1,572 results
Search Results
2. Biased Adjusted Poisson Ridge Estimators-Method and Application
- Author
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Pär Sjölander, Muhammad Qasim, Muhammad Amin, B. M. Golam Kibria, and Kristofer Månsson
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Mean squared error ,General Mathematics ,Maximum likelihood ,General Physics and Astronomy ,Regression estimator ,Poisson distribution ,Modified almost unbiased ridge estimators ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,Statistics ,Poisson regression ,0101 mathematics ,Mathematics ,010308 nuclear & particles physics ,010102 general mathematics ,Estimator ,Mean square error ,General Chemistry ,Ridge (differential geometry) ,Poisson ridge regression ,Multicollinearity ,Maximum likelihood estimator ,symbols ,General Earth and Planetary Sciences ,General Agricultural and Biological Sciences ,Research Paper - Abstract
Månsson and Shukur (Econ Model 28:1475–1481, 2011) proposed a Poisson ridge regression estimator (PRRE) to reduce the negative effects of multicollinearity. However, a weakness of the PRRE is its relatively large bias. Therefore, as a remedy, Türkan and Özel (J Appl Stat 43:1892–1905, 2016) examined the performance of almost unbiased ridge estimators for the Poisson regression model. These estimators will not only reduce the consequences of multicollinearity but also decrease the bias of PRRE and thus perform more efficiently. The aim of this paper is twofold. Firstly, to derive the mean square error properties of the Modified Almost Unbiased PRRE (MAUPRRE) and Almost Unbiased PRRE (AUPRRE) and then propose new ridge estimators for MAUPRRE and AUPRRE. Secondly, to compare the performance of the MAUPRRE with the AUPRRE, PRRE and maximum likelihood estimator. Using both simulation study and real-world dataset from the Swedish football league, it is evidenced that one of the proposed, MAUPRRE ($$ \hat{k}_{q4} $$ k ^ q 4 ) performed better than the rest in the presence of high to strong (0.80–0.99) multicollinearity situation.
- Published
- 2020
3. Global optimization in Hilbert space
- Author
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Benoît Chachuat, Boris Houska, Engineering & Physical Science Research Council (EPSRC), and Commission of the European Communities
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Technology ,Optimization problem ,Mathematics, Applied ,0211 other engineering and technologies ,CONVEX COMPUTATION ,010103 numerical & computational mathematics ,02 engineering and technology ,ELLIPSOIDS ,01 natural sciences ,90C26 ,93B40 ,Convergence analysis ,0102 Applied Mathematics ,Branch-and-lift ,CUT ,Mathematics ,65K10 ,021103 operations research ,Full Length Paper ,Operations Research & Management Science ,0103 Numerical and Computational Mathematics ,Bounded function ,Physical Sciences ,symbols ,49M30 ,Calculus of variations ,INTEGRATION ,SET ,Complexity analysis ,Complete search ,Operations Research ,General Mathematics ,APPROXIMATIONS ,Set (abstract data type) ,symbols.namesake ,Applied mathematics ,ALGORITHM ,0101 mathematics ,INTERSECTION ,Global optimization ,0802 Computation Theory and Mathematics ,Science & Technology ,Infinite-dimensional optimization ,Hilbert space ,Computer Science, Software Engineering ,Constraint (information theory) ,Computer Science ,Software - Abstract
We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-suboptimal global solution within finite run-time for any given termination tolerance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}ε>0. Finally, we illustrate these results for a problem of calculus of variations.
- Published
- 2017
4. d-Hermite rings and skew $$\textit{PBW}$$ PBW extensions
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Oswaldo Lezama and Claudia Gallego
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Hermite polynomials ,Rank (linear algebra) ,General Mathematics ,010102 general mathematics ,Short paper ,Skew ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,symbols.namesake ,Computational Theory and Mathematics ,Kronecker delta ,symbols ,Kronecker's theorem ,Finitely-generated abelian group ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this short paper we study the d-Hermite condition about stably free modules for skew $$\textit{PBW}$$ extensions. For this purpose, we estimate the stable rank of these non-commutative rings. In addition, and closely related with these questions, we will prove Kronecker’s theorem about the radical of finitely generated ideals for some particular types of skew $$\textit{PBW}$$ extensions.
- Published
- 2015
5. On Lacunas in the Spectrum of the Laplacian with the Dirichlet Boundary Condition in a Band with Oscillating Boundary
- Author
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Denis Borisov
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Statistics and Probability ,Applied Mathematics ,General Mathematics ,Operator (physics) ,Mathematical analysis ,Spectrum (functional analysis) ,Boundary (topology) ,Function (mathematics) ,symbols.namesake ,Amplitude ,Dirichlet boundary condition ,symbols ,Flat band ,Laplace operator ,Mathematics - Abstract
In this paper, we consider the Laplace operator in a flat band whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude.
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- 2021
6. Logarithmic Potential and Generalized Analytic Functions
- Author
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O.V. Nesmelova, Vladimir Gutlyanskiĭ, Vladimir Ryazanov, and A.S. Yefimushkin
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Statistics and Probability ,Dirichlet problem ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Harmonic (mathematics) ,Unit disk ,Sobolev space ,Riemann hypothesis ,symbols.namesake ,Harmonic function ,symbols ,Neumann boundary condition ,Analytic function ,Mathematics - Abstract
The study of the Dirichlet problem in the unit disk 𝔻 with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua [48] has been devoted to boundary-value problems (only with Holder continuous data) for the generalized analytic functions, i.e., continuous complex valued functions h(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form 𝜕zh + ah + b $$ \overline{h} $$ = c ; where it was assumed that the complex valued functions a; b and c belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. The present paper is a natural continuation of our previous articles on the Riemann, Hilbert, Dirichlet, Poincar´e and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic, and the so-called A−harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here, we extend the corresponding results to the generalized analytic functions h : D → ℂ with the sources g : 𝜕zh = g ∈ Lp, p > 2 , and to generalized harmonic functions U with sources G : △U = G ∈ Lp, p > 2. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary-value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary-values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincar´e problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media.
- Published
- 2021
7. The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation
- Author
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Ihyeok Seo and Yoonjung Lee
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symbols.namesake ,General Mathematics ,Open problem ,symbols ,Initial value problem ,Beta (velocity) ,Lambda ,Nonlinear Schrödinger equation ,Energy (signal processing) ,Mathematics ,Mathematical physics - Abstract
In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrodinger equation $$i\partial _{t}u+\Delta u=\lambda |x|^{-\alpha }|u|^{\beta }u$$ in $$H^1$$ . The well-posedness theory in $$H^1$$ has been intensively studied in recent years, but the currently known approaches do not work for the critical case $$\beta =(4-2\alpha )/(n-2)$$ . It is still an open problem. The main contribution of this paper is to develop the theory in this case.
