Banach—Saks Property and Property (β) in Cesàro Sequence Spaces

A new constant C(X) for any Banach space X is introduced. It is proved that C(X) < 2 implies the weak Banach—Saks property for the space X. In particular, C(ces_p) is found for Cesàro sequence space ces_p (1 < p < ). Moreover, it is shown that the space ces_p (1 < p < ) has property ().AMS Subject Classification (2000), 46E30, 46E40, 46B20Supported by NSF and ECF of China.     

A Remarkable q,t-Catalan Sequence and q-Lagrange Inversion

We introduce a rational function C _n(q, t) and conjecture that it always evaluates to a polynomial in q, t with non-negative integer coefficients summing to the familiar Catalan number . We give supporting evidence by computing the specializations and C _n (q) = C _n(q,1) = C _n(1,q). We show that, in fact, D _n(q) q-counts Dyck words by the major index and C _n(q) q-counts Dyck paths by area. We also show that C _n(q, t) is the coefficient of the elementary symmetric function e _nin a symmetric polynomial DH_n(x; q, t) which is the conjectured Frobenius characteristic of the module of diagonal harmonic polynomials. On the validity of certain conjectures this yields that C _n(q, t) is the Hilbert series of the diagonal harmonic alternants. It develops that the specialization DH_n(x; q, 1) yields a novel and combinatorial way of expressing the solution of the q-Lagrange inversion problem studied by Andrews [2], Garsia [5] and Gessel [11]. Our proofs involve manipulations with the Macdonald basis {P (x; q, t)} which are best dealt with in -ring notation. In particular we derive here the -ring version of several symmetric function identities.Work carried out under NSF grant support.     

The expected values of invariant polynomials with matrix argument of elliptical distributions

ThisresearchissupportedbytheChineseAcademyofSciences.1.IntroductionTherearemailypublicatiollsonmatrixvariatedistributions(ormatrixdistributionforsimplicity),inparticular,abollttheirexpectedvaluesofzonalpolynomialsoftheirquadraticforms(of.[12],[151,[171and…     

Modulation spaces and pseudodifferential operators

We use methods from time-frequency analysis to study boundedness and traceclass properties of pseudodifferential operators. As natural symbol classes, we use the modulation spaces onR ^2d , which quantify the notion of the time-frequency content of a function or distribution. We show that if a symbol lies in the modulation spaceM _,1 (R ^2d ), then the corresponding pseudodifferential operator is bounded onL ²(R ^d ) and, more generally, on the modulation spacesM _p,p (R ^d ) for 1p. If lies in the modulation spaceM _2,2 ^s (R ^2d )=L _s ^/2 (R ^2d )H ^s (R ^2d ), i.e., the intersection of a weightedL ²-space and a Sobolev space, then the corresponding operator lies in a specified Schatten class. These results hold for both the Weyl and the Kohn-Nirenberg correspondences. Using recent embedding theorems of Lipschitz and Fourier spaces into modulation spaces, we show that these results improve on the classical Calderòn-Vaillancourt boundedness theorem and on Daubechies' trace-class results.     

On the On-line Number of Snacks Problem

In the number of snacks problem (NSP), which was originally proposed by our team, an on-line player is given the task of deciding how many shares of snacks his noshery should prepare each day. The on-line player must make his decision and then finish the preparation before the customers come to his noshery for the snacks; in other words, he must make decision in an on-line fashion. His goal is to minimize the competitive ratio, defined as _: C_A()/C_OPT(), where denotes a sequence of numbers of customers, C _OPT() is the cost of satisfying by an optimal off-line algorithm, and C _A() is the cost of satisfying by an on-line algorithm. In this paper we give a competitive algorithm for on-line number of snacks problem P1, the Extreme Numbers Harmonic Algorithm(ENHA), with competitive ratio 1+p(M-m)/(M+m), where M and m are two extreme numbers of customers over the total period of the game, and p is a ratio concerning the cost of the two types of situations, and then prove that this competitive ratio is the best one if an on-line player chooses a fixed number of shares of snacks for any sequence of numbers of customers. We also discuss several variants of the NSP and give some results for it. Finally, we propose a conjecture for the on-line NSP.     

Приближение непреры вных функций средним и Эйлера

Let _n(1,f,x)=1/2 ⁿ _k=1 ⁿ C _k ⁿ S_k(x,f) denote the Euler means of the Fourier series of the 2-periodic functionf(x). For a function the main term of deviationf(x)– _n (1,f, x) is calculated. Asymptotically exact order of decrease of the upper bound of such deviations over the classH ⁽⁾ is also obtained.     

