Convergence of Appell Polynomials of Long Range Dependent Moving Averages in Martingale Differences

We study limit distribution of partial sums S_N,k(t) = _{s = 1} ^{[N t]} A_k(X_s) of Appell polynomials of the long-range dependent moving average process X_t> = _{i t} b_{t - i i}, where {_i} is a strictly stationary and weakly dependent martingale difference sequence, and b_i i^{d - 1} (0 < d < 1/2). We show that if k(1-2 d)<1, then suitably normalized partial sums S_N,k(t) converge in distribution to the kth order Hermite process. This result generalizes the corresponding results of Surgailis, and Avram and Taqqu obtained in the case of the i.i.d. sequence { _i}.     

A Distance-Regular Graph with Strongly Closed Subgraphs

Let be a distance-regular graph of diameter d, valency k and r := maxi | (c _i,b _i) = (c ₁,b ₁). Let q be an integer with r + 1 q d – 1.In this paper we prove the following results: Theorem 1 Suppose for any pair of vertices at distance q there exists a strongly closed subgraph of diameter q containing them. Then for any integer i with 1 i q and for any pair of vertices at distance i there exists a strongly closed subgraph of diameter i containing them. Theorem 2 If r 2, then c _2r+3 1.As a corollary of Theorem 2 we have d k ²(r + 1) if r 2.     

Exact Bounds on the Sizes of Covering Codes

A code c is a covering code of X with radius r if every element of X is within Hamming distance r from at least one codeword from c. The minimum size of such a c is denoted by c _r(X). Answering a question of Hämäläinen et al. [10], we show further connections between Turán theory and constant weight covering codes. Our main tool is the theory of supersaturated hypergraphs. In particular, for n > n ₀(r) we give the exact minimum number of Hamming balls of radius r required to cover a Hamming ball of radius r + 2 in {0, 1}ⁿ. We prove that c _r(B _n(0, r + 2)) = _{1 i r + 1} ( ^{(n + i – 1) / (r + 1) 2}) + n / (r + 1) and that the centers of the covering balls B(x, r) can be obtained by taking all pairs in the parts of an (r + 1)-partition of the n-set and by taking the singletons in one of the parts.     

Generalized perfect arrays and menon difference sets

Given an s ₁ × ... × s _rinteger-valued array A and a (0, 1) vector z = (z ₁, ..., z _r), form the array A from A by recursively adjoining a negative copy of the current array for each dimension i where z _i = 1. A is a generalized perfect array type z if all periodic autocorrelation coefficients of A are zero, except for shifts (u ₁, ..., u _r) where u _i, - 0 (mod s _i) for all i. The array is perfect if z = (0, ..., 0) and binary if the array elements are all ±1. A nontrivial perfect binary array (PBA) is equivalent to a Menon difference set in an abelian group.Using only elementary techniques, we prove various construction theorems for generalized perfect arrays and establish conditions on their existence. We show that a generalized PBA whose type is not (0, ..., 0) is equivalent to a relative difference set in an abelian factor group. We recursively construct several infinite families of generalized PBAs, and deduce nonexistence results for generalized PBAs whose type is not (0, ..., 0) from well-known nonexistence results for PBAs. A central result is that a PBA with 2^2y 3^2u elements and no dimension divisible by 9 exists if and only if no dimension is divisible by 2 ^y+2. The results presented here include and enlarge the set of sizes of all previously known generalized PBAs.     

Uniqueness of limit cycles for polynomial first-order differential equations

We study the uniqueness of limit cycles (periodic solutions that are isolated in the set of periodic solutions) in the scalar ODE in terms of {i_k}, {j_k}, {n_k}. Our main result characterizes, under some additional hypotheses, the exponents {i_k}, {j_k}, {n_k}, such that for any choice of the equation has at most one limit cycle. The obtained results have direct application to rigid planar vector fields, thus, planar systems of the form x^′=y+xR(x,y), y^′=−x+yR(x,y), where . Concretely, when the set has at least three elements (or exactly one) and another technical condition is satisfied, we characterize the exponents {i_k}, {j_k} such that the origin of the rigid system is a center for any choice of and also when there are no limit cycles surrounding the origin for any choice of .     

Über zweidimensionale diskrete Bewegungsgruppen der pseudoeuklidischen Ebene

Zusammenfassung Ausgehend von der BedingungD ₂.Pr>0U _r (P)P ^G ={P} von L. Fejes Tóth untersuchen wir in der pseudoeuklidischen Ebene zweidimensionale Bewegungsgruppen mit Drehungen aufD ₂-Diskretheit. Mit Hilfe eines ergodischen Maßes auf dem Torus zeigen wir die Nichtdiskretheit solcher Gruppen.