- Published
- 2021
8. New Computational Formulas for Special Numbers and Polynomials Derived from Applying Trigonometric Functions to Generating Functions
- Author
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Yilmaz Simsek and Neslihan Kilar
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Catalan number ,Pure mathematics ,Bernoulli's principle ,symbols.namesake ,General Mathematics ,Factorial number system ,Euler's formula ,symbols ,Stirling number ,Trigonometric functions ,Type (model theory) ,Mathematics - Abstract
The aim of this paper is to apply trigonometric functions with functional equations of generating functions. Using the resulted new equations and formulas from this application, we obtain many special numbers and polynomials such as the Stirling numbers, Bernoulli and Euler type numbers, the array polynomials, the Catalan numbers, and the central factorial numbers. We then introduce combinatorial sums related to these special numbers and polynomials. Moreover, we gave some remarks that relates our new findings from this paper to the relations found in earlier studies.
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- 2021
9. Existence and Uniqueness of the Global L1 Solution of the Euler Equations for Chaplygin Gas
- Author
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Zhen Wang, Tingting Chen, and Aifang Qu
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Continuous function ,General Mathematics ,Weak solution ,010102 general mathematics ,General Physics and Astronomy ,Euler system ,Absolute continuity ,Lebesgue integration ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,symbols ,Local boundedness ,Applied mathematics ,Initial value problem ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
In this paper, we establish the global existence and uniqueness of the solution of the Cauchy problem of a one-dimensional compressible isentropic Euler system for a Chaplygin gas with large initial data in the space L loc 1 . The hypotheses on the initial data may be the least requirement to ensure the existence of a weak solution in the Lebesgue measurable sense. The novelty and also the essence of the difficulty of the problem lie in the fact that we have neither the requirement on the local boundedness of the density nor that which is bounded away from vacuum. We develop the previous results on this degenerate system. The method used is Lagrangian representation, the essence of which is characteristic analysis. The key point is to prove the existence of the Lagrangian representation and the absolute continuity of the potentials constructed with respect to the space and the time variables. We achieve this by finding a property of the fundamental theorem of calculus for Lebesgue integration, which is a sufficient and necessary condition for judging whether a monotone continuous function is absolutely continuous. The assumptions on the initial data in this paper are believed to also be necessary for ruling out the formation of Dirac singularity of density. The ideas and techniques developed here may be useful for other nonlinear problems involving similar difficulties.
- Published
- 2021
10. On the pair correlations of powers of real numbers
- Author
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Christoph Aistleitner and Simon Baker
- Subjects
11K06, 11K60 ,General Mathematics ,Modulo ,FOS: Physical sciences ,0102 computer and information sciences ,Lebesgue integration ,01 natural sciences ,Combinatorics ,symbols.namesake ,Pair correlation ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Algebra over a field ,Classical theorem ,Mathematical Physics ,Real number ,Mathematics ,Sequence ,Mathematics - Number Theory ,Probability (math.PR) ,010102 general mathematics ,Mathematical Physics (math-ph) ,010201 computation theory & mathematics ,symbols ,Martingale (probability theory) ,Mathematics - Probability - Abstract
A classical theorem of Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In the present paper we extend Koksma's theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every $x>1$ the pair correlations of the fractional parts of $(x^n)_{n=1}^{\infty}$ are asymptotically Poissonian. The proof is based on a martingale approximation method., Version 2: some minor changes. The paper will appear in the Israel Journal of Mathematics
- Published
- 2021
11. Concomitants of Generalized Order Statistics from Bivariate Cambanis Family of Distributions Under a General Setting
- Author
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Haroon M. Barakat, M. A. Alawady, and M. A. Abd Elgawad
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Recurrence relation ,General Mathematics ,010102 general mathematics ,Order statistic ,Extension (predicate logic) ,Bivariate analysis ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Joint probability distribution ,Statistics ,symbols ,0101 mathematics ,Fisher information ,Divergence (statistics) ,Mathematics - Abstract
In this paper, we study the concomitants of m-generalized order statistics (m-GOSs) and m-dual generalized order statistics (m-DGOSs) from bivariate Cambanis family with nonzero parameter values as an extension of several recent papers. Moreover, we derive some information measures, namely the Shannon entropy, Kullback–Leibler (KL) divergence and Fisher information number (FIN) for the concomitants of m-GOSs, when $$m>-1,$$ and record values, for $$m=-1.$$ Also, the joint distribution of concomitants of m-GOSs and record values for this family are studied. Besides, some useful recurrence relations between moments of concomitants are obtained. Finally, the ordinary order statistics (OOSs), record values and sequential order statistics (SOSs) as subclasses of m-GOSs, as well as the progressive type II censored order statistics (POSs) as a more general subclass of GOSs, are separately discussed.
- Published
- 2021
12. Central limit theorems for the ℤ2-periodic Lorentz gas
- Author
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Damien Thomine, Françoise Pène, Laboratoire de Mathématiques de Bretagne Atlantique (LMBA), Université de Bretagne Sud (UBS)-Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS), Université de Brest (UBO), Laboratoire de Mathématiques d'Orsay (LMO), and Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Dynamical systems theory ,General Mathematics ,Lorentz transformation ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,010102 general mathematics ,Spectral properties ,Hölder condition ,Observable ,0102 computer and information sciences ,01 natural sciences ,Measure (mathematics) ,symbols.namesake ,Discrete time and continuous time ,010201 computation theory & mathematics ,symbols ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,Mathematical physics ,Central limit theorem - Abstract
This paper is devoted to the stochastic properties of dynamical systems preserving an infinite measure. More precisely we prove central limit theorems for Birkhoff sums of observables of ℤ2-extensions of dynamical systems (satisfying some nice spectral properties). The motivation of our paper is the ℤ2-periodic Lorentz process for which we establish a functional central limit theorem for Holder continuous observables (in discrete time as well as in continuous time).
- Published
- 2021
13. Approximating a common solution of extended split equality equilibrium and fixed point problems
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J. M. Ngnotchouye, F. U. Ogbuisi, and F. O. Isiogugu
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TheoryofComputation_MISCELLANEOUS ,Iterative method ,Applied Mathematics ,General Mathematics ,Numerical analysis ,Hilbert space ,TheoryofComputation_GENERAL ,Extension (predicate logic) ,Fixed point ,symbols.namesake ,Monotone polygon ,Convergence (routing) ,symbols ,Applied mathematics ,Equilibrium problem ,Mathematics - Abstract
In this paper, we study an extension of the split equality equilibrium problem called the extended split equality equilibrium problem. We give an iterative algorithm for approximating a solution of extended split equality equilibrium and fixed point problems and obtained a strong convergence result in a real Hilbert space. We further applied our result to solve extended split equality monotone variational inclusion and equilibrium problems. The result of this paper complements and extends results on split equality equilibrium problems in the literature.
- Published
- 2021
14. Mapped Regularization Methods for the Cauchy Problem of the Helmholtz and Laplace Equations
- Author
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Hojjatollah Shokri Kaveh and Hojjatollah Adibi
- Subjects
Cauchy problem ,Laplace transform ,General Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,General Physics and Astronomy ,General Chemistry ,Spectral galerkin ,Regularization (mathematics) ,Tikhonov regularization ,symbols.namesake ,Helmholtz free energy ,Singular value decomposition ,symbols ,General Earth and Planetary Sciences ,Applied mathematics ,Initial value problem ,General Agricultural and Biological Sciences ,Mathematics - Abstract
In this paper, Spectral Galerkin Method is applied for Cauchy problem of Helmholtz and Laplace equations in the regular domains. It is well known that these problems have severely ill-posed solutions. Accordingly, regularization methods are required to overcome the ill-posedness issue. In this paper, we utilize the regularization method based upon mapped methods. These methods include Tikhonov and truncated singular value decomposition methods and additionally several new filters of regularization which are introduced. Finally, some test examples are given to demonstrate the capability and efficiency of the proposed method.
