2,697 results
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2. Einstein’s Approach to Statistical Mechanics: The 1902–04 Papers
- Author
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Raúl Rechtman and Luca Peliti
- Subjects
Thermal equilibrium ,Basis (linear algebra) ,Physics - History and Philosophy of Physics ,Equations of motion ,Statistical and Nonlinear Physics ,Statistical mechanics ,01 natural sciences ,Sketch ,010305 fluids & plasmas ,Theoretical physics ,symbols.namesake ,0103 physical sciences ,symbols ,Einstein ,010306 general physics ,Focus (optics) ,Mathematical Physics ,Mathematics - Abstract
We summarize the papers published by Einstein in the Annalen der Physik in the years 1902-04 on the derivation of the properties of thermal equilibrium on the basis of the mechanical equations of motion and of the calculus of probabilities. We point out the line of thought that led Einstein to an especially economical foundation of the discipline, and to focus on fluctuations of the energy as a possible tool for establishing the validity of this foundation. We also sketch a comparison of Einstein's approach with that of Gibbs, suggesting that although they obtained similar results, they had different motivations and interpreted them in very different ways., Comment: 22 pages, submitted to JSP
- Published
- 2016
3. Bending Paper and the Möbius Strip
- Author
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Sören Bartels and Peter Hornung
- Subjects
symbols.namesake ,Mathematics::Combinatorics ,Mechanics of Materials ,Mathematics::Number Theory ,Mechanical Engineering ,Plate theory ,symbols ,General Materials Science ,Geometry ,Möbius strip ,Bending ,Nonlinear elasticity ,Mathematics - Abstract
We present some rigorous results about the bending behaviour of paper. By adapting these results to the Mobius strip, we obtain some qualitative properties of developable Mobius strips which minimize the bending energy. We also provide some numerical simulations which illustrate and strengthen the analytic results.
- Published
- 2014
4. On a Method of Solution of Systems of Fractional Pseudo-Differential Equations
- Author
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YangQuan Chen, Ravshan Ashurov, and Sabir Umarov
- Subjects
matrix symbol ,Differential equation ,Primary 35E15 ,33E12 ,01 natural sciences ,symbols.namesake ,solution operator ,Completeness (order theory) ,Mittag-Leffler function ,fractional order differential equation ,Applied mathematics ,fractional system of differential equations ,Uniqueness ,0101 mathematics ,Differential (infinitesimal) ,Secondary 35S10 ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Linear system ,system of differential equations ,pseudo-differential operator ,Pseudo-differential operator ,010101 applied mathematics ,Sobolev space ,35R11 ,symbols ,Analysis ,Research Paper - Abstract
This paper is devoted to the general theory of linear systems of fractional order pseudo-differential equations. Single fractional order differential and pseudo-differential equations are studied by many authors and several monographs and handbooks have been published devoted to its theory and applications. However, the state of systems of fractional order ordinary and partial or pseudo-differential equations is still far from completeness, even in the linear case. In this paper we develop a new method of solution of general systems of fractional order linear pseudo-differential equations and prove existence and uniqueness theorems in the special classes of distributions, as well as in the Sobolev spaces.
- Published
- 2021
5. Stationary distribution and density function expression for a stochastic SIQRS epidemic model with temporary immunity
- Author
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Baoquan Zhou, Yucong Dai, Daqing Jiang, and Tasawar Hayat
- Subjects
Lyapunov function ,Aerospace Engineering ,Temporary immunity ,Ocean Engineering ,Probability density function ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,Applied mathematics ,Ergodic theory ,Density function ,Ergodic stationary distribution ,Electrical and Electronic Engineering ,Logistic function ,010301 acoustics ,Mathematics ,Original Paper ,Stationary distribution ,Applied Mathematics ,Mechanical Engineering ,Stochastic SIQRS epidemic model ,Control and Systems Engineering ,Fokker–Planck equation ,symbols ,Epidemic model ,Deterministic system - Abstract
Recently, considering the temporary immunity of individuals who have recovered from certain infectious diseases, Liu et al. (Phys A Stat Mech Appl 551:124152, 2020) proposed and studied a stochastic susceptible-infected-recovered-susceptible model with logistic growth. For a more realistic situation, the effects of quarantine strategies and stochasticity should be taken into account. Hence, our paper focuses on a stochastic susceptible-infected-quarantined-recovered-susceptible epidemic model with temporary immunity. First, by means of the Khas’minskii theory and Lyapunov function approach, we construct a critical value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0^S$$\end{document}R0S corresponding to the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0$$\end{document}R0 of the deterministic system. Moreover, we prove that there is a unique ergodic stationary distribution if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0^S>1$$\end{document}R0S>1. Focusing on the results of Zhou et al. (Chaos Soliton Fractals 137:109865, 2020), we develop some suitable solving theories for the general four-dimensional Fokker–Planck equation. The key aim of the present study is to obtain the explicit density function expression of the stationary distribution under \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {R}}_0^S>1$$\end{document}R0S>1. It should be noted that the existence of an ergodic stationary distribution together with the unique exact probability density function can reveal all the dynamical properties of disease persistence in both epidemiological and statistical aspects. Next, some numerical simulations together with parameter analyses are shown to support our theoretical results. Last, through comparison with other articles, results are discussed and the main conclusions are highlighted.
- Published
- 2021
6. Dynamics and optimal control of a stochastic coronavirus (COVID-19) epidemic model with diffusion
- Author
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Zhouchao Wei and Yuxi Li
- Subjects
Original Paper ,Stationary distribution ,Computer simulation ,Reaction–diffusion ,Turing instability ,Applied Mathematics ,Mechanical Engineering ,COVID-19 ,Aerospace Engineering ,Ocean Engineering ,Optimal control ,Nonlinear system ,symbols.namesake ,Stochastic epidemic model ,Control and Systems Engineering ,Reaction–diffusion system ,Taylor series ,symbols ,Applied mathematics ,Uniqueness ,Electrical and Electronic Engineering ,Epidemic model ,Amplitude equations ,Mathematics - Abstract
In view of the facts in the infection and propagation of COVID-19, a stochastic reaction–diffusion epidemic model is presented to analyse and control this infectious diseases. Stationary distribution and Turing instability of this model are discussed for deriving the sufficient criteria for the persistence and extinction of disease. Furthermore, the amplitude equations are derived by using Taylor series expansion and weakly nonlinear analysis, and selection of Turing patterns for this model can be determined. In addition, the optimal quarantine control problem for reducing control cost is studied, and the differences between the two models are compared. By applying the optimal control theory, the existence and uniqueness of the optimal control and the optimal solution are obtained. Finally, these results are verified and illustrated by numerical simulation.
- Published
- 2021
7. Biased Adjusted Poisson Ridge Estimators-Method and Application
- Author
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Pär Sjölander, Muhammad Qasim, Muhammad Amin, B. M. Golam Kibria, and Kristofer Månsson
- Subjects
Mean squared error ,General Mathematics ,Maximum likelihood ,General Physics and Astronomy ,Regression estimator ,Poisson distribution ,Modified almost unbiased ridge estimators ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,Statistics ,Poisson regression ,0101 mathematics ,Mathematics ,010308 nuclear & particles physics ,010102 general mathematics ,Estimator ,Mean square error ,General Chemistry ,Ridge (differential geometry) ,Poisson ridge regression ,Multicollinearity ,Maximum likelihood estimator ,symbols ,General Earth and Planetary Sciences ,General Agricultural and Biological Sciences ,Research Paper - Abstract
Månsson and Shukur (Econ Model 28:1475–1481, 2011) proposed a Poisson ridge regression estimator (PRRE) to reduce the negative effects of multicollinearity. However, a weakness of the PRRE is its relatively large bias. Therefore, as a remedy, Türkan and Özel (J Appl Stat 43:1892–1905, 2016) examined the performance of almost unbiased ridge estimators for the Poisson regression model. These estimators will not only reduce the consequences of multicollinearity but also decrease the bias of PRRE and thus perform more efficiently. The aim of this paper is twofold. Firstly, to derive the mean square error properties of the Modified Almost Unbiased PRRE (MAUPRRE) and Almost Unbiased PRRE (AUPRRE) and then propose new ridge estimators for MAUPRRE and AUPRRE. Secondly, to compare the performance of the MAUPRRE with the AUPRRE, PRRE and maximum likelihood estimator. Using both simulation study and real-world dataset from the Swedish football league, it is evidenced that one of the proposed, MAUPRRE ($$ \hat{k}_{q4} $$ k ^ q 4 ) performed better than the rest in the presence of high to strong (0.80–0.99) multicollinearity situation.
