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## Exact Solution for Long-Term Size Exclusion Suspension-Colloidal Transport in Porous Media

**Long-term deep bed filtration in porous media with size exclusion particle capture mechanism** **is studied.**** For mono dispersed suspension and transport in porous media whit distributed pore sizes, the micro stochastic model allows for upscaling and the exact solution is derived for the obtained macro scale equation system.** **Results show that** **transient pore size distribution and nonlinear relation between the filtration coefficient and captured particle concentration during suspension filtration and retention are the main features of long-term deep bed filtration, which generalises the classical deep bed filtration model and its latter modifications. Furthermore, the exact solution demonstrates earlier breakthrough and lower breakthrough concentration for larger particles. Among all the pores with different sizes, the ones with intermediate sizes (between the minimum pore size and the particle size) vanish first. Total concentration of all the pores smaller than the particles** **turns to zero asymptotically when time tends to infinity, which corresponds to complete pluggi****ng of smaller pores.**

A mathematical model of particle motion in the filter based on mechanical-geometrical interaction of particles with a porous medium is considered. Near the wave front an asymptotic solution of a quasi-linear system of partial differential equations for suspended and deposited particle concentrations is constructed. For a linear filtration coefficient asymptotics is compared with the exact solution.

Previous empirical research has developed stochastic electoral models for Israel, Turkey, and other polities. The work suggests that convergence to an electoral center (often predicted by electoral models) is a nongeneric phenomenon. In an attempt to explain nonconvergence, a formal model based on intrinsic valence is presented. This theory showed that there are necessary and sufficient conditions for convergence. The necessary condition is that a convergence coefficient c is bounded above by the dimension w of the policy space, while a sufficient condition is that the coefficient is bounded above by 1. This coefficient is defined in terms of the difference in exogenous valences, the “spatial coefficient”, and the electoral variance. The theoretical model is then applied to empirical analyses of elections in the United States and Britain. These empirical models include sociodemographic valence and electoral perceptions of character trait. It is shown that the model implies convergence to positions close to the electoral origin. To explain party divergence, the model is then extended to incorporate activist valences. This extension gives a first-order balance condition that allows the party to calculate the optimal marginal condition to maximize vote share. We argue that the equilibrium positions of presidential candidates in US elections and by party leaders in British elections are principally due to the influence of activists, rather than the centripetal effect of the electorate.

We present an analytical description of the class of unsteady vortex surface waves generated by non- uniformly distributed, time-harmonic pressure. The fluid motion is described by an exact solution of the equations of hydrodynamics generalizing the Gerstner solution. The trajectories of the fluid particles are circumferences. The particles on a free surface rotate around circumferences of the same radii, with the centers of the circumferences lying on different horizons. A family of waves has been found in which a variable pressure acts on a limited section of the free surface. The law of external pressure distribution includes an arbitrary function. An example of the evolution of a non-uniform wave packet is considered. The wave and pressure profiles, as well as vorticity distribution are studied. It is shown that, in the case of a uniform traveling wave of external pressure, the Gerstner solution is valid but with a different form of the dispersion relation. A possibility of observing the studied waves in laboratory and in the real ocean is discussed.

In this paper we consider the nature of local Nash equilibrium (LNE) for a model

of the 2007 Duma election in Russia, using estimates of valence obtained from sociodemographic

variables.

We then extend this sociodemographic valence model by including institutional valences,

the approval by voters of the various institutions, including the President, the PrimeMinister,

the State Duma and the Federation Council.We show by simulation that the vote maximizing

LNE of this general stochastic model were not at the electoral origin. The dominant feature

of the election was the influence of approval or disapproval of President Putin on each voter’s

political choice.

The cell formation problem (CFP) is an NP-hard optimization problem considered for cell manufacturing systems. Because of its high computational complexity several heuristics have been developed for solving this problem. In this paper we present a branch and bound algorithm which provides exact solutions of the CFP. This algorithm finds optimal solutions for 13 problems of the 35 popular benchmark instances from the literature.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.