Peter Franek

Robust Satisfiability of Systems of Equations

We study the problem of robust satisfiability of systems of nonlinear equations, namely, whether for a given continuous function f:K → Rn on a finite simplicial complex K and α>0, it holds that each function g:K → Rn such that ║g−f║∞ ≤ α, has a root in K. Via a reduction to the extension problem of maps into a sphere, we particularly show that this problem is decidable in polynomial time for every fixed n, assuming dim K ≤ 2n−3. This is a substantial extension of previous computational applications of topological degree and related concepts in numerical and interval analysis. Via a reverse reduction, we prove that the problem is undecidable when dim K ≥ 2n−2, where the threshold comes from the stable range in homotopy theory. For the lucidity of our exposition, we focus on the setting when f is simplexwise linear. Such functions can approximate general continuous functions, and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings.

Satisfiability of systems of equations of real analytic functions is quasi-decidable

In this paper we consider the problem of checking whether a system of equations of real analytic functions is satisfiable, that is, whether it has a solution. We prove that there is an algorithm (possibly non-terminating) for this problem such that (1) whenever it terminates, it computes a correct answer, and (2) it always terminates when the input is robust. A system of equations of robust, if its satisfiability does not change under small perturbations. As a basic tool for our algorithm we use the notion of degree from the field of (differential) topology.

Effective topological degree computation based on interval arithmetic

We describe a new algorithm for calculating the topological degree deg (f, B, 0) where B \subseteq Rn is a product of closed real intervals and f : B \rightarrow Rn is a real-valued continuous function given in the form of arithmetical expressions. The algorithm cleanly separates numerical from combinatorial computation. Based on this, the numerical part provably computes only the information that is strictly necessary for the following combinatorial part, and the combinatorial part may optimize its computation based on the numerical information computed before. We also present computational experiments based on an implementation of the algorithm. Also, in contrast to previous work, the algorithm does not assume knowledge of a Lipschitz constant of the function f, and works for arbitrary continuous functions for which some notion of interval arithmetic can be defined.

Quasi-decidability of a Fragment of the First-Order Theory of Real Numbers

In this paper we consider a fragment of the first-order theory of the real numbers that includes systems of n equations in n variables, and for which all functions are computable in the sense that it is possible to compute arbitrarily close interval approximations. Even though this fragment is undecidable, we prove that—under the additional assumption of bounded domains—there is a (possibly non-terminating) algorithm for checking satisfiability such that (1) whenever it terminates, it computes a correct answer, and (2) it always terminates when the input is robust. A formula is robust, if its satisfiability does not change under small continuous perturbations. We also prove that it is not possible to generalize this result to the full first-order language—removing the restriction on the number of equations versus number of variables. As a basic tool for our algorithm we use the notion of degree from the field of topology.

Persistence of Zero Sets

We study robust properties of zero sets of continuous maps

Symmetries of Quasi-Values

f:X\to\mathbb{R}^n

Proving the existence of loops in robot trajectories

. Formally, we analyze the family

Invariant Operators of First Order Generalizing the Dirac Operator in 2 Variables

Z_r(f)=\{g^{-1}(0):\,\,\|g-f\|<r\}

Description of a Complex of Operators Acting Between Higher Spinor Modules

of all zero sets of all continuous maps

First-principles study of the electronic structure and exchange interactions in bcc europium

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