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2. Rebuttal of Donnelly's paper on the spectral gap
- Author
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Antoine Henrot, Mark S. Ashbaugh, Richard S. Laugesen, Department of Mathematics, University of Missouri Columbia, University of Missouri [Columbia] (Mizzou), University of Missouri System-University of Missouri System, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Department of Mathematics [Urbana], University of Illinois at Urbana-Champaign [Urbana], University of Illinois System-University of Illinois System, CORIDA, and Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est
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Discrete mathematics ,Sequence ,Conjecture ,General Mathematics ,010102 general mathematics ,Mathematics::History and Overview ,Mathematics::Spectral Theory ,01 natural sciences ,Domain (mathematical analysis) ,Computer Science::Computers and Society ,010101 applied mathematics ,symbols.namesake ,Physics::Popular Physics ,Dirichlet boundary condition ,Euclidean geometry ,symbols ,Calculus ,Convex body ,Quantitative Biology::Populations and Evolution ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Spectral gap ,0101 mathematics ,Mathematics ,Unit interval - Abstract
The spectral gap conjecture of M. van den Berg [2, formula (65)] asserts that λ2 − λ1 ≥ 3π for all convex euclidean domains of diameter 1, where λ1 and λ2 denote the first two eigenvalues of the Dirichlet Laplacian. Notice that equality holds for the 1-dimensional unit interval, which can be regarded also as a degenerate n-dimensional rectangular box. The gap estimate is conjectured to hold more generally for Schrodinger operators with convex potentials, under Dirichlet boundary conditions; see the work of S.-T. Yau and collaborators [9, 11]. This Schrodinger gap conjecture was proved some time ago in 1 dimension by R. Lavine [8], and more recently in all dimensions by B. Andrews and J. Clutterbuck [1]. The proof in this journal by H. Donnelly [3] of the original gap conjecture in 2 dimensions (for the Dirichlet Laplacian with zero potential) is not correct. The Editors of Mathematische Zeitschrift have asked us to describe the flaws in the proof, in order to clarify the state of the literature. Donnelly’s approach to the problem is a natural one: first perform a shape optimization to rule out a non-degenerate minimizing domain, and then analyze the spectral gap for a sequence of domains degenerating to an interval, with the help of results by D. Jerison [5]. (For some history on this approach, and on the gap conjecture more generally, see the report on the AIM meeting “Low Eigenvalues of Laplace and Schrodinger Operators” [10], especially page 12 of the open problems list.) The error lies in the proof of the shape optimization step, as we now explain. Donnelly wishes to prove that no minimizing domain can exist for
- Published
- 2011
3. Appearance of Temporal and Spatial Chaos in an Ecological System: A Mathematical Modeling Study
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S. N. Raw, B P Sarangi, P. Mishra, and B. Tiwari
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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.
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- 2021
4. Biased Adjusted Poisson Ridge Estimators-Method and Application
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Pär Sjölander, Muhammad Qasim, Muhammad Amin, B. M. Golam Kibria, and Kristofer Månsson
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Mean squared error ,General Mathematics ,Maximum likelihood ,General Physics and Astronomy ,Regression estimator ,Poisson distribution ,Modified almost unbiased ridge estimators ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,Statistics ,Poisson regression ,0101 mathematics ,Mathematics ,010308 nuclear & particles physics ,010102 general mathematics ,Estimator ,Mean square error ,General Chemistry ,Ridge (differential geometry) ,Poisson ridge regression ,Multicollinearity ,Maximum likelihood estimator ,symbols ,General Earth and Planetary Sciences ,General Agricultural and Biological Sciences ,Research Paper - Abstract
Månsson and Shukur (Econ Model 28:1475–1481, 2011) proposed a Poisson ridge regression estimator (PRRE) to reduce the negative effects of multicollinearity. However, a weakness of the PRRE is its relatively large bias. Therefore, as a remedy, Türkan and Özel (J Appl Stat 43:1892–1905, 2016) examined the performance of almost unbiased ridge estimators for the Poisson regression model. These estimators will not only reduce the consequences of multicollinearity but also decrease the bias of PRRE and thus perform more efficiently. The aim of this paper is twofold. Firstly, to derive the mean square error properties of the Modified Almost Unbiased PRRE (MAUPRRE) and Almost Unbiased PRRE (AUPRRE) and then propose new ridge estimators for MAUPRRE and AUPRRE. Secondly, to compare the performance of the MAUPRRE with the AUPRRE, PRRE and maximum likelihood estimator. Using both simulation study and real-world dataset from the Swedish football league, it is evidenced that one of the proposed, MAUPRRE ($$ \hat{k}_{q4} $$ k ^ q 4 ) performed better than the rest in the presence of high to strong (0.80–0.99) multicollinearity situation.
