In this paper we discuss the dissipative property of near-equilibrium classical solutions to the Cauchy problem of the Vlasov-Maxwell-Boltzmann System in the whole space @math when the positive charged ion flow provides a spatially uniform background. The most key point of studying this coupled degenerately dissipative system here is to establish the dissipation of the electromagnetic field which turns out to be of the regularity-loss type. Precisely, for the linearized non-homogeneous system, some @math energy functionals and @math time-frequency functionals which are equivalent with the naturally existing ones are designed to capture the optimal dissipation rate of the system, which in turn yields the optimal @math - @math type time-decay estimates of the corresponding linearized solution operator. These results show a special feature of the one-species Vlasov-Maxwell-Boltzmann system different from the case of two-species, that is, the dissipation of the magnetic field in one-species is strictly weaker than the one in two-species. As a by-product, the global existence of solutions to the nonlinear Cauchy problem is also proved by constructing some similar energy functionals but the time-decay rates of the obtained solution still remain open.
@math C(1+t)^ - 3 4 |u_0 |_ Z_1 L^2 @math | u |_ L^2_ ^2+ | a |^2 |k|^2 @math @math C(1+t)^ - 1 4 |u_0 |_ Z_1 L^2 @math | u |_ L^2_ ^2+|[ E , B ]|^2 @math @math @math | u |_ L^2_ ^2+|[ E , B ]|^2 @math @math @math b(t,x) @math u @math b(t,x) @math _1 @math _2 @math _4 @math E @math ^ lin ,2 (U(t)) @math (k) @math L^p @math L^q @math E u @math E ( u)^2 ,dxd @math (U(t))^ 1 2 (U(t)) @math E @math u @math (U(t)) @math @math $ We remark that this also has been observed in @cite_0 in the study of the one-species Vlasov-Poisson-Boltzmann system.
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!# @math ) 4 ' " + @math 4 ) @ & A B C D E . * . ! F ' G H ' IJ K L * M & + *) N z'cC g) ' &  p : q :  r Htd|#t9 ,s tvuw x yFz |;w]tS 9 x |;   C n+ U <dU' 4 2 Let @math be a primitive Hilbert modular form of parallel weight @math and level @math for the totally real field @math , and let @math be a rational prime coprime to @math . If @math is ordinary at @math and @math is a CM extension of @math of relative discriminant @math prime to @math , we give an explicit construction of the @math -adic Rankin-Selberg @math -function @math . When the sign of its functional equation is @math , we show, under the assumption that all primes @math are principal ideals of @math which split in @math , that its central derivative is given by the @math -adic height of a Heegner point on the abelian variety @math associated with @math . This @math -adic Gross--Zagier formula generalises the result obtained by Perrin-Riou when @math and @math satisfies the so-called Heegner condition. We deduce applications to both the @math -adic and the classical Birch and Swinnerton-Dyer conjectures for @math . Let T = (T*, T(Delta)) he a spherical latin bitrade. With each a = (a(1), a(2), a(3)) is an element of T* associate a set of linear equations Eq(T, a) of the form b(1) + b(2) = b(3), where b = (b(1), b(2), b(3)) runs through T* a . Assume a(1) = 0 = a(2) and a(3) = 1. Then Eq(T, a) has in rational numbers a unique solution b(i) = (b) over bar (i). Suppose that (b) over bar (i) not equal (c) over bar (i) for all b, c is an element of T* such that (b) over bar (i) not equal c(i) and i is an element of 1, 2, 3 . We prove that then T(Delta) can be interpreted as a dissection of an equilateral triangle. We also consider group modifications of latin bitrades and show that the methods for generating the dissections can be used for a proof that T* can be embedded into the operational table of a finite abelian group, for every spherical latin bitrade T. (C) 2009 Wiley Periodicals, Inc. J Combin Designs 18: 1-24, 2010 Consider a network of @math n processes, where each process inputs a @math d-dimensional vector of reals. All processes can communicate directly with others via reliable FIFO channels. We discuss two problems. The multidimensional Byzantine consensus problem, for synchronous systems, requires processes to decide on a single @math d-dimensional vector @math v?Rd, inside the convex hull of @math d-dimensional vectors that were input by the non-faulty processes. Also, the multidimensional Byzantine approximate agreement (MBAA) problem, for asynchronous systems, requires processes to decide on multiple @math d-dimensional vectors in @math Rd, all within a fixed Euclidean distance @math ∈ of each other, and inside the convex hull of @math d-dimensional vectors that were input by the non-faulty processes. We obtain the following results for the problems above, while tolerating up to @math f Byzantine failures in systems with complete communication graphs: (1) In synchronous systems, @math n>max 3f,(d+1)f is necessary and sufficient to solve the multidimensional consensus problem. (2) In asynchronous systems, @math n>(d+2)f is necessary and sufficient to solve the multidimensional approximate agreement problem. Our sufficiency proofs are constructive, giving explicit protocols for the problems. In particular, for the MBAA problem, we give two protocols with strictly different properties and applications.