- Published
- 2021
15. Rarefaction Wave Interaction and Shock-Rarefaction Composite Wave Interaction for a Two-Dimensional Nonlinear Wave System
- Author
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Sisi Xie and Geng Lai
- Subjects
Conservation law ,Equation of state ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Rarefaction ,01 natural sciences ,Shock (mechanics) ,010104 statistics & probability ,Nonlinear system ,Riemann hypothesis ,symbols.namesake ,Method of characteristics ,symbols ,Order (group theory) ,0101 mathematics ,Mathematics - Abstract
In order to construct global solutions to two-dimensional (2D for short) Riemann problems for nonlinear hyperbolic systems of conservation laws, it is important to study various types of wave interactions. This paper deals with two types of wave interactions for a 2D nonlinear wave system with a nonconvex equation of state: Rarefaction wave interaction and shock-rarefaction composite wave interaction. In order to construct solutions to these wave interactions, the authors consider two types of Goursat problems, including standard Goursat problem and discontinuous Goursat problem, for a 2D self-similar nonlinear wave system. Global classical solutions to these Goursat problems are obtained by the method of characteristics. The solutions constructed in the paper may be used as building blocks of solutions of 2D Riemann problems.
- Published
- 2021
16. Modified Extragradient Method for Pseudomonotone Variational Inequalities in Infinite Dimensional Hilbert Spaces
- Author
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Yeol Je Cho, Yi-bin Xiao, Dang Van Hieu, and Poom Kumam
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021103 operations research ,Weak convergence ,General Mathematics ,Operator (physics) ,0211 other engineering and technologies ,Hilbert space ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,symbols.namesake ,Convergence (routing) ,Variational inequality ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we prove the weak convergence of a modified extragradient algorithm for solving a variational inequality problem involving a pseudomonotone operator in an infinite dimensional Hilbert space. Moreover, we establish further the R-linear rate of the convergence of the proposed algorithm with the assumption of error bound. Several numerical experiments are performed to illustrate the convergence behaviour of the new algorithm in comparisons with others. The results obtained in the paper have extended some recent results in the literature.
- Published
- 2020
17. On Boundedness Property of Singular Integral Operators Associated to a Schrödinger Operator in a Generalized Morrey Space and Applications
- Author
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Thanh-Nhan Nguyen, Xuan Truong Le, and Ngoc Trong Nguyen
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Property (philosophy) ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,General Physics and Astronomy ,Function (mathematics) ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,Schrödinger equation ,010101 applied mathematics ,symbols.namesake ,Riesz transform ,Operator (computer programming) ,symbols ,0101 mathematics ,Schrödinger's cat ,Mathematics - Abstract
In this paper, we provide the boundedness property of the Riesz transforms associated to the Schrodinger operator $${\cal L} = \Delta + {\bf{V}}$$ in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces. The additional potential V considered in this paper is a non-negative function satisfying the suitable reverse Holder’s inequality. Our results are new and general in many cases of problems. As an application of the boundedness property of these singular integral operators, we obtain some regularity results of solutions to Schrodinger equations in the new Morrey space.
- Published
- 2020
18. Nψ,ϕ-type Quotient Modules over the Bidisk
- Author
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Chang Hui Wu and Tao Yu
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Essential spectrum ,Hardy space ,Characterization (mathematics) ,Type (model theory) ,01 natural sciences ,symbols.namesake ,Compact space ,Compression (functional analysis) ,0103 physical sciences ,Quotient module ,symbols ,010307 mathematical physics ,0101 mathematics ,Quotient ,Mathematics - Abstract
Let H2(ⅅ2) be the Hardy space over the bidisk ⅅ2, and let Mψ,ϕ = [(ψ(z) − ϕ(w))2] be the submodule generated by (ψ(z) − ϕ(w))2, where ψ(z) and ϕ(w) are nonconstant inner functions. The related quotient module is denoted by Nψ,ϕ = H2(ⅅ2) ⊖ Mψ,ϕ. In this paper, we give a complete characterization for the essential normality of Nψ,ϕ. In particular, if ψ(z)= z, we simply write Mψ,ϕ and Nψ,ϕ as Mϕ and Nϕ respectively. This paper also studies compactness of evaluation operators L(0)∣nϕ and R(0)ϕnϕ, essential spectrum of compression operator Sz on Nϕ, essential normality of compression operators Sz and Sw on Nϕ.
- Published
- 2020
19. More about singular traces on simply generated operator ideals
- Author
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Albrecht Pietsch
- Subjects
Large class ,Sequence ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,Extension (predicate logic) ,Space (mathematics) ,01 natural sciences ,symbols.namesake ,Operator (computer programming) ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
During half a century, singular traces on ideals of Hilbert space operators have been constructed by looking for linear forms on associated sequence ideals. Only recently, the author was able to eliminate this auxiliary step by directly applying Banach’s version of the extension theorem; see (Integral Equ. Oper. Theory 91, 21, 2019 and 92, 7, 2020). Of course, the relationship between the new approach and the older ones must be investigated. In the first paper, this was done for $${\mathfrak {L}}_{1,\infty } (H)$$ . To save space, such considerations were postponed in the second paper, which deals with a large class of principal ideals, called simply generated. This omission will now be rectified.
- Published
- 2020
20. Vector-valued q-variational inequalities for averaging operators and the Hilbert transform
- Author
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Tao Ma, Wei Liu, and Guixiang Hong
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Banach space ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,Variational inequality ,symbols ,010307 mathematical physics ,Hilbert transform ,0101 mathematics ,Martingale (probability theory) ,Mathematics - Abstract
Recently, the authors have established $$L^p$$ -boundedness of vector-valued q-variational inequalities for averaging operators which take values in the Banach space satisfying the martingale cotype q property in Hong and Ma (Math Z 286(1–2):89–120, 2017). In this paper, we prove that the martingale cotype q property is also necessary for the vector-valued q-variational inequalities, which was a question left open in the previous paper. Moreover, we also prove that the UMD property and the martingale cotype q property can be characterized in terms of vector valued q-variational inequalities for the Hilbert transform.
- Published
- 2020
21. Dini–Lipschitz functions for the quaternion linear canonical transform
- Author
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N. Safouane, Radouan Daher, Azzedine Achak, and A. Bouhlal
- Subjects
Pure mathematics ,symbols.namesake ,Fourier transform ,General Mathematics ,Computation ,symbols ,Image processing ,Equivalence (formal languages) ,Quaternion ,Singular integral operators ,Lipschitz continuity ,Interpolation theory ,Mathematics - Abstract
This paper is an exposition of some results on calculation of the K-functional which have number of applications of interpolation theory. In particular some recent problems in image processing and singular integral operators require the computation of suitable K-functionals. In this paper we will give some results concerning the equivalence of a K-functional and the modulus of smoothness constructed by the generalized Steklov function. We mention here that we have generalized the Steklov’s function for Fourier transform to quaternion linear canonical transform. This paper generalizes also Titchmarsh’s theorem for measurable sets from complex domain to hyper complex domain by using quaternion algebras, associated with the quaternion linear canonical transform.
