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2. Corrigendum to the papers on Exceptional orthogonal polynomials: J. Approx. Theory 182 (2014) 29–58, 184 (2014) 176–208 and 214 (2017) 9–48
- Author
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Antonio J. Durán
- Subjects
Numerical Analysis ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Hilbert space ,Approx ,symbols.namesake ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Orthogonal polynomials ,symbols ,Analysis ,Mathematics - Abstract
We complete a gap in the proof that exceptional polynomials are complete orthogonal systems in the associated Hilbert spaces.
- Published
- 2020
3. Erratum to the paper 'L∞(L∞)-boundedness and convergence of DG(p)-solutions for nonlinear conservation laws with boundary conditions'
- Author
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Christian Henke and Lutz Angermann
- Subjects
Conservation law ,Pure mathematics ,Lemma (mathematics) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Lebesgue integration ,Computational Mathematics ,Nonlinear system ,symbols.namesake ,Convergence (routing) ,symbols ,Boundary value problem ,Affine transformation ,Constant (mathematics) ,Mathematics - Abstract
In the paper (HA14), unfortunately, a computational error occurred in one estimate. Although the wrong estimate does not affect the main results, we want to present the necessary corrections. Essentially, Lemma 5.2 has to be corrected and, since it is used in the proof of Theorem 5.1, the proof of this theorem also requires an adaptation. (i) The corrected formulation of Lemma 5.2 is as follows. Lemma 5.2 For Lagrange finite elements with a shape-regular family of affine meshes { T n h } h>0 there is a constant C > 0 independent of q and h such that for all w ∈ Wh and q = 2m, m ∈N: CΛq−2 p (∇w,∇Ip h (wq−1))T ∫ T ‖∇w‖l2‖w‖ q−2 0,∞,T dx, ∀T ∈ T n h , (5.1) where Λp = ‖ ∑ndof i=1 |φi|‖0,∞,T is the Lebesgue constant.
- Published
- 2015
4. A weighted uniform $L^{p}$--estimate of Bessel functions: A note on a paper of Guo
- Author
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Krzysztof Stempak
- Subjects
symbols.namesake ,Cylindrical harmonics ,Bessel process ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Struve function ,Bessel polynomials ,symbols ,Calculus ,Bessel function ,Lommel function ,Mathematics - Published
- 2000
5. Remarks on DiPerna’s paper 'Convergence of the viscosity method for isentropic gas dynamics'
- Author
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Gui-Qiang Chen
- Subjects
Discrete mathematics ,Isentropic process ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Vacuum state ,Finite difference method ,Euler equations ,Binary entropy function ,symbols.namesake ,Riemann hypothesis ,Compact space ,Mathematics Subject Classification ,symbols ,Mathematics - Abstract
Concerns have been voiced about the correctness of certain technical points in DiPerna’s paper (Comm. Math. Phys. 91 (1983), 1–30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods. In [Di], DiPerna found a global entropy solution of the isentropic Euler equations for the following exponents in the equation of state for the pressure: γ = 1 + 2/(2m+ 1), m ≥ 2 integer. (1) He divided his arguments into the following two steps. 1. Compactness framework Assume that a sequence of approximate solutions (ρ (x, t),m (x, t)), 0 ≤ t ≤ T , satisfies: (i). There exists a constant C(T ) > 0, independent of > 0, such that 0 ≤ ρ (x, t) ≤ C, |m (x, t)/ρ (x, t)| ≤ C; (ii). For all weak entropy pairs (η, q) of the isentropic Euler equations, the measure sequence η(ρ ,m )t + q(ρ ,m )x is contained in a compact subset of H −1 loc (R× [0, T ]). If γ satisfies (1), then the sequence (ρ (x, t),m (x, t)) is compact in Lloc(R× [0, T ]). The reason for the restriction on the number γ is that, in such a case, any weak entropy function is a polynomial function of the Riemann invariants (w, z). This is the key step in DiPerna’s arguments and is also his main contribution to the compensated compactness method in this aspect. Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 35K55, 35L65; Secondary 76N15, 35L60, 65M06.
- Published
- 1997
6. Response to Bucy’s comment on a paper by Udwadia and Kalaba
- Author
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Firdaus E. Udwadia and Robert E. Kalaba
- Subjects
Rank (linear algebra) ,Independent equation ,General Mathematics ,Mathematical analysis ,General Engineering ,General Physics and Astronomy ,Equations of motion ,symbols.namesake ,Theory of equations ,Simultaneous equations ,Lagrange multiplier ,Costate equations ,symbols ,Applied mathematics ,Mathematics ,Numerical partial differential equations - Abstract
We disagree with Bucy’s comments (Bucy 1994) on our (Udwadia &; Kalaba 1992) paper. The reasons are as follows. 1. Just as the Gibbs-Appell equations (which use quasi-coordinates) are not the same as the Lagrange equations (which use Lagrange multipliers), the equations of motion obtained by us are not the same as the Gibbs-Appell equations. Some of the steps required to obtain the Gibbs-Appell equations are (Pars 1979; Whittaker 1937): (1) choice of quasi-coordinates; (2) elimination of certain quasicoordinates in terms of other preferred quasi-coordinates; (3) setting up of the Gibbs function; and (4) differentiation of the Gibbs function with respect to the preferred quasi-coordinates. None of these steps is used in obtaining our equations of motion. 2. Yet, the equations of motion obtained by us are equivalent to the Lagrange equations with multipliers (Kalaba et al. 1995) and to the Gibbs-Appell equations (Udwadia & Kalaba 1996). By equivalent we mean each set of equations implies the other. In fact all these different sets of equations are simply different, yet equivalent, ways of stating D’Alembert’s principle for bilinear constraints. Thus what applies to one set of equations applies to the others. Any deficiency in the equations derived in our paper (say regarding rank changes of the matrix A) is therefore present in the Gibbs-Appell equations and Lagrange’s equations as well, because of their equivalence.
