2,795 results
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2. Remarks on a paper about functional inequalities for polynomials and Bernoulli numbers
- Author
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Jens Schwaiger
- Subjects
Combinatorics ,Polynomial ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Discrete Mathematics and Combinatorics ,Arithmetic function ,Context (language use) ,Limit (mathematics) ,Function (mathematics) ,Bernoulli number ,Mathematics - Abstract
The authors of [KMM] consider a system of two functional inequalities for a function $$f : {\mathbb{R}} \rightarrow {\mathbb{R}}$$ , and they show that, if certain arithmetical conditions and inequalities for certain parameters are fulfilled, f has to be a polynomial provided that f is continuous at some point x0. This result is derived here under the weaker condition that for some x0 the limit $${\rm lim}_{x \rightarrow x_0} f(x)$$ exists. Moreover, another system of inequalities is given leading to the same result on the nature of f. The methods used also give natural explanations for the fact that Bernoulli numbers play an important role in this context.
- Published
- 2009
3. On elementary theories of linear elastic beams, plates and shells (review paper)
- Author
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Constantin Mitropoulos and Mahir Sayir
- Subjects
Dynamic problem ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Linear elasticity ,General Physics and Astronomy ,Elasticity (economics) ,Anisotropy ,Mathematics - Abstract
This paper presents a review of the elementary theories on the bending of straight and curved beams, on plates and shells, using asymptotic approximations of the basic linearized equations of elasticity in three dimensions. The maximun norm has been chosen to specify the orders of magnitude of the quantities involved. The expansions are given as usual in terms of the small geometrical parameter characterizing the thinness of the structure. Most of the ideas and results are well known. Nevertheless, in the cases where more than one small parameter may be involved, such as small curvatures (shallow structures) or the small loading parameter used to linearize the equations of elasticity, the discussion on the limits of validity of the different theories lead to some interesting newer aspects. Moreover, the main ideas presented in this paper concerning multiple parameter expansions may be applied to discuss the behaviour of the structures and to obtain valuable analytical results in more complicated situations such as moderate and strong anisotropy, dynamic problems, stability etc.
- Published
- 1980
4. Remarks on the paper ?asymptotics of the coefficients in Levy-Wiener theorems on absolute convergence of trigonometric series? of B. a. Rogozin
- Author
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L. Marki
- Subjects
General Mathematics ,Mathematical analysis ,Applied mathematics ,Absolute convergence ,Mathematics ,Trigonometric series - Published
- 1978
5. A functional equation of Ih-Ching Hsu
- Author
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John S. Lew
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Functional equation ,Short paper ,Discrete Mathematics and Combinatorics ,Mathematics - Abstract
A recent note of Ih-Ching Hsu poses an unsolved problem, to wit, the general solution of the functional equation g(x1, x2) + g(φ1(x1), φ2(x2)) = g(x1, φ2(x2)) + g(φ1(x1),x2), where the φi are given functions. This short paper obtains the general solution. It gives conditions which imply that anycontinuous solution has form g1(x1) + g2(x2).
- Published
- 1989
6. Smoothness of Generalized Solutions of the Neumann Problem for a Strongly Elliptic Differential-Difference Equation on the Boundary of Adjacent Subdomains
- Author
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D. A. Neverova
- Subjects
Statistics and Probability ,Smoothness (probability theory) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Neumann boundary condition ,Boundary (topology) ,Differential difference equations ,General Medicine ,Mathematics - Abstract
This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. Some results for these equations such as existence and smoothness of generalized solutions in certain subdomains of Q were obtained earlier. Nevertheless, the smoothness of generalized solutions of such problems can be violated near the boundary of these subdomains even for infinitely differentiable right-hand side. The subdomains are defined as connected components of the set that is obtained from the domain Q by throwing out all possible shifts of the boundary Q by vectors of a certain group generated by shifts occurring in the difference operators. For the one dimensional Neumann problem for differential-difference equations there were obtained conditions on the coefficients of difference operators, under which for any continuous right-hand side there is a classical solution of the problem that coincides with the generalized solution. 2 Also there was obtained the smoothness (in Sobolev spaces W k ) of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in subdomains excluding -neighborhoods of certain points. However, the smoothness (in Ho lder spaces) of generalized solutions of the second boundary-value problem for strongly elliptic differential-difference equations on the boundary of adjacent subdomains was not considered. In this paper, we study this question in Ho lder spaces. We establish necessary and sufficient conditions for the coefficients of difference operators that guarantee smoothness of the generalized solution on the boundary of adjacent subdomains for any right-hand side from the Ho lder space.
- Published
- 2022
7. On Lacunas in the Spectrum of the Laplacian with the Dirichlet Boundary Condition in a Band with Oscillating Boundary
- Author
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Denis Borisov
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,Operator (physics) ,Mathematical analysis ,Spectrum (functional analysis) ,Boundary (topology) ,Function (mathematics) ,symbols.namesake ,Amplitude ,Dirichlet boundary condition ,symbols ,Flat band ,Laplace operator ,Mathematics - Abstract
In this paper, we consider the Laplace operator in a flat band whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude.
- Published
- 2021
8. 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
9. Maximum entropy principle closure for 14-moment system for a non-polytropic gas
- Author
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Tommaso Ruggeri
- Subjects
Physics ,Equation of state ,Internal energy ,Applied Mathematics ,General Mathematics ,Principle of maximum entropy ,010102 general mathematics ,Mathematical analysis ,Closure (topology) ,Polytropic process ,01 natural sciences ,Boltzmann equation ,010305 fluids & plasmas ,Moment (mathematics) ,0103 physical sciences ,Initial value problem ,0101 mathematics - Abstract
In this paper, we consider a rarefied polyatomic gas with a non-polytropic equation of state. We use the variational procedure of maximum entropy principle (MEP) to obtain the closure of the binary hierarchy of 14 moments associated with the Boltzmann equation in which the distribution function depends also on the energy of internal modes. The closed partial differential system is symmetric hyperbolic and the Cauchy problem is well-posed. In the limiting case of polytropic gas in which the internal energy is a linear function of the temperature, we find, as a special case, the previous results of Pavic et al. (Physica A 392:1302–1317, 2013). This paper, therefore, completes the equivalence between the closure obtained in the phenomenological rational extended thermodynamics theory and the one obtained by the MEP for general non-polytropic gas.
- Published
- 2020
10. Rectifying and Osculating Curves on a Smooth Surface
- Author
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Absos Ali Shaikh and Pinaki Ranjan Ghosh
- Subjects
Applied Mathematics ,General Mathematics ,Numerical analysis ,010102 general mathematics ,Mathematical analysis ,Osculating curve ,01 natural sciences ,Smooth surface ,0103 physical sciences ,Mathematics::Metric Geometry ,Mathematics::Differential Geometry ,010307 mathematical physics ,Tangent vector ,0101 mathematics ,Invariant (mathematics) ,Mathematics ,Geodesic curvature ,Osculating circle - Abstract
The main motive of the paper is to look on rectifying and osculating curves on a smooth surface. In this paper we find the normal and geodesic curvature for a rectifying curve on a smooth surface and we also prove that geodesic curvature is invariant under the isometry of surfaces such that rectifying curves remain. We find a sufficient condition for which an osculating curve on a smooth surface remains invariant under isometry of surfaces and also we prove that the component of the position vector of an osculating curve α(s) on a smooth surface along any tangent vector to the surface at α(s) is invariant under such isometry.
