose X is a multidimensional diffusion process. Assume that at time zero the state of X is fully observed, but at time $T>0$ only linear combinations of its components are observed. That is, one only observes the vector $L X_T$ for a given matrix L. In this paper we show how samples from the conditioned process can be generated. The main contribution of this paper is to prove that guided proposals, introduced in [35], can be used in a unified way for both uniformly elliptic and hypo-elliptic diffusions, even when L is not the identity matrix. This is illustrated by excellent performance in two challenging cases: a partially observed twice-integrated diffusion with multiple wells and the partially observed FitzHugh–Nagumo model.
- Published
- 2020
11. Explicit Formula for Preimages of Relaxed One-Sided Lipschitz Mappings with Negative Lipschitz Constants: A Geometric Approach
- Author
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Boris S. Mordukhovich, Janosch Rieger, and Andrew S. Eberhard
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Pure 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 ,Characterization (mathematics) ,Lipschitz continuity ,01 natural sciences ,Image (mathematics) ,Metric (mathematics) ,0101 mathematics ,Variational analysis ,Variety (universal algebra) ,Extreme point ,Constant (mathematics) ,Mathematics - Abstract
This paper addresses Lipschitzian stability issues, that play an important role in both theoretical and numerical aspects of variational analysis, optimization, and their applications. We particularly concentrate on the so-called relaxed one-sided Lipschitz property of set-valued mappings with negative Lipschitz constants. This property has been much less investigated than more conventional Lipschitzian behavior, while being well recognized in a variety of applications. Recent work has revealed that set-valued mappings satisfying the relaxed one-sided Lipschitz condition with negative Lipschitz constant possess a localization property, that is stronger than uniform metric regularity, but does not imply strong metric regularity. The present paper complements this fact by providing a characterization, not only of one specific single point of a preimage, but of entire preimages of such mappings. Developing a geometric approach, we derive an explicit formula to calculate preimages of relaxed one-sided Lipschitz mappings between finite-dimensional spaces and obtain a further specification of this formula via extreme points of image sets.
- Published
- 2020
12. A family of measures of noncompactness in the Hölder space Cn,γ(R+) and its application to some fractional differential equations and numerical methods
- Author
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Hojjatollah Amiri Kayvanloo, Mahnaz Khanehgir, and Reza Allahyari
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Pure mathematics ,Applied Mathematics ,Numerical analysis ,Hölder condition ,Fixed-point theorem ,010103 numerical & computational mathematics ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Shaping ,Boundary value problem ,0101 mathematics ,Fractional differential ,Mathematics - Abstract
In this paper, we prove the existence of solutions for the following fractional boundary value problem c D α u ( t ) = f ( t , u ( t ) ) , α ∈ ( n , n + 1 ) , 0 ≤ t + ∞ , u ( 0 ) = 0 , u ′ ′ ( 0 ) = 0 , … , u ( n ) ( 0 ) = 0 , lim t → + ∞ c D α − 1 u ( t ) = β u ( ξ ) . The considerations of this paper are based on the concept of a new family of measures of noncompactness in the space of functions C n , γ ( R + ) satisfying the Holder condition and a fixed point theorem of Darbo type. We also provide an illustrative example in support of our existence theorems. Finally, to credibility, we apply successive approximation and homotopy perturbation method to find solution of the above problem with high accuracy.
- Published
- 2020
13. Uniqueness of meromorphic functions with their reduced linear c-shift operators sharing two or more values or sets
- Author
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Abhijit Banerjee and Saikat Bhattacharyya
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Pure mathematics ,Riemann sphere ,010103 numerical & computational mathematics ,Shared set ,01 natural sciences ,symbols.namesake ,Operator (computer programming) ,Meromorphic functions ,Uniqueness ,0101 mathematics ,Difference operator ,Meromorphic function ,Mathematics ,Algebra and Number Theory ,Functional analysis ,Applied Mathematics ,lcsh:Mathematics ,010102 general mathematics ,Function (mathematics) ,lcsh:QA1-939 ,Ordinary differential equation ,Content (measure theory) ,symbols ,Weighted sharing ,Analysis - Abstract
In the paper, we introduce a new notion of reduced linear c-shift operator $L _{c}^{r}\,f$Lcrf, and with the aid of this new operator, we study the uniqueness of meromorphic functions $f(z)$f(z) and $L_{c}^{r}\,f$Lcrf sharing two or more values in the extended complex plane. The results obtained in the paper significantly improve a number of existing results. Further, using the notion of weighted sharing of sets, we deal the same problem. We exhibit a handful number of examples to justify certain statements relevant to the content of the paper. We are also able to determine the form of the function that coincides with its reduced linear c-shift operator. At the end of the paper, we pose an open question for future research.
