5,356 results
Search Results
2. A Note on Recent Papers by Grafakos and Teschl, and Estrada
- Author
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Adam Nowak and Krzysztof Stempak
- Subjects
Hankel transform ,Partial differential equation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Function (mathematics) ,Transplantation ,symbols.namesake ,Radial function ,Fourier transform ,Fourier analysis ,symbols ,Analysis ,Mathematics - Abstract
We indicate how recent results of Grafakos and Teschl (J Fourier Anal Appl 19:167–179, 2013), and Estrada (J Fourier Anal Appl 20:301–320, 2014), relating the Fourier transform of a radial function in $$\mathbb R^n$$ and the Fourier transform of the same function in $$\mathbb R^{n+2}$$ and $$\mathbb R^{n+1}$$ , respectively, are located within known results on transplantation for Hankel transforms.
- Published
- 2014
3. Corrigendum to the paper 'Adjoining an Order Unit to a Matrix Ordered Space'
- Author
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Anil Kumar Karn
- Subjects
Pure mathematics ,General Mathematics ,Mathematical analysis ,Operator theory ,Potential theory ,Theoretical Computer Science ,Strictly convex space ,symbols.namesake ,Matrix (mathematics) ,Fourier analysis ,Ordered space ,symbols ,Order (group theory) ,Unit (ring theory) ,Analysis ,Mathematics - Abstract
An error has been detected (and also corrected) in Theorem 2.8 of the paper entitled “Adjoining an Order Unit to a Matrix Ordered Space” (Positivity, (2005)9: 207–223; DOI 10.1007/s11117-003-2778-5). Accordingly, some of the results of the paper have been modified. Also, a notion of C*-matricially, Riesz normed spaces has been introduced.
- Published
- 2007
4. A remark on a paper of F. Luca and A. Sankaranarayanan
- Author
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Imre Kátai
- Subjects
Set (abstract data type) ,Discrete mathematics ,symbols.namesake ,Number theory ,Statement (logic) ,General Mathematics ,Ordinary differential equation ,Multiplicative function ,Zero (complex analysis) ,symbols ,Calculus ,Euler's totient function ,Mathematics - Abstract
We generalize a result of F. Luca and A. Sankaranarayanan by proving that the set of n for which ϕ(1) + + ϕ(n) is squareful is of zero density. A similar statement holds for σ (n) instead of ϕ(n) and for some other multiplicative functions.
- Published
- 2008
5. Correction to the paper 'on the curvature of a generalization of a contact metric manifolds'
- Author
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L. Di Terlizzi
- Subjects
Christoffel symbols ,Riemann curvature tensor ,symbols.namesake ,Mean curvature flow ,General Mathematics ,Mathematical analysis ,symbols ,Curvature form ,Sectional curvature ,Curvature ,Ricci curvature ,Scalar curvature ,Mathematics - Abstract
We considered in Example 3.1 of the paper [1] an S-structure on R2n+s . We concluded that when s > 1 this manifold cannot be of constant φ-sectional curvature. Unfortunately this result is wrong. In fact, essentially due to a sign mistake in defining the φ-structure and a consequent transposition of the elements of the φ-basis (3.2), some of the Christoffel’s symbols were incorrect. In the present rectification, using a more slendler tecnique, we prove that our manifold is of constant φ-sectional curvature −3s and then it is η-Einstein.