- Published
- 2020
8. Ruelle Zeta Function from Field Theory
- Author
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Charles Hadfield, Santosh Kandel, and Michele Schiavina
- Subjects
Nuclear and High Energy Physics ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,01 natural sciences ,Interpretation (model theory) ,Ruelle zeta function ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Analytic torsion ,Field theory (psychology) ,Mathematics - Algebraic Topology ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics::Symplectic Geometry ,Equivalence (measure theory) ,Mathematical Physics ,Mathematical physics ,Mathematics ,Original Paper ,Partition function (quantum field theory) ,Conjecture ,010102 general mathematics ,37C30, 81T70, 81T45 ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Mathematics - Symplectic Geometry ,symbols ,Symplectic Geometry (math.SG) ,010307 mathematical physics ,Lagrangian - Abstract
We propose a field-theoretic interpretation of Ruelle zeta function and show how it can be seen as the partition function forBFtheory when an unusual gauge-fixing condition on contact manifolds is imposed. This suggests an alternative rephrasing of a conjecture due to Fried on the equivalence between Ruelle zeta function and analytic torsion, in terms of homotopies of Lagrangian submanifolds., Annales Henri Poincaré, 21 (12), ISSN:1424-0661, ISSN:1424-0637
- Published
- 2020
9. Stochastic contagion models without immunity: their long term behaviour and the optimal level of treatment
- Author
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Kovacevic, Raimund M.
- Subjects
Original Paper ,Contagion ,Markov chain ,Markov processes ,010102 general mathematics ,Markov process ,Management Science and Operations Research ,Decision problem ,01 natural sciences ,Noise (electronics) ,Term (time) ,010101 applied mathematics ,Stochastic differential equation ,symbols.namesake ,Asymptotic properties ,Disease control ,Simple (abstract algebra) ,symbols ,Applied mathematics ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
In this paper we analyze two stochastic versions of one of the simplest classes of contagion models, namely so-called SIS models. Several formulations of such models, based on stochastic differential equations, have been recently discussed in literature, mainly with a focus on the existence and uniqueness of stationary distributions. With applicability in view, the present paper uses the Fokker-Planck equations related to SIS stochastic differential equations, not only in order to derive basic facts, but also to derive explicit expressions for stationary densities and further characteristics related to the asymptotic behaviour. Two types of models are analyzed here: The first one is a version of the SIS model with external parameter noise and saturated incidence. The second one is based on the Kramers-Moyal approximation of the simple SIS Markov chain model, which leads to a model with scaled additive noise. In both cases we analyze the asymptotic behaviour, which leads to limiting stationary distributions in the first case and limiting quasistationary distributions in the second case. Finally, we use the derived properties for analyzing the decision problem of choosing the cost-optimal level of treatment intensity.
- Published
- 2018
10. Eventually Entanglement Breaking Markovian Dynamics: Structure and Characteristic Times
- Author
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Cambyse Rouzé, Daniel Stilck França, Eric P. Hanson, Apollo - University of Cambridge Repository, and Hanson, Eric [0000-0002-4675-4348]
- Subjects
Nuclear and High Energy Physics ,Structure (category theory) ,FOS: Physical sciences ,Quantum entanglement ,Quantum channel ,Computer Science::Digital Libraries ,01 natural sciences ,Separable space ,symbols.namesake ,quant-ph ,0103 physical sciences ,Statistical physics ,Ball (mathematics) ,010306 general physics ,Quantum ,Mathematical Physics ,Mathematics ,Original Paper ,Quantum Physics ,Mathematics::Operator Algebras ,Statistical and Nonlinear Physics ,ddc ,Discrete time and continuous time ,Poincaré conjecture ,symbols ,010307 mathematical physics ,Quantum Physics (quant-ph) - Abstract
Funder: Cantab Capital Institute for the Mathematics of Information, We investigate entanglement breaking times of Markovian evolutions in discrete and continuous time. In continuous time, we characterize which Markovian evolutions are eventually entanglement breaking, that is, evolutions for which there is a finite time after which any entanglement initially present has been destroyed by the noisy evolution. In the discrete-time framework, we consider the entanglement breaking index, that is, the number of times a quantum channel has to be composed with itself before it becomes entanglement breaking. The PPT2 conjecture is that every PPT quantum channel has an entanglement breaking index of at most 2; we prove that every faithful PPT quantum channel has a finite entanglement breaking index, and more generally, any faithful PPT CP map whose Hilbert–Schmidt adjoint is also faithful is eventually entanglement breaking. We also provide a method to obtain concrete bounds on this index for any faithful quantum channel. To obtain these estimates, we use a notion of robustness of separability to obtain bounds on the radius of the largest separable ball around faithful product states. We also extend the framework of Poincaré inequalities for non-primitive semigroups to the discrete setting to quantify the convergence of quantum semigroups in discrete time, which is of independent interest.
- Published
- 2020
11. Holism as the empirical significance of symmetries
- Author
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Henrique Gomes, Gomes, Henrique [0000-0002-9285-0090], and Apollo - University of Cambridge Repository
- Subjects
Thought experiment ,Philosophy of science ,Inertial frame of reference ,Gauge Theory ,Fields ,Holism ,Separability ,Paper in Philosophy of the Natural Sciences ,Supervenience ,5002 History and Philosophy Of Specific Fields ,Motion (physics) ,Philosophy of physics ,Philosophy ,Theoretical physics ,symbols.namesake ,History and Philosophy of Science ,50 Philosophy and Religious Studies ,Direct empirical significance ,Galileo (satellite navigation) ,symbols ,Mathematics - Abstract
Not all symmetries are on a par. For instance, within Newtonian mechanics, we seem to have a good grasp on the empirical significance of boosts, by applying it to subsystems. This is exemplified by the thought experiment known as Galileo’s ship: the inertial state of motion of a ship is immaterial to how events unfold in the cabin, but is registered in the values of relational quantities such as the distance and velocity of the ship relative to the shore. But the significance of gauge symmetries seems less clear. For example, can gauge transformations in Yang-Mills theory—taken as mere descriptive redundancy—exhibit a similar relational empirical significance as the boosts of Galileo’s ship? This question has been debated in the last fifteen years in philosophy of physics. I will argue that the answer is ‘yes’, but only for a finite subset of gauge transformations, and under special conditions. Under those conditions, we can mathematically identify empirical significance with a failure of supervenience: the state of the Universe is not uniquely determined by the intrinsic state of its isolated subsystems. Empirical significance is therefore encoded in those relations between subsystems that stand apart from their intrinsic states.
- Published
- 2021
- Full Text
- View/download PDF
12. Time series and fractal analyses of wheezing: a novel approach
- Author
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Vimal Raj, S. Sreejyothi, M. S. Swapna, A. Renjini, and S. Sankararaman
- Subjects
Time Factors ,Wavelet Analysis ,Biomedical Engineering ,Biophysics ,Lyapunov exponent ,Scientific Paper ,Fractal dimension ,symbols.namesake ,Fractal ,medicine ,Humans ,Nonlinear time series analysis ,Radiology, Nuclear Medicine and imaging ,Instrumentation ,Mathematics ,Respiratory Sounds ,Hurst exponent ,Principal Component Analysis ,medicine.diagnostic_test ,Radiological and Ultrasound Technology ,business.industry ,Respiration ,Spectral density ,Pattern recognition ,Signal Processing, Computer-Assisted ,Auscultation ,Sample entropy ,Wheeze ,Fractals ,Radiology Nuclear Medicine and imaging ,Principal component analysis ,symbols ,Artificial intelligence ,business ,Biotechnology - Abstract
Since the outbreak of the pandemic Coronavirus Disease 2019, the world is in search of novel non-invasive methods for safer and early detection of lung diseases. The pulmonary pathological symptoms reflected through the lung sound opens a possibility of detection through auscultation and of employing spectral, fractal, nonlinear time series and principal component analyses. Thirty-five signals of vesicular and expiratory wheezing breath sound, subjected to spectral analyses shows a clear distinction in terms of time duration, intensity, and the number of frequency components. An investigation of the dynamics of air molecules during respiration using phase portrait, Lyapunov exponent, sample entropy, fractal dimension, and Hurst exponent helps in understanding the degree of complexity arising due to the presence of mucus secretions and constrictions in the respiratory airways. The feature extraction of the power spectral density data and the application of principal component analysis helps in distinguishing vesicular and expiratory wheezing and thereby, giving a ray of hope in accomplishing an early detection of pulmonary diseases through sound signal analysis.
