Pollaczek polynomials and hypergeometric representation

The Pfaff-Euler Transform for hypergeometric ₂ F ₁-series is applied to provide a direct and elementary proof that the hypergeometric representation with algebraic parameters of Pollaczek polynomials are indeed polynomials. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33C45; Secondary—33C05     

On the symmetric square Unstable twisted characters

We provide a purely local computation of the (elliptic) twisted (by “transpose-inverse”) character of the representationπ=I(1) of PGL(3) over ap-adic field induced from the trivial representation of the maximal parabolic subgroup. This computation is independent of the theory of the symmetric square lifting of [IV] of automorphic and admissible representations of SL(2) to PGL(3). It leads — see [FK] — to a proof of the (unstable) fundamental lemma in the theory of the symmetric square lifting, namely that corresponding spherical functions (on PGL(2) and PGL(3)) are matching: they have matching orbital integrals. The new case in [FK] is the unstable one. A direct local proof of the fundamental lemma is given in [V].     

New transformations for elliptic hypergeometric series on the root system A n

Recently, Kajihara gave a Bailey-type transformation relating basic hypergeometric series on the root system A _n , with different dimensions n. We give, with a new, elementary proof, an elliptic extension of this transformation. We also obtain further Bailey-type transformations as consequences of our result, some of which are new also in the case of basic and classical hypergeometric series. 2000 Mathematics Subject Classification Primary—33D67; Secondary—11F50     

Generalized hypergeometric,digamma and trigamma distributions

Summary In this paper we introduce and study new probability distributions named “digamma” and “trigamma” defined on the set of all positive integers. They are obtained as limits of the zero-truncated Type B3 generalized hypergeometric distributions (inverse Pólya-Eggenberger or negative binomial beta distributions), and also by compounding the logarithmic series distributions. The family of digamma distributions has the logarithmic series as a limit and the trigamma as another limit. The trigamma distributions are very close to the zeta (Zipf) distributions. Thus, our new distributions are useful as substitutes of the logarithmic series when the observed frequency data have such a long tail that cannot be fitted by the latter distributions. In the beginning sections we summarize properties of the Type B3 generalized hypergeometric distributions. It is emphasized that the distributions are obtained by compounding a Poisson distribution by “gamma product-ratio” distributions.     

A gould type integral with respect to a multisubmeasure

In [GOULD, G. G.: Integration over vector-valued measures, Proc. London Math. Soc. (3) 15, (1965), 193–205], G. G. Gould introduced a type of integral of a bounded, real valued function with respect to a finite additive set function taking values in a Banach space, integral which is more general than the Lebesgue one. Recently, A. Precupanu and A. Croitoru gave the generalization, defining a Gould type integral for multimeasures with values in _kc(X), X being a Banach space ([PRECUPANU, A.—CROITORU, A.: A Gould type integral with respect to a multimeasure, I, An. Ştiinţ. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. 48 (2002), 165–200]). Taking as starting point this work and [PRECUPANU, A.—CROITORU, A.: A Gould type integral with respect to a multimeasure, II, An. Ştiinţ. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. 49 (2003), 183–207], we define here the notion of a Gould type integral with respect to a _bf(X)-valued multisubmeasure, pointing out important properties of it. We also establish that, even if we deal with multisubmeasures, the integral is still a multimeasure.      

r-entropy,equipartition, and Ornstein’s isomorphism theorem inR n

A new approach is given to the entropy of a probability-preserving group action (in the context ofZ and ofR _n ), by defining an approximate “r-entropy”, 0<r<1, and lettingr → 0. If the usual entropy may be described as the growth rate of the number of essential names, then ther-entropy is the growth rate of the number of essential “groups of names” of width≦r, in an appropriate sense. The approach is especially useful for actions of continuous groups. We apply these techniques to state and prove a “second order” equipartition theorem forZ _m ×R _n and to give a “natural” proof of Ornstein’s isomorphism theorem for Bernoulli actions ofZ _m ×R _n , as well as a characterization of such actions which seems to be the appropriate generalization of “finitely determined”.     