- Published
- 2020
11. Explicit transfer matrix for an incompressible orthotropic elastic layer and applications
- Author
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Vu Thi Ngoc Anh, L. T. Thang, Pham Chi Vinh, and Nguyen Thi Khanh Linh
- Subjects
Physics ,Lamb waves ,Composite plate ,Wave propagation ,Applied Mathematics ,General Mathematics ,Dispersion relation ,Mathematical analysis ,Reflection (physics) ,General Physics and Astronomy ,Boundary value problem ,Orthotropic material ,Transfer matrix - Abstract
In this paper, we establish transfer matrix for an incompressible orthotropic elastic layer. It is explicit and expressed compactly in terms of square brackets. This transfer matrix is a very convenient tool for solving various problems of wave propagation in layered elastic media including incompressible orthotropic layers. To prove this point, we apply it to investigate the reflection of SV-waves from an incompressible orthotropic layer overlying an incompressible orthotropic half-spaces and the propagation of Lamb waves in a composite plate consisting of two incompressible orthotropic layers. By using the obtained transfer matrix along with the effective boundary condition technique, we reduce the reflection of SV-waves from the layer to the reflection of SV-waves from the surface of half-space. The necessary and sufficient conditions for one or two reflected waves to exist have been established, and formulas for the reflection coefficients have been derived. Unlike the previously obtained formulas, the formulas derived in the present paper for the reflection coefficients are totally explicit. Employing the obtained transfer matrix, we arrive immediately at explicit dispersion equation of Lamb waves. Based on the obtained dispersion equation, it is shown numerically that for a two-layered plate with high-contrast material properties of the layers, the cutoff frequency of the first harmonic is close to zero. That means the low-frequency vibration spectrum of strongly inhomogeneous two-layered plates involves not only the fundamental bending mode, but also the first harmonic.
- Published
- 2021
12. Comparison of Asymptotic and Numerical Approaches to the Study of the Resonant Tunneling in Two-Dimensional Symmetric Quantum Waveguides of Variable Cross-Sections
- Author
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M. M. Kabardov, N. M. Sharkova, O. V. Sarafanov, and Boris Plamenevskii
- Subjects
Statistics and Probability ,Helmholtz equation ,Applied Mathematics ,General Mathematics ,Numerical analysis ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,010305 fluids & plasmas ,Matrix (mathematics) ,Resonator ,0103 physical sciences ,Boundary value problem ,0101 mathematics ,Wave function ,Quantum ,Quantum tunnelling ,Mathematics - Abstract
The waveguide considered coincides with a strip having two narrows of width e. An electron wave function satisfies the Dirichlet boundary value problem for the Helmholtz equation. The part of the waveguide between the narrows serves as a resonator, and conditions for the electron resonant tunneling may occur. In the paper, asymptotic formulas as e → 0 for characteristics of the resonant tunneling are used. The asymptotic results are compared with the numerical ones obtained by approximate calculation of the scattering matrix for energies in the interval between the second and third thresholds. The comparison allows us to state an interval of e, where the asymptotic and numerical approaches agree. The suggested methods can be applied to more complicated models than that considered in the paper. In particular, the same approach can be used for asymptotic and numerical analysis of the tunneling in three-dimensional quantum waveguides of variable cross-sections. Bibliography: 3 titles.
- Published
- 2019
13. Phaseless Sampling and Reconstruction of Real-Valued Signals in Shift-Invariant Spaces
- Author
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Junzheng Jiang, Qiyu Sun, and Cheng Cheng
- Subjects
FOS: Computer and information sciences ,Computer Science - Information Theory ,General Mathematics ,02 engineering and technology ,01 natural sciences ,symbols.namesake ,Robustness (computer science) ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Mathematics - Numerical Analysis ,0101 mathematics ,Invariant (mathematics) ,Sampling density ,Mathematics ,Box spline ,Partial differential equation ,Euclidean space ,Information Theory (cs.IT) ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,020206 networking & telecommunications ,Reconstruction algorithm ,Numerical Analysis (math.NA) ,Fourier analysis ,symbols ,Algorithm ,Analysis - Abstract
Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. In this paper, we consider phaseless sampling and reconstruction of real-valued signals in a high-dimensional shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. The determination of a signal in a shift-invariant space, up to a sign, by its magnitude measurements on the whole Euclidean space has been shown in the literature to be equivalent to its nonseparability. In this paper, we introduce an undirected graph associated with the signal in a shift-invariant space and use connectivity of the graph to characterize nonseparability of the signal. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that nonseparable signals in the shift-invariant space can be reconstructed in a stable way from their phaseless samples taken on that set. In this paper, we also propose a reconstruction algorithm which provides an approximation to the original signal when its noisy phaseless samples are available only. Finally, numerical simulations are performed to demonstrate the robustness of the proposed algorithm to reconstruct box spline signals from their noisy phaseless samples.
- Published
- 2018
14. Global well-posedness of 3D magneto-micropolar fluid equations with mixed partial viscosity near an equilibrium
- Author
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Yu-Zhu Wang and Weijia Li
- Subjects
Physics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,Bootstrapping (finance) ,01 natural sciences ,Stability (probability) ,Magnetic field ,Physics::Fluid Dynamics ,010101 applied mathematics ,Viscosity ,Initial value problem ,0101 mathematics ,Fluid equation ,Magneto ,Well posedness - Abstract
In this paper, we investigate the initial value problem for the 3D magneto-micropolar fluid equations with mixed partial viscosity. The main purpose of this paper is to establish global well-posedness of classical small solutions. More precisely, we prove that the global stability of perturbations near the steady solution is given by a background magnetic field. The proof is mainly based on the energy estimate and bootstrapping argument.
- Published
- 2021
15. Double phase problems with variable growth and convection for the Baouendi–Grushin operator
- Author
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Vicenţiu D. Rădulescu, Anouar Bahrouni, and Patrick Winkert
- Subjects
Convection ,Partial differential equation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,Differential operator ,01 natural sciences ,Term (time) ,010101 applied mathematics ,Nonlinear system ,Operator (computer programming) ,0101 mathematics ,Transonic ,Variable (mathematics) ,Mathematics - Abstract
In this paper we study a class of quasilinear elliptic equations with double phase energy and reaction term depending on the gradient. The main feature is that the associated functional is driven by the Baouendi–Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. We first establish some new qualitative properties of a differential operator introduced recently by Bahrouni et al. (Nonlinearity 32(7):2481–2495, 2019). Next, under quite general assumptions on the convection term, we prove the existence of stationary waves by applying the theory of pseudomonotone operators. The analysis carried out in this paper is motivated by patterns arising in the theory of transonic flows.
