Analytic classification of plane branches up to multiplicity 4

We perform the analytic classification of plane branches of multiplicity less than or equal to four. This is achieved by computing a Standard Basis for the modules of Kähler differentials of such branches by means of the algorithm developed in [Hefez, A., Hernandes, M.E., 2007a. Standard bases for local rings of branches and their module of differentials. J. Symbolic Comput. 42, 178–191] and then applying the classification method for plane branches given in [Hefez, A., Hernandes, M.E., 2007b. The analytic classification of plane branches. arXiv:0707.4502].     

A double look at duality

The two forms of duality that are encountered most frequently by the systems theorist-linear duality and convex duality-are examined. Linear duality has a strong algebraic characterization which extends to other structures such as groups and modules. Convex duality, on the other hand, capitalizes so strongly on the vector space structure that the resulting powerful theory (which is typically interpreted geometrically) loses the algebraic flavor of its roots. An algebraic characterization of convex duality is presented that generalizes the standard algebraic characterization of linear duality. This provides a link between the two forms of duality most important for the systems theorist. The algebraic and geometric interpretations together give a double view of duality as used in systems theory     

An inner product space on irreducible and synchronizable probabilistic finite state automata

Probabilistic finite state automata (PFSA) have found their applications in diverse systems. This paper presents the construction of an inner-product space structure on a class of PFSA over the real field via an algebraic approach. The vector space is constructed in a stationary setting, which eliminates the need for an initial state in the specification of PFSA. This algebraic model formulation avoids any reference to the related notion of probability measures induced by a PFSA. A formal language-theoretic and symbolic modeling approach is adopted. Specifically, semantic models are constructed in the symbolic domain in an algebraic setting. Applicability of the theoretical formulation has been demonstrated on experimental data for robot motion recognition in a laboratory environment.     

Integration of Dynamic Equation of Spring-Mass-Damper Systems via Chebyshev Polynomials

A numerical scheme based on Chebyshev polynomials for the determination of the response of spring-mass-damper systems is presented. The state vector of the differential equation of the spring-mass-damper system is expanded in terms of Chebyshev polynomials. This expansion reduces the original differential equations to a set of linear algebraic equations where the unknowns are the coefficient of Chebyshev polynomials. A formal procedure to generate the coefficient matrix and the right-hand side vector of this system of algebraic equations is discussed. The numerical efficiency of the proposed method is compared with that of Runge-Kutta method. It is shown that this scheme is accurate and is computationally efficient.     

基于分数阶微分的流线增强方法

为增强流线间强度对比,提高矢量场可视化效果,提出了一种基于分数阶微分滤波的流线增强方法.该方法通过构造分数阶微分算子,对卷积纹理在垂直矢量方向上进行分数阶微分滤波,增强了矢量线间的对比,改善了图像质量.实验结果表明,该方法能有效地增强矢量场流线可视化效果,为分析流场特征提供帮助.     

Solution modules and system equivalence

In this paper we consider a system as a relation between input, internal variables and output. This relation is given by the solution space of the system's equations. For time invariant linear systems in differential operator representation the solution space carries a K[s]-module structure defined by the ordinary differential operator. This algebraic structure is exploited systematically to develop a self-contained theory of strict system equivalence in time domain.

The module of free motions is considered as space of initial conditions. An algebraic characterization of systems having the same solution space is presented. System homomorphisms are defined as special K[s] homomorphisms between the solution modules. Two systems are called system-equivalent, if there exists a system-isomorphism between their solution spaces. It turns out that, this concept coincides with Rosenbrock's concept of strict. system equivalence. It. is shown that further concepts and results of linear system theory (construction of a state-space model in time domain, controllability and observability criteria, uniqueness theorem of linear realization theory) can be derived within this framework.     

Localization and parametrization of linear multidimensional control systems

We study the link existing between the parametrization of differential operators by potential-like arbitrary functions and the localization of differential modules, while applying these results to the parametrization of linear multidimensional control systems. We show that the localization of differential modules is a natural way to generalize some well-known results on transfer matrix, classically obtained by using Laplace transform, to time-varying ordinary differential control systems and to partial differential control systems with variable coefficients. In particular, we show that the parametrizations obtained by localization are simpler than those obtained by formal duality but are worse in the sense of Palamodov–Kashiwara's classification of differential modules.     