- Published
- 1996
7. An operator valued function space integral: A sequel to Cameron and Storvick’s paper
- Author
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D. L. Skoug and G. W. Johnson
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Multiple integral ,Integral representation theorem for classical Wiener space ,Mathematical analysis ,Riemann integral ,Riemann–Stieltjes integral ,Singular integral ,Fourier integral operator ,Volume integral ,symbols.namesake ,symbols ,Daniell integral ,Mathematics - Abstract
Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.
- Published
- 1971
8. Book Review: The lost notebook and other unpublished papers
- Author
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Richard Askey
- Subjects
symbols.namesake ,Applied Mathematics ,General Mathematics ,symbols ,Ramanujan's sum ,Mathematics - Published
- 1988
9. The Application of Basic Numbers to Bessel's and Legendre's Functions (Second Paper)
- Author
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F. H. Jackson
- Subjects
Bessel process ,Legendre wavelet ,General Mathematics ,Legendre's equation ,Legendre function ,symbols.namesake ,Bessel polynomials ,Struve function ,symbols ,Calculus ,Applied mathematics ,Legendre's constant ,Legendre polynomials ,Mathematics - Abstract
n/a
- Published
- 1905
10. ON THE DISTRIBUTION OF THE CORRELATION COEFFICIENT IN SMALL SAMPLES. APPENDIX II TO THE PAPERS OF 'STUDENT' AND R. A. FISHER. A COOPERATIVE STUDY
- Author
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H. E. Soper, A. W. Young, A. Lee, Karl Pearson, and B. M. Cave
- Subjects
Statistics and Probability ,Distribution (number theory) ,Correlation coefficient ,Intraclass correlation ,Applied Mathematics ,General Mathematics ,Fisher transformation ,Correlation ratio ,Agricultural and Biological Sciences (miscellaneous) ,Spearman's rank correlation coefficient ,Pearson product-moment correlation coefficient ,symbols.namesake ,Statistics ,symbols ,Statistics, Probability and Uncertainty ,General Agricultural and Biological Sciences ,Mathematics - Published
- 1917
11. Some remarks on A. C. Schaeffer's paper on Dirichlet series
- Author
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Hwa S. Hahn
- Subjects
symbols.namesake ,General Mathematics ,Dirichlet's principle ,Calculus ,symbols ,Applied mathematics ,30.24 ,Dirichlet's energy ,Dirichlet series ,Mathematics - Published
- 1960
12. Observations on a paper by Rosenblum
- Author
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S. Cater
- Subjects
Complex conjugate ,Applied Mathematics ,General Mathematics ,Hilbert space ,Uniform limit theorem ,Combinatorics ,symbols.namesake ,Operator (computer programming) ,Skew-Hermitian matrix ,Bounded function ,symbols ,Normal operator ,Complex number ,Mathematics - Abstract
M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.
- Published
- 1961
13. 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
14. On the Zeros of Dirichlet L-Functions.II (With Corrections to Ön the Zeros of Dirichlet L-Functions.I' and the Subsequent Papers)
- Author
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Akio Fujii
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Dirichlet L-function ,Dirichlet's energy ,Dirichlet eta function ,Class number formula ,symbols.namesake ,Dirichlet kernel ,Dirichlet's principle ,symbols ,General Dirichlet series ,Dirichlet series ,Mathematics - Published
- 1981
15. Correction to the Paper 'A Problem Concerning Orthogonal Polynomials'
- Author
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G. Szegö
- Subjects
Gegenbauer polynomials ,Applied Mathematics ,General Mathematics ,Discrete orthogonal polynomials ,Mathematical analysis ,Classical orthogonal polynomials ,Algebra ,symbols.namesake ,Wilson polynomials ,Orthogonal polynomials ,Hahn polynomials ,symbols ,Jacobi polynomials ,Koornwinder polynomials ,Mathematics - Published
- 1936
16. On the Distribution of the Correlation Coefficient in Small Samples. Appendix II to the Papers of 'Student' and R. A. Fisher
- Author
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Karl Pearson, A. W. Young, A. Lee, H. E. Soper, and B. M. Cave
- Subjects
Statistics and Probability ,Correlation coefficient ,Distribution (number theory) ,Intraclass correlation ,Applied Mathematics ,General Mathematics ,Fisher transformation ,Correlation ratio ,Agricultural and Biological Sciences (miscellaneous) ,Pearson product-moment correlation coefficient ,symbols.namesake ,Statistics ,symbols ,Statistics, Probability and Uncertainty ,General Agricultural and Biological Sciences ,Mathematics - Published
- 1917
17. Application of Bernoulli Polynomials of Negative Order to Differencing (Second Paper)
- Author
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B. F. Kimball
- Subjects
Classical orthogonal polynomials ,symbols.namesake ,Difference polynomials ,General Mathematics ,Discrete orthogonal polynomials ,Mathematical analysis ,symbols ,Applied mathematics ,Order (ring theory) ,Bernoulli process ,Mathematics ,Bernoulli polynomials - Published
- 1934
18. Application of the Theory of Relative Cyclic Fields to both Cases of Fermat's Last Theorem (Second Paper)
- Author
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H. S. Vandiver
- Subjects
Pure mathematics ,Fermat's little theorem ,Proofs of Fermat's little theorem ,Applied Mathematics ,General Mathematics ,Regular prime ,Fermat's theorem on sums of two squares ,Wieferich prime ,Fermat's factorization method ,symbols.namesake ,Fermat's theorem ,symbols ,Mathematics ,Fermat number - Published
- 1927
19. A Correction to the Paper 'On Effective Sets of Points in Relation to Integral Functions'
- Author
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V. Ganapathy Iyer
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,Line integral ,Riemann–Stieltjes integral ,Riemann integral ,Fourier integral operator ,Volume integral ,symbols.namesake ,Improper integral ,symbols ,Coarea formula ,Daniell integral ,Mathematics - Published
- 1938
20. A remark on Neuwirth and Newman’s paper: 'Positive 𝐻^{1/2} functions are constants'
- Author
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Shinji Yamashita
- Subjects
Combinatorics ,Lemma (mathematics) ,symbols.namesake ,Applied Mathematics ,General Mathematics ,Blaschke product ,symbols ,Function (mathematics) ,Absolute value (algebra) ,Boundary values ,Decomposition theorem ,Mathematical physics ,Mathematics - Abstract
PROOF. By a theorem of Rudin a function gEH' in U whose boundary values are real a.e. on I can be analytically continued to D [3, p. 59]. The lemma follows on applying Rudin's result to gi= (1/2) (fl+f2) and g2=(i/2) (fi-f2). PROOF OF THEOREM 1. By a well-known decomposition theorem [2, p. 87], f(z)=B(z)F2(Z), where B(z) is a Blaschke product and F(z) EH1. Since the boundary values of B (z) have absolute value one a.e. on K, we have a.e. on I, f(ei0)= |f(eio)I, or B(ei0)F2(ei0) = F2(ei0) |, and hence
- Published
- 1969
21. Volterra's Integral Equation of the Second Kind, with Discontinuous Kernel, Second Paper
- Author
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Griffith C. Evans
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,Summation equation ,Electric-field integral equation ,Integral equation ,Volterra integral equation ,symbols.namesake ,Integro-differential equation ,Kernel (statistics) ,Improper integral ,symbols ,Daniell integral ,Mathematics - Published
- 1911
22. Integers represented as the sum of one prime, two squares of primes and powers of 2
- Author
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Haiwei Sun and Guangshi Lü
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Short paper ,MathematicsofComputing_GENERAL ,Prime number ,Prime (order theory) ,Algebra ,symbols.namesake ,Integer ,symbols ,Idoneal number ,Prime power ,Sphenic number ,Mathematics - Abstract
In this short paper we prove that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and 83 83 powers of 2 2 .