- Published
- 2019
14. Extensions of linear operators from hyperplanes and strong uniqueness of best approximation in L(X,W)
- Author
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Paweł Wójcik
- Subjects
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
15. Existence of meromorphic solutions of some generalized Fermat functional equations
- Author
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Linlin Wu, Weiran Lü, Chun He, and Feng Lü
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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
16. Some Properties of Fuzzy Supra Soft Topological Spaces
- Author
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Alaa Mohamed Abd El-latif
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Statistics and Probability ,Numerical Analysis ,Pure mathematics ,Algebra and Number Theory ,Mathematics::General Mathematics ,Generalization ,Astrophysics::High Energy Astrophysical Phenomena ,Applied Mathematics ,010103 numerical & computational mathematics ,02 engineering and technology ,Topological space ,01 natural sciences ,Fuzzy logic ,Theoretical Computer Science ,Compact space ,Physics::Plasma Physics ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Geometry and Topology ,0101 mathematics ,Counterexample ,Mathematics - Abstract
In this paper, we introduce the notion of fuzzy supra soft topological spaces, which is a generalization to the notion of fuzzy soft topological spaces and supra soft topological spaces. Also, we consider the notion of fuzzy supra soft continuity as a generalization to fuzzy soft continuity, supported by examples and counterexamples. These examples illustrating the notions used in the paper are included. So we can see that all these concepts are independent from each other ordoes implies the other. Finally, as a direct application to fuzzy supra soft topological spaces, we introduce the notion of fuzzy supra soft compact (resp. fuzzy supra soft lindel ̈of) spaces to such spaces as a generalization to fuzzy soft compactness. Furthermore, we establish some interesting properties of this notion.
- Published
- 2019
17. Iterative criteria for identifying strong H-tensors
- Author
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Changfeng Ma and Baohua Huang
- Subjects
Numerical linear algebra ,Pure mathematics ,Applied Mathematics ,010103 numerical & computational mathematics ,computer.software_genre ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Positive definiteness ,Symmetric tensor ,Tensor ,0101 mathematics ,computer ,Mathematics ,Diagonally dominant matrix - Abstract
Strong H -tensors play an important role in the theories and applications of numerical linear algebra. It is necessary to identify whether a given tensor is a strong H -tensor or not. In this paper, we establish some iterative criteria for identifying strong H -tensors. These criteria depend on the elements of the tensors; therefore, they are easy to be verified. The results obtained in this paper extend the corresponding conclusions for strictly generalized diagonally dominant matrices. As an application, some sufficient conditions for the positive definiteness of an even-order real symmetric tensor are presented. Some numerical experiments show the feasibility and efficiency of the results which are obtained in this paper.
- Published
- 2019
18. The representation of the generalized linear Euler sums with parameters
- Author
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Huizeng Qin and Aijuan Li
- Subjects
Pure mathematics ,Applied Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Special values ,01 natural sciences ,Riemann zeta function ,Hurwitz zeta function ,symbols.namesake ,Digamma function ,Special functions ,Euler's formula ,symbols ,0101 mathematics ,Analysis ,Mathematics - Abstract
In this paper, by using residue method, we obtain the representations of some basic linear generalized Euler sums with parameters. Based on the linear generalized Euler sums with parameters, some new Euler sums are obtained and expressed in the closed forms. When the parameters of new Euler sums take special values, we can get some usual expressions of Euler sums. Moreover, the integrals of many special functions can be expressed as the Euler sums given in this paper.
- Published
- 2019
19. Effective Dimension of Some Weighted Pre-Sobolev Spaces with Dominating Mixed Partial Derivatives
- Author
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Art B. Owen
- Subjects
Numerical Analysis ,Pure mathematics ,Applied Mathematics ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Effective dimension ,01 natural sciences ,Quadrature (mathematics) ,Sobolev space ,Computational Mathematics ,FOS: Mathematics ,Partial derivative ,Mathematics - Numerical Analysis ,Quasi-Monte Carlo method ,Ball (mathematics) ,0101 mathematics ,Mathematics - Abstract
This paper considers two notions of effective dimension for quadrature in weighted pre-Sobolev spaces with dominating mixed partial derivatives. We begin by finding a ball in those spaces just barely large enough to contain a function with unit variance. If no function in that ball has more than $\varepsilon$ of its variance from ANOVA components involving interactions of order $s$ or more, then the space has effective dimension at most $s$ in the superposition sense. A similar truncation sense notion replaces the cardinality of the ANOVA component by the largest index it contains. Some Poincar\'e type inequalities are used to bound variance components by multiples of these space's squared norm and those in turn provide bounds on effective dimension. Very low effective dimension in the superposition sense holds for some spaces defined by product weights in which quadrature is strongly tractable. The superposition dimension is $O( \log(1/\varepsilon)/\log(\log(1/\varepsilon)))$ just like the superposition dimension used in the multidimensional decomposition method. Surprisingly, even spaces where all subset weights are equal, regardless of their cardinality or included indices, have low superposition dimension in this sense. This paper does not require periodicity of the integrands.
- Published
- 2019
20. Geometric properties of F-normed Orlicz spaces
- Author
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Yunan Cui, Paweł Kolwicz, Radosław Kaczmarek, and Henryk Hudzik
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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
21. Cramer’s rule for a system of quaternion matrix equations with applications
- Author
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Shaowen Yu, Qing-Wen Wang, and Guang-Jing Song
- Subjects
Computational Mathematics ,Pure mathematics ,Applied Mathematics ,010102 general mathematics ,Quaternion matrix ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Hermitian matrix ,Mathematics ,Cramer's rule - Abstract
In this paper, we investigate Cramer’s rule for the general solution to the system of quaternion matrix equations A 1 X B 1 = C 1 , A 2 X B 2 = C 2 , and Cramer’s rule for the general solution to the generalized Sylvester quaternion matrix equation A X B + C Y D = E , respectively. As applications, we derive the determinantal expressions for the Hermitian solutions to some quaternion matrix equations. The findings of this paper extend some known results in the literature.
