Padé approximations to certain power series

Summary An explicit identity involvingQ _n (q ⁱ z) (i = 0, 1,, 4) is shown, whereQ _n (z) is the denominator of thenth Padé approximant to the functionf(z) = _k=0 q ^1/2k(k–1 Z ^k . By using the Padé approximations, irrationality measures for certain values off(z) are also given.

     

Type II Hermite�CPad�� Approximations of Generalized Hypergeometric Series

We shall present short proofs for type II (simultaneous) Hermite–Padé approximations of the generalized hypergeometric and q-hypergeometric series

An Identity of Ramanujan and the Representation of Integers as Sums of Triangular Numbers

Let k be a positive number and t _k(n) denote the number of representations of n as a sum of k triangular numbers. In this paper, we will calculate t _2k (n) in the spirit of Ramanujan. We first use the complex theory of elliptic functions to prove a theta function identity. Then from this identity we derive two Lambert series identities, one of them is a well-known identity of Ramanujan. Using a variant form of Ramanujan's identity, we study two classes of Lambert series and derive some theta function identities related to these Lambert series . We calculate t ₁₂(n), t ₁₆(n), t ₂₀(n), t ₂₄(n), and t ₂₈(n) using these Lambert series identities. We also re-derive a recent result of H. H. Chan and K. S. Chua [6] about t ₃₂(n). In addition, we derive some identities involving the Ramanujan function (n), the divisor function ₁₁(n), and t ₂₄(n). Our methods do not depend upon the theory of modular forms and are somewhat more transparent.     

Asymptotic Formulae and Divisor Problems

We investigate the asymptotic behaviour of the summatory functions of _z(n, ), _k(n, ) z ⁽ⁿ⁾ and _k(n, ) z ⁽ⁿ⁾.     

On Marcel Riesz's Ultrahyperbolic Kernel

Let t = (t₁, …, t_n) be a point of ?ⁿ. We shall write . We put by definition R_α(u) = u^(α?n)/2/K_n(α); here α is a complex parameter, n the dimension of the space, and K_n(α) is a constant. First we evaluate □R_α(u) = R_α(u), where □ the ultrahyperbolic operator. Then we obtain the following results: R_?2k(u) = □^kδ; R₀(u) = δ; and □^kR_2k(u) = δ, k = 0, 1, …. The first result is the n-dimensional ultrahyperbolic correlative of the well-known one-dimensional formula . Equivalent formulas have been proved by Nozaki by a completely different method. The particular case µ = 1 was solved previously.     

On the code generated by the incidence matrix of points and k-spaces in and its dual

In this paper, we study the p-ary linear code C_k(n,q), q=p^h, p prime, h1, generated by the incidence matrix of points and k-dimensional spaces in PG(n,q). For kn/2, we link codewords of C_k(n,q)C_k(n,q) of weight smaller than 2q^k to k-blocking sets. We first prove that such a k-blocking set is uniquely reducible to a minimal k-blocking set, and exclude all codewords arising from small linear k-blocking sets. For k<n/2, we present counterexamples to lemmas valid for kn/2. Next, we study the dual code of C_k(n,q) and present a lower bound on the weight of the codewords, hence extending the results of Sachar [H. Sachar, The F_p span of the incidence matrix of a finite projective plane, Geom. Dedicata 8 (1979) 407–415] to general dimension.     

Equalities on the number of conjugacy classes of the Sylow p-subgroups of GL(n,q)

We denote by Gn the group of the upper unitriangular matrices over F_q, the finite field with q = p^t elements, and r(G_n) the number of conjugacy classes of G_n. In this paper, we obtain the value of r(G_n) modulo (q² -1)(q -1). We prove the following equalities     

Polynomial projectors preserving homogeneous partial differential equations

A polynomial projector Π of degree d on is said to preserve homogeneous partial differential equations (HPDE) of degree k if for every and every homogeneous polynomial of degree k, q(z)=∑_|α|=ka_αz^α, there holds the implication: q(D)f=0q(D)Π(f)=0. We prove that a polynomial projector Π preserves HPDE of degree if and only if there are analytic functionals with such that Π is represented in the following form

with some , where u_α(z)z^α/α!. Moreover, we give an example of polynomial projectors preserving HPDE of degree k (k1) without preserving HPDE of smaller degree. We also give a characterization of Abel–Gontcharoff projectors as the only Birkhoff polynomial projectors that preserve all HPDE.     

Finite and p-adic Polylogarithms

The finite nth polylogarithm li _n (z) /p(z) is defined as _k=1 ^p–1 z ^k /k ⁿ . We state and prove the following theorem. Let Li _k : _{p p} be the p-adic polylogarithms defined by Coleman. Then a certain linear combination F _n of products of polylogarithms and logarithms, with coefficients which are independent of p, has the property that p ^1–n DF _n (z) reduces modulo p>n+1 to li _n–1((z)), where D is the Cathelineau operator z(1–z)d/dz and is the inverse of the p-power map. A slightly modified version of this theorem was conjectured by Kontsevich. This theorem is used by Elbaz-Vincent and Gangl to deduce functional equations of finite polylogarithms from those of complex polylogarithms.     