- Published
- 2008
23. Higher order Turán inequalities for the Riemann $\xi$-function
- Author
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Dimitar K. Dimitrov, Fábio Rodrigues Lucas, Universidade Estadual Paulista (Unesp), and Universidade Estadual de Campinas (UNICAMP)
- Subjects
Degree (graph theory) ,Applied Mathematics ,General Mathematics ,Entire function ,Mathematical analysis ,Short paper ,Function (mathematics) ,Maclaurin coefficients ,Riemann ξ function ,Combinatorics ,Riemann hypothesis ,symbols.namesake ,Jensen polynomials ,symbols ,Order (group theory) ,Shape function ,Laguerre-Pólya class ,Turán inequalities ,Mathematics - Abstract
Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-27T11:25:28Z No. of bitstreams: 0Bitstream added on 2014-05-27T14:41:41Z : No. of bitstreams: 1 2-s2.0-79951846250.pdf: 494002 bytes, checksum: 56b6ee8beddda3e7dae971355d44a19f (MD5) Made available in DSpace on 2014-05-27T11:25:28Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-03-01 Item merged in doublecheck by Felipe Arakaki (arakaki@reitoria.unesp.br) on 2015-12-11T17:28:11Z Item was identical to item(s): 71803, 21370 at handle(s): http://hdl.handle.net/11449/72321, http://hdl.handle.net/11449/21804 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) The simplest necessary conditions for an entire function ψ(x) =∞ ∑ k=0 γk xk/k! to be in the Laguerre-Pólya class are the Turán inequalities γ2 k- γk+1γk-1 ≥ 0. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with ψ to be hyperbolic. The higher order Turán inequalities 4(γ2 n - γn-1γn+1)(γ2n +1 - γnγn+2) - (γnγn+1 - γn-1γn+2) 2 ≥ 0 are also necessary conditions for a function of the above form to belong to the Laguerre-Pólya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic. Pólya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turán inequalities hold for the coefficients of the Riemann ψ-function. In this short paper, we prove that the higher order Turán inequalities also hold for the ψ-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three. © 2010 American Mathematical Society. Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP Departamento de matemática Aplicada IMECC UNICAMP, 13083-859 Campinas, SP Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP FAPESP: 03/01874-2 FAPESP: 06/60420-0 CNPq: 305622/2009-9 CAPES: DGU-160
- Published
- 2011
24. Splines of the Fourth Order Approximation and the Volterra Integral Equations
- Author
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D.E. Zhilin, A.G. Doronina, and I. G. Burova
- Subjects
Polynomial ,Series (mathematics) ,General Mathematics ,Type (model theory) ,Integral equation ,Volterra integral equation ,symbols.namesake ,Continuation ,Computer Science::Graphics ,symbols ,Applied mathematics ,Focus (optics) ,Mathematics ,Interpolation - Abstract
This paper is a continuation of a series of papers devoted to the numerical solution of integral equations using local interpolation splines. The main focus is given to the use of splines of the fourth order of approximation. The features of the application of the polynomial and non-polynomial splines of the fourth order of approximation to the solution of Volterra integral equation of the second kind are discussed. In addition to local splines of the Lagrangian type, integro-differential splines are also used to construct computational schemes. The comparison of the solutions obtained by different methods is carried out. The results of the numerical experiments are presented.
- Published
- 2021
25. Stability and collapse of the Lyapunov spectrum for Perron–Frobenius operator cocycles
- Author
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Anthony Quas and Cecilia González-Tokman
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Dense set ,Applied Mathematics ,General Mathematics ,Blaschke product ,Banach space ,Lyapunov exponent ,Fixed point ,symbols.namesake ,Unit circle ,symbols ,Invariant measure ,Mathematics ,Analytic function - Abstract
In this paper, we study random Blaschke products, acting on the unit circle, and consider the cocycle of Perron-Frobenius operators acting on Banach spaces of analytic functions on an annulus. We completely describe the Lyapunov spectrum of these cocycles. As a corollary, we obtain a simple random Blaschke product system where the Perron-Frobenius cocycle has infinitely many distinct Lyapunov exponents, but where arbitrarily small natural perturbations cause a complete collapse of the Lyapunov spectrum, except for the exponent 0 associated with the absolutely continuous invariant measure. That is, under perturbations, the Lyapunov exponents become 0 with multiplicity 1, and $-\infty$ with infinite multiplicity. This is superficially similar to the finite-dimensional phenomenon, discovered by Bochi \cite{Bochi-thesis}, that away from the uniformly hyperbolic setting, small perturbations can lead to a collapse of the Lyapunov spectrum to zero. In this paper, however, the cocycle and its perturbation are explicitly described; and further, the mechanism for collapse is quite different. We study stability of the Perron-Frobenius cocycles arising from general random Blaschke products. We give a necessary and sufficient criterion for stability of the Lyapunov spectrum in terms of the derivative of the random Blaschke product at its random fixed point, and use this to show that an open dense set of Blaschke product cocycles have hyperbolic Perron-Frobenius cocycles. In the final part, we prove a relationship between the Lyapunov spectrum of a single cocycle acting on two different Banach spaces, allowing us to draw conclusions for the same cocycles acting on $C^r$ functions spaces.
