Modified tangential frequency filtering decomposition and its fourier analysis

In this paper, a modified tangential frequency filtering decomposition (MTFFD) preconditioner is proposed. The optimal order of the modification and the optimal relaxation parameter is determined by Fourier analysis. With the choice of optimal order of modification, the Fourier results show that the condition number of the preconditioned matrix is O(h^-\frac23){{\mathcal O}(h^{-\frac{2}{3}})}, and the spectrum distribution of the preconditioned matrix can be predicted by the Fourier results. The performance of MTFFD preconditioner is compared with tangential frequency filtering (TFFD) preconditioner on a variety of large sparse matrices arising from the discretization of PDEs with discontinuous coefficients. The numerical results show that the MTFFD preconditioner is much more efficient than the TFFD preconditioner.     

Large time asymptotics of the doubly nonlinear equation in the non-displacement convexity regime

We study the long-time asymptotics of the doubly nonlinear diffusion equation ${\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))}We study the long-time asymptotics of the doubly nonlinear diffusion equation r_t=div(|?r^m |^p-2 ?(r^m)){\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))} in \mathbbRⁿ{\mathbb{R}^n}, in the range \fracn-pn(p-1) < m < \fracn-p+1n(p-1){\frac{n-p}{n(p-1)} < m < \frac{n-p+1}{n(p-1)}} and 1 < p < ∞ where the mass of the solution is conserved, but the associated energy functional is not displacement convex. Using a linearisation of the equation, we prove an L ¹-algebraic decay of the non-negative solution to a Barenblatt-type solution, and we estimate its rate of convergence. We then derive the nonlinear stability of the solution by means of some comparison method between the nonlinear equation and its linearisation. Our results cover the exponent interval \frac2nn+1 < p < \frac2n+1n+1{\frac{2n}{n+1} < p < \frac{2n+1}{n+1}} where a rate of convergence towards self-similarity was still unknown for the p-Laplacian equation.     

Analysis of a new class of forward semi-Lagrangian schemes for the 1D Vlasov Poisson equations

The Vlasov equation is a kinetic model describing the evolution of a plasma which is a globally neutral gas of charged particles. It is self-consistently coupled with Poisson’s equation, which rules the evolution of the electric field. In this paper, we introduce a new class of forward semi-Lagrangian schemes for the Vlasov–Poisson system based on a Cauchy Kovalevsky (CK) procedure for the numerical solution of the characteristic curves. Exact conservation properties of the first moments of the distribution function for the schemes are derived and a convergence study is performed that applies as well for the CK scheme as for a more classical Verlet scheme. A L ¹ convergence of the schemes will be proved. Error estimates [in O(Dt²+h² + \frach²Dt){O\left(\Delta{t}^2+h^2 + \frac{h^2}{\Delta{t}}\right)} for Verlet] are obtained, where Δt and h = max(Δx, Δv) are the discretization parameters.     

Regularizing and decay rate estimates for solutions to the Cauchy problem of the Debye–Hückel system

In this paper we establish some regularizing and decay rate estimates for mild solutions of the Debye–Hückel system. We prove that if the initial data belong to the critical Lebesgue space L^\fracn2(\mathbbRⁿ){L^{\frac{n}{2}}(\mathbb{R}^{n})} , then the L ^q -norm ( \fracn2 ￡ q ￡ ￥{\frac{n}{2} \leq q \leq \infty}) of the βth order spatial derivative of mild solutions are majorized by K₁(K₂|b|)^|b|t^{-\frac|b|2-1+\fracn2q}{K_{1}(K_{2}|\beta|)^{|\beta|}t^{-\frac{|\beta|}{2}-1+\frac{n}{2q}}} for some constants K ₁ and K ₂. These estimates particularly imply that mild solutions are analytic in the space variable, and provide decay estimates in the time variable for higher-order derivatives of mild solutions. We also prove that similar estimates also hold for mild solutions whose initial data belong to the critical homogeneous Besov space [(B)\dot]^-2+\fracnp_p,￥(\mathbbRⁿ){\dot{B}^{-2+\frac{n}{p}}_{p,\infty}(\mathbb{R}^n)} ( \fracn2 < p < n{\frac{n}{2} < p < n}).     

