On a new procedure for finding nonclassical symmetries

A new technique for deriving the determining equations of nonclassical symmetries associated with a partial differential equation system is introduced. The problem is reduced to computing the determining equations of the classical symmetries associated with a related equation with coefficients which depend on the nonclassical symmetry operator. As a consequence, all the symbolic manipulation programs designed for the latter task can also be used to find the determining equations of the nonclassical symmetries, without any adaptation of the program. The algorithm was implemented as the MAPLE routine GENDEFNC and uses the MAPLE package DESOLV (authors Carminati and Vu). As an example, we consider the Huxley partial differential equation.     

[SADE] a Maple package for the symmetry analysis of differential equations

We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, nonclassical, Lie–Bäcklund and potential symmetries, invariant solutions, first-integrals, Nöther theorem for both discrete and continuous systems, solution of ordinary differential equations, order and dimension reductions using Lie symmetries, classification of differential equations, Casimir invariants, and the quasi-polynomial formalism for ODE's (previously implemented by the authors in the package QPSI) for the determination of quasi-polynomial first-integrals, Lie symmetries and invariant surfaces. Examples of use of the package are given.

Program summary

Program title: SADECatalogue identifier: AEHL_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEHL_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 27 704No. of bytes in distributed program, including test data, etc.: 346 954Distribution format: tar.gzProgramming language: MAPLE 13 and MAPLE 14Computer: PCs and workstationsOperating system: UNIX/LINUX systems and WINDOWSClassification: 4.3Nature of problem: Determination of analytical properties of systems of differential equations, including symmetry transformations, analytical solutions and conservation laws.Solution method: The package implements in MAPLE some algorithms (discussed in the text) for the study of systems of differential equations.Restrictions: Depends strongly on the system and on the algorithm required. Typical restrictions are related to the solution of a large over-determined system of linear or non-linear differential equations.Running time: Depends strongly on the order, the complexity of the differential system and the object computed. Ranges from seconds to hours.     

Statistical properties of controlled fluid flows with applications to control of mixing

In this paper, we study statistical properties of fluid flows that are actively controlled. Statistical properties such as Lagrangian and Eulerian time-averages are important flow quantities in fluid flows, particularly during mixing processes. Due to the assumption of incompressibility, the transformations in the state space can be described by a sequence of measure preserving transformations on a measure space. The classical Birkhoff's pointwise ergodic theorem does not necessarily apply in the context of sequences of transformations. We call B-regular a sequence for which this theorem holds. Motivated by mixing control concepts, we define three notions of asymptotic equivalence for sequences of transformations. We show an example in which Birkhoff's pointwise ergodic theorem does not hold even when a ‘strong’ asymptotic equivalence to a B-regular sequence is assumed. Under a ‘very strong’ asymptotic equivalence condition, we prove B-regularity. In the context of optimize-then-stabilize strategy for mixing control, we also prove that very strong asymptotic equivalence to a mixing sequence implies mixing. The mean ergodic theorem and the Poincare’ recurrence theorem are also proven for sequences of transformations under suitable asymptotic equivalence assumptions.     

Computing the regularization of a linear differential–algebraic system

We study the regularization problem for linear differential–algebraic systems. As an improvement of former results we show that any system can be regularized by a combination of state-space and input-space transformations, behavioral equivalence transformations and a reorganization of variables. The additional state feedback which is needed in earlier publications is shown to be superfluous. We provide an algorithmic procedure for the construction of the regularization and discuss computational aspects.     

Transformation groups for weakly non-linear periodic systems†

The method of infinitesimal transformations is applied to weakly non-linear periodic systems described by n first-order coupled differential equations. The transformation groups, which leave the equations invariant, are determined by a perturbation method, Those transformations roduce the systems to non-linear autonomous form. The equivalence of the results of the method, within the first-order approximation, to other averaging and perturbation techniques is illustrated by an example of a second-order system.     

A symbolic algorithm for computing recursion operators of nonlinear partial differential equations

A recursion operator is an integro-differential operator which maps a generalized symmetry of a nonlinear partial differential equation (PDE) to a new symmetry. Therefore, the existence of a recursion operator guarantees that the PDE has infinitely many higher-order symmetries, which is a key feature of complete integrability. Completely integrable nonlinear PDEs have a bi-Hamiltonian structure and a Lax pair; they can also be solved with the inverse scattering transform and admit soliton solutions of any order.