- Published
- 2020
22. The Wiener Measure on the Heisenberg Group and Parabolic Equations
- Author
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S. V. Mamon
- Subjects
Statistics and Probability ,Pure mathematics ,Semigroup ,Stochastic process ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Markov process ,01 natural sciences ,Measure (mathematics) ,010305 fluids & plasmas ,Nilpotent ,symbols.namesake ,0103 physical sciences ,Path integral formulation ,Lie algebra ,symbols ,Heisenberg group ,0101 mathematics ,Mathematics - Abstract
In this paper, we study questions related to the theory of stochastic processes on Lie nilpotent groups. In particular, we consider the stochastic process on the Heisenberg group H3(ℝ) whose trajectories satisfy the horizontal conditions in the stochastic sense relative to the standard contact structure on H3 (ℝ). It is shown that this process is a homogeneous Markov process relative to the Heisenberg group operation. There was found a representation in the form of a Wiener integral for a one-parameter linear semigroup of operators for which the Heisenberg sublaplacian generated by basis vector fields of the corresponding Lie algebra L(H3) is producing. The main method of solving the problem in this paper is using the path integrals technique, which indicates the common direction of further development of the results.
- Published
- 2020
23. KAM Tori for the Derivative Quintic Nonlinear Schrödinger Equation
- Author
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Guang Hua Shi and Dong Feng Yan
- Subjects
Kolmogorov–Arnold–Moser theorem ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mean value ,Zero (complex analysis) ,Torus ,Derivative ,01 natural sciences ,Quintic function ,010101 applied mathematics ,symbols.namesake ,symbols ,0101 mathematics ,Nonlinear Schrödinger equation ,Mathematical physics ,Mathematics - Abstract
This paper is concerned with one-dimensional derivative quintic nonlinear Schrodinger equation, $${\rm{i}}u_t-u_{xx}+{\rm{i}}(|u|^4u)_x=0, \;\; x\in\mathbb{T}.$$ The existence of a large amount of quasi-periodic solutions with two frequencies for this equation is established. The proof is based on partial Birkhoff normal form technique and an unbounded KAM theorem. We mention that in the present paper the mean value of u does not need to be zero, but small enough, which is different from the assumption (1.7) in Geng-Wu [J. Math. Phys., 53, 102702 (2012)].
- Published
- 2020
24. Purely Sequential and k-Stage Procedures for Estimating the Mean of an Inverse Gaussian Distribution
- Author
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Ajit Chaturvedi, Neeraj Joshi, and Sudeep R. Bapat
- Subjects
Statistics and Probability ,General Mathematics ,Closeness ,Inverse Gaussian distribution ,symbols.namesake ,Sample size determination ,Bounded function ,symbols ,Applied mathematics ,Stage (hydrology) ,Point estimation ,Scale parameter ,Expected loss ,Mathematics - Abstract
In the first part of this paper, we propose purely sequential and k-stage (k ≥ 3) procedures for estimation of the mean μ of an inverse Gaussian distribution having prescribed ‘proportional closeness’. The problem is constructed in such a manner that the boundedness of the expected loss is equivalent to the estimation of parameter with given ‘proportional closeness’. We obtain the associated second-order approximations for both the procedures. Second part of this paper deals with developing the minimum risk and bounded risk point estimation problems for estimating the mean μ of an inverse Gaussian distribution having unknown scale parameter λ. We propose an useful family of loss functions for both the problems and our aim is to control the associated risk functions. Moreover, we establish the failure of fixed sample size procedures to deal with these problems and hence propose purely sequential and k-stage (k ≥ 3) procedures to estimate the mean μ. We also obtain the second-order approximations associated with our sequential procedures. Further, we provide extensive sets of simulation studies and real data analysis to show the performances of our proposed procedures.
- Published
- 2020
25. Titchmarsh’s theorem and some remarks concerning the right-sided quaternion Fourier transform
- Author
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Azzedine Achak, Radouan Daher, Aziz Bouhlal, and N. Safouane
- Subjects
Pure mathematics ,Quaternion algebra ,General Mathematics ,010102 general mathematics ,Space (mathematics) ,Lipschitz continuity ,01 natural sciences ,Square (algebra) ,Function of several real variables ,010101 applied mathematics ,Translation operator ,symbols.namesake ,Fourier transform ,symbols ,0101 mathematics ,Quaternion ,Mathematics - Abstract
This paper is based mainly on Titchmarsh’s theorem (Introduction to the theory of Fourier integrals. Clarendon Press, Oxford, 1937, Theorem 84) in the one-dimensional case. Abilov et al. (Comput Math Math Phys 48:2146, 2008) proved two useful estimates for the Fourier transform in the space of square integral multivariable functions on certain classes of functions characterized by the generalized continuity modulus, and these estimates are proved by Abilovs for only two variables, using a translation operator. The purpose of this paper is to study these estimates for Quaternion Fourier transforms, also the functions satisfy Lipschitz conditions of certain orders. Thus we study the Quaternion Fourier transforms of Lipschitz function in the functions space $$L^r({\mathbb {R}}^{2},{\mathcal {H}})$$, where $${\mathcal {H}}$$ a quaternion algebra which will be specified in due course.
- Published
- 2020
26. Necessary optimality conditions for a semivectorial bilevel optimization problem using the kth-objective weighted-constraint approach
- Author
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Khadija Hamdaoui, Mohammed El Idrissi, and N. Gadhi
- Subjects
021103 operations research ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,02 engineering and technology ,Operator theory ,First order ,Mathematical proof ,01 natural sciences ,Bilevel optimization ,Potential theory ,Theoretical Computer Science ,Constraint (information theory) ,symbols.namesake ,Fourier analysis ,symbols ,Applied mathematics ,0101 mathematics ,Variational analysis ,Analysis ,Mathematics - Abstract
In this paper, we have pointed out that the proof of Theorem 11 in the recent paper (Lafhim in Positivity, 2019. https://doi.org/10.1007/s11117-019-00685-1 ) is erroneous. Using techniques from variational analysis, we propose other proofs to detect necessary optimality conditions in terms of Karush–Kuhn–Tucker multipliers. Our main results are given in terms of the limiting subdifferentials and the limiting normal cones. Completely detailed first order necessary optimality conditions are then given in the smooth setting while using the generalized differentiation calculus of Mordukhovich.