- Published
- 2020
- Full Text
- View/download PDF
13. Romberg extrapolation for Euler summation-based cubature on regular regions
- Author
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Willi Freeden and Christian Gerhards
- Subjects
Discrete mathematics ,Original Paper ,Extrapolation ,010103 numerical & computational mathematics ,01 natural sciences ,Romberg extrapolation ,Cubature ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Trapezoidal rule (differential equations) ,symbols.namesake ,Rate of convergence ,Simple (abstract algebra) ,Modeling and Simulation ,Romberg's method ,symbols ,General Earth and Planetary Sciences ,65D30 ,65B99 ,0101 mathematics ,Remainder ,Cube ,Euler summation ,Mathematics - Abstract
Romberg extrapolation is a long-known method to improve the convergence rate of the trapezoidal rule on intervals. For simple regions such as the cube \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,1]^q$$\end{document}[0,1]q it is directly transferable to cubature in q dimensions. In this paper, we formulate Romberg extrapolation for Euler summation-based cubature on arbitrary q-dimensional regular regions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}\subset \mathbb {R}^q$$\end{document}G⊂Rq and derive an explicit representation for the remainder term.
- Published
- 2017
14. Tikhonov regularization of a second order dynamical system with Hessian driven damping
- Author
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Szilárd László, Radu Ioan Boţ, and Ernö Robert Csetnek
- Subjects
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.
- Published
- 2020
15. Adaptive Time Propagation for Time-dependent Schrödinger equations
- Author
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Harald Hofstätter, Othmar Koch, Winfried Auzinger, and Michael Quell
- Subjects
Original Paper ,Time-dependent Schrödinger equations ,Applied Mathematics ,Estimator ,Splitting methods ,Kinetic energy ,01 natural sciences ,Schrödinger equation ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Integrator ,0103 physical sciences ,symbols ,Computational Science and Engineering ,Applied mathematics ,0101 mathematics ,010306 general physics ,Time variable ,Hamiltonian (quantum mechanics) ,Magnus-type integrators ,Mathematics ,Adaptive stepsize selection - Abstract
We compare adaptive time integrators for the numerical solution of linear Schrödinger equations where the Hamiltonian explicitly depends on time. The approximation methods considered are splitting methods, where the time variable is split off and advanced separately, and commutator-free Magnus-type methods. The time-steps are chosen adaptively based on asymptotically correct estimators of the local error in both cases. It is found that splitting methods are more efficient when the Hamiltonian naturally suggests a separation into kinetic and potential part, whereas Magnus-type integrators excel when the structure of the problem only allows to advance the time variable separately.
- Published
- 2020
16. 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
17. The domain interface method in non-conforming domain decomposition multifield problems
- Author
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Javier Oliver, M. Cafiero, A. Ferrer, J. C. Cante, and Oriol Lloberas-Valls
- Subjects
Discretization ,Interface (Java) ,Multiphysics ,Computational Mechanics ,Ocean Engineering ,010103 numerical & computational mathematics ,Mixed formulations ,01 natural sciences ,Domain decomposition methods ,symbols.namesake ,Non-conforming interface ,Polygon mesh ,0101 mathematics ,Mortar methods ,Mathematics ,Original Paper ,Delaunay triangulation ,Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Weak coupling techniques for non-matching meshes ,Lagrange multiplier ,symbols - Abstract
The Domain Interface Method (DIM) is extended in this contribution for the case of mixed fields as encountered in multiphysics problems. The essence of the non-conforming domain decomposition technique consists in a discretization of a fictitious zero-thickness interface as in the original methodology and continuity of the solution fields across the domains is satisfied by incorporating the corresponding Lagrange Multipliers. The multifield DIM inherits the advantages of its irreducible version in the sense that the connections between non-matching meshes, with possible geometrically non-conforming interfaces, is accounted by the automatic Delaunay interface discretization without considering master and slave surfaces or intermediate surface projections as done in many established techniques, e.g. mortar methods. The multifield enhancement identifies the Lagrange multiplier field and incorporates its contribution in the weak variational form accounting for the corresponding consistent stabilization term based on a Nitsche method. This type of constraint enforcement circumvents the appearance of instabilities when the Ladyzhenskaya---Babuska---Brezzi (LBB) condition is not fulfilled by the chosen discretization. The domain decomposition framework is assessed in a large deformation setting for mixed displacement/pressure formulations and coupled thermomechanical problems. The continuity of the mixed field is studied in well selected benchmark problems for both mixed formulations and the objectivity of the response is compared to reference monolithic solutions. Results suggest that the presented strategy shows sufficient potential to be a valuable tool in situations where the evolving physics at particular domains require the use of different spatial discretizations or field interpolations.
- Published
- 2016
18. d-Hermite rings and skew $$\textit{PBW}$$ PBW extensions
- Author
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Oswaldo Lezama and Claudia Gallego
- Subjects
Hermite polynomials ,Rank (linear algebra) ,General Mathematics ,010102 general mathematics ,Short paper ,Skew ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,symbols.namesake ,Computational Theory and Mathematics ,Kronecker delta ,symbols ,Kronecker's theorem ,Finitely-generated abelian group ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this short paper we study the d-Hermite condition about stably free modules for skew $$\textit{PBW}$$ extensions. For this purpose, we estimate the stable rank of these non-commutative rings. In addition, and closely related with these questions, we will prove Kronecker’s theorem about the radical of finitely generated ideals for some particular types of skew $$\textit{PBW}$$ extensions.
- Published
- 2015
19. Gibbs–Helmholtz equation and entropy
- Author
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Ernő Keszei
- Subjects
Free entropy ,Entropy (statistical thermodynamics) ,Short paper ,Enthalpy ,Thermodynamics ,General Chemistry ,Massieu function ,Biochemistry ,Gibbs free energy ,Compensation effect ,symbols.namesake ,symbols ,Gibbs–Helmholtz equation ,Statistical physics ,Mathematics - Abstract
This short paper deals with the close relation of the Gibbs–Helmholtz (G–H) equation to entropy. It is shown that the G–H equation is readily derived from the entropy equivalent of the Gibbs function, the Massieu function. This derivation is also compared to another simple and straightforward derivation which uses mostly pure mathematical operations for multivariate functions, without revoking the Massieu function. Finally, the so-called “compensation effect” is also treated in some detail, citing a critical paper which testifies that this effect is an artefact; it simply reflects the relation between the Gibbs function, the enthalpy, and the entropy.
- Published
- 2016
20. Kronecker delta method for testing independence between two vectors in high-dimension
- Author
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Julio Cesar Araujo Silva Junior, Yan Zhuang, and Ivair R. Silva
- Subjects
Statistics and Probability ,High-dimensional Data ,Estimator ,Regular Article ,Sample (statistics) ,Covariance ,Nominal level ,symbols.namesake ,Sample size determination ,Kronecker delta ,Statistics ,symbols ,Randomized testing ,Kronecker delta covariance structure ,Statistics, Probability and Uncertainty ,Multivariate Gaussian Vectors ,Independence (probability theory) ,Type I and type II errors ,Mathematics - Abstract
Conventional methods for testing independence between two Gaussian vectors require sample sizes greater than the number of variables in each vector. Therefore, adjustments are needed for the high-dimensional situation, where the sample size is smaller than the number of variables in at least one of the compared vectors. It is critical to emphasize that the methods available in the literature are unable to control the Type I error probability under the nominal level. This fact is evidenced through an intensive simulation study presented in this paper. To cover this lack, we introduce a valid randomized test based on the Kronecker delta covariance matrices estimator. As an empirical application, based on a sample of companies listed on the stock exchange of Brazil, we test the independence between returns of stocks of different sectors in the COVID-19 pandemic context.
