The large-time asymptotic behaviour of real-valued solutions of the pure initial-value problem for Burgers' equation ut + uuxuxx = 0, is studied. The initial data satisfy u0(x) ~ nx as |x|, where nR. There are two constants of the motion that affect the large-time behaviour:Hopf considered the case n = 0 (i.e. u0L1(R)), and the casesufficiently small was considered by Dix. Here we completely remove that smallness condition. When n < 1, we find an explicit function U(), depending only onand n, such thatuniformly in . When n1, there are two different functions U() that simultaneously attract the quantity t12u(t12, t), and each one wins in its own range of . Thus we give an asymptotic description of the solution in different regions and compute its decay rate in L. Sharp error estimates are proved.

Letbe an arbitrary non-empty bounded Lipschitz domain in RMRN. Given> 0, squeezeby the factorin the y-direction to obtain the squeezed domain := {(x, y) | (x, y)}. Letandbe positive constants. Consider the following semilinear damped wave equation on ,where is the exterior normal vector field on and G is an appropriate nonlinearity, which ensures that (W) generates a (local) flow ̃ on X := H1()L2(). We show that there is a closed subspace X0 of X and a flow ̃0 on X0 that is the limit flow of the family ̃,> 0. We show that, as 0, the family ̃ converges in some singular sense to ̃ and establish a technical singular asymptotic compactness property. As a corollary, we obtain an upper-semicontinuity result for global attractors of the family ̃, 0, generalizing results obtained previously by Hale and Raugel for domains that are ordinate sets of a positive function.

The results obtained here are also applied in our paper On a general Conley index continuation principle for singular perturbation problems to establish a singular Conley index continuation principle for damped wave equations on thin domains.

We consider two model reaction-diffusion systems of bistable type arising in the theory of phase transition; they appear in various physical contexts, such as thin magnetic films and diblock copolymers. We prove the convergence of the solution of these systems to the solution of free-boundary problems involving modified motion by mean curvature.

An integer n\geq2 is said to be a genus of a finite group G if there is a compact Riemann surface of genus n on which G acts as a group of automorphisms. In this paper, formulae are given for the minimum genus, minimum stable genus and the gap sequence, i.e., the (finite) set of non-genera, for a split metacyclic group of order pq, where p and q are primes. This information completely determines the genus spectrum for such groups.

In 1953, Pjateckii˘-S˘apiro has proved that there are infinitely many primes of the form [n^c] for 1 [less than] c [less than] {12\over11} (with an asymptotic result). This range, which measures our progress in the technique of exponential sums, has been improved by many authors. In this paper we obtain 1 [less than] c\leq{243\over205}.

All the Gabriel topologies on a Dubrovin valuation ring are classified in terms of its prime ideals. Furthermore, these Gabriel topologies are cogenerated by two kinds of indecomposable injective modules.

If R is a complete discrete valuation ring and M is a reduced, torsion-free R-module of rank \kappa, where \aleph_0 \leq \kappa < 2^ (\aleph_0), we show that M \prop\oplus_(\aleph_0) R \oplus C for some R-module C. As a consequence, it must be the case that M \prop M \oplus (\oplus{_\alpha}R), where \alpha \leq \aleph_0, and {\rm (End)_R}M/\rm (Fin)M has rank at least 2^ (\aleph_0), where Fin M denotes the set of endomorphisms of M with finite rank image.

We obtain the complex orbifold structure of the moduli space for one parameter equisymmetric Riemann surfaces of genus two. For each family, by using the orbifold structure, we obtain the points in the moduli corresponding to real algebraic curves and a special form for the period matrices of Riemann surfaces that admit an anticonformal involution. We describe the topological type of anti-conformal involutions admitted by surfaces of the families depending on the type of period matrix.

