A Geometrically Aberrant Banach Space with Uniformly Normal Structure

A.L. Browu [2] devised an equivalent norm ||·||_B on (l_2, ||·||) such that itsrestriction to M = { (α_1,α_2,…) ∈l_2, α_1 = 0} remains the original l_2-norm ||·||_2, and itsrestriction to M_k = span(e_l ,e_k), k≥3 is an l_p (k) -norm, where (e_k) is a naturalbasis on l_2 and p(k)→∞ as k-∞, p(3) = 16. J.R. Giles, B .Sims & S. Swamina-     

SOME PROPERTIES OF THE l_2-VALUED LONG JAMES BANACH SPACE J(η, l_2)

The main result of this paper is to show that the bidual J(η, l_2)~(**) of the long James type l_2-valued Banach space J(η, l_2) can be identified with transfinite matrices of scalars [(b_(α, i)i∈[0,ω)]_(α∈[0,n)) with some conditions and the norm of the element x~(**) in J(η, l_2)~(**) equals ■‖■b_(α,i)φ_(α,i)‖_(J(η_2)~(**)).     

ON INTERPOLATION BY S_3~1(△_mn~(1))(I)

§0 Introduction Let △_(mn)~(1) be a subdivision of D: [a,b]x[c,d] (Fig.1). l_1,_2be lengths of [a,b] and [c,d] respectively, and h_1=l_1/m, h_2=l_2/n,l=max(l_1,l_2). Suppose that S∈S1/3(△_(mn)~(1)). In §1-§5 we consider thefollowing interpolation problem:     

Some Theorems of Analytic Functions

1 Introduction We denote that: σ—the class of functions ω(z)=A_1z+A_2z~2+…regular in the unit disk such that sum from n=1 to ∞ (n|A_n|~2<∞);K_c— the class of close-to-convex function f(z),that is, if f(z)=α_1z+α_2z~2+…there exists a starlike function g(z) =b_1z+b_2z~2+…such that     

STABLE RECOVERY OF SIGNALS WITH THE HIGH ORDER D-RIP CONDITION

This paper establishes a high order condition on the restricted isometry property adapted to a frame D(D-RIP) for the signal recovery.It is shown that if the measurement matrix A satisfies the D-RIP condition δ_(tk)(1/2)((t-1)/t) for t1,then all signals f which are sparse in terms of a tight frame D can be recovered stably or exactly via the l_1-analysis model based on y =Af+z in l_2 and Dantzig selector bounded noise setting.     

CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS

In this paper, we are concerned with the following Hardy-Sobolev type system{(-?)~(α/2) u(x) =v~q(x)/|y|~(t_2) (-?)α/2 v(x) =u~p(x)/|y|~(t_1),x =(y, z) ∈(R ~k\{0}) × R~(n-k),(0.1)where 0 α n, 0 t_1, t_2 min{α, k}, and 1 p ≤τ_1 :=(n+α-2t_1)/( n-α), 1 q ≤τ_2 :=(n+α-2 t_2)/( n-α).We first establish the equivalence of classical and weak solutions between PDE system(0.1)and the following integral equations(IE) system{u(x) =∫_( R~n) G_α(x, ξ)v~q(ξ)/|η|t~2 dξ v(x) =∫_(R~n) G_α(x, ξ)(u~p(ξ))/|η|~(t_1) dξ,(0.2)where Gα(x, ξ) =(c n,α)/(|x-ξ|~(n-α))is the Green's function of(-?)~(α/2) in R~n. Then, by the method of moving planes in the integral forms, in the critical case p = τ_1 and q = τ_2, we prove that each pair of nonnegative solutions(u, v) of(0.1) is radially symmetric and monotone decreasing about the origin in R~k and some point z0 in R~(n-k). In the subcritical case (n-t_1)/(p+1)+(n-t_2)/(q+1) n-α,1 p ≤τ_1 and 1 q ≤τ_2, we derive the nonexistence of nontrivial nonnegative solutions for(0.1).     

数学问题解答

因而ω≥-1(等号时δ_1=δ_2=0对应x=y=1)故ω最小值为-1。852 设D,E,F分别是△ABC的三边BC,CA,AB上的点,l_1,l_2,l_3分别是过D垂直于BC,过E垂直于CA,过F垂直于AB的直线。证明:l_1,l_2,l_3共点的充分必要条件是     

对一道习题解法的探究

(一)题目:通过点(8,6)引四条直线与ox轴的夹角之比为1:2:3:4,已知第二条直线的方程为3x-4y=0,求其余三条直线的方程。 (华东师大数学系编《解析几何习题集》(以下简称甲书)P_71。18题;翟连林等主编《中学数学习题集第三册》(以下简称乙书)P230第7题。) (二)上述两书的解答乙书给出的解答如下: 设四条直线为l_1、l_2、k_3、l_4,倾斜角顺次为α、2α、3α、4α。由l_2的方程3x-4y=0(?)tg2α=3/4即2tgα/(1-tg~2α)=3/4(?)tgα=1/3,tga=-3(舍)(?)tg3α=13/9,tg4α=24/7∴l_1:y-6=(x-8)/3即x-3y+10=0     

Tensor products of complementary series of rank one Lie groups

We consider the tensor product π_α ? π_βof complementary series representations π_α and π_β of classical rank one groups SO_0(n, 1), SU(n, 1) and Sp(n, 1). We prove that there is a discrete component π_(α+β)for small parameters α and β(in our parametrization). We prove further that for SO_0(n, 1) there are finitely many complementary series of the form π_(α+β+2j,)j = 0, 1,..., k, appearing in the tensor product π_α ? π_βof two complementary series π_α and π_β, where k = k(α, β, n) depends on α, β and n.     