- Published
- 2019
27. Stability and instability results for Cauchy laminated Timoshenko-type systems with interfacial slip and a heat conduction of Gurtin–Pipkin’s law
- Author
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Aissa Guesmia
- Subjects
Polynomial ,Applied Mathematics ,General Mathematics ,General Physics and Astronomy ,Cauchy distribution ,Dissipation ,Type (model theory) ,Thermal conduction ,symbols.namesake ,Thermoelastic damping ,Fourier analysis ,Law ,symbols ,Variable (mathematics) ,Mathematics - Abstract
The subject of the present paper is to study the stability of a class of laminated Timoshenko-type systems in the whole line $$\mathbb {R}$$ combined with a heat conduction given by Gurtin–Pipkin’s law and acting only on one equation of the laminated Timoshenko-type system. The main result of this paper shows that the thermoelastic dissipation generated by Gurtin–Pipkin’s law is strong enough to stabilize the system at least polynomially, even if only the second or the third equation of the laminated Timoshenko-type system is controlled and the two other ones are free. When only the first equation of the laminated Timoshenko-type system is controlled, we give a necessary and sufficient condition for the polynomial stability. The polynomial decays in the $$L^2$$ -norm of the solution, and its higher-order derivatives with respect to the space variable are specified in terms of the regularity of the initial data and some connections between the coefficients. An application to the particular case of Timoshenko-type systems is also given. The proofs are based on the energy method and Fourier analysis combined with some well-chosen weight functions.
- Published
- 2021
28. Parallel shrinking inertial extragradient approximants for pseudomonotone equilibrium, fixed point and generalized split null point problem
- Author
-
Yasir Arfat, Parinya Sa Ngiamsunthorn, Poom Kumam, and Muhammad Aqeel Ahmad Khan
- Subjects
Sequence ,Current (mathematics) ,Inertial frame of reference ,Applied Mathematics ,General Mathematics ,Numerical analysis ,Hilbert space ,Fixed point ,Set (abstract data type) ,symbols.namesake ,symbols ,Applied mathematics ,Null point ,Mathematics - Abstract
This paper provides an iterative construction for a common solution associated with the pseudomonotone equilibrium problems, fixed point problem of a finite family $$\eta $$ -demimetric operators and the generalized split null point problem in Hilbert spaces. The sequence of approximants is a variant of the parallel shrinking extragradient algorithm with the inertial effect converging strongly to the optimal common solution under suitable set of control conditions. The viability of the approximants is demonstrated for various theoretical as well as numerical results. The results presented in this paper improve various existing results in the current literature.
- Published
- 2021
29. New solutions to Legendre’s incomplete elliptic integrals of the first and second kinds and p-elliptic integrals
- Author
-
Prateek Pralhad Kulkarni
- Subjects
Pure mathematics ,Series (mathematics) ,Applied Mathematics ,General Mathematics ,Numerical analysis ,symbols.namesake ,Special functions ,Euler's formula ,symbols ,Elliptic integral ,Inverse trigonometric functions ,Gamma function ,Legendre polynomials ,Mathematics - Abstract
The elliptic integrals are of interest in various disciplines. While series solutions do exist for complete elliptic integrals, there are no deduced series solutions for Incomplete elliptic integrals, in terms of the special functions. This paper provides novel solutions of the Legendre forms of incomplete elliptic integrals of the first and second kinds in terms of the Euler’s gamma functions. The paper also proposes new solutions to inverse trigonometric functions, which has never been known till date.
- Published
- 2021
30. Existence and Regularity of Weak Solutions for $$\psi $$-Hilfer Fractional Boundary Value Problem
- Author
-
J. Vanterler da C. Sousa, E. Capelas de Oliveira, and M. Aurora P. Pulido
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,01 natural sciences ,Fractional calculus ,010101 applied mathematics ,symbols.namesake ,symbols ,High Energy Physics::Experiment ,Integration by parts ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
In the present paper, we investigate the existence and regularity of weak solutions for $$\psi $$ -Hilfer fractional boundary value problem in $$\mathbb {C}^{\alpha ,\beta ;\psi }_{2}$$ and $$\mathcal {H}$$ (Hilbert space) spaces, using extension of the Lax–Milgram theorem. In this sense, to finalize the paper, we discuss the integration by parts for $$\psi $$ -Riemann–Liouville fractional integral and $$\psi $$ -Hilfer fractional derivative.
- Published
- 2021
31. Solution for nonvariational quasilinear elliptic systems via sub-supersolution technique and Galerkin method
- Author
-
Leandro S. Tavares, Francisco Julio S. A. Corrêa, and Gelson C. G. dos Santos
- Subjects
Change of variables ,Elliptic systems ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,General Physics and Astronomy ,Monotonic function ,Type (model theory) ,01 natural sciences ,Schrödinger equation ,010101 applied mathematics ,symbols.namesake ,symbols ,Dissipative system ,Applied mathematics ,0101 mathematics ,Galerkin method ,Schrödinger's cat ,Mathematics - Abstract
In this paper, we obtain the existence of positive solution for a system of quasilinear Schrodinger equations with concave nonlinearities which is related to several applications in Hydrodynamics, Heidelberg Ferromagnetism and Magnus Theory, Condensed Matter Theory, Dissipative Quantum Mechanics and nanotubes and fullerene-related structures. The quasilinear Schrodinger problem is studied by considering a suitable change of variables which transforms the original problem in to a semilinear one. By means of the several properties of the change of variables, constructions of suitable sub-supersolutions, monotonic iteration arguments and the Galerkin method, we obtain the existence of solution for the semilinear problem. The paper is divided in two parts. In the first one, we use the method of sub-supersolutions to obtain a solution for the problem. In the second part, we use the Galerkin method and a comparison argument to obtain a solution for the system considered. An important feature is that the sub-supersolution approach is rare in the literature for the type of problem considered here and the Galerkin method was not used to consider quasilinear Schrodinger equations.
- Published
- 2021
32. Resolvent Decomposition Theorems and Their Application in Denumerable Markov Processes with Instantaneous States
- Author
-
Anyue Chen
- Subjects
Statistics and Probability ,General Mathematics ,010102 general mathematics ,Probabilistic logic ,Markov process ,01 natural sciences ,Interpretation (model theory) ,Algebra ,010104 statistics & probability ,symbols.namesake ,symbols ,Decomposition (computer science) ,Countable set ,Uniqueness ,0101 mathematics ,Statistics, Probability and Uncertainty ,Resolvent ,Mathematics ,Analytic proof - Abstract
The basic aim of this paper is to provide a fundamental tool, the resolvent decomposition theorem, in the construction theory of denumerable Markov processes. We present a detailed analytic proof of this extremely useful tool and explain its clear probabilistic interpretation. We then apply this tool to investigate the basic problems of existence and uniqueness criteria for denumerable Markov processes with instantaneous states to which few results have been obtained even until now. Although the complete answers regarding these existence and uniqueness criteria will be given in a subsequent paper, we shall, in this paper, present part solutions of these very important problems that are closely linked with the subtle Williams S and N conditions.