- Published
- 2018
22. The representation of Euler sums with parameters
- Author
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Aijuan Li and Huizeng Qin
- Subjects
Pure mathematics ,Applied Mathematics ,Elementary arithmetic ,010102 general mathematics ,010103 numerical & computational mathematics ,Limiting ,01 natural sciences ,Riemann zeta function ,symbols.namesake ,symbols ,Base function ,Euler's formula ,0101 mathematics ,Representation (mathematics) ,Analysis ,Mathematics - Abstract
In this paper, by choosing different kernel functions and base functions, we obtain some Euler sums with parameters. Moreover, we also obtain the new Euler sums with parameters by differentiating, limiting and elementary arithmetic. Thus, more Euler sums with parameters can be obtained. Furthermore, some Euler sums given in this paper are closed forms.
- Published
- 2018
23. 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
24. 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
25. Periodic representations for quadratic irrationalities in the field of p-adic numbers
- Author
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Umberto Cerruti, Nadir Murru, and Stefano Barbero
- Subjects
Pure mathematics ,Algebra and Number Theory ,Applied Mathematics ,Field (mathematics) ,continued fractions ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Browkin algorithm ,Quadratic equation ,p–adic numbers ,0101 mathematics ,Browkin algorithm, continued fractions, p–adic numbers, quadratic irrationals ,quadratic irrationals ,Mathematics - Abstract
Continued fractions have been widely studied in the field of p p -adic numbers Q p \mathbb Q_p , but currently there is no algorithm replicating all the good properties that continued fractions have over the real numbers regarding, in particular, finiteness and periodicity. In this paper, first we propose a periodic representation, which we will call standard, for any quadratic irrational via p p -adic continued fractions, even if it is not obtained by a specific algorithm. This periodic representation provides simultaneous rational approximations for a quadratic irrational both in R \mathbb R and Q p \mathbb Q_p . Moreover given two primes p 1 p_1 and p 2 p_2 , using the Binomial transform, we are also able to pass from approximations in Q p 1 \mathbb {Q}_{p_1} to approximations in Q p 2 \mathbb {Q}_{p_2} for a given quadratic irrational. Then, we focus on a specific p p –adic continued fraction algorithm proving that it stops in a finite number of steps when processes rational numbers, solving a problem left open in a paper by Browkin [Math. Comp. 70 (2001), pp. 1281–1292]. Finally, we study the periodicity of this algorithm showing when it produces standard representations for quadratic irrationals.
- Published
- 2021
26. Exponential integrators preserving first integrals or Lyapunov functions for conservative or dissipative systems
- Author
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Xinyuan Wu and Yuwen Li
- Subjects
Lyapunov function ,Pure mathematics ,65L04, 65L05, 65M20, 65P10, 65Z05 ,Applied Mathematics ,010103 numerical & computational mathematics ,Numerical Analysis (math.NA) ,Exponential integrator ,01 natural sciences ,Exponential function ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Matrix (mathematics) ,Scheme (mathematics) ,Dissipative system ,symbols ,FOS: Mathematics ,Nabla symbol ,Differentiable function ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics - Abstract
In this paper, combining the ideas of exponential integrators and discrete gradients, we propose and analyze a new structure-preserving exponential scheme for the conservative or dissipative system $\dot{y} = Q(M y + \nabla U (y))$, where $Q$ is a $d\times d$ skew-symmetric or negative semidefinite real matrix, $M$ is a $d\times d$ symmetric real matrix, and $U : \mathbb{R}^d\rightarrow\mathbb{R}$ is a differentiable function. We present two properties of the new scheme. The paper is accompanied by numerical results that demonstrate the remarkable superiority of our new scheme in comparison with other structure-preserving schemes in the scientific literature.
- Published
- 2020
27. Solutions to the linear transpose matrix equations and their application in control
- Author
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Caiqin Song and Wenli Wang
- Subjects
0209 industrial biotechnology ,Pure mathematics ,Applied Mathematics ,Linear system ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Computational Mathematics ,Matrix (mathematics) ,symbols.namesake ,020901 industrial engineering & automation ,Kronecker delta ,Transpose ,symbols ,Limit (mathematics) ,0101 mathematics ,Coefficient matrix ,Eigenvalues and eigenvectors ,Parametric statistics ,Mathematics - Abstract
In this paper, we study the solutions to the linear transpose matrix equations $$AX+X^{T}B=C$$ and $$AX+X^{T}B=CY$$ , which have many important applications in control theory. By applying Kronecker map and Sylvester sum, we obtain some necessary and sufficient conditions for existence of solutions and the expressions of explicit solutions for the Sylvester transpose matrix equation $$AX+X^{T}B=C$$ . Our conditions only need to check the eigenvalue of $$ B^{T}A^{-1}$$ , and, therefore, are simpler than those reported in the paper (Piao et al. in J Frankl Inst 344:1056–1062, 2007). The corresponding algorithms permit the coefficient matrix C to be any real matrix and remove the limit of $$C=C^{T}$$ in Piao et al. Moreover, we present the solvability and the expressions of parametric solutions for the generalized Sylvester transpose matrix equation $$AX+X^{T}B=CY$$ using an alternative approach. A numerical example is given to demonstrate that the introduced algorithm is much faster than the existing method in the paper (De Teran and Dopico in 434:44–67;2011). Finally, the continuous zeroing dynamics design of time-varying linear system is provided to show the effectiveness of our algorithm in control.