- Published
- 2009
6. On the Paper 'A Note on Spaces of Absolutely Convergent Fourier Transforms' by Björn G.Walther (this Issue). Letter to the Editor
- Author
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S. V. Kislyakov
- Subjects
Algebra ,symbols.namesake ,Letter to the editor ,Partial differential equation ,Fourier transform ,Fourier analysis ,Applied Mathematics ,General Mathematics ,Mathematics education ,symbols ,Absolute convergence ,Analysis ,Mathematics - Published
- 2014
7. Appearance of Temporal and Spatial Chaos in an Ecological System: A Mathematical Modeling Study
- Author
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S. N. Raw, B P Sarangi, P. Mishra, and B. Tiwari
- Subjects
Patter formulation ,Computer science ,General Mathematics ,General Physics and Astronomy ,Lyapunov exponent ,01 natural sciences ,Stability (probability) ,symbols.namesake ,Quantitative Biology::Populations and Evolution ,Statistical physics ,0101 mathematics ,Bifurcation ,Hopf bifurcation ,Computer simulation ,Phase portrait ,Turing instability ,010102 general mathematics ,Time evolution ,General Chemistry ,Function (mathematics) ,010101 applied mathematics ,symbols ,Chaos ,Mutual interference ,General Earth and Planetary Sciences ,General Agricultural and Biological Sciences ,Research Paper - Abstract
The ecological theory of species interactions rests largely on the competition, interference, and predator–prey models. In this paper, we propose and investigate a three-species predator–prey model to inspect the mutual interference between predators. We analyze boundedness and Kolmogorov conditions for the non-spatial model. The dynamical behavior of the system is analyzed by stability and Hopf bifurcation analysis. The Turing instability criteria for the Spatio-temporal system is estimated. In the numerical simulation, phase portrait with time evolution diagrams shows periodic and chaotic oscillations. Bifurcation diagrams show the very rich and complex dynamical behavior of the non-spatial model. We calculate the Lyapunov exponent to justify the dynamics of the non-spatial model. A variety of patterns like interference, spot, and stripe are observed with special emphasis on Beddington–DeAngelis function response. These complex patterns explore the beauty of the spatio-temporal model and it can be easily related to real-world biological systems.
- Published
- 2021
8. A remark to a paper of Kato and Ikebe
- Author
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Wolf von Wahl
- Subjects
Pure mathematics ,General Mathematics ,Mathematical analysis ,Algebraic geometry ,Sobolev space ,symbols.namesake ,Number theory ,Operator (computer programming) ,Square-integrable function ,symbols ,Order (group theory) ,Element (category theory) ,Schrödinger's cat ,Mathematics - Abstract
This paper deals with Schrodinger operators as they were treated by Kato - Ikebe [3]. It is shown that every element of the domain of definition of the adjoint of such an operator has locally square integrable distributional derivatives up to the order 2. For this the potential of the Schrodinger operator must fulfil a local Stummel condition; if the potential is only locally square integrable a somewhat weaker statement is possible for three dimensions (see remark 2 at the end of this paper).
- Published
- 1977
9. The greatest mathematical paper of all time
- Author
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A. J. Coleman
- Subjects
Weyl group ,Pure mathematics ,General Mathematics ,Cartan decomposition ,Killing form ,Kac–Moody algebra ,Affine Lie algebra ,Algebra ,symbols.namesake ,History and Philosophy of Science ,symbols ,Cartan matrix ,Lie theory ,Mathematics::Representation Theory ,E8 ,Mathematics - Abstract
Why do I think that Z.v.G.II was an epoch-making paper? (1) It was the paradigm for subsequent efforts to classify the possible structures for any mathematical object. Hawkins [15] documents the fact that Killing’s paper was the immediate inspiration for the work of Cartan, Molien, and Maschke on the structure of linearassociative algebras which culminated in Wedderburn’s theorems. Killing’s success was certainly an example which gave Richard Brauer the will to persist in the attempt to classify simple groups. (2) Weyl’s theory of the representation of semi-simple Lie groups would have been impossible without ideas, results, and methods originated by Killing in Z.v.G.II. Weyl’s fusion of global and local analysis laid the basis for the work of Harish-Chandra and the flowering of abstract harmonic analysis. (3) The whole industry of root systems evinced in the writings of I. Macdonald, V. Kac, R. Moody, and others started with Killing. For the latest see [21]. (4) The Weyl group and the Coxeter transformation are in Z.v.G.II. There they are realized not as orthogonal motions of Euclidean space but as permutations of the roots. In my view, this is the proper way to think of them for general Kac-Moody algebras. Further, the conditions for symmetrisability which play a key role in Kac’s book [17] are given on p. 21 of Z.v.G.II. (5) It was Killing who discovered the exceptional Lie algebra E8, which apparently is the main hope for saving Super-String Theory—not that I expect it to be saved! (6) Roughly one third of the extraordinary work of Elie Cartan was based more or less directly on Z.v.G.II.