- Published
- 2021
26. On Some Properties of the New Generalized Fractional Derivative with Non-Singular Kernel
- Author
-
Khalid Hattaf
- Subjects
Lyapunov function ,Article Subject ,Non singular ,General Mathematics ,Science and engineering ,General Engineering ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010305 fluids & plasmas ,Fractional calculus ,010101 applied mathematics ,symbols.namesake ,Exponential stability ,Kernel (statistics) ,0103 physical sciences ,QA1-939 ,symbols ,Applied mathematics ,TA1-2040 ,0101 mathematics ,Mathematics - Abstract
This paper presents some new formulas and properties of the generalized fractional derivative with non-singular kernel that covers various types of fractional derivatives such as the Caputo–Fabrizio fractional derivative, the Atangana–Baleanu fractional derivative, and the weighted Atangana–Baleanu fractional derivative. These new properties extend many recent results existing in the literature. Furthermore, the paper proposes some interesting inequalities that estimate the generalized fractional derivatives of some specific functions. These inequalities can be used to construct Lyapunov functions with the aim to study the global asymptotic stability of several fractional-order systems arising from diverse fields of science and engineering.
- Published
- 2021
27. On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations
- Author
-
Thomas Y. Hou, De Huang, and Jiajie Chen
- Subjects
symbols.namesake ,Mathematics - Analysis of PDEs ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,FOS: Mathematics ,symbols ,Finite time ,Analysis of PDEs (math.AP) ,Mathematics ,Euler equations - Abstract
We present a novel method of analysis and prove finite time asymptotically self-similar blowup of the De Gregorio model \cite{DG90,DG96} for some smooth initial data on the real line with compact support. We also prove self-similar blowup results for the generalized De Gregorio model \cite{OSW08} for the entire range of parameter on $\mathbb{R}$ or $S^1$ for H\"older continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically self-similar singularity into the problem of establishing the nonlinear stability of an approximate self-similar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate self-similar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations., Comment: Added discussion in Section 2.3 and made some minor edits. Main paper 57 pages, Supplementary material 29 pages. In previous arXiv versions, the hyperlinks of the equation number in the main paper are linked to the supplementary material, which is fixed in this version
- Published
- 2021
28. 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
29. EXISTENCE OF SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS WITH FRACTIONAL-ORDER DERIVATIVE TERMS
- Author
-
Ai Sun, Tongxiang Li, Qingchun Yuan, and You-Hui Su
- Subjects
Computer simulation ,Iterative method ,General Mathematics ,010102 general mathematics ,Fixed-point theorem ,Derivative ,Function (mathematics) ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,Green's function ,symbols ,Applied mathematics ,Point (geometry) ,0101 mathematics ,Mathematics - Abstract
The study in this paper is made on the nonlinear fractional differential equation whose nonlinearity involves the explicit fractional order D0+β u(t). The corresponding Green's function is derived first, and then the completely continuous operator is proved. Besides, based on the Schauder's fixed point theorem and the Krasnosel'skii's fixed point theorem, the sufficient conditions for at least one or two existence of positive solutions are established. Furthermore, several other sufficient conditions for at least three, n or 2n-1 positive solutions are also obtained by applying the generalized AveryHenderson fixed point theorem and the Avery-Peterson fixed point theorem. Finally, several simulation examples are provided to illustrate the main results of the paper. In particularly, a novel efficient iterative method is employed for simulating the examples mentioned above, that is, the interesting point of this paper is that the approximation graphics for the solutions are given by using the iterative method.
- Published
- 2021
30. Infinitely many solutions for a class of fractional Robin problems with variable exponents
- Author
-
Ramzi Alsaedi
- Subjects
Class (set theory) ,Work (thermodynamics) ,General Mathematics ,variational methods ,robin ,Mathematics::Spectral Theory ,Type (model theory) ,variable exponents ,Euler equations ,symbols.namesake ,Continuation ,fracional sobolev spaces ,Operator (computer programming) ,QA1-939 ,symbols ,Applied mathematics ,Boundary value problem ,Mathematics ,Variable (mathematics) - Abstract
In this paper, we are concerned with a class of fractional Robin problems with variable exponents. Their main feature is that the associated Euler equation is driven by the fractional $ p(\cdot)- $Laplacian operator with variable coefficient while the boundary condition is of Robin type. This paper is a continuation of the recent work established by A. Bahrouni, V. Radulescu and P. Winkert [ 5 ].
