Courbure de Ricci et géométrie conforme

Résumé On connaÎt l'intérÊt porté sur les liaisons entre courbure de Ricci et géométrie conforme d'une variété riemannienne. Ici on étudie une fonctionnelle attachée aux déformations conformes du tenseur d'Einstein d'une variété riemannienne compacte (M, g) de dimension n > 2, en obtenant certains résultats globaux concernant la géométrie et la topologie de (M, g), en particulier une condition pour que (M, g) soit isométrique à une sphère euclidienne. On étudie, ensuite, la courbure de Ricci du fibré tangent en cercles T_cV d'une variété riemannienne (V, g) de dimension 2, fibré muni de la métrique g^s de Sasaki, en obtenant, d'une part, des resultats, soit locaux soit globaux, sur la géométrie conforme de (T_cV, g^s) et, d'autre part, des conditions d'isométrie, soit de (V, g) soit de (T_cV, g^s), à certaines variétés standard. Certains des résultats de ce travail ont été annoncés dans une note aux Comptes Rendus de l'Académie des Sciences de Paris.Travail exécuté d'après le programme du «Gruppo Nazionale per le strutture algebriche e geometriche e loro Applicazioni, C.N.R. Italia».Alla memoria della mia adorata madre     

On statistical convergence of vector-valued sequences associated with multiplier sequences

We introduce vector-valued sequence spaces w _∞(F, Q, p, u), w ₁(F, Q, p, u), w ₀(F, Q, p, u), S _u ^q , and S _0u ^q using a sequence of modulus functions and a multiplier sequence u = (u _k ) of nonzero complex numbers. We give some relations for these sequence spaces. It is also shown that if a sequence is strongly u _q -Cesàro summable with respect to the modulus function, then it is u _q -statistically convergent. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 125–131, January, 2006.     

The diophantine equation x 2 + 2 a · 17 b = y n

Let ?, ? be the sets of all integers and positive integers, respectively. Let p be a fixed odd prime. Recently, there have been many papers concerned with solutions (x, y, n, a, b) of the equation x ² + 2 ^a p ^b = y ⁿ , x, y, n ε ?, gcd(x, y) = 1, n ? 3, a, b ε ?, a ? 0, b ? 0. And all solutions of it have been determined for the cases p = 3, p = 5, p = 11 and p = 13. In this paper, we mainly concentrate on the case p = 3, and using certain recent results on exponential diophantine equations including the famous Catalan equation, all solutions (x, y, n, a, b) of the equation x ²+2 ^a · 17 ^b = y ⁿ , x, y, n ε ?, gcd(x, y) = 1, n ? 3, a, b ∈ ?, a ? 0, b ? 0, are determined.     

Graphic sequences with a realization containing a generalized friendship graph

Gould, Jacobson and Lehel [R.J. Gould, M.S. Jacobson, J. Lehel, Potentially G-graphical degree sequences, in: Y. Alavi, et al. (Eds.), Combinatorics, Graph Theory and Algorithms, vol. I, New Issues Press, Kalamazoo, MI, 1999, pp. 451-460] considered a variation of the classical Turán-type extremal problems as follows: for any simple graph H, determine the smallest even integer σ(H,n) such that every n-term graphic sequence π=(d₁,d₂,…,d_n) with term sum σ(π)=d₁+d₂+?+d_n≥σ(H,n) has a realization G containing H as a subgraph. Let F_t,r,k denote the generalized friendship graph on kt−kr+r vertices, that is, the graph of k copies of K_t meeting in a common r set, where K_t is the complete graph on t vertices and 0≤r≤t. In this paper, we determine σ(F_t,r,k,n) for k≥2, t≥3, 1≤r≤t−2 and n sufficiently large.     

On vectorial addition chains

We prove the equivalence between the problem of computing a multinominal x₁^n₁·x₂^n₂···x_k^n_k, posed by R. E. Bellman (Amer. Math. Monthly70 1963, 765), and the problem of computing simultaneously the monomials x^n₁,…, x^n_k, posed by D. E. Knuth (“Seminumerical Algorithms”, Sect. 4.6.3, Addison-Wesley, Reading, Mass 1969.     

Hauptprobleme der Unternehmensforschung

Zusammenfassung Die Haupt — Problemkreise der Unternehmensforschung, nämlich Lagerhaltungsprobleme, Zuordnungs- oder Zuteilungsprobleme, Wartezeit — Probleme, Ersatzstellungs- und Wartungsprobleme, sowie Konkurrenz-probleme werden besprochen und ihre Lösungsmöglichkeiten kurz skizziert. Dabei wird die Bedeutung elektronischer Datenverarbeitungsmaschinen zur Lösung solcher Probleme besonders hervorgehoben.

