Hall–Littlewood functions and the Rogers–Ramanujan identities

We prove an identity for Hall–Littlewood symmetric functions labelled by the Lie algebra A₂. Through specialization this yields a simple proof of the A₂ Rogers–Ramanujan identities of Andrews, Schilling and the author.     

Studying Baskakov–Durrmeyer operators and quasi-interpolants via special functions

We prove that the kernels of the Baskakov–Durrmeyer and the Szász–Mirakjan–Durrmeyer operators are completely monotonic functions. We establish a Bernstein type inequality for these operators and apply the results to the quasi-interpolants recently introduced by Abel. For the Baskakov–Durrmeyer quasi-interpolants, we give a representation as linear combinations of the original Baskakov–Durrmeyer operators and prove an estimate of Jackson–Favard type and a direct theorem in terms of an appropriate K-functional.     

Conservation laws for Camassa–Holm equation, Dullin–Gottwald–Holm equation and generalized Dullin–Gottwald–Holm equation

In this paper we construct the conservation laws for the Camassa–Holm equation, the Dullin–Gottwald–Holm equation (DGH) and the generalized Dullin–Gottwald–Holm equation (generalized DGH). The variational derivative approach is used to derive the conservation laws. Only first order multipliers are considered. Two multipliers are obtained for the Camassa–Holm equation. For the DGH and generalized DGH equations the variational derivative approach yields two multipliers; thus two conserved vectors are obtained.     

Applications of Feller–Reuter–Riley Transition Functions

A Feller–Reuter–Riley function is a Markov transition function whose corresponding semigroup maps the set of the real-valued continuous functions vanishing at infinity into itself. The aim of this paper is to investigate applications of such functions in the dual problem, Markov branching processes, and the Williams-matrix. The remarkable property of a Feller–Reuter–Riley function is that it is a Feller minimal transition function with a stable q-matrix. By using this property we are able to prove that, in the theory of branching processes, the branching property is equivalent to the requirement that the corresponding transition function satisfies the Kolmogorov forward equations associated with a stable q-matrix. It follows that the probabilistic definition and the analytic definition for Markov branching processes are actually equivalent. Also, by using this property, together with the Resolvent Decomposition Theorem, a simple analytical proof of the Williams' existence theorem with respect to the Williams-matrix is obtained. The close link between the dual problem and the Feller–Reuter–Riley transition functions is revealed. It enables us to prove that a dual transition function must satisfy the Kolmogorov forward equations. A necessary and sufficient condition for a dual transition function satisfying the Kolmogorov backward equations is also provided.     

The Gerber–Shiu discounted penalty functions for a risk model with two classes of claims

In this paper, we consider the ruin problems for a risk model involving two independent classes of insurance risks. We assume that the claim number processes are independent Poisson and generalized Erlang(n) processes, respectively. When the generalized Lundberg equation has distinct roots with positive real parts, both of the Gerber–Shiu discounted penalty functions with zero initial surplus and the Laplace transforms of the Gerber–Shiu discounted penalty functions are obtained. Finally, some explicit expressions for the Gerber–Shiu discounted penalty functions with positive initial surplus are given when the claim size distributions belong to the rational family.     

Fourier–Bessel-type series: The fourth-order case

The structured higher-order Bessel-type linear ordinary differential equations were first discovered in 1994. There is a denumerable infinity of these higher-order equations, all of then of even-order.These differential equations possess many of the properties of the classical second-order Bessel differential equation, but these higher-order cases bring remarkable new analytic structures. In many ways it is sufficient to study the properties of the fourth-order Bessel-type differential equation to be able to assess the corresponding properties of the sixth-and higher-order cases.This paper follows a number of earlier papers devoted to the study of the fourth-order case. These publications show the connections between the special function properties of solutions of the differential equation, and the properties of linear differential operators generated by the associated linear differential expression in certain weighted Lebesgue, and Lebesgue–Stieltjes function spaces.To follow the earlier papers on the study of the fourth-order Bessel-type differential equation, this present paper determines the form of the Fourier–Bessel-type series which best extends the classical theory of the second-order Fourier–Bessel series.In fact the Fourier–Bessel-type series are based on a new orthogonal system in terms of the regular eigensolutions of the fourth-order Bessel-type equation. The corresponding eigenvalues are obtained by restricting the spectral parameter to the zeros of an analytic function arising already in the Dini boundary conditions.     

Finitistic dimension and Igusa–Todorov algebras

The notion of Igusa–Todorov algebras is introduced in connection with the (little) finitistic dimension conjecture, and the conjecture is proved for those algebras. Such algebras contain many known classes of algebras over which the finitistic dimension conjecture holds, e.g., algebras with the representation dimension at most 3, algebras with radical cube zero, monomial algebras and left serial algebras, etc. It is an open question whether all artin algebras are Igusa–Todorov. We provide some methods to construct many new classes of (2-)Igusa–Todorov algebras and thus obtain many algebras such that the finitistic dimension conjecture holds. In particular, we show that the class of 2-Igusa–Todorov algebras is closed under taking endomorphism algebras of projective modules. Hence, if all quasi-hereditary algebras are 2-Igusa–Todorov, then all artin algebras are 2-Igusa–Todorov by [V. Dlab, C.M. Ringel, Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring, Proc. Amer. Math. Soc. 107 (1) (1989) 1–5] and have finite finitistic dimension.     

Fully polynomial-time approximation schemes for time–cost tradeoff problems in series–parallel project networks

We consider the deadline problem and budget problem of the nonlinear time–cost tradeoff project scheduling model in a series–parallel activity network. We develop fully polynomial-time approximation schemes for both problems using K-approximation sets and functions, together with series and parallel reductions.     