Simultaneous and non-simultaneous blow-up for heat equations with coupled nonlinear boundary fluxes

This paper deals with simultaneous and non-simultaneous blow-up for heat equations coupled via nonlinear boundary fluxes

Homogenization for strongly anisotropic nonlinear elliptic equations

In this paper we present homogenization results for elliptic degenerate differential equations describing strongly anisotropic media. More precisely, we study the limit as e? 0 \epsilon \to 0 of the following Dirichlet problems with rapidly oscillating periodic coefficients:¶¶ . \cases {{ -div(\alpha(\frac{x}{\epsilon}}, \nabla u) A(\frac{x}{\epsilon}) \nabla u) = f(x) \in L^{\infty}(\Omega) \atop u = 0 su \eth\Omega\ } ¶¶where, p > 1,     a: \Bbb Rⁿ ×\Bbb Rⁿ ? \Bbb R,     a(y,x) ? áA(y)x,x?^p/2-1, A ? M^{n ×n}(\Bbb R) p>1, \quad \alpha : \Bbb R^n \times \Bbb R^n \to \Bbb R, \quad \alpha(y,\xi) \approx \langle A(y)\xi,\xi \rangle ^{p/2-1}, A \in M^{n \times n}(\Bbb R) , A being a measurable periodic matrix such that A^t(x) = A(x) 3 0A^t(x) = A(x) \ge 0 almost everywhere.¶¶The anisotropy of the medium is described by the following structure hypothesis on the matrix A:¶¶l^2/p(x) |x|² ￡ áA(x)x,x? ￡ L ^2/p(x) |x|², \lambda^{2/p}(x) |\xi|^2 \leq \langle A(x)\xi,\xi \rangle \leq \Lambda ^{2/p}(x) |\xi|^2, ¶¶where the weight functions l \lambda and L \Lambda (satisfying suitable summability assumptions) can vanish or blow up, and can also be "moderately" different. The convergence to the homogenized problem is obtained by a classical compensated compactness argument, that had to be extended to two-weight Sobolev spaces.     

On the Crank-Nicolson scheme once again

Let (t_j)_{j ? \mathbbN}{\left(\tau_j\right)_{j\in\mathbb{N}}} be a sequence of strictly positive real numbers, and let A be the generator of a bounded analytic semigroup in a Banach space X. Put A_n=?_j=1ⁿ(I+\frac12 t_jA) (I-\frac12 t_jA)^-1{A_n=\prod_{j=1}^n\left(I+\frac{1}{2} \tau_jA\right) \left(I-\frac{1}{2} \tau_jA\right)^{-1}}, and let x ? X{x\in X}. Define the sequence (x_n)_{n ? \mathbbN} ì X{\left(x_n\right)_{n\in\mathbb{N}}\subset X} by the Crank–Nicolson scheme: x _n  = A _n x. In this paper, it is proved that the Crank–Nicolson scheme is stable in the sense that sup_{n ? \mathbbN}||A_nx|| < ￥{\sup_{n\in\mathbb{N}}\left\Vert A_nx\right\Vert<\infty}. Some convergence results are also given.     

Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials

We prove that max |p′(x)|, where p runs over the set of all algebraic polynomials of degree not higher than n ≥ 3 bounded in modulus by 1 on [−1, 1], is not lower than ( n - 1 ) \mathord

Asymptotic Behavior of Variable-Coefficient Toeplitz Determinants

For a continuous function s\sigma defined on [0,1]×\mathbbT[0,1]\times\mathbb{T}, let \ops\op\sigma stand for the (n+1)×(n+1)(n+1)\times(n+1) matrix whose (j,k)(j,k)-entries are equal to \frac1 2pò₀^2p s( \fracjn,e^iq) e^-i(j-k)q  dq,        j,k = 0,1,...,n . \displaystyle \frac{1} {2\pi}\int_0^{2\pi} \sigma \left( \frac{j}{n},e^{i\theta}\right) e^{-i(j-k)\theta} \,d\theta, \qquad j,k =0,1,\dots,n~. These matrices can be thought of as variable-coefficient Toeplitz matrices or as the discrete analogue of pseudodifferential operators. Under the assumption that the function s\sigma possesses a logarithm which is sufficiently smooth on [0,1]×\mathbbT[0,1]\times\mathbb{T}, we prove that the asymptotics of the determinants of \ops\op\sigma are given by det[\ops] ~ G[s]⁽ⁿ⁺¹⁾E[s]     \text as   n?￥ , \det \left[\op\sigma\right] \sim G[\sigma]^{(n+1)}E[\sigma] \quad \text{ as \ } n\to\infty~, where G[s]G[\sigma] and E[s]E[\sigma] are explicitly determined constants. This formula is a generalization of the Szegö Limit Theorem. In comparison with the classical theory of Toeplitz determinants some new features appear.     

