Partitions and sums and products—Two counterexamples

A negative answer is provided to a question of Erdös. Specifically, a two celled partition of N is provided with the property that neither cell includes an infinite set together with all finite products and pairwise sums from that set. Also, a seven celled partition of N is provided with the property that no cell includes an infinite set together with all pairwise products and pairwise sums from that set.     

Chords and disjoint paths in matroids

A chord of a circuit C of a matroid M on E is a cell e ? S\C such that C spans e. Menger's theorem gives necessary and sufficient conditions for a cell of a graphic matroid to be a chord of some circuit. We extend this result to a large class of matroids and find all minimal counterexamples. The theorem is used to obtain results on disjoint paths and to characterize a class of matroid sums.     

On properties of cell matrices

In this paper properties of cell matrices are studied. A determinant of such a matrix is given in a closed form. In the proof a general method for determining a determinant of a symbolic matrix with polynomial entries, based on multivariate polynomial Lagrange interpolation, is outlined. It is shown that a cell matrix of size n>1 has exactly one positive eigenvalue. Using this result it is proven that cell matrices are (Circum-)Euclidean Distance Matrices ((C)EDM), and their generalization, k-cell matrices, are CEDM under certain natural restrictions. A characterization of k-cell matrices is outlined.     

Categorification of (induced) cell modules and the rough structure of generalised Verma modules

This paper presents categorifications of (right) cell modules and induced cell modules for Hecke algebras of finite Weyl groups. In type A we show that these categorifications depend only on the isomorphism class of the cell module, not on the cell itself. Our main application is multiplicity formulas for parabolically induced modules over a reductive Lie algebra of type A, which finally determines the so-called rough structure of generalised Verma modules. On the way we present several categorification results and give a positive answer to Kostant's problem from [A. Joseph, Kostant's problem, Goldie rank and the Gelfand-Kirillov conjecture, Invent. Math. 56 (3) (1980) 191-213] in many cases. We also present a general setup of decategorification, precategorification and categorification.     

Generalized Latin rectangles I: Construction and decomposition

A (p, q, x)-latin rectangle is a rectangular matrix with x symbols in each cell such that each symbol occurs at most p times in each row and at most q times in each column. We exploit the close correspondence between (p, q, x)-latin rectangles and equitable edge-colourings of certain graphs. This paper contains results on existence and various forms of decomposition of such rectangles. In a sequel paper embedding is considered.     

Modeling the dynamical impact of HIV on the immune system: Viral clearance,infection, and AIDS

A phenomenological model simulating the time-dependent consequences of the HIV challenge on the immune system is presented. One of the important features of the model is its ability to handle T helper cell production and apoptosis (genetically determined suicide). The values of the independent, generally time-dependent, model parameters were chosen to be compatible with known experimental data. A new approach to the numerical solution of the resulting coupled, nonlinear model equations is presented, and simulations of a typical viral challenge that is cleared and one that leads to infection and AIDS are exhibited.It is shown that a change in the saturated value of a single model parameter is sufficient to change a simulated challenge on its way to being cleared into one that leads to infection instead (and vice versa). If the saturated values of all of the independent model parameters are known at the beginning of a challenge, the outcome of the challenge can be predicted in advance. If the virulence of the HIV strain (defined in this paper) is above a critical threshold at inoculation, infection will result regardless of the initial viral load. This latter result could explain why accidental HIV contaminated needle sticks sometime result in infection regardless of the counter-measures undertaken.A model simulating the time evolution of the collapse of the T helper cell density leading to AIDS is introduced. This model consists of immunological and mathematical parts and is compatible with experimental data. The immediate cause of the beginning of this collapse is postulated to be a spontaneous mutation of the virus into a more virulent form that not only leads to an explosion in the viral load but also to a dramatic increase in the level of induced apoptosis of T helper cells. The results of this model are consistent with the known experimental behavior of the viral load and T helper cell densities in the final stage of HIV infection.     

Collapsing unicolored graphs

A graph is said to be unicolored if it is colored by nonnegative integers so that adjacent points have colors that differ in absolute value by one. A unicolored graph is collapsible if it has a 1-factor that does not contain a 1-factor of any bicolored cycle. We show that a regular CW complex K cell collapses to a subcomplex O if and only if its relative unicolored incidence graph collapses. We consider the 1-factors and the bicolored cycles of unicolored incidence graphs and their relationship to the relative homology and homotopy properties of the pair of cell complexes.     

