On the estimates of the order of finitelya k -determined map-germs

In this paper we deal with the estimates of the order of determinacy of finitely determined map-germs. This problem has been noted by mathematicians for more than ten years. But the results given by some authors for the estimates of the order of determinacy are stated with respect toa, while those given in this paper are stated with respect toa _k , wherek may be any non-negative integer, and we havea ₀=a in particular. This paper was completed during the author's visit to the Institute of Mathematics, Now I'd like to express my heart-felt thanks to the Institute for its financial support and Prof. Li Peixin for his instruction.     

Simultaneous dominance representation of multiple posets

We characterize the codominance pairs-pairs of posets that admit simultaneous dominance representations in the (x, y)-and (–x, y)-coordinate systems-and present a linear algorithm to recognize them and construct codominance representations. We define dominance polysemy as a generalization of codominance and describe several related problems and preliminary results.This author's research, supported in part by NSF grant CCR-9300079, also appears in his doctoral thesis [15], written, at the Johns Hopkins University under the supervision of Professors Edward R. Scheinerman and Michael T. Goodrich.This author's research, supported in part by an NSERC operating grant and an FCAR team grant, was performed in part during a visit at INRIA Sophia Antipolis.     

Singular integrals with rough kernels along real-analytic submanifolds in R n

L ^p mapping properties will be established in this paper for singular Radon transforms with rough kernels defined by translated of a real-analytic submanifold in R ⁿ .Work in this paper was done during the second author's visit at the Department of Mathematics, University of Pittsburgh.Supported in part by NSF Grant DMS-9622979.     

On perfectt-shift codes in abelian groups

LetG be a finite abelian group,t a positive integer. Thet-shift sphere with centerx G is the setS _t (x)={±ix|i=1,...,t}. At-shift code is a subsetX ofG such that the setsS _t (x) (x X) have size 2t and are disjoint. Clearly, the sphere packing bound: 2t|X|+1|G| holds for anyt-shift codeX. Aperfect t-shift code is at-shift codeX with 2t|X|+1=|G|. A necessary and sufficient condition for the existence of a perfectt-shift code in a finite abelian group is known fort-1, 2. In this paper, we determine finite abelian groups in which there exists a perfectt-shift code fort=3, 4.This research was completed during the author's visit at the Institute for System Analysis, Moscow, as a Heizaemon Honda fellow of the Japan Association for Mathematical Sciences.     

Exit times from cones in ℝ n of Brownian motion

Summary For a fairly general class of cones inn dimensions (n3) we determine the corresponding distributions of Brownian first exit times. Asymptotic results may then be read off.This paper is a generalization of the author's Ph.D. dissertation completed in May 1984 at the Massachusetts Institute of Technology under the supervision of Professor R.M. DudleyThis research was supported in part by NSF grant DMS-8301367     

Invariant curves of one-step methods

This paper considers the invariant sets of numerical one-step integration methods in a neighbourhood of a hyperbolic periodic solution of a nonlinear ODE. Using results from the dynamical systems theory it was possible to show that for the usual one-step methods the invariant sets areC ^k-circles (closed curves) for small enough stepsizeh. Here we give a direct proof for that and also show that they areO(h ^p)C ^k-close to the true periodic trajectory, wherep is the order of the method.Most of this work was done during the author's visit at the Mittag-Leffler Institute, Djursholm, Sweden, academic year 1985–86.     

Clebsch-Gordan coefficients for the quantum algebrasu q (2)

A simple method for studying the Clebsch-Gordan coefficients for the algebrasu _q (2) is discussed.Institute of Physics, Academy of Sciences of Azerbaidzhan; I. V. Kurchatov Institute of Atomic Energy. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98, No. 1, pp. 3–11, January, 1994.     

Harnack inequality for some classes of Markov processes

In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps. Mathematics Subject Classification (2000): Primary 60J45, 60J75, Secondary 60J25.This work was completed while the authors were in the Research in Pairs program at the Mathematisches Forschungsinstitut Oberwolfach. We thank the Institute for the hospitality.The research of this author is supported in part by NSF Grant DMS-9803240.The research of this author is supported in part by MZT grant 0037107 of the Republic of Croatia.     

On totally positive units of real holomorphy rings

In this paper we will show that every totally positive unit of the real holomorphy ring of a formally real field is a sum of 2n-th powers of totally positive units for all natural numbersn. Moreover, in the casen=1 we give a bound on the number of summands required in such a representation. This research was supported by the Alexander-von-Humboldt-Foundation and carried out during the author's stay at the Institute for Advanced Studies at the Hebrew University of Jerusalem.     

On a Decomposition of Complete Graphs

It is shown, among other results, that for any prime power q, the complete graph on 1+q+q ²+q ³ vertices can be decomposed into a union of 1+q Siamese Strongly Regular Graphs S R G(1+q+q ²+q ³,q+q ²,q–1,q+1) sharing 1+q ² cliques of size 1+q. Acknowledgments.The authors are indebted to a referee for a very extensive report and for many suggestions which improved the presentation of the paper tremendously.AMS Subject Numbers: 05B05, 05B20, 05E30This work was completed while the first author was on sabbatical leave visiting Institute for studies in theoretical Physics and Mathematics, (IPM), in Tehran, Iran. Support and hospitality is appreciated. Supported by an NSERC operating grant.     

Interpolation, correlation identities, and inequalities for infinitely divisible variables

We present an interpolation formula for the expectation of functions of infinitely divisible (i.d.) variables. This is then applied to study the association problem for i.d. vectors and to present new covariance expansions and correlation inequalities. Acknowledgements and Notes. The research of C. Houdré was supported in part by an NSF Mathematical Sciences Post-Doctoral Fellowship and by an NSF-NATO Postdoctoral Fellowship and by the NSF grant No. DMS-98032039. This research was completed while V. Pérez-Abreu was visiting the Georgia Institute of Technology.     