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On the basis of the conditionD ₂.Pr>0;U _r (P)P ^G ={P} by L. Fejes Tóth we test two-dimensional motion groups with rotations of the pseudoeuclidean plane forD ₂-discreteness. Using an ergodic measure on the torus we will show that these groups aren'tD ₂-discrete.

     

A bipartite analogue of Dilworth’s theorem for multiple partial orders

Let r be a fixed positive integer. It is shown that, given any partial orders <₁, …, <_r on the same n-element set P, there exist disjoint subsets A,BP, each with at least n^1−o(1) elements, such that one of the following two conditions is satisfied: (1) there is an such that every element of A is larger than every element of B in the partial order <_i, or (2) no element of A is comparable with any element of B in any of the partial orders <₁, …, <_r. As a corollary, we obtain that any family C of n convex compact sets in the plane has two disjoint subfamilies A,BC, each with at least n^1−o(1) members, such that either every member of A intersects all members of B, or no member of A intersects any member of B.     

On the optimum capacity of capacity expansion problems

In this paper we consider problems of the following type: Let E = { e ₁, e ₂,..., e _n } be a finite set and be a family of subsets of E. For each element e _i in E, c _i is a given capacity and _i is the cost of increasing capacity c _i by one unit. It is assumed that we can expand the capacity of each element in E so that the capacity of family can be expanded to a level r. For each r, let f (r) be the efficient function with respect to the capacity r of family , and be the cost function for expanding the capacity of family to r. The goal is to find the optimum capacity value r ^* and the corresponding expansion strategy so that the pure efficency function is the largest. Firstly, we show that this problem can be solved efficiently by figuring out a series of bottleneck capacity expansion problem defined by paper (Yang and Chen, Acta Math Sci 22:207–212, 2002) if f (r) is a piecewise linear function. Then we consider two variations and prove that these problems can be solved in polynomial time under some conditions. Finally the optimum capacity for maximum flow expansion problem is discussed. We tackle it by constructing an auxiliary network and transforming the problem into a maximum cost circulation problem on the auxiliary network.     

A generalization of Meshulam’s theorem on subsets of finite abelian groups with no 3-term arithmetic progression

Let r ₁, …, r _s be non-zero integers satisfying r ₁ + ⋯ + r _s = 0. Let G be a finite abelian group with k _i |k _i-1(2 ≤ i ≤ n), and suppose that (r _i , k ₁) = 1(1 ≤ i ≤ s). Let denote the maximal cardinality of a set which contains no non-trivial solution of r ₁ x ₁ + ⋯ + r _s x _s = 0 with . We prove that . We also apply this result to study problems in finite projective spaces.      

A new characterization of Bergman–Schatten spaces and a duality result

Let denote the space of all upper triangular matrices A such that lim_r→1⁻(1−r²)(A*C(r))^′_B(ℓ²)=0. We also denote by the closed Banach subspace of consisting of all upper triangular matrices whose diagonals are compact operators. In this paper we give a duality result between and the Bergman–Schatten spaces . We also give a characterization of the more general Bergman–Schatten spaces , 1p<∞, in terms of Taylor coefficients, which is similar to that of M. Mateljevic and M. Pavlovic [M. Mateljevic, M. Pavlovic, L^p-behaviour of the integral means of analytic functions, Studia Math. 77 (1984) 219–237] for classical Bergman spaces.     

On the weighted Kneser-Poulsen conjecture

Suppose that p = (p1, p2, …, pN) and q = (q1, q2, …, qN) are two configurations in , which are centers of balls B ^d (p _i , r _i ) and B ^d (q _i , r _i ) of radius r _i , for i = 1, …, N. In [9] it was conjectured that if the pairwise distances between ball centers p are contracted in going to the centers q, then the volume of the union of the balls does not increase. For d = 2 this was proved in [1], and for the case when the centers are contracted continuously for all d in [2]. One extension of the Kneser-Poulsen conjecture, suggested in [6], was to consider various Boolean expressions in the unions and intersections of the balls, called flowers, where appropriate pairs of centers are only permitted to increase, and others are only permitted to decrease. Again under these distance constraints, the volume of the flower was conjectured to change in a monotone way. Here we show that these generalized Kneser-Poulsen flower conjectures are equivalent to an inequality between certain integrals of functions (called flower weight functions) over , where the functions in question are constructed from maximum and minimum operations applied to functions each being radially symmetric monotone decreasing and integrable. Research supported in part by NSF Grant No. DMS-0209595.     