- Published
- 2020
16. Stability of periodic traveling waves for nonlocal dispersal cooperative systems in space–time periodic habitats
- Author
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Xiongxiong Bao
- Subjects
Physics ,Weight function ,Current (mathematics) ,Applied Mathematics ,General Mathematics ,Space time ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,01 natural sciences ,010101 applied mathematics ,Stability theory ,Initial value problem ,Uniform boundedness ,Uniqueness ,0101 mathematics ,Weighted space - Abstract
The current paper is devoted to the study of the stability of space–time periodic traveling wave solutions and positive space–time periodic entire solutions of nonlocal dispersal cooperative systems in space–time periodic habitats. We first show the existence, uniqueness and stability of positive space–time periodic entire solution $${\mathbf {u}}^{*}(t,x)$$ for such nonlocal dispersal cooperative system. The existence of space–time periodic traveling wave solution connecting $${\mathbf {0}}$$ and positive space–time periodic entire solution $${\mathbf {u}}^{*}(t,x)$$ has been established by Bao, Shen and Shen (Commun. Pure Appl. Anal. 18: 361–396, 2019). In this paper, by using comparison principle and a weight function, we further show that the space–time periodic traveling wave solution for nonlocal dispersal cooperative system is asymptotically stable, as long as the initial value is uniformly bounded in a weighted space.
- Published
- 2020
17. Periodic traveling wavefronts of a multi-type SIS epidemic model with seasonality
- Author
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Haiqin Zhao and Yumeng Gu
- Subjects
Wavefront ,education.field_of_study ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Population ,General Physics and Astronomy ,Type (model theory) ,Stability (probability) ,Coincidence ,Distribution (mathematics) ,Uniqueness ,Epidemic model ,education ,Mathematics - Abstract
This paper is concerned with a time-periodic and nonlocal system arising from the spread of a deterministic epidemic in multi-types of population by incorporating a seasonal variation. The existence of the critical wave speed of the periodic traveling wavefronts and its coincidence with the spreading speed were proved in Wu et al. (J Math Anal Appl 463:111–133, 2018). In this paper, we prove the uniqueness and stability of all non-critical periodic wavefronts. Of particular interest is the influences of time-periodicity on the spreading speed in one-dimensional case. It turns out that, in comparison with the autonomous case, the periodicity of the infection rate increases the spreading speed, while the periodicity of the combined death/emigration/recovery rate for infectious individuals decreases the spreading speed. We also find that the contact distribution increases the spreading speed.
- Published
- 2020
18. Mean Value Property of Harmonic Functions on the Tetrahedral Sierpinski Gasket
- Author
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Kui Yao, Hua Qiu, and Yipeng Wu
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,020206 networking & telecommunications ,02 engineering and technology ,01 natural sciences ,Domain (mathematical analysis) ,Sierpinski triangle ,symbols.namesake ,Fourier transform ,Harmonic function ,Fourier analysis ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Tetrahedron ,0101 mathematics ,Symmetry (geometry) ,Laplace operator ,Analysis ,Mathematics - Abstract
In this paper, we study the mean value property for both the harmonic functions and the functions in the domain of the Laplacian on the tetrahedral Sierpinski gasket. This paper is a continuation of the work of Strichartz and the first author (Qiu and Strichartz, J Fourier Anal Appl 19:943–966, 2013)where the same property on p.c.f. self-similar sets with Dihedral-3 symmetry was considered.
- Published
- 2018
19. A 2 $$\times $$ × 2 simple model in which the sub-shock exists when the shock velocity is slower than the maximum characteristic velocity
- Author
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Tommaso Ruggeri and Shigeru Taniguchi
- Subjects
Physics ,Entropy principle ,Applied Mathematics ,General Mathematics ,Numerical analysis ,010102 general mathematics ,Mathematical analysis ,Characteristic velocity ,Critical value ,Lambda ,01 natural sciences ,Convexity ,Hyperbolic systems ,010305 fluids & plasmas ,Entropy inequality ,0103 physical sciences ,0101 mathematics - Abstract
For a generic hyperbolic system of balance laws, the shock-structure solution is not continuous and a discontinuous part (sub-shock) arises when the velocity of the front s is greater than a critical value. In particular, for systems compatible with the entropy principle, continuous shock-structure solutions cannot exist when s is larger than the maximum characteristic velocity evaluated in the unperturbed state $$s >\lambda ^{\max }_0 $$ . This is the typical situation of systems of Rational Extended Thermodynamics (ET). Nevertheless, in principle, sub-shocks may exist also for s smaller than $$\lambda ^{\max }_0 $$ . This was proved with a simple example in a recent paper by Taniguchi and Ruggeri (Int J Non-Linear Mech 99:69, 2018). In the present paper, we offer another simple case that satisfies all requirements of ET, that is, the entropy inequality, convexity of the entropy, sub-characteristic condition and Shizuta-Kawashima condition, however, there exists a sub-shock with s slower than $$\lambda ^{\max }_0 $$ . Therefore there still remains an open question which other property makes the systems coming from ET have this beautiful property that the sub-shock exists only for s greater than the unperturbed maximum characteristic velocity.
- Published
- 2018
20. Quenching phenomenon for a parabolic MEMS equation
- Author
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Qi Wang
- Subjects
Microelectromechanical systems ,Quenching ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Zero (complex analysis) ,Lambda ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Dirichlet boundary condition ,Bounded function ,Domain (ring theory) ,symbols ,Initial value problem ,0101 mathematics ,Mathematics - Abstract
This paper deals with the electrostatic MEMS-device parabolic equation $${u_t} - \Delta u = \frac{{\lambda f(x)}}{{{{(1 - u)}^p}}}$$ in a bounded domain Ω of ℝ N , with Dirichlet boundary condition, an initial condition u0(x) ∈ [0, 1) and a nonnegative profile f, where λ > 0, p > 1. The study is motivated by a simplified micro-electromechanical system (MEMS for short) device model. In this paper, the author first gives an asymptotic behavior of the quenching time T* for the solution u to the parabolic problem with zero initial data. Secondly, the author investigates when the solution u will quench, with general λ, u0(x). Finally, a global existence in the MEMS modeling is shown.
- Published
- 2018
21. On Short-Wave Diffraction by an Elongated Body. Numerical Experiments
- Author
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N. Ya. Kirpichnikova, M. M. Popov, and N. M. Semtchenok
- Subjects
Statistics and Probability ,Diffraction ,Field (physics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Rotational symmetry ,01 natural sciences ,Fock space ,Boundary layer ,Exact solutions in general relativity ,0103 physical sciences ,0101 mathematics ,Convex function ,Asymptotic expansion ,010301 acoustics ,Mathematics - Abstract
The paper is a continuation of previous papers of the authors dealing with the exploration of shortwave diffraction by smooth and strictly convex bodies of revolution (the axisymmetric case). In these problems, the boundary layer method contains two large parameters: one is the Fock parameter M and the second is Λ that characterizes the oblongness of the scatterer. This naturally gives the possibility of using the two-scaled asymptotic expansion, where both M and Λ are regarded as independent. The approximate formulas for the wave field in this situation depend on the mutual strength between the large parameters and may vary. In the paper, we carry out numerical experiments with our formulas, in the case where the Fock analytical solution is in good coincidence with the exact solution of a model problem, in order to examine the influence of the parameter Λ on the wave field. It follows from our numerical experiments that the influence of the oblongness of the scatterer on the wave field is really insignificant if the method of Leontovich–Fock parabolic equation does not meet mathematical difficulties.
