Thomas decompositions of parametric nonlinear control systems

This paper presents an algorithmic method to study structural properties of nonlinear control systems in dependence of parameters. The result consists of a description of parameter configurations which cause different control-theoretic behaviour of the system (in terms of observability, flatness, etc.). The constructive symbolic method is based on the differential Thomas decomposition into disjoint simple systems, in particular its elimination properties

Algorithmic Thomas Decomposition of Algebraic and Differential Systems

In this paper, we consider systems of algebraic and non-linear partial differential equations and inequations. We decompose these systems into so-called simple subsystems and thereby partition the set of solutions. For algebraic systems, simplicity means triangularity, square-freeness and non-vanishing initials. Differential simplicity extends algebraic simplicity with involutivity. We build upon the constructive ideas of J. M. Thomas and develop them into a new algorithm for disjoint decomposition. The given paper is a revised version of a previous paper and includes the proofs of correctness and termination of our decomposition algorithm. In addition, we illustrate the algorithm with further instructive examples and describe its Maple implementation together with an experimental comparison to some other triangular decomposition algorithms.Comment: arXiv admin note: substantial text overlap with arXiv:1008.376

Recent progress in an algebraic analysis approach to linear systems

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

Refining Argon Trap Trace Analysis - Crucial Features and Characterization Tools for Reliable Routine Measurements

Argon Trap Trace Analysis resolves 39Ar concentrations at the 10−16 level. Perfect selectivity is achieved by exploiting the isotopic shift and a high number of photon scattering processes for detection. However, good measurement precision requires high count rates of39Ar atoms. Therefore, this work presents crucial features and characterization tools for reliable routine measurements. A new instrument for absolute quantization of the metastable 40Ar flux is developed and put into operation. The magneto-optical trap loading rate is evaluated and established as a standard characterization tool. It is part of the new collimator alignment procedure that resulted in a factor of 1.5 higher 39Ar count rate. Other improved alignment procedures are presented. In addition, a new laboratory control and monitoring system for stable day-to-day measurements is developed and installed. The second part focuses on the study of sources of metastable atoms. Two radio frequency (RF) antennas and two cooling devices are presented and their main features are analyzed. A new LN2-dewar for cooling reduces the amount of liquid nitrogen consumed by a factor of 3 to 5. In combination with the established RF antenna, increasing the source pressure to 2.0 × 10−5 mbar leads to 50% higher metastable flux, while the atoms slow down longitudinally

A parallel evolutionary approach to solving systems of equations in polycyclic groups

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.</jats:p

Lagrangian constraints and differential Thomas decomposition

publisher: Elsevier articletitle: Lagrangian constraints and differential Thomas decomposition journaltitle: Advances in Applied Mathematics articlelink: http://dx.doi.org/10.1016/j.aam.2015.09.005 content_type: article copyright: Crown copyright © 2015 Published by Elsevier Inc. All rights reserved

Conley: Computing connection matrices in Maple

In this work we announce the Maple package conley to compute connection and C-connection matrices. conley is based on our abstract homological algebra package homalg. We emphasize that the notion of braids is irrelevant for the definition and for the computation of such matrices. We introduce the notion of triangles that suffices to state the definition of (C)-connection matrices. The notion of octahedra, which is equivalent to that of braids is also introduced.Comment: conley is based on the package homalg: math.AC/0701146, corrected the false "counter example