Showing papers in "Analysis and Applications in 2011"
TL;DR: In this article, the authors considered a model class of second order, linear, parametric, elliptic PDE's in a bounded domain D with coefficients depending on possibly countably many parameters and showed that the dependence of the solution on the parameters in the diffusion coefficient is analytically smooth.
Abstract: Parametric partial differential equations are commonly used to model physical systems. They also arise when Wiener chaos expansions are used as an alternative to Monte Carlo when solving stochastic elliptic problems. This paper considers a model class of second order, linear, parametric, elliptic PDE's in a bounded domain D with coefficients depending on possibly countably many parameters. It shows that the dependence of the solution on the parameters in the diffusion coefficient is analytically smooth. This analyticity is then exploited to prove that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations can be simultaneously approximated by multivariate polynomials (in the parameters) with coefficients taking values in the Hilbert space of weak solutions of the elliptic problem with a controlled number of terms N. The convergence rate in terms of N does not depend on the number of parameters in V which may be countable, therefore breaking the curse of dimensionality. The discretization of the coefficients from a family of continuous, piecewise linear finite element functions in D is shown to yield finite dimensional approximations whose convergence rate in terms of the overall number Ndof of degrees of freedom is the minimum of the convergence rates afforded by the best N-term sequence approximations in the parameter space and the rate of finite element approximations in D for a single instance of the parametric problem.
426 citations
TL;DR: In this paper, the global existence of solution to the Cauchy problem in the whole space is proved in some suitable weighted Sobolev spaces for hard potential when the solution is a small perturbation of a global equilibrium.
Abstract: As a continuation of our series works on the Boltzmann equation without angular cutoff assumption, in this part, the global existence of solution to the Cauchy problem in the whole space is proved in some suitable weighted Sobolev spaces for hard potential when the solution is a small perturbation of a global equilibrium.
72 citations
TL;DR: In this article, the existence of a renormalized solution for a class of nonlinear parabolic problems whose prototype is where QT = Ω × (0, T), Ω is an open and bounded subset of ℝN, N ≥ 2, T > 0, Δp is the so called p-Laplace operator,, c ∈ (Lr(QT))N with,, b ∈ LN+2, 1,QT), f ∈ l1(QTs), g ∈(Lp'(Qts
Abstract: In this paper, we prove, the existence of a renormalized solution for a class of nonlinear parabolic problems whose prototype is where QT = Ω × (0, T), Ω is an open and bounded subset of ℝN, N ≥ 2, T > 0, Δp is the so called p-Laplace operator, , c ∈ (Lr(QT))N with , , b ∈ LN+2, 1(QT), f ∈ L1(QT), g ∈ (Lp'(QT))N and u0 ∈ L1(Ω).
44 citations
TL;DR: This work considers a fully online regression algorithm associated with a general convex loss function and Gaussian kernels with changing variances and shows that the achieved learning rates can be of polynomial decays.
Abstract: We consider a fully online regression algorithm associated with a general convex loss function and Gaussian kernels with changing variances. Error analysis is conducted in a setting with samples drawn from a non-identical sequence of probability measures. When a fixed Gaussian is used, it was known that the learning ability of induced algorithms is weak. By allowing varying Gaussians, we show that the achieved learning rates can be of polynomial decays.
23 citations
TL;DR: In this article, the authors established the rate of decay to 0 and studied the oscillation properties of solutions to the scalar second order ODE: u″ + c|u′|αu′ + |u|βu = 0, where c, α, β are positive constants.
Abstract: We establish the rate of decay to 0 and we study the oscillation properties of solutions to the scalar second order ODE: u″ + c|u′|αu′ + |u|βu = 0, where c, α, β are positive constants. Various extensions (forced equation, system) are considered.
22 citations
TL;DR: In this paper, the authors studied the behavior of the solutions when the viscosity goes to zero and established the regularity of the solution of the viscous and inviscid quasigeotrophic equations.
