Inverting Onto Functions and Polynomial Hierarchy

In this paper we construct an oracle under which the polynomial hierarchy is infinite but there are non-invertible polynomial time computable multivalued onto functions

Multiple Hamiltonian structure of Bogoyavlensky-Toda lattices

This paper is mainly a review of the multi--Hamiltonian nature of Toda and generalized Toda lattices corresponding to the classical simple Lie groups but it includes also some new results. The areas investigated include master symmetries, recursion operators, higher Poisson brackets, invariants and group symmetries for the systems. In addition to the positive hierarchy we also consider the negative hierarchy which is crucial in establishing the bi--Hamiltonian structure for each particular simple Lie group. Finally, we include some results on point and Noether symmetries and an interesting connection with the exponents of simple Lie groups. The case of exceptional simple Lie groups is still an open problem.Comment: 65 pages, 67 reference

Nondeterministic functions and the existence of optimal proof systems

We provide new characterizations of two previously studied questions on nondeterministic function classes: Q1: Do nondeterministic functions admit efficient deterministic refinements? Q2: Do nondeterministic function classes contain complete functions? We show that Q1 for the class is equivalent to the question whether the standard proof system for SAT is p-optimal, and to the assumption that every optimal proof system is p-optimal. Assuming only the existence of a p-optimal proof system for SAT, we show that every set with an optimal proof system has a p-optimal proof system. Under the latter assumption, we also obtain a positive answer to Q2 for the class . An alternative view on nondeterministic functions is provided by disjoint sets and tuples. We pursue this approach for disjoint -pairs and its generalizations to tuples of sets from and with disjointness conditions of varying strength. In this way, we obtain new characterizations of Q2 for the class . Question Q1 for is equivalent to the question of whether every disjoint -pair is easy to separate. In addition, we characterize this problem by the question of whether every propositional proof system has the effective interpolation property. Again, these interpolation properties are intimately connected to disjoint -pairs, and we show how different interpolation properties can be modeled by -pairs associated with the underlying proof system

On Soliton-type Solutions of Equations Associated with N-component Systems

The algebraic geometric approach to $N$-component systems of nonlinear integrable PDE's is used to obtain and analyze explicit solutions of the coupled KdV and Dym equations. Detailed analysis of soliton fission, kink to anti-kink transitions and multi-peaked soliton solutions is carried out. Transformations are used to connect these solutions to several other equations that model physical phenomena in fluid dynamics and nonlinear optics.Comment: 43 pages, 16 figure

Degenerate Frobenius manifolds and the bi-Hamiltonian structure of rational Lax equations

The bi-Hamiltonian structure of certain multi-component integrable systems, generalizations of the dispersionless Toda hierarchy, is studies for systems derived from a rational Lax function. One consequence of having a rational rather than a polynomial Lax function is that the corresponding bi-Hamiltonian structures are degenerate, i.e. the metric which defines the Hamiltonian structure has vanishing determinant. Frobenius manifolds provide a natural setting in which to study the bi-Hamiltonian structure of certain classes of hydrodynamic systems. Some ideas on how this structure may be extanded to include degenerate bi-Hamiltonian structures, such as those given in the first part of the paper, are given.Comment: 28 pages, LaTe