Real tropicalization and negative faces of the Newton polytope

In this work, we explore the relation between the tropicalization of a real semi-algebraic set $S = \{ f_1 < 0, \dots , f_k < 0\}$ defined in the positive orthant and the combinatorial properties of the defining polynomials $f_1, \dots, f_k$. We describe a cone that depends only on the face structure of the Newton polytopes of $f_1, \dots ,f_k$ and the signs attained by these polynomials. This cone provides an inner approximation of the real tropicalization, and it coincides with the real tropicalization if $S = \{ f < 0\}$ and the polynomial $f$ has generic coefficients. Furthermore, we show that for a maximally sparse polynomial $f$ the real tropicalization of $S = \{ f < 0\}$ is determined by the outer normal cones of the Newton polytope of $f$ and the signs of its coefficients. Our arguments are valid also for signomials, that is, polynomials with real exponents defined in the positive orthant

Discrete calculus with cubic cells on discrete manifolds

This work is thought as an operative guide to discrete exterior calculus (DEC), but at the same time with a rigorous exposition. We present a version of (DEC) on cubic cell, defining it for discrete manifolds. An example of how it works, it is done on the discrete torus, where usual Gauss and Stokes theorems are recovered

Voronoi Cells in Metric Algebraic Geometry of Plane Curves

Voronoi cells of varieties encode many features of their metric geometry. We prove that each Voronoi or Delaunay cell of a plane curve appears as the limit of a sequence of cells obtained from point samples of the curve. We use this result to study metric features of plane curves, including the medial axis, curvature, evolute, bottlenecks, and reach. In each case, we provide algebraic equations defining the object and, where possible, give formulas for the degrees of these algebraic varieties. We show how to identify the desired metric feature from Voronoi or Delaunay cells, and therefore how to approximate it by a finite point sample from the variety.Comment: 23 pages, 14 figure

Identifying parameter regions for multistationarity

Mathematical modelling has become an established tool for studying the dynamics of biological systems. Current applications range from building models that reproduce quantitative data to identifying systems with predefined qualitative features, such as switching behaviour, bistability or oscillations. Mathematically, the latter question amounts to identifying parameter values associated with a given qualitative feature. We introduce a procedure to partition the parameter space of a parameterized system of ordinary differential equations into regions for which the system has a unique or multiple equilibria. The procedure is based on the computation of the Brouwer degree, and it creates a multivariate polynomial with parameter depending coefficients. The signs of the coefficients determine parameter regions with and without multistationarity. A particular strength of the procedure is the avoidance of numerical analysis and parameter sampling. The procedure consists of a number of steps. Each of these steps might be addressed algorithmically using various computer programs and available software, or manually. We demonstrate our procedure on several models of gene transcription and cell signalling, and show that in many cases we obtain a complete partitioning of the parameter space with respect to multistationarity.Comment: In this version the paper has been substantially rewritten and reorganised. Theorem 1 has been reformulated and Corollary 1 adde

Multirepresentations and multiconstraints approach to the numerical synthesis of serial kinematic structures of manipulators

This paper presents a set of algorithms for the synthesis of kinematic structures of serial manipulators using multiple constraint formulation and provides a performance comparison of different kinematic representations, the Denavit-Hartenberg notation, the Product of Exponentials (screws), and Roll-Pitch-Yaw angles with translation parameters. Synthesis is performed for five given tasks, and both revolute and prismatic joints can be synthesized. Two different non-linear programming optimization algorithms were used to support the findings. The results are compared and discussed. Data show that the choice of the constraint design method has a significant impact on the success rate of optimization convergence. The choice of representation has a lower impact on convergence, but there are differences in the optimization time and the length of the designed manipulators. Furthermore, the best results are obtained when multiple methodologies are used in combination. An arbitrary manipulator was designed and assembled based on a trajectory in the collision environment to demonstrate the advantages of the proposed methodology. The input/output data and synthesis methodology algorithms are provided through an open repository.Web of Science10689516893

Extended finite element methods for approximation of singularities

Tato doktorská práce je zaměřena na řešení problému proudění podzemní vody v porézním prostředí, které je ovlivněno přítomností vrtů či studní. Model proudění je sestaven na základě konceptu redukce dimenzí, který je hojně využíván při modelování rozpukaného porézního prostředí, především granitů. Vrty jsou modelovány jako 1d objekty, které protínají blok horniny. Propojení těchto domén v redukovaném modelu způsobuje singularity v řešení v okolí vrtů. Vrty i porézní médium jsou síťovány nezávisle na sobě což vede k výpočetním sítím kombinujícím elementy různých dimenzí.Jádrem doktorské práce je pak vývoj specializované metody konečných prvků pro výše popsaný model. Pro umožnění propojení sítí různých dimenzí a pro zpřesnění aproximace singularit v okolí vrtů je použita rozšířená metoda konečných prvků (XFEM), v rámci níž jsou navrženy nové typy obohacení konečně-prvkové aproximace. Metoda XFEM je nejprve aplikována v modelu pro tlak, dále je navrženo obohacení pro rychlost a metoda je použita ve smíšeném modelu, jehož řešením jsou rychlost i tlak.Doktorská práce se dále detailně věnuje numerickým aspektům v metodě XFEM, a to především adaptivním kvadraturám, volbě velikosti obohacené oblasti nebo podmíněnosti výsledného lineárního systému. Vlastnosti navržené XFEM metody a optimální konvergence jsou ověřeny na sérii numerických experimentů. Praktickým výstupem doktorské práce je implementace metody XFEM jako součásti open-source softwaru Flow123d.In this doctoral thesis, a model of groundwater flow in porous media intersected with wells (boreholes, channels) is developed. The model is motivated by the reduced dimension approach which is being often used in fractured porous media problems, especially in granite rocks. The wells are modeled as lower dimensional 1d objects and they intersect the surrounding bulk rock domains. The coupling between the wells and the rock then causes a singular behaviour of the solution in the higher dimensional domains in the vicinity of the cross-sections. The domains are discretized separately resulting in an incompatible mesh of combined dimensions.The core contribution of this work is in the developement of a specialized finite element method for such model. Different Extended finite element methods (XFEM) are studied and new enrichments are suggested to better approximate the singularities and to enable the coupling of the wells with the higher dimensional domains. At first the XFEM is applied in a pressure model, later an enrichment for velocity is suggested and the XFEM is used in a mixed model, solving both velocity and pressure.Different numerical aspects of the XFEM is studied in details, including an adaptive quadrature strategy, a proper choice of the enrichment zone or a conditioning of the resulting linear system. The properties of the suggested XFEM are validated on a set of numerical tests and the optimal convergence rate is demonstrated. The XFEM is implemented as a part of the open-source software Flow123d