Abstract: In this article, we consider the barotropic quasigeostrophic equation of the ocean in the context of the β-plane approximation and small viscosity (see, e.g., [21, 22]). The aim is to study the behavior of the solutions when the viscosity goes to zero. To avoid the substantial complications due to the corners (see, e.g., [25]) which will be addressed elsewhere, we assume periodicity in one direction (0y). The behavior of the solution in the boundary layers at x = 0, 1 necessitate the introduction of several correctors, solving various analogues of the Prandtl equation. Convergence is obtained at all orders even in the nonlinear case. We also establish as an auxiliary result, the regularity of the solutions of the viscous and inviscid quasigeotrophic equations.
21 citations
TL;DR: In this article, the authors apply complex inversion formula for the $\mathcal{L}_2$-transform and also show some applications of the $L 2 -transform for solving of singular integral equation with trigonometric kernel and system of partial fractional differential equations.
Abstract: In this paper, the authors apply complex inversion formula for the $\mathcal{L}_2$-transform and also show some applications of the $\mathcal{L}_2$-transform for solving of singular integral equation with trigonometric kernel and system of partial fractional differential equations.
19 citations
TL;DR: In this article, the authors considered a stochastic ratio-dependent predator-prey model and proved the existence, uniqueness and positivity of the solutions, and studied the boundedness of moments of population.
Abstract: In this paper, we consider a stochastic ratio-dependent predator-prey model. We firstly prove the existence, uniqueness and positivity of the solutions. Then, the boundedness of moments of population are studied. Finally, we show the upper-growth rates and exponential death rates of population under some conditions.
15 citations
TL;DR: In this article, a variational formulation of the existence problem for the traveling wave solution is presented. But the main objective is to obtain exact and approximate traveling wave solutions with error estimates.
Abstract: Reaction-diffusion systems arise in many different areas of the physical and biological sciences, and traveling wave solutions play special roles in some of these applications. In this paper, we develop a variational formulation of the existence problem for the traveling wave solution. Our main objective is to use this variational formulation to obtain exact and approximate traveling wave solutions with error estimates. As examples, we look at the Fisher equation, the Nagumo equation, and an equation with a fourth-degree nonlinearity. Also, we apply the method to the multi-component Lotka–Volterra competition-diffusion system.
13 citations
TL;DR: In this paper, the logarithmic derivative l(x) of an entire function of genus p and having only non-positive zeros is represented in terms of a Stieltjes function.
Abstract: The logarithmic derivative l(x) of an entire function of genus p and having only non-positive zeros is represented in terms of a Stieltjes function. As a consequence, (-1)p(xml(x))(m+p) is a completely monotonic function for all m ≥ 0. This generalizes earlier results on complete monotonicity of functions related to Euler's psi-function. Applications to Barnes' multiple gamma functions are given.
12 citations
TL;DR: In this paper, the (n, p)-type boundary value problem of nonlinear fractional differential equation was considered and the authors derived an interval of λ such that any λ lying in this interval has multiple positive solutions.
Abstract: In this paper, we consider the (n, p)-type boundary value problem of nonlinear fractional differential equation \begin{array}{rcl} \mathbf{D}_{0+}^\alpha u(t) + \lambda f(t,u(t)) & = & 0,\quad 0 < t < 1 \\[3pt] u^{(j)}(0) & = & 0,\quad 0\le j\le n-2, \\[3pt] u^{(p)}(1) & = & 0, \end{array} where λ is a parameter, α ∈ (n - 1, n] is a real number and n ≥ 3, 1 ≤ p ≤ α - 1 is fixed and integer, $\mathbf{D}_{0+}^\alpha$ is the Riemann–Liouville's fractional derivative, and f is continuous and semipositone. We derive an interval of λ such that any λ lying in this interval, the semipositone boundary value problem has multiple positive solutions.
TL;DR: In this paper, existence results for periodic solutions of second-order differential inclusions systems with (q, p)-Laplacian Laplacians were obtained for a class of nonautonomous systems.