- Published
- 2019
33. An effective Chebotarev density theorem for families of number fields, with an application to $$\ell $$-torsion in class groups
- Author
-
Lillian B. Pierce, Caroline L. Turnage-Butterbaugh, and Melanie Matchett Wood
- Subjects
Discrete mathematics ,Mathematics - Number Theory ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Algebraic number field ,01 natural sciences ,Riemann hypothesis ,symbols.namesake ,Arbitrarily large ,Number theory ,Discriminant ,Field extension ,0103 physical sciences ,FOS: Mathematics ,symbols ,Torsion (algebra) ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Dedekind zeta function ,Mathematics - Abstract
We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal $L$-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal $L$-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of $L$-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for $\ell$-torsion in class groups, for all integers $\ell \geq 1$, applicable to infinite families of fields of arbitrarily large degree., Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.02008
- Published
- 2019
34. Concomitants of Generalized Order Statistics from Huang–Kotz Farlie–Gumble–Morgenstern Bivariate Distribution: Some Information Measures
- Author
-
Haroon M. Barakat, M. A. Abd Elgawad, M. A. Alawady, and Shengwu Xiong
- Subjects
Recurrence relation ,Exponential distribution ,General Mathematics ,010102 general mathematics ,Order statistic ,Bivariate analysis ,Extension (predicate logic) ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Joint probability distribution ,Product (mathematics) ,Statistics ,symbols ,0101 mathematics ,Fisher information ,Mathematics - Abstract
In this paper, we study the concomitants of m-dual generalized order statistics (and consequently m-generalized order statistics) from Huang–Kotz Farlie–Gumble–Morgenstern bivariate distribution as an extension of several recent papers. Some well-known information measures, the Shannon entropy, the Kullback–Leibler distance and the Fisher information number, are derived. Moreover, some useful recurrence relations between single and product moments of concomitants are obtained. Finally, an application of these results is given for bivariate generalized exponential distribution.
- Published
- 2019
35. Courant-sharp Robin eigenvalues for the square: the case with small Robin parameter
- Author
-
Katie Gittins, Bernard Helffer, Université de Neuchâtel (Université de Neuchâtel), Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), and Helffer, Bernard
- Subjects
Spectral theory ,General Mathematics ,Courant-sharp ,[MATH] Mathematics [math] ,01 natural sciences ,Domain (mathematical analysis) ,Square (algebra) ,Mathematics - Spectral Theory ,symbols.namesake ,Robin eigenvalues ,0103 physical sciences ,FOS: Mathematics ,[MATH.MATH-SP] Mathematics [math]/Spectral Theory [math.SP] ,Neumann boundary condition ,square ,[MATH]Mathematics [math] ,0101 mathematics ,Spectral Theory (math.SP) ,Eigenvalues and eigenvectors ,Mathematics ,35P99, 58J50, 58J37 ,010102 general mathematics ,Mathematical analysis ,Mathematics::Spectral Theory ,Robin boundary condition ,Number theory ,Dirichlet boundary condition ,symbols ,010307 mathematical physics ,[MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP] - Abstract
International audience; This article is the continuation of our first work on the determination of the cases where there is equality in Courant's Nodal Domain theorem in the case of a Robin boundary condition (with Robin parameter h). For the square, our first paper focused on the case where h is large and extended results that were obtained by Pleijel, Bérard-Helffer, for the problem with a Dirichlet boundary condition. There, we also obtained some general results about the behaviour of the nodal structure (for planar domains) under a small deformation of h, where h is positive and not close to 0. In this second paper, we extend results that were obtained by Helffer-Persson-Sundqvist for the Neumann problem to the case where h > 0 is small. MSC classification (2010): 35P99, 58J50, 58J37.
- Published
- 2019
36. Extremal decomposition of a multidimensional complex space for five domains
- Author
-
Yaroslav Zabolotnii and I. V. Denega
- Subjects
Statistics and Probability ,Pure mathematics ,Geometric function theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Unit circle ,Complex space ,Product (mathematics) ,Green's function ,0103 physical sciences ,Simply connected space ,Decomposition (computer science) ,symbols ,0101 mathematics ,Quadratic differential ,Mathematics - Abstract
The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains containing points of a unit circle and the power γ of the inner radius of a domain containing the origin. The problem was formulated in 1994 in Dubinin’s paper in the journal “Russian Mathematical Surveys” in the list of unsolved problems and then repeated in his monograph in 2014. Currently, it is not solved in general. In this paper, we obtained a solution of the problem for five simply connected domains and power γ ∈ (1; 2:57] and generalized this result to the case of multidimensional complex space.
- Published
- 2019
37. Quantum Markov States and Quantum Hidden Markov States
- Author
-
Z. I. Bezhaeva and V. I. Oseledets
- Subjects
Statistics and Probability ,Discrete mathematics ,Markov chain ,Applied Mathematics ,General Mathematics ,Markov process ,Function (mathematics) ,State (functional analysis) ,Mathematical proof ,Tree (graph theory) ,symbols.namesake ,symbols ,Hidden Markov model ,Quantum ,Mathematics - Abstract
In a previous paper (Funct. Anal. Appl., 3 (2015), 205–209), we defined quantum Markov states. Here we recall this definition and present a proof of the results from that paper (which are given there without proofs). We give a definition of a quantum hidden Markov state generated by a function of a quantum Markov process and show how it is related to other definitions of such states. Our definitions work for quantum Markov fields on ℤN and on graphs. We consider an example with the Cayley tree.
- Published
- 2019
38. On Justification of the Asymptotics of Eigenfunctions of the Absolutely Continuous Spectrum in the Problem of Three One-Dimensional Short-Range Quantum Particles with Repulsion
- Author
-
S. B. Levin, A. M. Budylin, and I. V. Baibulov
- Subjects
Statistics and Probability ,Scattering ,Applied Mathematics ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Spectrum (functional analysis) ,Eigenfunction ,Absolute continuity ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,0103 physical sciences ,symbols ,0101 mathematics ,Quantum ,Schrödinger's cat ,Resolvent ,Mathematics ,Mathematical physics - Abstract
The present paper offers a new approach to the construction of the coordinate asymptotics of the kernel of the resolvent of the Schrodinger operator in the scattering problem of three onedimensional quantum particles with short-range pair potentials. Within the framework of this approach, the asymptotics of eigenfunctions of the absolutely continuous spectrum of the Schrodinger operator can be constructed. In the paper, the possibility of a generalization of the suggested approach to the case of the scattering problem of N particles with arbitrary masses is discussed.
- Published
- 2019
39. Strong Convergence of the Euler-Maruyama Method for Nonlinear Stochastic Convolution Itô-Volterra Integral Equations with Constant Delay
- Author
-
Zhan Wen Yang, Shu Fang Ma, and Jian Fang Gao
- Subjects
Statistics and Probability ,General Mathematics ,010102 general mathematics ,Superconvergence ,01 natural sciences ,Integral equation ,Volterra integral equation ,Euler–Maruyama method ,Convolution ,010104 statistics & probability ,Nonlinear system ,symbols.namesake ,Kernel (image processing) ,symbols ,Applied mathematics ,0101 mathematics ,Constant (mathematics) ,Mathematics - Abstract
This paper mainly focuses on the strong convergence of the Euler-Maruyama method for nonlinear stochastic convolution Ito-Volterra integral equations with constant delay. It is well known that the strong approximation of the Ito integral usually leads to 0.5-order approximation for stochastic problems. However, in this paper, we will show that 1-order strong superconvergence can be obtained for nonlinear stochastic convolution Ito-Volterra integral equations with constant delay under some mild conditions on the kernel of the diffusion term. Finally, some numerical experiments are given to illustrate our theoretical results.
