Limiting process for integral functionals of a wiener process on a cylinder

We prove that integral functionals, whose integrands are bounded functions of a Wiener process on a cylinder, weakly converge to the processw ₁(τ(t)), τ(t) = β₁ t + (β₂ − β₁)mes {s:w ₂(s)≥0,s<t}, wherew ₁(t andw ₂(t) are independent one-dimensional Wiener processes, β₁ and β₂ are nonrandom values, and β₂≥β₁≥0. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 765–768, June, 1994.     

Adaptive policies for time-varying stochastic systems under discounted criterion

We consider a class of time-varying stochastic control systems, with Borel state and action spaces, and possibly unbounded costs. The processes evolve according to a discrete-time equation x _{n + 1}=G _n (x _n , a _n , ξ_n), n=0, 1, … , where the ξ_n are i.i.d. ℜ^k-valued random vectors whose common density is unknown, and the G _n are given functions converging, in a restricted way, to some function G _∞ as n→∞. Assuming observability of ξ_n, we construct an adaptive policy which is asymptotically discounted cost optimal for the limiting control system x _n+1=G _∞ (x _n , a _n , ξ_n).     

Two finiteness theorems for periodic tilings of d-dimensional euclidean space

Consider the d -dimensional euclidean space E ^d . Two main results are presented: First, for any N∈ N, the number of types of periodic equivariant tilings that have precisely N orbits of (2,4,6, . . . ) -flags with respect to the symmetry group Γ , is finite. Second, for any N∈ N, the number of types of convex, periodic equivariant tilings that have precisely N orbits of tiles with respect to the symmetry group Γ , is finite. The former result (and some generalizations) is proved combinatorially, using Delaney symbols, whereas the proof of the latter result is based on both geometric arguments and Delaney symbols. <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>20n2p143.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>no <sectionname> </lsiheader> Received September 5, 1996, and in revised form January 6, 1997.     

The Characterization for a Graphic Sequence to have a Realization Containing K 1,1,s

A graphic sequence π?=?(d ₁, d ₂, . . . , d _n ) is said to be potentially K _1,1,s -graphic if there is a realization of π containing K _1,1,s as a subgraph, where K _1,1,s is the 1?× 1?× s complete 3-partite graph. In this paper, a simple characterization of potentially K _1,1,s -graphic sequences for s?≥ 2 and n?≥ 3s?+?1 is obtained. This characterization implies Lai’s conjecture on σ(K _1,1,s , n), which was confirmed by J.H. Yin, J.S. Li and W.Y. Li, and the values of σ(K _2,s , n) for s?≥ 4 and n?≥ 3s?+?1, where K _2,s is the 2?× s complete bipartite graph.     

On linear widths of classesH ω

We obtain new results related to the estimation of the linear widths λ _N and λ ^N in the spacesC andL _p for the classesH ^ω (in particular, forH ^α, 0<α<1). Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1255–1264, September, 1996.     

Kp,q-factorization of complete bipartite graphs

Let K _m,nbe a complete bipartite graph with two partite sets having m and n vertices, respectively. A K _p,q-factorization of K _m,n is a set of edge-disjoint K _p,q-factors of K _m,n which partition the set of edges of K _m,n. When p = 1 and q is a prime number, Wang, in his paper “On K _1,k -factorizations of a complete bipartite graph” (Discrete Math, 1994, 126: 359—364), investigated the K _1,q -factorization of K _m,nand gave a sufficient condition for such a factorization to exist. In the paper “K _1,k -factorizations of complete bipartite graphs” (Discrete Math, 2002, 259: 301—306), Du and Wang extended Wang’s result to the case that q is any positive integer. In this paper, we give a sufficient condition for K _m,n to have a K _p,q-factorization. As a special case, it is shown that the Martin’s BAC conjecture is true when p : q = k : (k+ 1) for any positive integer k.     

Strong laws for exponential order statistics and spacings

We obtain upper and lower almost sure asymptotic bounds for spacings η _{K, N} − η _{K + 1, N} , where η _{1, N} > η _{2, N} > ⋯ > η _{N, N} are the order statistics of independent exponentially distributed random variables η ₁, η ₂, ..., η _N with mean 1; here N → ∞, and K = 1, 2, ... is fixed. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 477–491, October–December, 2006.     