- Published
- 2021
21. Variable dispersion beta regressions with parametric link functions
- Author
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Fábio M. Bayer and Diego Ramos Canterle
- Subjects
Statistics and Probability ,05 social sciences ,Logit ,Estimator ,Score ,050109 social psychology ,Regression analysis ,050105 experimental psychology ,62J99, 62-07 ,symbols.namesake ,Distribution (mathematics) ,Covariate ,symbols ,Applied mathematics ,0501 psychology and cognitive sciences ,Statistics, Probability and Uncertainty ,Fisher information ,Statistics - Methodology ,Mathematics ,Parametric statistics - Abstract
This paper presents a new class of regression models for continuous data restricted to the interval $(0,1)$, such as rates and proportions. The proposed class of models assumes a beta distribution for the variable of interest with regression structures for the mean and dispersion parameters. These structures consider covariates, unknown regression parameters, and parametric link functions. Link functions depend on parameters that model the relationship between the random component and the linear predictors. The symmetric and assymetric Aranda-Ordaz link functions are considered in details. Depending on the parameter values, these link functions refer to particular cases of fixed links such as logit and complementary log-log functions. Joint estimation of the regression and link function parameters is performed by maximum likelihood. Closed-form expressions for the score function and Fisher's information matrix are presented. Aspects of large sample inferences are discussed, and some diagnostic measures are proposed. A Monte Carlo simulation study is used to evaluate the finite sample performance of point estimators. Finally, a practical application that employs real data is presented and discussed., Comment: Accepted paper
- Published
- 2017
22. Power spectral density analysis for nonlinear systems based on Volterra series
- Author
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Yan Zhao, Xianghong Xu, and Penghui Wu
- Subjects
Frequency response ,Partial differential equation ,Applied Mathematics ,Mechanical Engineering ,Gaussian ,Volterra series ,Spectral density ,Exponential function ,Nonlinear system ,symbols.namesake ,Mechanics of Materials ,symbols ,Applied mathematics ,Random vibration ,Mathematics - Abstract
A consequence of nonlinearities is a multi-harmonic response via a mono-harmonic excitation. A similar phenomenon also exists in random vibration. The power spectral density (PSD) analysis of random vibration for nonlinear systems is studied in this paper. The analytical formulation of output PSD subject to the zero-mean Gaussian random load is deduced by using the Volterra series expansion and the conception of generalized frequency response function (GFRF). For a class of nonlinear systems, the growing exponential method is used to determine the first 3rd-order GFRFs. The proposed approach is used to achieve the nonlinear system’s output PSD under a narrow-band stationary random input. The relationship between the peak of PSD and the parameters of the nonlinear system is discussed. By using the proposed method, the nonlinear characteristics of multi-band output via single-band input can be well predicted. The results reveal that changing nonlinear system parameters gives a one-of-a-kind change of the system’s output PSD. This paper provides a method for the research of random vibration prediction and control in real-world nonlinear systems.
- Published
- 2021
23. Asymptotic behavior of dependence measures for Ornstein-Uhlenbeck model based on long memory processes
- Author
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Janusz Gajda and Agnieszka Wyłomańska
- Subjects
Basis (linear algebra) ,Order (ring theory) ,Ornstein–Uhlenbeck process ,Measure (mathematics) ,Stable distribution ,symbols.namesake ,Autocovariance ,Range (mathematics) ,Wiener process ,symbols ,General Earth and Planetary Sciences ,Applied mathematics ,General Environmental Science ,Mathematics - Abstract
In this paper, we study the long memory property of two processes based on the Ornstein-Uhlenbeck model. Their are extensions of the Ornstein-Uhlenbeck system for which in the classic version we replace the standard Brownian motion (or other L$$\acute{e}$$ e ´ vy process) by long range dependent processes based on $$\alpha -$$ α - stable distribution. One way of characterizing long- and short-range dependence of second order processes is in terms of autocovariance function. However, for systems with infinite variance the classic measure is not defined, therefore there is a need to consider alternative measures on the basis of which the long range dependence can be recognized. In this paper, we study three alternative measures adequate for $$\alpha -$$ α - stable-based processes. We calculate them for examined processes and indicate their asymptotic behavior. We show that one of the analyzed Ornstein-Uhlenbeck process exhibits long memory property while the second does not. Moreover, we show the ratio of two introduced measures is limited which can be a starting point to introduction of a new estimation method of stability index for analyzed Ornstein-Uhlenbeck processes.
- Published
- 2021
24. Uniform Polynomial Decay and Approximation in Control of a Family of Abstract Thermoelastic Models
- Author
-
S. Nafiri
- Subjects
Polynomial (hyperelastic model) ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Discretization ,Mathematical analysis ,Finite difference ,Finite element method ,symbols.namesake ,Thermoelastic damping ,Exponential stability ,Control and Systems Engineering ,Dirichlet boundary condition ,Bounded function ,symbols ,Mathematics - Abstract
In this paper, we consider the approximation of abstract thermoelastic models. It is by now well known that approximated systems are not in general uniformly exponentially or polynomially stable with respect to the discretization parameter, even if the continuous system has this property. Our goal in this paper is to study the uniform exponential/polynomial stability of a sequence of a system of weakly coupled thermoelastic models. We prove that when $0\leqslant \beta
- Published
- 2021
25. Asymptotic distribution of the partition crank
- Author
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Aaron Kriegman, Asimina Hamakiotes, and Wei-Lun Tsai
- Subjects
Crank ,Partition function (quantum field theory) ,Algebra and Number Theory ,Mathematics - Number Theory ,Mathematics::Number Theory ,Nuclear Theory ,Function (mathematics) ,Mathematics::Numerical Analysis ,Ramanujan's sum ,Combinatorics ,Equidistributed sequence ,symbols.namesake ,Number theory ,FOS: Mathematics ,symbols ,Partition (number theory) ,Asymptotic formula ,Number Theory (math.NT) ,Mathematics - Abstract
The partition crank is a statistic on partitions introduced by Freeman Dyson to explain Ramanujan's congruences. In this paper, we prove that the crank is asymptotically equidistributed modulo Q, for any odd number Q. To prove this, we obtain effective bounds on the error term from Zapata Rolon's asymptotic estimate for the crank function. We then use those bounds to prove the surjectivity and strict log-subadditivity of the crank function., 14 pages 1) Some clarifications and corrections have been made. 2) This paper will appear in Ramanujan Journal
- Published
- 2021
26. On Potential Theory of Markov Processes with Jump Kernels Decaying at the Boundary
- Author
-
Renming Song, Zoran Vondraček, and Panki Kim
- Subjects
60J45, 60J50, 60J75 ,Boundary (topology) ,01 natural sciences ,Potential theory ,Dirichlet distribution ,010104 statistics & probability ,symbols.namesake ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,0101 mathematics ,Harnack's inequality ,Mathematical physics ,Mathematics ,Functional analysis ,Euclidean space ,Probability (math.PR) ,010102 general mathematics ,Jump processes ,Jumping kernel with boundary part ,Harnack inequality ,Carleson estimate ,Boundary Harnack principle ,16. Peace & justice ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Harmonic function ,symbols ,Mathematics - Probability ,Analysis ,Kernel (category theory) ,Analysis of PDEs (math.AP) - Abstract
Motivated by some recent potential theoretic results on subordinate killed L\'evy processes in open subsets of the Euclidean space, we study processes in an open set $D\subset {\mathbb R}^d$ defined via Dirichlet forms with jump kernels of the form $J^D(x,y)=j(|x-y|)\mathcal{B}(x,y)$ and critical killing functions. Here $j(|x-y|)$ is the L\'evy density of an isotropic stable process (or more generally, a pure jump isotropic unimodal L\'evy process) in $\mathbb{R}^d$. The main novelty is that the term $\mathcal{B}(x,y)$ tends to 0 when $x$ or $y$ approach the boundary of $D$. Under some general assumptions on $\mathcal{B}(x,y)$, we construct the corresponding process and prove that non-negative harmonic functions of the process satisfy the Harnack inequality and Carleson's estimate. We give several examples of boundary terms satisfying those assumptions. The examples depend on four parameters, $\beta_1, \beta_2, \beta_3$, $\beta_4$, roughly governing the decay of the boundary term near the boundary of $D$. In the second part of this paper, we specialise to the case of the half-space $D=\mathbb{R}_+^d=\{x=(\widetilde{x},x_d):\, x_d>0\}$, the $\alpha$-stable kernel $j(|x-y|)=|x-y|^{-d-\alpha}$ and the killing function$\kappa(x)=c x_d^{-\alpha}$, $\alpha\in (0,2)$, where $c$ is a positive constant. Our main result in this part is a boundary Harnack principle which says that, for any $p>(\alpha-1)_+$, there are values of the parameters $\beta_1, \beta_2, \beta_3$, $\beta_4$, and the constant $c$ such that non-negative harmonic functions of the process must decay at the rate $x_d^p$ if they vanish near a portion of the boundary. We further show that there are values of the parameters $\beta_1, \beta_2, \beta_3$, $\beta_4$, for which the boundary Harnack principle fails despite the fact that Carleson's estimate is valid., Comment: This is a corrected version of the published paper https://doi.org/10.1007/s11118-021-09947-8. Proofs of Lemmas 9.3 and 9.4 had minor gaps (estimates of integrals I_2 and IV in Lemma 9.3 and integral IV in Lemma 9.4 were incorrect). These are now fixed. We also corrected some typos
- Published
- 2021
27. Modified wavelet method for solving multitype variable-order fractional partial differential equations generated from the modeling of phenomena
- Author
-
Yadollah Ordokhani, Haniye Dehestani, and Mohsen Razzaghi
- Subjects
Statistics and Probability ,Numerical Analysis ,Partial differential equation ,Applied Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,Derivative ,Differential operator ,Computer Science Applications ,Burgers' equation ,symbols.namesake ,Algebraic equation ,Wavelet ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Signal Processing ,symbols ,Applied mathematics ,Klein–Gordon equation ,Analysis ,Information Systems ,Mathematics ,Variable (mathematics) - Abstract
The aim of this paper is to introduce a new wavelet method for presenting approximate solutions of multitype variable-order (VO) fractional partial differential equations arising from the modeling of phenomena. In specific, this paper focuses on the numerical solution of the VO-fractional mobile-immobile advection-dispersion equation, Klein Gordon equation and Burgers equation. These equations are converted into a system of algebraic equations with the assistance of the bivariate Genocchi wavelet functions, their operational matrices, and the variable-order fractional Caputo derivative operator. Also, we present a new technique to get the operational matrix of integration and VO-fractional derivative. The modified operational matrices for solving the proposed equations are powerful and effective. So that, the accuracy of these matrices directly affects the implementation process. Finally, we consider numerical examples to confirm the superiority of the scheme, and for each example, exhibit the results through graphs and tables.
