On Nonlinear Eigen-problems of Quasi-linear Elliptic Operators

In this paper, we study the following Eigen-problem {-\frac{∂}{∂x_i}(a_{ij}(x, u)\frac{∂u}{∂x_j}) + \frac{1}{2}a_{iju}(x,u)\frac{∂u}{∂x_i}\frac{∂u}{∂x_j} + h(x)u = μμ\frac{n+2}{n-2} \quad in Ω \qquad (0.1) u = 0 \quad on ∂Ω u > 0 \quad in Ω ⊂ R^n under some assumptions. First. we minimize I(u) = \frac{1}{2}∫_Ωa_{ij}(x, u)\frac{∂u}{∂x_i}\frac{∂u}{∂x_j} + h(x)u² over E_α = {u ∈ H¹_0(Ω); ∫_Ωu^α = 1} ( 2 < α < N = \frac{2n}{n-2}) to give a H¹_0-solution U_α of the perturbation problems of (0.1). Since I is not differentiable in H¹_0(Ω), the key point is the estimate of U_α. Then, we derive local uniform bounds of (U_α) and give a 'bad' solution of (0.1). Last, we remove the singular points of the 'bad' solution to obtain a solution of (0.1), our result is a extension of that of Brezis & Nirenberg.     

More,or less,trivial matrix functional equations

Various functional equations satisfied by one or two (N × N)-matrices \({\mathbf{F}(z) }\) and \({\mathbf{G}(z) }\) depending on the scalar variable z are investigated, with N an arbitrary positive integer. Some of these functional equations are generalizations to the matrix case (N > 1) of well-known functional equations valid in the scalar (N = 1) case, such as \({\mathbf{F}(x) \, \mathbf{F}(y) = \, \mathbf{F}(x y) \, \rm and \, \mathbf{G}({\it x}) \, \mathbf{G}({\it y}) = \mathbf{G}({\it x+y}) }\); others—such as \({\mathbf{G}(y) \, \mathbf{F}(x) = \mathbf{F}(x) \, \mathbf{G}(xy) }\)—possess nontrivial solutions only in the matrix case (N > 1), namely their scalar (N = 1) counterparts only feature quite trivial solutions. It is also pointed out that if two (N × N)-matrices \({\mathbf{F}(x) \, \rm and \, \mathbf{G}({\it y})}\) satisfy the triplet of functional equations written above—and nontrivial examples of such matrices are exhibited— then they also satisfy an endless hierarchy of matrix functional relations involving an increasing number of scalar independent variables, the first items of which read \({\mathbf{F}(x_{1}) \, \mathbf{G}(y_{1}) \, \mathbf{F} (x_{2}) = \mathbf{F}(x_{1} x_{2}) \, \mathbf{G } (x_{2} y_{1}) \, \rm and \, \mathbf{G}({\it y}_{1}) \, \mathbf{F} ({\it x}_{1}) \, \mathbf{G} ({\it y}_{2}) = \mathbf{F} ({\it x}_{1}) \, \mathbf{G} ({\it x}_{1} {\it y}_{1}+{\it y}_{2}) }\).     

系数在模李超代数~$W(m,3,\underline{1})$ 上的~$\frak{gl}(2,\mathbb{F})$ 的一维上同调

研究了系数在模李超代数~$W(m,3,\underline{1})$ 上的~$\frak{gl}(2,\mathbb{F})$ 的一维上同调, 其中~$\mathbb{F}$ 是一个素特征的代数闭域且~$\frak{gl}(2,\mathbb{F})$ 是系数在~$\mathbb{F}$ 上的~$2\times 2$ 阶矩阵李代数. 计算出所有~$\frak{gl}(2,\mathbb{F})$ 到模李超代数~$W(m,3,\underline{1})$ 的子模的导子和内导子. 从而一维上同调~$\textrm{H}^{1}(\frak{gl}(2,\mathbb{F}),W(m,3,\underline{1}))$ 可以完全用矩阵的形式表示.     