Control of stochastic distributed-parameter systems

A scheme is proposed for the feedback control of distributed-parameter systems with unknown boundary and volume disturbances and observation errors. The scheme consists of employing a nonlinear filter in the control loop such that the controller uses the optimal estimates of the state of the system. A theoretical comparison of feedback proportional control of a styrene polymerization reactor with and without filtering is presented. It is indicated how an approximate filter can be constructed, greatly reducing the amount of computing required.Notation a(t) l-vector noisy dynamic input to system - A(t, a) l-vector function - A frequency factor for first-order rate law (5.68×10⁶ sec^–1) - b distance to the centerline between two coil banks in the reactor (4.7 cm) - B k-vector function defining the control action - c(, ) concentration of styrene monomer, molel ^–1 - C _p heat capacity (0.43 cal · g^–1 · K^–1) - C _ij constants in approximate filter, Eqs. (49)–(52) - E activation energy (20330 cal · mole^–1) - expectation operator - f(t,...) n-vector function - g ₀,g ₁(t,...) n-vector functions - h(t, u) m-vector function relating observations to states - H(t) function defined in Eq. (36) - k dimensionality of control vectorv(x, t) - k _i constants in approximate filter, Eqs. (49)–(52) - K dimensionless proportional gain - l dimensionality of dynamic inputa(t) - m dimensionality of observation vectory(t) - n dimensionality of state vectoru(x, t) - P ^(vv)(x, s, t) n×n matrix governed by Eq. (9) - P ^(va)(x, t) n×l matrix governed by Eq. (10) - P ^(aa)(t) l×l matrix governed by Eq. (11) - q _i (t) diagonal elements ofm×m matrixQ(x, s, t) - Q(x, s, t) m×m weighting matrix - R universal gas constant (1.987 cal · mole^–1 · K^–1) - R(x, s, t) n×n weighting matrix - R _i (t) n×n weighting matrix - s dimensionless spatial variable - S(x, s, t) matrix defined in Eq. (11) - t dimensionless time variable - T(, ) temperature, K - u(x, t) n-dimensional state vector - u _c (t) wall temperature - u _d desired value ofu ₁(1,t) - u _c * reference control value ofu _c - u _c ^max maximum value ofu _c - u _c ^min minimum value of _c - v(x, t) k-dimensional control vector - W(t) l×l weighting matrix - x dimensionless spatial variable - y(t) m-dimensional observation vector - _i constants in approximate filter, Eqs. (49)–(52) - dimensionless parameter defined in Eq. (29) - H heat of reaction (17500 cal · mole^–1) - dimensionless activation energy, defined in Eq. (29) - (x) Dirac delta function - (x, t) m-dimensional observation noise - thermal conductivity (0.43×10^–3 cal · cm^–1 · sec^–1 · K^–1) - density (1 g · cm^–3) - time, sec - dimensionless parameter defined in Eq. (29) - spatial variable, cm - * reference value - ^ estimated value     

A Sharp Threshold for k‐Colorability

Let k be a fixed integer and f_k(n, p) denote the probability that the random graph G(n, p) is k‐colorable. We show that for k≥3, there exists d_k(n) such that for any ϵ>0, (1) As a result we conclude that for sufficiently large n the chromatic number of G(n, d/n) is concentrated in one value for all but a small fraction of d>1. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 14, 63–70, 1999     

Turán problems for integer‐weighted graphs

A multigraph is (k,r)‐dense if every k‐set spans at most r edges. What is the maximum number of edges ex_?(n,k,r) in a (k,r)‐dense multigraph on n vertices? We determine the maximum possible weight of such graphs for almost all k and r (e.g., for all r>k³) by determining a constant m=m(k,r) and showing that ex_?(n,k,r)=m +O(n), thus giving a generalization of Turán's theorem. We find exact answers in many cases, even when negative integer weights are also allowed. In fact, our main result is to determine the maximum weight of (k,r)‐dense n‐vertex multigraphs with arbitrary integer weights with an O(n) error term. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 195–225, 2002     

Non-uniform Turán-type problems

Given positive integers n,k,t, with 2?k?n, and t<2^k, let m(n,k,t) be the minimum size of a family F of (nonempty distinct) subsets of [n] such that every k-subset of [n] contains at least t members of F, and every (k-1)-subset of [n] contains at most t-1 members of F. For fixed k and t, we determine the order of magnitude of m(n,k,t). We also consider related Turán numbers T_?r(n,k,t) and T_r(n,k,t), where T_?r(n,k,t) (T_r(n,k,t)) denotes the minimum size of a family such that every k-subset of [n] contains at least t members of F. We prove that T_?r(n,k,t)=(1+o(1))T_r(n,k,t) for fixed r,k,t with and n→∞.     

