Clear thinking will never be out of date

The paper expounds the need to cultivate a breed of applied mathematician, called a generalist, with enough breadth and flexibility to cope with many‐faceted challenges such as occur in robotics, biomedical imaging, etc. The attributes of such a generalist are outlined and the importance of a sound training in fundamentals is highlighted.     

Narayana numbers and Schur–Szeg? composition

In the present paper we find a new interpretation of Narayana polynomials N_n(x) which are the generating polynomials for the Narayana numbers where stands for the usual binomial coefficient, i.e. . They count Dyck paths of length n and with exactly k peaks, see e.g. [R.A. Sulanke, The Narayana distribution, in: Lattice Path Combinatorics and Applications (Vienna, 1998), J. Statist. Plann. Inference 101 (1–2) (2002) 311–326 (special issue)] and they appeared recently in a number of different combinatorial situations, see for e.g. [T. Doslic, D. Syrtan, D. Veljan, Enumerative aspects of secondary structures, Discrete Math. 285 (2004) 67–82; A. Sapounakis, I. Tasoulas, P. Tsikouras, Counting strings in Dyck paths, Discrete Math. 307 (2007) 2909–2924; F. Yano, H. Yoshida, Some set partitions statistics in non-crossing partitions and generating functions, Discrete Math. 307 (2007) 3147–3160]. Strangely enough Narayana polynomials also occur as limits as n→∞ of the sequences of eigenpolynomials of the Schur–Szeg? composition map sending (n−1)-tuples of polynomials of the form (x+1)ⁿ⁻¹(x+a) to their Schur–Szeg? product, see below. We present below a relation between Narayana polynomials and the classical Gegenbauer polynomials which implies, in particular, an explicit formula for the density and the distribution function of the asymptotic root-counting measure of the polynomial sequence {N_n(x)}.     

On eigenvalues of rectangular matrices

Given a (k+1)-tuple A,B ₁, ..., B _k of m×n matrices with m ≤ n, we call the set of all k-tuples of complex numbers {λ ₁, ..., λ k} such that the linear combination A+λ ₁ B ₁+λ ₂ B ₂+ ... +λ _k B _k has rank smaller than m the eigenvalue locus of the latter pencil. Motivated primarily by applications to multiparameter generalizations of the Heine-Stieltjes spectral problem, we study a number of properties of the eigenvalue locus in the most important case k = n−m+1.     

Forces on a porous particle in an oscillating flow

We investigate theoretically forces acting on a porous particle in an oscillating viscous incompressible flow. We use the unsteady equations describing the creeping flow, namely the Stokes equations exterior to the particle and the Darcy or Brinkman equations inside it. The effect of particle permeability and oscillation frequency on the flow and forces is expressed via the Brinkman parameter beta = a/square root(k) and the frequency parameter Y = square root(a(2)omega/2nu) = a/delta, respectively. Here a is particle radius, k is its permeability, omega is the angular frequency, delta is the thickness of Stokes layer (penetration depth) and nu is the fluid kinematic viscosity. It is shown that the oscillations interact with permeable structure of the particle and affect both the Stokes-like viscous drag and the added mass force components.     

Poisson geometry of directed networks in a disk

We investigate Poisson properties of Postnikov’s map from the space of edge weights of a planar directed network into the Grassmannian. We show that this map is Poisson if the space of edge weights is equipped with a representative of a 6-parameter family of universal quadratic Poisson brackets and the Grassmannian is viewed as a Poisson homogeneous space of the general linear group equipped with an appropriate R-matrix Poisson–Lie structure. We also prove that the Poisson brackets on the Grassmannian arising in this way are compatible with the natural cluster algebra structure.      

On the Derivatives of Quaternionic Functions Along Two-Dimensional Planes

In a previous paper we introduced the concept of a two-dimensional directional derivative of a quaternionic function along a two-dimensional plane. In this paper we provide a deeper analysis of its properties, as well as of its relations with hyperholomorphic functions, with holomorphic maps of two complex variables and with Cullen-regular functions. Received: October, 2007. Accepted: February, 2008.