Sign analysis technique for predicting system behavior

Sign analysis is a technique for deriving the behavior of a system when it is subjected to a disturbance. the technique is for systems whose mathematical models are of the form of algebraic equations. the technique is based on generating sign combinations of total differentials of system parameters in table form from closed form algebraic functions which model the system. Sign combinations are then analyzed to predict system behavior with respect to a disturbance from the equilibrium state. the technique may be applied to total differentials of gains as well. © 1992 John Wiley & Sons, Inc.     

Elementary differentials,their graphs and codes

When designing high-order one-step numerical methods like Runge-Kutta, Rosenbrock, or ABC-schemes for solving ordinary differential equations, one has to take into account many tens and hundreds of elementary differentials. Graphical representation of the latter in use nowadays does not solve the problem of the computerization of the enormous amount of manual work. We have proposed a simple and intuitive way of digital coding of elementary differentials or their graphs. Algorithms for generation, analysis, and synthesis of such codes were developed and implemented in a computer program, which computes tables of codes for elementary differentials of any order together with their multiplicities and gamma-factors.     

逻辑系统的代数状态空间方法的基础、现状及其应用

逻辑系统指自变量只取有限个值的动态系统.包括2值的经典逻辑(或布尔逻辑)、k值逻辑、(一般)有限值逻辑.近年来,利用矩阵半张量积发展起来的逻辑动态系统的代数状态空间方法得到长足的进展和普遍的重视.同时,它被广泛应用于许多工程问题或理论研究中.它类似于Rn上由微分或差分方程描述的动态系统的Kalman状态空间方法,为逻辑系统的分析与控制设计提供了一个便捷的平台.本文首先对该方法作一简要介绍,然后,对该新兴学科分支的现状作一评述.最后,详细介绍该方法目前的应用以及其更广泛的应用前景.     

New results in realization theory for linear time-varying analytic systems

The realization of linear time-varying systems specified by an analytic weighting pattern is approached in a novel manner using an algebraic framework defined over the ring of analytic functions. Realizations are given by a state representation consisting of a first-order vector differential equation and an output equation, both with analytic coefficients. Various new criteria for realizability are derived, including conditions given in terms of the finiteness of modules over the ring of analytic functions generated by the elementary rows or columns of a (generalized) Hankel matrix. These results are related to local criteria for realizability specified in terms of the rank of matrix functions, as developed in the work of Silverman and Meadows [5], [8], [9] and Kalman [7]. It is shown that the construction of minimal realizations reduces to the problem of computing a basis for a finite free module defined over the ring of analytic functions. A minimal realization algorithm is then derived using a constructive procedure for computing bases for finite free modules over a Bezout domain. The Silverman-Meadows realization algorithm [5] is a special case of the procedure given here. In the last part of the paper, the realization algorithm is applied to the problem of system reduction.     

Vector space formulation of probabilistic finite state automata

This paper develops a vector space model of a class of probabilistic finite state automata (PFSA) that are constructed from finite-length symbol sequences. The vector space is constructed over the real field, where the algebraic operations of vector addition and the associated scalar multiplication operations are defined on a probability measure space, and implications of these algebraic operations are interpreted. The zero element of this vector space is semantically equivalent to a PFSA, referred to as symbolic white noise. A norm is introduced on the vector space of PFSA, which provides a measure of the information content. An application example is presented in the framework of pattern recognition for identification of robot motion in a laboratory environment.     