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- 2020
5. On the Characterizations of Wave Front Sets in Terms of the Short-Time Fourier Transform
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Stevan Pilipović and Bojan Prangoski
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Wavefront ,General Mathematics ,010102 general mathematics ,Short paper ,Mathematical analysis ,Short-time Fourier transform ,02 engineering and technology ,01 natural sciences ,Sobolev space ,symbols.namesake ,020303 mechanical engineering & transports ,Fourier transform ,0203 mechanical engineering ,symbols ,0101 mathematics ,Mathematics - Abstract
© 2019, Pleiades Publishing, Ltd. It is well known that the classical and Sobolev wave fronts were extended to nonequivalent global versions by the use of the short-time Fourier transform. In this very short paper, we give complete characterizations of the former wave front sets in terms of the short-time Fourier transform.
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- 2019
6. Derived Non-archimedean analytic Hilbert space
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Mauro Porta, Jorge António, Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)
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Pure mathematics ,Fiber (mathematics) ,General Mathematics ,010102 general mathematics ,Short paper ,Formal scheme ,Hilbert space ,Space (mathematics) ,01 natural sciences ,symbols.namesake ,Mathematics - Algebraic Geometry ,Mathematics::Category Theory ,0103 physical sciences ,Localization theorem ,FOS: Mathematics ,symbols ,010307 mathematical physics ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,0101 mathematics ,Algebraic Geometry (math.AG) ,Quotient ,Mathematics - Abstract
In this short paper we combine the representability theorem introduced in [17, 18] with the theory of derived formal models introduced in [2] to prove the existence representability of the derived Hilbert space RHilb(X) for a separated k-analytic space X. Such representability results relies on a localization theorem stating that if X is a quasi-compact and quasi-separated formal scheme, then the \infty-category Coh^+(X^rig) of almost perfect complexes over the generic fiber can be realized as a Verdier quotient of the \infty-category Coh^+(X). Along the way, we prove several results concerning the the \infty-categories of formal models for almost perfect modules on derived k-analytic spaces., 28 pages
- Published
- 2019
7. Tikhonov regularization of a second order dynamical system with Hessian driven damping
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Szilárd László, Radu Ioan Boţ, and Ernö Robert Csetnek
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Hessian matrix ,General Mathematics ,0211 other engineering and technologies ,Dynamical Systems (math.DS) ,02 engineering and technology ,Dynamical system ,01 natural sciences ,Hessian-driven damping ,90C26 ,Tikhonov regularization ,symbols.namesake ,34G25, 47J25, 47H05, 90C26, 90C30, 65K10 ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Mathematics - Dynamical Systems ,Mathematics - Optimization and Control ,Mathematics ,65K10 ,021103 operations research ,Full Length Paper ,47J25 ,47H05 ,010102 general mathematics ,Hilbert space ,90C30 ,Function (mathematics) ,Convex optimization ,Optimization and Control (math.OC) ,Second order dynamical system ,34G25 ,symbols ,Fast convergence methods ,Convex function ,Software - Abstract
We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system with Hessian driven damping and a Tikhonov regularization term in connection with the minimization of a smooth convex function in Hilbert spaces. We obtain fast convergence results for the function values along the trajectories. The Tikhonov regularization term enables the derivation of strong convergence results of the trajectory to the minimizer of the objective function of minimum norm.
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- 2020
8. Global optimization in Hilbert space
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Benoît Chachuat, Boris Houska, Engineering & Physical Science Research Council (EPSRC), and Commission of the European Communities
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Technology ,Optimization problem ,Mathematics, Applied ,0211 other engineering and technologies ,CONVEX COMPUTATION ,010103 numerical & computational mathematics ,02 engineering and technology ,ELLIPSOIDS ,01 natural sciences ,90C26 ,93B40 ,Convergence analysis ,0102 Applied Mathematics ,Branch-and-lift ,CUT ,Mathematics ,65K10 ,021103 operations research ,Full Length Paper ,Operations Research & Management Science ,0103 Numerical and Computational Mathematics ,Bounded function ,Physical Sciences ,symbols ,49M30 ,Calculus of variations ,INTEGRATION ,SET ,Complexity analysis ,Complete search ,Operations Research ,General Mathematics ,APPROXIMATIONS ,Set (abstract data type) ,symbols.namesake ,Applied mathematics ,ALGORITHM ,0101 mathematics ,INTERSECTION ,Global optimization ,0802 Computation Theory and Mathematics ,Science & Technology ,Infinite-dimensional optimization ,Hilbert space ,Computer Science, Software Engineering ,Constraint (information theory) ,Computer Science ,Software - Abstract
We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-suboptimal global solution within finite run-time for any given termination tolerance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}ε>0. Finally, we illustrate these results for a problem of calculus of variations.