Some linked subsets of posets

Recently Frank Tall has introduced the notion of a poset being “endowed” which is used in a forcing and reflection type proof to show that certain properties are preserved by forcing with such a poset. Subsequently Bill Fleissner introduced the property of a boolean algebra having a “lynx” and used this in an “Axiom and combinatorics” type proof of the same result. The two notions are basically the same thing. In this paper we prove that some of the usual Cohen algebras have lynxes, that it is consistent that nearly all of them have lynxes and investigate more generally the notion of lynxes. Research supported by NSERC of Canada.     

A hypergeometric approach,via linear forms involving logarithms,to criteria for irrationality of Euler’s constant

Using an integral of a hypergeometric function, we give necessary and sufficient conditions for irrationality of Euler’s constant γ. The proof is by reduction to known irrationality criteria for γ involving a Beukers-type double integral. We show that the hypergeometric and double integrals are equal by evaluating them. To do this, we introduce a construction of linear forms in 1, γ, and logarithms from Nesterenko-type series of rational functions. In the Appendix, S. Zlobin gives a change-of-variables proof that the series and the double integral are equal.      

A simple proof of the stability criterion of Gray and Griffeath

Summary Gray and Griffeath studied attractive nearest neighbor spin systems on the integers having “all 0's” and “all 1's” as traps. Using the contour method, they established a necessary and sufficient condition for the stability of the “all 1's” equilibrium under small perturbations. In this paper we use a renormalized site construction to give a much simpler proof. Our new approach can be used in many situations as a substitute for the contour method. Partially supported by a grant from the National Science Foundation Partially supported by the Army Research Office through the Mathematical Sciences Institute at Cornell University     

CALCULUS ON CANTOR TRIADIC SET （Ⅱ）—EXCEPTIONAL SET

In this paper the derivative and its exceptional set according to the alternatively jumping function and self-similar function respectively are discussed. And in the appendix, a proof of the statement 1.2(2) of “Calculus on Cantor triadic set (Ⅱ)—derivative” which is important to the discussion of exceptional set is given.     

Twisted convolution and Riesz means

We improve a result of Thangavelu on Riesz means associated with the twisted Laplacian. Askey—Wainger estimates for Laguerre functions are the main tools used. Research supported in part by KBN grants and by ECC under program “Fourier Analysis” contract no: CIPDCT 940001.     

Some curious q-series expansions and beta integral evaluations

We deduce several curious q-series expansions by applying inverse relations to certain identities for basic hypergeometric series. After rewriting some of these expansions in terms of q-integrals, we obtain, in the limit q→ 1, some curious beta-type integral evaluations which appear to be new. Dedicated to Dick Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—15A09, 33D15, 33E20; Secondary—05A30 M. Schlosser was fully supported by an APART fellowship of the Austrian Academy of Sciences.     

Two faces of mathematics lessons in Korea: conventional lessons and innovative lessons

This study identifies the characteristics of mathematics classrooms in Korea. First, conventional Korean mathematics lessons are analyzed from the perspective of “theory of variation”. Second, an innovative lesson for gifted children is reported in detail and analyzed from the perspective of “Lakatos’ proofs and refutations”. Third, the classroom characteristics identified in both the conventional lessons and the innovative lesson are interpreted in terms of the underlying cultural values that they share with other East Asian countries. The study concludes that although the two faces of Korean mathematics lessons look different, they may flow from the same “heart”—that of the common Confucian heritage culture culture, and in particular East Asian pragmatism.     

On the partial regularity of energy-minimizing, area-preserving maps

We prove a partial regularity assertion for a Lipschitz continuous mapping u in the plane that minimizes an appropriate convex (or quasiconvex) energy functional, under the “hard” constraint that det D u = 1 a.e. The primary technical assumption is that u be nondegenerate, meaning that, locally, at least one of its partial derivatives is bounded away from zero a.e. The method of proof is to convert to a related minimization problem for a generating function w, the advantage being that we now have the “soft” constraint . Received March 16, 1999 / Accepted April 23, 1999     

On the interior regularity of weak solutions to the non-stationary Stokes system

In this paper, we prove that any weak solution to the non-stationary Stokes system in 3D with right hand side -div f satisfying (1.4) below, belongs to C( ]0, T[; C ^α (Ω)). The proof is based on Campanato-type inequalities and the existence of a local pressure introduced in Wolf [13]. Proc. Conference “Variational analysis and PDE’s”. Intern. Centre “E. Majorana”, Erice, July 5–14, 2006.     