- Published
- 2020
28. Lifting modules with finite internal exchange property and direct sums of hollow modules
- Author
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Yosuke Kuratomi
- Subjects
Pure mathematics ,Algebra and Number Theory ,Property (philosophy) ,Computer Science::Information Retrieval ,Applied Mathematics ,010102 general mathematics ,Astrophysics::Instrumentation and Methods for Astrophysics ,Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) ,010103 numerical & computational mathematics ,01 natural sciences ,Dual (category theory) ,Perfect ring ,Decomposition (computer science) ,Computer Science::General Literature ,0101 mathematics ,Mathematics - Abstract
A module [Formula: see text] is said to be lifting if, for any submodule [Formula: see text] of [Formula: see text], there exists a decomposition [Formula: see text] such that [Formula: see text] and [Formula: see text] is a small submodule of [Formula: see text]. A lifting module is defined as a dual concept of the extending module. A module [Formula: see text] is said to have the finite internal exchange property if, for any direct summand [Formula: see text] of [Formula: see text] and any finite direct sum decomposition [Formula: see text], there exists a direct summand [Formula: see text] of [Formula: see text] [Formula: see text] such that [Formula: see text]. This paper is concerned with the following two fundamental unsolved problems of lifting modules: “Classify those rings all of whose lifting modules have the finite internal exchange property” and “When is a direct sum of indecomposable lifting modules lifting?”. In this paper, we prove that any [Formula: see text]-square-free lifting module over a right perfect ring satisfies the finite internal exchange property. In addition, we give some necessary and sufficient conditions for a direct sum of hollow modules over a right perfect ring to be lifting with the finite internal exchange property.
- Published
- 2020
29. Generalized fractional integral inequalities of Hermite–Hadamard type for harmonically convex functions
- Author
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Hüseyin Budak, Dafang Zhao, Artion Kashuri, Muhammad Ali, and [Belirlenecek]
- Subjects
Pure mathematics ,Algebra and Number Theory ,Partial differential equation ,Hermite polynomials ,Functional analysis ,lcsh:Mathematics ,Applied Mathematics ,Hermite-Hadamard inequalities ,010102 general mathematics ,Harmonically convex functions ,010103 numerical & computational mathematics ,Extension (predicate logic) ,Type (model theory) ,lcsh:QA1-939 ,01 natural sciences ,Hermite–Hadamard inequalities ,Hadamard transform ,Ordinary differential equation ,Generalized fractional integral ,0101 mathematics ,Convex function ,Analysis ,Mathematics - Abstract
WOS: 000522451500001 In this paper, we establish inequalities of Hermite-Hadamard type for harmonically convex functions using a generalized fractional integral. The results of our paper are an extension of previously obtained results (Iscan in Hacet. J. Math. Stat. 43(6):935-942, 2014 and Iscan and Wu in Appl. Math. Comput. 238:237-244, 2014). We also discuss some special cases for our main results and obtain new inequalities of Hermite-Hadamard type. Special Soft Science Research Projects of Technological Innovation in Hubei Province [2019ADC46]; Fundamental Research Funds for Central UniversitiesFundamental Research Funds for the Central Universities [2019B44914]; Key Projects of Education Commission of Hubie Province of China [D20192501]; Natural Science Foundation of Jiangsu ProvinceJiangsu Planned Projects for Postdoctoral Research FundsNatural Science Foundation of Jiangsu Province [BK20180500]; National Key Research and Development Program of China [2018YFC1508100]; National Natural Science Foundation of ChinaNational Natural Science Foundation of China [11971241] This work was supported in part by Special Soft Science Research Projects of Technological Innovation in Hubei Province (2019ADC46), the Fundamental Research Funds for Central Universities (2019B44914), Key Projects of Education Commission of Hubie Province of China (D20192501), the Natural Science Foundation of Jiangsu Province (BK20180500), the National Key Research and Development Program of China (2018YFC1508100) and partially supported by the National Natural Science Foundation of China (11971241).