A straightforward method for the symbolic computation of polynomial recursion operators of nonlinear PDEs in (1+1) dimensions is presented. Based on conserved densities and generalized symmetries, a candidate recursion operator is built from a linear combination of scaling invariant terms with undetermined coefficients. The candidate recursion operator is substituted into its defining equation and the resulting linear system for the undetermined coefficients is solved.

The method is algorithmic and is implemented in Mathematica. The resulting symbolic package PDERecursionOperator.m can be used to test the complete integrability of polynomial PDEs that can be written as nonlinear evolution equations. With PDERecursionOperator.m, recursion operators were obtained for several well-known nonlinear PDEs from mathematical physics and soliton theory.     

Basic Results in Automatic Transformations of Shared Memory Parallel Programs into Sequential Programs

We study the problem of developing a set of syntax-driven transformations for automatic translation of shared memory parallel programs into sequential programs. The result is a sequential program that is semantically equivalent to the original program. Consequently, the problem of debugging parallel programs is reduced to the problem of debugging sequential programs. Moreover, the efficiency of parallel programs can be increased by sequentializing code segments that include extra parallelism.The main difficulty in developing such a system is to preserve the fairness property of any actual parallel execution, which states that no process can wait forever unserved. Thus, non termination of the sequential version (namely an infinite loop) is allowed only if there is at least one fair parallel execution that does not halt as well (i.e., a process that executes an infinite loop whose termination is not dependent on any other process).We show that it is sufficient to consider the case of two sequential programs executed in parallel in order to solve the general case. We then describe several types of transformations and check their ability to preserve fairness. The results, with regards to the existence of such a transformation for general parallel programs are not conclusive; however, we do show that restricted cases (which are likely to appear in the reality) can be sequentialized using this set of transformations.     

FracSym: Automated symbolic computation of Lie symmetries of fractional differential equations

In this paper, we present an algorithm for the systematic calculation of Lie point symmetries for fractional order differential equations (FDEs) using the method as described by Buckwar & Luchko (1998) and Gazizov, Kasatkin & Lukashchuk (2007, 2009, 2011). The method has been generalised here to allow for the determination of symmetries for FDEs with n$n$ independent variables and for systems of partial FDEs. The algorithm has been implemented in the new MAPLE package FracSym (Jefferson and Carminati 2013) which uses routines from the MAPLE symmetry packages DESOLVII (Vu, Jefferson and Carminati, 2012) and ASP (Jefferson and Carminati, 2013). We introduce FracSym by investigating the symmetries of a number of FDEs; specific forms of any arbitrary functions, which may extend the symmetry algebras, are also determined. For each of the FDEs discussed, selected invariant solutions are then presented.     

关于非线性系统的对程性、可达性与可控性*

本文讨论对称及广义对称非线性系统的可达性与可控性。文中证明了如果广义对称系统在一点的邻域可控,那么系统在包含该点群作用轨道的一个开集上是可控的。利用广义对称性还给出了控制律的原则算法。对于对称系统,给出了可控性的充要条件。     

Upper Bounds for the Average Probabilities of Difference Characteristics of Block Ciphers with Alternation of Markov Transformations and Generalized Markov Transformations

A new method is proposed to find the average probabilities of differential characteristics of block ciphers, which allows us to use branch number even in case where the cipher is not Markov and has different operations in key adder. The upper bounds are obtained for the average probabilities of differential characteristics of block ciphers with alternation of Markov transformations and generalized Markov transformations.     

A transformation system for concurrent processes

Program transformation techniques have been extensively studied in the framework of functional and logic languages, where they were applied mainly to obtain more efficient and readable programs. All these works are based on the Unfold/Fold program transformation method developed by Burstall and Darlington in the context of their recursive equational language. The use of Unfold/Fold based transformations for concurrent languages is a relevant issue that has not yet received an adequate attention. In this paper we define a transformation methodology for CCS. We give a set of general rules which are a specialization of classical program transformation rules, such as Fold and Unfold. Moreover, we define the general form of other rules, “oriented” to the goal of a transformation strategy, and we give conditions for the correctness of these rules. We prove that a strategy using the general rules and a set of goal oriented rules is sound, i.e. it transforms CCS programs into equivalent ones. We show an example of application of our method. We define a strategy to transform, if possible, a full CCS program into an equivalent program whose semantics is a finite transition system. We show that, by means of our methodology, we are able to a find finite representations for a class of CCS programs which is larger than the ones handled by the other existing methods. Our transformational approach can be seen as unifying in a common framework a set of different techniques of program analysis. A further advantage of our approach is that it is based only on syntactic transformations, thus it does not requires any semantic information. Received: 24 April 1997 / 19 November 1997     

