Almost maximal spaces

A topological space X is called almost maximal if it is without isolated points and for every x∈X, there are only finitely many ultrafilters on X converging to x. We associate with every countable regular homogeneous almost maximal space X a finite semigroup Ult(X) so that if X and Y are homeomorphic, Ult(X) and Ult(Y) are isomorphic. Semigroups Ult(X) are projectives in the category F of finite semigroups. These are bands decomposing into a certain chain of rectangular components. Under MA, for each projective S in F, there is a countable almost maximal topological group G with Ult(G) isomorphic to S. The existence of a countable almost maximal topological group cannot be established in ZFC. However, there are in ZFC countable regular homogeneous almost maximal spaces X with Ult(X) being a chain of idempotents.     

Entropy and ergodic probability for differentiable dynamical systems and their bundle extensions

We answer a problem of Liao [S.T. Liao, Standard systems of differential equations and obstruction sets—from linearity to perturbations, in: System Researches, Proceedings Dedicated to the 85th Anniversary of Qian Xue-Sen, Zhejiang Education Press, Hangzhou, China, 1996, pp. 279-290 (in Chinese)]: A C¹ vector field or a C¹ diffeomorphism on an n-dimensional manifold has equal entropy with that of its bundle extensions. We also prove that each ergodic probability with simple Lyapunov spectrum has at most n²n! covering probabilities on each bundle extension.     

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Krichever-Novikov algebras for more than two points

Krichever-Novikov algebras of meromorphic vector fields with more than two poles on higher genus Riemann surfaces are introduced. The structure of these algebras and their induced modules of forms of weight is studied.     

The twisted SU(N) group. On the C *-algebra C(S μ U(N))

It is proved that C(S U(N)) is a type I C ^*-algebra.     

Nonlinear stability and convergence of finite-difference methods for the “good” Boussinesq equation

Summary The good Boussinesq equationu _tt =–u _xxxx +u _xx +(u ²) _xx has recently been found to possess an interesting soliton-interaction mechanism. In this paper we study the nonlinear stability and the convergence of some simple finite-difference schemes for the numerical solution of problems involving the good Boussinesq equation. Numerical experimentas are also reported.     

A shape-optimization technique for the capillary surface problem

Summary The problem of the construction of an equilibrium surface taking the surface tension into account leads to Laplace-Young equation which is a nonlinear elliptic free-boundary problem. In contrast to Orr et al. where an iterative technique is used for direct solution of the equation for problems with simple geometry, we propose here an alternative approach based on shape optimization techniques. The shape of the domain of the liquid is varied to attain the optimality condition. Using optimal control theory to derive expressions for the gradient, a numerical scheme is proposed and simple model problems are solved to validate the scheme.     

Finding numerical solution of linear partial differential equations of first order with real coefficients

Applying Bittner's operational calculus we present a method to give approximate solutions of linear partial differential equations of first order