On the isolated singularities of the solutions of the Gaussian curvature equation for nonnegative curvature

The precise asymptotic behaviour of the solutions to the two-dimensional curvature equation Δu=k(z)e2u with e2u∈L¹ for bounded nonnegative curvature functions −k(z) near isolated singularities is obtained.     

Weak convergence of currents

We give certain conditions to guarantee weak convergence u _k T _k → uT, where u _k , u are plurisubharmonic functions and T _k , T are positive closed currents. As applications we obtain that convergence in capacity of plurisubharmonic functions u _k implies weak convergence of the complex Monge-Ampère measures (dd ^c u _k ) ⁿ if all of the plurisubharmonic functions u _k are bounded below by one of some sorts of plurisubharmonic functions.     

The inverse of a matrix polynomial

An explicit representation is obtained for P(z)^?1 when P(z) is a complex n×n matrix polynomial in z whose coefficient of the highest power of z is the identity matrix. The representation is a sum of terms involving negative powers of z?λ for each λ such that P(λ) is singular. The coefficients of these terms are generated by sequences u_k, v_k of 1×n and n×1 vectors, respectively, which satisfy u₁≠0, v₁≠0, ∑^k?1_h=0(1?h!)u_k?hP^(h)(λ)=0, ∑^k?1_h=0(1?h!)P^(h)(λ)v_k?h=0, and certain orthogonality relations. In more general cases, including that when P(z) is analytic at λ but not necessarily a polynomial, the terms in the representation involving negative powers of z?λ provide the principal part of the Laurent expansion for P(z)^?1 in a punctured neighborhood of z=λ.     

Analytic approximations of plurisubharmonic singularities

We study several classes of isolated singularities of plurisubharmonic functions that can be approximated by analytic singularities with control over their residual Monge–Ampère masses. They are characterized in terms of Green functions for Demailly’s approximations, relative types, and valuations. Furthermore, the classes are shown to appear when studying graded families of ideals of analytic functions and the corresponding asymptotic multiplier ideals.     

Sur un probleme aux valeurs propres non line aire

In this paper, we give some sufficient conditions for which the differential operatorP(λ=P ₀+λP ₁+...+λ ^m?1 P _m?1+λ ^m , depending polynomially on the complex parameter λ, verifies the following statement: there exists λ₀ ∈ ?,u _o=0,u ₀ ∈ ?(? ⁿ ) a Schwartz space of rapidly decreasing functions, such thatP(λ₀)u ₀=0-.     

Almost everywhere divergent subsequences of Fourier sums of functions from φ(L) ∩ H 1 ω

For a gap sequence of natural numbers {n _k } _k=1 ^∞ , for a nondecreasing function φ: [0,+∞) → [0,+∞) such that φ(u) = o(u ln ln u) as u → ∞, and a modulus of continuity satisfying the condition (ln k)^?1 = O(ω(n _k ^?1 )), we present an example of a function F ∈ φ(L) ∩ H ₁ ^ω with an almost everywhere divergent subsequence {S _{n k} (F, x)} of the sequence of partial sums of the trigonometric Fourier series of the function F.     

Valuations and multiplier ideals

We present a new approach to the study of multiplier ideals in a local, two-dimensional setting. Our method allows us to deal with ideals, graded systems of ideals and plurisubharmonic functions in a unified way. Among the applications are a formula for the complex integrability exponent of a plurisubharmonic function in terms of Kiselman numbers, and a proof of the openness conjecture by Demailly and Kollár. Our technique also yields new proofs of two recent results: one on the structure of the set of complex singularity exponents for holomorphic functions; the other by Lipman and Watanabe on the realization of ideals as multiplier ideals.

     

Lelong numbers with respect to regular plurisubharmonic weights

Generalized Lelong numbers v(T, φ) due to Demailly are specified for the case of positive closed currents T = dd ^cu and plurisubharmonic weights φ with multicircled asymptotics. Explicit formulas for these values are obtained in terms of the directional Lelong numbers of the functions u and the Newton diagrams of φ. An extension of Demailly’s approximation theorem is proved as well.     