Abstract: Some existence results are obtained for periodic solutions of nonautonomous second-order differential inclusions systems with (q, p)-Laplacian.
TL;DR: In this paper, an m-layer peiodic Lotka-Volterra network-like multidirectional food chain with harvesting terms is proposed by applying Mawhin's continuation theorem of coincidence degree theory and some skills of the inequalities.
Abstract: An m-layer peiodic Lotka–Volterra network-like multidirectional food-chain with harvesting terms is proposed in this paper By applying Mawhin's continuation theorem of coincidence degree theory and some skills of the inequalities, sufficient conditions which guarantee the existence of positive periodic solutions of the system are obtained An example is given to illustrate the effectiveness of our results
TL;DR: In this article, a generalized Journe wavelet set was proposed and a class of non-MSF wavelet sets were constructed using these wavelets, including the one constructed by Vyas [Bull. Polish Acad. Sci. Math.
Abstract: Dai and Larson [Mem. Amer. Math. Soc.134 (1998), no. 640] obtained a family of wavelet sets using the Journe wavelet set. In this paper, we expand this family and call its members to be generalized Journe wavelet sets. Furthermore, with the help of these wavelet sets, we provide a class of non-MSF wavelets which includes the one constructed by Vyas [Bull. Polish Acad. Sci. Math.57 (2009) 33–40]. Most of these non-MSF wavelets are found to be non-MRA.
TL;DR: It is shown how stability can be recovered in a wide class of error bounds developed in the context of statistical learning theory expressed in terms of functionals of the regression function, for instance, its norm in a reproducing kernel Hilbert space or other functional space.
Abstract: We consider a wide class of error bounds developed in the context of statistical learning theory which are expressed in terms of functionals of the regression function, for instance, its norm in a reproducing kernel Hilbert space or other functional space. These bounds are unstable in the sense that a small perturbation of the regression function can induce an arbitrary large increase of the relevant functional and make the error bound useless. Using a known result involving Fano inequality, we show how stability can be recovered.
TL;DR: In this article, the stability of impulsive functional differential equations with infinite delays is investigated using Lyapunov functions and the Razumikhin technique, and a new theorem on the uniform asymptotic stability and global stability for such differential equations is obtained.
Abstract: In this paper, the stability of impulsive functional differential equations with infinite delays are investigated. By using Lyapunov functions and the Razumikhin technique, a new theorem on the uniform asymptotic stability and global asymptotic stability for such differential equations is obtained. An example is given to illustrate the feasibility of the result.
TL;DR: In this article, the authors investigated the magnetic Rayleigh problem, where a semi-infinite flat plate is moving with a power-law velocity in a non-Newtonian power law fluid (Ostwald-de Wael model).
Abstract: We investigate in the present paper the magnetic Rayleigh problem, where a semi-infinite flat plate is moving with a power-law velocity, in a non-Newtonian power-law fluid (Ostwald–de Wael model). The non-stationary flow of this electrically conducting fluid in a transverse magnetic field is then analyzed. The solutions of this problem are obtained by means of similarity techniques. The main goal, is to investigate existence, uniqueness and behavior of such solutions, according to the values of the physical parameters.
TL;DR: In this paper, a temporal linear semi-implicit approximation of the two-dimensional Rayleigh-Benard convection problem is considered. And the authors prove that the stationary statistical properties as well as the global attractors of this linear semiimplicit scheme converge to those of the 2D Rayleigh -Benard problem as the time step approaches zero.
Abstract: In this article, we consider a temporal linear semi-implicit approximation of the two-dimensional Rayleigh–Benard convection problem. We prove that the stationary statistical properties as well as the global attractors of this linear semi-implicit scheme converge to those of the 2D Rayleigh–Benard problem as the time step approaches zero.
TL;DR: In this article, a new two-dimensional nonlinear membrane plate theory is derived via a formal asymptotic procedure for a family of hyperelastic nonlinear materials proposed by Ciarlet and Geymonat, whose stored energy function is polyconvex and becomes infinite when the determinant of the deformation gradient tends to zero, and can be adjusted to arbitrary Lame constants.