The standard H∞ problem in systems with a singleinput delay

The standard H_∞ problem is solved for LTI systems with a single, pure input lag. The solution is based on state-space analysis, mixing a finite-dimensional and an abstract evolution model. Utilizing the relatively simple structure of these distributed systems, the associated operator Riccati equations are reduced to a combination of two algebraic Riccati equations and one differential Riccati equation over the delay interval. The results easily extend to finite time and time-varying problems where the algebraic Riccati equations are substituted by differential Riccati equations over the process time duration     

Macaulay inverse systems revisited

Since its original publication in 1916 under the title The Algebraic Theory of Modular Systems, the book (Macaulay, 1916) by Macaulay has attracted a lot of scientists with a view towards pure mathematics (Eisenbud, 1996) or applications to control theory (Oberst, 1990) through the last chapter dealing with the so-called inverse system. The basic intuitive idea is the parallel existing between ideals in polynomial rings and systems of partial differential (PD) equations in one unknown with constant coefficients.A first purpose of this paper is thus to extend these results to arbitrary systems of PD equations by exhibiting a link with the formal theory of systems of PD equations ( [Pommaret, 1994], [Seiler, 2009] and [Spencer, 1965]) where concepts such as formal integrability and involution are superseding the H-bases of Macaulay.The second idea is to transfer the properties of ideals to their residue modules, in particular to extend to differential modules the unmixedness assumption of Macaulay. For this we use extensively the results of modern algebraic analysis ( [Bjork, 1993], [Kashiwara, 1995], [Palamodov, 1970], [Pommaret, 2001] and [Pommaret, 2005]), revisiting in particular the concept of purity by means of localization techniques. Accordingly, this paper can also be considered as a refinement and natural continuation of Pommaret (2007).Finally, following again Macaulay in the differential setting, the cornerstone and main novelty of the paper is to replace the socle of a module by the top of the corresponding dual system in order to be able to look for generators by using known arguments of algebraic geometry such as Nakayama’s lemma (Kunz, 1985).Many explicit examples are provided in order to illustrate the main constructive results that provide new hints for applying computer algebra to algebraic analysis (Quadrat, 2009). This paper is an extended version of a lecture given at the “Applications of Computer Algebra” meeting ACA 2008, held at RISC-Linz, Austria, on July 27-30.     

Algebraic Modeling of an ad Hoc Network for Mobile Computing

We develop an algebraic model of an ad hoc network [9] for mobile computing. The ad hoc network's structure and functionality is based (in our approach) on a pathset algebra. A pathset is the set of all paths in the network having specific source(s) and destination(s). The pathsets' set refers to a multi-digraph that constitutes the physical infrastructure of an ad hoc network. The applications, actions, and/or services (referred to further under the general term services) are organized as multi-agent systems. The formal model of every multi-agent system is based on a modal algebra. An additional function (or special operation) couples the two algebras specifying thus the interface of their interoperability. This operation is analogous to the scalar multiplication of a vector space on a field.     

Improved Half Vector Space Light Transport

In this paper, we present improvements to half vector space light transport (HSLT) [ KHD14 ], which make this approach more practical, robust for difficult input geometry, and faster. Our first contribution is the computation of half vector space ray differentials in a different domain than the original work. This enables a more uniform stratification over the image plane during Markov chain exploration. Furthermore, we introduce a new multi chain perturbation in half vector space, which, if combined appropriately with half vector perturbation, makes the mutation strategy both more robust to geometric configurations with fine displacements and faster due to reduced number of ray casts. We provide and analyze the results of improved HSLT and discuss possible applications of our new half vector ray differentials.     

Internal model based tracking and disturbance rejection for stable well-posed systems

In this paper we solve the tracking and disturbance rejection problem for infinite-dimensional linear systems, with reference and disturbance signals that are finite superpositions of sinusoids. We explore two approaches, both based on the internal model principle. In the first approach, we use a low gain controller, and here our results are a partial extension of results by Hämäläinen and Pohjolainen. In their papers, the plant is required to have an exponentially stable transfer function in the Callier-Desoer algebra, while in this paper we only require the plant to be well-posed and exponentially stable. These conditions are sufficiently unrestrictive to be verifiable for many partial differential equations in more than one space variable. Our second approach concerns the case when the second component of the plant transfer function (from control input to tracking error) is positive. In this case, we identify a very simple stabilizing controller which is again an internal model, but which does not require low gain. We apply our results to two problems involving systems modeled by partial differential equations: the problem of rejecting external noise in a model for structure/acoustics interactions, and a similar problem for two coupled beams.