B-splines and Hermite–Padé approximants to the exponential function

This paper is the continuation of a work initiated in [P. Sablonnière, An algorithm for the computation of Hermite–Padé approximations to the exponential function: divided differences and Hermite–Padé forms. Numer. Algorithms 33 (2003) 443–452] about the computation of Hermite–Padé forms (HPF) and associated Hermite–Padé approximants (HPA) to the exponential function. We present an alternative algorithm for their computation, based on the representation of HPF in terms of integral remainders with B-splines as Peano kernels. Using the good properties of discrete B-splines, this algorithm gives rise to a great variety of representations of HPF of higher orders in terms of HPF of lower orders, and in particular of classical Padé forms. We give some examples illustrating this algorithm, in particular, another way of constructing quadratic HPF already described by different authors. Finally, we briefly study a family of cubic HPF.     

Uniformly Gâteaux smooth approximations on

Every Lipschitz mapping from c₀(Γ) into a Banach space Y can be uniformly approximated by Lipschitz mappings that are simultaneously uniformly Gâteaux smooth and C^∞-Fréchet smooth.     

Discrete -convex extremal distributions: Theory and applications

Given a nondegenerate moment space with s fixed moments, explicit formulas for the discrete s-convex extremal distribution have been derived for s=1,2,3 (see [M. Denuit, Cl. Lefèvre, Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences, Insurance Math. Econom. 20 (1997) 197–214]). If s=4, only the maximal distribution is known (see [M. Denuit, Cl. Lefèvre, M. Mesfioui, On s-convex stochastic extrema for arithmetic risks, Insurance Math. Econom. 25 (1999) 143–155]). This work goes beyond this limitation and proposes a method for deriving explicit expressions for general nonnegative integer s. In particular, we derive explicitly the discrete 4-convex minimal distribution. For illustration, we show how this theory allows one to bound the probability of extinction in a Galton–Watson branching process. The results are also applied to derive bounds for the probability of ruin in the compound binomial and Poisson insurance risk models.     

On the Convergence of Bounded J-Fractions on the Resolvent Set of the Corresponding Second Order Difference Operator

We study connections between continued fractions of type J and spectral properties of second order difference operators with complex coefficients. It is known that the convergents of a bounded J-fraction are diagonal Padé approximants of the Weyl function of the corresponding difference operator and that a bounded J-fraction converges uniformly to the Weyl function in some neighborhood of infinity. In this paper we establish convergence in capacity in the unbounded connected component of the resolvent set of the difference operator and specify the rate of convergence. Furthermore, we show that the absence of poles of Padé approximants in some subdomain implies already local uniform convergence. This enables us to verify the Baker–Gammel–Wills conjecture for a subclass of Weyl functions. For establishing these convergence results, we study the ratio and the nth root asymptotic behavior of Padé denominators of bounded J-fractions and give relations with the Green function of the unbounded connected component of the resolvent set. In addition, we show that the number of “spurious” Padé poles in this set may be bounded.     

Quadratic Hermite–Padé polynomials associated with the exponential function

The asymptotic behavior of quadratic Hermite–Padé polynomials associated with the exponential function is studied for n→∞. These polynomials are defined by the relation

(*)

p_n(z)+q_n(z)e^z+r_n(z)e^2z=O(z³ⁿ⁺²) as z→0,

where O(·) denotes Landau's symbol. In the investigation analytic expressions are proved for the asymptotics of the polynomials, for the asymptotics of the remainder term in (*), and also for the arcs on which the zeros of the polynomials and of the remainder term cluster if the independent variable z is rescaled in an appropriate way. The asymptotic expressions are defined with the help of an algebraic function of third degree and its associated Riemann surface. Among other possible applications, the results form the basis for the investigation of the convergence of quadratic Hermite–Padé approximants, which will be done in a follow-up paper.     

Highly connected coloured subgraphs via the regularity lemma

For integers , n≥k and r≥s, let m(n,r,s,k) be the largest (in order) k-connected component with at most s colours one can find in any r-colouring of the edges of the complete graph K_n on n vertices. Bollobás asked for the determination of m(n,r,s,k).Here, bounds are obtained in the cases s=1,2 and k=o(n), which extend results of Liu, Morris and Prince. Our techniques use Szemerédi’s Regularity Lemma for many colours.We shall also study a similar question for bipartite graphs.     

Second-order symmetric duality with cone constraints

Wolfe and Mond–Weir type second-order symmetric duals are formulated and appropriate duality theorems are established under η-bonvexity/η-pseudobonvexity assumptions. This formulation removes several omissions in an earlier second-order primal dual pair introduced by Devi [Symmetric duality for nonlinear programming problems involving η-bonvex functions, European J. Oper. Res. 104 (1998) 615–621].     

