Die Bedeutung von Eilhard Alfred Mitscherlich für die Entwicklung der Bodenkunde

Eilhard Alfred Mitscherlich (1874 – 1956)wurde am 29. August 1874 als Sohn von Alfred Mitscherlich, Professor der Chirurgie, und Valeska geb. Ackermann in Berlin geboren. Nach dem Abitur im Frühjahr 1895 studierte er in Kiel zwei Semester Landwirtschaft bei Hermann Rodewald (1856-1938) und Physik bei seinem Schwager Hermann Ebert

Impacts of global change on water-related sectors and society in a trans-boundary central European river basin – Part 1: Project framework and impacts on agriculture

Central Europe, the focus region of this study, is a region in transition, climatically from maritime to continental and politically from formerly more planning-oriented to more market-oriented management regimes, and in terms of climate change from regions of increasing precipitation in the west and north of Europe to regions of decreasing precipitation in central and southern Europe. The Elbe basin, a trans-boundary catchment flowing from the Czech Republic through Germany into the North Sea, was selected to investigate the possible impacts of global change on crop yields and water resources in this region. For technical reasons, the paper has been split into two parts, the first showing the overall model concept, the model set-up for the agricultural sector, and first results linking eco-hydrological and agro-economic tools for the German part of the basin. The second part describes the model set-up for simulating water supply and demand linking eco-hydrological and water management tools for the entire basin including the Czech part

Certification of Bounds of Non-linear Functions: the Templates Method

The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of inequalities. We introduce an approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis). This algorithm consists in bounding some of the constituents of the function by suprema of quadratic forms with a well chosen curvature. This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation. We illustrate the efficiency of our framework with various examples from the literature and discuss the interfacing with Coq.Comment: 16 pages, 3 figures, 2 table

Finding polynomial loop invariants for probabilistic programs

Quantitative loop invariants are an essential element in the verification of probabilistic programs. Recently, multivariate Lagrange interpolation has been applied to synthesizing polynomial invariants. In this paper, we propose an alternative approach. First, we fix a polynomial template as a candidate of a loop invariant. Using Stengle's Positivstellensatz and a transformation to a sum-of-squares problem, we find sufficient conditions on the coefficients. Then, we solve a semidefinite programming feasibility problem to synthesize the loop invariants. If the semidefinite program is unfeasible, we backtrack after increasing the degree of the template. Our approach is semi-complete in the sense that it will always lead us to a feasible solution if one exists and numerical errors are small. Experimental results show the efficiency of our approach.Comment: accompanies an ATVA 2017 submissio

Fast construction of irreducible polynomials over finite fields

International audienceWe present a randomized algorithm that on input a finite field $K$ with $q$ elements and a positive integer $d$ outputs a degree $d$ irreducible polynomial in $K[x]$. The running time is $d^{1+o(1)} \times (\log q)^{5+o(1)}$ elementary operations. The $o(1)$ in $d^{1+o(1)}$ is a function of $d$ that tends to zero when $d$ tends to infinity. And the $o(1)$ in $(\log q)^{5+o(1)}$ is a function of $q$ that tends to zero when $q$ tends to infinity. In particular, the complexity is quasi-linear in the degree $d$

On the Generation of Positivstellensatz Witnesses in Degenerate Cases

One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some linear equality (Positivstellensatz). This produces a witness for the desired property, from which it is reasonably easy to obtain a formal proof of the property suitable for a proof assistant such as Coq. The problem of finding a witness reduces to a feasibility problem in semidefinite programming, for which there exist numerical solvers. Unfortunately, this problem is in general not strictly feasible, meaning the solution can be a convex set with empty interior, in which case the numerical optimization method fails. Previously published methods thus assumed strict feasibility; we propose a workaround for this difficulty. We implemented our method and illustrate its use with examples, including extractions of proofs to Coq.Comment: To appear in ITP 201