线性约束下Hermite-广义反Hamilton矩阵的最佳逼近问题

本文利用对称向量与反对称向量的特征性质,给出了约束矩阵集合非空的充分必要条件及矩阵的一般表达式.运用空间分解理论和闭凸集上的逼近理论,得到了任一n阶复矩阵在约束矩阵集合中的惟一最佳逼近解.     

(Bounded) continuous cohomology and Gromov��s proportionality principle

Let X be a topological space, and let C*(X) be the complex of singular cochains on X with coefficients in ${\mathbb{R}}$ . We denote by ${C^{\ast}_{c}(X) \subseteq C^{\ast}(X)}$ the subcomplex given by continuous cochains, i.e. by such cochains whose restriction to the space of simplices (endowed with the compact-open topology) defines a continuous real function. We prove that at least for ??reasonable?? spaces the inclusion ${C^{\ast}_{c}(X) \hookrightarrow C^{\ast}(X)}$ induces an isomorphism in cohomology, thus answering a question posed by Mostow. We also prove that this isomorphism is isometric with respect to the L ^??-norm on cochains defined by Gromov. As an application, we clarify some details of Gromov??s original proof of the proportionality principle for the simplicial volume of Riemannian manifolds, also providing a self-contained exposition of Gromov??s argument.     

The Existence and Uniqueness of the Weak Solution for the Evolutionary Electrochemical Machining Problem

A time dependent electrochemical machining problem, in which the cathode is fed towards the anode with a constant velocity, is studied. We prove the existence and uniqueness of the weak solution for the problem under the assumption that the cathode is C^{1+β} for some β ∈ (0,1).     

C1,α-partial Regularity of Nonlinear Parabolic Systems

We prove C^{1,α}-partial regularity of weak solution of nonlinear parabolic systems u^i_t - D_αA^α_i(x, t, u, Du) = B_i(x, t, u, Du), \quad i=1,…, N under the main assumption that A^α_i and B_i, satisfy the natural growth condition.     

C1,α Regularity of Viscosity Solutions of Fully Nonlinear Elliptic PDE Under Natural Structure Conditions

In this paper we are concemed with fully nonlinear elliptic equation F(x, u, Du, D²u) = 0. We establish the interior Lipschitz continuity and C^{1,α} regularity of viscosity solutions under natural structure conditions without differentiating the equation as usual, especially we give a new analytic Harnack inequality approach to C^{1,α} estimate for viscosity solutions instead of the geometric approach given by L. Caffarelli \& L. Wang and improve their results. Our structure conditions are rather mild.     

W2,ploc(\Omega)\cap C1,α(\bar Ω) Viscosity Solutions of Neumann Problems for Fully Nonlinear Elliptic Equations

In this paper we study fully nonlinear elliptic equations F(D²u, x) = 0 in Ω ⊂ R^n with Neumann boundary conditions \frac{∂u}{∂v} = a(x)u under the rather mild structure conditions and without the concavity condition. We establish the global C^{1,Ω} estimates and the interior W^{2,p} estimates for W^{2,q}(Ω) solutions (q > 2n) by introducing new independent variables, and moreover prove the existence of W^{2,p}_{loc}(Ω)∩ C^{1,α}(\bar \Omega} viscosity solutions by using the accretive operator methods, where p E (0, 2), α ∈ (0, 1}.     

ON THE APPROXIMAMATION OF DIFFERENTIABLE FUNCTIONS BY EULER MEANS

Let f∈C_(2π)~r.Denote by _n(f,x)the n-th Euler mean of f(x).This paper gives theasympto ic representations of the deviation _n(f,x)-f(x)and the quantity | _a(f,x)-f(x)|.Additionally,some applications of these asymptotic representations are obtained.     

$$L^{p}$$-Norms a nd Mahler’s Measure of Poly nomials o n the n-Dime nsio nal Torus

We prove Nikol’skii type inequalities that, for polynomials on the n-dimensional torus \(\mathbb {T}^n\), relate the \(L^p\)-norm with the \(L^q\)-norm (with respect to the normalized Lebesgue measure and \(0 <p <q < \infty \)). Among other things, we show that \(C=\sqrt{q/p}\) is the best constant such that \(\Vert P\Vert _{L^q}\le C^{\text {deg}(P)} \Vert P\Vert _{L^p}\) for all homogeneous polynomials P on \(\mathbb {T}^n\). We also prove an exact inequality between the \(L^p\)-norm of a polynomial P on \(\mathbb {T}^n\) and its Mahler measure M(P), which is the geometric mean of |P| with respect to the normalized Lebesgue measure on \(\mathbb {T}^n\). Using extrapolation, we transfer this estimate into a Khintchine–Kahane type inequality, which, for polynomials on \(\mathbb {T}^n\), relates a certain exponential Orlicz norm and Mahler’s measure. Applications are given, including some interpolation estimates.     