On the real part of ultraflat sequences of unimodular polynomials: Consequences implied by the resolution of the phase problem

 Let P _n (z)=∑ _k=0 ⁿ a _k,n z ^k ℂ[z] be a sequence of unimodular polynomials (|a _k,n |=1 for all k, n) which is ultraflat in the sense of Kahane, i.e.,

A constructive approach for the lower bounds on the Ramsey numbers R (s,t)

Graph G is a (k, p)‐graph if G does not contain a complete graph on k vertices K_k, nor an independent set of order p. Given a (k, p)‐graph G and a (k, q)‐graph H, such that G and H contain an induced subgraph isomorphic to some K_k?1‐free graph M, we construct a (k, p + q ? 1)‐graph on n(G) + n(H) + n(M) vertices. This implies that R (k, p + q ? 1) ≥ R (k, p) + R (k, q) + n(M) ? 1, where R (s, t) is the classical two‐color Ramsey number. By applying this construction, and some its generalizations, we improve on 22 lower bounds for R (s, t), for various specific values of s and t. In particular, we obtain the following new lower bounds: R (4, 15) ≥ 153, R (6, 7) ≥ 111, R (6, 11) ≥ 253, R (7, 12) ≥ 416, and R (8, 13) ≥ 635. Most of the results did not require any use of computer algorithms. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 231–239, 2004     

On vector space partitions and uniformly resolvable designs

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 vector in V _n (q) is contained in exactly one subspace in . A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the same size. A partition of V _n (q) containing a _i subspaces of dimension n _i for 1 ≤ i ≤ k induces a uniformly resolvable design on q ⁿ points with a _i parallel classes with block size , 1 ≤ i ≤ k, and also corresponds to a factorization of the complete graph into -factors, 1 ≤ i ≤ k. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable designs on q ⁿ points where corresponding partitions of V _n (q) do not exist. A. D. Blinco—Part of this research was done while the author was visiting Illinois State University.     

Some Upper Bounds on the Covering Radii of Linear Codes Over Fq and Their Applications

We show that the covering radius R of an [n,k,d] code over F_q is bounded above by R n-n _q(k, d/q). We strengthen this bound when R d and find conditions under which equality holds.As applications of this and other bounds, we show that all binary linear codes of lengths up to 15, or codimension up to 9, are normal. We also establish the normality of most codes of length 16 and many of codimension 10. These results have applications in the construction of codes that attain t[n,k,/it>], the smallest covering radius of any binary linear [n,k].We also prove some new results on the amalgamated direct sum (ADS) construction of Graham and Sloane. We find new conditions assuring normality of the ADS; covering radius 1 less than previously guaranteed for ADS of codes with even norms; good covering codes as ADS without the hypothesis of normality, from concepts p- stable and s- stable; codes with best known covering radii as ADS of two, often cyclic, codes (thus retaining structure so as to be suitable for practical applications).     

Distortion Properties of Holomorphic Functions of Several Complex Variables

Let Q(D) be a class of functions q, q(0) = 0, |q(z)| < 1 holomorphic in the Reinhardt domain D ? C ⁿ, a and b — arbitrary fixed numbers satisfying the condition — 1 ≤ b < a ≤ 1. ??(a, b; D) — the class of functions p such that p ? ??(a, b; D) iff for some q ? Q(D) and every z ? D. S*(a, b; D) — the class of functions f such that f ? S*(a, g; D) iff S^c(a, b; D) — the class of functions q such that q ? S^c(a, b; D) iff , where p ε ??(a, b; D) and K is an operator of the form for z=z₁,z₂,…z_n. The author obtains sharp bounds on |p(z)|, f(z)| g(z)| as well as sharp coefficient inequalities for functions in ??(a, b; D), S*(a, b; D) and S^c(a, b; D).     

Rate of expansion of an inhomogeneous branching process of brownian particles

Summary Let X be the (B ⁰, {q _n (x)})-branching diffusion where B ⁰is the exp -subprocess of BM(R¹) and q _n (x) is the probability that a particle dying at x produces n offspring, q ₀ q ₁0. Put m(x) = nq _n (x). We assume q _n , n2, m and k are all continuous (but m is not necessarily bounded). If k(x)m(x)0 as ¦x¦, then we prove that R _t /t( ₂/2)^1/2, as t, a.s. and in mean (of any order) where R _t is the position of the rightmost particle at time t and ₀ is the largest eigenvalue of (1/2)d ²/dx ² + Q, Q(x) = k(x)(m(x)–1).This work was supported in part by a grant from the National Science Foundation # MCS-8201470.