- Published
- 2020
30. Alternative representations of the normal cone to the domain of supremum functions and subdifferential calculus
- Author
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Marco A. López, Rafael Correa, Abderrahim Hantoute, Universidad de Alicante. Departamento de Matemáticas, and Laboratorio de Optimización (LOPT)
- Subjects
Statistics and Probability ,Pure mathematics ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,Subderivative ,Type (model theory) ,01 natural sciences ,Domain (mathematical analysis) ,Estadística e Investigación Operativa ,FOS: Mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,Subdifferentials ,Optimality conditions ,Numerical Analysis ,46N10, 52A41, 90C25 ,021103 operations research ,Applied Mathematics ,Supremum of convex functions ,Function (mathematics) ,Infimum and supremum ,Convex optimization ,Effective domain ,Optimization and Control (math.OC) ,Geometry and Topology ,Convex function ,Normal cone ,Analysis - Abstract
The first part of the paper provides new characterizations of the normal cone to the effective domain of the supremum of an arbitrary family of convex functions. These results are applied in the second part to give new formulas for the subdifferential of the supremum function, which use both the active and nonactive functions at the reference point. Only the data functions are involved in these characterizations, the active ones from one side, together with the nonactive functions multiplied by some appropriate parameters. In contrast with previous works in the literature, the main feature of our subdifferential characterization is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of this domain) does not appear. A new type of optimality conditions for convex optimization is established at the end of the paper. Research supported by ANID (Fondecyt 1190012 and 1190110), Proyecto CMM ANID PIA AFB170001, MICIU of Spain and Universidad de Alicante (Contract Beatriz Galindo BEA-GAL 18/00205), and Research Project PGC2018-097960-B-C21 from MICINN, Spain. The research of the third author is also supported by the Australian ARC -Discovery Projects DP 180100602.
- Published
- 2020
- Full Text
- View/download PDF
31. 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
32. 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
33. Approximation of Functions Belonging to Generalized Hölder’s Class $$H_{\alpha }^{(\omega )}[0,1)$$ by First Kind Chebyshev Wavelets and Its Applications in the Solution of Linear and Nonlinear Differential Equations
- Author
-
Indra Bhan and Shyam Lal
- Subjects
Pure mathematics ,Differential equation ,Applied Mathematics ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Chebyshev filter ,010101 applied mathematics ,Computational Mathematics ,Wavelet ,Exact solutions in general relativity ,Collocation method ,Initial value problem ,0101 mathematics ,Galerkin method ,Mathematics - Abstract
In this paper, two new estimators $$E^{(1)}_{2^{k-1},0}$$ and $$ E^{(2)}_{2^{k-1},M}$$ of a function f belonging to generalized Holder’s class $$H_{\alpha }^{(\omega )}[0,1)$$ by first kind Chebyshev wavelets have been determined. These estimators are new and best possible approximations in wavelet analysis. Applying this algorithm, a solution of linear and non-linear second order differential equations have been obtained. These solutions are approximately the same as the exact solution of the differential equation. Two non-linear initial value problems are solved by Chebyshev wavelet method when Collocation method, Galerkin’s method and other known methods do not work. This is a significant achievement of this paper in wavelet analysis.
- Published
- 2019
34. Taylor spectrum approach to Brownian-type operators with quasinormal entry
- Author
-
Jan Stochel, Sameer Chavan, Zenon Jan Jabłoński, and Il Bong Jung
- Subjects
subnormal operator ,Pure mathematics ,Applied Mathematics ,010102 general mathematics ,Spectrum (functional analysis) ,Primary 47B20, 47B37 Secondary 44A60 ,moment problems ,Triangular matrix ,Cauchy distribution ,010103 numerical & computational mathematics ,Taylor’s spectrum ,Characterization (mathematics) ,Type (model theory) ,01 natural sciences ,m-isometry ,Functional Analysis (math.FA) ,Linear map ,Mathematics - Functional Analysis ,upper triangular 2×2 block matrix ,Operator (computer programming) ,FOS: Mathematics ,0101 mathematics ,Algebraic number ,linear operator pencil ,Mathematics - Abstract
In this paper, we introduce operators that are represented by upper triangular $2\times 2$ block matrices whose entries satisfy some algebraic constraints. We call them Brownian-type operators of class $\mathcal Q,$ briefly operators of class $\mathcal Q.$ These operators emerged from the study of Brownian isometries performed by Agler and Stankus via detailed analysis of the time shift operator of the modified Brownian motion process. It turns out that the class $\mathcal Q$ is closely related to the Cauchy dual subnormality problem which asks whether the Cauchy dual of a completely hyperexpansive operator is subnormal. Since the class $\mathcal Q$ is closed under the operation of taking the Cauchy dual, the problem itself becomes a part of a more general question of investigating subnormality in this class. This issue, along with the analysis of nonstandard moment problems, covers a large part of the paper. Using the Taylor spectrum technique culminates in a full characterization of subnormal operators of class $\mathcal Q.$ As a consequence, we solve the Cauchy dual subnormality problem for expansive operators of class $\mathcal Q$ in the affirmative, showing that the original problem can surprisingly be extended to a class of operators that are far from being completely hyperexpansive. The Taylor spectrum approach turns out to be fruitful enough to allow us to characterize other classes of operators including $m$-isometries. We also study linear operator pencils associated with operators of class $\mathcal Q$ proving that the corresponding regions of subnormality are closed intervals with explicitly described endpoints., 4 figures
