Multinomial expansions and the pythagorean theorem

An addition formula for homogeneous polynomials is used to obtain a generalization of the Pythagorean theorem and a new view of the multinomial expansion.

     

A uniqueness theorem with moving targets without counting multiplicity

In this paper, we prove a uniqueness theorem for holomorphic curves with moving targets without counting multiplicity.

     

Counting problems relating to a theorem of Dirichlet

We study weighted versions of Dirichlet's theorem on the probability that two integers, taken at random, are relatively prime. This leads to a uniform approach in the study of several counting problems in discrete and computational geometry relating to incidences between points and lines.     

A local limit theorem for closed geodesics and homology

In this paper, we study the distribution of closed geodesics on a compact negatively curved manifold. We concentrate on geodesics lying in a prescribed homology class and, under certain conditions, obtain a local limit theorem to describe the asymptotic behaviour of the associated counting function as the homology class varies.

     

Sullivan's local Euler characteristic theorem

Using a certain cell decomposition of a closed neighborhood of a point a in a real analytic set A and the orientability modulo 2 of A ([1,3.7] or [5,7.3]), we obtain a short proof, by counting cells, of D. Sullivan's theorem ([9]) that X(A,A {a})) is odd.Research partially supported by NSF Grant GP29321.     

A central limit theorem for Ramanujan’s tau function

A central limit theorem is established for the absolute value of the modular Fourier-coefficient function defined by Ramanujan, and for that of the error term in the formula counting representations of integers as sums of twenty-four squares, in which the function appears.     

Upward Morley's theorem downward

By a celebrated theorem of Morley, a theory T is ?₁‐categorical if and only if it is κ‐categorical for all uncountable κ. In this paper we are taking the first steps towards extending Morley's categoricity theorem “to the finite”. In more detail, we are presenting conditions, implying that certain finite subsets of certain ?₁‐categorical T have at most one n‐element model for each natural number (counting up to isomorphism, of course).     

A Sturmian separation theorem for symplectic difference systems

We establish a Sturmian separation theorem for conjoined bases of 2n-dimensional symplectic difference systems. In particular, we show that the existence of a conjoined basis without focal points in a discrete interval (0,N+1] implies that any other conjoined basis has at most n focal points (counting multiplicities) in this interval.     

The algebra of set functions II: An enumerative analogue of Hall’s theorem for bipartite graphs

Triesch (1997) [25] conjectured that Hall’s classical theorem on matchings in bipartite graphs is a special case of a phenomenon of monotonicity for the number of matchings in such graphs. We prove this conjecture for all graphs with sufficiently many edges by deriving an explicit monotonic formula counting matchings in bipartite graphs.This formula follows from a general duality theory which we develop for counting matchings. Moreover, we make use of generating functions for set functions as introduced by Lass [20], and we show how they are useful for counting matchings in bipartite graphs in many different ways.     

The algebra of set functions I: The product theorem and duality

We give a comprehensive introduction to the algebra of set functions and its generating functions. This algebraic tool allows us to formulate and prove a product theorem for the enumeration of functions of many different kinds, in particular injective functions, surjective functions, matchings and colourings of the vertices of a hypergraph. Moreover, we develop a general duality theory for counting functions.     

A fixed point theorem including the last theorem of Poincaré

Poincaré's last theorem is the most famous among those theorems which are not subsumed by the Lefschetz fixed point theorem. A fixed point theorem is proved directly and constructively which in a special case reduces to the last theorem of Poincaré.     

Second main theorem and uniqueness theorem with moving targets on parabolic manifolds

In this paper, with the motivation from Diophantine approximation, a truncated second main theorem is established for meromorphic maps from M   into P(V)$\mathbb{P}(V)$ with moving targets _gj:M→P(V^?)$g_j:M\to \mathbb{P}(V^{?})$, 1≤j≤q$1\le j\le q$, where M is a parabolic manifold and V is a Hermitian vector space. As an application of this second main theorem, a uniqueness theorem without counting multiplicities is given.     

A minimax theorem

A minimax theorem is proved, under a continuity condition complementing the celebrated von Neumann–Sion's minimax theorem. The result is based on an intersection theorem which may be of interest on its own right.     

Hurwitz' theorem implies Rouché's theorem

It is well known that Hurwitz's theorem is easily proved from Rouché's theorem. We show that conversely, Rouché's theorem is readily proved from Hurwitz' theorem. Since Hurwitz' theorem is easily proved from the formula giving the number of roots of an analytic function, our result thus gives also a simple proof of Rouché's theorem.     

A stable theorem of the alternative: an extension of the gordan theorem

A theorem with a number of equivalent alternatives is proposed as an extension of the classical Gordan theorem of the alternative. The theorem can handle nonzero unrestricted variables which cannot be directly treated by ordinary theorems of the alternative. Like the Gordan theorem, the extended theorem has the stability feature that small perturbations in the data will not invalidate an alternative that is in force. The theorem has useful applications in establishing the boundedness and uniqueness of feasible points of polyhedral sets and of solutions to linear programming problems.     

内函数逼近定理及上溢原理的推广及应用

在κ-饱和的非标准模型中,首先推广了内函数逼近定理,并用这一推广定理证明了著名的A sco li定理;其次将上(下)溢原理推广到一般的定向集上,并证明了拓扑空间中单子的一些性质,给出了无穷小延伸定理的一个简单证明.     

On the Skitovich-Darmois theorem and Heyde theorem in a Banach space

According to the well-known Skitovich-Darmois theorem, the independence of two linear forms of independent random variables with nonzero coefficients implies that the random variables are Gaussian variables. This result was generalized by Krakowiak for random variables with values in a Banach space in the case where the coefficients of forms are continuous invertible operators. In the first part of the paper, we give a new proof of the Skitovich-Darmois theorem in a Banach space. Heyde proved another characterization theorem similar to the Skitovich-Darmois theorem, in which, instead of the independence of linear forms, it is supposed that the conditional distribution of one linear form is symmetric if the other form is fixed. In the second part of the paper, we prove an analog of the Heyde theorem in a Banach space. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1234–1242, September, 2008.     

Analytic proofs of a network feasibility theorem and a theorem of Fulkerson

We provide an elementary proff of Fulkerson's theorem which gives the permutation matrices as extreme points of a certain unbounded convex polyhedron. An adaptation of the proof also establishes an analogous feasibility theorem for network flows which has Fulkerson's theorem as a corollary.     

Splitting theorem, Poincaré-Hopf theorem and jumping nonlinear problems

In this paper, we establish a Gromoll-Meyer splitting theorem and a shifting theorem for J∈C^2-0(E,R) and by using the finite-dimensional approximation, mollifiers and Morse theory we generalize the Poincaré-Hopf theorem to J∈C¹(E,R) case. By combining the Poincaré-Hopf theorem and the splitting theorem, we study the existence of multiple solutions for jumping nonlinear elliptic equations.     

Noethers theorem

Summary We shall here generalise a geometrical version of Noethers theorem due to F. Severi. This is essentially done by replacing the notion of virtual multiplicity by that of «imaginary multiplicity» to be introduced below. The proof of this generalised Noethers theorem is based on the use of the resultant of the two curves in question.