- Published
- 2021
28. Ruelle Zeta Functions of Hyperbolic Manifolds and Reidemeister Torsion
- Author
-
Werner E. G. Müller
- Subjects
Pure mathematics ,Fundamental group ,010102 general mathematics ,Holomorphic function ,Hyperbolic manifold ,01 natural sciences ,Ruelle zeta function ,symbols.namesake ,Differential geometry ,Fourier analysis ,0103 physical sciences ,symbols ,Torsion (algebra) ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Representation (mathematics) ,Mathematics - Abstract
This paper is concerned with the behavior of twisted Ruelle zeta functions of compact hyperbolic manifolds at the origin. Fried proved that for an orthogonal acyclic representation of the fundamental group of a compact hyperbolic manifold, the twisted Ruelle zeta function is holomorphic at $$s=0$$ s = 0 and its value at $$s=0$$ s = 0 equals the Reidemeister torsion. He also established a more general result for orthogonal representations, which are not acyclic. The purpose of the present paper is to extend Fried’s result to arbitrary finite dimensional representations of the fundamental group. The Reidemeister torsion is replaced by the complex-valued combinatorial torsion introduced by Cappell and Miller.
- Published
- 2021
29. Hypotheses testing and posterior concentration rates for semi-Markov processes
- Author
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Nikolaos Limnios, Vlad Stefan Barbu, Ghislaine Gayraud, Irene Votsi, Le Mans Université (UM), Université de Technologie de Compiègne (UTC), Université de Rouen Normandie (UNIROUEN), and Normandie Université (NU)
- Subjects
Statistics and Probability ,Hellinger distance ,Posterior probability ,Markov process ,Mathematics - Statistics Theory ,semi-Markov kernel ,Statistics Theory (math.ST) ,Space (mathematics) ,01 natural sciences ,Bayesian nonparametrics ,010104 statistics & probability ,symbols.namesake ,0502 economics and business ,Prior probability ,FOS: Mathematics ,Countable set ,State space ,Applied mathematics ,testing procedure ,0101 mathematics ,050205 econometrics ,Statistical hypothesis testing ,Mathematics ,posterior concentration rate ,05 social sciences ,[STAT.TH]Statistics [stat]/Statistics Theory [stat.TH] ,semi-Markov kernels ,robust statistical tests ,symbols ,Bayesian nonparametric statistics ,posterior concentration rates ,semi-Markov processes - Abstract
In this paper, we adopt a nonparametric Bayesian approach and investigate the asymptotic behavior of the posterior distribution in continuous-time and general state space semi-Markov processes. In particular, we obtain posterior concentration rates for semi-Markov kernels. For the purposes of this study, we construct robust statistical tests between Hellinger balls around semi-Markov kernels and present some specifications to particular cases, including discrete-time semi-Markov processes and countable state space Markov processes. The objective of this paper is to provide sufficient conditions on priors and semi-Markov kernels that enable us to establish posterior concentration rates.
- Published
- 2021
30. Long-Time Behavior of a Reaction–Diffusion Model with Strong Allee Effect and Free Boundary: Effect of Protection Zone
- Author
-
Chengxia Lei and Ningkui Sun
- Subjects
symbols.namesake ,Partial differential equation ,Ordinary differential equation ,Reaction–diffusion system ,Mathematical analysis ,symbols ,Boundary (topology) ,Analysis ,Mathematics ,Allee effect - Abstract
This paper concerns the effect of the (separated/connected) protection zone for the evolution of an endangered species on the reaction–diffusion equation with the strong Allee effect and the free boundary. First, we describe the long-time dynamical behavior of the system of two types protection zones with the same length. Furthermore, the asymptotic profiles of solutions and the asymptotic spreading speed are estimated when spreading happens. Our results, together with those in previous papers Du et al. (J Differ Equ 266:7327–7356, 2019), Du and Lou (J Eur Math Soc 17:2673–2724, 2015) on two other closely related models, show that the protection zone and the free boundary play an important role in the evolution of the endangered species.
- Published
- 2021
31. The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation
- Author
-
Ihyeok Seo and Yoonjung Lee
- Subjects
symbols.namesake ,General Mathematics ,Open problem ,symbols ,Initial value problem ,Beta (velocity) ,Lambda ,Nonlinear Schrödinger equation ,Energy (signal processing) ,Mathematics ,Mathematical physics - Abstract
In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrodinger equation $$i\partial _{t}u+\Delta u=\lambda |x|^{-\alpha }|u|^{\beta }u$$ in $$H^1$$ . The well-posedness theory in $$H^1$$ has been intensively studied in recent years, but the currently known approaches do not work for the critical case $$\beta =(4-2\alpha )/(n-2)$$ . It is still an open problem. The main contribution of this paper is to develop the theory in this case.
- Published
- 2021
32. Euler angles and numerical representation of the railroad track geometry
- Author
-
Ahmed A. Shabana and Hao Ling
- Subjects
Differential equation ,Mechanical Engineering ,Frenet–Serret formulas ,Computational Mechanics ,Geometry ,Curvature ,Track (rail transport) ,Euler angles ,symbols.namesake ,Intersection ,symbols ,Track geometry ,Tangent vector ,Mathematics - Abstract
The geometry description plays a central role in many engineering applications and directly influences the quality of the computer simulation results. The geometry of a space curve can be completely defined in terms of two parameters: the horizontal and vertical curvatures, or equivalently, the curve curvature and torsion. In this paper, distinction is made between the track angle and space-curve bank angle, referred to in this paper as the Frenet bank angle. In railroad vehicle systems, the track bank angle measures the track super-elevation required to define a balance speed and achieve a safe vehicle operation. The formulation of the track space-curve differential equations in terms of Euler angles, however, shows the dependence of the Frenet bank angle on two independent parameters, often used as inputs in the definition of the track geometry. This paper develops the general differential equations that govern the track geometry using the Euler angle sequence adopted in practice. It is shown by an example that a curve can be twisted and vertically elevated but not super-elevated while maintaining a constant vertical-development angle. The continuity conditions at the track segment transitions are also examined. As discussed in the paper, imposing curvature continuity does not ensure continuity of the tangent vectors at the curve/spiral intersection. Several curve geometries that include planar and helix curves are used to explain some of the fundamental issues addressed in this study.