- Published
- 1989
10. 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
- Subjects
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
11. Note on a paper of B. Grünbaum on acyclic colorings
- Author
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Gerd Wegner
- Subjects
Discrete mathematics ,symbols.namesake ,General Mathematics ,symbols ,Algebra over a field ,Arithmetic ,Notation ,Group theory ,Planar graph ,Mathematics - Abstract
The aim of this short note is to improve some recent results of B. Grunbaum by some remarks. We use Grunbaum's notations.
- Published
- 1973
12. An addendum to the paper ‘Dynamic response of an infinite plate subjected to a steadily moving transverse force’
- Author
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David H. Y. Yen and Clifford C. Chou
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,General Physics and Astronomy ,Addendum ,Transverse force ,Integral transform ,Physics::Fluid Dynamics ,symbols.namesake ,Classical mechanics ,Mach number ,Plate theory ,symbols ,Initial value problem ,Supersonic speed ,Mathematics - Abstract
The dynamic response of an infinite, elastic plate to a steadily moving transverse force is studied under the so-called improved plate theory. Analytic solutions for the subsonic and the intersonic cases are obtained using the method of integral transforms. The usual difficulty of the nonuniqueness of the steady-state solutions is circumvented by formulating the problems as initial value problems and then seeking the large time limits of the solutions of such initial value problems. The subsonic and intersonic solutions obtained here are similar to that obtained previously for the supersonic case. However, in contrast with the supersonic solution, which contains two families of Mach lines, the subsonic solution contains no Mach lines and the intersonic solution contains only one family of Mach lines.
- Published
- 1974
13. Note on a paper by C. C. Brown
- Author
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S. J. Bernau
- Subjects
Statistics and Probability ,Discrete mathematics ,symbols.namesake ,Sequence ,General Mathematics ,Hilbert space ,symbols ,Statistics, Probability and Uncertainty ,Lambda ,Analysis ,Self-adjoint operator ,Mathematics - Abstract
Let H be a real or complex Hilbert space and let L p(1≦p
- Published
- 1969
14. Note on my paper 'a simple proof for von Neumann's minimax theorem'
- Author
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I. Joó
- Subjects
Pure mathematics ,General Mathematics ,Minimax theorem ,symbols.namesake ,Von Neumann's theorem ,Parthasarathy's theorem ,Von Neumann algebra ,Calculus ,symbols ,Danskin's theorem ,Abelian von Neumann algebra ,Affiliated operator ,Analytic proof ,Mathematics - Published
- 1984
15. x + 674 pp. $26.50 hardback, $14.50 paperbackE.C. Zeeman, Catastrophe Theory: Selected Papers 1972–1977, Addison-Wesley Publishing Co. (1977)
- Author
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Robert Rosen
- Subjects
Pharmacology ,Zeeman effect ,business.industry ,General Mathematics ,General Neuroscience ,Philosophy ,Immunology ,General Biochemistry, Genetics and Molecular Biology ,symbols.namesake ,Computational Theory and Mathematics ,Publishing ,symbols ,Catastrophe theory ,General Agricultural and Biological Sciences ,business ,General Environmental Science ,Mathematical physics - Published
- 1979
16. Note to the paper by R. Hempel, A. M. Hinz, and H. Kalf: On the essential spectrum of Schr�dinger operators with spherically symmetric potentials
- Author
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J. Weidmann
- Subjects
symbols.namesake ,General Mathematics ,Essential spectrum ,Mathematical analysis ,symbols ,Schrödinger's cat ,Mathematical physics ,Mathematics - Published
- 1987
17. Extinction probabilities in branching processes: A note on holgate and Lakhani's paper
- Author
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D. J. Daley
- Subjects
Pharmacology ,Mathematical and theoretical biology ,Extinction ,General Mathematics ,General Neuroscience ,Immunology ,General Medicine ,Poisson distribution ,General Biochemistry, Genetics and Molecular Biology ,Combinatorics ,Branching (linguistics) ,symbols.namesake ,Computational Theory and Mathematics ,symbols ,General Agricultural and Biological Sciences ,General Environmental Science ,Branching process ,Mathematics - Abstract
Within the class of offspring distributions with given meanm>1 and probability of no offspringp o, the probabilityq of ultimate extinction in a Galton-Watson branching process starting from one individual satisfiesp 0