On a conjecture of Zaremba

LetN _C (x) be the number of integersm≤x such that there is an integera with 1≤a<m, (a, m)=1 and all partial quotients in the continued fraction expansion ofa/m are at mostC. We prove for allx≥1 that $$N_c (x) > {1 \mathord{\left/ {\vphantom {1 {\sqrt {2C} x^{{1 \mathord{\left/ {\vphantom {1 {2(1 - 1/C^2 )}}} \right. \kern-\nulldelimiterspace} {2(1 - 1/C^2 )}}} }}} \right. \kern-\nulldelimiterspace} {\sqrt {2C} x^{{1 \mathord{\left/ {\vphantom {1 {2(1 - 1/C^2 )}}} \right. \kern-\nulldelimiterspace} {2(1 - 1/C^2 )}}} }}$$ .     

Some existence and nonexistence theorems for compact graphs of constant mean curvature with boundary in parallel planes

Given H≥0 and bounded convex curves α₁, ...,⇌_n, α in the plane z=0 bounding domains D₁, …, D_n, D, respectively, with if i ∈ j and with D_i ⊂ D, we obtain several results proving the existence of a constanth depending only on H and on the geometry of the curves α_i, α such that the Dirichlet problem for the constant mean curvature H equation: where may accept or not a solution.     

The Bargmann Transform and Windowed Fourier Localization

We consider the relationship between Gabor-Daubechies windowed Fourier localization operators and Berezin-Toeplitz operators T _φ, using the Bargmann isometry β. For “window” w a finite linear combination of Hermite functions, and a very general class of functions φ, we prove an equivalence of the form by obtaining the exact formulas for the operator C and the linear differential operator D.     

Entropy solutions of anisotropic elliptic nonlinear obstacle problem with measure data

We prove the existence of an entropy solution for a class of nonlinear anisotropic elliptic unilateral problem associated to the following equation $$\begin{aligned} -\sum _{i=1}^{N} \partial _i a_i(x,u, \nabla u) -\sum _{i=1}^{N}\partial _{i}\phi _{i}( u)=\mu , \end{aligned}$$where the right hand side $$\mu $$ belongs to $$L^{1}(\Omega )+ W^{-1, \vec {p'}}(\Omega )$$. The operator $$-\sum _{i=1}^{N} \partial _i a_i(x,u, \nabla u) $$ is a Leray–Lions anisotropic operator and $$\phi _{i} \in C^{0}({\mathbb {R}}, {\mathbb {R}})$$.     

Sign-changing solutions and multiplicity results for elliptic problems via lower and upper solutions

In the first part of this work, we recall variational methods related to invariant sets in ${C^1_0}$ . In the second part of the work, we consider an elliptic Dirichlet problem in a situation where the origin is a solution around which the nonlinearity has a slope between two consecutive eigenvalues of order larger than 2 and near + infinity the slope of the nonlinearity is smaller than the first eigenvalue. Then we discuss the conditions needed near - infinity in order to ensure the existence of a positive solution and two sign-changing solutions.     

The Whitney extension problem and Lipschitz selections of set-valued mappings in jet-spaces

We study a variant of the Whitney extension problem (1934) for the space . We identify with a space of Lipschitz mappings from into the space of polynomial fields on equipped with a certain metric. This identification allows us to reformulate the Whitney problem for as a Lipschitz selection problem for set-valued mappings into a certain family of subsets of . We prove a Helly-type criterion for the existence of Lipschitz selections for such set-valued mappings defined on finite sets. With the help of this criterion, we improve estimates for finiteness numbers in finiteness theorems for due to C. Fefferman.