- Published
- 2021
33. An asymptotic analysis for a generalized Cahn–Hilliard system with fractional operators
- Author
-
Pierluigi Colli, Gianni Gilardi, and Jürgen Sprekels
- Subjects
Pure mathematics ,Asymptotic analysis ,010102 general mathematics ,Hilbert space ,Type (model theory) ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,symbols.namesake ,Projection (relational algebra) ,Mathematics (miscellaneous) ,Operator (computer programming) ,Bounded function ,symbols ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
In the recent paper “Well-posedness and regularity for a generalized fractional Cahn–Hilliard system” (Colli et al. in Atti Accad Naz Lincei Rend Lincei Mat Appl 30:437–478, 2019), the same authors have studied viscous and nonviscous Cahn–Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers $$A^{2r}$$ A 2 r and $$B^{2\sigma }$$ B 2 σ (in the spectral sense) of general linear operators A and B, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space $$L^2(\Omega )$$ L 2 ( Ω ) , for some bounded and smooth domain $$\Omega \subset {{\mathbb {R}}}^3$$ Ω ⊂ R 3 , and have compact resolvents. Existence, uniqueness, and regularity results have been proved in the quoted paper. Here, in the case of the viscous system, we analyze the asymptotic behavior of the solution as the parameter $$\sigma $$ σ appearing in the operator $$B^{2\sigma }$$ B 2 σ decreasingly tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of B appears.
- Published
- 2021
34. Modularity of the displacement coefficients and complete plate theories in the framework of the consistent-approximation approach
- Author
-
Michael Meyer-Coors, Reinhold Kienzler, and Patrick Schneider
- Subjects
Linear elasticity ,General Physics and Astronomy ,02 engineering and technology ,01 natural sciences ,Displacement (vector) ,010305 fluids & plasmas ,symbols.namesake ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Mechanics of Materials ,0103 physical sciences ,Plate theory ,Taylor series ,symbols ,Applied mathematics ,General Materials Science ,Boundary value problem ,Reduction (mathematics) ,Series expansion ,Variable (mathematics) ,Mathematics - Abstract
Starting from the three-dimensional theory of linear elasticity, we arrive at the exact plate problem by the use of Taylor series expansions. Applying the consistent-approximation approach to this problem leads to hierarchic generic plate theories. Mathematically, these plate theories are systems of partial-differential equations (PDEs), which contain the coefficients of the series expansions of the displacements (displacement coefficients) as variables. With the pseudo-reduction method, the PDE systems can be reduced to one main PDE, which is entirely written in the main variable, and several reduction PDEs, each written in the main variable and several non-main variables. So, after solving the main PDE, the reduction PDEs can be solved by insertion of the main variable. As a great disadvantage of the generic plate theories, there are fewer reduction PDEs than non-main variables so that not all of the latter can be determined independently. Within this paper, a modular structure of the displacement coefficients is found and proved. Based on it, we define so-called complete plate theories which enable us to determine all non-main variables independently. Also, a scheme to assemble Nth-order complete plate theories with equations from the generic plate theories is found. As it turns out, the governing PDEs from the complete plate theories fulfill the local boundary conditions and the local form of the equilibrium equations a priori. Furthermore, these results are compared with those of the classical theories and recently published papers on the consistent-approximation approach.
- Published
- 2021
35. A detailed note on the finite-buffer queueing system with correlated batch-arrivals and batch-size-/phase-dependent bulk-service
- Author
-
A. D. Banik, Souvik Ghosh, Herwig Bruneel, and Joris Walraevens
- Subjects
PROBABILITIES ,Consecutive customer loss (CCL) ,Technology and Engineering ,STRATEGIES ,Performance ,Computation ,0211 other engineering and technologies ,Phase (waves) ,Markov process ,02 engineering and technology ,Management Science and Operations Research ,01 natural sciences ,Finite-buffer queue ,Theoretical Computer Science ,Management Information Systems ,010104 statistics & probability ,symbols.namesake ,Markovian ,QUEUES ,service process (MSP) ,INPUT ,Batch-size-dependent bulk service ,Applied mathematics ,Markovian arrival process ,Batch Markovian arrival process (BMAP) ,0101 mathematics ,CONSECUTIVE CUSTOMER LOSSES ,Mathematics ,Service (business) ,Queueing theory ,021103 operations research ,measures ,EFFICIENT COMPUTATIONAL ANALYSIS ,Queueing system ,Service process ,MODEL ,Computational Theory and Mathematics ,symbols - Abstract
This paper analyzes a finite-buffer queueing system, where customers arrive in batches and the accepted customers are served in batches by a single server. The service is assumed to be dependent on the batch-size and follows a general bulk service rule. The inter-arrival times of batches are assumed to be correlated and they are represented through the batch Markovian arrival process (BMAP). Computation procedure of the queue-length distributions at the post-batch-service completion, an arbitrary, and the pre-batch-arrival epochs are discussed. Various performance measures along with the consecutive customer loss probabilities are studied considering batch-size-dependent renewal service time distributions. Further, the above finite-buffer bulk-service queueing model is also investigated considering correlated batch-service times which are presented through the Markovian service process (MSP). The phase-dependent consecutive loss probabilities for the correlated batch-service times are determined. In the form of tables and graphs, a variety of numerical results for different batch-service time distributions are presented in this paper.
- Published
- 2021
36. Extinction and Non-extinction of Solutions to a Fast Diffusion p-Laplace Equation with Logarithmic Non-linearity
- Author
-
Xiumei Deng and Jun Zhou
- Subjects
Laplace's equation ,0209 industrial biotechnology ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Logarithm ,010102 general mathematics ,Mathematical analysis ,02 engineering and technology ,01 natural sciences ,Domain (mathematical analysis) ,symbols.namesake ,020901 industrial engineering & automation ,Control and Systems Engineering ,Extinction (optical mineralogy) ,Bounded function ,Dirichlet boundary condition ,symbols ,0101 mathematics ,Diffusion (business) ,Energy (signal processing) ,Mathematics - Abstract
This paper deals with a class of fast diffusion p-Laplace equation with logarithmic non-linearity in a bounded smooth domain with homogeneous Dirichlet boundary condition. By using energy estimates and some ordinary differential inequalities, we study the conditions on extinction and non-extinction of global solutions. The results of this paper extend and complete the previous studies on this equation.