- Published
- 1969
18. A remark on my paper ?On the Saito-Kurokawa lifting?
- Author
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I. I. Piatetski-Shapiro
- Subjects
Algebra ,symbols.namesake ,Automorphic L-function ,General Mathematics ,Artin L-function ,Eisenstein series ,Langlands–Shahidi method ,symbols ,Automorphic form ,Mathematics - Published
- 1984
19. Note on the paper of H. Amato und G. Mensch: Rank restriction on the quadratic form in indefinite quadratic programming
- Author
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Götz Uebe
- Subjects
General Mathematics ,Quadratic function ,Management Science and Operations Research ,Isotropic quadratic form ,Legendre symbol ,Combinatorics ,Definite quadratic form ,Algebra ,symbols.namesake ,Quadratic form ,symbols ,Binary quadratic form ,Quadratic field ,Quadratic programming ,Software ,Mathematics - Published
- 1972
20. 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
- Subjects
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
21. Remark on the papers 'Certain properties of class functions and interpolation problems' and 'An analog of the poisson-jensen formula with a double integral'
- Author
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V. P. Kabaila
- Subjects
Pure mathematics ,symbols.namesake ,Class (set theory) ,Number theory ,General Mathematics ,Ordinary differential equation ,Multiple integral ,Mathematical analysis ,Class function ,symbols ,Poisson distribution ,Mathematics ,Interpolation - Published
- 1973
22. d-Hermite rings and skew $$\textit{PBW}$$ PBW extensions
- Author
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Oswaldo Lezama and Claudia Gallego
- Subjects
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
23. A procedure for measuring the flexibility of single wood-pulp fibres
- Author
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J. W. Provan, C. Biggs, M. Tchepel, and A. Nishida
- Subjects
Materials science ,Softwood ,Polymers and Plastics ,Confocal laser scanning microscope ,General Mathematics ,Pulp (paper) ,Second moment of area ,Young's modulus ,engineering.material ,Condensed Matter Physics ,Black spruce ,Biomaterials ,symbols.namesake ,Mechanics of Materials ,Solid mechanics ,Ceramics and Composites ,symbols ,engineering ,Composite material ,Elastic modulus - Abstract
A way of determining the flexibility of wood-pulp fibres is developed, which involves i) a precise measurement of the topology of single-fibres by using a confocal laser scanning microscope and ii) the measurement of the elastic modulus of the fibres by using a single-fibre fatigue cell. Reported in this paper are the initial results of tests carried out on black spruce fibres, which have been subjected to three different levels of mechanical refining energy, namely ∼1100, 2300, and 3500 kWh/t. It is found that the fibre flexibility rises significantly between the first and second energy levels, but it does not change to the same degree between the second and third ones. The described procedure of measuring the flexibility of fibres may be used to establish the appropriate refiner energy necessary for the production of a specific grade of paper.