Pairwise “Orthogonal generalized room squares” and equidistant permutation arrays

A generalized Room square S(r, λ; v) is an r × r array such that every cell in the array contains a subset of a v-set V. This subset could of course be the empty set. The array has the property that every element of V is contained precisely once in every row and column and that any two distinct elements of V are contained in precisely λ common cells. In this paper we define pairwise orthogonal generalized Room squares and give a construction for these using finite projective geometries. This is another generalization of the concept of pairwise orthogonal latin squares. We use these orthogonal arrays to construct permutations having a constant Hamming distance.     

On Howell designs

A Howell Design of type H(s, 2n) consists of a square of side s such that each cell is either empty or contains an unordered pair of integers taken from amongst 1, 2, 3, …, 2n provided: (1) each integer from 1 to 2n appears exactly once in each row and each column of the square and (2) every unordered pair appears at most once in a cell of the square. It is easily seen that for a Howell Design to exist that n ? s ? 2n ? 1. This paper presents a large number of constructions of Howell Designs and some existence theorems.     

Finite difference on grids with nearly uniform cell area and line spacing for the wave equation on complex domains

A finite difference time-dependent numerical method for the wave equation, supported by recently derived novel elliptic grids, is analyzed. The method is successfully applied to single and multiple two-dimensional acoustic scattering problems including soft and hard obstacles with complexly shaped boundaries. The new grids have nearly uniform cell area (J-grids) and nearly uniform grid line spacing (αγ-grids). Numerical experiments reveal the positive impact of these two grid properties on the scattered field convergence to its harmonic steady state. The restriction imposed by stability conditions on the time step size is relaxed due to the near uniformity cell areas and grid line spacing. As a consequence, moderately large time steps can be used for relatively fine spatial grids resulting in greater accuracy at a lower computational cost. Also, numerical solutions for wave problems inside annular regions of complex shapes are obtained. The use of the new grids results in late time stability in contrast with other classical finite difference time-dependent methods.     

The existence of Howell designs of even side

A Howell design of side s and order 2n, or more briefly, an H(s, 2n), is an s × s array in which each cell either is empty or contains an unordered pair of elements from some 2n-set, say X, such that (i) each row and column is Latin (that is, every element of X is in precisely one cell of each row and column) and (ii) any unordered pair of elements of X is in at most one cell of the array. A necessary condition for the existence of an H(s, 2n) is that n = 0 or n ? s ? 2n ?1. An $\text{H}{}^1\text{(s, 2n)}$ is an H(s, 2n) in which there is a subset of X, say Y, of cardinality 2n ? s such that no pair of elements from Y is in any cell of the array. In this paper it is shown that if s is an even positive integer, if s and n satisfy the necessary condition and if (s, 2n) ≠ (2, 4) or (6, 12), then there is an $\text{H}{}^1\text{(s, 2n)}$; furthermore, there is no H(2, 4) nor any $\text{H}{}^1\text{(6,12)}$ though there is an H(6, 12).     

Extreme values for characteristic radii of a Poisson-Voronoi Tessellation

A homogeneous Poisson-Voronoi tessellation of intensity γ is observed in a convex body W. We associate to each cell of the tessellation two characteristic radii: the inradius, i.e. the radius of the largest ball centered at the nucleus and included in the cell, and the circumscribed radius, i.e. the radius of the smallest ball centered at the nucleus and containing the cell. We investigate the maximum and minimum of these two radii over all cells with nucleus in W. We prove that when \(\gamma \rightarrow \infty \) , these four quantities converge to Gumbel or Weibull distributions up to a rescaling. Moreover, the contribution of boundary cells is shown to be negligible. Such approach is motivated by the analysis of the global regularity of the tessellation. In particular, consequences of our study include the convergence to the simplex shape of the cell with smallest circumscribed radius and an upper-bound for the Hausdorff distance between W and its so-called Poisson-Voronoi approximation.     

The number of distinct symbols in sections of rectangular arrays

We investigate transversals of rectangular arrays. For positive integers m and n, where 2?m?n an m by n array consists of mn cells arranged in m rows and n columns. Each cell contains one symbol. When m=n we speak of an array of order n. A section in the array consists of m cells, one from each row and no two from the same column. A transversal is a section whose m symbols are distinct. A partial transversal is a subset of a transversal. We investigate the existence in an array of a section with many different symbols, in particular the existence of a transversal.     