- Published
- 2021
37. Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: measures
- Author
-
Bjoern Bringmann
- Subjects
Computer Science::Machine Learning ,Statistics and Probability ,FOS: Physical sciences ,Computer Science::Digital Libraries ,01 natural sciences ,Statistics::Machine Learning ,010104 statistics & probability ,symbols.namesake ,Mathematics - Analysis of PDEs ,Singularity ,Gaussian free field ,FOS: Mathematics ,Statistical physics ,0101 mathematics ,Gibbs measure ,Mathematical Physics ,Mathematics ,Partial differential equation ,35L15, 60H30 ,Series (mathematics) ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Mathematical Physics (math-ph) ,Hartree ,Absolute continuity ,Wave equation ,Modeling and Simulation ,Computer Science::Mathematical Software ,symbols ,Mathematics - Probability ,Analysis of PDEs (math.AP) - Abstract
In this two-paper series, we prove the invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree nonlinearity. The main novelty is the singularity of the Gibbs measure with respect to the Gaussian free field. The singularity has several consequences in both measure-theoretic and dynamical aspects of our argument. In this paper, we construct and study the Gibbs measure. Our approach is based on earlier work of Barashkov and Gubinelli for the $\Phi^4_3$-model. Most importantly, our truncated Gibbs measures are tailored towards the dynamical aspects in the second part of the series. In addition, we develop new tools dealing with the non-locality of the Hartree interaction. We also determine the exact threshold between singularity and absolute continuity of the Gibbs measure depending on the regularity of the interaction potential., Comment: Added uniqueness statement and corrected typos
- Published
- 2021
38. On primitive elements of algebraic function fields and models of $$X_0(N)$$
- Author
-
Iva Kodrnja and Goran Muić
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,010102 general mathematics ,Modular form ,0102 computer and information sciences ,Modular forms, Modular curves, Birational equivalence, Primitive elements ,01 natural sciences ,11F11, 11F23 ,Separable space ,Mathematics - Algebraic Geometry ,symbols.namesake ,Continuation ,Number theory ,010201 computation theory & mathematics ,Fourier analysis ,Field extension ,FOS: Mathematics ,symbols ,Algebraic function ,Number Theory (math.NT) ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
This paper is a continuation of our previous works where we study maps from $X_0(N)$, $N \ge 1$, into $\mathbb P^2$ constructed via modular forms of the same weight and criteria that such a map is birational (see [12]). In the present paper our approach is based on the theory of primitive elements in finite separable field extensions. We prove that in most of the cases the constructed maps are birational, and we consider those such that the resulting equation of the image in $\mathbb P^2$ is simplest possible., Comment: arXiv admin note: text overlap with arXiv:1305.2428
- Published
- 2021
39. On the pair correlations of powers of real numbers
- Author
-
Christoph Aistleitner and Simon Baker
- Subjects
11K06, 11K60 ,General Mathematics ,Modulo ,FOS: Physical sciences ,0102 computer and information sciences ,Lebesgue integration ,01 natural sciences ,Combinatorics ,symbols.namesake ,Pair correlation ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Algebra over a field ,Classical theorem ,Mathematical Physics ,Real number ,Mathematics ,Sequence ,Mathematics - Number Theory ,Probability (math.PR) ,010102 general mathematics ,Mathematical Physics (math-ph) ,010201 computation theory & mathematics ,symbols ,Martingale (probability theory) ,Mathematics - Probability - Abstract
A classical theorem of Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In the present paper we extend Koksma's theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every $x>1$ the pair correlations of the fractional parts of $(x^n)_{n=1}^{\infty}$ are asymptotically Poissonian. The proof is based on a martingale approximation method., Version 2: some minor changes. The paper will appear in the Israel Journal of Mathematics
- Published
- 2021
40. 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
41. Automorphic Schwarzian equations and integrals of weight 2 forms
- Author
-
Abdellah Sebbar and Hicham Saber
- Subjects
Pure mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,Differential equation ,11F03, 11F11, 34M05 ,010102 general mathematics ,Modular form ,0102 computer and information sciences ,01 natural sciences ,symbols.namesake ,Number theory ,010201 computation theory & mathematics ,Fourier analysis ,Eisenstein series ,FOS: Mathematics ,Pi ,symbols ,Equivariant map ,Number Theory (math.NT) ,0101 mathematics ,Meromorphic function ,Mathematics - Abstract
In this paper, we investigate the non-modular solutions to the Schwarz differential equation $\{f,\tau \}=sE_4(\tau)$ where $E_4(\tau)$ is the weight 4 Eisenstein series and $s$ is a complex parameter. In particular, we provide explicit solutions for each $s=2\pi^2(n/6)^2$ with $n\equiv 1\mod 12$. These solutions are obtained as integrals of meromorphic weight 2 modular forms. As a consequence, we find explicit solutions to the differential equation $\displaystyle y''+\frac{\pi^2n^2}{36}\,E_4\,y=0$ for each $n\equiv 1\mod 12$ generalizing the work of Hurwitz and Klein on the case $n=1$. Our investigation relies on the theory of equivariant functions on the complex upper half-plane. This paper supplements a previous work where we determine all the parameters $s$ for which the above Schwarzian equation has a modular solution., Comment: 20 pages
- Published
- 2021
42. A parameter-free total Lagrangian smooth particle hydrodynamics algorithm applied to problems with free surfaces
- Author
-
Javier Bonet, Antonio J. Gil, Chun Hean Lee, Jibran Haider, and Kenny W.Q. Low
- Subjects
Fluid Flow and Transfer Processes ,Numerical Analysis ,Conservation law ,Discretization ,Computational Mechanics ,01 natural sciences ,Projection (linear algebra) ,010305 fluids & plasmas ,010101 applied mathematics ,Smoothed-particle hydrodynamics ,Computational Mathematics ,symbols.namesake ,TA ,Rate of convergence ,Inviscid flow ,Modeling and Simulation ,0103 physical sciences ,Jacobian matrix and determinant ,symbols ,0101 mathematics ,Hamiltonian (quantum mechanics) ,Algorithm ,Civil and Structural Engineering ,Mathematics - Abstract
This paper presents a new Smooth Particle Hydrodynamics computational framework for the solution of inviscid free surface flow problems. The formulation is based on the Total Lagrangian description of a system of first-order conservation laws written in terms of the linear momentum and the Jacobian of the deformation. One of the aims of this paper is to explore the use of Total Lagrangian description in the case of large deformations but without topological changes. In this case, the evaluation of spatial integrals is carried out with respect to the initial undeformed configuration, yielding an extremely efficient formulation where the need for continuous particle neighbouring search is completely circumvented. To guarantee stability from the SPH discretisation point of view, consistently derived Riemann-based numerical dissipation is suitably introduced where global numerical entropy production is demonstrated via a novel technique in terms of the time rate of the Hamiltonian of the system. Since the kernel derivatives presented in this work are fixed in the reference configuration, the non-physical clumping mechanism is completely removed. To fulfil conservation of the global angular momentum, a posteriori (least-squares) projection procedure is introduced. Finally, a wide spectrum of dedicated prototype problems is thoroughly examined. Through these tests, the SPH methodology overcomes by construction a number of persistent numerical drawbacks (e.g. hour-glassing, pressure instability, global conservation and/or completeness issues) commonly found in SPH literature, without resorting to the use of any ad-hoc user-defined artificial stabilisation parameters. Crucially, the overall SPH algorithm yields equal second order of convergence for both velocities and pressure.
- Published
- 2021
43. The direct method of Lyapunov for nonlinear dynamical systems with fractional damping
- Author
-
André Schmidt, Remco I. Leine, and Matthias Hinze
- Subjects
Lyapunov function ,State variable ,Invariance principle ,Dynamical systems theory ,Differential equation ,Applied Mathematics ,Mechanical Engineering ,Direct method ,Aerospace Engineering ,Ocean Engineering ,01 natural sciences ,010305 fluids & plasmas ,Fractional calculus ,Nonlinear system ,symbols.namesake ,Control and Systems Engineering ,0103 physical sciences ,symbols ,Applied mathematics ,Electrical and Electronic Engineering ,010301 acoustics ,Mathematics - Abstract
In this paper, we introduce a generalization of Lyapunov’s direct method for dynamical systems with fractional damping. Hereto, we embed such systems within the fundamental theory of functional differential equations with infinite delay and use the associated stability concept and known theorems regarding Lyapunov functionals including a generalized invariance principle. The formulation of Lyapunov functionals in the case of fractional damping is derived from a mechanical interpretation of the fractional derivative in infinite state representation. The method is applied on a single degree-of-freedom oscillator first, and the developed Lyapunov functionals are subsequently generalized for the finite-dimensional case. This opens the way to a stability analysis of nonlinear (controlled) systems with fractional damping. An important result of the paper is the solution of a tracking control problem with fractional and nonlinear damping. For this problem, the classical concepts of convergence and incremental stability are generalized to systems with fractional-order derivatives of state variables. The application of the related method is illustrated on a fractionally damped two degree-of-freedom oscillator with regularized Coulomb friction and non-collocated control., Bundesministerium für Bildung und Forschung, Projekt DEAL
- Published
- 2020
44. On periodic solutions for one-phase and two-phase problems of the Navier–Stokes equations
- Author
-
Thomas Eiter, Yoshihiro Shibata, and Mads Kyed
- Subjects
Pure mathematics ,symbols.namesake ,Mathematics (miscellaneous) ,Fourier transform ,Bounded function ,Free boundary problem ,symbols ,Boundary value problem ,Ball (mathematics) ,Navier–Stokes equations ,Parabolic partial differential equation ,Resolvent ,Mathematics - Abstract
This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier–Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasilinear systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value 0, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal $$L_p$$ L p –$$L_q$$ L q regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, which is obtained by the systematic use of $${\mathcal R}$$ R -solvers developed in Shibata (Diff Int Eqns 27(3–4):313–368, 2014; On the $${{\mathcal {R}}}$$ R -bounded solution operators in the study of free boundary problem for the Navier–Stokes equations. In: Shibata Y, Suzuki Y (eds) Springer proceedings in mathematics & statistics, vol. 183, Mathematical Fluid Dynamics, Present and Future, Tokyo, Japan, November 2014, pp 203–285, 2016; Comm Pure Appl Anal 17(4): 1681–1721. 10.3934/cpaa.2018081, 2018; $${{\mathcal {R}}}$$ R boundedness, maximal regularity and free boundary problems for the Navier Stokes equations, Preprint 1905.12900v1 [math.AP] 30 May 2019) to the resolvent problem for the linearized equations and the transference theorem obtained in Eiter et al. ($${{\mathcal {R}}}$$ R -solvers and their application to periodic $$L_p$$ L p estimates, Preprint in 2019) for the $$L_p$$ L p boundedness of operator-valued Fourier multipliers. These approaches are the novelty of this paper.
