On the log-convexity of combinatorial sequences

This paper is devoted to the study of the log-convexity of combinatorial sequences. We show that the log-convexity is preserved under componentwise sum, under binomial convolution, and by the linear transformations given by the matrices of binomial coefficients and Stirling numbers of two kinds. We develop techniques for dealing with the log-convexity of sequences satisfying a three-term recurrence. We also introduce the concept of $q$-log-convexity and establish the connection with linear transformations preserving the log-convexity. As applications of our results, we prove the log-convexity and $q$-log-convexity of many famous combinatorial sequences of numbers and polynomials.Comment: 25 pages, final version, to appear in Advances in Applied Mathematic

Strong q-log-convexity of the Eulerian polynomials of Coxeter groups

In this paper we prove the strong $q$-log-convexity of the Eulerian polynomials of Coxeter groups using their exponential generating functions. Our proof is based on the theory of exponential Riordan arraya and a criterion for determining the strong $q$-log-convexity of polynomials sequences, whose generating functions can be given by the continued fraction. As consequences, we get the strong $q$-log-convexity the Eulerian polynomials of type $A_n,B_n$, their $q$-analogous and the generalized Eulerian polynomials associated to the arithmetic progression $\{a,a+d,a+2d,a+3d,\ldots\}$ in a unified manner.Comment: 13page

Positivity of Toeplitz determinants formed by rising factorial series and properties of related polynomials

In this note we prove positivity of Maclaurin coefficients of polynomials written in terms of rising factorials and arbitrary log-concave sequences. These polynomials arise naturally when studying log-concavity of rising factorial series. We propose several conjectures concerning zeros and coefficients of a generalized form of those polynomials. We also consider polynomials whose generating functions are higher order Toeplitz determinants formed by rising factorial series. We make three conjectures about these polynomials. All proposed conjectures are supported by numerical evidence.Comment: 11 pages; no figure

Imaginary projections of polynomials

We introduce the imaginary projection of a multivariate polynomial $f \in \mathbb{C}[\mathbf{z}]$ as the projection of the variety of $f$ onto its imaginary part, $\mathcal{I}(f) \ = \ \{\text{Im}(\mathbf{z}) \, : \, \mathbf{z} \in \mathcal{V}(f) \}$. Since a polynomial $f$ is stable if and only if $\mathcal{I}(f) \cap \mathbb{R}_{>0}^n \ = \ \emptyset$, the notion offers a novel geometric view underlying stability questions of polynomials. We show that the connected components of the complement of the closure of the imaginary projections are convex, thus opening a central connection to the theory of amoebas and coamoebas. Building upon this, the paper establishes structural properties of the components of the complement, such as lower bounds on their maximal number, proves a complete classification of the imaginary projections of quadratic polynomials and characterizes the limit directions for polynomials of arbitrary degree.Comment: Revised version, 21 page

On unimodality problems in Pascal's triangle

Many sequences of binomial coefficients share various unimodality properties. In this paper we consider the unimodality problem of a sequence of binomial coefficients located in a ray or a transversal of the Pascal triangle. Our results give in particular an affirmative answer to a conjecture of Belbachir et al which asserts that such a sequence of binomial coefficients must be unimodal. We also propose two more general conjectures.Comment: 12 pages, 2 figure

Contributions of Issai Schur to Analysis

The name Schur is associated with many terms and concepts that are widely used in a number of diverse fields of mathematics and engineering. This survey article focuses on Schur's work in analysis. Here too, Schur's name is commonplace: The Schur test and Schur-Hadamard multipliers (in the study of estimates for Hermitian forms), Schur convexity, Schur complements, Schur's results in summation theory for sequences (in particular, the fundamental Kojima-Schur theorem), the Schur-Cohn test, the Schur algorithm, Schur parameters and the Schur interpolation problem for functions that are holomorphic and bounded by one in the unit disk. In this survey, we discuss all of the above mentioned topics and then some, as well as some of the generalizations that they inspired. There are nine sections of text, each of which is devoted to a separate theme based on Schur's work. Each of these sections has an independent bibliography. There is very little overlap. A tenth section presents a list of the papers of Schur that focus on topics that are commonly considered to be analysis. We begin with a review of Schur's less familiar papers on the theory of commuting differential operators

