On the Number of k‐Dominating Independent Sets

We study the existence and the number of k‐dominating independent sets in certain graph families. While the case namely the case of maximal independent sets—which is originated from Erd?s and Moser—is widely investigated, much less is known in general. In this paper we settle the question for trees and prove that the maximum number of k‐dominating independent sets in n‐vertex graphs is between and if , moreover the maximum number of 2‐dominating independent sets in n‐vertex graphs is between and . Graph constructions containing a large number of k‐dominating independent sets are coming from product graphs, complete bipartite graphs, and finite geometries. The product graph construction is associated with the number of certain Maximum Distance Separable (MDS) codes.     

The Lp dual Minkowski problem for p > 1 and q > 0

General $L_p$ dual curvature measures have recently been introduced by Lutwak, Yang and Zhang [24]. These new measures unify several other geometric measures of the Brunn–Minkowski theory and the dual Brunn–Minkowski theory. $L_p$ dual curvature measures arise from qth dual intrinsic volumes by means of Alexandrov-type variational formulas. Lutwak, Yang and Zhang [24] formulated the $L_p$ dual Minkowski problem, which concerns the characterization of $L_p$ dual curvature measures. In this paper, we solve the existence part of the $L_p$ dual Minkowski problem for $p>1$ and $q>0$, and we also discuss the regularity of the solution.     

IR Tools: a MATLAB package of iterative regularization methods and large-scale test problems

Numerical Algorithms - This paper describes a new MATLAB software package of iterative regularization methods and test problems for large-scale linear inverse problems. The software package, called...     

A new approach to constant term identities and Selberg-type integrals

Selberg-type integrals that can be turned into constant term identities for Laurent polynomials arise naturally in conjunction with random matrix models in statistical mechanics. Built on a recent idea of Karasev and Petrov we develop a general interpolation based method that is powerful enough to establish many such identities in a simple manner. The main consequence is the proof of a conjecture of Forrester related to the Calogero–Sutherland model. In fact we prove a more general theorem, which includes Aomoto's constant term identity at the same time. We also demonstrate the relevance of the method in additive combinatorics.     

Strong Asymptotics of the Orthogonal Polynomials with Respect to a Measure Supported on the Plane

We consider the orthogonal polynomials with respect to the measure over the whole complex plane. We obtain the strong asymptotic of the orthogonal polynomials in the complex plane and the location of their zeros in a scaling limit where n grows to infinity with N . The asymptotics are described in terms of three (probability) measures associated with the problem. The first measure is the limit of the counting measure of zeros of the polynomials, which is captured by the g‐function much in the spirit of ordinary orthogonal polynomials on the real line. The second measure is the equilibrium measure that minimizes a certain logarithmic potential energy, supported on a region K of the complex plane. The third measure is the harmonic measure of K ^c with a pole at ∞ . This appears as the limit of the probability measure given (up to the normalization constant) by the squared modulus of the n^th orthogonal polynomial times the orthogonality measure, i.e., The compact region K that is the support of the second measure undergoes a topological transition under the variation of the parameter in a double scaling limit near the critical point given by we observe the Hastings‐McLeod solution to Painlevé II in the asymptotics of the orthogonal polynomials. © 2014 Wiley Periodicals, Inc.     

3-Nets realizing a group in a projective plane

In a projective plane $\mathit{PG}(2,\mathbb{K})$ defined over an algebraically closed field $\mathbb{K}$ of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky (Compos. Math. 140:1614–1624, 2004), arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky (Adv. Math. 219:672–688, 2008), comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzúa’s 3-nets (Adv. Geom. 10:287–310, 2010) realizing the quaternion group of order 8 are the unique sporadic examples. If p is larger than the order of the group, the above classification holds in characteristic p>0 apart from three possible exceptions $\rm{Alt}_{4}$ , $\rm{Sym}_{4}$ , and $\rm{Alt}_{5}$ . Motivation for the study of finite 3-nets in the complex plane comes from the study of complex line arrangements and from resonance theory; see (Falk and Yuzvinsky in Compos. Math. 143:1069–1088, 2007; Miguel and Buzunáriz in Graphs Comb. 25:469–488, 2009; Pereira and Yuzvinsky in Adv. Math. 219:672–688, 2008; Yuzvinsky in Compos. Math. 140:1614–1624, 2004; Yuzvinsky in Proc. Am. Math. Soc. 137:1641–1648, 2009).     

Inversion of the Short-Time Fourier Transform Using Riemannian Sums

The inversion formula for the short-time Fourier transform is usually considered in the weak sense, or only for specific combinations of window functions and function spaces such as L₂ and modulation spaces. In the present note the Riemannian sums of the inverse short-time Fourier transform are investigated. Under some conditions on the window functions we prove that the Riemannian sums converge to f in the modulation spaces and inWiener amalgam norms, hence also in the L_p sense.     

Location-routing: Issues,models and methods

This paper is a survey of location-routing: a relatively new branch of locational analysis that takes into account vehicle routing aspects. We propose a classification scheme and look at a number of problem variants. Both exact and heuristic algorithms are investigated. Finally, some suggestions for future research are presented.     

Extension of a theorem of Fejér to double Fourier-Stieltjes series

A theorem of Fejér states that if a periodic function F is of bounded variation on the closed interval [0, 2π], then the nth partial sum of its formally differentiated Fourier series divided by n converges to π^-1[F(x+0)-F(x-0)] at each point x. The generalization of this theorem for Fourier-Stieltjes series of (nonperiodic) functions of bounded variation is also well known. The aim of the present article is to extend these results to the (m, n)th rectangular partial sum of double Fourier or Fourier-Stieltjes series of a function F(x, y) of bounded variation over the closed square [0, 2π]×[0, 2π] in the sense of Hardy and Krause. As corollaries, we also obtain the following results:

The maximal Fejér operator on real Hardy spaces

We prove that the maximal Fej'er operator is not bounded on the real Hardy spaces H ¹, which may be considered over and . We also draw corollaries for the corresponding Hardy spaces over ² and ². This revised version was published online in August 2006 with corrections to the Cover Date.