Let B(H) denote the C*-algebra of all bounded linear operators on a separable Hilbert space H. For A,B∈B(H), the chordal transform f_(A,B), as an operator on B(H), is defined by {f_(A,B)(X)=(\vert A^*\vert ^2+I)^(-1/2){\delta_(A,B)(X)(\vert B\vert ^2+I)^(-1/2)}, where {\delta_(A,B)} is the generalized derivation defined on B(H) by {\delta _(A,B)(X)=AX-XB}. Orthogonality of the range and the kernel of f_(A,B), with respect to the unitarily invariant norms \vert \vert \vert .\vert \vert \vert , are discussed. It is shown that if A, B are self-adjoint, then {\vert \vert \vert f_(A,B)(X)\vert \vert \vert \le \vert \vert \vert X\vert \vert \vert for all X. Related norm inequalities comparing f_(A,B) and {\delta _(A,B) are also given.

In this paper, relative pictures are used to analyze a certain family of exponent-sum two equations over groups.

The purpose of this paper is to study an optimal control problem associated with the thermally coupled Navier-Stokes equations. An existence result for this problem is obtained. The most important result of this paper is the proof of the existence and regularity of a solution of the adjoint system. By defining several functions, this system (which is not a divergence free one) is replaced by a divergence free system.

The N-soliton solution of a generalised Vakhnenko equation is found, where N is an arbitrary positive integer. The solution, which is obtained by using a blend of transformations of the independent variables and Hirota's method, is expressed in terms of a Moloney & Hodnett (1989) type decomposition. Different types of soliton are possible, namely loops, humps or cusps. Details of the different types of interactions between solitons, including resonant soliton interactions, are discussed in detail for the case N=2. A proof of the ‘N-soliton condition’ is given in the Appendix.

We study the conditions which force a semiperfect ring to admit a Nakayama permutation of its basic idempotents. We also give a few necessary and sufficient conditions for a semiperfect ring R, which cogenerates every 2-generated right R-module, to be right pseudo-Frobenius.

We provide sufficient conditions on a positive finite rotation invariant Borel measure on {𝔻}^n which guarantee that the analogue of the Carleson measure theorem remains valid for Bergman spaces of holomorphic and n-harmonic functions on {𝔻}^n generated by the measure.

This paper considers the long-time behavior of solutions for the Cauchy problem of a class of second-order nonlinear differential equations: -x''+ f(t,x,x')x'+g(x)=h(t). Under appropriate conditions it is shown that the solutions of the problem possess some dichotomy properties.

Let Ea(u,v) be the extremal algebra determined by two hermitians u and v with u^2=v^2=1. We show that: Ea(u,v)={f+gu:f,g\in C(𝕋)} , where [] is the unit circle; Ea(u,v) is C^*-equivalent to C^*({\cal G}), where {\cal G} is the infinite dihedral group; most of the hermitian elements k of Ea(u,v) have the property that k^n is hermitian for all odd n but for no even n; any two hermitian words in {\cal G} generate an isometric copy of Ea(u,v) in Ea(u,v).

We apply a previously obtained ansatz for extremal Kähler metrics to show that if a manifold admits a Hodge metric with constant scalar curvature, then the total space of the projectivization of a line bundle with first Chern class equal to the Kähler class of the metric admits a one-parameter family of extremal Kähler metrics. This generalizes earlier constructions.

A compact Klein surface X is a compact surface with a dianalytic structure. Such a surface is said to be q-hyperelliptic if it admits an involution \phi , that is an order two automorphism, such that X/ < \phi > has algebraic genus q. A Klein surface of genus 1 and k boundary components is a k-bordered torus.

By means of NEC groups, q-hyperelliptic k-bordered tori are studied and a geometrical description of their associated Teichmüller spaces is given.

We compute explicitly the isomorphic structure of the normalized unit group of an abelian group ring under some minimal natural restrictions on the group basis and the coefficient ring. This enlarges affirmations due to Chatzidakis-Pappas (J. London Math. Soc., 1991) and Mollov (Publ. Math. Debrecen, 1971).

We calculate the height of quantum determinantal ideals in the algebra of quantum matrices, and also the Gelfand-Kirillov, Krull and Classical Krull dimensions of the corresponding quantum determinantal factors.