---

Summary Operations research deals with five main groups of problems, namely Inventory problems, Allocations problems, Waiting-time problems, Replacement problems, as well as with Competitive problems. The nature and the means of solutions are explained. Special emphasis is put on the fact that electronic data processing machines are first of all suited to handle these problems.

---

Résumé Les Problèmes de la recherche opérationelle peuvent être attribués à cinq classes principales de problèmes, c'est-à-dire des problèmes de stock, des problèmes de répartition, des problèmes d'attente, des problèmes de replacement, et enfin des problèmes de concurrence. La nature et les moyens de solution en sont expliqués, et une importance spéciale est attribuée au fait que ces problèmes peuvent, en pratique, aisément être résolus par le moyen de machines à calcul éléctroniques. Il en résulte la nécessité croissante de donner aux économistes des cours de programmation de ces machines.

     

Nonperiodicity implies unique composition for self-similar translationally finite Tilings

Let be a translationally finite self-similar tiling of R ^d . We prove that if is nonperiodic, then it has the unique composition property. More generally, has the unique composition property modulo the group of its translation symmetries. <lsiheader> <onlinepub>7 August, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>20n2p265.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>no <sectionname> </lsiheader> Received August 7, 1996, and in revised form December 18, 1996.     

The Basis Number of Some Special Non-Planar Graphs

The basis number of a graph G was defined by Schmeichel to be the least integer h such that G has an h-fold basis for its cycle space. He proved that for m, n 5, the basis number b(K _m,n ) of the complete bipartite graph K _m,n is equal to 4 except for K _6,10, K _5,n and K _6,n with n = 5, 6, 7, 8. We determine the basis number of some particular non-planar graphs such as K _5,n and K _6,n , n = 5, 6, 7, 8, and r-cages for r = 5, 6, 7, 8, and the Robertson graph.     

An Inequality on the Size of a Set in a Cartesian Product

Let Ω be a finite subset of the Cartesian productW₁  ×   × W_nof n sets. ForA    {1, 2, , n }, denote by Ω_Athe projection ofΩ onto the Cartesian product of W_i, i   A. Generalizing an inequality given in an article by Shen, we prove that | Ω |² ≤  |Ω_A1 || Ω_Ak| provided that { A₁, , A_k} is a double cover of {1, 2, , n }. This inequality is applied to give some bounds on the numbers of special subgraphs of a graph.     

Prime and composite integers close to powers of a number

We prove that, for any real numbers ξ ≠ 0 and ν, the sequence of integer parts [ξ2 ⁿ  + ν], n = 0, 1, 2, . . . , contains infinitely many composite numbers. Moreover, if the number ξ is irrational, then the above sequence contains infinitely many elements divisible by 2 or 3. The same holds for the sequence [ξ( − 2) ⁿ  + ν _n ], n = 0, 1, 2, . . . , where ν ₀, ν ₁, ν ₂, . . . all lie in a half open real interval of length 1/3. For this, we show that if a sequence of integers x ₁, x ₂, x ₃, . . . satisfies the recurrence relation x _n+d  = cx _n  + F(x _n+1, . . . , x _n+d-1) for each n  ≥  1, where c ≠ 0 is an integer, F(z₁,...,z_d-1) ? \mathbb Z[z₁,...,z_d-1],{F(z_1,\dots,z_{d-1}) \in \mathbb {Z}[z_1,\dots,z_{d-1}],} and lim _{n→ ∞}|x _n | = ∞, then the number |x _n | is composite for infinitely many positive integers n. The proofs involve techniques from number theory, linear algebra, combinatorics on words and some kind of symbolic computation modulo 3.     

On nonrational divisors over non-Gorenstein terminal singularities

Let (X, o) be a germ of a 3-dimensional terminal singularity of index m ≥ 2. If (X, o) has type cAx/4, cD/3-3, cD/2-2, or cE/2, then we assume that the standard equation of X in ℂ⁴/ℤ _m is nondegenerate with respect to its Newton diagram. Let π: Y → X be a resolution. We show that there are at most 2 nonrational divisors E _i , i = 1, 2, on Y such that π(E _i ) = o and the discrepancy a(E _i , X) is at most 1. When such divisors exist, we describe them as exceptional divisors of certain blowups of (X, o) and study their birational type. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 169–184, 2005.     