- Published
- 2006
24. On Lacunas in the Spectrum of the Laplacian with the Dirichlet Boundary Condition in a Band with Oscillating Boundary
- Author
-
Denis Borisov
- Subjects
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.
- Published
- 2021
25. Logarithmic Potential and Generalized Analytic Functions
- Author
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O.V. Nesmelova, Vladimir Gutlyanskiĭ, Vladimir Ryazanov, and A.S. Yefimushkin
- Subjects
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
26. The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation
- Author
-
Ihyeok Seo and Yoonjung Lee
- Subjects
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
27. 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
- Subjects
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.
- Published
- 2021
28. Existence and Uniqueness of the Global L1 Solution of the Euler Equations for Chaplygin Gas
- Author
-
Zhen Wang, Tingting Chen, and Aifang Qu
- Subjects
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
29. On the pair correlations of powers of real numbers
- Author
-
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
30. Concomitants of Generalized Order Statistics from Bivariate Cambanis Family of Distributions Under a General Setting
- Author
-
Haroon M. Barakat, M. A. Alawady, and M. A. Abd Elgawad
- Subjects
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
31. Central limit theorems for the ℤ2-periodic Lorentz gas
- Author
-
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
32. Approximating a common solution of extended split equality equilibrium and fixed point problems
- Author
-
J. M. Ngnotchouye, F. U. Ogbuisi, and F. O. Isiogugu
- Subjects
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
33. Mapped Regularization Methods for the Cauchy Problem of the Helmholtz and Laplace Equations
- Author
-
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
34. 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
35. Global well-posedness for fractional Navier-Stokes equations in variable exponent Fourier-Besov-Morrey spaces
- Author
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Jiecheng Chen and Muhammad Zainul Abidin
- Subjects
Physics ,Variable exponent ,General Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,General Physics and Astronomy ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,symbols.namesake ,Fourier transform ,symbols ,Exponent ,0101 mathematics ,Navier–Stokes equations ,Well posedness - Abstract
In this paper we study the Cauchy problem of the incompressible fractional Navier-Stokes equations in critical variable exponent Fourier-Besov-Morrey space $${\cal F}\dot {\cal N}_{p\left( \cdot \right),h\left( \cdot \right),q}^{s\left( \cdot \right)}\left( {{\mathbb{R}^3}} \right)$$ with $$s\left( \cdot \right) = 4 - 2\alpha - {3 \over {p\left( \cdot \right)}}$$ . We prove global well-posedness result with small initial data belonging to $${\cal F}\dot {\cal N}_{p\left( \cdot \right),h\left( \cdot \right),q}^{4 - 2\alpha - {3 \over {p\left( \cdot \right)}}}\left( {{\mathbb{R}^3}} \right)$$ The result of this paper extends some recent work.
- Published
- 2020
36. An Improved Noise Quantum Annealing Method for TSP
- Author
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Zhijie Huang and Yumin Dong
- Subjects
Physics and Astronomy (miscellaneous) ,Pauli matrices ,Basis (linear algebra) ,010308 nuclear & particles physics ,Computer science ,General Mathematics ,Quantum annealing ,Context (language use) ,01 natural sciences ,Travelling salesman problem ,symbols.namesake ,Exact algorithm ,Distance matrix ,0103 physical sciences ,symbols ,010306 general physics ,Algorithm ,Quantum tunnelling - Abstract
Traveling Salesman Problem (TSP) is a combinatorial optimization problem, which has NP-Complete (NPC) complexity. At present, there is no exact algorithm to solve similar problems, only an approximate algorithm to simplify the problem can be found. For example, quantum annealing algorithm (QA) can play such a role. QA transforms the distance matrix sum of TSP into Pauli matrix, adds a transverse magnetic field to realize the quantum tunneling effect, reduces the energy needed to cross the barrier, and reduces the number of iterations to find the optimal solution. However, although the QA has fewer iteration steps, it usually produces errors. In this context, this paper’s INQA improves the QA. In this paper, the theoretical basis of quantum annealing algorithm is introduced, aiming at the shortcomings of quantum annealing algorithm in solving TSP problem, a new algorithm INQA is proposed, and it is verified by experiments. In the case of high initial temperature, the INQA has larger search range and more active search. Meanwhile, with the decrease of temperature, the search range is reduced, and the accuracy of the algorithm is improved. In fact, QA in this paper is a quantum annealing method for simulation.
