Singularities and nonhyperbolic manifolds do not coincide

We consider the billiard flow of elastically colliding hard balls on the flat $\nu$-torus ($\nu\ge 2$), and prove that no singularity manifold can even locally coincide with a manifold describing future non-hyperbolicity of the trajectories. As a corollary, we obtain the ergodicity (actually the Bernoulli mixing property) of all such systems, i.e. the verification of the Boltzmann-Sinai Ergodic Hypothesis.Comment: Final version, to appear in Nonlinearit

The quantum group, Harper equation and the structure of Bloch eigenstates on a honeycomb lattice

The tight-binding model of quantum particles on a honeycomb lattice is investigated in the presence of homogeneous magnetic field. Provided the magnetic flux per unit hexagon is rational of the elementary flux, the one-particle Hamiltonian is expressed in terms of the generators of the quantum group $U_q(sl_2)$. Employing the functional representation of the quantum group $U_q(sl_2)$ the Harper equation is rewritten as a systems of two coupled functional equations in the complex plane. For the special values of quasi-momentum the entangled system admits solutions in terms of polynomials. The system is shown to exhibit certain symmetry allowing to resolve the entanglement, and basic single equation determining the eigenvalues and eigenstates (polynomials) is obtained. Equations specifying locations of the roots of polynomials in the complex plane are found. Employing numerical analysis the roots of polynomials corresponding to different eigenstates are solved out and the diagrams exhibiting the ordered structure of one-particle eigenstates are depicted.Comment: 11 pages, 4 figure

Algorithm Diversity for Resilient Systems

Diversity can significantly increase the resilience of systems, by reducing the prevalence of shared vulnerabilities and making vulnerabilities harder to exploit. Work on software diversity for security typically creates variants of a program using low-level code transformations. This paper is the first to study algorithm diversity for resilience. We first describe how a method based on high-level invariants and systematic incrementalization can be used to create algorithm variants. Executing multiple variants in parallel and comparing their outputs provides greater resilience than executing one variant. To prevent different parallel schedules from causing variants' behaviors to diverge, we present a synchronized execution algorithm for DistAlgo, an extension of Python for high-level, precise, executable specifications of distributed algorithms. We propose static and dynamic metrics for measuring diversity. An experimental evaluation of algorithm diversity combined with implementation-level diversity for several sequential algorithms and distributed algorithms shows the benefits of algorithm diversity