Bibasic analystic functions and discrete ‘bibasic’ hypergeometric series

This paper deals with a study of a class of functions called bibasis analytic functions. Using discrete powerz ⁽ⁿ⁾, discrete bibasic hypergeometric functions have been introduced.     

Poincaré series

Let N_α denote the number of solutions to the congruence F(x_i,..., x_m) ≡ 0 (mod p^α) for a polynomial F(x_i,..., x_m) with integral p-adic coefficients. We examine the series \(\varphi (t) = \sum\nolimits_{\alpha = 0}^\infty {N_{\alpha ^{t^\alpha } } } \) . called the Poincaré series for the polynomial F. In this work we prove the rationality of the series ?(t) for a class of isometrically equivalent polynomials of m variables, m ≥ 2, containing the sum of two forms ?_n(x, y) + ?_n+1(x, y) respectively of degrees n and n+1, n ≥ 2. In particular the Poincaré series for any third degree polynomial F₃(x, y) (over the set of unknowns) with integral p-adic coefficients is a rational function of t.     

Arithmetic–geometric discrete systems

Analyzing a consensus in opinion dynamics one meets interesting open problems concerning the iteration of various means. Whereas in two dimensions the consensus can be computed by the arithmetic-geometric mean of Gauss not very much is known in higher dimensions.     

On discrete ℓ1-regularization

A real m × n matrix A and a vector y?∈?? ^m determine the discrete l ¹-regularization (DLR) problem 0.1 $$ \min \left\{\mbox{\,}|y-Ax|_1+\rho |x|_1:\,x\in\mathbb{R}^n \right\}, $$ where | · |₁ denotes the l ¹-norm of a vector and ρ?≥?0 is a nonnegative parameter. In this paper, we provide a detailed analysis of this problem which include a characterization of all solutions to (0.1), remarks about the geometry of the solution set and an effective iterative algorithm for numerical solution of (0.1). We are specially interested in the behavior of the solution of (0.1) as a function of ρ and in this regard, we prove in general the existence of critical values of ρ between which the l ¹-norm of any solution remains constant. These general remarks are significantly refined when A is a strictly totally positive (STP) matrix. The importance of STP matrices is well-established [5, 14]. Under this setting, the relationship between the number of nonzero coordinates of a distinguished solution of the DLR problem is precisely explained as a function of the regularization parameter for a certain class of vectors in ? ^m . Throughout our analysis of the DLR problem, we emphasize the importance of the dual maximum problem by demonstrating that any solution of it leads to a solution of the DLR problem, and vice versa.     

Nörlund summability of Fourier series and its conjugate series

Summary In this paper the author has obtained two theorems for the N?rlund summability of Fourier series and its conjugate series respectively under very general conditions.     

Sperner’s lemma and zero point theorems on a discrete simplex and a discrete simplotope

We show two discrete zero point theorems that are derived from Sperner’s lemma and a Sperner-like theorem (van der Laan and Talman [Math. Oper. Res. (1982)] [5]; Freund [Math. Oper. Res. (1986)]) [3]. Applications to economic and game models are also presented.     

Θ-summability of Fourier series

A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ we show that if the maximal Fejér operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means of these Fourier series is bounded from H _p to L _p (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function f∈L ₁ converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallée-Poussin, Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.     

Computation of Hilbert-Poincaré series

We describe a new algorithm for computing standard and multi-graded Hilbert-Poincaré series of a monomial ideal. We compare it with different strategies along with implementation details and timing data.     

Remarks on ω-domination of discrete subspaces

Abstract

Given a space X, we will say that a class of subsets of X is dominated by a class ? if for any A ∈ , there exists a B ∈ ? such that A ? . In particular, all (closed) discrete subsets of X are countably dominated (which we frequently abbreviate as ω-dominated) if, for any (closed) discrete set D ? X, there exists a countable set B ? X such that D ? . In this paper, we investigate the topological properties of spaces in which (closed) discrete subspaces are dominated either by countable subsets or by Lindelöf subspaces.     

A new discrete Hopf–Rinow theorem

We prove a version of the Hopf–Rinow theorem with respect to path metrics on discrete spaces. The novel aspect is that we do not a priori assume local finiteness but isolate a local finiteness type condition, called essentially locally finite, that is indeed necessary. As a side product we identify the maximal weight, called the geodesic weight, generating the path metric in the situation when the space is complete with respect to any of the equivalent notions of completeness proven in the Hopf–Rinow theorem. As an application we characterize the graphs for which the resistance metric is a path metric induced by the graph structure.     

On certain Schlömilch-type series

By using a form of the Poisson summation formula together with a generalization due to Srivastava and Exton of the discontinuous integral of Weber and Schafheitlin and the (heretofore not readily available) cosine transform of the hypergeometric function ₁F₂[−b²t²] (b > 0), several new Schlömilch and Fourier-type series are evaluated. By specialization of the latter series numerous results appearing in the literature are obtained in a unified way.     

Complex series for 1/π

Many series for 1/?? were discovered since the appearance of S. Ramanujan??s famous paper ??Modular equations and approximation to ???? published in 1914. Almost all these series involve only real numbers. Recently, in an attempt to prove a series for 1/?? discovered by Z.-W.?Sun, the authors found that a series for 1/?? involving complex numbers is needed. In this article, we illustrate a method that would allow us to prove series of this type.     

“Spatial nodes” in discrete location problems

Discrete location models often assume an underlying network where demands originate at point nodes. To apply these models to planar regions with continuously distributed demand, the region is usually partitioned into zones and the demand from each zone is assumed to originate at a point, usually the zone centroid. Thus, the point node in the underlying network represents a spatial zone with a finite area. This paper examines the effect of approximating these spatial nodes by point nodes. In some problem scenarios, the approximation does not affect the solution. However, especially when the locational criterion includes the consideration of intra-zonal travel cost variances (e.g. travel time variance) and demands may originate anywhere within zones of nonzero area, point nodes do not give an accurate evaluation of the performance of a locational design. To illustrate the application of the concept of spatial nodes, a model is formulated for locatingp (fire-fighting) units in a region having continuously distributed demand with the objective of minimizing a nonlinear function of arrival times of the first and second closest units to any (fire) incident. A heuristic site-substitution procedure is presented that solves the formulated model.     

Dynamical properties of discrete Lotka–Volterra equations

A discrete version of the Lotka–Volterra differential equations for competing population species is analyzed in detail in much the same way as the discrete form of the logistic equation has been investigated as a source of bifurcation phenomena and chaotic dynamics. It is found that in addition to the logistic dynamics – ranging from very simple to manifestly chaotic regimes in terms of governing parameters – the discrete Lotka–Volterra equations exhibit their own brands of bifurcation and chaos that are essentially two-dimensional in nature. In particular, it is shown that the system exhibits “twisted horseshoe” dynamics associated with a strange invariant set for certain parameter ranges.     

Convolution operators in discrete Cesàro spaces

Archiv der Mathematik - The discrete Cesàro spaces $${{\text {ces(p )}}}, 1&lt; p &lt; \infty ,$$ are well known and arise from the classical sequence spaces $$\ell ^p,1&lt;\infty...     

Boundary value problems of discrete generalized Emden-Fowler equation

By using the critical point theory, some sufficient conditions for the existence of the solutions to the boundary value problems of a discrete generalized Emden-Fowler equation are obtained. In a special case, a sharp condition is obtained for the existence of the boundary value problems of the above equation. For a linear case, by the discrete variational theory, a necessary and sufficient condition for the existence, uniqueness and multiplicity of the solutions is also established.