On the distribution of poles of Padé approximants to the Z-transform of complex Gaussian white noise

In the application of Padé methods to signal processing a basic problem is to take into account the effect of measurement noise on the computed approximants. Qualitative deterministic noise models have been proposed which are consistent with experimental results. In this paper the Padé approximants to the Z-transform of a complex Gaussian discrete white noise process are considered. Properties of the condensed density of the Padé poles such as circular symmetry, asymptotic concentration on the unit circle and independence on the noise variance are proved. An analytic model of the condensed density of the Padé poles for all orders of the approximants is also computed. Some Monte Carlo simulations are provided.     

Legendre modified moments for Euler's constant

Polynomial moments are often used for the computation of Gauss quadrature to stabilize the numerical calculation of the orthogonal polynomials, see [W. Gautschi, Computational aspects of orthogonal polynomials, in: P. Nevai (Ed.), Orthogonal Polynomials-Theory and Practice, NATO ASI Series, Series C: Mathematical and Physical Sciences, vol. 294. Kluwer, Dordrecht, 1990, pp. 181–216 [6]; W. Gautschi, On the sensitivity of orthogonal polynomials to perturbations in the moments, Numer. Math. 48(4) (1986) 369–382 [5]; W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Statist. Comput. 3(3) (1982) 289–317 [4]] or numerical resolution of linear systems [C. Brezinski, Padé-type approximation and general orthogonal polynomials, ISNM, vol. 50, Basel, Boston, Stuttgart, Birkhäuser, 1980 [3]]. These modified moments can also be used to accelerate the convergence of sequences to a real or complex numbers if the error satisfies some properties as done in [C. Brezinski, Accélération de la convergence en analyse numérique, Lecture Notes in Mathematics, vol. 584. Springer, Berlin, New York, 1977; M. Prévost, Padé-type approximants with orthogonal generating polynomials, J. Comput. Appl. Math. 9(4) (1983) 333–346]. In this paper, we use Legendre modified moments to accelerate the convergence of the sequence H_n-log(n+1) to the Euler's constant γ. A formula for the error is given. It is proved that it is a totally monotonic sequence. At last, we give applications to the arithmetic property of γ.     

Scalar and matrix Riemann–Hilbert approach to the strong asymptotics of Padé approximants and complex orthogonal polynomials with varying weight

We describe methods for the derivation of strong asymptotics for the denominator polynomials and the remainder of Padé approximants for a Markov function with a complex and varying weight. Two approaches, both based on a Riemann–Hilbert problem, are presented. The first method uses a scalar Riemann–Hilbert boundary value problem on a two-sheeted Riemann surface, the second approach uses a matrix Riemann–Hilbert problem. The result for a varying weight is not with the most general conditions possible, but the loss of generality is compensated by an easier and transparent proof.     

A filtration of -Catalan numbers

The operator of F. Bergeron, Garsia, Haiman and Tesler [F. Bergeron, A. Garsia, M. Haiman, G. Tesler, Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions, Methods Appl. Anal. 6 (1999) 363–420] acting on the k-Schur functions [L. Lapointe, A. Lascoux, J. Morse, Tableaux atoms and a new Macdonald positivity conjecture, Duke Math. J. 116 (2003) 103–146; L. Lapointe, J. Morse, Schur functions analogs for a filtration of the symmetric functions space, J. Combin. Theory Ser. A 101 (2003) 191–224; L. Lapointe, J. Morse, Tableaux on k+1-cores, reduced words for affine permutations and k-Schur expansion, J. Combin. Theory Ser. A 112 (2005) 44–81] indexed by a single column has a coefficient in the expansion which is an analogue of the (q,t)-Catalan number with a level k. When k divides n we conjecture a representation theoretical model in this case such that the graded dimensions of the module are the coefficients of the (q,t)-Catalan polynomials of level k. When the parameter t is set to 1, the Catalan numbers of level k are shown to count the number of Dyck paths that lie below a certain Dyck path with q counting the area of the path.     

Ordinal machines and admissible recursion theory

We generalize standard Turing machines, which work in time ω on a tape of length ω, to α-machines with time α and tape length α, for α some limit ordinal. We show that this provides a simple machine model adequate for classical admissible recursion theory as developed by G. Sacks and his school. For α an admissible ordinal, the basic notions of α-recursive or α-recursively enumerable are equivalent to being computable or computably enumerable by an α-machine, respectively. We emphasize the algorithmic approach to admissible recursion theory by indicating how the proof of the Sacks–Simpson theorem, i.e., the solution of Post’s problem in α-recursion theory, could be based on α-machines, without involving constructibility theory.     

Über ein Turánsches problem für radiale, positiv definite Funktionen, II

Turán's problem is to determine the greatest possible value of the integral for positive definite functions f(x), , supported in a given convex centrally symmetric body , . We consider the problem for positive definite functions of the form f(x)=(x₁), , with supported in [0,π], extending results of our first paper from two to arbitrary dimensions.Our two papers were motivated by investigations of Professor Y. Xu and the 2nd named author on, what they called, ℓ-1 summability of the inverse Fourier integral on . Their investigations gave rise to a pair of transformations (h_d,m_d) on which they studied using special functions, in particular spherical Bessel functions.To study the d-dimensional Turán problem, we had to extend relevant results of B. & X., and we did so using again Bessel functions. These extentions seem to us to be equally interesting as the application to Turán's problem.     

