Related papers: Algorithms yield upper bounds in differential alge…
Border bases can be considered to be the natural extension of Gr\"obner bases that have several advantages. Unfortunately, to date the classical border basis algorithm relies on (degree-compatible) term orderings and implicitly on reduced…
$f,g_1,...,g_m$ be elements of the polynomial ring $\mathbb{R}[x_1,...,x_n]$. The paper deals with the general problem of computing a lower bound for $f$ on the subset of $\mathbb{R}^n$ defined by the inequalities $g_i\ge 0$, $i=1,...,m$.…
Algorithms typically come with tunable parameters that have a considerable impact on the computational resources they consume. Too often, practitioners must hand-tune the parameters, a tedious and error-prone task. A recent line of research…
We show how any PAC learning algorithm that works under the uniform distribution can be transformed, in a blackbox fashion, into one that works under an arbitrary and unknown distribution $\mathcal{D}$. The efficiency of our transformation…
Partial Differential Equations (PDEs) are fundamental tools for modeling physical phenomena, yet most PDEs of practical interest cannot be solved analytically and require numerical approximations. The feasibility of such numerical methods,…
We prove the first, even super-polynomial, lower bounds on the size of tropical (min,+) and (max,+) circuits approximating given optimization problems. Many classical dynamic programming (DP) algorithms for optimization problems are pure in…
We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases.…
The coalgebraic $\mu$-calculus provides a generic semantic framework for fixpoint logics over systems whose branching type goes beyond the standard relational setup, e.g. probabilistic, weighted, or game-based. Previous work on the…
The works presented in this habilitation concern the algorithmics of polynomials. This is a central topic in computer algebra, with numerous applications both within and outside the field - cryptography, error-correcting codes, etc. For…
Efficient computation of trajectories of switched affine systems becomes possible, if for any such hybrid system, we can manage to efficiently compute the sequence of switching times. Once the switching times have been computed, we can…
We develop a fast algorithm for computing the bound of an Ore polynomial over a skew field, under mild conditions. As an application, we state a criterion for deciding whether a bounded Ore polynomial is irreducible, and we discuss a…
Many proofs in discrete mathematics and theoretical computer science are based on the probabilistic method. To prove the existence of a good object, we pick a random object and show that it is bad with low probability. This method is…
A solution for Smale's 17th problem, for the case of systems with bounded degree was recently given. This solution, an algorithm computing approximate zeros of complex polynomial systems in average polynomial time, assumed infinite…
Polynomial optimization problems often arise in sequences indexed by dimension, and it is of interest to compute bounds on the optimal values of all problems in the sequence. Examples include certifying inequalities between symmetric…
On the basis of additive schemes (splitting schemes) we construct efficient numerical algorithms to solve approximately the initial-boundary value problems for systems of time-dependent partial differential equations (PDEs). In many applied…
New algorithms are devised for finding the maxima of multidimensional point samples, one of the very first problems studied in computational geometry. The algorithms are very simple and easily coded and modified for practical needs. The…
Ordinary differential equations that model technical systems often contain states, that are considered dangerous for the system. A trajectory that reaches such a state usually indicates a flaw in the design. In this paper, we present and…
Given a non-zero polynomial $f$ in a polynomial ring $R$ with coefficients in a finite field of prime characteristic $p$, we present an algorithm to compute a differential operator $\delta$ which raises $1/f$ to its $p$th power. For some…
A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not…
We introduce linear programs encoding regular expressions of finite languages. We show that, given a language, the optimum value of the associated linear program is a lower bound on the size of any regular expression of the language.…