Bilinear Relative Equilibria of Identical Point Vortices

A new class of bilinear relative equilibria of identical point vortices in which the vortices are constrained to be on two perpendicular lines, conveniently taken to be the x- and y-axes of a Cartesian coordinate system, is introduced and studied. In the general problem we have m vortices on the y-axis and n on the x-axis. We define generating polynomials q(z) and p(z), respectively, for each set of vortices. A second-order, linear ODE for p(z) given q(z) is derived. Several results relating the general solution of the ODE to relative equilibrium configurations are established. Our strongest result, obtained using Sturm’s comparison theorem, is that if p(z) satisfies the ODE for a given q(z) with its imaginary zeros symmetric relative to the x-axis, then it must have at least n?m+2 simple, real zeros. For m=2 this provides a complete characterization of all zeros, and we study this case in some detail. In particular, we show that, given q(z)=z ²+η ², where η is real, there is a unique p(z) of degree n, and a unique value of η ²=A _n , such that the zeros of q(z) and p(z) form a relative equilibrium of n+2 point vortices. We show that $A_{n} \approx\frac{2}{3}n + \frac{1}{2}$ , as n→∞, where the coefficient of n is determined analytically, the next-order term numerically. The paper includes extensive numerical documentation on this family of relative equilibria.     

On graph measures in banach spaces and description of essential spectra of multidimensional transport equation

In this paper, we define new measures called respectively graph measure of noncompactness and graph measure of weak noncompactness. Moreover, we apply the obtained results to discuss the incidence of some perturbation results realized in [2] on the behavior of essential spectra of such closed densely defined linear operators on Banach spaces. These results are exploited to investigate the essential spectra of a multidimensional neutron transport operator on L₁ spaces.     

Spectral Analysis of a Transport Operator Arising in Growing Cell Populations

The present paper is concerned with the spectral analysis of a transport-like operator derived from a model introduced by Rotenberg describing the growth of a cell population. Each cell of this population is distinguished by its degree of maturity μ and its maturation velocity v. The biological boundaries of μ = 0 and μ = a (a > 0) are fixed and tightly coupled through mitosis. At mitosis daughter cells and mother cells are related by a general reproduction rule which covers all known biological ones. We first discuss in detail the spectrum of the streaming operator for smooth and partly smooth boundary conditions. Next, we discuss the existence and nonexistence of eigenvalues of the transport operator in the half plane {λ ∈ ℂ : Reλ > where denotes the spectral bound of the streaming operator. In particular, the strict monotonicity of the leading eigenvalue (when it exists) of the transport operator with respect to different parameters of the equation is also considered. We close the paper by describing in detail the various essential spectra of the transport operator for wide classes of collision and boundary operators.     

A secure authentication scheme for VANETs with batch verification

Wireless Networks - Vehicular Ad-hoc Networks (VANETs) will start becoming deployed within the next decade. Among other benefits, it is expected that VANETs will support applications and services...     

A kinetic study and application of electro-Fenton process for the remediation of aqueous environment containing toluene in a batch reactor

In this work, the degradation and mineralization of toluene (TOL) in synthetic wastewater were studied by electro-Fenton process in a batch reactor. Also the impact of operational factors such as applied current, electrolysis time, concentration of TOL and Ferrous ion have been explored on the removal of toluene. In optimum conditions, the removal efficiency of TOL and total organic carbon (TOC) were 94.5 and 32.3% after 60 min of reaction. Based on the kinetic study the pseudo first-order rate constant for the removal of toluene and TOC was obtained at k = 5.90 × 10^?2 and 9.8 × 10^?3 min^?1, respectively.