Construction of Some Optimal Ternary Linear Codes and the Uniqueness of [294, 6, 195; 3]-Codes Meeting the Griesmer Bound

Let n_q(k, d) denote the smallest value of n for which there exists an [n, k, d; q]-code. It is known (cf. (J. Combin. Inform. Syst. Sci.18, 1993, 161–191)) that (1) n₃(6, 195) {294, 295}, n₃(6, 194) {293, 294}, n₃(6, 193) {292, 293}, n₃(6, 192) {290, 291}, n₃(6, 191) {289, 290}, n₃(6, 165) {250, 251} and (2) there is a one-to-one correspondence between the set of all nonequivalent [294, 6, 195; 3]-codes meeting the Griesmer bound and the set of all {v₂ + 2v₃ + v₄, v₁ + 2v₂ + v₃; 5, 3}-minihypers, where v_i = (3ⁱ − 1)/(3 − 1) for any integer i ≥ 0. The purpose of this paper is to show that (1) n₃(6, 195) = 294, n₃(6, 194) = 293, n₃(6, 193) = 292, n₃(6, 192) = 290, n₃(6, 191) = 289, n₃(6, 165) = 250 and (2) a [294, 6, 195; 3]-code is unique up to equivalence using a characterization of the corresponding {v₂ + 2v₃ + v₄, v₁ + 2v₂ + v₃; 5, 3}-minihypers.     

On the Degree of Approximation by Manifolds of Finite Pseudo-Dimension

The pseudo-dimension of a real-valued function class is an extension of the VC dimension for set-indicator function classes. A class of finite pseudo-dimension possesses a useful statistical smoothness property. In [10] we introduced a nonlinear approximation width = which measures the worst-case approximation error over all functions by the best manifold of pseudo-dimension n . In this paper we obtain tight upper and lower bounds on ρ _n (W ^r,d _p , L _q ) , both being a constant factor of n ^-r/d , for a Sobolev class W ^r,d _p , . As this is also the estimate of the classical Alexandrov nonlinear n -width, our result proves that approximation of W ^r,d _p by the family of manifolds of pseudo-dimension n is as powerful as approximation by the family of all nonlinear manifolds with continuous selection operators. March 12, 1997. Dates revised: August 26, 1997, October 24, 1997, March 16, 1998, June 15, 1998. Date accepted: June 25, 1998.     

An extremal set theoretical characterization of some steiner systems

Letn, k, t be integers,n>k>t≧0, and letm(n, k, t) denote the maximum number of sets, in a family ofk-subsets of ann-set, no two of which intersect in exactlyt elements. The problem of determiningm(n, k, t) was raised by Erdős in 1975. In the present paper we prove that ifk≦2t+1 andk−t is a prime, thenm(n, k, t)≦( _t ⁿ )( _k ^2k-t-1 )/( _t ^2k-t-1 ). Moreover, equality holds if and only if an (n, 2k−t−1,t)-Steiner system exists. The proof uses a linear algebraic approach.     

A Generalization of the Erdos - Szekeres Theorem to Disjoint Convex Sets

Let F denote a family of pairwise disjoint convex sets in the plane. F is said to be in convex position if none of its members is contained in the convex hull of the union of the others. For any fixed k≥ 3 , we estimate P _k (n) , the maximum size of a family F with the property that any k members of F are in convex position, but no n are. In particular, for k=3 , we improve the triply exponential upper bound of T. Bisztriczky and G. Fejes Tóth by showing that P ₃ (n) < 16 ⁿ . <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>19n3p437.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader> Received March 27, 1997, and in revised form July 10, 1997.     

Méthode de mesure de la chaleur spécifique par température croissante ou décroissante

Résumé L'obtention des valeurs expérimentales de l'entropie, à l'équilibre, à partir des grandeurs thermiques, est rendue compliquée pour un supraconducteur se trouvant dans une phase mélangée, mixte ou intermédiaire, par le comportement irréversible observé en général dans les échantillons réels. Il est possible de réduire, voire supprimer l'incertitude dans l'interprétation des résultats si l'on détermine les variations d'entropie non seulement pour des températures et champs magnétiques croissants, mais également pour des températures et champs décroissants. On décrit une méthode de mesure de la chaleur spécifique, valable pourT<10°K, inspirée de la méthode d'impulsion d'une part, en ce sens qu'elle a l'avantage de travailler sur des états de quasi équilibre en température du système, et de la méthode de refroidissement continu d'autre part, puisqu'elle utilise une perte thermique permettant la mesure par température décroissante. Le modèle théorique correspondant au dispositif expérimental est traité complètement du point de vue mathématique. Les erreurs systématiques propres à la méthode peuvent être réduites à moins de 0.5% et des variations de température de quelques millidegrés peuvent être encore mesurées avec une précision de l'ordre du pourcent. Le procédé convient donc bien à l'étude détaillée des transitions de phase supraconductrices.