Abstract: A new two-dimensional nonlinear membrane plate theory is derived via a formal asymptotic procedure for a family of hyperelastic nonlinear materials proposed by Ciarlet and Geymonat [11], whose stored energy function is polyconvex and becomes infinite, when the determinant of the deformation gradient tends to zero, and can be adjusted to arbitrary Lame constants.
TL;DR: In this paper, the uniform asymptotics of a system of polynomials orthogonal on [-1, 1] with weight function w(x) = exp{-1/(1 - x2)μ}, 0 < μ < 1/2, via the Riemann-Hilbert approach are studied.
Abstract: We study the uniform asymptotics of a system of polynomials orthogonal on [-1, 1] with weight function w(x) = exp{-1/(1 - x2)μ}, 0 < μ < 1/2, via the Riemann–Hilbert approach. These polynomials belong to the Szego class. In some earlier literature involving Szego class weights, Bessel-type parametrices at the endpoints ±1 are used to study the uniform large degree asymptotics. Yet in the present investigation, we show that the original endpoints ±1 of the orthogonal interval are to be shifted to the MRS numbers ±βn, depending on the polynomial degree n and serving as turning points. The parametrices at ±βn are constructed in shrinking neighborhoods of size 1 - βn, in terms of the Airy function. The polynomials exhibit a singular behavior as compared with the classical orthogonal polynomials, in aspects such as the location of the extreme zeros, and the approximation away from the orthogonal interval. The singular behavior resembles that of the typical non-Szego class polynomials, cf. the Pollaczek polynomials. Asymptotic approximations are obtained in overlapping regions which cover the whole complex plane. Several large-n asymptotic formulas for πn(1), i.e. the value of the nth monic polynomial at 1, and for the leading and recurrence coefficients, are also derived.
TL;DR: In this article, the authors proposed a method to use the National Natural Science Foundation of China [10871025] and the Fundamental Research Funds for the Certral Universities of China (FRLU) of China.
Abstract: National Natural Science Foundation of China [10871025]; University of China; Fundamental Research Funds for the Certral Universities of China [2011121040]
TL;DR: The nonlinear max-product Bernstein operator has been shown to preserve the quasi-concavity of positive functions in this article, and the uniform estimate of the order O[nω1(f;1/n)2 + ω 1(f, 1/n), while near to the endpoints 0 and 1, the better pointwise estimate is obtained.
Abstract: In this paper, we find large classes of positive functions, others than those in [1], having even a Jackson-type estimate, ω1(f;1/n), in approximation by the nonlinear max-product Bernstein operator. The uniform estimate of the order O[nω1(f;1/n)2 + ω1(f;1/n)] is achieved, while near to the endpoints 0 and 1, the better pointwise estimate of the order is obtained. Finally, we prove that besides the preservation of quasi-convexity found in [1], the nonlinear max-product Bernstein operator preserves the quasi-concavity too.
TL;DR: In this paper, the existence of a determined open interval of positive eigenvalues for which the problem admits at least three non-zero weak solutions was proved for an eigenvalue Neumann problem.
Abstract: In this note we obtain a multiplicity result for an eigenvalue Neumann problem. Precisely, a recent critical point result for differentiable functionals is exploited, in order to prove the existence of a determined open interval of positive eigenvalues for which the problem admits at least three non-zero weak solutions.
TL;DR: In this article, the Sierpinski gasket was used to study the nonlinear elliptic equation Δu(x) + a(x,u,x) = g(x),f(u,u) with zero Dirichlet boundary condition and established the existence of infinitely many weak solutions.
Abstract: We study the nonlinear elliptic equation Δu(x) + a(x)u(x) = g(x)f(u(x)) on the Sierpinski gasket and with zero Dirichlet boundary condition. By extending a method introduced by Faraci and Kristaly in the framework of Sobolev spaces to the case of function spaces on fractal domains, we establish the existence of infinitely many weak solutions.