- Published
- 2017
22. Boundary value problem for one evolution equation
- Author
-
Sherif Amirov
- Subjects
Cauchy problem ,0209 industrial biotechnology ,Independent equation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,02 engineering and technology ,Mixed boundary condition ,01 natural sciences ,Elliptic boundary value problem ,020901 industrial engineering & automation ,Free boundary problem ,lcsh:Q ,Cauchy boundary condition ,Boundary value problem ,0101 mathematics ,lcsh:Science ,Constant (mathematics) ,Mathematics - Abstract
The aim of the paper is to investigate the boundary value problem of the evolution equation Lu = K (x,t) ut - Δu + a (x,t) u = f (x,t). The characteristic property of this type of equations is the failure of the Petrovski’s “A” condition when coefficients are constant [1]. In this case, Cauchy problem is incorrect in the sense of Hadamard. Hence in this paper, the space, guaranteeing the correctness of the boundary value problem in the sense of Hadamard, is selected by adding some additional conditions to the coefficients of the equation.
- Published
- 2017
23. On the $$\mathbb {K}$$ K -Riemann integral and Hermite–Hadamard inequalities for $$\mathbb {K}$$ K -convex functions
- Author
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Andrzej Olbryś
- Subjects
Mathematics(all) ,Pure mathematics ,Hermite polynomials ,Mathematics::Complex Variables ,Generalization ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Riemann integral ,01 natural sciences ,Convexity ,010101 applied mathematics ,symbols.namesake ,Hadamard transform ,symbols ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Convex function ,Mathematics - Abstract
In the present paper we introduce a notion of the \(\mathbb {K}\)-Riemann integral as a natural generalization of a usual Riemann integral and study its properties. The aim of this paper is to extend the classical Hermite–Hadamard inequalities to the case when the usual Riemann integral is replaced by the \(\mathbb {K}\)-Riemann integral and the convexity notion is replaced by \(\mathbb {K}\)-convexity.
- Published
- 2017
24. Fractional Action Cosmology with Variable Order Parameter
- Author
-
Rami Ahmad El-Nabulsi
- Subjects
Physics ,Formalism (philosophy of mathematics) ,Physics and Astronomy (miscellaneous) ,010308 nuclear & particles physics ,General Mathematics ,0103 physical sciences ,Mathematical analysis ,Applied mathematics ,010303 astronomy & astrophysics ,01 natural sciences ,Equations for a falling body ,Cosmology ,Fractional calculus - Abstract
Fractional action cosmology with variable order parameter was constructed in this paper. Starting from a fractional weighted action which generalizes the fractional actionlike variational approach, a large number of cosmological dynamical equations are obtained depending on the mathematical type of the fractional order parameter. Through this paper, we selected two independent types which result on a number of cosmological scenarios and we discussed their dynamical consequences. It was observed that the present fractional cosmological formalism holds a large family of solutions and offers new features not found in the standard formalism and in many fundamental research papers.
- Published
- 2017
25. A revised pre-order principle and set-valued Ekeland variational principles with generalized distances
- Author
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Jing Hui Qiu
- Subjects
Pure mathematics ,021103 operations research ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,0211 other engineering and technologies ,Regular polygon ,02 engineering and technology ,01 natural sciences ,Vector optimization ,Variational principle ,0101 mathematics ,Variational analysis ,Mathematics - Abstract
In my former paper “A pre-order principle and set-valued Ekeland variational principle” (see [J. Math. Anal. Appl., 419, 904–937 (2014)]), we established a general pre-order principle. From the pre-order principle, we deduced most of the known set-valued Ekeland variational principles (denoted by EVPs) in set containing forms and their improvements. But the pre-order principle could not imply Khanh and Quy’s EVP in [On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings, J. Glob. Optim., 49, 381–396 (2011)], where the perturbation contains a weak τ-function, a certain type of generalized distances. In this paper, we give a revised version of the pre-order principle. This revised version not only implies the original pre-order principle, but also can be applied to obtain the above Khanh and Quy’s EVP. In particular, we give several new set-valued EVPs, where the perturbations contain convex subsets of the ordering cone and various types of generalized distances.
- Published
- 2017
26. Limit Distributions of a Random Term of a Variational Series
- Author
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Yu. N. Blagoveschenskiy
- Subjects
Statistics and Probability ,Distribution (mathematics) ,Series (mathematics) ,Applied Mathematics ,General Mathematics ,Order statistic ,Mathematical analysis ,Sample (statistics) ,Limit (mathematics) ,Random variable ,Normal limit ,Mathematics ,Term (time) - Abstract
There are a lot of papers devoted to the study of order statistics, in particular, the asymptotics of the kth term of a variational series for a sample of size n of independent identically distributed random variables with distribution F(x) with different relations between k and n. In the “central” part, where min(k, n − k) → ∞, the normal limit distribution dominates. In the present paper we study the asymptotic behavior of a random term of the variational series: its number ν is a random variable, taking values 1, 2,…, n with equal probabilities. Under mild assumptions on the density of underlying distribution, we find all possible limit distributions for the νth term of variational series.
- Published
- 2016
27. Nonlinear stability of rarefaction waves for one-dimensional compressible Navier–Stokes equations for a reacting mixture
- Author
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Zefu Feng and Zheng Xu
- Subjects
Physics ,Cauchy problem ,Partial differential equation ,Differential equation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,General Physics and Astronomy ,01 natural sciences ,Compressible flow ,010101 applied mathematics ,Fluid dynamics ,Initial value problem ,0101 mathematics ,Adiabatic process ,Navier–Stokes equations - Abstract
In this paper, we study the long-time behavior toward rarefaction waves for the Cauchy problem to a one-dimensional Navier–Stokes equations for a reacting mixture. It is shown that under the condition adiabatic exponent $$\gamma $$ is close to 1, the global stability is established. In this paper, the initial perturbation can be large.
- Published
- 2019
28. Viscoelastic versus frictional dissipation in a variable coefficients plate system with time-varying delay
- Author
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Jianghao Hao and Peipei Wang
- Subjects
0209 industrial biotechnology ,Work (thermodynamics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Ode ,General Physics and Astronomy ,02 engineering and technology ,Function (mathematics) ,Dissipation ,01 natural sciences ,Viscoelasticity ,Exponential function ,020901 industrial engineering & automation ,Relaxation (physics) ,0101 mathematics ,Variable (mathematics) ,Mathematics - Abstract
In this paper, we are concerned with variable coefficients plate system subjected to three partially distributed feedbacks: time-varying delay, frictional and viscoelastic dissipations. This work is devoted to, without any prior quantification of both decay rate of relaxation function and growth rate of frictional dissipation near the origin, establish a general decay result which corresponds to a certainly stable ODE. Our result extends the decay result obtained for some kind of problems with finite history to problem with infinite history. Moreover, this paper allows a wider class of kernels of infinite history, and the usual exponential and polynomial decay rates are only special cases. The proof is based on the multiplier method and some techniques about convex functionals.