A Remark on the Growth of the Denominators of Convergents

For log\frac1+?52 ￡ l_* ￡ l^* < ￥{\rm log}\frac{1+\sqrt{5}}{2}\leq \lambda_\ast \leq \lambda^\ast < \infty , let E(λ_*, λ^*) be the set {x ? [0,1): liminf_{n ? ￥}\fraclogq_n(x)n=l_*, limsup_{n ? ￥}\fraclogq_n(x)n=l^*}. \left\{x\in [0,1):\ \mathop{\lim\inf}_{n \rightarrow \infty}\frac{\log q_n(x)}{n}=\lambda_{\ast}, \mathop{\lim\sup}_{n \rightarrow \infty}\frac{\log q_n(x)}{n}=\lambda^{\ast}\right\}. It has been proved in [1] and [3] that E(λ_*, λ^*) is an uncountable set. In the present paper, we strengthen this result by showing that dimE(l_*, l^*) 3 \fracl_* -log\frac1+?522l^*\dim E(\lambda_{\ast}, \lambda^{\ast}) \ge \frac{\lambda_{\ast} -\log \frac{1+\sqrt{5}}{2}}{2\lambda^{\ast}}     

Blow up of subcritical quantities at the first singular time of the mean curvature flow

Consider a family of smooth immersions F(·,t) : Mⁿ? \mathbbRⁿ⁺¹{F(\cdot,t)\,:\,{M^n\to \mathbb{R}^{n+1}}} of closed hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} moving by the mean curvature flow \frac?F(p,t)?t = -H(p,t)·n(p,t){\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)}, for t ? [0,T){t\in [0,T)}. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example ò₀^tò_Ms\frac|A|^{n + 2} log (2 + |A|) dmds,{\int_{0}^{t}\int_{M_{s}}\frac{{\vert{\it A}\vert}^{n + 2}}{ log (2 + {\vert{\it A}\vert})}} d\mu ds, blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.     

On the sum of a prime and a k-th power of prime in short intervals

Let H_k\mathcal{H}_{k} denote the set {n∣2|n, n\not o 1 (mod p)n\not\equiv 1\ (\mathrm{mod}\ p) ∀ p>2 with p−1|k}. We prove that when X^{\frac1120(1-\frac12k) +e}\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{2k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n ? \allowbreak H_k ?(X, X+H]n\in\allowbreak {\mathcal{H}_{k} \cap (X, X+H]} can be represented as the sum of a prime and a k-th power of prime for k≧3. Moreover, when X^{\frac1120(1-\frac1k) +e}\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n∈(X,X+H] can be represented as the sum of a prime and a k-th power of integer for k≧3.     

Optimization of the Hermitian and Skew-Hermitian Splitting Iteration for Saddle-Point Problems

We study the asymptotic rate of convergence of the alternating Hermitian/skew-Hermitian iteration for solving saddle-point problems arising in the discretization of elliptic partial differential equations. By a careful analysis of the iterative scheme at the continuous level we determine optimal convergence parameters for the model problem of the Poisson equation written in div-grad form. We show that the optimized convergence rate for small mesh parameter h is asymptotically 1–O(h ^1/2). Furthermore we show that when the splitting is used as a preconditioner for a Krylov method, a different optimization leading to two clusters in the spectrum gives an optimal, h-independent, convergence rate. The theoretical analysis is supported by numerical experiments.This revised version was published online in October 2005 with corrections to the Cover Date.     

Remarks on a sine polynomial

Let n ≥ 0 be an integer. Then we have for ${x\in(0,\pi)}Let n ≥ 0 be an integer. Then we have for x ? (0,p){x\in(0,\pi)} :