Primitive Illumination Systems for Families of Convex Bodies in the Plane

Let F= {C₁,C₂,...,C} be a family of ndisjoint convex bodies in the plane. We say that a set Vof exterior light sources illuminates F, if for every boundary point of any member of Fthere is a point in Vsuch that is visible from ,i.e. the open line segment joining and is disjoint from F. An illumination system Vis called primitive if no proper subset of Villuminates F. Let p_max(F) denote the maximum number of points forming a primitive illumination system for F, and letp_max(n) denote the minimum of F) taken over all families Fconsisting of ndisjoint convex bodies in the plane. The aim of this paper is to investigate the quantities p_max(F) and p_max(n).     

Remarks on Linear Homeomorphisms of Function Spaces

It is proved that whenever X and Y are completely regular -spaces of pointwise countable type and the spaces C _p(X) and C _p(Y) of real-valued continuous functions on X and Y, respectively, endowed with the topology of pointwise convergence, are linearly homeomorphic, the X is locally compact iff Y is locally compact. This extends the McCoy and Ntantu result.     

A Coding Theory Bound and Zero-Sum Square Matrices

For a code C=C(n,M) the level k code of C, denoted C _k , is the set of all vectors resulting from a linear combination of precisely k distinct codewords of C. We prove that if k is any positive integer divisible by 8, and n=k, M=k2k then there is a codeword in C _k whose weight is either 0 or at most . In particular, if <(4–2)²/48 then there is a codeword in C _k whose weight is n/2–(n). The method used to prove this result enables us to prove the following: Let k be an integer divisible by p, and let f(k,p) denote the minimum integer guaranteeing that in any square matrix over Z _p , of order f(k,p), there is a square submatrix of order k such that the sum of all the elements in each row and column is 0. We prove that lim inf f(k,2)/k<3.836. For general p we obtain, using a different approach, that f(k,p)p( ^{k / ln k )(1+ o} _k ⁽¹⁾⁾.     

Linear foliations ofT n

We present a definition of diophantine matrix and use this concept to distinguish two classes of minimal linear foliations ofT ⁿ, the diophantine and the Liouville one. Let _p , 1pn–1, denote a minimal (all leaves are dense) linearp-dimensional foliation ofT ⁿ, andH ^om(T ⁿ, _p ), 1mp, the cohomology group of type (0,m) of the foliated manifold (T ⁿ, _p ). Our main result is the computation of these groups.H ^om(T ⁿ, _p ) is isomorphic to if _p is diophantine and is an infinite dimensional non-Hausdorff vector space if _p is Liouville. Some of these groups were computed before, see [4], [6] and [9].     

The Embedding Theorem for Cantor Varieties

Let m and n be fixed integers, with 1 m < n. A Cantor variety C _m,n is a variety of algebras with m n-ary and n m-ary basic operations which is defined in a signature ={g₁,...,g_m,f₁,...,f_n} by the identities f_ig₁x₁,...,x_n),...,g_mx₁,...,x_n) = x_i, _i=1,...,n, g_jf₁x₁,...,x_m),...,f_nx₁,...,x_m)) = x_j, _j=1,...,m. We prove the following: (a) every partial C _m,n-algebra A is isomorphically embeddable in the algebra G= A; S(A) of C _m,n; (b) for every finitely presented algebra G= A; S in C _m,n, the word problem is decidable; (c) for finitely presented algebras in _{C m}, the occurrence problem is decidable; (d) _{C m,n} has a hereditarily undecidable elementary theory.     

Clean triangulations

A polyhedron on a surface is called a clean triangulation if each face is a triangle and each triangle is a face. LetS _p (resp.N _p ) be the closed orientable (resp. nonorlentable) surface of genusp. If (S) is the smallest possible number of triangles in a clean triangulation ofS, the results are: (N ₁)=20, (S ₁)=24, lim(S _p )p ^–1=4, lim(N _p )p ^–1=2 forp.     

You can recognize the shape of a figure from its shadows!