- Published
- 2019
29. The asymptotic solution of the ion-damped acoustic-gravity wave equation
- Author
-
John A. Adam
- Subjects
Physics ,Applied Mathematics ,General Mathematics ,Atmospheric wave ,Mathematical analysis ,General Physics and Astronomy ,Equations of motion ,Wave equation ,symbols.namesake ,Gravitational field ,symbols ,Wavenumber ,Vector field ,Gravity wave ,Lorentz force - Abstract
In his classic work Hydrodynamics, Horace Lamb devoted a significant amount of effort to the mathematical analysis of atmospheric waves, i.e. waves in a compressible medium under the influence of a local gravitational field and a variable temperature (and hence sound speed) profile. In so doing, he derived equations for both the divergence and the curl of the velocity field, yielding expressions of considerable mathematical beauty and complexity. Eight decades later, Derek Moore and Edward Spiegel extended Lamb’s analysis to include an arbitrary external applied force in the equations of motion. By a suitable choice of such force, the governing wave equations for a wide variety of Lorenz force/Coriolis force-induced wave motions can be derived. However, they chose to investigate the radiation field resulting from the application of a concentrated vertical force. In so doing, they were able to utilize an important theorem by James Lighthill concerning the asymptotic radiation field from a source with compact support, using the concept of a wavenumber surface. This paper has two main components: the first is to extend the work of Moore and Spiegel to ion-damped acoustic-gravity waves in the ionospheric F-region, based on the seminal work of C. H. Liu and K. C. Yeh. The second part is a consequence of the first: the corresponding wavenumber surface becomes complex, and so Lighthill’s method has to be modified to account for the effects of this. Furthermore, a significant inconsistency in the formulation of the physical problem by Liu and Yeh has been corrected and the corresponding derivations have been reformulated. Some graphical information has also been provided in special cases to illustrate the comparative effects of damping on the asymptotic behaviour of the acoustic-gravity radiation field. A final feature of the paper is that the equations derived are very general and can provide the basis for investigation of more realistic atmospheric temperature profiles in future work.
- Published
- 2019
30. Incompressible inviscid limit of the viscous two-fluid model with general initial data
- Author
-
Fucai Li and Young-Sam Kwon
- Subjects
Physics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,Interval (mathematics) ,Two-fluid model ,Wave equation ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Inviscid flow ,Convergence (routing) ,Compressibility ,Limit (mathematics) ,0101 mathematics - Abstract
In this paper, we study the incompressible inviscid limit of the viscous two-fluid model in the whole space $${\mathbb {R}}^3$$ with general initial data in the framework of weak solutions. By applying the refined relative entropy method and carrying out the detailed analysis on the oscillations of the densities and the velocity, we prove rigorously that the weak solutions of the compressible two-fluid model converge to the strong solution of the incompressible Euler equations in the time interval provided that the latter exists. Moreover, thanks to the Strichartz’s estimates of linear wave equations, we also obtain the convergence rates. The main ingredient of this paper is that our wave equations include the oscillations caused by the two different densities and the velocity and we also give an detailed analysis on the effect of the oscillations on the evolution of the solutions.
- Published
- 2019
31. Control of the radiative heating of a semi-transparent body
- Author
-
Hawraa Nabolsi, Ali Wehbe, and Luc Paquet
- Subjects
Physics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,02 engineering and technology ,021001 nanoscience & nanotechnology ,Coupling (probability) ,01 natural sciences ,Implicit function theorem ,Robin boundary condition ,Distribution (mathematics) ,Variational inequality ,Radiative transfer ,Boundary value problem ,0101 mathematics ,0210 nano-technology ,Absolute zero - Abstract
In a preceding paper, we have studied the radiative heating of a semi-transparent body $$\varOmega $$ (e.g., glass) by a black radiative source S surrounding it, black source at absolute uniform temperature u(t) at time t between time 0 and time $$t_\mathrm{f}$$ , the final time of the radiative heating. This problem has been modeled by an appropriate coupling between quasi-steady radiative transfer boundary value problems with nonhomogeneous reflectivity boundary conditions (one for each wavelength band in the semi-transparent electromagnetic spectrum of the glass) and a nonlinear heat conduction evolution equation with a nonlinear Robin boundary condition which takes into account those wavelengths for which the glass behaves like an opaque body. In the present paper, u being considered as the control variable, we want to adjust the absolute temperature distribution $$(x,t) \mapsto T(x,t)$$ inside the semi-transparent body $$\varOmega $$ near a desired temperature distribution $$T_\mathrm{d}(\cdot ,\cdot )$$ during the time interval of radiative heating $$]0,t_\mathrm{f}[$$ by acting on u, the purpose being to deform $$\varOmega $$ to manufacture a new object. In this respect, we introduce the appropriate cost functional and the set of admissible controls $$U_\mathrm{ad}$$ , for which we prove the existence of optimal controls. Introducing the state space and the state equation, a first-order necessary condition for a control $$u:t \mapsto u(t)$$ to be optimal is then derived in the form of a Variational Inequality by using the implicit function theorem and the adjoint problem. We close this paper by some numerical considerations.
- Published
- 2018
32. Lower Bounds for Haar Projections: Deterministic Examples
- Author
-
Tino Ullrich and Andreas Seeger
- Subjects
Mathematics::Functional Analysis ,Basis (linear algebra) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Classical Analysis and ODEs ,Probabilistic logic ,Haar ,010103 numerical & computational mathematics ,Triebel–Lizorkin space ,01 natural sciences ,Sobolev space ,Computational Mathematics ,Range (mathematics) ,Mathematics - Classical Analysis and ODEs ,Simple (abstract algebra) ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,46E35, 46B15, 42C40 ,Besov space ,Applied mathematics ,0101 mathematics ,Analysis ,Mathematics - Abstract
In a previous paper by the authors, the existence of Haar projections with growing norms in Sobolev–Triebel–Lizorkin spaces has been shown via a probabilistic argument. This existence was sufficient to determine the precise range of Triebel–Lizorkin spaces for which the Haar system is an unconditional basis. The aim of the present paper is to give simple deterministic examples of Haar projections that show this growth behavior in the respective range of parameters.
- Published
- 2016
33. Existence and non-existence of positive solutions of Sturm–Liouville BVPs for ODEs on whole line
- Author
-
Yuji Liu
- Subjects
Differential equation ,General Mathematics ,Numerical analysis ,010102 general mathematics ,Mathematical analysis ,Ode ,Fixed-point theorem ,Sturm–Liouville theory ,Algebraic geometry ,01 natural sciences ,010101 applied mathematics ,Singular solution ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
This paper is concerned with a boundary value problem of second order singular differential equations on whole line. Sufficient conditions to guarantee existence and non-existence of positive solutions are established. Our results improve some theorems in known papers but the methods used are different. We give two examples to illustrate main theorems.