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- 2017
9. d-Hermite rings and skew $$\textit{PBW}$$ PBW extensions
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Oswaldo Lezama and Claudia Gallego
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Hermite polynomials ,Rank (linear algebra) ,General Mathematics ,010102 general mathematics ,Short paper ,Skew ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,symbols.namesake ,Computational Theory and Mathematics ,Kronecker delta ,symbols ,Kronecker's theorem ,Finitely-generated abelian group ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this short paper we study the d-Hermite condition about stably free modules for skew $$\textit{PBW}$$ extensions. For this purpose, we estimate the stable rank of these non-commutative rings. In addition, and closely related with these questions, we will prove Kronecker’s theorem about the radical of finitely generated ideals for some particular types of skew $$\textit{PBW}$$ extensions.
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- 2015
10. Phase portraits of separable quadratic systems and a bibliographical survey on quadratic systems
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Jaume Llibre and Tao Li
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Pure mathematics ,Class (set theory) ,Poincaré compactification ,Phase portrait ,General Mathematics ,010102 general mathematics ,Quadratic function ,01 natural sciences ,Separable space ,Quadratic system ,symbols.namesake ,Quadratic equation ,Separable system ,Poincaré conjecture ,symbols ,Compactification (mathematics) ,0101 mathematics ,Quadratic differential ,Mathematics - Abstract
Although planar quadratic differential systems and their applications have been studied in more than one thousand papers, we still have no complete understanding of these systems. In this paper we have two objectives. First we provide a brief bibliographical survey on the main results about quadratic systems. Here we do not consider the applications of these systems to many areas as in Physics, Chemist, Economics, Biology, … Second we characterize the new class of planar separable quadratic polynomial differential systems. For such class of systems we provide the normal forms which contain one parameter, and using the Poincare compactification and the blow up technique, we prove that there exist 10 non-equivalent topological phase portraits in the Poincare disc for the separable quadratic polynomial differential systems.
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- 2021
11. On the singular value decomposition over finite fields and orbits of GU×GU
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Robert M. Guralnick
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Pure mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Unitary state ,Nilpotent matrix ,symbols.namesake ,Finite field ,Character (mathematics) ,Kronecker delta ,Singular value decomposition ,Linear algebra ,symbols ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of GU m ( q ) × GU n ( q ) on M m × n ( q 2 ) (which is the analog of the singular value decomposition). The proof involves Kronecker’s theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form A A ∗ and A ∗ A over a finite field and A A ⊤ and A ⊤ A over arbitrary fields.
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- 2021
12. A generalization of the Freidlin–Wentcell theorem on averaging of Hamiltonian systems
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Yichun Zhu
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Pure mathematics ,Girsanov theorem ,Weak convergence ,General Mathematics ,010102 general mathematics ,Identity matrix ,Differential operator ,01 natural sciences ,Hamiltonian system ,010101 applied mathematics ,symbols.namesake ,Matrix (mathematics) ,Compact space ,Wiener process ,symbols ,0101 mathematics ,Mathematics - Abstract
In this paper, we generalize the classical Freidlin-Wentzell’s theorem for random perturbations of Hamiltonian systems. In (Probability Theory and Related Fields 128 (2004) 441–466), M.Freidlin and M.Weber generalized the original result in the sense that the coefficient for the noise term is no longer the identity matrix but a state-dependent matrix and taking the drift term into consideration. In this paper, We generalize the result by adding a state-dependent matrix that converges uniformly to 0 on any compact sets as ϵ tends to 0 to a state-dependent noise and considering the drift term which contains two parts, the state-dependent mapping and a state-dependent mapping that converges uniformly to 0 on any compact sets as ϵ tends to 0. In the proof, we adapt a new way to prove the weak convergence inside the edge by constructing an auxiliary process and modify the proof in (Probability Theory and Related Fields 128 (2004) 441–466) when proving gluing condition.
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- 2021
13. The integrals and integral transformations connected with the joint vector Gaussian distribution
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N. F. Kako and V. S. Mukha
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010302 applied physics ,Distribution (number theory) ,General Mathematics ,Gaussian ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,01 natural sciences ,symbols.namesake ,Computational Theory and Mathematics ,0103 physical sciences ,symbols ,0101 mathematics ,Joint (geology) ,Mathematics - Abstract
In many applications it is desirable to consider not one random vector but a number of random vectors with the joint distribution. This paper is devoted to the integral and integral transformations connected with the joint vector Gaussian probability density function. Such integral and transformations arise in the statistical decision theory, particularly, in the dual control theory based on the statistical decision theory. One of the results represented in the paper is the integral of the joint Gaussian probability density function. The other results are the total probability formula and Bayes formula formulated in terms of the joint vector Gaussian probability density function. As an example the Bayesian estimations of the coefficients of the multiple regression function are obtained. The proposed integrals can be used as table integrals in various fields of research.