- Published
- 2020
37. 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
- Subjects
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
38. 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
39. 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
40. More about singular traces on simply generated operator ideals
- Author
-
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
41. 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
42. 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
43. The Wiener Measure on the Heisenberg Group and Parabolic Equations
- Author
-
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
44. 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
45. 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
46. Titchmarsh’s theorem and some remarks concerning the right-sided quaternion Fourier transform
- Author
-
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
47. Necessary optimality conditions for a semivectorial bilevel optimization problem using the kth-objective weighted-constraint approach
- Author
-
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
48. A weak integral condition and its connections with existence and uniqueness of solutions for some abstract Cauchy problems in Banach spaces
- Author
-
Donal O'Regan and Constantin Buse
- Subjects
Physics ,Measurable function ,010505 oceanography ,Semigroup ,General Mathematics ,010102 general mathematics ,Banach space ,Hilbert space ,Duality (optimization) ,Cauchy distribution ,01 natural sciences ,Combinatorics ,symbols.namesake ,symbols ,Uniqueness ,0101 mathematics ,0105 earth and related environmental sciences ,Counterexample - Abstract
In control theory, problems occur regarding the behavior of solutions of some abstract Cauchy problems like that 0.1$$\begin{aligned} \begin{array}{ll} u'(t) =&{} -A(u(t))-f(t)b,\quad t\in \mathbb {R} \\ u(\infty )=&{}\lim \nolimits _{t\rightarrow \infty }u(t)=0. \end{array} \end{aligned}$$Here A generates a strongly continuous semigroup $$\mathbf{T}=\{T(t)\}$$ acting on a complex Banach space X, f is a complex valued measurable function defined on $$\mathbb {R}_+$$ verifying a certain integral condition (as in Theorem 4.1 below), $$b\in X$$ is a randomly chosen vector and the limit is considered in the norm of X. We prove that the Cauchy Problem (0.1) has at least one solution (that is unique when X is a complex Hilbert space) provided the semigroup $$\mathbf{T}$$ is $$\Phi $$-weakly stable, that is, for every $$x\in X$$ and $$x'\in X'$$ of norms less than or equal to 1 the map $$t\mapsto \Phi (|\langle e^{tA}x, x'\rangle |$$ belongs to $$L^1(\mathbb {R}_+)$$. Concrete examples and even the expression of solutions are also provided in this paper. Here $$\Phi $$ is a given N-function, $$X'$$ denotes the strong dual of X and $$\langle \cdot , \cdot \rangle $$ denotes the duality map between X and $$X'$$. It is known (Storozhuk in Sib Math J 51:330–337, 2010) that the uniform spectral bound $$s_0(A)$$ is negative whenever the semigroup $$\mathbf{T}$$ generated of A is $$\Phi $$-weakly stable for the above $$\Phi $$. We complete this result by proving that if the semigroup is $$\Phi $$-weakly stable then there exists a positive number $$\nu $$ such that $$s_0(A)\le -\nu $$. An implicit expression of $$\nu $$, in terms of $$\Phi $$ and $$\mathbf{T}$$, is also given. The condition that $$\Phi $$ is positive near to 0 is necessary in the proofs. A counterexample showing this is provided in the last section of the paper.
- Published
- 2019
49. Stability and instability results for Cauchy laminated Timoshenko-type systems with interfacial slip and a heat conduction of Gurtin–Pipkin’s law
- Author
-
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
50. 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
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