- Published
- 2019
40. Unbounded asymptotic equivalences of operator nets with applications
- Author
-
Niyazi Anıl Gezer and Nazife Erkurşun-Özcan
- Subjects
Pure mathematics ,021103 operations research ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,02 engineering and technology ,Operator theory ,01 natural sciences ,Potential theory ,Theoretical Computer Science ,symbols.namesake ,Operator (computer programming) ,General theory ,Fourier analysis ,Lattice (order) ,symbols ,Equivalence relation ,0101 mathematics ,Mathematics::Representation Theory ,Martingale (probability theory) ,Analysis ,Mathematics - Abstract
Present paper deals with applications of asymptotic equivalence relations on operator nets. These relations are defined via unbounded convergences on vector lattices. Given two convergences $$\mathfrak {c}$$ and $$\mathfrak {d}$$ on a vector lattice, we study $$\mathfrak {d}$$ -asymptotic properties of operator nets formed by $$\mathfrak {c}$$ -continuous operators. Asymptotic equivalences are known to be useful and extremely important tools to study infinite behaviors of strongly convergent operator nets and continuous semigroups. After giving a general theory, paper focuses on $$\mathfrak {d}$$ -martingale and $$\mathfrak {d}$$ -Lotz–Rabiger nets.
- Published
- 2019
41. Continuability and Boundedness of Solutions for a Kind of Nonlinear Delay Integrodifferential Equations of the Third Order
- Author
-
Timur Ayhan and Cemil Tunç
- Subjects
Statistics and Probability ,Lyapunov function ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,01 natural sciences ,010305 fluids & plasmas ,Nonlinear system ,symbols.namesake ,Third order ,0103 physical sciences ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In the paper, we consider a nonlinear integrodifferential equation of the third order with delay. We establish sufficient conditions guaranteeing the global existence and boundedness of the solutions of the analyzed equation. We use the Lyapunov second method to prove the main result. An example is also given to illustrate the applicability of our result. The result of this paper is new and improves previously known results.
- Published
- 2018
42. Grüss type and related integral inequalities in probability spaces
- Author
-
László Horváth
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Inequality ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,MathematicsofComputing_NUMERICALANALYSIS ,Mathematics::Classical Analysis and ODEs ,Hilbert space ,Complex valued ,Type inequality ,Type (model theory) ,symbols.namesake ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,symbols ,Discrete Mathematics and Combinatorics ,Mathematics ,media_common - Abstract
In this paper we study Gruss type inequalities for real and complex valued functions in probability spaces. Some earlier Gruss type inequalities are extended and refined. Our approach leads to new integral inequalities which are interesting in their own right. As an application, we give a Gruss type inequality for normal operators in a Hilbert space. Similar results are obtained only for self-adjoint operators in earlier papers.
- Published
- 2018
43. Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I
- Author
-
Robert E. Gaunt
- Subjects
Statistics and Probability ,Mathematics(all) ,General Mathematics ,Stein’s method ,Computer Science::Digital Libraries ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Variance-gamma approximation ,Applied mathematics ,0101 mathematics ,Gaussian process ,Mathematics ,Laplace transform ,010102 general mathematics ,Distributional transformation ,Stein's method ,Rate of convergence ,Variance-gamma distribution ,Distribution (mathematics) ,Laplace's method ,symbols ,Statistics, Probability and Uncertainty ,Random variable - Abstract
The variance-gamma (VG) distributions form a four-parameter family that includes as special and limiting cases the normal, gamma and Laplace distributions. Some of the numerous applications include financial modelling and approximation on Wiener space. Recently, Stein’s method has been extended to the VG distribution. However, technical difficulties have meant that bounds for distributional approximations have only been given for smooth test functions (typically requiring at least two derivatives for the test function). In this paper, which deals with symmetric variance-gamma (SVG) distributions, and a companion paper (Gaunt 2018), which deals with the whole family of VG distributions, we address this issue. In this paper, we obtain new bounds for the derivatives of the solution of the SVG Stein equation, which allow for approximations to be made in the Kolmogorov and Wasserstein metrics, and also introduce a distributional transformation that is natural in the context of SVG approximation. We apply this theory to obtain Wasserstein or Kolmogorov error bounds for SVG approximation in four settings: comparison of VG and SVG distributions, SVG approximation of functionals of isonormal Gaussian processes, SVG approximation of a statistic for binary sequence comparison, and Laplace approximation of a random sum of independent mean zero random variables.
- Published
- 2018
44. On the Bessel–Wright Transform
- Author
-
Ilyes Karoui, Lazhar Dhaouadi, and Ahmed Fitouhi
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,010103 numerical & computational mathematics ,Eigenfunction ,Differential operator ,01 natural sciences ,symbols.namesake ,Operator (computer programming) ,Schwartz space ,symbols ,Differentiable function ,0101 mathematics ,Invariant (mathematics) ,Bessel function ,Lommel function ,Mathematics - Abstract
In the present paper, we consider a class of second-order singular differential operators which generalize the well-known Bessel differential operator. The associated eigenfunctions are the Bessel–Wright functions. These functions can be obtained by the action of the Riemann–Liouville operator on the normalized Bessel functions. We introduce a Bessel–Wright transform with Bessel–Wright functions as kernel which is connected to the classical Bessel–Fourier transform via the dual of the Riemann–Liouville operator. The Bessel–Wright transform leaves invariant the Schwartz space and sends the set of functions indefinitely differentiable with compact support into the Paley–Wiener space. We conclude the paper by proving two variants of the inversion formulas.
- Published
- 2018
45. $$L^p$$ L p Sobolev Regularity for a Class of Radon and Radon-Like Transforms of Various Codimension
- Author
-
Michael Greenblatt
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,020206 networking & telecommunications ,Resolution of singularities ,02 engineering and technology ,Codimension ,Surface (topology) ,01 natural sciences ,Measure (mathematics) ,Sobolev space ,Polyhedron ,symbols.namesake ,Fourier transform ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,0101 mathematics ,Oscillatory integral ,Analysis ,Mathematics - Abstract
In the paper (Greenblatt in J Funct Anal, https://doi.org/10.1016/j.jfa.2018.05.014 , 2018) the author proved $$L^p$$ Sobolev regularity results for averaging operators over hypersurfaces and connected them to associated Newton polyhedra. In this paper, we use rather different resolution of singularities techniques along with oscillatory integral methods applied to surface measure Fourier transforms to prove $$L^p$$ Sobolev regularity results for a class of averaging operators over surfaces which can be of any codimension.
- Published
- 2018
46. Phaseless Sampling and Reconstruction of Real-Valued Signals in Shift-Invariant Spaces
- Author
-
Junzheng Jiang, Qiyu Sun, and Cheng Cheng
- Subjects
FOS: Computer and information sciences ,Computer Science - Information Theory ,General Mathematics ,02 engineering and technology ,01 natural sciences ,symbols.namesake ,Robustness (computer science) ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Mathematics - Numerical Analysis ,0101 mathematics ,Invariant (mathematics) ,Sampling density ,Mathematics ,Box spline ,Partial differential equation ,Euclidean space ,Information Theory (cs.IT) ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,020206 networking & telecommunications ,Reconstruction algorithm ,Numerical Analysis (math.NA) ,Fourier analysis ,symbols ,Algorithm ,Analysis - Abstract
Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. In this paper, we consider phaseless sampling and reconstruction of real-valued signals in a high-dimensional shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. The determination of a signal in a shift-invariant space, up to a sign, by its magnitude measurements on the whole Euclidean space has been shown in the literature to be equivalent to its nonseparability. In this paper, we introduce an undirected graph associated with the signal in a shift-invariant space and use connectivity of the graph to characterize nonseparability of the signal. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that nonseparable signals in the shift-invariant space can be reconstructed in a stable way from their phaseless samples taken on that set. In this paper, we also propose a reconstruction algorithm which provides an approximation to the original signal when its noisy phaseless samples are available only. Finally, numerical simulations are performed to demonstrate the robustness of the proposed algorithm to reconstruct box spline signals from their noisy phaseless samples.