Monochromatic Hamiltonian t‐tight Berge‐cycles in hypergraphs

In any r‐uniform hypergraph for 2 ≤ t ≤ r we define an r‐uniform t‐tight Berge‐cycle of length ?, denoted by C_?^{(r, t)}, as a sequence of distinct vertices v₁, v₂, … , v_?, such that for each set (v_i, v_{i + 1}, … , v_{i + t ? 1}) of t consecutive vertices on the cycle, there is an edge E_i of that contains these t vertices and the edges E_i are all distinct for i, 1 ≤ i ≤ ?, where ? + j ≡ j. For t = 2 we get the classical Berge‐cycle and for t = r we get the so‐called tight cycle. In this note we formulate the following conjecture. For any fixed 2 ≤ c, t ≤ r satisfying c + t ≤ r + 1 and sufficiently large n, if we color the edges of K_n^(r), the complete r‐uniform hypergraph on n vertices, with c colors, then there is a monochromatic Hamiltonian t‐tight Berge‐cycle. We prove some partial results about this conjecture and we show that if true the conjecture is best possible. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 34–44, 2008     

On the convergence of the LJ search method

The convergence of the Luus-Jaakola search method for unconstrained optimization problems is established.Notation E _n Euclideann-space - f Gradient off(x) - ² f Hessian matrix - (·) ^T Transpose of (·) - I Index set {1, 2, ...,n} - [x _i1 ^*(j) ] Point around which search is made in the (j + 1)th iteration, i.e., [x _1l ^*(j) ,x _2l ^*(j) ,...,x _n1 ^*(j) ] - r _i ⁽ⁱ⁾ Range ofx _il ^*(i) in the (j + 1)th iteration - l ₁ min_i {r _i ⁽⁰⁾ } - l ₂ min_i {r _i ⁽⁰⁾ } - A _j Region of search in thejth iteration, i.e., {x E _n:x _il ^*(j-1) –0.5r _i ^(j-1) x _ix _il ^*(j-1) +0.5r _i ^(j-1) ,i I} - S _j Closed sphere with center origin and radius _j - Reduction factor in each iteration - 1– - (·) Gamma function Many discussions with Dr. S. N. Iyer, Professor of Electrical Engineering, College of Engineering, Trivandrum, India, are gratefully acknowledged. The author has great pleasure to thank Dr. K. Surendran, Professor, Department of Electrical Engineering, P.S.G. College of Technology, Coimbatore, India, for suggesting this work.     

Topological classification of holomorphic, semi-hyperbolic germs, in ``Quasi-Absence' of resonances

We give the classification, under topological conjugacy, of invertible holomorphic germs f:, with λ₁, . . . ,λ_n eigenvalues of d f₀, and |λ_i|≠1 for i=2, . . . ,n while λ₁ is a root of the unity, in the suitable hypothesis of ``quasi-absence' of resonances (i.e., assuming that for r_i≥0 and i=2, . . . ,n, with ).     

Partitions of finite vector spaces into subspaces

Let V_n(q) denote a vector space of dimension n over the field with q elements. A set of subspaces of V_n(q) is a partition of V_n(q) if every nonzero element of V_n(q) is contained in exactly one element of . Suppose there exists a partition of V_n(q) into x_i subspaces of dimension n_i, 1 ≤ i ≤ k. Then x₁, …, x_k satisfy the Diophantine equation . However, not every solution of the Diophantine equation corresponds to a partition of V_n(q). In this article, we show that there exists a partition of V_n(2) into x subspaces of dimension 3 and y subspaces of dimension 2 if and only if 7x + 3y = 2ⁿ ? 1 and y ≠ 1. In doing so, we introduce techniques useful in constructing further partitions. We also show that partitions of V_n(q) induce uniformly resolvable designs on qⁿ points. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 329–341, 2008     

A theorem on arrangements of lines in the plane

LetA be an arrangement ofn lines in the plane. IfR ₁, …,R _r arer distinct regions ofA, andR _i is ap _i-gon (i=1, …,r) then we show that . Further we show that for allr this bound is the best possible ifn is sufficiently large. Financial support for this research was provided by the Carnegie Trust for the Universities of Scotland.     

An Erdo&#x030B;s–Ko–Rado Theorem for Direct Products

Letn_i, k_ibe positive integers,i=1, ..., d,satisfyingn_i≥2k_i.LetX₁, ..., X_dbe pairwise disjoint sets with |X_i| =n_i.Letbe the family of those (k₁+···+k_d)-element sets which have exactlyk_ielements inX_i, i=1,..., d.It is shown that if⊂is an intersecting family then ||/||≤max_ik_i/n_i,and this is best possible. The proof is algebraic, although in thed=2 case a combinatorial argument is presented as well.     