- Published
- 2019
35. 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
36. Continued fraction expansions for the Lambert $$\varvec{W}$$ W function
- Author
-
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
37. Generalized Pseudospectral Method and Zeros of Orthogonal Polynomials
- Author
-
Oksana Bihun and Clark Mourning
- Subjects
Pure mathematics ,Polynomial ,Hermite polynomials ,Article Subject ,Degree (graph theory) ,Physics ,QC1-999 ,Applied Mathematics ,010102 general mathematics ,General Physics and Astronomy ,010103 numerical & computational mathematics ,Differential operator ,01 natural sciences ,Matrix (mathematics) ,33C45, 33C47, 26C10, 65L60 ,Mathematics - Classical Analysis and ODEs ,Orthogonal polynomials ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,0101 mathematics ,Algebraic number ,Real line ,Mathematics - Abstract
Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization is based on a modification of pseudospectral matrix representations of linear differential operators proposed in the paper, which allows these representations to depend on two, rather than one, sets of interpolation nodes. The identities hold for every polynomial family $\{p_\nu(x)\}_{\nu=0}^\infty$ orthogonal with respect to a measure supported on the real line that satisfies some standard assumptions, as long as the polynomials in the family satisfy differential equations $\mathcal{A} p_\nu(x) =q_\nu(x) p_\nu(x)$, where $\mathcal{A}$ is a linear differential operator and each $q_\nu(x)$ is a polynomial of degree at most $n_0 \in \mathbb{N}$; $n_0$ does not depend on $\nu$. The proposed identities generalize known identities for classical and Krall orthogonal polynomials, to the case of the nonclassical orthogonal polynomials that belong to the class described above. The generalized pseudospectral representations of the differential operator $\mathcal{A}$ for the case of the Sonin-Markov orthogonal polynomials, also known as generalized Hermite polynomials, are presented. The general result is illustrated by new algebraic relations satisfied by the zeros of the Sonin-Markov polynomials., Comment: This version contains minor improvements to the exposition to match the published version of the paper
- Published
- 2018
38. Weighted G-Drazin inverses and a new pre-order on rectangular matrices
- Author
-
Marina Lattanzi, Néstor Thome, and Carmen Coll
- Subjects
G-Drazin inverse ,Pure mathematics ,G-Drazin partial order ,Applied Mathematics ,Mathematics::Rings and Algebras ,020206 networking & telecommunications ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Square matrix ,Minus partial order ,Weighted Drazin pre-order ,Algebra ,Computational Mathematics ,Weighted Drazin inverse ,0202 electrical engineering, electronic engineering, information engineering ,Order (group theory) ,0101 mathematics ,MATEMATICA APLICADA ,Mathematics - Abstract
[EN] This paper deals with weighted G-Drazin inverses, which is a new class of matrices introduced to extend (to the rectangular case) G-Drazin inverses recently considered by Wang and Liu for square matrices. First, we define and characterize weighted G-Drazin inverses. Next, we consider a new pre-order defined on complex rectangular matrices based on weighted G-Drazin inverses. Finally, we characterize this pre-order and relate it to the minus partial order and to the weighted Drazin pre-order. (C) 2017 Elsevier Inc. All rights reserved., This paper was partially supported by Universidad Nacional de La Pampa, Facultad de Ingenieria, grant resol. no. 155/14. The first and third authors were partially supported by Ministerio de Economia y Competitividad of Spain (grant no. DGI MTM2013-43678-P) and the third author was also partially supported by Ministerio de Economia y Competitividad of Spain (Red de Excelencia MTM2015-68805-REDT).
- Published
- 2018
39. Extended Semismooth Newton Method for Functions with Values in a Cone
- Author
-
Séverine Bernard, Alain Pietrus, Catherine Cabuzel, Silvère Paul Nuiro, and Bernard, Séverine
- Subjects
Pure mathematics ,021103 operations research ,Generalized Jacobian ,Partial differential equation ,Applied Mathematics ,0211 other engineering and technologies ,[MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM] ,Context (language use) ,010103 numerical & computational mathematics ,02 engineering and technology ,Function (mathematics) ,01 natural sciences ,symbols.namesake ,Cone (topology) ,Convergence (routing) ,symbols ,Convex cone ,0101 mathematics ,Newton's method ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
This paper deals with variational inclusions of the form $0 \in K-f(x)$ where $f : \mathbb{R}^{n} \rightarrow \mathbb{R} ^{m}$ is a semismooth function and $K$ is a nonempty closed convex cone in $\mathbb{R}^{m}$ . We show that the previous problem can be solved by a Newton-type method using the Clarke generalized Jacobian of $f$ . The results obtained in this paper extend those obtained by Robinson in the famous paper (Robinson in Numer. Math. 19:341–347, 1972). We provide a semilocal method with a superlinear convergence that is new in the context of semismooth functions. Finally, numerical results are also given to illustrate the convergence.