- Published
- 2021
14. A Numerical Approach for Evaluating the Time-Dependent Distribution of a Quasi Birth-Death Process
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Birgit Sollie, Michel Mandjes, and Mathematics
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Statistics and Probability ,Mathematical optimization ,General Mathematics ,0211 other engineering and technologies ,Markov process ,Context (language use) ,02 engineering and technology ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,SDG 3 - Good Health and Well-being ,0101 mathematics ,Mathematics ,021103 operations research ,Series (mathematics) ,Markov chain ,Model selection ,Quasi birth-death processes ,Maximum likelihood estimation ,Uniformization (probability theory) ,Quasi-birth–death process ,symbols ,Matrix exponential ,Time-dependent probabilities ,Erlang distribution - Abstract
This paper considers a continuous-time quasi birth-death (qbd) process, which informally can be seen as a birth-death process of which the parameters are modulated by an external continuous-time Markov chain. The aim is to numerically approximate the time-dependent distribution of the resulting bivariate Markov process in an accurate and efficient way. An approach based on the Erlangization principle is proposed and formally justified. Its performance is investigated and compared with two existing approaches: one based on numerical evaluation of the matrix exponential underlying the qbd process, and one based on the uniformization technique. It is shown that in many settings the approach based on Erlangization is faster than the other approaches, while still being highly accurate. In the last part of the paper, we demonstrate the use of the developed technique in the context of the evaluation of the likelihood pertaining to a time series, which can then be optimized over its parameters to obtain the maximum likelihood estimator. More specifically, through a series of examples with simulated and real-life data, we show how it can be deployed in model selection problems that involve the choice between a qbd and its non-modulated counterpart.
- Published
- 2022
15. On Some Properties of the New Generalized Fractional Derivative with Non-Singular Kernel
- Author
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Khalid Hattaf
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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
16. Stationary Wavelet with Double Generalised Rayleigh Distribution
- Author
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Hassan M. Aljohani
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021103 operations research ,Article Subject ,Computer science ,Rayleigh distribution ,General Mathematics ,0211 other engineering and technologies ,General Engineering ,Wavelet transform ,Markov chain Monte Carlo ,02 engineering and technology ,Inverse problem ,Engineering (General). Civil engineering (General) ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Noise ,Wavelet ,Multicollinearity ,Gaussian noise ,QA1-939 ,symbols ,TA1-2040 ,0101 mathematics ,Algorithm ,Mathematics - Abstract
Statistics are mathematical tools applying scientific investigations, such as engineering and medical and biological analyses. However, statistical methods are often improved. Nowadays, statisticians try to find an accurate way to solve a problem. One of these problems is estimation parameters, which can be expressed as an inverse problem when independent variables are highly correlated. This paper’s significant goal is to interpret the parameter estimates of double generalized Rayleigh distribution in a regression model using a wavelet basis. It is difficult to use the standard version of the regression methods in practical terms, which is obtained using the likelihood. Since a noise level usually makes the result of estimation unstable, multicollinearity leads to various estimates. This kind of problem estimates that features of the truth are complicated. So it is reasonable to use a mixed method that combines a fully Bayesian approach and a wavelet basis. The usual rule for wavelet approaches is to choose a wavelet basis, where it helps to compute the wavelet coefficients, and then, these coefficients are used to remove Gaussian noise. Recovering data is typically calculated by inverting the wavelet coefficients. Some wavelet bases have been considered, which provide a shift-invariant wavelet transform, simultaneously providing improvements in smoothness, in recovering, and in squared-error performance. The proposed method uses combining a penalized maximum likelihood approach, a penalty term, and wavelet tools. In this paper, real data are involved and modeled using double generalized Rayleigh distributions, as they are used to estimate the wavelet coefficients of the sample using numerical tools. In practical applications, wavelet approaches are recommended. They reduce noise levels. This process may be useful since the noise level is often corrupted in real data, as a significant cause of most numerical estimation problems. A simulation investigation is studied using the MCMC tool to estimate the underlying features as an essential task statistics.