- Published
- 2021
31. On Admissible Locations of Transonic Shock Fronts for Steady Euler Flows in an Almost Flat Finite Nozzle with Prescribed Receiver Pressure
- Author
-
Zhouping Xin and Beixiang Fang
- Subjects
35A01, 35A02, 35B20, 35B35, 35B65, 35J56, 35L65, 35L67, 35M30, 35M32, 35Q31, 35R35, 76L05, 76N10 ,Shock (fluid dynamics) ,Astrophysics::High Energy Astrophysical Phenomena ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Nozzle ,Mathematical analysis ,Boundary (topology) ,Euler system ,01 natural sciences ,Physics::Fluid Dynamics ,010104 statistics & probability ,symbols.namesake ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Free boundary problem ,Euler's formula ,symbols ,Boundary value problem ,0101 mathematics ,Transonic ,Astrophysics::Galaxy Astrophysics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
This paper concerns the existence of transonic shock solutions to the 2-D steady compressible Euler system in an almost flat finite nozzle ( in the sense that it is a generic small perturbation of a flat one ), under physical boundary conditions proposed by Courant-Friedrichs in \cite{CourantFriedrichs1948}, in which the receiver pressure is prescribed at the exit of the nozzle. In the resulting free boundary problem, the location of the shock-front is one of the most desirable information one would like to determine. However, the location of the normal shock-front in a flat nozzle can be anywhere in the nozzle so that it provides little information on the possible location of the shock-front when the nozzle's boundary is perturbed. So one of the key difficulties in looking for transonic shock solutions is to determine the shock-front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution will be taken as an initial approximation for the transonic shock solution. In this paper, a sufficient condition in terms of the geometry of the nozzle and the given exit pressure is derived which yields the existence of the solutions to the proposed free boundary problem. Once an initial approximation is obtained, a further nonlinear iteration could be constructed and proved to lead to a transonic shock solution., 53 pages
- Published
- 2020
32. Area‐Minimizing Currents mod 2 Q : Linear Regularity Theory
- Author
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Jonas Hirsch, Camillo De Lellis, Salvatore Stuvard, and Andrea Marchese
- Subjects
Pure mathematics ,multiple valued functions, Dirichlet integral, regularity theory, area minimizing currents mod(p), minimal surfaces, linearization ,Generalization ,General Mathematics ,Dimension (graph theory) ,area minimizing currents mod(p) ,linearization ,minimal surfaces ,Dirichlet integral ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Mathematics - Analysis of PDEs ,Mod ,FOS: Mathematics ,49Q15, 49Q05, 49N60, 35B65, 35J47 ,0101 mathematics ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Codimension ,regularity theory ,symbols ,multiple valued functions ,Analysis of PDEs (math.AP) - Abstract
We establish a theory of $Q$-valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents $\mathrm{mod}(p)$ when $p=2Q$, and to establish a first general partial regularity theorem for every $p$ in any dimension and codimension., 37 pages. First part of a two-papers work aimed at establishing a first general partial regularity theory for area minimizing currents modulo p, for any p and in any dimension and codimension. v3 is the final version, to appear on Comm. Pure Appl. Math
- Published
- 2020
33. On Class of Fractional-Order Chaotic or Hyperchaotic Systems in the Context of the Caputo Fractional-Order Derivative
- Author
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Ameth Ndiaye and Ndolane Sene
- Subjects
Equilibrium point ,Class (set theory) ,Article Subject ,Phase portrait ,General Mathematics ,Chaotic ,Context (language use) ,Lyapunov exponent ,01 natural sciences ,Stability (probability) ,010305 fluids & plasmas ,Nonlinear Sciences::Chaotic Dynamics ,symbols.namesake ,0103 physical sciences ,QA1-939 ,symbols ,Order (group theory) ,Applied mathematics ,010301 acoustics ,Mathematics - Abstract
In this paper, we consider a class of fractional-order systems described by the Caputo derivative. The behaviors of the dynamics of this particular class of fractional-order systems will be proposed and experienced by a numerical scheme to obtain the phase portraits. Before that, we will provide the conditions under which the considered fractional-order system’s solution exists and is unique. The fractional-order impact will be analyzed, and the advantages of the fractional-order derivatives in modeling chaotic systems will be discussed. How the parameters of the model influence the considered fractional-order system will be studied using the Lyapunov exponents. The topological changes of the systems and the detection of the chaotic and hyperchaotic behaviors at the assumed initial conditions and the considered fractional-order systems will also be investigated using the Lyapunov exponents. The investigations related to the Lyapunov exponents in the context of the fractional-order derivative will be the main novelty of this paper. The stability analysis of the model’s equilibrium points has been focused in terms of the Matignon criterion.
- Published
- 2020
34. Infinite-dimensional stochastic differential equations and tail $\sigma$-fields II: the IFC condition
- Author
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Hirofumi Osada, Hideki Tanemura, and Yosuke Kawamoto
- Subjects
General Mathematics ,Weak solution ,Universality (philosophy) ,Dirichlet distribution ,Symmetry (physics) ,Primary 60K35, Secondary 60H10, 82C22, 60B20 ,symbols.namesake ,Stochastic differential equation ,symbols ,Applied mathematics ,Uniqueness ,Random matrix ,Mathematics - Probability ,Brownian motion ,Computer Science::Information Theory ,Mathematics - Abstract
In a previous report, the second and third authors gave general theorems for unique strong solutions of infinite-dimensional stochastic differential equations (ISDEs) describing the dynamics of infinitely many interacting Brownian particles. One of the critical assumptions is the \lq\lq IFC" condition. The IFC condition requires that, for a given weak solution, the scheme consisting of the finite-dimensional stochastic differential equations (SDEs) related to the ISDEs exists. Furthermore, the IFC condition implies that each finite-dimensional SDE has unique strong solutions. Unlike other assumptions, the IFC condition is challenging to verify, and so the previous report only verified solution for solutions given by quasi-regular Dirichlet forms. In the present paper, we provide a sufficient condition for the IFC requirement in more general situations. In particular, we prove the IFC condition without assuming the quasi-regularity or symmetry of the associated Dirichlet forms. As an application of the theoretical formulation, the results derived in this paper are used to prove the uniqueness of Dirichlet forms and the dynamical universality of random matrices., Comment: This paper is a continuation of "Infinite-dimensional stochastic differential equations and tail $\sigma $-fields", which published in Probability Theory and Related Fields, https://doi.org/10.1007/s00440-020-00981-y. This paper will be published in Journal of the Mathematical Society of Japan
- Published
- 2022
35. Minimization arguments in analysis of variational-hemivariational inequalities
- Author
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Weimin Han and Mircea Sofonea
- Subjects
Applied Mathematics ,General Mathematics ,Hilbert space ,Structure (category theory) ,General Physics and Astronomy ,Contrast (statistics) ,010103 numerical & computational mathematics ,Fixed point ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,Contact mechanics ,Compact space ,symbols ,Applied mathematics ,Minification ,0101 mathematics ,Mathematics - Abstract
In this paper, an alternative approach is provided in the well-posedness analysis of elliptic variational–hemivariational inequalities in real Hilbert spaces. This includes the unique solvability and continuous dependence of the solution on the data. In most of the existing literature on elliptic variational–hemivariational inequalities, well-posedness results are obtained by using arguments of surjectivity for pseudomonotone multivalued operators, combined with additional compactness and pseudomonotonicity properties. In contrast, following (Han in Nonlinear Anal B Real World Appl 54:103114, 2020; Han in Numer Funct Anal Optim 42:371–395, 2021), the approach adopted in this paper is based on the fixed point structure of the problems, combined with minimization principles for elliptic variational–hemivariational inequalities. Consequently, only elementary results of functional analysis are needed in the approach, which makes the theory of elliptic variational–hemivariational inequalities more accessible to applied mathematicians and engineers. The theoretical results are illustrated on a representative example from contact mechanics.