Dynamics of a Mutualistic Model with Diffusion

This article is concerned with a system of semilinear parabolic equations with no-flux boundary condition in a mutualistic ecological model. Stability result of the equilibrium about relevant ODE problem is proved by discussing its Jacobian matrix, we give two priori estimates and prove that the model is permanent when ε_1+ ε_2 ≠ 0. Moreover sufficient conditions for the global asymptotical stability of the unique positive equilibrium of the model are obtained. Nonexistence of nonconstant positive steady states of the model is also given. When ε_1+ε_2= 0, grow up property is derived if the geometric mean of the interaction coefficients is greater than 1(α_1α_2 1),while if the geometric mean of the interaction coefficients is less than 1(α_1α_2 1), there exists a global solution. Finally, numerical simulations are given.     

非线性期望和非线性估价的次可加性

证明了在适当的条件下F-期望的次可加性蕴涵条件F-期望的次可加性,F-估价的次可加性蕴涵Ft-相容非线性估价的次可加性.     

具有吸收和非线性边界流的非线性扩散方程的爆破估计

本文研究一类具有非线性吸收和非线性边界流的非线性扩散方程，建立了解的爆破速率估计，所得结果依赖于模型中三种非线性机制之间的相互作用。     

某些非线性积分不等式

本文考虑一定有序局部紧空间上的非线性积分不等式，并给出其对于一定非线性积分方程的应用．     

Nonlinear Lagrangians for Nonlinear Programming Based on Modified Fischer-Burmeister NCP Functions

This paper proposes nonlinear Lagrangians based on modified Fischer-Burmeister NCP functions for solving nonlinear programming problems with inequality constraints. The convergence theorem shows that the sequence of points generated by this nonlinear Lagrange algorithm is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions, and the error bound of solution, depending on the penalty parameter, is also established. It is shown that the condition number of the nonlinear Lagrangian Hessian at the optimal solution is proportional to the controlling penalty parameter. Moreover, the paper develops the dual algorithm associated with the proposed nonlinear Lagrangians. Numerical results reported suggest that the dual algorithm based on proposed nonlinear Lagrangians is effective for solving some nonlinear optimization problems.     

Nonlinear Kolmogorov widths

We investigate a generalization of width in the sense of Kolmogorov suitable for estimating bestm-term approximations. We generalize Carl's inequality which gives lower estimate of Kolmogorov widths in terms of entropy numbers. Application of these new inequalities gives some progress in the problem of estimating bestm-term trigonometric approximations of multivariate functions.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 891–902, June, 1998.This research was supported by the National Science Foundation under grant No. DMS 9622925 and by the ONR foundation under grant No. N00014-96-1-1003.     

Degenerate Nonlinear Boundary Value Problems for Fully Nonlinear Elliptic Equations of Second Order

In this paper, we will consider the degenerate nonlinear boundary value problems for nonlinear elliptic equations. We extend Lieberman & Trudinger's results from nondegenerate oblique derivative boundary values to degenerate boundary values.     

A Nonlinear Diffusion System with Coupled Nonlinear Boundary Flux

This paper studies a nonlinear diffusion system with coupled nonlinear boundary flux and two kinds of inner sources （positive for the first and negative for the second）, where the four nonlinear mechanisms are described by eight nonlinear parameters. The critical exponent of the system is determined by a complete classification of the eight nonlinear parameters, which is represented via the characteristic algebraic system introduced to the problem.     

Doubly Nonlinear Degenerate Parabolic Systems with Coupled Nonlinear Boundary Conditions

In this paper, we study the global existence and the global nonexistence of doubly nonlinear degenerate parabolic systems with nonlinear boundary conditions. We first prove a local existence result by the regularization method. Next, we construct a weak comparison principle. Then we discuss the large time behavior of solutions by using a modified upper and lower solution methods and constructing various upper and lower solutions. Necessary and sufficient conditions on the global existence of all positive (weak) solutions are obtained.     

KKT Conditions for Rank-Deficient Nonlinear Least-Square Problems with Rank-Deficient Nonlinear Constraints

In nonlinear least-square problems with nonlinear constraints, the function , where f ₂ is a nonlinear vector function, is to be minimized subject to the nonlinear constraints f ₁(x)=0. This problem is ill-posed if the first-order KKT conditions do not define a locally unique solution. We show that the problem is ill-posed if either the Jacobian of f ₁ or the Jacobian of J is rank-deficient (i.e., not of full rank) in a neighborhood of a solution satisfying the first-order KKT conditions. Either of these ill-posed cases makes it impossible to use a standard Gauss–Newton method. Therefore, we formulate a constrained least-norm problem that can be used when either of these ill-posed cases occur. By using the constant-rank theorem, we derive the necessary and sufficient conditions for a local minimum of this minimum-norm problem. The results given here are crucial for deriving methods solving the rank-deficient problem.     

非线性评估方法研究

研究了评估中评价指标间的非线性特征对评估的影响,提出了对非线性特征明显的因素进行修正的方法和思路,最后针对院校实验室评估进行应用举例.