- Published
- 2021
17. Fourier restriction in low fractal dimensions
- Author
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Bassam Shayya
- Subjects
Conjecture ,Measurable function ,Characteristic function (probability theory) ,General Mathematics ,Second fundamental form ,010102 general mathematics ,42B10, 42B20 (Primary), 28A75 (Secondary) ,0102 computer and information sciences ,Function (mathematics) ,Lebesgue integration ,01 natural sciences ,Measure (mathematics) ,Combinatorics ,symbols.namesake ,Hypersurface ,Mathematics - Classical Analysis and ODEs ,010201 computation theory & mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,0101 mathematics ,Mathematics - Abstract
Let $S \subset \Bbb R^n$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\Bbb R^n$. If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^q(X)} \leq C \| f \|_{L^p(S)}$ holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set $X$: there is a number $0 < \alpha \leq n$ such that $|X \cap B_R| \leq c \, R^\alpha$ for all balls $B_R$ in $\Bbb R^n$ of radius $R \geq 1$. On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^q$ against the measure $\chi_X dx$. Our approach consists of replacing the characteristic function $\chi_X$ of $X$ by an appropriate weight function $H$, and studying the resulting estimate in three different regimes: small values of $\alpha$, intermediate values of $\alpha$, and large values of $\alpha$. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted H\"{o}lder-type inequality that holds for general non-negative Lebesgue measurable functions on $\Bbb R^n$, and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du-Zhang theorem in the range $0 < \alpha < n/2$., Comment: 31 pages. Minor revision
- Published
- 2021
18. Existence and Uniqueness of the Global L1 Solution of the Euler Equations for Chaplygin Gas
- Author
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Zhen Wang, Tingting Chen, and Aifang Qu
- 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
19. On the pair correlations of powers of real numbers
- Author
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Christoph Aistleitner and Simon Baker
- Subjects
11K06, 11K60 ,General Mathematics ,Modulo ,FOS: Physical sciences ,0102 computer and information sciences ,Lebesgue integration ,01 natural sciences ,Combinatorics ,symbols.namesake ,Pair correlation ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Algebra over a field ,Classical theorem ,Mathematical Physics ,Real number ,Mathematics ,Sequence ,Mathematics - Number Theory ,Probability (math.PR) ,010102 general mathematics ,Mathematical Physics (math-ph) ,010201 computation theory & mathematics ,symbols ,Martingale (probability theory) ,Mathematics - Probability - Abstract
A classical theorem of Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In the present paper we extend Koksma's theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every $x>1$ the pair correlations of the fractional parts of $(x^n)_{n=1}^{\infty}$ are asymptotically Poissonian. The proof is based on a martingale approximation method., Version 2: some minor changes. The paper will appear in the Israel Journal of Mathematics
- Published
- 2021
20. Concomitants of Generalized Order Statistics from Bivariate Cambanis Family of Distributions Under a General Setting
- Author
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Haroon M. Barakat, M. A. Alawady, and M. A. Abd Elgawad
- 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
21. Sharp Hardy Identities and Inequalities on Carnot Groups
- Author
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Guozhen Lu, Nguyen Lam, and Joshua Flynn
- Subjects
Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Statistical and Nonlinear Physics ,01 natural sciences ,03 medical and health sciences ,symbols.namesake ,0302 clinical medicine ,symbols ,030212 general & internal medicine ,0101 mathematics ,Carnot cycle ,Mathematics ,media_common - Abstract
In this paper we establish general weighted Hardy identities for several subelliptic settings including Hardy identities on the Heisenberg group, Carnot groups with respect to a homogeneous gauge and Carnot–Carathéodory metric, general nilpotent groups, and certain families of Hörmander vector fields. We also introduce new weighted uncertainty principles in these settings. This is done by continuing the program initiated by [N. Lam, G. Lu and L. Zhang, Factorizations and Hardy’s-type identities and inequalities on upper half spaces, Calc. Var. Partial Differential Equations 58 2019, 6, Paper No. 183; N. Lam, G. Lu and L. Zhang, Geometric Hardy’s inequalities with general distance functions, J. Funct. Anal. 279 2020, 8, Article ID 108673] of using the Bessel pairs introduced by [N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Math. Surveys Monogr. 187, American Mathematical Society, Providence, 2013] to obtain Hardy identities. Using these identities, we are able to improve significantly existing Hardy inequalities in the literature in the aforementioned subelliptic settings. In particular, we establish the Hardy identities and inequalities in the spirit of [H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 1997, 443–469] and [H. Brezis and M. Marcus, Hardy’s inequalities revisited. Dedicated to Ennio De Giorgi, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 1–2, 217–237] in these settings.