- Published
- 2016
34. Strong Convergence Theorem on Split Equilibrium and Fixed Point Problems in Hilbert Spaces
- Author
-
Xiaoying Gong, Shinmin Kang, and Shenghua Wang
- Subjects
021103 operations research ,Iterative method ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,0211 other engineering and technologies ,Hilbert space ,02 engineering and technology ,Fixed point ,01 natural sciences ,Set (abstract data type) ,symbols.namesake ,Convergence (routing) ,symbols ,Applied mathematics ,Equilibrium problem ,Common element ,0101 mathematics ,Mathematics - Abstract
In this paper, we propose an iterative algorithm to find the common element of set of solutions of a split equilibrium problem and set of fixed points of an asymptotically nonexpansive mapping in Hilbert spaces. The new method is used to prove the strong convergence for the result of this paper. The result extends the corresponding one in the literature.
- Published
- 2016
35. A Projection-Type Method for Set Valued Variational Inequality Problems on Hadamard Manifolds
- Author
-
Chandal Nahak and S. Jana
- Subjects
021103 operations research ,General Mathematics ,Hadamard three-lines theorem ,010102 general mathematics ,Mathematical analysis ,0211 other engineering and technologies ,Solution set ,02 engineering and technology ,01 natural sciences ,Projection (linear algebra) ,Hadamard's inequality ,Set (abstract data type) ,Hadamard transform ,Variational inequality ,Applied mathematics ,Vector field ,0101 mathematics ,Mathematics - Abstract
In this paper, we develop a projection-type algorithm for set-valued variational inequalities on Hadamard manifolds. The proposed method is well defined whether the solution set of the problem is non-empty or not. Under pseudomonotonicity assumptions on the underlying vector field, our method is convergent to a solution of the given set-valued variational inequality. The results presented in this paper generalize and improve some known results introduced by Tang et al. (Optimization 64(5):1081–1096, 2015).
- Published
- 2016
36. The Length of an Extremal Network in a Normed Space: Maxwell Formula
- Author
-
Denis Petrovich Ilyutko, Igor Nikonov, and A. G. Bannikova
- Subjects
Statistics and Probability ,Pure mathematics ,Extremal length ,Euclidean space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,0102 computer and information sciences ,01 natural sciences ,010201 computation theory & mathematics ,Norm (mathematics) ,0101 mathematics ,Mathematics ,Normed vector space - Abstract
In the present paper we consider local minimal and extremal networks in normed spaces. It is well known that in the case of the Euclidean space these two classes coincide and the length of a local minimal network can be found by using only the coordinates of boundary vertices and the directions of boundary edges (the Maxwell formula). Moreover, as was shown by Ivanov and Tuzhilin [15], the length of a local minimal network in the Euclidean space can be found by using the coordinates of boundary vertices and the structure of the network. In the case of an arbitrary norm there are local minimal networks that are not extremal networks, and an analogue of the formula mentioned above is only true for extremal networks; this is the main result of the paper. Moreover, we generalize the Maxwell formula for the case of extremal networks in normed spaces and give an explicit construction of norming functionals used in the formula for several normed spaces.
- Published
- 2016
37. On the number of limit cycles of a Z 4-equivariant quintic near-Hamiltonian system
- Author
-
Mao An Han and Xianbo Sun
- Subjects
Hopf bifurcation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Heteroclinic bifurcation ,Upper and lower bounds ,Quintic function ,Hamiltonian system ,symbols.namesake ,Limit cycle ,symbols ,Equivariant map ,Limit (mathematics) ,Mathematics - Abstract
In this paper, we study the number of limit cycles of a near-Hamiltonian system having Z4-equivariant quintic perturbations. Using the methods of Hopf and heteroclinic bifurcation theory, we find that the perturbed system can have 28 limit cycles, and its location is also given. The main result can be used to improve the lower bound of the maximal number of limit cycles for some polynomial systems in a previous work, which is the main motivation of the present paper.
- Published
- 2015
38. Analytical solution for the electric field in Hall plates
- Author
-
Dorel Homentcovschi and Romeo Bercia
- Subjects
Physics ,Integrable system ,Applied Mathematics ,General Mathematics ,020208 electrical & electronic engineering ,Mathematical analysis ,General Physics and Astronomy ,Conformal map ,02 engineering and technology ,Condensed Matter::Mesoscopic Systems and Quantum Hall Effect ,021001 nanoscience & nanotechnology ,Domain (mathematical analysis) ,Quadrature (mathematics) ,Electric field ,0202 electrical engineering, electronic engineering, information engineering ,Gravitational singularity ,0210 nano-technology ,Voltage ,Analytic function - Abstract
The paper provides a closed-form expression (the General Formula) for the electric field in a Hall plate. The parameters entering in this formula are the images of the contacts extremities in the conformal map into the upper half-plane (canonical domain) and also a number of real unknown constants equal to the number of contacts. The paper gives also formulas for computing the voltages and currents at contacts of the Hall plate. Since the unknown constants are connected in a linear manner with the terminal voltages and the currents, the General Formula can be used for modeling a large variety of Hall devices. Finally, the explicit calculations require the quadrature of some analytic functions having integrable singularities.
- Published
- 2018
39. Two-dimensional strain gradient damage modeling: a variational approach
- Author
-
Emilio Barchiesi, Anil Misra, and Luca Placidi
- Subjects
Karush–Kuhn–Tucker conditions ,Deformation (mechanics) ,Applied Mathematics ,General Mathematics ,Linear elasticity ,Mathematical analysis ,Isotropy ,General Physics and Astronomy ,02 engineering and technology ,Dissipation ,01 natural sciences ,010101 applied mathematics ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Damage mechanics ,0101 mathematics ,Galerkin method ,Energy functional ,Mathematics - Abstract
In this paper, we formulate a linear elastic second gradient isotropic two-dimensional continuum model accounting for irreversible damage. The failure is defined as the condition in which the damage parameter reaches 1, at least in one point of the domain. The quasi-static approximation is done, i.e., the kinetic energy is assumed to be negligible. In order to deal with dissipation, a damage dissipation term is considered in the deformation energy functional. The key goal of this paper is to apply a non-standard variational procedure to exploit the damage irreversibility argument. As a result, we derive not only the equilibrium equations but, notably, also the Karush–Kuhn–Tucker conditions. Finally, numerical simulations for exemplary problems are discussed as some constitutive parameters are varying, with the inclusion of a mesh-independence evidence. Element-free Galerkin method and moving least square shape functions have been employed.
- Published
- 2018
40. Monostable traveling waves for a time-periodic and delayed nonlocal reaction–diffusion equation
- Author
-
Shi-Liang Wu and Panxiao Li
- Subjects
Physics ,Time periodic ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,General Physics and Astronomy ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Multivibrator ,Exponential stability ,Population model ,Reaction–diffusion system ,Traveling wave ,Uniqueness ,0101 mathematics - Abstract
This paper is concerned with a time-periodic and delayed nonlocal reaction–diffusion population model with monostable nonlinearity. Under quasi-monotone or non-quasi-monotone assumptions, it is known that there exists a critical wave speed $$c_*>0$$ such that a periodic traveling wave exists if and only if the wave speed is above $$c_*$$ . In this paper, we first prove the uniqueness of non-critical periodic traveling waves regardless of whether the model is quasi-monotone or not. Further, in the quasi-monotone case, we establish the exponential stability of non-critical periodic traveling fronts. Finally, we illustrate the main results by discussing two types of death and birth functions arising from population biology.