---

Summary It is difficult to obtain equilibrium values of entropy from measured thermal quantities, for superconductors in a mixed phase, because of irreversible behavior generally observed in real samples. It is possible to reduce, even suppress, the uncertainty arising from interpretation of results, if entropy variations are determined not only for increasing magnetic field and temperature, but also for decreasing field and temperature. A method to measure the specific heat is described, which is valid forT<10°K. As in the case of the heat burst method, the advantage is that the determination occurs from quasi-equilibrium states in temperature of the system. In addition, a heat leak allows us also to make measurements with decreasing temperature, as in the cooling curve method. The theoretical model corresponding to the experimental arrangement is thoroughly analysed from a mathematical point of view. Systematic errors, arising from the method itself, may be reduced to less than 0,5% and temperature variations of a few milli-degrees can still be measured with a precision of the order of 1%. This procedure is then well adapted to the detailed study of superconducting phase transitions.

     

General local convergence theory for a class of iterative processes and its applications to Newton’s method

General local convergence theorems with order of convergence r≥1$r\ge 1$ are provided for iterative processes of the type _xn+1=T_xn$x_{n+1}=T{x}_n$, where T:D⊂X→X$T:D\subset X\to X$ is an iteration function in a metric space X$X$. The new local convergence theory is applied to Newton iteration for simple zeros of nonlinear operators in Banach spaces as well as to Schröder iteration for multiple zeros of polynomials and analytic functions. The theory is also applied to establish a general theorem for the uniqueness ball of nonlinear equations in Banach spaces. The new results extend and improve some results of [K. Do?ev, Über Newtonsche Iterationen, C. R. Acad. Bulg. Sci. 36 (1962) 695–701; J.F. Traub, H. Wo?niakowski, Convergence and complexity of Newton iteration for operator equations, J. Assoc. Comput. Mach. 26 (1979) 250–258; S. Smale, Newton’s method estimates from data at one point, in: R.E. Ewing, K.E. Gross, C.F. Martin (Eds.), The Merging of Disciplines: New Direction in Pure, Applied, and Computational Mathematics, Springer, New York, 1986, pp. 185–196; P. Tilli, Convergence conditions of some methods for the simultaneous computation of polynomial zeros, Calcolo 35 (1998) 3–15; X.H. Wang, Convergence of Newton’s method and uniqueness of the solution of equations in Banach space, IMA J. Numer. Anal. 20 (2000) 123–134; I.K. Argyros, J.M. Gutiérrez, A unified approach for enlarging the radius of convergence for Newton’s method and applications, Nonlinear Funct. Anal. Appl. 10 (2005) 555–563; M. Giusti, G. Lecerf, B. Salvy, J.-C. Yakoubsohn, Location and approximation of clusters of zeros of analytic functions, Found. Comput. Math. 5 (3) (2005) 257–311], and others.     

Optimal interpolation in spaces of Lorentz-Zygmund type

The Lorentz-Zygmund spaces, introduced by C. Bennett and K. Rudnick in [BR], are generalized by taking the exterior norm in arbitrary rearrangement invariant spaceE instead of onlyL _r-spaces. On the spacesL _p,α,E thus obtained, we study all operatorsT of two weak types (a, b) and (p, q) with 1≤a<p≤∞, 1≤b<q≤∞, and prove thatT:L _p,α,E→L _q,α−1,E. Moreover, for any set of parametersp, q, α, E, we construct the smallest possible spaceB _q,α,E such thatT:L _p,α,E→B _q,α,E and the largest possible spaceA _p,α,E such thatT:A _p,α,E→L _q,α−1,E. For spaces of all three types, we find their fundamental functions and Boyd indices, state various embeddings, equivalences and other properties. The research was supported by the Center of Scientific Absorption of the Ministry of Absorption of the State of Israel.     

Intersection Properties of Families of Convex (n,d)-Bodies

We study a multicomponent generalization of Helly's theorem. An (n,d)-body K is an ordered n -tuple of d -dimensional sets, K= < K ₁ , . . . ,K _n > . A family of (n,d)-bodiesis weakly intersecting if there exists an n -point p = < p ₁ , . . . , p _n > such that for every there exists an index 1 for which p _i ∈ K _i . A family of (n,d)-bodies is strongly intersecting if there exists an index i such that . The main question addressed in this paper is: What is the smallest number H(n,d), such that for every finite family of convex (n,d)-bodies, if every H(n,d) of them are strongly intersecting, then the entire family is weakly intersecting? We establish some basic facts about H(n,d) , and also prove an upper bound . In addition, we introduce and discuss two interesting related questions of a combinatorial-topological nature. Received February 9, 1996, and in revised form November 6, 1996, December 16, 1996, and January 7, 1998.     