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- 2020
45. Self-similar Asymptotics for a Modified Maxwell–Boltzmann Equation in Systems Subject to Deformations
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Alexander Bobylev, Juan J. L. Velázquez, and Alessia Nota
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010102 general mathematics ,Mathematical analysis ,Complex system ,FOS: Physical sciences ,Second moment of area ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Collision ,01 natural sciences ,Maxwell–Boltzmann distribution ,Boltzmann equation ,symbols.namesake ,Mathematics - Analysis of PDEs ,Norm (mathematics) ,0103 physical sciences ,FOS: Mathematics ,symbols ,Higher order moments ,010307 mathematical physics ,0101 mathematics ,Mathematical Physics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
In this paper we study a generalized class of Maxwell-Boltzmann equations which in addition to the usual collision term contains a linear deformation term described by a matrix A. This class of equations arises, for instance, from the analysis of homoenergetic solutions for the Boltzmann equation considered by many authors since 1950s. Our main goal is to study a large time asymptotics of solutions under assumption of smallness of the matrix A. The main result of the paper is formulated in Theorem 2.1. Informally stated, this Theorem says that, for sufficiently small norm of A, any non-negative solution with finite second moment tends to a self-similar solution of relatively simple form for large values of time. This is what we call "the self-similar asymptotics". We also prove that the higher order moments of the self-similar profile are finite under further smallness condition on the matrix A., Comment: 38 pages
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- 2020
46. On Boundedness Property of Singular Integral Operators Associated to a Schrödinger Operator in a Generalized Morrey Space and Applications
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Thanh-Nhan Nguyen, Xuan Truong Le, and Ngoc Trong Nguyen
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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.
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- 2020
47. Diffusion-Probabilistic Least Mean Square Algorithm
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Chun Meng, Sihai Guan, and Bharat B. Biswal
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0209 industrial biotechnology ,Computational complexity theory ,Logarithm ,Applied Mathematics ,Gaussian ,Posterior probability ,Probabilistic logic ,02 engineering and technology ,Stability (probability) ,Noise (electronics) ,Least mean squares filter ,symbols.namesake ,020901 industrial engineering & automation ,Signal Processing ,symbols ,Hardware_ARITHMETICANDLOGICSTRUCTURES ,Algorithm ,Mathematics - Abstract
In this paper, a novel diffusion estimation algorithm is proposed from a probabilistic perspective by combining the diffusion strategy and the probabilistic least mean square (LMS) at all distributed network nodes. The proposed method, namely diffusion-probabilistic LMS (DPLMS), is more robust to the input signal and impulsive noise than previous algorithms like the diffusion sign-error LMS (DSE-LMS), diffusion robust variable step-size LMS (DRVSSLMS), and diffusion least logarithmic absolute difference (DLLAD) algorithms. Instead of minimizing the estimation error, the DPLMS algorithm is based on approximating the posterior distribution with an isotropic Gaussian distribution. In this paper, the stability of the mean estimation error and the computational complexity of the DPLMS algorithm are analyzed theoretically. Simulation experiments are conducted to explore the mean estimation error for the DPLMS algorithm with varied conditions for input signals and impulsive interferences, compared to the DSE-LMS, DRVSSLMS, and DLLAD algorithms. Both results from the theoretical analysis and simulation suggest that the DPLMS algorithm has superior performance than the DSE-LMS, DRVSSLMS, and DLLAD algorithms when estimating the unknown linear system under the changeable impulsive noise environments.
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- 2020
48. An Additive Approximation to Multiplicative Noise
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Ruanui Nicholson and Jari P. Kaipio
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FOS: Computer and information sciences ,Statistics and Probability ,Computational complexity theory ,Machine Learning (stat.ML) ,02 engineering and technology ,Poisson distribution ,Multiplicative noise ,Normal distribution ,symbols.namesake ,Statistics - Machine Learning ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,Mathematics - Optimization and Control ,Mathematics ,Applied Mathematics ,Multiplicative function ,Condensed Matter Physics ,Term (time) ,Noise ,Optimization and Control (math.OC) ,Modeling and Simulation ,symbols ,020201 artificial intelligence & image processing ,Geometry and Topology ,Computer Vision and Pattern Recognition ,Deconvolution - Abstract
Multiplicative noise models are often used instead of additive noise models in cases in which the noise variance depends on the state. Furthermore, when Poisson distributions with relatively small counts are approximated with normal distributions, multiplicative noise approximations are straightforward to implement. There are a number of limitations in the existing approaches to deal with multiplicative errors, such as positivity of the multiplicative noise term. The focus in this paper is on large dimensional (inverse) problems for which sampling-type approaches have too high computational complexity. In this paper, we propose an alternative approach utilising the Bayesian framework to carry out approximative marginalisation over the multiplicative error by embedding the statistics in an additive error term. The Bayesian framework allows the statistics of the resulting additive error term to be found based on the statistics of the other unknowns. As an example, we consider a deconvolution problem on random fields with different statistics of the multiplicative noise. Furthermore, the approach allows for correlated multiplicative noise. We show that the proposed approach provides feasible error estimates in the sense that the posterior models support the actual image.
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- 2020
49. Efficient parametric estimation for a signal-plus-noise Gaussian model from discrete time observations
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Khalil El Waled, Dominique Dehay, Vincent Monsan, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Qassim University [Kingdom of Saudi Arabia], Université Houphouët-Boigny, Abidjan,Côte d'Ivoire, Université de Cocody, AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)
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Statistics and Probability ,Hellinger distance ,Gaussian ,Low frequency sampling ,Minimax efficiency ,01 natural sciences ,Noise (electronics) ,010104 statistics & probability ,symbols.namesake ,0502 economics and business ,Applied mathematics ,0101 mathematics ,050205 econometrics ,Mathematics ,Bayes estimator ,05 social sciences ,Asymptotic properties of estimators ,Triangular Gaussian array ,[STAT.TH]Statistics [stat]/Statistics Theory [stat.TH] ,Bayesian estimation ,Maximum likelihood estimation ,Minimax ,High frequency sampling ,Discrete time and continuous time ,Gaussian noise ,symbols ,Gaussian network model ,Triangular array - Abstract
International audience; This paper deals with the parametric inference for integrated continuous time signals embeddedin an additive Gaussian noise and observed at deterministic discrete instants which arenot necessarily equidistant. The unknown parameter ismultidimensional and compounded ofa signal-of-interest parameter and a variance parameter of the noise.We state the consistencyand the minimax efficiency of the maximum likelihood estimator and of the Bayesian estimatorwhen the time of observation tends to infinity and the delays between two consecutiveobservations tend to 0 or are only bounded. The class of signals in consideration containsamong others, almost periodic signals and also non-continuous periodic signals. Howeverthe problem of frequency estimation is not considered here. Furthermore, in this paper thesignal-plus-noise discretely observed in time model is considered as a particular case of amore general model of independent Gaussian observations forming a triangular array.
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- 2020
50. Linear interval parametric approach to testing pseudoconvexity
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Milan Hladík, Iwona Skalna, and Lubomir V. Kolev
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Hessian matrix ,021103 operations research ,Control and Optimization ,Applied Mathematics ,0211 other engineering and technologies ,02 engineering and technology ,Interval (mathematics) ,Function (mathematics) ,Management Science and Operations Research ,Computer Science Applications ,symbols.namesake ,Pseudoconvexity ,symbols ,Applied mathematics ,Affine transformation ,Differentiable function ,Affine arithmetic ,Mathematics ,Parametric statistics - Abstract
The recent paper ( DOI: 10.1007/s10898-017-0537-6 ) suggests various practical tests (sufficient conditions) for checking pseudoconvexity of a twice differentiable function on an interval domain. The tests were implemented using interval extensions of the gradient and the Hessian of the function considered. In this paper, we present an alternative approach which is based on the use of more accurate affine form enclosures and affine arithmetic. We modify the tests to work with linear interval parametric enclosures of the gradients and the Hessians. We also present computational complexity results, showing that performing some tests exactly is NP-hard. It is shown by numerical experiments on random and benchmark data that the new approach results in more efficient tests for checking pseudoconvexity, however, at the expense of higher computation time.
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- 2020
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