- Published
- 2018
41. Three-dimensional coupled thermoelastodynamic stress and flux induced wave propagation for isotropic half-space with scalar potential functions
- Author
-
Yazdan Hayati and Morteza Eskandari-Ghadi
- Subjects
Physics ,Partial differential equation ,Applied Mathematics ,General Mathematics ,Traction (engineering) ,Mathematical analysis ,General Physics and Astronomy ,Scalar potential ,02 engineering and technology ,Half-space ,021001 nanoscience & nanotechnology ,Integral transform ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Ordinary differential equation ,Boundary value problem ,Cylindrical coordinate system ,0210 nano-technology - Abstract
An asymmetric three-dimensional thermoelastodynamic wave propagation with scalar potential functions is presented for an isotropic half-space, in such a way that the wave may be originated from an arbitrary either traction or heat flux applied on a patch at the free surface of the half-space. The displacements, stresses and temperature are presented within the framework of Biot’s coupled thermoelasticity formulations. By employing a complete representation for the displacement and temperature fields in terms of two scalar potential functions, the governing equations of coupled thermoelasticity are uncoupled into a sixth- and a second-order partial differential equation in cylindrical coordinate system. By virtue of Fourier expansion and Hankel integral transforms, the angular and radial variables are suppressed respectively, and a $$6{\mathrm{th}}$$ - and a $$2{\mathrm{nd}}$$ -order ordinary differential equation in terms of depth are received, which are solved readily, from which the displacement, stresses and temperature fields are derived in transformed space by satisfying both the regularity and boundary conditions. By applying the inverse Hankel integral transforms, the displacements and temperature are numerically evaluated to determine the solutions in the real space. The numerical evaluations are done for three specific cases of vertical and horizontal time-harmonic patch traction and a constant heat flux passing through a circular disc on the surface of the half-space. It has been previously proved that the potential functions used in this paper are applicable from elastostatics to thermoelastodynamics. Thus, the analytical solutions presented in this paper are verified by comparing the results of this study with two specific problems reported in the literature, which are an elastodynamic problem and an axisymmetric quasi-static thermoelastic problem. To show the accuracy of numerical results, the solution of this study is also compared with the solution for elastodynamics exists in the literature for surface excitation, where a very good agreement is achieved. The formulations presented in this study may be used as benchmark for other related researches and it may be implemented in the related boundary integral equations.
- Published
- 2018
42. Viscosity iterative algorithm for variational inequality problems and fixed point problems of strict pseudo-contractions in uniformly smooth Banach spaces
- Author
-
Gang Cai
- Subjects
Pure mathematics ,Iterative method ,Applied Mathematics ,General Mathematics ,Viscosity (programming) ,Variational inequality ,Mathematical analysis ,Convergence (routing) ,Banach space ,Countable set ,Uniformly convex space ,Fixed point ,Mathematics - Abstract
The purpose of this paper is to study a new viscosity iterative algorithm based on a generalized contraction for finding a common element of the set of solutions of a general variational inequality problem for finite inversely strongly accretive mappings and the set of common fixed points for a countable family of strict pseudo-contractions in uniformly smooth Banach spaces. We prove some strong convergence theorems under some suitable conditions. The results obtained in this paper improve and extend the recent ones announced by many others in the literature.
- Published
- 2015
43. Cauchy problems of pseudo-parabolic equations with inhomogeneous terms
- Author
-
Zhongping Li and Wanjuan Du
- Subjects
Life span ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,Fujita exponent ,Mathematical analysis ,Mathematics::Analysis of PDEs ,General Physics and Astronomy ,Cauchy distribution ,Infinity ,Parabolic partial differential equation ,Term (time) ,Initial value problem ,Critical exponent ,media_common ,Mathematics - Abstract
This paper deals with Cauchy problems of pseudo-parabolic equations with inhomogeneous terms. The aim of the paper is to study the influence of the inhomogeneous term on the asymptotic behavior of solutions. We at first determine the critical Fujita exponent and then give the secondary critical exponent on the decay asymptotic behavior of an initial value at infinity. Furthermore, the precise estimate of life span for the blow-up solution is obtained. Our results show that the asymptotic behavior of solutions is seriously affected by the inhomogeneous term.
- Published
- 2015
44. Cubature and Quadrature Formulas of High Order of Approximation
- Author
-
D. A. Silaev
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Mathematics::Numerical Analysis ,Quadrature (mathematics) ,Smoothing spline ,Spline (mathematics) ,Orders of approximation ,Simply connected space ,Locally integrable function ,Polar coordinate system ,High order ,Mathematics - Abstract
This paper is concerned with the use of semilocal smoothing splines (or S-splines) for constructing cubature formulas. Such a spline is a piecewise-polynomial function. The first coefficients of each of the polynomials are determined by the smooth joint conditions, and the remaining ones, by the least-squares method. Previous studies were concerned with splines of degree 3 and 5. In the present paper, we consider S-splines of degree n (n = 9, 10). Of special importance for calculation of integrals are the S-splines of class C 0 (the continuous ones). Such splines are employed in building quadrature and cubature formulas of high order of approximation for calculation of one-, two-, and three-dimensional integrals in a simply connected domain to 10th and 11th orders of approximation. The integrable function is assumed to lie in the class C (n+1) (n = 9, 10) in a somewhat larger domain than the original one (in which the integration takes place). It is also assumed that the boundary of the domain is given parametrically. This makes it possible to take into account, with high order of accuracy, the boundary of the domain. The corresponding convergence rates are estimates. A similar approach is also capable of building formulas for integration of smooth functions in multidimensional domains.
- Published
- 2015
45. Morozov-type discrepancy principle for nonlinear ill-posed problems under η-condition
- Author
-
M Thamban Nair
- Subjects
Linear map ,Well-posed problem ,Tikhonov regularization ,Nonlinear system ,General Mathematics ,Mathematical analysis ,Applied mathematics ,Special case ,Lipschitz continuity ,Regularization (mathematics) ,Mathematics - Abstract
For proving the existence of a regularization parameter under a Morozov- type discrepancy principle for Tikhonov regularization of nonlinear ill-posed problems, it is required to impose additional nonlinearity assumptions on the forward operator. Lipschitz continuity of the Freechet derivative and requirement of the Lipschitz constant to depend on a source condition is one such restriction (Ramlau P, Numer. Funct. Anal. Optim. 23(1&22 )( 2003) 147-172). Another nonlinearity condition considered by Scherzer (Computing, 51 (1993) 45-60) was by requiring the forward operator to be close to a linear operator in a restricted sense. A seemingly natural nonlinear assump- tion which appears in many applications which attracted attention in various contexts of the study of nonlinear problems is the so-called η-condition. However, a Morozov-type discrepancy principle together with η-condition does not seem to have been studied, except in a recent paper by the author (Bull. Aust. Math. Soc. 79 (2009) 337-342), where error estimates under a general source condition is derived, by assuming the existence of the parameter. In this paper, the existence of the parameter satisfying a Morozov-type discrepancy principle is proved under the η-condition on the forward operator, by assuming the source condition as in the papers of Scherzer (Computing, 51 (1993) 45-60) and Ramlau (Numer. Funct. Anal. Optim. 23(1&22 )( 2003) 147-172). This source condition is, in fact, a special case of the source condition in the author's paper (Bull. Aust. Math. Soc. 79 (2009) 337-342).