     

具有凹凸非线性项和变号位势函数拟线性椭圆系统解的多重结果

研究拟线性椭圆系统(?)的非平凡非负解或正解的多重性,这里Ω(?)R~N是具有光滑边界(?)Ω的有界域,1≤qp~*/p~*-q,其中当N≤p时,p~*=+∞,而当1相似文献   

The Core-EP,Weighted Core-EP Inverse of Matrices and Constrained Systems of Linear Equations

We study the constrained systemof linear equations Ax=b,x∈R(A^k)for A∈C^n×nand b∈Cn,k=Ind(A).When the system is consistent,it is well known that it has a unique A^Db.If the system is inconsistent,then we seek for the least squares solution of the problem and consider min_x∈R(A^k)||b?Ax||2,where||·||2 is the 2-norm.For the inconsistent system with a matrix A of index one,it was proved recently that the solution is A^■b using the core inverse A^■of A.For matrices of an arbitrary index and an arbitrary b,we show that the solution of the constrained system can be expressed as A^■b where A^■is the core-EP inverse of A.We establish two Cramer’s rules for the inconsistent constrained least squares solution and develop several explicit expressions for the core-EP inverse of matrices of an arbitrary index.Using these expressions,two Cramer’s rules and one Gaussian elimination method for computing the core-EP inverse of matrices of an arbitrary index are proposed in this paper.We also consider the W-weighted core-EP inverse of a rectangular matrix and apply the weighted core-EP inverse to a more general constrained system of linear equations.     

COMMUTATORS OF MULTIPLIER OPERATORS

Denote M~l={ω∈C~∞(R~K\{0}:|ω~((β))(ξ)|≤C_β|ξ|~(l-|β|)},l is an integer.R_((-α))~((m))is the n-foldcomposition of Taylor series remainder operator,m=(m_1,…,m_n)∈Z~n.Z is the set ofnon-negative integers,α∈(R~K)n.DenoteThe main results are as follows:(i) If γ_1,γ_2∈Z~K and l is an integer such that |γ_1|+|γ_2|+l=|m|=m_1+…+m_n,0≤|γ_1|≤{m_4},and ω∈M~l,then we havewhereis a conseant.(ii)In the same sense of notation as in (i),but now|m|=1,we havewhereThese results extend the corresponding ones given by coifman-Meyer in [4] andCohen,J.in [2],and,in a sense,extend those given by Calderón,A.P.in [1].     

Invertible Composition Operators: The Product of a Composition Operator with the Adjoint of a Composition Operator

In this paper, we study the product of a composition operator \(C_{\varphi }\) with the adjoint of a composition operator \(C^{*}_{\psi }\) on the Hardy space \(H^2(\mathbb {D})\) . The order of the product gives rise to two different cases. We completely characterize when the operator \(C_{\varphi }C^{*}_{\psi }\) is invertible, isometric, and unitary and when the operator \(C^{*}_{\psi }C_{\varphi }\) is isometric and unitary. If one of the inducing maps \(\varphi \) or \(\psi \) is univalent, we completely characterize when \(C^{*}_{\psi }C_{\varphi }\) is invertible.     

Global Boundedness in a Two-Competing-Species Chemotaxis System with Two Chemicals

This paper deals with a two-competing-species chemotaxis system with two different chemicals

$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad}l} \displaystyle u_{t}=\Delta u-\chi_{1}\nabla \cdot (u\nabla v)+\mu_{1} u(1-u-a _{1}w), & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle \tau v_{t}=\Delta v-v+w, & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle w_{t}=\Delta w-\chi_{2}\nabla \cdot (w\nabla z)+\mu_{2}w(1-a_{2}u-w), & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle \tau z_{t}=\Delta z-z+u, & (x,t)\in \varOmega \times (0,\infty ), \end{array}\displaystyle \right . \end{aligned}$$

under homogeneous Neumann boundary conditions in a smooth bounded domain \(\varOmega \subset \mathbb{R}^{n}\) \((n\geq 1)\) with the nonnegative initial data \((u_{0},\tau v_{0},w_{0},\tau z_{0})\in C^{0}(\overline{\varOmega }) \times W^{1,\infty }(\varOmega )\times C^{0}(\overline{\varOmega })\times W ^{1,\infty }(\varOmega )\), where \(\tau \in \{0,1\}\) and the parameters \(\chi_{i},\mu_{i},a_{i}\) (\(i=1,2\)) are positive. When \(\tau =0\), based on some a priori estimates and Moser-Alikakos iteration, it is shown that regardless of the size of initial data, the system possesses a unique globally bounded classical solution for any positive parameters if \(n=2\). On the other hand, when \(\tau =1\), relying on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a unique globally bounded classical solution provided that \(n\geq 1\) and there exists \(\theta_{0}>0\) such that \(\frac{\chi_{2}}{ \mu_{1}}<\theta_{0}\) and \(\frac{\chi_{1}}{\mu_{2}}<\theta_{0}\).

     

On Weyl group equivariant maps

We prove an equivariant analogue of Chevalley's isomorphism theorem for polynomial, or maps.