Spectral order and isotonic differential operators of Laguerre-Polya type

The spectral order on \bR^n induces a natural partial ordering on the manifold \calH_{n} of monic hyperbolic polynomials of degree $n$. We show that all differential operators of Laguerre-P\'olya type preserve the spectral order. We also establish a global monotony property for infinite families of deformations of these operators parametrized by the space \li of real bounded sequences. As a consequence, we deduce that the monoid \calA' of linear operators that preserve averages of zero sets and hyperbolicity consists only of differential operators of Laguerre-P\'olya type which are both extensive and isotonic. In particular, these results imply that any hyperbolic polynomial is the global minimum of its \calA'-orbit and that Appell polynomials are characterized by a global minimum property with respect to the spectral order.Comment: Final version, to appear in Ark. Mat.; 21 pages, no figures, LaTeX2

A family of projectively natural polygon iterations

The pentagram map was invented by Richard Schwartz in his search for a projective-geometric analogue of the midpoint map. It turns out that the dynamical behavior of the pentagram map is totally different from that of the midpoint map. Recently, Schwartz has constructed a related map, the projective heat map, which empirically exhibits similar dynamics as the midpoint map. In this paper, we will demonstrate that there is a one-parameter family of maps which behaves a lot like Schwartz's projective heat map.Comment: 24 pages, 10 figure

On a Stirling-Whitney-Riordan triangle

Based on the Stirling triangle of the second kind, the Whitney triangle of the second kind and one triangle of Riordan, we study a Stirling-Whitney-Riordan triangle $[T_{n,k}]_{n,k}$ satisfying the recurrence relation: \begin{eqnarray*} T_{n,k}&=&(b_1k+b_2)T_{n-1,k-1}+[(2\lambda b_1+a_1)k+a_2+\lambda( b_1+b_2)] T_{n-1,k}+\\ &&\lambda(a_1+\lambda b_1)(k+1)T_{n-1,k+1}, \end{eqnarray*} where initial conditions $T_{n,k}=0$ unless $0\le k\le n$ and $T_{0,0}=1$. We prove that the Stirling-Whitney-Riordan triangle $[T_{n,k}]_{n,k}$ is $\textbf{x}$-totally positive with $\textbf{x}=(a_1,a_2,b_1,b_2,\lambda)$. We show that the row-generating function $T_n(q)$ has only real zeros and the Tur\'{a}n-type polynomial $T_{n+1}(q)T_{n-1}(q)-T^2_n(q)$ is stable. We also present explicit formulae for $T_{n,k}$ and the exponential generating function of $T_n(q)$ and give a Jacobi continued fraction expansion for the ordinary generating function of $T_n(q)$. Furthermore, we get the $\textbf{x}$-Stieltjes moment property and $3$-$\textbf{x}$-log-convexity of $T_n(q)$ and show that the triangular convolution $z_n=\sum_{i=0}^nT_{n,i}x_iy_{n-i}$ preserves Stieltjes moment property of sequences. Finally, for the first column $(T_{n,0})_{n\geq0}$, we derive some properties similar to those of $(T_n(q))_{n\geq0}.$Comment: To appear in Journal of Algebraic Combinatoric

Gauges, Loops, and Polynomials for Partition Functions of Graphical Models

Graphical models represent multivariate and generally not normalized probability distributions. Computing the normalization factor, called the partition function, is the main inference challenge relevant to multiple statistical and optimization applications. The problem is of an exponential complexity with respect to the number of variables. In this manuscript, aimed at approximating the PF, we consider Multi-Graph Models where binary variables and multivariable factors are associated with edges and nodes, respectively, of an undirected multi-graph. We suggest a new methodology for analysis and computations that combines the Gauge Function technique with the technique from the field of real stable polynomials. We show that the Gauge Function has a natural polynomial representation in terms of gauges/variables associated with edges of the multi-graph. Moreover, it can be used to recover the Partition Function through a sequence of transformations allowing appealing algebraic and graphical interpretations. Algebraically, one step in the sequence consists in application of a differential operator over gauges associated with an edge. Graphically, the sequence is interpreted as a repetitive elimination of edges resulting in a sequence of models on decreasing in size graphs with the same Partition Function. Even though complexity of computing factors in the sequence models grow exponentially with the number of eliminated edges, polynomials associated with the new factors remain bi-stable if the original factors have this property. Moreover, we show that Belief Propagation estimations in the sequence do not decrease, each low-bounding the Partition Function.Comment: 18 pages, 3 figure