Convergence of Gaussian Quadrature and Lagrange Interpolation in Haar Systems

Consider a Markov system of functions whose linear span is dense with respect to the uniform norm in the space of the continuous functions on a finite interval. Gaussian rules are those which correctly integrate as many successive basis functions as possible with the lesser number of nodes. In this paper we provide a simple proof of the fact that such rules converge for all bounded Riemann-Stieltjes integrable functions. The proposed proof is also valid for any sequence of quadrature rules with positive coefficients which converge for the basis functions. Taking the nodes of the Gaussian rules as nodes for Lagrange interpolation, we give a sufficient condition for the convergence in L ₂-norm of such processes for bounded Riemann-Stieltjes integrable functions.     

A short proof of a general multiplier rule

A general multiplier rule obtained by Hestenes is shown to follow directly from the Lagrange multiplier rule by a simple compactness argument. A similar simplification is effected in the proof of Gittleman's extension of Hestenes' rule.     

A note on time decay for the nonlinear beam equation

In this note we obtain another form of Morawetz-type identity by Lagrange method and present a simple proof of the time decay estimates for the solution of the nonlinear beam equation, for a nonlinear term with repelling sign, as t→∞. As an immediate consequence, we also get the time-integrability of the local energy of the solution for the beam equation.     

Cramer法则在数值计算中的若干应用

用Cramer法则给出了Lagrange插值公式和Newton插值公式的简洁证明,同时得到了Vandermonde矩阵的逆矩阵的LU分解.     

An elementary proof of the Lagrange multiplier theorem in normed linear spaces

We present an elementary proof of the Lagrange multiplier theorem for optimization problems with equality constraints in normed linear spaces. Most proofs in the literature rely on advanced concepts and results, such as the implicit function theorem and the Lyusternik theorem. By contrast, the proof given in this article employs only basic results from linear algebra, the critical-point condition for unconstrained minima and the fact that a continuous function attains its minimum over a closed ball in the finite-dimensional space.     

关于Shimoji and Takahashi一个定理的简单证明

目的是在一般Banach空间中建立一个无限可数多个非扩张映像族的公共不动点的强收敛定理．使用一些新的分析技巧,尤其是Dotson引理和W映像技巧,给出了Shimoji and Takahashi的一个定理的简单证明．     

Global properties of integrable Hamiltonian systems

This paper deals with Lagrangian bundles which are symplectic torus bundles that occur in integrable Hamiltonian systems. We review the theory of obstructions to triviality, in particular monodromy, as well as the ensuing classification problems which involve the Chern and Lagrange class. Our approach, which uses simple ideas from differential geometry and algebraic topology, reveals the fundamental role of the integer affine structure on the base space of these bundles. We provide a geometric proof of the classification of Lagrangian bundles with fixed integer affine structure by their Lagrange class.      

A Short Proof of Krattenthaler Formulas

With an effort to investigate a unified approach to the Lagrange inverse Krattenthaler established operator method we finally found a general pair of inverse relations, called the Krattenthaler forumlas. The present paper presents a very short proof of this formula via Lagrange interpolation. Further, our method of proof declares that the Krattenthaler result is unique in the light of Lagrange interpolation. Received April 9, 1999, Accepted December 7, 2000     

对一道数学分析题证明方法的改进

某文献在处理一道关于高阶导数的应用问题时,反复利用Rolle定理来证明高阶导数为零．考虑到这种做法过于繁琐,遂通过对其证明方法的改进,综合使用Lagrange中值定理和Taylor公式,使该问题的解决获得简化．     

A remark on Burnside's theorem on matrix algebras

We present a simple proof of Burnside's theorem using techniques from elementary linear algebra.     

Stability of the determination of the surface impedance of an obstacle from the scattering amplitude

We prove a stability estimate of logarithmic type for the inverse problem consisting in the determination of the surface impedance of an obstacle from the scattering amplitude. We present a simple and direct proof, which is essentially based on an elliptic Carleman inequality. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Tau Function for the Schlesinger Equation of Isomonodromic Deformations

The tau function for the Schlesinger equation of isomonodromic deformations is represented as the result of successively applied elementary gauge transformations; this, in particular, suggests a simple proof for the Miwa theorem about the tau function.     

Lagrange Inversion Theorem for Dirichlet series

We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [5].     

分部积分公式的一种证明

运用阿贝尔分部求和公式,给出定积分分部积分公式的一种新证明．     

An alternative approach to the Faber–Krahn inequality for Robin problems

We give a simple proof of the Faber–Krahn inequality for the first eigenvalue of the p-Laplace operator with Robin boundary conditions. The techniques introduced allow to work with much less regular domains by using test function arguments. We substantially simplify earlier proofs, and establish the sharpness of the inequality for a larger class of domains at the same time.     

Transport and instability for perfect fluids

Incompressible perfect fluids are described by the Euler equations. We provide a new simple proof for well-posedness for velocities in and linear and nonlinear instability results using transport techniques. The results have an important consequence: the topology of is too fine for interesting questions about large time behavior. Received: 14 September 2001 / Published online: 4 April 2002     

Capital budgeting with benders decomposition

Capital budgeting problems with different interest rates for borrowing and lending and with possible limits on borrowing are applied to dual and primal decomposition. While the former fails if a dual gap exists, the latter becomes attractive. The paper elaborates dual and primal decomposition to capital budgeting models and discusses variants of the Benders scheme. A computer implementation is described and results of extensive computer runs with different strategies are reported which give proof of the efficiency of the implemented decomposition procedure.     

Lagrange Inversion: When and How

The aim of the present paper is to show how the Lagrange Inversion Formula (LIF) can be applied in a straight-forward way i) to find the generating function of many combinatorial sequences, ii) to extract the coefficients of a formal power series, iii) to compute combinatorial sums, and iv) to perform the inversion of combinatorial identities. Particular forms of the LIF are studied, in order to simplify the computation steps. Some examples are taken from the literature, but their proof is different from the usual, and others are new.      

Diffusive limit of a tagged particle in asymmetric simple exclusion processes

Invariance principles are proved under diffusive scaling for the centered position of a tagged particle in the simple exclusion process with asymmetric nonzero drift jump probabilities in dimensions d ≥ 3. The method of proof is by martin‐gale techniques which rely on the fact that symmetric random walks are transient in high dimensions. © 2000 John Wiley & Sons, Inc.     

哑运算的概率解释

利用随机变量的矩以及期望运算,给出了哑运算一种简单、自然的概率解释,并且得到了Abel恒等式的一个广泛哑运算证明.