- Published
- 2017
40. On Hermitian Solutions of the Split Quaternion Matrix Equation $$AXB+CXD=E$$ A X B + C X D = E
- Author
-
Yong Tian, Shi-Fang Yuan, Yi-Bin Yu, and Qing-Wen Wang
- Subjects
Pure mathematics ,Applied Mathematics ,010102 general mathematics ,Mathematics::Differential Geometry ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Hermitian matrix ,Split-quaternion ,Mathematics - Abstract
In this paper, we discuss Hermitian solutions of split quaternion matrix equation $$AXB+CXD=E,$$ where X is an unknown split quaternion Hermitian matrix, and A, B, C, D, E are known split quaternion matrices with suitable size. The objective of this paper is to establish a necessary and sufficient condition for the existence of a solution and a solution formulas. Moreover, we provide numerical algorithms and numerical examples to exemplify the results.
- Published
- 2017
41. On rational functions without Froissart doublets
- Author
-
Ana C. Matos, Bernhard Beckermann, and George Labahn
- Subjects
Pure mathematics ,Applied Mathematics ,Numerical analysis ,010102 general mathematics ,Zero (complex analysis) ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Derivative ,Rational function ,01 natural sciences ,Computational Mathematics ,Matrix (mathematics) ,41A21, 65F22 ,Euclidean geometry ,FOS: Mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Condition number ,Mathematics - Abstract
In this paper we consider the problem of working with rational functions in a numeric environment. A particular problem when modeling with such functions is the existence of Froissart doublets, where a zero is close to a pole. We discuss three different parameters which allow one to monitor the absence of Froissart doublets for a given general rational function. These include the euclidean condition number of an underlying Sylvester-type matrix, a parameter for determing coprimeness of two numerical polynomials and bounds on the spherical derivative. We show that our parameters sharpen those found in a previous paper by two of the autours., 18 pages, 1 figure
- Published
- 2017
42. Minimizers for nonconvex variational problems in the plane via convex/concave rearrangements
- Author
-
Dean A. Carlson
- Subjects
Convex hull ,Convex analysis ,Pure mathematics ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Convex set ,Proper convex function ,010103 numerical & computational mathematics ,Subderivative ,01 natural sciences ,Convex polytope ,Convex combination ,0101 mathematics ,Absolutely convex set ,Analysis ,Mathematics - Abstract
Recently, A. Greco utilized convex rearrangements to present some new and interesting existence results for noncoercive functionals in the calculus of variations. Moreover, the integrands were not necessarily convex. In particular, using convex rearrangements permitted him to establish the existence of convex minimizers essentially considering the uniform convergence of the minimizing sequence of trajectories and the pointwise convergence of their derivatives. The desired lower semicontinuity property is now a consequence of Fatou's lemma. In this paper we point out that such an approach was considered in the late 1930's in a series of papers by E.J. McShane for problems satisfying the usual coercivity condition. In addition, we will update some hypotheses that McShane made by making use of a result due to T.S. Angell, concerning property (D) on the avoidance of the Lavrentiev phenomenon.
- Published
- 2017
43. Boundedness of operators arising from Schwarz BVP in modified local Morrey-type spaces
- Author
-
Tuğçe Ünver, Vagif S. Guliyev, Kerim Koca, R. Ch. Mustafayev, Kırşehir Ahi Evran Üniversitesi, Fen-Edebiyat Fakültesi, Matematik Bölümü, and Kırıkkale Üniversitesi
- Subjects
Mathematics::Functional Analysis ,Numerical Analysis ,Pure mathematics ,Class (set theory) ,Mathematics::Complex Variables ,Applied Mathematics ,Hardy inequalities ,010102 general mathematics ,Mathematical analysis ,Mathematics::Classical Analysis and ODEs ,010103 numerical & computational mathematics ,Type (model theory) ,Operator theory ,Sublinear operators ,01 natural sciences ,Computational Mathematics ,local Morrey-type spaces ,0101 mathematics ,Complex plane ,Unit (ring theory) ,Analysis ,Mathematics - Abstract
Unver Yildiz, Tugce/0000-0003-0414-8400; Yildiz, Tugce Unver/0000-0003-0414-8400; Mustafayev, Rza/0000-0002-2806-9646 WOS: 000406237200009 In this paper, we prove the boundedness of a class of operators arising from Schwarz BVP in modified local Morrey-type spaces in the unit disc of the complex plane. Ministry of Education and Science of the Russian FederationMinistry of Education and Science, Russian Federation [02. a03.21.0008] We thank the anonymous referee for his remarks, which have improved the final version of this paper. The research of V.S. Guliyev was partially supported by the Ministry of Education and Science of the Russian Federation (the Agreement number No: 02. a03.21.0008).
- Published
- 2017
44. Birkhoff–James orthogonality of linear operators on finite dimensional Banach spaces
- Author
-
Debmalya Sain
- Subjects
Pure mathematics ,Applied Mathematics ,Operator (physics) ,010102 general mathematics ,Linear operators ,Banach space ,Zero (complex analysis) ,010103 numerical & computational mathematics ,01 natural sciences ,Orthogonality ,0101 mathematics ,Symmetry (geometry) ,Analysis ,Mathematics - Abstract
In this paper we characterize Birkhoff–James orthogonality of linear operators defined on a finite dimensional real Banach space X . We also explore the left symmetry of Birkhoff–James orthogonality of linear operators defined on X . Using some of the related results proved in this paper, we finally prove that T ∈ L ( l p 2 ) ( p ≥ 2 , p ≠ ∞ ) is left symmetric with respect to Birkhoff–James orthogonality if and only if T is the zero operator.
