arXiv subjects
Publications and source records attributed to Michael.
We consider magnetic geodesic flows on the 2-torus. We prove that the question of existence of polynomial in momenta first integrals on one energy level leads to a Semi-Hamiltonian system of quasi-linear equations, i.e. in the hyperbolic regions the system has Riemann invariants and can be written in conservation laws form.
The aim of this paper is to consider quasi-linear systems which are not in the form of evolution equations. We propose new condition of Richness or Semi-Hamiltonicity for such a system and prove that the blow up analysis along characteristic curves can be performed for it in an analogous manner. This opens a possibility to use this ansatz also for geometric problems. We apply the results to the problem of Polynomial integral for geodesic flows on the 2-torus.
We study periodic solutions for a quasi-linear system, which is the so called dispersionless Lax reduction of the Benney moments chain. This question naturally arises in search of integrable Hamiltonian systems of the form $ H=p^2/2+u(q,t) $ Our main result classifies completely periodic solutions for 3 by 3 system. We prove that the only periodic solutions have the form of traveling waves, so in particular, the potential $u$ is a function of a linear combination of $t$ and $q$. This result implies that the there are no nontrivial cases of existence of the fourth power integral of motion for $H$: if it exists, then it is equal necessarily to the square of the quadratic one. Our method uses two new general observations. The first is the genuine non-linearity of the maximal and minimal eigenvalues for the system. The second observation uses the compatibility conditions of Gibonns-Tsarev in order to give certain exactness for the system in Riemann invariants. This exactness opens a possibility to apply the Lax analysis of blow up of smooth solutions, which usually does not work for systems of higher order.
Codd [Codd 1970] wrote the first paper in which the model of a relational database was proposed. Adleman [Adleman 1994] wrote the first paper in which DNA strands in a test tube were used to solve an instance of the Hamiltonian path problem. From [Adleman 1994], it is obviously indicated that for storing information in molecules of DNA allows for an information density of approximately 1 bit per cubic nm (nanometer) and a dramatic improvement over existing storage media such as video tape which store information at a density of approximately 1 bit per 1012 cubic nanometers. This paper demonstrates that biological operations can be applied to construct bio-molecular databases where data records in relational tables are encoded as DNA strands. In order to achieve the goal, DNA algorithms are proposed to perform eight operations of relational algebra (calculus) on bio-molecular relational databases, which include Cartesian product, union, set difference, selection, projection, intersection, join and division. Furthermore, this work presents clear evidence of the ability of molecular computing to perform data retrieval operations on bio-molecular relational databases.