On convergence of a generalized Pál interpolation process

Let f∈C _[−1,1] ^″ (r≥1) and R_n(f,α,β,x) be the generalized Pál interpolation polynomials satisfying the conditions R_n(f,α,β,x_k)=f(x_k),R_n ^′(f,α,β,x_k)=f′(x_k)(k=1,2,…,n), where {x_k} are the roots of n-th Jacobi polynomial P_n(α,β,x),α,β>−1 and {x _k ^″ } are the roots of (1−x²)P_n″(α,β,x). In this paper, we prove that holds uniformly on [0,1]. In Memory of Professor M. T. Cheng Supported by the Science Foundation of CSBTB and the Natural Science Foundatioin of Zhejiang.     

The Dirichlet Problems for a Class of Fully Nonlinear Elliptic Equations Relative to the Eigenvalues of the Hessian

ln this paper we discuss the Dirichlet problems for a class of fully nonliucar elliptic equations F(D² u) = ψ(x, u)(ψ(x, u, ∇u)) \quad in Q u = φ(x) \quad on ∂Ω where F is represented by a symmetric function f(λ_1, …, λ_n) of the eigenvalues (λ_1, …,λ_n) of the Hessian D²u. This result extends the works of Caffarelli L., Nirenberg L., Spruck L. [2] to more general cases.     

3$\times$3阶上三角型算子矩阵的可能谱

Let H1, H2 and H3 be infinite dimensional separable complex Hilbert spaces. We denote by M（D,V,F） a 3×3 upper triangular operator matrix acting on Hi ＋H2＋ H3 of theform M（D,E,F）=（A D F 0 B F 0 0 C）.For given A ∈ B（H1）, B ∈ B（H2） and C ∈ B（H3）, the sets ∪D,E,F^σp（M（D,E,F））,∪D,E,F ^σr（M（D,E,F））,∪D,E,F ^σc（M（D,E,F）） and ∪D,E,F σ（M（D,E,F）） are characterized, where D ∈ B（H2,H1）, E ∈B（H3, H1）, F ∈ B（H3,H2） and σ（·）, σp（·）, σr（·）, σc（·） denote the spectrum, the point spectrum, the residual spectrum and the continuous spectrum, respectively.     

Nonexcellence of the Function Field of the Product of Two Conics

Let k₀ be a field, k₀ ≠ 2, and α, β 2-fold Pfister forms over k₀. Denote by [α], [β] the classes of the corresponding quaternion algebras in ₂Brk₀, and by X_α, X_β the corresponding projective k₀-conics. Suppose ([α] + [β]) = 4. We construct a field F over k₀ such that the field extension F(X_α × X_β)/F is not excellent. Moreover, we find a 2-fold Pfister form γ over F such that ([α ] +[β ] + [γ]) = 4 and the homology group of the complex

An Extremal Problem with Excluded Subposet in the Boolean Lattice

Let 2[n] be the poset of all subsets of {1,...,n} ordered by inclusion. A poset P is said to be contained in a subposet F of 2[n] as a subposet if there exists a subposet P' of F that is isomorphic to P. In this paper we will give an estimate on the size of a maximally sized subposet of 2[n] under the assumption that the subposet does not contain Y(u,v) = ({a1,...,au, b1,...,bv}, {a1< ··· < au, au< b1, au< b2,...,au< bv}) as a subposet.     

严格上三角矩阵李代数上的积零导子

设$F$ 为域, $n\geq 3$, $\bf{N}$$(n,\mathbb{F})$ 为域$\mathbb{F}$ 上所有$n\times n$ 阶严格上三角矩阵构成的严格上三角矩阵李代数, 其李运算为$[x,y]=xy-yx$. $\bf{N}$$(n, \mathbb{F})$ 上一线性映射$\varphi$ 称为积零导子,如果由$[x,y]=0, x,y\in \bf{N}$$(n,\mathbb{F})$,总可推出 $[\varphi(x), y]+[x,\varphi(y)]=0$. 本文证明 $\bf{N}$$(n,\mathbb{F})$上一线性映射 $\varphi$ 为积零导子当且仅当 $\varphi$ 为$\bf{N}$$(n,\mathbb{F})$ 上内导子, 对角线导子, 极端导子, 中心导子和标量乘法的和.     