Representations of the compact quantum groupSU q(2) and geometrical quantization

The Souriau—Kostant method of geometrical quantization is used to construct infinite-dimensional irreducible unitary representations of the algebra of functions of the compact quantum groupSU _q(2). The generalization to the case of the quantum groupSU _q (n) is discussed.V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 100, No. 2, pp. 163–172, August, 1994.     

The Dvoretzky-Wald-Wolfowitz theorem and purification in atomless finite-action games

In 1951, Dvoretzky, Wald and Wolfowitz (henceforth DWW) showed that corresponding to any mixed strategy into a finite action space, there exists a pure-strategy with an identical integral with respect to a finite set of atomless measures. DWW used their theorem for purification: the elimination of randomness in statistical decision procedures and in zero-sum two-person games. In this short essay, we apply a consequence of their theorem to a finite-action setting of finite games with incomplete and private information, as well as to that of large games. In addition to simplified proofs and conceptual clarifications, the unification of results offered here re-emphasizes the close connection between statistical decision theory and the theory of games.A first draft of this paper was completed when Khan and Rath were visiting the Institute for Mathematical Sciences at the National University of Singapore in August 2003; they thank the Institute for supporting their visit. A preliminary version was presented at the Midwest Economic Theory Conference held at Indiana University, Bloomington in October 2003; the authors are grateful to Eric Balder, Robert Becker, William Thomson and Myrna Wooders for questions and helpful suggestions.     

Supersimplicity and quadratic extensions

An elliptic curve over a supersimple field with exactly one extension of degree 2 has an s-generic point. A. Martin-Pizarro would like to express his gratitude to the organisation of the Florida Logic year, in which he took part. Martin-Pizarro research was supported by a DFG-Forschungsstipendium MA3310/1-1. F. O. Wagner is a Membre junior de l’Institut Universitaire de France. Work done during the semester on Model Theory and Applications in Algebra and Analysis at the Isaac Newton Institute for the Mathematical Sciences, whose hospitality is gratefully acknowledged.     

On the value of a logarithmic-trigonometric integral

A closed expression is derived for the integral ₀ ^/2 log ⁿ cosxlog ^p sinxdx, wheren andp are non-negative integers. As already remarked by Nielsen in a monograph on the generalized polylogarithms published early in this century, this integral is equal to times a homogeneous polynomial in (q) (the Riemann zeta function for integer arguments) and log 2, with rational coefficients. Explicit expressions for the integral are given for 0<n4, 0p4, most of which have been found from the general formula by means of a computer.Part of this work was done during the author's leave of absence at the Institute of Theoretical Physics at McGill University, Montreal, Quebec, Canada. This work was supported in part by the National Research Council of Canada.     

N-person differential games governed by infinite-dimensional systems

This paper discussesN-person differential games governed by infinite-dimensional systems. The minimax principle, which is a necessary condition for the existence of open-loop equilibrium strategies, is proved. For linear-quadraticN-person differential games, global necessary and sufficient conditions for the existence of open-loop and closed-loop equilibrium strategies are derived.This work was supported by the Science Fund of the Chinese Academy of Sciences and the Research Foundation of Purdue University.The problems discussed in this paper were proposed by Professor G. Chen, during the author's visit to Pensylvania State University, and were completed at Purdue University. The author would like to thank Professors L. D. Berkovitz and G. Chen for their hospitality.     

Exact calculability,semigroup of representations,and the stability property for representations of the algebra of functions on the quantum groupSU q(2)

An operation of the coproduct of representations of a bialgebra is defined. The coproduct operation of representations for the Hopf algebra of functions on the quantum groupSU _q (2) is investigated. For this Hopf algebra a representation II called the stable representation is constructed. This representation is invariant with respect to the coproduct with an arbitrary representation. An expression for the trace in the representation II is derived. The invariant Voronovich integal inSU _q (2) takes the form fdµ=tr(fcc ^*).V. A. Steklov Mathematics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 2, pp. 163–178, November, 1994.     

Notes on binary codes related to the O(5, q) generalized quadrangle for odd q

In this note we determine the dimensions of the binary codes spanned by the lines or by the point neighborhoods in the generalized quadrangle Sp(4, q) and its dual O(5, q), where q is odd. Several more general results are given. As a side result we find that if a square generalized quadrangle of odd order has an antiregular point, then all of its points are antiregular.On leave from the Indian Statistical Institute, Calcutta; research supported by a grant from NWO.     

On the code generated by the incidence matrix of points and hyperplanes in <Emphasis Type="Italic">PG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">q</Emphasis>) and its dual

In this paper, we study the p-ary linear code C(PG(n,q)), q = p ^h , p prime, h ≥ 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n,q), and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n,q) and we exclude all possible codewords arising from small linear blocking sets. We also look at the dual code of C(PG(n,q)) and we prove that finding the minimum weight of the dual code can be reduced to finding the minimum weight of the dual code of points and lines in PG(2,q). We present an improved upper bound on this minimum weight and we show that we can drop the divisibility condition on the weight of the codewords in Sachar’s lower bound (Geom Dedicata 8:407–415, 1979). G. Van de Voorde’s research was supported by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen).     

On subsequences of the Haar system inL p [0, 1], (1<p<∞)

We show that ifX is the closed linear span inL _p [0,1] of a subsequence of the Haar system, thenX is isomorphic either tol _p or toL _p [0,1], [1<p<∞]. We give criteria to determine which of these cases holds; for a given subsequence, this is independent ofp. This is part of the second author's Ph.D. dissertation, written at the University of Alberta under the supervision of J. L. B. Galmen. The first author's research was partially supported by NRC A7552.