In 1957, Steinhaus conjectured that a chain of regular tetrahedra, meeting face-to-face and forming a closed loop, does not exist; this was later proven by Świerczkowski. We show that modifying the statement by requiring the tetrahedra of a chain to be isosceles results in the first examples of closed chains with all tetrahedra being congruent. As a result, we provide a census of toroidal polyhedra arising from closed chains consisting of up to 20 isosceles tetrahedra. Moreover, we establish the existence of an infinite family of toroidal polyhedra emerging from chains of isosceles tetrahedra. Additionally, we generalise the notion of chains of isosceles tetrahedra and therefore introduce clusters of isosceles tetrahedra along with a combinatorial tool called tetrahedral symbol to describe them. Finally, we exploit our methods to construct clusters of isosceles tetrahedra that yield polyhedra of higher genera.

In 1957, Steinhaus proved that a chain of regular tetrahedra, meeting face-to-face and forming a closed loop does not exist. Over the years, various modifications of this statement have been considered and analysed. Weakening the statement by only requiring the tetrahedra of a chain to be wild, i.e. having all faces congruent, results in various examples of such chains. In this paper, we elaborate on the construction of these chains of wild tetrahedra. We therefore introduce the notions of chains and clusters of wild tetrahedra and relate these structures to simplicial surfaces. We establish that clusters and chains of wild tetrahedra can be described by polyhedra in Euclidean 3-space. As a result, we present methods to construct toroidal polyhedra arising from chains and provide a census of such toroidal polyhedra consisting of up to 20 wild tetrahedra. Here, we classify toroidal polyhedra with respect to self-intersections and reflection symmetries. We further prove the existence of an infinite family of toroidal polyhedra emerging from chains of wild tetrahedra and present clusters of wild tetrahedra that yield polyhedra of higher genera.

An algebraic approach to the design of resource-efficient carbon-reinforced concrete structures is presented. Interdisciplinary research in the fields of mathematics and algebra on the one hand and civil engineering and concrete structures on the other can lead to fruitful interactions and can contribute to the development of resource-efficient and sustainable concrete structures. Textile-reinforced concrete (TRC) using non-crimp fabric carbon reinforcement enables very thin and lightweight constructions and thus requires new construction strategies and new manufacturing methods. Algebraic methods applied to topological interlocking contribute to modular, reusable, and hence resource-efficient TRC structures. A modular approach to construct new interlocking blocks by combining different Platonic and Archimedean solids is presented. In particular, the design of blocks that can be decomposed into various n-prisms is the focus of this paper. It is demonstrated that the resulting blocks are highly versatile and offer numerous possibilities for the creation of interlocking assemblies, and a rigorous proof of the interlocking property is outlined.

Carbonbeton bietet gegenüber der klassischen Stahlbetonbauweise neue Konstruktionsansätze. Um diese auszuschöpfen, werden Entwurfsstrategien entwickelt, welche auf eine materialminimierte Bauweise und auf maschinengestützte Fertigungsmethoden abzielen. Die Inspiration für innovative Strukturen, deren Bewertung sowie der Gewinn eines tiefergehenden Verständnisses für Material-, Bruch- und Verbundverhalten des Werkstoffs können durch Simulationen unterstützt werden. Dieser Beitrag bietet einen Überblick über verschiedene numerische Methoden, die den Prozess von der Ideenfindung, über den Übertrag auf den Werkstoff Carbonbeton und dessen Untersuchung und Bewertung unterstützen sollen. Das umfasst zum einen Methoden, die Inspiration für Geometrie- und Bewehrungsentwürfe aus der Mathematik und der Botanik ableiten sowie Modellreduktionsstrategien, um unterschiedlichste Geometrien effizient zusammenzusetzen und numerisch zu testen. Zum anderen werden verschiedene Mehrskalenmethoden und Material- und Schädigungsmodelle präsentiert, die dazu dienen, das Materialverhalten zuverlässig voraussagen zu können. Weiterhin wird eine Methode zur automatischen Rissdetektion vorgestellt.

Let G be a classical group of dimension d and let a = (a(1), ..., a(d)) be differential indeterminates over a differential field F of characteristic zero with algebraically closed field of constants C. Further let A(a) be a generic element in the Lie algebra g(F < a >) of G obtained from parametrizing a basis of g with the indeterminates a. It is known (cf. [5]) that the differential Galois group of y' = A(a)y over F < a > is G(C). In this paper we construct a differential field extension r of F < a > such that the field of constants of L is C, the differential Galois group of y' = A(a)y over L, is still the full group G(C) and A(a) is gauge equivalent over L to a matrix in normal form which we introduced in [13]. We also consider specializations of the coefficients of A(a).