Regularity Results for Minimizers of Certain Functional Having Nonquadratic Growth with Obstacles

We prove partial regularity for minimizers of degenerate variational integrals ∫_Ω F(x, u, Du)dx with obstacles of either the form (i) μ_f = {u ∈ H^{1,m} (Ω,\mathbb{R}^N)|u^N ≥ f_1(u¹, ... ,u^{N-1}) + f_2(x) a.e.} or (ii) μ_N = {u ∈ H^{1,m}(Ω,\mathbb{R}^N)|u^i(x) ≥ h^i(x), a.e.; i=1, ... ,N} The typical mode of variational integrals is given by ∫_Ω [a^{αβ}(x, u)b_{ij}(x, u)D_αu^i D_βu^i]^{\frac{m}{2}}dx, m ≥ 2     

Characterizations of Riemannian space forms, Einstein spaces and conformally flat spaces

In a recent paper the first author introduced two sequences of Riemannian invariants on a Riemannian manifold , denoted respectively by and , which trivially satisfy . In this article, we completely determine the Riemannian manifolds satisfying the condition . By applying the notions of these -invariants, we establish new characterizations of Einstein and conformally flat spaces; thus generalizing two well-known results of Singer-Thorpe and of Kulkarni.

     

The -invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings

Let be a regular local ring and let be a filtration of ideals in such that is a Noetherian ring with . Let and let be the -invariant of . Then the theorem says that is a principal ideal and for all if and only if is a Gorenstein ring and . Hence , if is a Gorenstein ring, but the ideal is not principal.

     

A New Characterization of Gundersen’s Example of Two Meromorphic Functions Sharing Four Values

Let ? and g be nonconstant meromophic functions sharing four values IM and satisfying ?^?1({a}) ? ^?1({b}) for two values a, b not shared by ? and g. Then either ? = T o g with a Möbius transformation T or \(f=L\ {\rm o}\ \hat{f}\ {\rm o}\ h\) and \(g=L\ {\rm o}\ \hat{g}\ {\rm o}\ h\) , where \(\hat{f}(z)=({\rm exp}\ z+1)/({\rm exp}\ z-1)^2\) and \(\hat{g}(z)=({\rm exp}\ z+1)^2/(8({\rm exp}\ z-1))\) are the functions in Gundersen’s example [1], L is a Möbius transformation and h is an entire function.     

On the Cauchy Problem and Initial Trace for Nonlinear Filtration Type with Singularity

In this paper, we consider the Cauchy problem \frac{∂u}{∂t} = Δφ(u) in R^N × (0, T] u(x,0} = u_0(x) in R^N where φ ∈ C[0,∞) ∩ C¹(0,∞), φ(0 ) = 0 and (1 - \frac{2}{N})^+ < a ≤ \frac{φ'(s)s}{φ(s)} ≤ m for some a ∈ ((1 - \frac{2}{n})^+, 1), s > 0. The initial value u_0 (z) satisfies u_0(x) ≥ 0 and u_0(x) ∈ L¹_{loc}(R^N). We prove that, under some further conditions, there exists a weak solution u for the above problem, and moreover u ∈ C^{α, \frac{α}{2}}_{x,t_{loc}} (R^N × (0, T]) for some α > 0.     

Notes on combinatorial set theory

We shall prove some unconnected theorems: (1) (G.C.H.) \omega _{\alpha + 1} \to \left( {\omega _\alpha + \xi } \right)_2^2 when ℵ_α is regular, │ξ│⁺<ωα. (2) There is a Jonsson algebra in ℵ_α+n, and \aleph _{a + n} \not \to \left[ {\aleph _{a + n} } \right]_{\aleph _{a + n} }^{n + 1} if 2^{\aleph _{ - - } } = \aleph _{a + n} \cdot (3) If λ>ℵ₀ is a strong limit cardinal, then among the graphs with ≦λ vertices each of valence <λ there is a universal one. (4)(G.C.H.) If f is a set mapping on \omega _{a + 1} (ℵ_α regular) │f(x)∩f(y│<ℵ_α, then there is a free subset of order-type ζ for every ζ<ω_α+1.     