- Published
- 2021
22. 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
23. Characterization of the dissipative structure for the symmetric hyperbolic system with non-symmetric relaxation
- Author
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Yoshihiro Ueda
- Subjects
Lyapunov function ,Physics ,General Mathematics ,010102 general mathematics ,Non symmetric ,Mathematics::Analysis of PDEs ,Characterization (mathematics) ,01 natural sciences ,Hyperbolic systems ,010101 applied mathematics ,symbols.namesake ,Classical mechanics ,Symmetric property ,symbols ,Dissipative system ,Relaxation (physics) ,Relaxation matrix ,0101 mathematics ,Analysis - Abstract
This paper is concerned with the dissipative structure for the linear symmetric hyperbolic system with non-symmetric relaxation. If the relaxation matrix of the system has symmetric property, Shizuta and Kawashima in 1985 introduced the suitable stability condition called Classical Stability Condition in this paper, and Umeda, Kawashima and Shizuta in 1984 analyzed the dissipative structure of the standard type. On the other hand, Ueda, Duan and Kawashima in 2012 and 2018 focused on the system with non-symmetric relaxation, and got the partial result which is the extension of known results. Furthermore, they argued the new dissipative structure called the regularity-loss type. In this situation, our purpose of this paper is to extend the stability theory introduced by Shizuta and Kawashima in 1985 and Umeda, Kawashima and Shizuta in 1984 for our general system.
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- 2021
24. Inhomogeneous Vector Riemann Boundary Value Problem and Convolutions Equation on a Finite Interval
- Author
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A. F. Voronin
- Subjects
Work (thermodynamics) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Interval (mathematics) ,Wiener algebra ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Riemann problem ,Matrix function ,symbols ,Order (group theory) ,Boundary value problem ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
In this paper, we develop a new method for studying the inhomogeneous vector Riemann–Hilbert boundary value problem (which is also called the Riemann boundary value problem) in the Wiener algebra of order two. The method consists in reducing the Riemann problem to a truncated Wiener–Hopf equation (to a convolution equation on a finite interval). The idea of the method was proposed by the author in a previous work. Here the method is applied to the inhomogeneous Riemann boundary value problem and to matrix functions of a more general form. The efficiency of the method is shown in the paper: new sufficient conditions for the existence of a canonical factorization of the matrix function in the Wiener algebra of order two are obtained. In addition, it was established that for the correct solvability of the inhomogeneous vector Riemann boundary value problem, it is necessary and sufficient to prove the uniqueness of the solution to the corresponding truncated homogeneous Wiener–Hopf equation.
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- 2021
25. EXISTENCE OF SOLUTIONS TO FRACTIONAL DIFFERENTIAL EQUATIONS WITH FRACTIONAL-ORDER DERIVATIVE TERMS
- Author
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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
26. 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
27. 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
28. On Admissible Locations of Transonic Shock Fronts for Steady Euler Flows in an Almost Flat Finite Nozzle with Prescribed Receiver Pressure
- Author
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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
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- 2020
29. 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
30. On Bourgain’s Counterexample for the Schrödinger Maximal Function
- Author
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Lillian B. Pierce
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,02 engineering and technology ,021001 nanoscience & nanotechnology ,01 natural sciences ,symbols.namesake ,symbols ,Maximal function ,0101 mathematics ,0210 nano-technology ,Schrödinger's cat ,Counterexample ,Mathematics - Abstract
This paper provides a rigorous derivation of a counterexample of Bourgain, related to a well-known question of pointwise a.e. convergence for the solution of the linear Schrödinger equation, for initial data in a Sobolev space. This counterexample combines ideas from analysis and number theory, and the present paper demonstrates how to build such counterexamples from first principles, and then optimize them.
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- 2020
31. 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.
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- 2020
32. 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
33. 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
34. 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
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- 2020
35. 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.
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- 2020
36. For Most Frequencies, Strong Trapping Has a Weak Effect in Frequency‐Domain Scattering
- Author
-
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
37. Optimal-rate finite-element solution of Dirichlet problems in curved domains with straight-edged tetrahedra
- Author
-
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
38. More about singular traces on simply generated operator ideals
- Author
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Albrecht Pietsch
- Subjects
Large class ,Sequence ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,Extension (predicate logic) ,Space (mathematics) ,01 natural sciences ,symbols.namesake ,Operator (computer programming) ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
During half a century, singular traces on ideals of Hilbert space operators have been constructed by looking for linear forms on associated sequence ideals. Only recently, the author was able to eliminate this auxiliary step by directly applying Banach’s version of the extension theorem; see (Integral Equ. Oper. Theory 91, 21, 2019 and 92, 7, 2020). Of course, the relationship between the new approach and the older ones must be investigated. In the first paper, this was done for $${\mathfrak {L}}_{1,\infty } (H)$$ . To save space, such considerations were postponed in the second paper, which deals with a large class of principal ideals, called simply generated. This omission will now be rectified.