- Published
- 2022
- Full Text
- View/download PDF
36. On moderate deviations in Poisson approximation
- Author
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Qingwei Liu and Aihua Xia
- Subjects
Statistics and Probability ,Random graph ,Matching (graph theory) ,Distribution (number theory) ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Poisson distribution ,01 natural sciences ,Birthday problem ,Normal distribution ,010104 statistics & probability ,symbols.namesake ,FOS: Mathematics ,Rare events ,symbols ,Applied mathematics ,Moderate deviations ,0101 mathematics ,Statistics, Probability and Uncertainty ,Primary 60F05, secondary 60E15 ,Mathematics - Probability ,Mathematics - Abstract
In this paper, we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of Poisson distribution than {those} of normal distribution. We then show the moderate deviations in Poisson approximation generally require an adjustment and, with suitable adjustment, we establish better error estimates of the moderate deviations in Poisson approximation than those in \cite{CFS}. Our estimates contain no unspecified constants and are easy to apply. We illustrate the use of the theorems in six applications: Poisson-binomial distribution, matching problem, occupancy problem, birthday problem, random graphs and 2-runs. The paper complements the works of \cite{CC92,BCC95,CFS}., 29 pages and 5 figures
- Published
- 2020
37. For Most Frequencies, Strong Trapping Has a Weak Effect in Frequency‐Domain Scattering
- Author
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David Lafontaine, Euan A. Spence, and Jared Wunsch
- Subjects
Helmholtz equation ,Applied Mathematics ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Measure (mathematics) ,010104 statistics & probability ,symbols.namesake ,Helmholtz free energy ,Frequency domain ,symbols ,Scattering theory ,0101 mathematics ,Laplace operator ,Mathematics ,Resolvent - Abstract
It is well known that when the geometry and/or coefficients allow stable trapped rays, the solution operator of the Helmholtz equation (a.k.a. the resolvent of the Laplacian) grows exponentially through a sequence of real frequencies tending to infinity. In this paper we show that, even in the presence of the strongest-possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. One significant application of this result is in the convergence analysis of several numerical methods for solving the Helmholtz equation at high frequency that are based on a polynomial-growth assumption on the solution operator (e.g. $hp$-finite elements, $hp$-boundary elements, certain multiscale methods). The result of this paper shows that this assumption holds, even in the presence of the strongest-possible trapping, for most frequencies.
- Published
- 2020
38. Optimal-rate finite-element solution of Dirichlet problems in curved domains with straight-edged tetrahedra
- Author
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Vitoriano Ruas
- Subjects
Applied Mathematics ,General Mathematics ,Mathematical analysis ,010103 numerical & computational mathematics ,Finite element solution ,01 natural sciences ,Dirichlet distribution ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Tetrahedron ,symbols ,0101 mathematics ,Mathematics - Abstract
In a series of papers published since 2017 the author introduced a simple alternative of the $n$-simplex type, to enhance the accuracy of approximations of second-order boundary value problems subject to Dirichlet boundary conditions, posed on smooth curved domains. This technique is based upon trial functions consisting of piecewise polynomials defined on straight-edged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast, the test functions are defined by the standard degrees of freedom associated with the underlying method for polytopic domains. While the mathematical analysis of the method for Lagrange and Hermite methods for two-dimensional second- and fourth-order problems was carried out in earlier paper by the author this paper is devoted to the study of the three-dimensional case. Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are proved for a tetrahedron-based Lagrange family of finite elements. Novel error estimates in the $L^2$-norm, for the class of problems considered in this work, are also proved. A series of numerical examples illustrates the potential of the new technique. In particular, its superior accuracy at equivalent cost, as compared to the isoparametric technique, is highlighted.
- Published
- 2020
39. Linearization Method of Nonlinear Magnetic Levitation System
- Author
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Shengya Meng, Fanwei Meng, and Dini Wang
- Subjects
0209 industrial biotechnology ,Article Subject ,General Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,02 engineering and technology ,symbols.namesake ,020901 industrial engineering & automation ,Linearization ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,QA1-939 ,0202 electrical engineering, electronic engineering, information engineering ,Taylor series ,Applied mathematics ,Mathematics ,General Engineering ,Process (computing) ,Engineering (General). Civil engineering (General) ,Magnetic levitation system ,Nonlinear system ,Nonlinear model ,Maglev ,Control system ,symbols ,020201 artificial intelligence & image processing ,TA1-2040 - Abstract
Linearized model of the system is often used in control design. It is generally believed that we can obtain the linearized model as long as the Taylor expansion method is used for the nonlinear model. This paper points out that the Taylor expansion method is only applicable to the linearization of the original nonlinear function. If the Taylor expansion is used for the derived nonlinear equation, wrong results are often obtained. Taking the linearization model of the maglev system as an example, it is shown that the linearization should be carried out with the process of equation derivation. The model is verified by nonlinear system simulation in Simulink. The method in this paper is helpful to write the linearized equation of the control system correctly.