Spectral Criteria of Abstract Sequences; Application to Difference Equations

We suppose that M is a closed subspace of l ^∞(J, X), the space of all bounded sequences {x(n)} _n?J ? X, where J ? {Z⁺,Z} and X is a complex Banach space. We define the M-spectrum σ_M (u) of a sequence u ? l ^∞(J,X). Certain conditions will be supposed on both M and σ_M (u) to insure the existence of u ? M. We prove that if u is ergodic, such that σ_M (u,) is at most countable and, for every λ ? σ_M (u), the sequence e^?iλnu(n) is ergodic, then u ? M. We apply this result to the operator difference equationu(n + 1) = Au(n) + ψ(n), n ? J,and to the infinite order difference equation Σ ^r _k=1 a_k (u(n + k) ? u(n)) + Σ _{s ? Z}?(n ? s)u(s) = h(n), n?J, where ψ?l ^∞(Z,X) such that ψ| _J ? M, A is the generator of a C ₀-semigroup of linear bounded operators {T(t)} _t>0 on X, h ? M, ? ? l ¹(Z) and a_k ?C. Certain conditions will be imposed to guarantee the existence of solutions in the class M.     

Discrete approximating operators on function algebras

We give a new presentation and various extensions of one theorem of Somorjai. For any sequence of operatorsL _n , given byL _n f=∑ _k=1 ⁿ f(z _n,k )l _n,k withz _{n, k} ∈T andl _{n, k} ∈A(T), there exists a functionf∈A(T) such thatL _n f does not converge tof.     

Chain sequences and symmetric generalized orthogonal polynomials

In this paper, we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials. We use this to study the orthogonal polynomials K_n^(λ,M,k) associated with the probability measure dφ(λ,M,k;x), which is the Gegenbauer measure of parameter λ+1 with two additional mass points at ±k. When k=1 we obtain information on the polynomials K_n^(λ,M) which are the symmetric Koornwinder polynomials. Monotonicity properties of the zeros of K_n^(λ,M,k) in relation to M and k are also given.     

Some inverse results for Hill's equation

We consider Hill's equation y″+(λ−q)y=0 where q∈L¹[0,π]. We show that if l_n—the length of the n-th instability interval—is of order O(n^−(k+2)) then the real Fourier coefficients a_n^k,b_n^k of q^(k)—k-th derivative of q—are of order O(n⁻²), which implies that q^(k) is absolutely continuous almost everywhere for k=0,1,2,….     

Szegö kernel asymptotics and Morse inequalities on CR manifolds

Let X be an abstract compact orientable CR manifold of dimension ${2n-1, n\,\geqslant\,2}$ , and let L ^k be the k-th tensor power of a CR complex line bundle L over X. We assume that condition Y(q) holds at each point of X. In this paper we obtain a scaling upper-bound for the Szegö kernel on (0, q)-forms with values in L ^k , for large k. After integration, this gives weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities. We apply the strong Morse inequalities to the embedding of some convex–concave manifolds.     

On the Perron-Frobenius eigenvector for nonnegative integral matrices whose largest eigenvalue is integral

We first derive the bound |det(λI − A)|⩽λ^k − λ^k₀ (λ₀⩽λ), where A is a k × k nonnegative real matrix and λ₀ is the spectral radius of A. If A is irreducible and integral, and its largest nonnegative eigenvalue is an integer n, then we use this inequality to derive the upper bound n^k−1 on the components of the smallest integer eigenvector corresponding to n. Finer information on the components is also derived.     

An extremal problem on non-full colorable graphs

For a given graph G of order n, a k-L(2,1)-labelling is defined as a function f:V(G)→{0,1,2,…k} such that |f(u)-f(v)|?2 when d_G(u,v)=1 and |f(u)-f(v)|?1 when d_G(u,v)=2. The L(2,1)-labelling number of G, denoted by λ(G), is the smallest number k such that G has a k-L(2,1)-labelling. The hole index ρ(G) of G is the minimum number of integers not used in a λ(G)-L(2,1)-labelling of G. We say G is full-colorable if ρ(G)=0; otherwise, it will be called non-full colorable. In this paper, we consider the graphs with λ(G)=2m and ρ(G)=m, where m is a positive integer. Our main work generalized a result by Fishburn and Roberts [No-hole L(2,1)-colorings, Discrete Appl. Math. 130 (2003) 513-519].     