- Published
- 2015
46. Homogeneous Spaces with Inner Metric and with Integrable Invariant Distributions
- Author
-
V. V. Gorbatsevich and V. N. Berestovskii
- Subjects
Statistics and Probability ,Pure mathematics ,Algebra and Number Theory ,Integrable system ,Applied Mathematics ,General Mathematics ,Subalgebra ,Mathematical analysis ,Carnot group ,Lie group ,Algebra ,Homogeneous ,Symmetric space ,Lie algebra ,Heisenberg group ,Locally compact space ,Invariant (mathematics) ,Mathematical Physics ,Analysis ,Mathematics - Abstract
This paper is a survey of results (partly obtained by the authors) on homogeneous spaces of Lie groups \(G\) with a compact stabilizer subgroup \(H\), on which every \(G\)-invariant distribution is integrable. It is proved that the condition of integrability is necessary and sufficient for every invariant inner metric to be (holonomic) Finsler on such a space. As a corollary of the obtained results, we assert that the class of homogeneous spaces with invariant non-holonomic Riemannian metrics (in other terms, sub-Riemannian or Carnot–Caratheodory metrics), which were actively studied last 3 decades after Gromov’s work, is rather broad. On the other hand, the class of homogeneous spaces with integrable invariant distributions includes Cartan’s symmetric spaces as well as isotropy irreducible, in particular, strictly isotropy irreducible, homogeneous spaces, which have been classified in simply connected case in the papers by Wang and Ziller (respectively, by Manturov, Wolf and Kramer). Special attention is paid to the case, when the Lie groups \(G\) and \(H\) are connected. Then the integrability condition of the invariant distributions is equivalent to a purely algebraic condition, that for the Lie algebra \(h\) of the subgroup \(H\), any \(ad(h)\)-invariant vector subspace in the Lie algebra \(g\) of the Lie group \(G\) is a Lie subalgebra; such Lie subalgebra \(h\subset g\) is called a strong subalgebra. The first author proved that a simply connected and compact space \(G/H\) with this condition is isomorphic to a direct product of strictly isotropy irreducible homogeneous spaces. In line with this, the second author recently found several non-compact simply connected homogeneous spaces with this condition, which are not isomorphic to such direct products. These results are naturally related to the structure questions of a class of general homogeneous locally compact spaces with an inner metric. This class is exactly the closure in the Gromov–Hausdorff sense of the class of homogeneous manifolds with an inner metric. Any such manifold is isometric to some homogeneous manifold \(G/H\) with \(G\)-invariant (may be, non-holonomic) Finsler metric. The authors give fairly detailed survey of the existing methods of the search of geodesics, i.e., locally shortest arcs, on such manifolds (particularly, with invariant non-holonomic Riemannian metrics), non-holonomic metric geometry and its relations with the geometric group theory, \(CR\)-manifolds, thermodynamics, etc. Some unsolved problems are suggested.
- Published
- 2015
47. Monostatic SAR with Fold/Cusp Singularities
- Author
-
Raluca Felea and Cliff Nolan
- Subjects
Synthetic aperture radar ,Partial differential equation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Microlocal analysis ,Geometry ,Fourier integral operator ,symbols.namesake ,Operator (computer programming) ,Inflection point ,Fourier analysis ,symbols ,Gravitational singularity ,Analysis ,Mathematics - Abstract
This paper analyses the image that one obtains by backprojecting synthetic aperture RADAR data collected on a flight-track with inflection points. The result is that one obtains artefacts that are of equal strength as the bona-fide part of the image. Furthermore, we obtain a weak normal form for operators associated to a fold/cusp canonical relation, which appears for our forward operator. Therefore this paper should be of use to researchers in different fields where such a structure arises.
- Published
- 2015
48. On Transforms of Divergence-Free and Curl-Free Fields, Associated with Inverse Problems
- Author
-
M. N. Demchenko
- Subjects
Statistics and Probability ,Curl (mathematics) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Vector field ,Riemannian manifold ,Inverse problem ,Lipschitz continuity ,Unitary state ,Self-adjoint operator ,Mathematics - Abstract
The M- and N-transforms acting, respectively, on divergence-free and curl-free vector fields on a Riemannian manifold with boundary are investigated. These transforms arise in studying the inverse problems of electrodynamics and elasticity theory. A divergence-free field is mapped by M to a field that is tangential to equidistants of the boundary. The N-transform maps a curl-free field to a field that is normal to equidistants. In preceding papers, the operators M and N were considered in the case of smooth equidistants, which is realized in a sufficiently small near-boundary layer. This allows one to consider transforms of fields supported in such a layer; it was proved that M and N are unitary in the corresponding spaces with L2-norms. In one of the papers, the case of fields on the whole manifold was considered, but almost all equidistants were assumed to be Lipschitz surfaces. It was proved that M is coisometric (i.e., the adjoint operator is isometric). In the present paper, the same result is obtained for both transforms in the general case with no constraints on equidistants at all.
- Published
- 2015
49. Spectrum and stability analysis for a transmission problem in thermoelasticity with a concentrated mass
- Author
-
Genqi Xu and Zhong-Jie Han
- Subjects
Multiplier (Fourier analysis) ,Thermoelastic damping ,Transmission (telecommunications) ,Applied Mathematics ,General Mathematics ,Frequency domain ,Mathematical analysis ,Spectrum (functional analysis) ,General Physics and Astronomy ,Exponential decay ,Stability (probability) ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, a transmission problem between elastic and thermoelastic material is considered. Assume that these two materials are connected by a vibrating concentrated mass. By a detailed spectral analysis, the asymptotic expressions of the eigenvalues of the system are obtained, and based on which, the Riesz basis property of the eigenvectors is deduced. It is proved that the total energy of this system cannot achieve exponential decay. However, by the frequency domain method together with some multiplier techniques, the polynomial decay of the system is showed and the optimal decay rate is estimated. Finally, some numerical simulations are given to support the results obtained in this paper.
- Published
- 2015
50. Periodic solutions to nonlinear wave equations with spatially dependent coefficients
- Author
-
Jinhai Chen
- Subjects
Inverse function theorem ,Nonlinear system ,Nonlinear wave equation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Spectrum (functional analysis) ,General Physics and Astronomy ,Boundary value problem ,Uniqueness ,D'Alembert operator ,Eigenvalues and eigenvectors ,Mathematics - Abstract
This paper investigates the existence and uniqueness of weak solutions to a periodic boundary value problem for a system of nonlinear wave equations with spatially dependent coefficients. Priori estimates of weak solutions are also established for the periodic boundary value problem. The arguments rely on spectral properties of the corresponding wave operator and a global inverse function theorem. The results presented in this paper extend the ones known in the literature in that eigenvalues of nonlinear perturbing terms appeared in the system of nonlinear wave equations can be chosen from the spectrum of the underlying wave operator.
- Published
- 2015
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