Bounds of the Estrada index of graphs

Let G be a graph of order n and let λ1,λ2,...,λn be its eigenvalues. The Estrada index[2] of G is defined as EE = EE(G) =∑n i=1 eλi. In this paper, new bounds for EE are established, as well as some relations between EE and graph energy E.     

Mahler measures in a field are dense modulo 1

Let K be a number field. We prove that the set of Mahler measures M(α), where α runs over every element of K, modulo 1 is everywhere dense in [0, 1], except when or , where D is a positive integer. In the proof, we use a certain sequence of shifted Pisot numbers (or complex Pisot numbers) in K and show that the corresponding sequence of their Mahler measures modulo 1 is uniformly distributed in [0, 1]. Received: 24 March 2006     

The Theory of Filled Function Method for Finding Global Minimizers of Nonlinearly Constrained Minimization Problems

This paper is an extension of [1]. In this paper the descent and ascent segments are introduced to replace respectively the descent and ascent directions in [1] and are used to extend the concepts of S-basin and basin of a minimizer of a function. Lemmas and theorems similar to those in [1] are proved for the filled function $$P(x,r,p)= \frac{1}{r+F(x)}exp(-|x-x^*_1|^2/\rho^2),$$ which is the same as that in [1], where $x^*_1$ is a constrained local minimizer of the problem (0.3) below and $$F(x)=f(x)+\sum^{m'}_{i=1}\mu_i|c_i(x)|+ \sum^m_{i=m'+1}\mu_i max(0, -c_i(x))$$ is the exact penalty function for the constrained minimization problem$\mathop{\rm min}\limits_x f(x)$,subject to $$c_i(x) = 0 , i = 1, 2, \cdots, m',$$ $$c_i(x) \ge 0 , i = m'+1, \cdots, m,$$ where $μ_i＞0 \ (i=1, 2, \cdots, m)$ are sufficiently large. When $x^*_1$ has been located, a saddle point or a minimizer $\hat{x}$ of $P(x,r,\rho)$ can be located by using the nonsmooth minimization method with some special termination principles. The $\hat{x}$ is proved to be in a basin of a lower minimizer $x^*_2$ of $F(x)$, provided that the ratio $\rho^2/[r+F(x^*_1)]$ is appropriately small. Thus, starting with $\hat{x}$ to minimize $F(x)$, one can locate $x^*_2$. In this way a constrained global minimizer of (0.3) can finally be found and termination will happen.     

The Jump Conditions for Second Order Quasilinear Degenerate Parabolic Equations

ln this paper we consider the model problem for a second order quasilinear degenerate parabolic equation {D_xG(u) = t^{2N-1}D²_xK(u) + t^{N-1}D_x,F(u) \quad for \quad x ∈ R,t > 0 u(x,0) = A \quad for \quad x < 0, u(x,0) = B \quad for \quad x > 0 where A < B, and N > O are given constants; K(u) =^{def} ∫^u_Ak(s)ds, G(u)=^{def} ∫^u_Ag(s)ds, and F(u) =^{def} ∫^u_Af(s)ds are real-valued absolutely continuous functions defined on [A, B] such that K(u) is increasing, G(u) strictly increasing, and \frac{F(B)}{G(B)}G(u) - F(u) nonnegative on [A, B]. We show that the model problem has a unique discontinuous solution u_0 (x, t) when k(s) possesses at least one interval of degeneracy in [A, B] and that on each curve of discontinuity, x = z_j(t) =^{def} s_jt^N, where s_j= const., j=l,2, …, u_0(x, t) must satisfy the following jump conditions, 1°. u_0(z_j(t) - 0, t) = a_j, u_0 (z_j(t) + 0, t) = b_j, and u_0(z_j(t) - 0, t) = [a_j, b_j] where {[a_j, b_j]; j = 1, 2, …} is the collection of all intervals of degeneracy possessed by k (s) in [A, B], that is, k(s) = 0 a. e. on [a_j, b_j], j = 1, 2, …, and k(s) > 0 a. e. in [A, B] \U_j[a_j, b_j], and 2°. (z_j(t)G(u_0(x, t)) + t^{2N-1}D_xK(u_0(x, t)) + t^{N-1}F(u_0(x, t)))|\frac{s=s_j+0}{s=s_j-0} = 0