A generalization of ekellandz’s variational principle

This paper presents a vector valued variational principle by using a general concept of ?-efficiency and a nonconvex separation theorem     

The local–global principle for symmetric determinantal representations of smooth plane curves

A smooth plane curve is said to admit a symmetric determinantal representation if it can be defined by the determinant of a symmetric matrix with entries in linear forms in three variables. We study the local–global principle for the existence of symmetric determinantal representations of smooth plane curves over a global field of characteristic different from two. When the degree of the plane curve is less than or equal to three, we relate the problem of finding symmetric determinantal representations to more familiar Diophantine problems on the Severi–Brauer varieties and mod 2 Galois representations, and prove that the local–global principle holds for conics and cubics. We also construct counterexamples to the local–global principle for quartics using the results of Mumford, Harris, and Shioda on theta characteristics.     

Further generalization of Fermat’s principle

From Maxwell’ s equations for electromagnetic fields, time-averaged energy flow density vector of stable monochromatic linearly polarized light in an isotropic insulative nonmagnetic medium is deduced. By the introduction of time-averaged energy flow density rays and the definition of new generalized refractive indexn _G1, Fermat’s principle of geometric optics is further generalized and its application conditions are discussed. The generalized Fermat' s principle can be used to describe stable transmission of light in a medium with variable refractive index. The necessary and sufficient conditions of a nondivergent and nonfocusing light beam are derived from this Fermat’s principle. Project supported by the National Natural Science Foundation of China (Grant No. 69789801) and Guangdong Provincial Natural Science Foundation of China.     

A generalization of Müller’s iteration method based on standard information

For finding a root of a function f, Müler’s method is a root-finding algorithm using three values of f in every step. The natural values available are values of f and values of its first number of derivatives, called standard information. Based on standard information, we construct an iteration method with maximal order of convergence. It is a natural generalization of Müller’s iteration method. This work was partially supported by National Natural Science Foundation of China (Grant No. 10471128), NSFC (Grant No. 10731060).     

A determinantal approach to Sheffer–Appell polynomials via monomiality principle

In this article, the Sheffer and Appell polynomials are combined to introduce the family of Sheffer–Appell polynomials by using operational methods. The determinantal definition and other properties of the Sheffer–Appell polynomials are established. As particular cases of these polynomials, the Sheffer–Bernoulli and Sheffer–Euler polynomials are introduced and their determinantal definitions are obtained. The operational correspondence between the Appell and Sheffer–Appell polynomials is used to derive the results for the Sheffer–Appell polynomials. Certain results for the Hermite–Appell and Laguerre–Appell polynomials are also obtained.     

A generalization of Fulton’s conjecture for arbitrary groups

We prove a generalization of Fulton??s conjecture which relates intersection theory on an arbitrary flag variety to invariant theory.     

A generalization of Poincaré's theorem for recurrence systems

A new proof and a genuine generalization to systems of first order equations is given from Poincaré classical theorem on ratio asymptotics of solutions of higher order recurrence equations. The asymptotic behavior of a fundamental system of solutions is obtained.     

A determinantal version of the frobenius-könig theorem

Let F be a field and let {d ₁,…,d_k } be a set of independent indeterminates over F. Let A(d ₁,…,d_k ) be an n × n matrix each of whose entries is an element of F or a sum of an element of F and one of the indeterminates in {d ₁,…,d_k }. We assume that no d ¹ appears twice in A(d ₁,…,d_k ). We show that if det A(d ₁,…,d_k ) = 0 then A(d ₁,…,d_k ) must contain an r × s submatrix B, with entries in F, so that r + s = n + p and rank B ? p ? 1: for some positive integer p.     

A generalization of Hardy’s operator and an asymptotic Müntz–Szász Theorem

We prove an asymptotic version of the Muntz–Szasz theorem, and use it to prove that all monomial operators are vanishing preserving.     

A generalization of Chebyshev’s inequality for Hilbert-space-valued random elements

We obtain a new generalization of Chebyshev’s inequality for random vectors. Then we extend this result to random elements taking values in a separable Hilbert space.     

A generalization of lebesgue’s convergence criterion for fourier series

For generalized Dirichlet integrals of type \(\int_0^1\ f(x)(e^{irx}-e^{airx}){dx\over x}\) a somewhat sharpened version of Lebesgue’s well-known convergence criterion for Fourier series is proven. Integrals of this type are of importance in the general theory of eigenfunction expansions inaugurated by investigations of Birkhoff, Tamarkin, and Stone.     

A generalization of Bitsadze–Samarskii problem for mixed type equation

We study a boundary-value problem with Bitsadze–Samarskii conditions on boundary characteristic on a special inner curve and on a segment of degeneration of mixed type equation. Its solvability is proved by method of integral equations, and uniqueness of solution is established by means of the maximum principle.     

A generalization of Ohnoʼs relation for multiple zeta values

In this paper, we prove that certain parametrized multiple series satisfy the same relation as Ohno?s relation for multiple zeta values. This result gives us a generalization of Ohno?s relation for multiple zeta values. By virtue of this generalization, we obtain a certain equivalence between the above relation among the parametrized multiple series and a subfamily of the relation. As applications of the above results, we obtain some results on multiple zeta values.     

A theorem of Michael Mürmann revisited

A fundamental theorem of Mürmann [2] characterizing equilibrium distributions of physical clusters is reconsidered. We recover this result by means of the integration by parts formula approach to Gibbs processes due to Nguyen Xuan Xanh and Hans Zessin [4]. Dedicated to Reinhard Lang on the occasion of his 60 th birthday.     

A reflection principle for real—Analytic submanifolds of complex spaces

We give an invariant nondegeneracy condition for CR-maps between generic submanifolds in different dimensions and use it to prove a reftection principle for these maps.