Classroom note: Evaluating a class of series using Taylor's theorem

A class of infinite series is evaluated with the aid of Taylor's theorem and a comparison is made with other methods.     

Weyl's theorem in several variables

In this note we consider Weyl's theorem and Browder's theorem in several variables. The main result is as follows. Let T be a doubly commuting n-tuple of hyponormal operators acting on a complex Hilbert space. If T has the quasitriangular property, i.e., the dimension of the left cohomology for the Koszul complex Λ(T−λ) is greater than or equal to the dimension of the right cohomology for Λ(T−λ) for all λ∈Cn, then ‘Weyl's theorem’ holds for T, i.e., the complement in the Taylor spectrum of the Taylor Weyl spectrum coincides with the isolated joint eigenvalues of finite multiplicity.     

A Herbrand-Ribet theorem for function fields

We prove a function field analogue of the Herbrand-Ribet theorem on cyclotomic number fields. The Herbrand-Ribet theorem can be interpreted as a result about cohomology with μ _p -coefficients over the splitting field of μ _p , and in our analogue both occurrences of μ _p are replaced with the \mathfrakp\mathfrak{p}-torsion scheme of the Carlitz module for a prime \mathfrakp\mathfrak{p} in F _q [t].     

Interpolation theorem for a continuous function on orientations of a simple graph

Let G be a simple graph. A function f from the set of orientations of G to the set of non-negative integers is called a continuous function on orientations of G if, for any two orientations O ₁ and O ₂ of G, |f(O ₁)-f(O ₂)| 1 whenever O ₁ and O ₂ differ in the orientation of exactly one edge of G. We show that any continuous function on orientations of a simple graph G has the interpolation property as follows:If there are two orientations O ₁ and O ₂ of G with f(O ₁) = p and f(O ₂) = q, where p<q, then for any integer k such that p < k < q, there are at least m orientations O of G satisfying f(O) = k, where m equals the number of edges of G.It follows that some useful invariants of digraphs including the connectivity, the arc-connectivity and the absorption number, etc., have the above interpolation property on the set of all orientations of G.     

A uniqueness theorem for meromorphic mappings in several complex variables

In this paper, we give a uniqueness theorem for meromorphic mappings from ?n into ?N(?) with rank≥μregardless of multiplicities.     

A filled function method for finding a global minimizer of a function of several variables

The concept of a filled function is introduced. We construct a particular filled function and analyze its properties. An algorithm for global minimization is generated based on the concept and properties of the filled function. Some typical examples with 1 to 10 variables are tested and computational results show that in most cases this algorithm works better than the tunneling algorithm. The advantages and disadvantages are analyzed and further research directions are discussed.     

Lusin's theorem for derivatives with respect to a continuous function

For a nowhere constant continuous function on a real interval and for a Borel measure on , we give simple necessary and sufficient conditions guaranteeing, for any Borel function on , the existence of a continuous function on such that the derivative of with respect to is, almost everywhere, equal to .

     

Global maximum seeking algorithm for a function of several variables

Translated from Vychislitel'nye Kompleksy i Modelirovanie Slozhnykh Sistem, pp. 124–134, Moscow State University, 1989.     

The second main theorem for algebroid functions of several complex variables

The work of the first author was partially supported by the National Natural Science Foundation of China and the second one's by U.P.G.C. of Hong Kong     

A boundary uniqueness theorem for holomorphic functions of several complex variables

If D ? Cⁿ is a region with a smooth boundary and M ? ?D is a smooth manifold such that for some point p ∈ M the complex linear hull of the tangent plane T_p(M) coincides with Cⁿ, then for each functionf ε A(D) the conditionf¦M=0 implies thatf=0 in D.     

Bohr's power series theorem in several variables

Generalizing a classical one-variable theorem of Bohr, we show that if an -variable power series has modulus less than in the unit polydisc, then the sum of the moduli of the terms is less than in the polydisc of radius .

     

The Bloch theorem in several complex variables

The background theory for the Bloch theorem is generalized to several complex variables. This work involves study of the Bergman kernel functions in order to extend work of Landau and Bonk. The main conclusion is an estimate for Bloch’s constant for mappings of domains of the first classical type. In the special case of then-dimensional ball, the estimate of Bloch’s constant coincides with that of Liu.     

An extension of flanders theorem to several matrices

The Flanders Theorem relates the matrices AB and BA and provides a necessary and sufficient condition for the consistency of the matrix system P = AB Q = BA In this paper, we generalize the Flanders condition for several matrices.     

An extension of flanders theorem to several matrices

The Flanders Theorem relates the matrices AB and BA and provides a necessary and sufficient condition for the consistency of the matrix system P = ABQ = BA In this paper, we generalize the Flanders condition for several matrices.