Computer algebra can answer various questions about partial differential equations using symbolic algorithms. However, the inclusion of data into equations is rare in computer algebra. Therefore, recently, computer algebra models have been combined with Gaussian processes, a regression model in machine learning, to describe the behavior of certain differential equations under data. While it was possible to describe polynomial boundary conditions in this context, we extend these models to analytic boundary conditions. Additionally, we describe the necessary algorithms for Gröbner and Janet bases of Weyl algebras with certain analytic coefficients. Using these algorithms, we provide examples of divergence-free flow in domains bounded by analytic functions and adapted to observations.

There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of ordinary or partial differential equations. Furthermore, we show how a combination of this geometric theory with (differential) algebraic tools allows us to make parts of the theory algorithmic. Our three main results are firstly a proof that even in the case of partial differential equations regular points are generic. Secondly, we present an algorithm for the effective detection of all singularities at a given order or, more precisely, for the determination of a regularity decomposition. Finally, we give a rigorous definition of a regular differential equation, a notoriously difficult notion ubiquitous in the geometric theory of differential equations, and show that our algorithm extracts from each prime component a regular differential equation. Our main tools are on the one hand the algebraic resp. differential Thomas decomposition and on the other hand the Vessiot theory of differential equations.

The GVW algorithm computes simultaneously Gröbner bases of a given ideal and of the syzygy module of the given generating set. In this work, we discuss an extension of it to involutive bases. Pommaret bases play here a special role in several respects. We distinguish between a fully involutive GVW algorithm which determines involutive bases for both the given ideal and the syzygy module and a semi-involutive version which computes for the syzygy module only an ordinary Gröbner basis. A prototype implementation of the developed algorithms in Maple is described.

Retaining the combinatorial Euclidean structure of a regular icosahedron, namely the 20 equiangular (planar) triangles, the 30 edges of length 1, and the 12 different vertices together with the incidence structure, we investigate variations of the regular icosahedron admitting self-intersections of faces. We determine all rigid equivalence classes of these icosahedra with non-trivial automorphism group and find one curve of flexible icosahedra. Visualisations and explicit data for this paper are available under http://algebra.data.rwth-aachen.de/Icosahedra/visualplusdata.html.

This paper applies the Thomas decomposition technique to nonlinear control systems, in particular to the study of the dependence of the system behavior on parameters. Thomas’ algorithm is a symbolic method which splits a given system of nonlinear partial differential equations into a finite family of so-called simple systems which are formally integrable and define a partition of the solution set of the original differential system. Different simple systems of a Thomas decomposition describe different structural behavior of the control system in general. The paper gives an introduction to the Thomas decomposition method and shows how notions such as invertibility, observability and flat outputs can be studied. A Maple implementation of Thomas’ algorithm is used to illustrate the techniques on explicit examples.

For a wide class of polynomially nonlinear systems of partial differential equations we suggest an algorithmic approach that combines differential and difference algebra to analyze s(trong)-consistency of finite difference approximations. Our approach is applicable to regular solution grids. For the grids of this type we give a new definition of s-consistency for finite difference approximations which generalizes our definition given earlier for Cartesian grids. The algorithmic verification of s-consistency presented in the paper is based on the use of both differential and difference Thomas decomposition. First, we apply the differential decomposition to the input system, resulting in a partition of its solution space. Then, to the output subsystem that contains a solution of interest we apply a difference analogue of the differential Thomas decomposition which allows to check the s-consistency. For linear and some quasi-linear differential systems one can also apply difference bases for the s-consistency analysis. We illustrate our methods and algorithms by a number of examples, which include Navier-Stokes equations for viscous incompressible flow.

For a wide class of polynomially nonlinear systems of partial differential equations we suggest an algorithmic approach to the s(trong)-consistency analysis of their finite difference approximations on Cartesian grids. First we apply the differential Thomas decomposition to the input system, resulting in a partition of the solution set. We consider the output simple subsystem that contains a solution of interest. Then, for this subsystem, we suggest an algorithm for verification of s-consistency for its finite difference approximation. For this purpose we develop a difference analogue of the differential Thomas decomposition, both of which jointly allow to verify the s-consistency of the approximation. As an application of our approach, we show how to produce s-consistent difference approximations to the incompressible Navier-Stokes equations including the pressure Poisson equation.

We present the Maple package TDDS (Thomas Decomposition of Differential Systems). Given a polynomially nonlinear differential system, which in addition to equations may contain inequations, this package computes a decomposition of it into a finite set of differentially triangular and algebraically simple subsystems whose subsets of equations are involutive. Usually the decomposed system is substantially easier to investigate and solve both analytically and numerically. The distinctive property of a Thomas decomposition is disjointness of the solution sets of the output subsystems. Thereby, a solution of a well-posed initial problem belongs to one and only one output subsystem. The Thomas decomposition is fully algorithmic. It allows to perform important elements of algebraic analysis of an input differential system such as: verifying consistency, i.e., the existence of solutions; detecting the arbitrariness in the general analytic solution; given an additional equation, checking whether this equation is satisfied by all common solutions of the input system; eliminating a part of dependent variables from the system if such elimination is possible; revealing hidden constraints on dependent variables, etc. Examples illustrating the use of the package are given.