Askey–Wilson type integrals associated with root systems

A new formula for an Askey–Wilson type integral associated with the root system F ₄ is studied. A simple proof of the evaluation formula for the original Askey–Wilson integral is also stated. 2000 Mathematics Subject Classification Primary—33D67, 33D60 This work was supported in part by Grant-in-Aid for Scientific Research (C) No. 15540045 from the Ministry of Education, Culture, Sports, Science and Technology (Japan).     

A generalization of the Bombieri–Pila determinant method

The so-called determinant method was developed by Bombieri and Pila in 1989 for counting integral points of bounded height on affine plane curves. In this paper, we give a generalization of that method to varieties of higher dimension, yielding a proof of Heath-Brown’s “Theorem 14” by real-analytic considerations alone. Bibliography: 11 titles.     

Righting the early history of computing,or how sausage was made

The short answer to the question just posed seems to be, “Not much.” Since I have given “the long answer” elsewhere,¹² I can summarize it here. Berg could see no point in writing Bromley. What could he write to someone he believed guilty of plagiarism? What could such a letter accomplish? He did, however, write to New York University Press; to all the universities involved, and to the Works’ English publisher (Pickering and Chatto), who said they passed the letter on to Campbell-Kelly (30 June 1990); to a great many professional societies in Australia, England, and the United States; to a great many governmental agencies and some politicians in those countries; to some publications, both academic and popular; to the Pope and several cardinals; and to a miscellany of other individuals. Generally, those in the best position to do something—for example, the three universities involved —did not even answer Berg’s letter. Others often did answer, but their answer was generally that they were in no position to do anything. That was how matters stood when I published my first article on “the Berg Affair”.₁₂ Its publication finally roused those best positioned to answer. Late in 1993, Galler, Bromley, and Campbell-Kelly wrote letters to the editor of Accountability in Research criticizing me for not getting their side of the story before I published Berg’s. Campbell-Kelly threatened the journal’s publisher with a lawsuit if I (or it) did not retract. The three also provided some insight into what their explanation of events might be. Bromley, though listed prominently in ads for the Works, claimed to have had only a small part, merely advising Campbell-Kelly on selection and arrangement of the papers printed in Volumes 2 and 3. Campbell-Kelly confirmed that Bromley took no part in the detailed editing or in the provision of documents. That work was performed by one C.J.D. (“Jim”) Roberts, a “London-based independent scholar” who was “editorial consultant to the Works” (and, apparently, worked directly under Campbell-Kelly). Roberts seems to deserve more public credit than he has so far received. According to Campbell-Kelly, it was Roberts who, making a systematic search for unknown holdings of Babbage, turned up the original of the letter to Quetelet by writing the Royal Library (one “tiny triumph” among many). Campbell-Kelly also claimed that neither he nor Roberts knew of Berg’s prior discovery.     

If a finite extension of a Bernoulli shift has no finite rotation factors,it is Bernoulli

We show here that a finite extension of a Bernoulli shift either has a finite rotation factor or is Bernoulli. The proof lifts to this more general case the “nesting” technique we used previously to prove this for two point extensions. This work was supported by the Miller Institute for Basic Research.     

A Variational Proof of Alexandrov’s Convex Cap Theorem

We give a variational proof of the existence and uniqueness of a convex cap with the given metric on the boundary. The proof uses the concavity of the total scalar curvature functional (also called Hilbert-Einstein functional) on the space of generalized convex caps. As a by-product, we prove that generalized convex caps with the fixed metric on the boundary are globally rigid, that is uniquely determined by their curvatures. Research for this article was supported by the DFG Research Unit 565 “Polyhedral Surfaces”.