Utilizing Shelve Slots: Sufficiency Conditions for Some Easy Instances of Hard Problems

In this paper we present a minimal set of conditions sufficient to assure the existence of a solution to a system of nonnegative linear diophantine equations. More specifically, suppose we are given a finite item set U = {u₁, u₂, . . . , u_k} together with a "size" v_i ≡ v(u_i) ∈ Z⁺, such that v_i ≠ v_j for i ≠ j, a "frequency" a_i ≡ a(u_i) ∈ Z⁺, and a positive integer (shelf length) L ∈ Z⁺ with the following conditions: (i) L = ∏ⁿ_j=1p_j(p_j ∈ Z⁺ ∀j, p_j ≠ p_l for j ≠ l) and v_i = ∏ _j∈Aip_j, A_i ⊆ {l, 2, . . . , n} for i = 1, . . . , n; (ii) (A_i\{⋂^k_j=1A_j}) ∩ (A_l\{⋂^k_j=1A_j}) = ⊘∀i ≠ l. Note that v_i|L (divides L) for each i. If for a given m ∈ Z⁺, ∑ⁿ_i=1a_iv_i = mL (i.e., the total size of all the items equals the total length of the shelf space), we prove that conditions (i) and (ii) are sufficient conditions for the existence of a set of integers {b₁₁, b₁₂, . . . , b_1m, b₂₁, . . . , b_n1, . . . , b_nm}⊆ N such that ∑^m_j=1b_ij = a_i, i = 1, . . . , k, and ∑^k_i=1b_ijv_i = L, j =1, . . . , m (i.e., m shelves of length L can be fully utilized). We indicate a number of special cases of well known NP-complete problems which are subsequently decided in polynomial time.     

Local and global norm comparison theorems for solutions to the nonhomogeneous A-harmonic equation

We establish local and global norm inequalities for solutions of the nonhomogeneous A-harmonic equation A(x,g+du)=h+d^?v for differential forms. As applications of these inequalities, we prove the Sobolev-Poincaré type imbedding theorems and obtain Lp-estimates for the gradient operator ∇ and the homotopy operator T from the Banach space Ls(D,Λl) to the Sobolev space W1,s(D,Λl−1), l=1,2,…,n. These results can be used to study both qualitative and quantitative properties of solutions of the A-harmonic equations and the related differential systems.     

Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ballB⊂C ⁿ with its relative logarithmic capacity inC ⁿ with respect to the same ballB. An analogous comparison inequality for Borel subsets of euclidean balls of any generic real subspace ofC ⁿ is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of plurisubharmonic lemniscates associated to the Lelong class of plurisubharmonic functions of logarithmic singularities at infinity onC ⁿ as well as the Cegrell class of plurisubharmonic functions of bounded Monge-Ampère mass on a hyperconvex domain Ω⊂(C ⁿ . Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of plurisubharmonic functions. This work was partially supported by the programmes PARS MI 07 and AI.MA 180.     

О раэложении мероморфных функций специального вида на простейщие дроби

The paper is devoted to study the entire functions L(λ) with simple real zeros λ_k, k = 1, 2, ..., that admit an expansion of Krein’s type: $$\frac{1}{{\mathcal{L}(\lambda )}} = \sum\limits_{k = 1}^\infty {\frac{{c_k }}{{\lambda - \lambda _k }}} ,\sum\limits_{k = 1}^\infty {\left| {c_k } \right| < \infty } .$$ We present a criterion for these expansions in terms of the sequence {L′ (λ _k )} _k=1 ^∞ . We show that this criterion is applicable to certain classes of meromorphic functions and make more precise a theorem of Sedletski? on the annihilating property in L ² systems of exponents.     

Gargano,Lewinter and Malerba问题的解

§1　IntroductionLet G be a graph with vertex-set V(G) ={ v1 ,v2 ,...,vn} .A labeling of G is a bijectionL:V(G)→{ 1,2 ,...,n} ,where L (vi) is the label of a vertex vi.A labeled graph is anordered pair (G,L) consisting of a graph G and its labeling L.Definition1.An increasing nonconsecutive path in a labeled graph(G,L) is a path(u1 ,u2 ,...,uk) in G such thatL(ui) + 1相似文献