- Published
- 2018
47. Well-posedness and asymptotic behaviour of a wave equation with non-monotone memory kernel
- Author
-
Genqi Xu and Rongsheng Mu
- Subjects
Lyapunov function ,Semigroup ,Function space ,Applied Mathematics ,General Mathematics ,General Physics and Astronomy ,Monotonic function ,Function (mathematics) ,symbols.namesake ,Monotone polygon ,Exponential stability ,Kernel (statistics) ,symbols ,Applied mathematics ,Mathematics - Abstract
In this paper, we study the well-posedness and stability of a wave equation with infinitely structural memory, herein the memory kernel function does not satisfy the monotonicity. For the model, the history function space setting is a main difficulty because the usual space setting will lead the shift semigroup to be a unbounded semigroup. In the present paper, we modify the history function space setting and prove the well-posedness of the system. Further we study the stability of the system via Lyapunov function method. By constructing appropriate Lyapunov function, we show that the energy function of the system decays exponentially if the memory kernel function satisfies some conditions. Finally, we give an example of the memory kernel function that is not monotone but satisfies all conditions proposed in the present paper.
- Published
- 2021
48. Eigenfunction Expansions of Ultradifferentiable Functions and Ultradistributions. III. Hilbert Spaces and Universality
- Author
-
Aparajita Dasgupta and Michael Ruzhansky
- Subjects
Pure mathematics ,CONVOLUTION ,General Mathematics ,Structure (category theory) ,Boundary (topology) ,Type (model theory) ,Universality ,01 natural sciences ,Mathematics - Spectral Theory ,symbols.namesake ,Mathematics - Analysis of PDEs ,Primary 46F05 ,Tensor (intrinsic definition) ,0103 physical sciences ,FOS: Mathematics ,DISTRIBUTIONS ,Secondary 22E30 ,0101 mathematics ,Spectral Theory (math.SP) ,Mathematics ,Hilbert spaces ,Sequence ,Applied Mathematics ,010102 general mathematics ,Hilbert space ,Universality (philosophy) ,Eigenfunction ,Sequence spaces ,Smooth functions ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Mathematics and Statistics ,Physics and Astronomy ,Komatsu classes ,symbols ,Tensor representations ,010307 mathematical physics ,Primary 46F05, Secondary 22E30 ,Analysis ,Analysis of PDEs (math.AP) - Abstract
In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary., 23 pages
- Published
- 2021
49. The Computational Complexity of Plethysm Coefficients
- Author
-
Christian Ikenmeyer and Nick Fischer
- Subjects
FOS: Computer and information sciences ,Pure mathematics ,Rank (linear algebra) ,Computational complexity theory ,Geometric complexity theory ,General Mathematics ,Computational Complexity (cs.CC) ,68Q17, 05E10 ,Matrix multiplication ,Theoretical Computer Science ,Computer Science - Computational Complexity ,Computational Mathematics ,symbols.namesake ,Computational Theory and Mathematics ,Kronecker delta ,FOS: Mathematics ,symbols ,Computer Science::Symbolic Computation ,Tensor ,Representation Theory (math.RT) ,F.1.3 ,Time complexity ,Discrete tomography ,Mathematics - Representation Theory ,Mathematics - Abstract
In two papers, Bürgisser and Ikenmeyer (STOC 2011, STOC 2013) used an adaption of the geometric complexity theory (GCT) approach by Mulmuley and Sohoni (Siam J Comput 2001, 2008) to prove lower bounds on the border rank of the matrix multiplication tensor. A key ingredient was information about certain Kronecker coefficients. While tensors are an interesting test bed for GCT ideas, the far-away goal is the separation of algebraic complexity classes. The role of the Kronecker coefficients in that setting is taken by the so-called plethysm coefficients: These are the multiplicities in the coordinate rings of spaces of polynomials. Even though several hardness results for Kronecker coefficients are known, there are almost no results about the complexity of computing the plethysm coefficients or even deciding their positivity.In this paper, we show that deciding positivity of plethysm coefficients is -hard and that computing plethysm coefficients is #-hard. In fact, both problems remain hard even if the inner parameter of the plethysm coefficient is fixed. In this way, we obtain an inner versus outer contrast: If the outer parameter of the plethysm coefficient is fixed, then the plethysm coefficient can be computed in polynomial time. Moreover, we derive new lower and upper bounds and in special cases even combinatorial descriptions for plethysm coefficients, which we consider to be of independent interest. Our technique uses discrete tomography in a more refined way than the recent work on Kronecker coefficients by Ikenmeyer, Mulmuley, and Walter (Comput Compl 2017). This makes our work the first to apply techniques from discrete tomography to the study of plethysm coefficients. Quite surprisingly, that interpretation also leads to new equalities between certain plethysm coefficients and Kronecker coefficients.
- Published
- 2020
50. New criteria for Vandiver’s conjecture using Gauss sums – Heuristics and numerical experiments
- Author
-
Georges Gras
- Subjects
Discrete mathematics ,Conjecture ,Mathematics::Number Theory ,General Mathematics ,Modulo ,010102 general mathematics ,Galois group ,Order (ring theory) ,Cyclotomic field ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Gauss sum ,symbols ,0101 mathematics ,Bernoulli number ,Mathematics ,Counterexample - Abstract
The link between Vandiver’s conjecture and Gauss sums is well known since the papers of Iwasawa (Symposia Mathematica, vol 15, Academic Press, pp 447–459, 1975), Thaine (Mich Math J 42(2):311–344, 1995; Trans Am Math Soc 351(12):4769–4790, 1999) and Angles and Nuccio (Acta Arith 142(3):199–218, 2010). This conjecture is required in many subjects and we shall give such examples of relevant references. In this paper, we recall our interpretation of Vandiver’s conjecture in terms of minus part of the torsion of the Galois group of the maximal abelian p-ramified pro-p-extension of the p-th cyclotomic field (Sur la p-ramification abelienne (1984) vol. 20, pp. 1–26). Then we provide a specific use of Gauss sums of characters of order p of $${\mathbb {F}}_\ell ^\times $$ and prove new criteria for Vandiver’s conjecture to hold (Theorem 2 (a) using both the sets of exponents of p-irregularity and of p-primarity of suitable twists of the Gauss sums, and Theorem 2 (b) which does not need the knowledge of Bernoulli numbers or cyclotomic units). We propose in §5.2 new heuristics showing that any counterexample to the conjecture leads to excessive constraints modulo p on the above twists as $$\ell $$ varies and suggests analytical approaches to evidence. We perform numerical experiments to strengthen our arguments in the direction of the very probable truth of Vandiver’s conjecture and to inspire future research. The calculations with their PARI/GP programs are given in appendices.
- Published
- 2020
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