- Published
- 2020
40. Adaptive ADI Numerical Analysis of 2D Quenching-Type Reaction: Diffusion Equation with Convection Term
- Author
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Xiaoliang Zhu and Yongbin Ge
- Subjects
Article Subject ,Discretization ,General Mathematics ,Numerical analysis ,Degenerate energy levels ,General Engineering ,Finite difference ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010305 fluids & plasmas ,010101 applied mathematics ,Nonlinear system ,symbols.namesake ,Alternating direction implicit method ,0103 physical sciences ,Reaction–diffusion system ,QA1-939 ,Taylor series ,symbols ,Applied mathematics ,TA1-2040 ,0101 mathematics ,Mathematics - Abstract
An adaptive high-order difference solution about a 2D nonlinear degenerate singular reaction-diffusion equation with a convection term is initially proposed in the paper. After the first and the second central difference operator approximating the first-order and the second-order spatial derivative, respectively, the higher-order spatial derivatives are discretized by applying the Taylor series rule and the temporal derivative is discretized by using the Crank–Nicolson (CN) difference scheme. An alternating direction implicit (ADI) scheme with a nonuniform grid is built in this way. Meanwhile, accuracy analysis declares the second order in time and the fourth order in space under certain conditions. Sequentially, the high-order scheme is performed on an adaptive mesh to demonstrate quenching behaviors of the singular parabolic equation and analyse the influence of combustion chamber size on quenching. The paper displays rationally that the proposed scheme is practicable for solving the 2D quenching-type problem.
- Published
- 2020
41. Trace finite element methods for surface vector-Laplace equations
- Author
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Thomas Jankuhn and Arnold Reusken
- Subjects
Partial differential equation ,Discretization ,Applied Mathematics ,General Mathematics ,Tangent ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Lagrange multiplier ,Norm (mathematics) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,FOS: Mathematics ,symbols ,65N30, 65N12, 65N15 ,Applied mathematics ,Vector field ,Penalty method ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics - Abstract
In this paper we analyze a class of trace finite element methods for the discretization of vector-Laplace equations. A key issue in the finite element discretization of such problems is the treatment of the constraint that the unknown vector field must be tangential to the surface (‘tangent condition’). We study three different natural techniques for treating the tangent condition, namely a consistent penalty method, a simpler inconsistent penalty method and a Lagrange multiplier method. The main goal of the paper is to present an analysis that reveals important properties of these three different techniques for treating the tangent constraint. A detailed error analysis is presented that takes the approximation of both the geometry of the surface and the solution of the partial differential equation into account. Error bounds in the energy norm are derived that show how the discretization error depends on relevant parameters such as the degree of the polynomials used for the approximation of the solution, the degree of the polynomials used for the approximation of the level set function that characterizes the surface, the penalty parameter and the degree of the polynomials used for the approximation of the Lagrange multiplier.
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- 2020
42. The Lane-Emden equation with variable double-phase and multiple regime
- Author
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Vicenţiu D. Rădulescu and Claudianor O. Alves
- Subjects
Variable exponent ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,Mathematical proof ,Supercritical fluid ,symbols.namesake ,Mathematics - Analysis of PDEs ,Criticality ,Feature (computer vision) ,Dirichlet boundary condition ,FOS: Mathematics ,symbols ,Lane–Emden equation ,Analysis of PDEs (math.AP) ,Variable (mathematics) ,Mathematics - Abstract
We are concerned with the study of the Lane-Emden equation with variable exponent and Dirichlet boundary condition. The feature of this paper is that the analysis that we develop does not assume any subcritical hypotheses and the reaction can fulfill a mixed regime (subcritical, critical and supercritical). We consider the radial and the nonradial cases, as well as a singular setting. The proofs combine variational and analytic methods with a version of the Palais principle of symmetric criticality., The final version this paper will be published in Proc. AMS
- Published
- 2020
43. Null controllability of semi-linear fourth order parabolic equations
- Author
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K. Kassab, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), and Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)
- Subjects
Null controllability ,Observability ,Global Carleman estimate ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Null (mathematics) ,Exact controllability ,01 natural sciences ,Parabolic partial differential equation ,Dirichlet distribution ,Domain (mathematical analysis) ,010101 applied mathematics ,Controllability ,symbols.namesake ,Linear and semi-linear fourth order parabolic equation ,Bounded function ,MSC : 35K35, 93B05, 93B07 ,Neumann boundary condition ,symbols ,[MATH]Mathematics [math] ,0101 mathematics ,Mathematics - Abstract
International audience; In this paper, we consider a semi-linear fourth order parabolic equation in a bounded smooth domain Ω with homogeneous Dirichlet and Neumann boundary conditions. The main result of this paper is the null controllability and the exact controllability to the trajectories at any time T > 0 for the associated control system with a control function acting at the interior.; Dans ce papier, on considère uneéquation parabolique semi-linéaire de quatrième ordre dans un domaine borné régulier Ω avec des conditions aux limites de type Dirichlet et Neumann homogènes. Le résultat principal de ce papier concerne la contrôlabilitéà zéro et la contrôlabilité exacte pour tout T > 0 du système de contrôle associé avec un contrôle agissantà l'interieur.
- Published
- 2020
44. On Multiscale RBF Collocation Methods for Solving the Monge–Ampère Equation
- Author
-
Qiuyan Xu and Zhiyong Liu
- Subjects
Collocation ,Article Subject ,General Mathematics ,Direct method ,General Engineering ,Boundary (topology) ,Monge–Ampère equation ,010103 numerical & computational mathematics ,Engineering (General). Civil engineering (General) ,01 natural sciences ,Dirichlet distribution ,010101 applied mathematics ,Discrete system ,symbols.namesake ,Nonlinear system ,QA1-939 ,symbols ,Applied mathematics ,Radial basis function ,TA1-2040 ,0101 mathematics ,Mathematics - Abstract
This paper considers some multiscale radial basis function collocation methods for solving the two-dimensional Monge–Ampère equation with Dirichlet boundary. We discuss and study the performance of the three kinds of multiscale methods. The first method is the cascadic meshfree method, which was proposed by Liu and He (2013). The second method is the stationary multilevel method, which was proposed by Floater and Iske (1996), and is used to solve the fully nonlinear partial differential equation in the paper for the first time. The third is the hierarchical radial basis function method, which is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the first two methods, the hierarchical radial basis function method can not only solve the present problem on a single level with higher accuracy and lower computational cost but also produce highly sparse nonlinear discrete system. These observations are obtained by taking the direct approach of numerical experimentation.