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- 2020
39. On Some Local Asymptotic Properties of Sequences with a Random Index
- Author
-
Yu. V. Yakubovich, O. V. Rusakov, and B. A. Baev
- Subjects
Rademacher distribution ,Hurst exponent ,Pure mathematics ,Fractional Brownian motion ,Stochastic process ,General Mathematics ,010102 general mathematics ,Poisson distribution ,01 natural sciences ,010305 fluids & plasmas ,Cox process ,symbols.namesake ,0103 physical sciences ,symbols ,0101 mathematics ,Telegraph process ,Random variable ,Mathematics - Abstract
Random sequences with random or stochastic indices controlled by a doubly stochastic Poisson process are considered in this paper. A Poisson stochastic index process (PSI-process) is a random process with the continuous time ψ(t) obtained by subordinating a sequence of random variables (ξj), j = 0, 1, …, by a doubly stochastic Poisson process Π1(tλ) via the substitution ψ(t) = $${{\xi }_{{{{\Pi }_{1}}(t\lambda )}}}$$ , t $$ \geqslant $$ 0, where the random intensity λ is assumed independent of the standard Poisson process Π1. In this paper, we restrict our consideration to the case of independent identically distributed random variables (ξj) with a finite variance. We find a representation of the fractional Ornstein–Uhlenbeck process with the Hurst exponent H ∈ (0, 1/2) introduced and investigated by R. Wolpert and M. Taqqu (2005) in the form of a limit of normalized sums of independent identically distributed PSI-processes with an explicitly given distribution of the random intensity λ. This fractional Ornstein–Uhlenbeck process provides a local, at t = 0, mean-square approximation of the fractional Brownian motion with the same Hurst exponent H ∈ (0, 1/2). We examine in detail two examples of PSI-processes with the random intensity λ generating the fractional Ornstein–Uhlenbeck process in the Wolpert and Taqqu sense. These are a telegraph process arising when ξ0 has a Rademacher distribution ±1 with the probability 1/2 and a PSI-process with the uniform distribution for ξ0. For these two examples, we calculate the exact and the asymptotic values of the local modulus of continuity for a single PSI-process over a small fixed time span.
- Published
- 2020
40. Adaptive ADI Numerical Analysis of 2D Quenching-Type Reaction: Diffusion Equation with Convection Term
- Author
-
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. On Solvability of One Singular Equation of Peridynamics
- Author
-
A. V. Yuldasheva
- Subjects
Partial differential equation ,Peridynamics ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Function (mathematics) ,01 natural sciences ,Volterra integral equation ,010305 fluids & plasmas ,symbols.namesake ,0103 physical sciences ,Displacement field ,Solid mechanics ,symbols ,Applied mathematics ,0101 mathematics ,Laplace operator ,Mathematics - Abstract
In the classical theory of solid mechanics, the behavior of solids is described by partial differential equations (PDE) through Newton’s second law of motion. However, when spontaneous cracks and fractures exist, such PDE models are inadequate to characterize the discontinuities of physical quantities such as the displacement field. Recently, a peridynamic continuum model was proposed which only involves the integration over the differences of the displacement field. A linearized peridynamic model can be described by the integro-differential equation with initial values. In this paper, we study the well-posedness and regularity of a linearized peridynamic model with singular kernel. The novelty of the paper is that the singular kernel is represented as the Laplacian of a regular function. This let to convert the model to an operator valued Volterra integral equation. Then the existence and regularity of the solution of the peridynamics problem are established through the study of the Volterra integral equation.
- Published
- 2020
42. Vector-valued q-variational inequalities for averaging operators and the Hilbert transform
- Author
-
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
43. Graphons, permutons and the Thoma simplex: three mod‐Gaussian moduli spaces
- Author
-
Ashkan Nikeghbali, Valentin Féray, Pierre-Loïc Méliot, Universität Zürich [Zürich] = University of Zurich (UZH), Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), University of Zurich, and Méliot, Pierre‐Loïc
- Subjects
Pure mathematics ,General Mathematics ,Gaussian ,340 Law ,610 Medicine & health ,0102 computer and information sciences ,01 natural sciences ,symbols.namesake ,510 Mathematics ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Limit (mathematics) ,0101 mathematics ,Concentration inequality ,ComputingMilieux_MISCELLANEOUS ,2600 General Mathematics ,Mathematics ,Central limit theorem ,Random graph ,Simplex ,Probability (math.PR) ,010102 general mathematics ,Observable ,10003 Department of Banking and Finance ,Moduli space ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,10123 Institute of Mathematics ,010201 computation theory & mathematics ,symbols ,Mathematics - Probability - Abstract
In this paper, we show how to use the framework of mod-Gaussian convergence in order to study the fluctuations of certain models of random graphs, of random permutations and of random integer partitions. We prove that, in these three frameworks, a generic homogeneous observable of a generic random model is mod-Gaussian under an appropriate renormalisation. This implies a central limit theorem with an extended zone of normality, a moderate deviation principle, an estimate of the speed of convergence, a local limit theorem and a concentration inequality. The universal asymptotic behavior of the observables of these models gives rise to a notion of mod-Gaussian moduli space., Comment: New version: the paper has been slightly shortened, and a few references were added. 52 pages, 13 figures
- Published
- 2020
44. Trace finite element methods for surface vector-Laplace equations
- Author
-
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
45. Schrödinger Quantization of Infinite-Dimensional Hamiltonian Systems with a Nonquadratic Hamiltonian Function