Complexity of the min–max (regret) versions of min cut problems

This paper investigates the complexity of the min–max and min–max regret versions of the min s–t cut and min cut problems. Even if the underlying problems are closely related and both polynomial, the complexities of their min–max and min–max regret versions, for a constant number of scenarios, are quite contrasted since they are respectively strongly NP-hard and polynomial. However, for a non-constant number of scenarios, these versions become strongly NP-hard for both problems. In the interval scenario case, min–max versions are trivially polynomial. Moreover, for min–max regret versions, we obtain the same contrasted results as for a constant number of scenarios: min–max regret min s–t cut is strongly NP-hard whereas min–max regret min cut is polynomial.     

Central limit theorem and the bootstrap for -statistics of strongly mixing data

The asymptotic normality of U-statistics has so far been proved for iid data and under various mixing conditions such as absolute regularity, but not for strong mixing. We use a coupling technique introduced in 1983 by Bradley [R.C. Bradley, Approximation theorems for strongly mixing random variables, Michigan Math. J. 30 (1983),69–81] to prove a new generalized covariance inequality similar to Yoshihara’s [K. Yoshihara, Limiting behavior of U-statistics for stationary, absolutely regular processes, Z. Wahrsch. Verw. Gebiete 35 (1976), 237–252]. It follows from the Hoeffding-decomposition and this inequality that U-statistics of strongly mixing observations converge to a normal limit if the kernel of the U-statistic fulfills some moment and continuity conditions.The validity of the bootstrap for U-statistics has until now only been established in the case of iid data (see [P.J. Bickel, D.A. Freedman, Some asymptotic theory for the bootstrap, Ann. Statist. 9 (1981), 1196–1217]. For mixing data, Politis and Romano [D.N. Politis, J.P. Romano, A circular block resampling procedure for stationary data, in: R. Lepage, L. Billard (Eds.), Exploring the Limits of Bootstrap, Wiley, New York, 1992, pp. 263–270] proposed the circular block bootstrap, which leads to a consistent estimation of the sample mean’s distribution. We extend these results to U-statistics of weakly dependent data and prove a CLT for the circular block bootstrap version of U-statistics under absolute regularity and strong mixing. We also calculate a rate of convergence for the bootstrap variance estimator of a U-statistic and give some simulation results.     

The Fréchet and limiting subdifferentials of integral functionals on the spaces

A new approach to computing the Fréchet subdifferential and the limiting subdifferential of integral functionals is proposed. Thanks to this way, we obtain formulae for computing the Fréchet and limiting subdifferentials of the integral functional , uL₁(Ω,E). Here is a measured space with an atomless σ-finite complete positive measure, E is a separable Banach space, and . Under some assumptions, it turns out that these subdifferentials coincide with the Fenchel subdifferential of F.     

Optimality conditions for multiple objective fractional subset programming with ( , ρ,σ,θ )-V-type-I and related non-convex functions

In this paper, we introduce a new class of generalized convex n-set functions, called ( , ρ,σ,θ)-V-Type-I and related non-convex functions, and then establish a number of parametric and semi-parametric sufficient optimality conditions for the primal problem under the aforesaid assumptions. This work partially extends an earlier work of [G.J. Zalmai, Efficiency conditions and duality models for multiobjective fractional subset programming problems with generalized ( , α, ρ, θ)-V-convex functions, Comput. Math. Appl. 43 (2002) 1489–1520] to a wider class of functions.     

On reality and asymptotics of zeros of -Hankel transforms

We give sufficient conditions which guarantee that the finite q-Hankel transforms have only real zeros which satisfy some asymptotic relations. The study is carried out using two different techniques. The first is by a use of Rouché's theorem and the other is by applying a theorem of Hurwitz and Biehler. In every study further restrictions are imposed on q(0,1). We compare the results via some interesting applications involving second and third q-Bessel functions as well as q-trigonometric functions.     

An algorithmic sign-reversing involution for special rim-hook tableaux

Eğecioğlu and Remmel [Linear Multilinear Algebra 26 (1990) 59–84] gave an interpretation for the entries of the inverse Kostka matrix K⁻¹ in terms of special rim-hook tableaux. They were able to use this interpretation to give a combinatorial proof that KK⁻¹=I but were unable to do the same for the equation K⁻¹K=I. We define an algorithmic sign-reversing involution on rooted special rim-hook tableaux which can be used to prove that the last column of this second product is correct. In addition, following a suggestion of Chow [preprint, math.CO/9712230, 1997] we combine our involution with a result of Gasharov [Discrete Math. 157 (1996) 193–197] to give a combinatorial proof of a special case of the (3+1)-free Conjecture of Stanley and Stembridge [J. Combin. Theory Ser. A 62 (1993) 261–279].