On graphs without crowns with regularμ-subgraphs

Anm-crown is the complete tripartite graphK _{1, 1,m} with parts of order 1, 1,m, and anm-claw is the complete bipartite graphK _1,m with parts of order 1,m, wherem ≥ 3. A vertexa of a graph Γ is calledweakly reduced iff the subgraph {x є Γ ‖a ^⊥ =x ^⊥} consists of one vertex. A graph Γ is calledweakly reduced iff all its vertices are weakly reduced. In the present paper we classify all connected weakly reduced graphs without 3-crowns, all of whose μ-subgraphs are regular graphs of constant nonzero valency. In particular, we generalize the characterization of Grassman and Johnson graphs due to Numata, and the characterization of connected reduced graphs without 3-claws due to Makhnev. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 874–881, June, 2000. This research was supported by the Russian Foundation for Basic Research under grant No. 99-01-00462.     

Über die allgemeine lineare Differentialgleichung

Résumé Anmerkung der Redaktion. HerrFréchet (Paris) schreibt uns zu der vorliegenden Arbeit: L'intégration d'une équation linéaire d'ordren (1) est, comme on sait équivalente à celle d'un système canonique linéaireS den équations du premier ordre àn inconnues. Or le problème consistant à donner une expression formelle (et rigoureuse) de la solution générale deS (quand par exemple les coefficients deS sont continus¹⁾), a été résolu par plusieurs auteurs. Chacune des solutions se trouve exprimée, comme dans l'expression (2) de M. Ta Li, par une série d'intégrales multiples d'ordre croissant portant sur des polynomes de degrés croissants par rapport aux coefficients (fonctions dex) deS. C'est le résultat obtenu d'abord parPeano, Math. Ann. vol. 32, 1888, p. 459, puis indépendamment parBaker dans le cas de coefficients analytiques au moyen de la théorie des matrices (Proceedings London Math. Soc., vol. 34, 1902, p. 354 et 356, vol. 35, 1903, p. 333–378; vol. 2, sér. 2, 1904, p. 293–296; Phil. Trans. Royal Soc., A. Vol. 216, 1915–16, p. 129 à 186). Ce résultat s'obtient pour ainsi dire sans calcul comme l'a signaléM. Hostinský (Le problème de Cauchy pour les équations différentielles linéaires, Publ. Fac. Sc. Univ. Masaryk, Brno, Nr. 230, 1936) par une simple application de la méthode des approximations successives. Il en résulte immédiatement la formule correspondante pour l'intégrale générale de l'équation générale linéaire d'ordren (voir par exemple: Leçons sur les équations différentielles, parM. Fréchet, p. 126–129; [voir aussi p. 109–111], chez Tournier et Constant, Paris, 1938). Mais dans l'expression obtenue, il y a des simplifications à faire. La formule (2) de M.Ta Li a le grand intérêt de se présenter toutes simplifications faites.     

Lattice Polytopes with Distinct Pair-Sums

Let P be a lattice polytope in R ⁿ , and let P \cap Z ⁿ = {v ₁ ,\ldots,v _N } . If the N + N\choose 2 points 2v ₁ ,\ldots, 2v _N ;v ₁ +v ₂ ,\ldots, v _N-1 + v _N are distinct, we say that P is a ``distinct pair-sum' or ``dps' polytope. We show that if P is a dps polytope in R ⁿ , then N≤ 2 ⁿ , and, for every n , we construct dps polytopes in R ⁿ which contain 2 ⁿ lattice points. We also discuss the relation between dps polytopes and the study of sums of squares of real polynomials. Received November 10, 2000, and in revised form June 28, 2001. Online publication November 2, 2001.     

PATHS AND CYCLES EMBEDDING ON FAULTY ENHANCED HYPERCUBE NETWORKS

Let Qn,k(n≥3,1≤k≤n-1) be an n-dimensional enhanced hypercube which is an attractive variant of the hypercube and can be obtained by adding some complementary edges,fv and fe be the numbers of faulty vertices and faulty edges,respectively.In this paper,we give three main results.First,a fault-free path P [u,v] of length at least 2n-2fv-1(respectively,2n-2fv-2) can be embedded on Qn,k with fv+fe≤n-1 when d Qn,k(u,v) is odd(respectively,d Qn,k(u,v) is even).Secondly,an Qn,k is(n-2) edgefault-free hyper Hamiltonian-laceable when n(≥3) and k have the same parity.Lastly,a fault-free cycle of length at least 2n-2fv can be embedded on Qn,k with fe≤n-1 and fv+fe≤2n-4.     

All directed BIBDs with k = 3 exist

A directed BIBD with parameters ($\text{υ, b, r, k, λ}{}^1$) is a BIBD with parameters ($\text{υ, b, r, k, 2λ}{}^1$) in which each ordered pair of varieties occurs together in exactly $\text{λ}{}^1$ blocks. It is shown that $\text{λ}{}^1\text{υ(υ ? 1) ≡ 0}$ (mod 3) is a necessary and sufficient condition for the existence of a directed ($\text{υ, b, r, k, λ}{}^1$) BIBD with k = 3.