Abstract:
We propose the ``Competing Salesmen Problem'' (CSP), a 2-player competitive version of the classical Traveling Salesman Problem. This problem arises when considering two competing salesmen instead of just one. The concern for a shortest tour is replaced by the necessity to reach any of the customers before the opponent does. In particular, we consider the situation where players take turns, moving along one edge at a time within a graph G=(V,E). The set of customers is given by a subset V_C V of the vertices. At any given time, both players know of their opponent's position. A player wins if he is able to reach a majority of the vertices in V_C before the opponent does. We prove that the CSP is PSPACE-complete, even if the graph is bipartite, and both players start at distance 2 from each other. We show that the starting player may lose the game, even if both players start from the same vertex. For bipartite graphs, we show that the starting player always can avoid a loss. We also show that the second player can avoid to lose by more than one customer, when play takes place on a graph that is a tree T, and V_C consists of leaves of T. For the case where T is a star and V_C consists of n leaves of T, we give a simple and fast strategy which is optimal for both players. If V_C consists not only of leaves, the situation is more involved.

Abstract:
Motion of chemically driven droplets is analyzed by applying a solvability condition of perturbed hydrodynamic equations affected by the adsorbate concentration. Conditions for traveling bifurcation analogous to a similar transition in activator-inhibitor systems are obtained. It is shown that interaction of droplets leads to either scattering of mobile droplets or formation of regular patterns, respectively, at low or high adsorbate diffusivity. The same method is applied to droplets running on growing terrace edges during surface freezing.

Abstract:
This paper studies the traveling wave solutions to a three species competition cooperation system. The existence of the traveling waves is investigated via monotone iteration method. The upper and lower solutions come from either the waves of KPP equation or those of certain Lotka Volterra system. We also derive the asymptotics and uniqueness of the wave solutions. The results are then applied to a Lotka Volterra system with spatially averaged and temporally delayed competition.

Abstract:
This paper explains the uniqueness of positive steady state of general Lotka-Volterra competition model of two species of animals in the same environment.

Abstract:
These lecture notes give a short review of methods such as the matrix ansatz, the additivity principle or the macroscopic fluctuation theory, developed recently in the theory of non-equilibrium phenomena. They show how these methods allow to calculate the fluctuations and large deviations of the density and of the current in non-equilibrium steady states of systems like exclusion processes. The properties of these fluctuations and large deviation functions in non-equilibrium steady states (for example non-Gaussian fluctuations of density or non-convexity of the large deviation function which generalizes the notion of free energy) are compared with those of systems at equilibrium.

Abstract:
This paper is concerned with the traveling wave solutions of an integro-difference competition system, of which the purpose is to model the coinvasion-coexistence process of two competitors with age structure. The existence of nontrivial traveling wave solutions is obtained by constructing generalized upper and lower solutions. The asymptotic and nonexistence of traveling wave solutions are proved by combining the theory of asymptotic spreading with the idea of contracting rectangle.

Abstract:
This paper is concerned with the traveling wave solutions of delayed reaction-diffusion systems. By using Schauder's fixed point theorem, the existence of traveling wave solutions is reduced to the existence of generalized upper and lower solutions. Using the technique of contracting rectangles, the asymptotic behavior of traveling wave solutions for delayed diffusive systems is obtained. To illustrate our main results, the existence, nonexistence and asymptotic behavior of positive traveling wave solutions of diffusive Lotka-Volterra competition systems with distributed delays are established. The existence of nonmonotone traveling wave solutions of diffusive Lotka-Volterra competition systems is also discussed. In particular, it is proved that if there exists instantaneous self-limitation effect, then the large delays appearing in the intra-specific competitive terms may not affect the existence and asymptotic behavior of traveling wave solutions.

Abstract:
A new asymptotic method is presented for the analysis of the traveling waves in the one-dimensional reaction-diffusion system with the diffusion with a finite velocity and Kolmogorov-Petrovskii-Piskunov kinetics. The analysis makes use of the path-integral approach, scaling procedure and the singular perturbation techniques involving the large deviations theory for the Poisson random walk. The exact formula for the position and speed of reaction front is derived. It is found that the reaction front dynamics is formally associated with the relativistic Hamiltonian/Lagrangian mechanics.

Abstract:
The stress analysis of pressurized circumferential pipe weldments under steady state creep is considered. The creep response of the material is governed by Norton's law. Numerical and analytical solutions are obtained by means of perturbation method, the unperturbed solution corresponds to the stress field in a homogeneous pipe. The correction terms are treated as stresses defined with the help of an auxiliary linear elastic problem. Exact expressions for jumps of hoop and radial stresses at the interface are obtained. The proposed technique essentially simplifies parametric analysis of multi-material components.

Abstract:
The study of biological systems dynamics requires elucidation of the transitions of steady states. A “small perturbation” approach can provide important information on the “steady state” of a biological system. In our experiments, small perturbations were generated by applying a series of repeating small doses of ultraviolet radiation to a human keratinocyte cell line, HaCaT. The biological response was assessed by monitoring the gene expression profiles using cDNA microarrays. Repeated small doses (10 J/m2) of ultraviolet B (UVB) exposure modulated the expression profiles of two groups of genes in opposite directions. The genes that were up-regulated have functions mainly associated with anti-proliferation/anti-mitogenesis/apop？tosis,and the genes that were down-regulated were mainly related to proliferation/mitogenesis/anti-apoptosis？.For both groups of genes, repetition of the small doses of UVB caused an immediate response followed by relaxation between successive small perturbations. This cyclic pattern was suppressed when large doses (233 or 582.5 J/m2) of UVB were applied. Our method and results contribute to a foundation for computational systems biology, which implicitly uses the concept of steady state.