Marcinkiewicz–Zygmund inequalities

We study a generalization of the classical Marcinkiewicz–Zygmund inequalities. We relate this problem to the sampling sequences in the Paley–Wiener space and by using this analogy we give sharp necessary and sufficient computable conditions for a family of points to satisfy the Marcinkiewicz–Zygmund inequalities.     

Linear continuous operators for the Stieltjes moment problem in Gelfand–Shilov spaces

We obtain linear continuous operators providing a solution to the Stieltjes moment problem in the framework of Gelfand–Shilov spaces of rapidly decreasing smooth functions. The construction rests on an interpolation procedure due to R. Estrada for general rapidly decreasing smooth functions, and adapted by S.-Y. Chung, D. Kim and Y. Yeom to the case of Gelfand–Shilov spaces. It requires a linear continuous version of the so-called Borel–Ritt–Gevrey theorem in asymptotic theory.     

Gabor frames and directional time–frequency analysis

We introduce a directionally sensitive time–frequency decomposition and representation of functions. The coefficients of this representation allow us to measure the “amount” of frequency a function (signal, image) contains in a certain time interval, and also in a certain direction. This has been previously achieved using a version of wavelets called ridgelets [E.J. Candès, Harmonic analysis of neural networks, Appl. Comput. Harmon. Anal. 6 (1999) 197–218. [2]; E.J. Candès, D.L. Donoho, New tight frames of curvelets and optimal representations of objects with piesewise-C² singularities, Comm. Pure Appl. Math. 57 (2004) 219–266. [3]] but in this work we discuss an approach based on time–frequency or Gabor elements. For such elements, a Parseval formula and a continuous frame-type representation together with boundedness properties of a semi-discrete frame operator are obtained. Spaces of functions tailored to measure quantitative properties of the time–frequency–direction analysis coefficients are introduced and some of their basic properties are discussed. Applications to image processing and medical imaging are presented.     

On Classes of Generalized Convex Functions, Gordan–Farkas Type Theorems, and Lagrangian Duality

In this paper, we introduce several classes of generalized convex functions already discussed in the literature and show the relation between these classes. Moreover, a Gordan–Farkas type theorem is proved for all these classes and it is shown how these theorems can be used to verify strong Lagrangian duality results in finite-dimensional optimization.     

The Bishop–Phelps–Bollobás theorem fails for bilinear forms on

In this paper we show that the Bishop–Phelps–Bollobás theorem fails for bilinear forms on l₁×l₁, while it holds for linear operators from l₁ to l_∞.     

Meromorphic functions with bounded boundary rotation

In this article, we generalize the class of meromorphic functions with bounded boundary rotation and related classes. Characterizations and some properties of these classes of functions are given.     

Numerical solution of the nonlinear Klein–Gordon equation using radial basis functions

The nonlinear Klein–Gordon equation is used to model many nonlinear phenomena. In this paper, we propose a numerical scheme to solve the one-dimensional nonlinear Klein–Gordon equation with quadratic and cubic nonlinearity. Our scheme uses the collocation points and approximates the solution using Thin Plate Splines (TPS) radial basis functions (RBF). The implementation of the method is simple as finite difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme.     

On complexity of Ehrenfeucht–Fraïssé games

In this paper, we initiate the study of Ehrenfeucht–Fraïssé games for some standard finite structures. Examples of such standard structures are equivalence relations, trees, unary relation structures, Boolean algebras, and some of their natural expansions. The paper concerns the following question that we call the Ehrenfeucht–Fraïssé problem. Given nω as a parameter, and two relational structures and from one of the classes of structures mentioned above, how efficient is it to decide if Duplicator wins the n-round EF game ? We provide algorithms for solving the Ehrenfeucht–Fraïssé problem for the mentioned classes of structures. The running times of all the algorithms are bounded by constants. We obtain the values of these constants as functions of n.     

Asymptotic Gauss–Jacobi quadrature error estimation for Schwarz–Christoffel integrals

Numerical conformal mapping packages based on the Schwarz–Christoffel formula have been in existence for a number of years. Various authors, for good reasons of practical efficiency, have chosen to use composite n-point Gauss–Jacobi rules for the estimation of the Schwarz–Christoffel path integrals. These implementations rely on an ad hoc, but experimentally well-founded, heuristic for selecting the spacing of the integration end-points relative to the position of the nearby integrand singularities. In the present paper we derive an explicitly computable estimate, asymptotic as n→∞, for the relevant Gauss–Jacobi quadrature error. A numerical example illustrates the potential accuracy of the estimate even at low values of n. It is apparent that the error estimate will allow the adaptive construction of composite rules in a manner that is more efficient than has been possible hitherto.     

Hyers–Ulam–Rassias Stability of a Jensen Type Functional Equation

In this paper we solve the Jensen type functional equation (1.1). Likewise, we investigate the Hyers–Ulam–Rassias stability of this equation.     

MEROMORPHIC FUNCTIONS WITH BOUNDED BOUNDARY ROTATION

In this article, we generalize the class of meromorphic functions with bounded boundary rotation and related classes. Characterizations and some properties of these classes of functions are given.     

Solitons and periodic solutions for the Rosenau–KdV and Rosenau–Kawahara equations

In this work we use the sine–cosine and the tanh methods for solving the Rosenau–KdV and Rosenau–Kawahara equations. The two methods reveal solitons and periodic solutions. The study confirms the power of the two schemes.