?k=0n (( 2n+1) || (n-k ))\fracsin((2k+1)x)2k+1 ￡ \frac8n  n!(2n+1)!!.\sum_{k=0}^n { 2n+1 \choose n-k }\frac{\sin((2k+1)x)}{2k+1}\leq\frac{8^n \, n!}{(2n+1)!!}.      15. On the Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with K-Quasiconformal Boundary      Abdullah Cavus  Fahreddin G. Abdullayev 《逼近论及其应用》2001,17(1):97-105 Let G be a finite domain in the complex plane with K-quasicon formal boundary, z 0 be an arbitrary fixed point in G and p>0. Let jp ( z ): = òx0 x [ f( z) ]2/8 dz\varphi _p \left( z \right): = \int_{x_0 }^x {\left[ {\phi \left( \zeta \right)} \right]^{2/8} } d\zeta , and let \iintc | jp ( z ) - Px1 (z) |p d0x \iint\limits_c {\left| {\varphi _p \left( z \right) - P_x^1 (z)} \right|^p d0_x } in the class \mathop ?n \mathop \prod \limits_n of all polynomials of degree [`(G)]\bar G in case of $p > 2 - \frac{{K^2 + 1}}{{2K^4 }}$p > 2 - \frac{{K^2 + 1}}{{2K^4 }} .      16. Best constant in a three-parameter Poincaré inequality      I. V. Gerasimov  A. I. Nazarov 《Journal of Mathematical Sciences》2011,179(1):80-99 We study the problem of finding the best constant in the generalized Poincaré inequality lpqr = min\frac|| y￠ ||Lp[0,1]|| y ||Lp[0,1],        ò01 | y(t) |r - 2y(t)dt = 0, {{\rm{\lambda }}_{pqr}} = \min \frac{{\left\| {y'} \right\|{L_p}[0,1]}}{{\left\| y \right\|{L_p}[0,1]}},\quad \quad \mathop {\int }\limits_0^1 {\left| {y(t)} \right|^{r - 2}}y(t)dt = 0,      17. The Jacobi-Dunkl transform on ℝ and the convolution product on new spaces of distributions      Hassen Ben Mohamed 《The Ramanujan Journal》2010,21(2):145-171 In this work, we consider the Jacobi-Dunkl operator Λ α,β , a 3 b 3 \frac-12\alpha\geq\beta\geq\frac{-1}{2} , a 1 \frac-12\alpha\neq\frac{-1}{2} , on ℝ. The eigenfunction Yla,b\Psi_{\lambda}^{\alpha,\beta} of this operator permits to define the Jacobi-Dunkl transform. The main idea in this paper is to introduce and study the Jacobi-Dunkl transform and the Jacobi-Dunkl convolution product on new spaces of distributions      18. Explicit relation between the solutions of the heat and the Hermite heat equation      Bang-He Li 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,72(3):959-968 There are lots of results on the solutions of the heat equation \frac?u?t = \mathop?ni=1\frac?2?x2iu,\frac{\partial u}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\frac{\partial^2}{\partial x^{2}_{i}}u, but much less on those of the Hermite heat equation \frac?U?t = \mathop?ni=1(\frac?2?x2i - x2i) U\frac{\partial U}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\left(\frac{\partial^2}{\partial x^{2}_{i}} - x^{2}_{i}\right) U due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005).      19. Basics of a generalized Wiman–Valiron theory for monogenic Taylor series of finite convergence radius      D. Constales  R. De Almeida  R. S. Kraußhar 《Mathematische Zeitschrift》2010,266(3):665-681 In this paper, we develop the basic concepts for a generalized Wiman–Valiron theory for Clifford algebra valued functions that satisfy inside an n + 1-dimensional ball the higher dimensional Cauchy-Riemann system ${\frac{\partial f}{\partial x_0} + \sum_{i=1}^n e_i\frac{\partial f}{\partial x_i}=0}In this paper, we develop the basic concepts for a generalized Wiman–Valiron theory for Clifford algebra valued functions that satisfy inside an n + 1-dimensional ball the higher dimensional Cauchy-Riemann system \frac?f?x0 + ?i=1n ei\frac?f?xi=0{\frac{\partial f}{\partial x_0} + \sum_{i=1}^n e_i\frac{\partial f}{\partial x_i}=0} . These functions are called monogenic or Clifford holomorphic inside the ball. We introduce growth orders, the maximum term and a generalization of the central index for monogenic Taylor series of finite convergence radius. Our goal is to establish explicit relations between these entities in order to estimate the asymptotic growth behavior of a monogenic function in a ball in terms of its Taylor coefficients. Furthermore, we exhibit a relation between the growth order of such a function f and the growth order of its partial derivatives.      20. Estimate for the remainder in the Weyl asymptotics of the spectrum of the Maxwell operator in Lipschitz domains      N. A. Veniaminov 《Journal of Mathematical Sciences》2010,169(1):46-63 We study the asymptotics of the spectrum of the Maxwell operator M in a bounded Lipschitz domain W ì \mathbbR3 \Omega \subset {\mathbb{R}^3} under the condition of the perfect conductivity of the boundary ∂Ω. We obtain the following estimate for the remainder in the Weyl asymptotic expansion of the counting function N(λ,M) of positive eigenvalues of the Maxwell operator M: N( l, M ) = \frac\textmeas W3p2l3( 1 + O( l - 2 / 5 ) ), N\left( {\lambda, M} \right) = \frac{{{\text{meas }}\Omega }}{{3{\pi^2}}}{\lambda^3}\left( {1 + O\left( {{\lambda^{{{{ - 2}} \left/ {5} \right.}}}} \right)} \right),      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

