A bound for the theorem of de Franchis

We give a bound on the number of nonconstant holomorphic maps between compact Riemann surfaces of genera .

     

Some remarks on the representation theorem of moisil

In this paper, using the representation theorem of Moisil (see [2]) the author introduces and examines the concept of representability of Lukasiewicz algebras. The results and notations used are from [1], [2].     

Some remarks on the folk theorem in game theory

It is argued that although the pathological multiplicity of Nash equilibria of super games stated by the folk theorem can be removed by introducing limited observations into super games with a continuum of players, the consideration of super games in terms of the Nash equilibrium concept involves a more fundamental and conceptual difficulty.     

Some remarks on a three critical points theorem

Some remarks on a strict minimax inequality, which plays a fundamental role in Ricceri's three critical points theorem, are presented. As a consequence, some recent applications of Ricceri's theorem to nonlinear boundary value problems are revisited by obtaining more precise conclusions.     

Some remarks on Rademacher's theorem in infinite dimensions

We provide an infinite dimensional version of Rademacher's theorem in a linear space provided with a bounded Radon measure . The underlying concepts of the Lipschitz property and differentiability hold -almost everywhere and only in the linear subspace of directions along which is quasiinvariant. The particular case where (X, ) is the Wiener space (and for which the subspace of quasiinvariance coincides with the Cameron-Martin space) was proved in Enchev and Stroock (1993).Partially supported by the fund for promotion of research at the Technion.     

Some remarks on a theorem on sets with connected sections

In this paper we give a complete answer to a question which naturally arises in comparing classical results of F. E. Browder and K. Fan on sets with convex sections with a recent result of B. Ricceri on sets with connected sections.     

Some remarks on Vinogradov's mean value theorem and Tarry's problem

LetW(k, 2) denote the, least numbers for which the system of equations has a solution with . We show that for largek one hasW(k, 2)1/2k ²(logk+loglogk+O(1)), and moreover that whenK is large, one hasW(k, 2)1/2k(k+1)+1 for at least one valuek in the interval [K, K ^3/4+]. We show also that the leasts for which the expected asymptotic formula holds for the number of solutions of the above system of equations, inside a box, satisfiessk ²(logk+O(loglogk).Research supported in part by NSF grant DMS-9303505, an Alfred P. Sloan Research Fellowship, and a Fellowship from the David and Lucile Packard Foundation.     

Some remarks on a probability limit theorem for continued fractions

It is shown that if a certain condition on the variances of the partial sums is satisfied then a theorem of Philipp and Stout, which implies the asymptotic fluctuation results known for independent random variables, can be applied to some quantities related to continued fractions. Previous results on the behavior of the approximation by the continued fraction convergents to a random real number are improved.

     

Some remarks concerning the individual ergodic theorem of information theory

Let (X,μ, T) be an ergodic dynamic system and let ξ = (C₁, C₂, ...) be a discrete decomposition of X. Conditions are considered for the existence almost everywhere of $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\left| {\log \mu (C_{\xi ^n } (x))} \right|,$$ whereC _ξn(x) is the element of the decomposition ξⁿ = ξ V T ξ V ... < T^n-1ξ containing x. It is proved that the condition H(ξ) < ∞ is close to being necessary. If T is a Markov automorphism and ξ is the decomposition into states, then the limit exists, even if H(ξ) = ∞, and is equal to the entropy of the chain.