- Author
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N. N. Shamarov and Oleg G. Smolyanov
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Hamiltonian mechanics ,Pure mathematics ,Lebesgue measure ,General Mathematics ,010102 general mathematics ,Convex set ,Space (mathematics) ,01 natural sciences ,Measure (mathematics) ,010305 fluids & plasmas ,Hamiltonian system ,symbols.namesake ,Fourier transform ,0103 physical sciences ,symbols ,0101 mathematics ,Hamiltonian (control theory) ,Mathematics - Abstract
According to a theorem of Andre Weil, there does not exist a standard Lebesgue measure on any infinite-dimensional locally convex space. Because of that, Schrodinger quantization of an infinite-dimensional Hamiltonian system is often defined using a sigma-additive measure, which is not translation-invariant. In the present paper, a completely different approach is applied: we use the generalized Lebesgue measure, which is translation-invariant. In implicit form, such a measure was used in the first paper published by Feynman (1948). In this situation, pseudodifferential operators whose symbols are classical Hamiltonian functions are formally defined as in the finite-dimensional case. In particular, they use unitary Fourier transforms which map functions (on a finite-dimensional space) into functions. Such a definition of the infinite-dimensional unitary Fourier transforms has not been used in the literature.
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- 2020
46. 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)
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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.
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- 2020
47. Mappings with finite length distortion and prime ends on Riemann surfaces
- Author
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Sergei Volkov and I Vladimir Ryazanov
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Statistics and Probability ,Pure mathematics ,Series (mathematics) ,Generalization ,Applied Mathematics ,General Mathematics ,Riemann surface ,010102 general mathematics ,Boundary (topology) ,02 engineering and technology ,01 natural sciences ,Prime (order theory) ,010305 fluids & plasmas ,Sobolev space ,Distortion (mathematics) ,symbols.namesake ,020303 mechanical engineering & transports ,0203 mechanical engineering ,0103 physical sciences ,Euclidean geometry ,symbols ,0101 mathematics ,Mathematics - Abstract
The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings in the Sobolev classes (mappings with generalized derivatives) on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class of FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio-Ryazanov-Srebro-Yakubov at 2004 and then included in the known book of these authors at 2009 on the modern mapping theory. As was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes, because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasiisometries. We prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extensions to the boundary of the mappings with finite length distortion between domains on Riemann surfaces by Caratheodory prime ends. The criterion for the continuous extension of the inverse mapping to the boundary is turned out to be the very simple condition on the integrability of the dilatations in the first power. The criteria for the continuous extension of the direct mappings to the boundary have a much more refined nature. One of such criteria is the existence of a majorant for the dilatation in the class of functions with finite mean oscillation, i.e., having a finite mean deviation from its mean value over infinitesimal disks centered at boundary points. As consequences, the corresponding criteria for a homeomorphic extension of mappings with finite length distortion to the closures of domains by Caratheodory prime ends are obtained.
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- 2020
48. On Multiscale RBF Collocation Methods for Solving the Monge–Ampère Equation
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Qiuyan Xu and Zhiyong Liu
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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.
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- 2020
49. Stein–Weiss inequalities with the fractional Poisson kernel
- Author
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Guozhen Lu, Chunxia Tao, Lu Chen, and Zhao Liu
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Pure mathematics ,Inequality ,Mathematics::Complex Variables ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Poisson kernel ,Type inequality ,01 natural sciences ,Hang ,symbols.namesake ,symbols ,0101 mathematics ,media_common ,Mathematics - Abstract
In this paper, we establish the following Stein–Weiss inequality with the fractional Poisson kernel. Then we prove that there exist extremals for the Stein–Weiss inequality (⋆), and that the extremals must be radially decreasing about the origin. We also provide the regularity and asymptotic estimates of positive solutions to the integral systems which are the Euler–Lagrange equations of the extremals to the Stein–Weiss inequality (⋆) with the fractional Poisson kernel. Our result is inspired by the work of Hang, Wang and Yan, where the Hardy–Littlewood–Sobolev type inequality was first established when γ=2 and α=β=0. The proof of the Stein–Weiss inequality (⋆) with the fractional Poisson kernel in this paper uses recent work on the Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel by Chen, Lu and Tao, and the present paper is a further study in this direction.
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- 2020
50. The Wiener Measure on the Heisenberg Group and Parabolic Equations
- Author
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S. V. Mamon
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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.
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- 2020
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