On the Asymptotics of the Meixner—Pollaczek Polynomials and Their Zeros

An infinite asymptotic expansion is derived for the Meixner—Pollaczek polynomials M _n (nα;δ, η) as n→∞ , which holds uniformly for -M≤α≤ M , where M can be any positive number. This expansion involves the parabolic cylinder function and its derivative. If α _{n, s} denotes the s th zero of M _n (nα;δ, η) , counted from the right, and if α˜ _n,s denotes its s th zero counted from the left, then for each fixed s , three-term asymptotic approximations are obtained for both α _n,s and α˜ _n,s as n→∞ . December 28, 1998. Date revised: June 4, 1999. Date accepted: September 6, 1999.     

Characteristic features of the dynamics of the Ginzburg-Landau equation in a plane domain

We study the boundary value problem w_t=ℵ₀Δw+ℵ₁w-ℵ₂w|w|²,w|∂Ω₀=0 in the domain Ω₀={(x,y):0 ≤ x ≤ l₁,0 ≤ y ≤ l₂}. Here, w is a complex-valued function, Δ is the laplace operator, and ℵ_j, j=0,1,2, are complex constants withRe ℵ_j > 0. We show that under a rather general choice of the parameters l₁ and l₂, the number of stable invariant tori in the problem, as well as their dimensions, grows infinitely asRe ℵ₀ → 0 andRe ℵ₀ → 0. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 2, pp. 205–220, November, 2000.     

A note on the three versions of distributional chaos

The concept of distributional chaos was introduced by Schweizer et al. [Schweizer B, Sklar A, Smítal J. Measures of chaos and a spectral decomposition of dynamical systems on the interval. Tran Amer Math Soc 1994;344:737–854.] for a continuous selfmap on an interval. However, it turns out that, for a continuous selfmap on a compact metric space, three mutually nonequivalent versions of distributional chaos, DC1–DC3, can be discussed. In this paper, we consider a continuous map f : X → X, where X is a compact metric space, and show that DC1 (resp. DC2) is an iteration invariant, that is, for any integer N > 0, f is DC1 (resp. DC2) if and only if f^N is also DC1(resp. DC2). As applications, we show that the following statements hold:

• (1)Let G be a graph and f : G → G a continuous map. Then f is DC1 if and only if f is DC2.
• (2)For a continuous selfmap f on a tree T, these three versions of distributional chaos, DC1 − DC3 are mutually equivalent.

Furthermore, we present two examples which show that DC3 may be an iteration invariant. We will also discuss and partly solve the problem.     

Group divisible designs with block size four and type gum1—III

We deal with group divisible designs (GDDs) that have block size four and group type $g^u{m}^1$, where $g\equiv 2$ or 4 (mod 6). We show that the necessary conditions for the existence of a 4‐GDD of type $g^u{m}^1$ are sufficient when $g$ = 14, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 58, 62, 68, 76, 88, 92, 100, 104, 116, 124, 136, 152, 160, 176, 184, 200, 208, 224, 232, 248, 272, 304, 320, 368, 400, 448, 464 and 496. Using these results we go on to show that the necessary conditions are sufficient for $g=2^t{q}^s$, $q$ = 19, 23, 25, 29, 31, $s$, $t=1,2,\dots$, as well as for $g=2^{t}q$, $q$ = 2, 5, 7, 11, 13, 17, $t=1,2,\dots$, with possible exceptions $56^9{m}^1$, $80^9{m}^1$ and $112^9{m}^1$ for a few large values of $m$.     

Points surrounding the origin

Suppose d > 2, n > d+1, and we have a set P of n points in d-dimensional Euclidean space. Then P contains a subset Q of d points such that for any p ∈ P, the convex hull of Q∪{p} does not contain the origin in its interior. We also show that for non-empty, finite point sets A ₁, ..., A _d+1 in ℝ ^d , if the origin is contained in the convex hull of A _i ∪ A _j for all 1≤i<j≤d+1, then there is a simplex S containing the origin such that |S∩A _i |=1 for every 1≤i≤d+1. This is a generalization of Bárány’s colored Carathéodory theorem, and in a dual version, it gives a spherical version of Lovász’ colored Helly theorem. Dedicated to Imre Bárány, Gábor Fejes Tóth, László Lovász, and Endre Makai on the occasion of their sixtieth birthdays. Supported by the Norwegian research council project number: 166618, and BK 21 Project, KAIST. Part of the research was conducted while visiting the Courant Institute of Mathematical Sciences. Supported by NSF Grant CCF-05-14079, and by grants from NSA, PSC-CUNY, the Hungarian Research Foundation OTKA, and BSF.     

