On the Glivenko-Cantelli theorem

Summary Various generalizations of the classical Glivenko-Cantelli theorem are proved. In particular, we have strived for as general results as possible for theoretical distributions on euclidean spaces, which are absolutely continuous with respect to Lebesgue measure.     

On a generalization of the Gallai-Sylvester theorem

It is shown that ifS ^d , affS=aff ^d , and every hyperplane spanned by (a subset of)S misses fewer thank points ofS(k2), then (a) #Skm ifd=2m–1 is odd and (b) #Skm+1 ifd=2m is even. We also fully describe the extreme sets for which equality holds in (a) or in (b). For oddd the proofs are later modified to purely algebraic ones, and carry over to , where is an arbitrary field. For evend, (b) is generally not true when , but we prove some weaker inequalities that do hold over arbitrary fields.This is part of a Ph.D. thesis, supervised by Professor Micha A. Perles at the Hebrew University of Jerusalem. This research was supported in part by the Landau Center for Mathematical Research.     

On a generalization of the Keldysh theorem

The Keldysh theorem for an elliptic equation with characteristic parabolic degeneration is generalized for the case of an elliptic equation of the second-order canonical form with order and type degeneration. The criteria under which the Dirichlet or Keldysh problems are correct are given in a one-sided neighborhood of the degeneration segment, enabling one to write the criteria in a single form. Moreover, some cases are pointed out in which it is even nessesary to give a criterion in the neighborhood because it is impossible to establish it on the segment of degeneracy of the equation.     

On a generalization of Rubin's theorem

The work is devoted to the calculation of asymptotic value of the choice number of the complete r‐partite graph K_{m* r} = K_{m, …, m} with equal part size m. We obtained the asymptotics in the case lnr = o(lnm). The proof generalizes the classical result of A.L. Rubin for the case r = 2. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 226–234, 2011 67: 226‐234, 2011     

On a generalization of a theorem of Lyapunov

A constructive and simple proof is given of a generalization of a result of Lyapunov on the convexity of the range of vector measure.     

On generalization of Menshoff's theorem

The condition providing the analyticity for continuous locally univalent functions of complex variables is established. This generalizes the classical Menshoff result on homeomorphic mappings preserving infinitestimal circles.     

On a generalization of Schur theorem concerning resultants

Periodica Mathematica Hungarica - Let K be a field and put $${mathcal {A}}:={(i,j,k,m)in mathbb {N}^{4}:;ile j;text{ and };mle k}$$ . For any given $$Ain {mathcal {A}}$$ we consider...     

On a generalization of Albert’s theorem

It is proved by A. A. Albert that in an ordered division ring, any element algebraic over the center is central. In this paper, we shall investigate the following problem. LetD be an ordered division ring. Suppose every element inD is left algebraic over a maximal subfieldK. Does it follow thatD=K? We prove that the answers are affirmative in some cases.     

On a generalization of a theorem of B. Segre

An extension is given of Segre's generalization of Menelaus' theorem to an arbitrary collection of lines.     

On a generalization of a theorem of B. tyler

LetB be an arbitrary normal matrix, satisfying some conditions. AbsoluteB-summability factors in a sequence for Cesàro methodC ^α if α≧1 or α=0 and absolute convergence factors in a sequence forC ^α if 0<α<1 are obtained.     

On one generalization of a DiPerna and Majda theorem

The weak limits of sequences {f(u^ν)}_ν∈? where u^ν's are vector‐valued µ‐measurable functions defined on a compact set Ω and f is (possibly) discontinuous are investigated. As shown by the author (J. Conv. Anal. (to appear)), they are described in terms of integral formulae involving parametrized measures independent of f, similarly as in the classical theorem by Young and its generalization due to DiPerna and Majda. In the present paper we describe the supports of the involved parametrized measures. Copyright © 2006 John Wiley & Sons, Ltd.     

On a metric generalization of ramsey’s theorem

An increasing sequence of realsx=〈x _i :i<ω〉 is simple if all “gaps”x _i +1−x _i are different. Two simple sequencesx andy are distance similar ifx _i +1−x _i <x _j +1−x _j if and only ify _i +1−y _i <y _j +1−y _j for alli andj. Given any bounded simple sequencex and any coloring of the pairs of rational numbers by a finite number of colors, we prove that there is a sequencey distance similar tox all of whose pairs are of the same color. We also consider many related problems and generalizations. Partially supported by OTKA-4269. Partially supported by NSF grant STC-91-19999. Partially supported by OTKA-T-020914, NSF grant CCR-9424398 and PSC-CUNY Research Award 663472.     

On a generalization of Jentzsch’s theorem

Let E be a compact subset of with connected, regular complement and let G(z) denote Green’s function of Ω with pole at ∞. For a sequence (p_n)_nΛ of polynomials with degp_n=n, we investigate the value-distribution of p_n in a neighbourhood U of a boundary point z₀ of E if G(z) is an exact harmonic majorant of the subharmonic functions

in . The result holds for partial sums of power series, best polynomial approximations, maximally convergent polynomials and can be extended to rational functions with a bounded number of poles.