In work of C. Méray [Mér80] and C. Riquier [Riq10] in the second half of the 19th century a generalization of the Cauchy-Kovalevskaya Theorem was obtained.Riquier's Existence Theorem asserts the existence of analytic solutions to systems of PDEs of a certain class (cf.also [Tho28,Tho34], [Rit34, Chap.IX], [Rit50, Chap.VIII]).The equations are assumed to be solved for certain distinct partial derivatives and their right hand sides are analytic functions of z 1 , . . ., z n and of partial derivatives of u 1 , . . ., u m which are ranked lower than the ones on the respective left hand side with respect to a certain kind of total ordering.Moreover, the system is supposed to incorporate all integrability conditions in some sense discussed below.These notes consist of two sections following the Introduction.Section 2 treats the problems outlined above for systems of linear PDEs, whereas Section 3 is dedicated to the more general case of systems of nonlinear PDEs.The discussion of the linear case leads to the notion of Janet basis.A basic variant of an algorithm computing Janet bases is outlined in Subsection 2.2, which builds on a method for partitioning certain sets of monomials into disjoint cones, as introduced in Subsection 2.1.The concept of Thomas decomposition is central for the nonlinear case.It is introduced for algebraic systems in Subsection 3.1 and is then adapted to differential systems in Subsection 3.2.The final Subsection 3.3 explains how to apply the Thomas decomposition technique for eliminating unknown functions from a system of nonlinear PDEs.It is a non-trivial task to include here all relevant references.Among the most important historical ones we select: C. Méray [Mér80], C.

In this paper we show how to compute algorithmically the full set of algebraically independent constraints for singular mechanical and field-theoretical models with polynomial Lagrangians. If a model under consideration is not singular as a whole but has domains of dynamical (field) variables where its Lagrangian becomes singular, then our approach allows to detect such domains and compute the relevant constraints. In doing so, we assume that the Lagrangian of a model is a differential polynomial and apply the differential Thomas decomposition algorithm to the Euler-Lagrange equations.

AbstractThe Anshel–Anshel–Goldfeld (AAG) key exchange protocol is based upon the multiple conjugacy problem for a finitely-presented group. The hardness in breaking this protocol relies on the supposed difficulty in solving the corresponding equations for the conjugating element in the group. Two such protocols based on polycyclic groups as a platform were recently proposed and were shown to be resistant to length-based attack. In this article we propose a parallel evolutionary approach which runs on multicore high-performance architectures. The approach is shown to be more efficient than previous attempts to break these protocols, and also more successful. Comprehensive data of experiments run with a GAP implementation are provided and compared to the results of earlier length-based attacks. These demonstrate that the proposed platform is not as secure as first thought and also show that existing measures of cryptographic complexity are not optimal. A more accurate alternative measure is suggested. Finally, a linear algebra attack for one of the protocols is introduced.

Similarly to the correspondence between radical ideals of a polynomial ring and varieties in algebraic geometry, a correspondence between radical differential ideals and their analytic solution sets has been established in differential algebra. This tutorial discusses aspects of this correspondence involving symbolic computation. In particular, an introduction to the Thomas decomposition method is given. It splits a system of polynomially nonlinear partial differential equations into finitely many so-called simple differential systems whose solution sets form a partition of the original solution set. The power series solutions of each simple system can be determined in a straightforward way. Conversely, certain sets of analytic functions admit an implicit description in terms of partial differential equations and inequations. Strategies for solving related differential elimination problems and applications to symbolic solving of differential equations are presented. A Maple implementation of the Thomas decomposition method is freely available.

This paper addresses systems of linear functional equations from an algebraic point of view. We give an introduction to and an overview of recent work by a small group of people including the author of this article on effective methods which determine structural properties of such systems. We focus on parametrizability of the behavior, i.e., the set of solutions in an appropriate signal space, which is equivalent to controllability in many control-theoretic situations. Flatness of the linear system corresponds to the existence of an injective parametrization. Using an algebraic analysis approach, we associate with a linear system a module over a ring of operators. For systems of linear partial differential equations we choose a ring of differential operators, for multidimensional discrete linear systems a ring of shift operators, for linear differential time-delay systems a combination of those, etc. Rings of these kinds are Ore algebras, which admit Janet basis or Gröbner basis computations. Module theory and homological algebra can then be applied effectively to study a linear system via its system module, the interpretation depending on the duality between equations and solutions. In particular, the problem of computing bases of finitely generated free modules (i.e., of computing flat outputs for linear systems) is addressed for different kinds of algebras of operators, e.g., the Weyl algebras. Some work on computer algebra packages, which have been developed in this context, is summarized.