The theorem of the arithmetic and geometric means and its application to a problem in gas dynamics

Zusammenfassung Durch das Theorem der arithmetischen und geometrischen Mitte kann man den maximalen Wert des Produktes von Variablen, die einer linearen Beschränkung unterworfen sind, feststellen. Die vorliegende Arbeit untersucht einige einfache Versionen des Theorems. Die allgemeinste Version ergibt den maximalen Wert des Produktes vonn Variablen, wenn jede Variable zu einer höheren Potenz erhoben wird und die Variablen einer Gruppe vonr linearen Beschränkungen unterworfen sind (r<n). Das Theorem wird dann auf das Problem des maximalen totalen Druckwiedergewinns über einem Stosswellensystem angewandt und schliesslich wird daraus ein ziemlich allgemeines Theorem über das adiabatische Fliessen eines Gases hergeleitet.

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List of Symbols a _ij given positive constants - b , - f _i , - g _i , - h , - k , - m the arithmetic mean ofn variables, see Equation (2) - M the Mach number - n the number of variables - P _T the total pressure - q the geometric mean ofn variables, see Equation (1) - r the number of constraining conditions on the function to be maximised - R the gas constant - S the entropy - x _i for the Oswatitsch analysis, otherwisex _i represents any variable - y _i - ratio of specific heats - _j Lagrange multiplier - _i given positive constant - W _i shock wave angle of (i–1)th shock     

A queueing network with a single cyclically roving server

A queueingnetwork that is served by asingle server in a cyclic order is analyzed in this paper. Customers arrive at the queues from outside the network according to independent Poisson processes. Upon completion of his service, a customer mayleave the network, berouted to another queue in the network orrejoin the same queue for another portion of service. The single server moves through the different queues of the network in a cyclic manner. Whenever the server arrives at a queue (polls the queue), he serves the waiting customers in that queue according to some service discipline. Both the gated and the exhaustive disciplines are considered. When moving from one queue to the next queue, the server incurs a switch-over period. This queueing network model has many applications in communication, computer, robotics and manufacturing systems. Examples include token rings, single-processor multi-task systems and others. For this model, we derive the generating function and the expected number of customers present in the network queues at arbitrary epochs, and compute the expected values of the delays observed by the customers. In addition, we derive the expected delay of customers that follow a specific route in the network, and we introduce pseudo-conservation laws for this network of queues.Summary of notation B_i, B _i ^* (s) service time of a customer at queue i and its LST - b_i, b_i ⁽²⁾ mean and second moment of B_i - R_i, R _i ^* (s) duration of switch-over period from queue i and its LST - r_i, r_i mean and second moment of R_i - r, r⁽²⁾ mean and second moment of _i ^N =₁R_i - _i external arrival rate of type-i customers - _i total arrival rate into queue i - _i utilization of queue i; _i=_i - system utilization _i ^N =_1i - c=E[C] the expected cycle length - X _i ^j number of customers in queue j when queue i is polled - X_i=X _i ⁱ number of customers residing in queue i when it is polled - f_i(j) - X _i ^* number of customers residing in queue i at an arbitrary moment - Y_i the duration of a service period of queue i - W_i,T_i the waiting time and sojourn time of an arbitary customer at queue i - F^*(z₁, z₂,..., z_N) GF of number of customers present at the queues at arbitrary moments - F_i(z₁, z₂,..., z_N) GF of number of customers present at the queues at polling instants of queue i - ¯F_i(z₁, z₂,...,z_N) GF of number of customers present at the queues at switching instants of queue i - V_i(z₁, z₂,..., z_N) GF of number of customers present at the queues at service initiation instants at queue i - ¯V_i(z₁,z₂,...,z_N) GF of number of customers present at the queues at service completion instants at queue i The work of this author was supported by the Bernstein Fund for the Promotion of Research and by the Fund for the Promotion of Research at the Technion.Part of this work was done while H. Levy was with AT&T Bell Laboratories.     

Transversals of families in complete lattices,and torsion in product modules

Suppose L is a complete lattice containing no copy of the power-set 2 and no uncountable well-ordered chains. It is shown that for any family of nonempty subsets , one can choose elements p _i X _i so that _A p _i majorizes all elements of all but finitely many of the X _i . Ring-theoretic consequences are deduced: for instance, the direct product of a family of torsion modules over a commutative Noetherian integral domain R is torsion if and only if some element of R annihilates all but finitely many of the modules.