Joint continuity of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">h</Emphasis></Subscript><Emphasis Type="Italic">C</Emphasis>-functions with values in moore spaces

We introduce the notion of categorical cliquish mapping and show that, for each K _h C-mapping f: X × Y → Z, where X is a topological space, Y is a space with the first axiom of countability, and Z is a Moore space, with categorical-cliquish horizontal y-sections f _y , the sets C _y (f) are residual G _δ-type sets in X for every y ∈ Y. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 11, pp. 1539–1547, November, 2008.     

On Point Sets with Many Unit Distances in Few Directions

We study the problem of the maximum number of unit distances among n points in the plane, under the additional restriction that we count only those unit distances that occur in a fixed set of k directions, taking the maximum over all sets of n points and all sets of k directions. We prove that, for fixed k and sufficiently large n > n ₀ (k) , the extremal sets are essentially sections of lattices, bounded by edges parallel to the k directions and of equal length. <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p355.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader> Received January 10, 1997, and in revised form May 16, 1997.     

Calcul stochastique dépendant d'un paramètre

Résumé Le développement du calcul stochastique au cours des dix dernières années a rendu indispensable la mise à jour de l'article [3] de C. Doléans-Dade sur les intégrales stochastiques dépendant d'un paramètre, qui se limitait aux martingales de carré intégrable. Les deux auteurs se sont mis au travail indépendamment l'un de l'autre sur cette question, pour découvrir qu'ils avaient démontré beaucoup de résultats communs. Ayant confronté leurs rédactions, et constaté que leurs points de vue étaient légèrement différents, ils n'ont pu se résoudre à refondre le tout en un article incolore. On trouvera donc ci-dessous deux parties I et II, dont la première suit à peu près le canevas de l'article initial de M. Yor, à l'exception de la section 5, qui est une véritable synthèse des résultats obtenus par les deux auteurs sur les intégrales stochastiques dépendant d'un paramètre. La seconde partie correspond à peu près à l'article initial de C. Stricker, allégé des résultats communs, et présenté dans un ordre différent de l'ordre primitif.Les résultats que nous présentons peuvent rendre service à des utilisateurs du calcul stochastique pourvus d'un bagage technique relativement limité. C'est pourquoi nous avons donné des énoncés simples, rejetant dans les «remarques» divers raffinements. En particulier, la série des remarques entre astérisques^*..._* développe une extension des résultats énoncés, qui a son intérêt, mais qui aurait trop alourdi le texte principal.Enfin, nous remercions vivement P.A. Meyer dont les nombreuses remarques nous ont conduit à la version définitive de ce travail.     

Hopf Algebras of Type <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>, Twistings and the FRT-construction

We study pointed Hopf algebras of the form U(R _Q ), (Faddeev et al., Quantization of Lie groups and Lie algebras. Algebraic Analysis, vol. I, Academic, Boston, MA, pp. 129–139, 1988; Faddeev et al., Quantum groups. Braid group, knot theory and statistical mechanics. Adv. Ser. Math. Phys., vol. 9, World Science, Teaneck, NJ, pp. 97–110, 1989; Larson and Towber, Commun. Algebra 19(12):3295–3345, 1991), where R _Q is the Yang–Baxter operator associated with the multiparameter deformation of GL _n supplied in Artin et al. (Commun. Pure Appl. Math. 44:8–9, 879–895, 1991) and Sudbery (J. Phys. A, 23(15):697–704, 1990). We show that U(R _Q ) is of type A _n in the sense of Andruskiewitsch and Schneider (Adv. Math. 154:1–45, 2000; Pointed Hopf algebras. Recent developments in Hopf Algebras Theory, MSRI Series, Cambridge University Press, Cambridge, 2002). We consider the non-negative part of U(R _Q ) and show that for two sets of parameters, the corresponding Hopf sub-algebras can be obtained from each other by twisting the multiplication if and only if they possess the same groups of grouplike elements. We exhibit families of finite-dimensional Hopf algebras arising from U(R _Q ) with non-isomorphic groups of grouplike elements. We then discuss the case when the quantum determinant is central in A(R _Q ) and show that under some assumptions on the group of grouplike elements, two finite-dimensional Hopf algebras U(R _Q ), U(R _Q′) can be obtained from each other by twisting the comultiplication if and only if In the last part we show that U _Q is always a quotient of a double crossproduct. I wish to thank UIC, where some of the work was done, for hospitality.