Existence regions of positive periodic solutions for a discrete hematopoiesis model with unimodal production functions

A discrete model describing the increase and decrease of blood cells is considered in this paper. This hematopoiesis model is a discretization of a delay differential equation with unimodal production function whose coefficients and delays are periodic discrete functions with ω-period. This paper is concerned with the existence of positive ω-periodic solutions. Our results are proved by using the well-known continuation theorem of coincidence degree theory. The existence range of the positive ω-periodic solutions is also clarified. A concrete example and its simulation are also given to illustrate our result. Finally, we examine how positive numbers and coefficients making up our model influence the upper and lower limits of blood cell counts.     

A variational approach to the local character of G-closure: the convex case

This article is devoted to characterize all possible effective behaviors of composite materials by means of periodic homogenization. This is known as a G-closure problem. Under convexity and p  -growth conditions (p>1$p>1$), it is proved that all such possible effective energy densities obtained by a Γ-convergence analysis, can be locally recovered by the pointwise limit of a sequence of periodic homogenized energy densities with prescribed volume fractions. A weaker locality result is also provided without any kind of convexity assumption and the zero level set of effective energy densities is characterized in terms of Young measures. A similar result is given for cell integrands which enables to propose new counter-examples to the validity of the cell formula in the nonconvex case and to the continuity of the determinant with respect to the two-scale convergence.     

On the existence of a travelling wave solution for a model of actin-based bacterial movement

Some bacteria like Listeria monocytogenes, Shigella and Rickettsia Rickettsii can move inside the host cell thanks to an actin tail. In this paper a one-dimensional model for the motion of these bacteria, introduced by B. Bazaliy, Y. Bazaliy, and A. Friedman (2007) in [2], is studied. In particular, the model is a system of partial differential equation with two moving boundaries for which we prove the existence of a travelling wave solution.     

Partitions and sums of integers with repetition

A partition of N is called “admissible” provided some cell has arbitrarily long arithmetic progressions of even integers in a fixed increment. The principal result is that the statement “Whenever {A_i}_{i < r} is an admissible partition of N, there are some i < r and some sequence 〈x_n〉_{n < ω} of distinct members of N such that x_n + x_m?A_i whenever {m, n} ? ω″ is true when r = 2 and false when r ? 3.     

The development of a cellular automaton-finite volume model for dendritic growth

A cellular automaton to track the solid–liquid interface movement is linked to finite volume computations of solute diffusion to simulate the behavior of dendritic structures in binary alloys during solidification. A significant problem encountered in the CA formulation has been the presence of artificial anisotropy in growth kinetics introduced by a Cartesian CA grid. A new technique to track the interface movement is proposed to model dendritic growth in different crystallographic orientations while reducing the anisotropy due to grid orientation. The model stability with respect to the numerical parameters (cell size and time step) for various operating conditions is examined. A method for generating an operating window in Δt and Δx has been identified, in which the model gives a grid-independent set of results for calculated dendrite tip radius and tip undercooling. Finally, the model is compared to published experimental and analytical results for both directional and equiaxed growth conditions.     

Local duality for equitable partitions of a Hamming space

The Hamming space Qn is the set of binary words of length n. A partition (C₁,C₂,…,Cr) of Qn with quotient matrix B=[bij]_r×r is equitable if for all i and j, any word in the cell Ci has exactly bij neighbors in the cell Cj. In this paper, we provide an explicit formula relating the local spectrum of cells in the face to that in the orthogonal face.     

2-resonance of plane bipartite graphs and its applications to boron-nitrogen fullerenes

A set H of disjoint faces of a plane bipartite graph G is a resonant pattern if G has a perfect matching M such that the boundary of each face in H is an M-alternating cycle. An elementary result was obtained [Discrete Appl. Math. 105 (2000) 291-311]: a plane bipartite graph is 1-extendable if and only if every face forms a resonant pattern. In this paper we show that for a 2-extendable plane bipartite graph, any pair of disjoint faces form a resonant pattern, and the converse does not necessarily hold. As an application, we show that all boron-nitrogen (B-N) fullerene graphs are 2-resonant, and construct all the 3-resonant B-N fullerene graphs, which are all k-resonant for any positive integer k. Here a B-N fullerene graph is a plane cubic graph with only square and hexagonal faces, and a B-N fullerene graph is k-resonant if any disjoint faces form a resonant pattern. Finally, the cell polynomials of 3-resonant B-N fullerene graphs are computed.