On the Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with K-Quasiconformal Boundary

Let G be a finite domain in the complex plane with K-quasicon formal boundary, z ₀ be an arbitrary fixed point in G and p>0. Let j_p ( z ): = ò_x0 ^x [ f( z) ]^2/8 dz\varphi _p \left( z \right): = \int_{x_0 }^x {\left[ {\phi \left( \zeta \right)} \right]^{2/8} } d\zeta , and let \iint_c | j_p ( z ) - P_x¹ (z) |^p d0_x \iint\limits_c {\left| {\varphi _p \left( z \right) - P_x^1 (z)} \right|^p d0_x } in the class \mathop ?_n \mathop \prod \limits_n of all polynomials of degree [`(G)]\bar G in case of $p > 2 - \frac{{K^2 + 1}}{{2K^4 }}$p > 2 - \frac{{K^2 + 1}}{{2K^4 }} .     

Best constant in a three-parameter Poincaré inequality

We study the problem of finding the best constant in the generalized Poincaré inequality

The Jacobi-Dunkl transform on ℝ and the convolution product on new spaces of distributions

In this work, we consider the Jacobi-Dunkl operator Λ _α,β , a 3 b 3 \frac-12\alpha\geq\beta\geq\frac{-1}{2} , a 1 \frac-12\alpha\neq\frac{-1}{2} , on ℝ. The eigenfunction Y_l^a,b\Psi_{\lambda}^{\alpha,\beta} of this operator permits to define the Jacobi-Dunkl transform. The main idea in this paper is to introduce and study the Jacobi-Dunkl transform and the Jacobi-Dunkl convolution product on new spaces of distributions     

Explicit relation between the solutions of the heat and the Hermite heat equation

There are lots of results on the solutions of the heat equation \frac?u?t = \mathop?ⁿ_i=1\frac?²?x²_iu,\frac{\partial u}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\frac{\partial^2}{\partial x^{2}_{i}}u, but much less on those of the Hermite heat equation \frac?U?t = \mathop?ⁿ_i=1(\frac?²?x²_i - x²_i) U\frac{\partial U}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\left(\frac{\partial^2}{\partial x^{2}_{i}} - x^{2}_{i}\right) U due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005).     

Basics of a generalized Wiman–Valiron theory for monogenic Taylor series of finite convergence radius

In this paper, we develop the basic concepts for a generalized Wiman–Valiron theory for Clifford algebra valued functions that satisfy inside an n + 1-dimensional ball the higher dimensional Cauchy-Riemann system ${\frac{\partial f}{\partial x_0} + \sum_{i=1}^n e_i\frac{\partial f}{\partial x_i}=0}In this paper, we develop the basic concepts for a generalized Wiman–Valiron theory for Clifford algebra valued functions that satisfy inside an n + 1-dimensional ball the higher dimensional Cauchy-Riemann system \frac?f?x₀ + ?_i=1ⁿ e_i\frac?f?x_i=0{\frac{\partial f}{\partial x_0} + \sum_{i=1}^n e_i\frac{\partial f}{\partial x_i}=0} . These functions are called monogenic or Clifford holomorphic inside the ball. We introduce growth orders, the maximum term and a generalization of the central index for monogenic Taylor series of finite convergence radius. Our goal is to establish explicit relations between these entities in order to estimate the asymptotic growth behavior of a monogenic function in a ball in terms of its Taylor coefficients. Furthermore, we exhibit a relation between the growth order of such a function f and the growth order of its partial derivatives.     

Estimate for the remainder in the Weyl asymptotics of the spectrum of the Maxwell operator in Lipschitz domains

We study the asymptotics of the spectrum of the Maxwell operator M in a bounded Lipschitz domain W ì \mathbbR³ \Omega \subset {\mathbb{R}^3} under the condition of the perfect conductivity of the boundary ∂Ω. We obtain the following estimate for the remainder in the Weyl asymptotic expansion of the counting function N(λ,M) of positive eigenvalues of the Maxwell operator M: