A Product Formula for Jackson Integral Associated with the Root System F 4

A multiple summation formula for the root system F ₄ originated from Bailey's very-well-poised ₆₆ summation theorem is studied. The multiple sum in this paper is included in a classification list (M. Ito, Compositio Math., 129 (2001), 325–340) of Jackson integral associated with irreducible reduced root system which contains sums investigated by Gustafson, Macdonald, Aomoto, and the author.     

An Expansion Formula for q-Series and Applications

In this paper the author proves a q-expansion formula which utilizes the Leibniz formula for the q-differential operator. This expansion leads to new proofs of the Rogers–Fine identity, the nonterminating ₆₅ summation formula, and Watson's q-analog of Whipple's theorem. Andrews' identities for sums of three squares and sums of three triangular numbers are also derived. Other identities of Andrews and new identities for Hecke type series are also discussed.     

A Simple Proof of the Borel Extension Theorem and Weak Compactness of Operators

Let T be a locally compact Hausdorff space and let C ₀(T) be the Banach space of all complex valued continuous functions vanishing at infinity in T, provided with the supremum norm. Let X be a quasicomplete locally convex Hausdorff space. A simple proof of the theorem on regular Borel extension of X-valued -additive Baire measures on T is given, which is more natural and direct than the existing ones. Using this result the integral representation and weak compactness of a continuous linear map u: C ₀(T) X when c ₀ X are obtained. The proof of the latter result is independent of the use of powerful results such as Theorem 6 of [6] or Theorem 3 (vii) of [13].     

Infinite Families of Exact Sums of Squares Formulas,Jacobi Elliptic Functions,Continued Fractions,and Schur Functions

In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's 4 and 8 squares identities to 4n ² or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions. In addition, we extend Jacobi's special analysis of 2 squares, 2 triangles, 6 squares, 6 triangles to 12 squares, 12 triangles, 20 squares, 20 triangles, respectively. Our 24 squares identity leads to a different formula for Ramanujan's tau function (n), when n is odd. These results, depending on new expansions for powers of various products of classical theta functions, arise in the setting of Jacobi elliptic functions, associated continued fractions, regular C-fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. The Schur function form of these infinite families of identities are analogous to the -function identities of Macdonald. Moreover, the powers 4n(n + 1), 2n ² + n, 2n ² – n that appear in Macdonald's work also arise at appropriate places in our analysis. A special case of our general methods yields a proof of the two Kac–Wakimoto conjectured identities involving representing a positive integer by sums of 4n ² or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson's C nonterminating ₆₅ summation theorem, and Andrews' basic hypergeometric series proof of Jacobi's 2, 4, 6, and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n ² and n(n + 1) squares. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Sierpinski (1907), Uspensky (1913, 1925, 1928), Bulygin (1914, 1915), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Bell (1919), Estermann (1936), Rankin (1945, 1962), Lomadze (1948), Walton (1949), Walfisz (1952), Ananda-Rau (1954), van der Pol (1954), Krätzel (1961, 1962), Bhaskaran (1969), Gundlach (1978), Kac and Wakimoto (1994), and, Liu (2001). We list these authors by the years their work appeared.     

On the Weak Limiting Behavior of Almost Surely Convergent Row Sums from Infinite Arrays of Rowwise Independent Random Elements in Banach Spaces

For an array {V _nk ,k1,n1} of rowwise independent random elements in a real separable Banach space with almost surely convergent row sums , we provide criteria for S _n –A _n to be stochastically bounded or for the weak law of large numbers to hold where {A _n ,n1} is a (nonrandom) sequence in .     

Approximation Representations for Δ<Subscript>2</Subscript> Reals

We study ₂ reals x in terms of how they can be approximated symmetrically by a computable sequence of rationals. We deal with a natural notion of approximation representation and study how these are related computationally for a fixed x. This is a continuation of earlier work; it aims at a classification of ₂ reals based on approximation and it turns out to be quite different than the existing ones (based on information content etc.)Mathematics Subject Classification (2000): 03F60, 03D30     

C k -B-splines with Square Support on a Three-Direction Mesh of the Plane

Let be the uniform triangulation generated by the usual three-directional mesh of the plane and let ₁ be the unit square consisting of two triangles of . We study the space of piecewise polynomial functions in C ^k (R ²) with support ₁ having a sufficiently high degree n, which are symmetrical with respect to the first diagonal of ₁. Such splines are called ₁-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k0, we prove the existence of ₁-splines of class C ^k and minimal degree. These splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines, and we give an example.     

Dn Basic Hypergeometric Series

We study multiple series extensions of basic hypergeometric series related to the root system D_n. We make a small change in the notation used for C_n and D_n series to bring them closer to A_n series. This allows us to combine the three types of series, and get D_n extensions of the following classical summation and transformation theorems: The q-Pfaff-Saalschütz summation, Rogers' _{6 5} sum, the q-Gauss summation, q-Chu-Vandermonde summations, Watson's q-analogue of Whipple's transformation, and the q-Dougall summation theorem. We also define A_n and C_n extensions of the Rogers-Selberg function, and prove a reduction formula for both of them. This generalizes some work of Andrews. We use some techniques originally developed to study multiple basic hypergeometric series related to the root system A_n (U(n + 1) basic hypergeometric series).     

A simple technique for computing theH ∞-norm of polynomials

In this note, computation of theH -norm of polynomials is considered. It is shown that direct computation of theH -norm of polynomials, based on the definition of the norm, results in a simple and inexpensive technique for computing the norm.     

ℓ1-Rigid Graphs

An ₁-graph is a graph whose nodes can be labeled by binary vectors in such a way that the Hamming distance between the binary addresses is, up to scale, the distance in the graph between the corresponding nodes. We show that many interesting graphs are ₁-rigid, i.e., that they admit an essentially unique such binary labeling.     

Existence and construction of Δ1-splines of class Ck on a three-directional mesh

Let ^* be the equilateral triangulation of the plane and let ₁ ^* be the equilateral triangle formed by four triangles of ^*. We study the space of piecewise polynomial functions in C ^k (R ²) with support ₁ ^*, having a sufficiently high degree n and which are invariant with respect to the group of symmetries of ₁ ^*. Such splines are called ₁ ^*-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k0, we prove the existence of ₁ ^*-splines of class C ^k and minimal degree, but these splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines.     

Asymptotic Analysis and Solution of a Finite-Horizon H ∞ Control Problem for Singularly-Perturbed Linear Systems with Small State Delay

A finite-horizon H state-feedback control problem for singularly-perturbed linear time-dependent systems with a small state delay is considered. Two approaches to the asymptotic analysis and solution of this problem are proposed. In the first approach, an asymptotic solution of the singularly-perturbed system of functional-differential equations of Riccati type, associated with the original H problem by the sufficient conditions of the existence of its solution, is constructed. Based on this asymptotic solution, conditions for the existence of a solution of the original H problem, independent of the small parameter of singular perturbations, are derived. A simplified controller with parameter-independent gain matrices, solving the original H problem for all sufficiently small values of this parameter, is obtained. In the second approach, the original H problem is decomposed into two lower-dimensional parameter-independent H subproblems, the reduced-order (slow) and the boundary-layer (fast) subproblems; controllers solving these subproblems are constructed. Based on these controllers, a composite controller is derived, which solves the original H problem for all sufficiently small values of the singular perturbation parameter. An illustrative example is presented.     

Transfer Function Optimization Procedure for the H 2/H ∞ Problem

In this paper, the H ₂/H problem is considered in a transfer-function setting, i.e., without a priori chosen bounds on the controller order. An optimization procedure is described which is based on a parametrization of all feasible descending directions stemming from a given point of the feasible transfer-function set. A search direction at each such point can be obtained on the basis of the solution of a convex finite-dimensional problem which can be converted into a LMI problem. Moving along the chosen direction in each step, the procedure in question generates a sequence of feasible points whose cost functional values converge to the optimal value of the H ₂/H problem. Moreover, this sequence of feasible points is shown to converge in the sense of a weighted H ₂ norm; and it does so to the solution of the H ₂/H problem whenever such a solution exists.     

H∞-Controllers for Time-Delay Systems Using Linear Matrix Inequalities

In this paper, H -control design is developed for nominally linear systems with input as well as state delays. Both stability and H -norm bound conditions are established for asymptotically stable controlled systems. Necessary and sufficient conditions for feedback control synthesis are established first by using two forms; the first has one term representing pure state feedback, and the second has two terms comprising pure state feedback plus delayed state feedback. Then, the corresponding synthesis conditions for the cases of static-output feedback and observer-based feedback controllers are developed. The results are cast conveniently into a linear matrix inequality (LMI) framework, which can be solved numerically by efficient interior-point methods. With the aid of the LMI control toolbox software, the theoretical work is illustrated by computer simulation of numerous examples.     

On the gi/m/∞ service system with batch arrivals and different types of service distributions

We study the infinite-server system with batch arrivals ands different types of customers. With probabilityp _i an arriving customer is of typei (i=1,..., s) and requires an exponentially distributed service time with parameter _i (G ^GI /M ₁ ...M _s /). For theGI ^GI /M ₁...M _s / system it is shown that the binomial moments of thes-variate distribution of the number of type-i customers in the system at batch arrival epochs are determined by a recurrence relation and, in steady state, can be computed recursively. Furthermore, forG ^GI /M ₁...M _s /, relations between the distributions (and their binomial moments) of the system size vector at batch arrival and random epochs are given. Thus, earlier results by Takács [14], Gastwirth [9], Holman et al. [11], Brandt et al. [3] and Franken [6] are generalized.     

Ternary Code Construction of Unimodular Lattices and Self-Dual Codes over ℤ6

We revisit the construction method of even unimodular lattices using ternary self-dual codes given by the third author (M. Ozeki, in Théorie des nombres, J.-M. De Koninck and C. Levesque (Eds.) (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 772–784), in order to apply the method to odd unimodular lattices and give some extremal (even and odd) unimodular lattices explicitly. In passing we correct an error on the condition for the minimum norm of the lattices of dimension a multiple of 12. As the results of our present research, extremal odd unimodular lattices in dimensions 44, 60 and 68 are constructed for the first time. It is shown that the unimodular lattices obtained by the method can be constructed from some self-dual ₆-codes. Then extremal self-dual ₆-codes of lengths 44, 48, 56, 60, 64 and 68 are constructed.