We develop a mathematical model for the disease which can be transmitted via vector and through blood transfusion in host population. The host population is structured by the chronological age. We assume that the instantaneous death and infection rates depend on the age. Applying semigroup theory and so forth, we investigate the existence of equilibria. We also discuss local stability of steady states.

The main theme of this paper is the approximation on the sphere by weighted sums of spherical harmonics. We give necessary and sufficient conditions on the weights for convergence in both the continuous and the $L^P$ cases. Approximation by spherical convolution is a particular and important case that fits into our setting.

The problem of potential flow of a second-order fluid around an ellipsoid is solved, and the flow and stress fields are computed. The flow fields are determined by the harmonic potential but the stress fields depend on viscosity and the parameters of the second-order fluid. The stress fields on the surface of a tri-axial ellipsoid depend strongly on the ratios of principal axes and are such as to suggest the formation of gas bubble with a round flat nose and two-dimensional cusped trailing edge. A thin flat trailing edge gives rise to a large stress which makes the thin trailing edge thinner.

A dynamic model for multi-rigid-body systems which consists of interconnected rigid bodies based on particle dynamics and a recursive approach is presented. The method uses the concepts of linear and angular momentums to generate the rigid body equations of motion in terms of the Cartesian coordinates of a dynamically equivalent constrained system of particles, without introducing any rotational coordinates and the corresponding rotational transformation matrix. For the open-chain system, the equations of motion are generated recursively along the serial chains. A closed-chain system is transformed to open-chain by cutting suitable kinematical joints and introducing cut-joint constraints. An example is chosen to demonstrate the generality and simplicity of the developed formulation.

Consider the initial boundary value problem for the equation $u_t=-L\left(t\right)u$, $u\left(1\right)=w$ on an interval $\left[0,1\right]$ for $t>0$, where $w\left(x\right)$ is a given function in $L^2\left(\Omega \right)$ and $\Omega$ is a bounded domain in ${ℝ}^n$ with a smooth boundary $\partial \Omega$. $L$ is the unbounded, nonnegative operator in $L^2\left(\Omega \right)$ corresponding to a selfadjoint, elliptic boundary value problem in $\Omega$ with zero Dirichlet data on $\partial \Omega$. The coefficients of $L$ are assumed to be smooth and dependent of time. It is well known that this problem is ill-posed in the sense that the solution does not depend continuously on the data. We impose a bound on the solution at $t=0$ and at the same time allow for some imprecision in the data. Thus we are led to the constrained problem. There is built an approximation solution, found error estimate for the applied method, given preliminary error estimates for the approximate method.

The distributions of products and ratios of random variables are of interest in many areas of the sciences. In this paper, the exact distributions of the product $\left|XY\right|$ and the ratio $\left|X/Y\right|$ are derived when $X$ and $Y$ are Laplace and Bessel function random variables distributed independently of each other.

An evolutionary distribution (ED), denoted by $z\left(\mathbf{x},t\right)$, is a distribution of density of phenotypes over a set of adaptive traits $\mathbf{x}$. Here $\mathbf{x}$ is an $n$-dimensional vector that represents the adaptive space. Evolutionary interactions among phenotypes occur within an ED and between EDs. A generic approach to modeling systems of ED is developed. With it, two cases are analyzed. (1) A predator prey inter-ED interactions either with no intra-ED interactions or with cannibalism and competition (both intra-ED interactions). A predator prey system with no intra-ED interactions is stable. Cannibalism destabilizes it and competition strengthens its stability. (2) Mixed interactions (where phenotypes of one ED both benefit and are harmed by phenotypes of another ED) produce complete separation of phenotypes on one ED from the other along the adaptive trait. Foundational definitions of ED, adaptive space, and so on are also given. We argue that in evolutionary context, predator prey models with predator saturation make less sense than in ecological models. Also, with ED, the dynamics of population genetics may be reduced to an algebraic problem. Finally, extensions to the theory are proposed.

We study the asymptotic behaviour of solutions to the linear evolution problem for clamped curved rods with the small thickness $\epsilon$ under minimal regularity assumptions on the geometry. In addition, nonconstant density of the curved rods is considered.