- Published
- 2020
45. Difference gap functions and global error bounds for random mixed equilibrium problems
- Author
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Jen-Chih Yao, Xiaolong Qin, Vo Minh Tam, and Nguyen Van Hung
- Subjects
Class (set theory) ,symbols.namesake ,General Mathematics ,Hilbert space ,symbols ,Applied mathematics ,Function (mathematics) ,Type (model theory) ,Global error ,Mathematics - Abstract
The aim of this paper is to study the difference gap (in short, D-gap) function and error bounds for a class of the random mixed equilibrium problems in real Hilbert spaces. Firstly, we consider regularized gap functions of the Fukushima type and Moreau-Yosida type. Then difference gap functions are established by using these terms of regularized gap functions. Finally, the global error bounds for random mixed equilibrium problems are also developed. The results obtained in this paper are new and extend some corresponding known results in literatures. Some examples are given for the illustration of our results.
- Published
- 2020
46. EXISTENCE OF SOLUTIONS FOR DUAL SINGULAR INTEGRAL EQUATIONS WITH CONVOLUTION KERNELS IN CASE OF NON-NORMAL TYPE
- Author
-
Pingrun Li
- Subjects
General Mathematics ,010102 general mathematics ,Singular integral ,Type (model theory) ,01 natural sciences ,Integral equation ,Dual (category theory) ,Convolution ,010101 applied mathematics ,Riemann hypothesis ,symbols.namesake ,Fourier transform ,symbols ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
This paper is devoted to the study of dual singular integral equations with convolution kernels in the case of non-normal type. Via using the Fourier transforms, we transform such equations into Riemann boundary value problems. To solve the equation, we establish the regularity theory of solvability. The general solutions and the solvable conditions of the equation are obtained. Especially, we investigate the asymptotic property of solutions at nodes. This paper will have a significant meaning for the study of improving and developing complex analysis, integral equations and Riemann boundary value problems.
- Published
- 2020
47. Stochastic Wiener filter in the white noise space
- Author
-
Daniel Alpay and Ariel Pinhas
- Subjects
wiener filter ,lcsh:T57-57.97 ,General Mathematics ,Wiener filter ,Hilbert space ,Banach space ,White noise ,Operator theory ,Space (mathematics) ,symbols.namesake ,stochastic distribution ,Optimization and Control (math.OC) ,Bounded function ,lcsh:Applied mathematics. Quantitative methods ,FOS: Mathematics ,symbols ,Applied mathematics ,white noise space ,wick product ,Wick product ,Mathematics - Optimization and Control ,Mathematics - Abstract
In this paper we introduce a new approach to the study of filtering theory by allowing the system's parameters to have a random character. We use Hida's white noise space theory to give an alternative characterization and a proper generalization to the Wiener filter over a suitable space of stochastic distributions introduced by Kondratiev. The main idea throughout this paper is to use the nuclearity of this space in order to view the random variables as bounded multiplication operators (with respect to the Wick product) between Hilbert spaces of stochastic distributions. This allows us to use operator theory tools and properties of Wiener algebras over Banach spaces to proceed and characterize the Wiener filter equations under the underlying randomness assumptions.
- Published
- 2020
48. A Viscosity Iterative Algorithm Technique for Solving a General Equilibrium Problem System
- Author
-
Hamid Reza Sahebi, Mahdi Azhini, and masoumeh cheraghi
- Subjects
Sequence ,symbols.namesake ,General equilibrium theory ,Iterative method ,Semigroup ,Applied Mathematics ,General Mathematics ,Viscosity (programming) ,Convergence (routing) ,Hilbert space ,symbols ,Applied mathematics ,Mathematics - Abstract
In the recent decade, a considerable number of Equilibrium problems havebeen solved successfully based on the iteration methods. In this paper, we suggest a viscosity iterative algorithm for nonexpansive semigroup in the framework of Hilbert space. We prove that, the sequence generated by this algorithm under the certain conditions imposed on parameters strongly convergence to a common solution of general equilibrium problem system. Results presented in this paper extend and unify the previously known results announced by many other authors. Further, we give some numerical examples to justify our main results.
- Published
- 2019
49. Periodic solutions of a class of third-order differential equations with two delays depending on time and state
- Author
-
Djoudi Ahcene, Bouakkaz Ahlème, Khemis Rabah, and Ardjouni Abdelouaheb
- Subjects
symbols.namesake ,Third order ,Differential equation ,General Mathematics ,Green's function ,symbols ,Applied mathematics ,Fixed-point theorem ,State (functional analysis) ,Uniqueness ,Contraction principle ,Stability (probability) ,Mathematics - Abstract
The goal of the present paper is to establish some new results on the existence, uniqueness and stability of periodic solutions for a class of third order functional differential equations with state and time-varying delays. By Krasnoselskii's fixed point theorem, we prove the existence of periodic solutions and under certain sufficient conditions, the Banach contraction principle ensures the uniqueness of this solution. The results obtained in this paper are illustrated by an example.
- Published
- 2019
50. The study of the solution of a Fredholm-Volterra integral equation by Picard operators
- Author
-
Maria Dobritoiu
- Subjects
Mathematics::Functional Analysis ,General Mathematics ,Data dependence ,Mathematics::Classical Analysis and ODEs ,Fredholm integral equation ,Integral equation ,Stability (probability) ,Volterra integral equation ,symbols.namesake ,symbols ,Order (group theory) ,Applied mathematics ,Uniqueness ,Mathematics - Abstract
In this paper we will use the Picard operators technique, in order to establish the existence and uniqueness, data dependence and Gronwall-type results for the solutions of a Fredholm-Volterra functional-integral equation. The paper ends with a result of the Ulam-Hyers stability of this integral equation.
- Published
- 2019
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