- Published
- 2017
45. Algebraic hyperstructures associated to biological inheritance
- Author
-
M. Al Tahan and Bijan Davvaz
- Subjects
Statistics and Probability ,Pure mathematics ,Inheritance Patterns ,010103 numerical & computational mathematics ,0102 computer and information sciences ,Models, Biological ,01 natural sciences ,General Biochemistry, Genetics and Molecular Biology ,symbols.namesake ,Inheritance (object-oriented programming) ,Simple (abstract algebra) ,Hyperstructure ,0101 mathematics ,Connection (algebraic framework) ,Algebraic number ,Relation (history of concept) ,Mathematics ,General Immunology and Microbiology ,Applied Mathematics ,Epistasis, Genetic ,General Medicine ,Algebra ,010201 computation theory & mathematics ,Modeling and Simulation ,Mendelian inheritance ,symbols ,General Agricultural and Biological Sciences - Abstract
After introducing the notion of hyperstructures about 80 years ago, a number of researches on its applications have been done. This paper can be considered as one of its applications on Biology. It is well known that biological inheritance is related to Mathematics in the sense of probability. In this paper, we consider another relationship between inheritance and Mathematics in which we present a connection between it and hyperstructures. First, we present examples of five different types of Non- Mendelian inheritance and study, for the first time, their relation with hyperstucture theory. Then we make some hypothetical crosses for the n- hybrid case for both simple and incomplete inheritances and study their relations with hyperstructures.
- Published
- 2017
46. A System of Periodic Discrete-time Coupled Sylvester Quaternion Matrix Equations
- Author
-
Zhuo-Heng He and Qing-Wen Wang
- Subjects
Sylvester matrix ,Pure mathematics ,Algebra and Number Theory ,Generalized inverse ,Quaternion algebra ,Rank (linear algebra) ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Expression (computer science) ,01 natural sciences ,Sylvester's law of inertia ,Discrete time and continuous time ,0101 mathematics ,Sylvester equation ,Mathematics - Abstract
We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled Sylvester matrix equations AkXk + YkBk = Mk, CkXk+1 + YkDk = Nk (k = 1, 2) over the quaternion algebra to be consistent in terms of ranks and generalized inverses of the coefficient matrices. We also give an expression of the general solution to the system when it is solvable. The findings of this paper generalize some known results in the literature.
- Published
- 2017
47. Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients
- Author
-
Markus Bachmayr, Albert Cohen, Giovanni Migliorati, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and European Project: 338977,EC:FP7:ERC,ERC-2013-ADG,BREAD(2014)
- Subjects
Pure mathematics ,Holomorphic function ,010103 numerical & computational mathematics ,01 natural sciences ,symbols.namesake ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Taylor series ,Mathematics - Numerical Analysis ,0101 mathematics ,Legendre polynomials ,Cauchy's integral formula ,Mathematics ,parametric PDEs ,Numerical Analysis ,Applied Mathematics ,Numerical Analysis (math.NA) ,n-term approximation ,010101 applied mathematics ,Computational Mathematics ,Elliptic curve ,Modeling and Simulation ,Bounded function ,Norm (mathematics) ,affine coefficients ,symbols ,Jacobi polynomials ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,Analysis ,Analysis of PDEs (math.AP) - Abstract
We consider the linear elliptic equation − div(a∇u) = f on some bounded domain D, where a has the affine form a = a(y) = ā + ∑j≥1yjψj for some parameter vector y = (yj)j ≥ 1 ∈ U = [−1,1]N. We study the summability properties of polynomial expansions of the solution map y → u(y) ∈ V := H01(D) . We consider both Taylor series and Legendre series. Previous results [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11–47] show that, under a uniform ellipticity assuption, for any 0
- Published
- 2016
48. Derivatives of functions of eigenvalues and eigenvectors for symmetric matrices
- Author
-
Yongzeng Lai and Yongjia Xu
- Subjects
Pure mathematics ,Matrix differential equation ,021103 operations research ,Applied Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Eigenvalues and eigenvectors of the second derivative ,Algebra ,symbols.namesake ,Jacobi eigenvalue algorithm ,Matrix function ,symbols ,Symmetric matrix ,0101 mathematics ,Divide-and-conquer eigenvalue algorithm ,Analysis ,Eigenvalues and eigenvectors ,Eigenvalue perturbation ,Mathematics - Abstract
This paper discusses the differentiability of a class of functions associated with eigenvalues and eigenvectors of symmetric matrices. Recursive style formulas of partial derivatives for this class of functions are derived and higher order derivatives can be easily obtained from these formulas. Meanwhile, some interesting characteristics of multiple eigenvalues are revealed. Examples involving inverse eigenvalue problems and primary matrix functions are given to illustrate the applications of the results obtained in this paper.
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- 2016
49. Weyl and Bernstein numbers of embeddings of Sobolev spaces with dominating mixed smoothness
- Author
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Van Kien Nguyen
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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
50. Equilibrium problems in weakly admissible external fields created by pointwise charges
- Author
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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)
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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
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