Let C ₁ and C ₂ be convex closed domains in the plane with C ² boundaries C ₁ and C ₂ intersecting each other in nonzero angles. Assume the two strictly convex bodies F ₁ and F ₂ with C ² boundaries in the interior of C ₁C ₂ subtend equal visual angles at each point of C ₁ and C ₂. Then F ₁ and F ₂ coincide. Generalizations are also discussed.Supported by the Hungarian NSF, OTKA Nr. T4427, W015425 and F016226.     

A Theorem of Liouville Type for p-Harmonic Morphisms

In this article we prove a Liouville type theorem for p-harmonic morphisms. We show that if : MNis a p-harmonic morphism (p2) from a complete noncompact Riemannian manifold Mof nonnegative Ricci curvature into a Riemannian manifold Nof nonpositive scalar curvature such that the p-energy E _p (), or (2p–2)-energy E _2p–2() is finite, then is constant.     

Upper tails for triangles

With ξ the number of triangles in the usual (Erd?s‐Rényi) random graph G(m,p), p > 1/m and η > 0, we show (for some C_η > 0) This is tight up to the value of C_η. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 452–459, 2012     

A polynomial-time algorithm for a class of linear complementarity problems

Given ann × n matrixM and ann-dimensional vectorq, the problem of findingn-dimensional vectorsx andy satisfyingy = Mx + q, x 0,y 0,x _i y _i = 0 (i = 1, 2,,n) is known as a linear complementarity problem. Under the assumption thatM is positive semidefinite, this paper presents an algorithm that solves the problem in O(n ³ L) arithmetic operations by tracing the path of centers,{(x, y) S: x _i y _i = (i = 1, 2,,n) for some > 0} of the feasible regionS = {(x, y) 0:y = Mx + q}, whereL denotes the size of the input data of the problem.     

Random continued fractions and inverse Gaussian distribution on a symmetric cone

In this paper we introduce the inverse Gaussian and Wishart distributions on the cone of real (n, n) symmetric positive definite matricesH _n ⁺ () and more generally on an irreducible symmetric coneC. Then we study the convergence of random continued fractions onH _n ⁺ () andC by means of real Lagrangians forH _n ⁺ () and by new algebraic identities on symmetric cones forC. Finally we get a characterization of the inverse Gaussian distribution onH _n ⁺ () andC.     

Asymptotic Density and the Asymptotics of Partition Functions

Let A be a set of positive integers with gcd (A) = 1, and let p _A (n) be the partition function of A. Let c ₀ = 2/3. If A has lower asymptotic density and upper asymptotic density , then lim inf log p _A (n)/c ₀ n and lim sup log p _A (n)/c ₀ n . In particular, if A has asymptotic density > 0, then log p _A (n) c₀n. Conversely, if > 0 and log p _A (n) c ₀ n, then the set A has asymptotic density .     

Ein konvergentes Iterationsverfahren zur Bestimmung der Nullstellen eines Polynoms

Summary In order to determine the roots of a polynomialp, a sequence of numbers {x _k} is constructed such that the associated sequence {|p(x _k)|} decreases monotonically. To determine a new iteration pointx _k+1 such that |p(x _k+1)|<-|p(x _k)| ( is a positive real constant, <1, depending only on the degree ofp), we determine a circleK aroundx _k which contains no root ofp and compute the values ofp atN points which are distributed equally on the circumference ofK (N again depends only on the degree ofp); at least one of theN points is shown to satisfy the given condition. Computing the function values by means of Fourier synthesis according to Cooley-Tukey [2] and combining our iteration step with the normal step of the method of Nickel [1], we obtain a numerically safe and fast algorithm for determining the roots of arbitrary polynomials.     

Some properties related to torsion of a two-dimensional surface

We consider a two-dimensional surface F² of class C³ in Euclidean spaceE ⁿ ,n4. We introduce the concepts of the normal curvature vectork _N (x; t) and the Euler curvature vectorv _N (x; t) of the normalp _N (x;t) and the Euler _N (x;t) torsion of surface F² at the point x in the tangent directiont. We show that these magnitudes are characteristics of surface F² at the point x in the directiont, and derive formulas for their calculation. We establish necessary and sufficient conditions under which the directions of vectorsk _N (x; t) andp _N (x; t) are parallel displaced in the normal fiber bundle on F² from the pointx ¯F ² in the directiont. In particular, the following assertion holds: the direction of the Euler curvature vectorp _N (x; t) is parallel displaced in the normal fiber bundle on F² along any geodesic on F² if and only ifv _N (x; t) 0, x F ². t.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 45–52, 1990.