An extension of Wolovich's definition of equivalence of linearsystems

Wolovich's classical definition of equivalence for linear systems (1974) is extended to the generalized study of linear systems. It is shown that the resulting equivalence is an alternative characterization of the notion of full system equivalence underlying its fundamental role in the generalized study of linear systems     

Canonical transformation and stabilization of generalized Hamiltonian systems

This paper introduces generalized canonical transformations for generalized Hamiltonian systems which convert a generalized Hamiltonian system into another one, and preserve the generalized Hamiltonian structure of the original. As in classical mechanics, it is expected that canonical transformations will provide new insights and fundamental tools for both analysis and synthesis of those systems. Firstly, the class of generalized canonical transformations and some of their properties are indicated. Secondly, it is shown how to stabilize the generalized Hamiltonian systems using canonical transformations. In addition, some examples illustrate how such transformations are utilized for control systems design.     

Additional transformations for multiple‐level escape statements

Earlier work suggests that program transformations can simplify program verification. A given program containing complex language features is transformed into a semantically equivalent program containing only simpler language features. The transformed program is proven using a set of proof rules for only the simpler features. That approach was illustrated by transforming a given program that may contain multiple‐level escape statements within nested loops into an equivalent program that contains no escape statements. This paper gives additional transformations, which map a given program that may contain multiple‐level escape statements to a semantically equivalent program (TP) that contains only single‐level escape statements. The proof of TP uses proof rules for single‐level escape statements, or the earlier transformations further map TP to a program with no escape statements, whose proof uses proof rules for loops without escape statements. This paper also discusses escape statements where the number of levels is determined at run‐time. Copyright © 2002 John Wiley & Sons, Ltd.     

POINCARÉ CODE: A package of open-source implements for normalization and computer algebra reduction near equilibria of coupled ordinary differential equations

The Poincaré code is a Maple project package that aims to gather significant computer algebra normal form (and subsequent reduction) methods for handling nonlinear ordinary differential equations. As a first version, a set of fourteen easy-to-use Maple  commands is introduced for symbolic creation of (improved variants of Poincaré’s) normal forms as well as their associated normalizing transformations. The software is the implementation by the authors of carefully studied and followed up selected normal form procedures from the literature, including some authors’ contributions to the subject. As can be seen, joint-normal-form programs involving Lie-point symmetries are of special interest and are published in CPC Program Library for the first time, Hamiltonian variants being also very useful as they lead to encouraging results when applied, for example, to models from computational physics like Hénon–Heiles.     

A DSL for modeling application-specific functionalities of business applications

Models have been widely used in the information system development process. Models are not just means for system analysis and documentation. They may be also transformed into system implementation, primarily program code. Generated program code of screen forms and transaction programs mainly implements generic functionalities that can be expressed by simple retrieval, insertion, update, or deletion operations over database records. Besides the program code of generic functionalities, each application usually includes program code for specific business logic that represents application-specific functionalities, which may include complex calculations, as well as a series of database operations. There is a lack of domain-specific and tool-supported techniques for specification of such application-specific functionalities at the level of platform-independent models (PIMs). In this paper, we propose an approach and a domain-specific language (DSL), named IIS^⁎CFuncLang, aimed at enabling a complete specification of application-specific functionalities at the PIM level. We have developed algorithms for transformation of IIS^⁎CFuncLang specifications into executable program code, such as PL/SQL program code. In order to support specification of application-specific functionalities using IIS^⁎CFuncLang, we have also developed appropriate tree-based and textual editors. The language, editors, and the transformations are embedded into a Model-Driven Software Development tool, named Integrated Information Systems CASE (IIS^⁎Case). IIS^⁎Case supports platform-independent design and automated prototyping of information systems, which allows us to verify and test our approach in practice.     

Mechanical Control Systems and Kinematic Systems

The aim of this technical note is to analyze the equivalence between the second-order equations describing the dynamics of mechanical systems, and the associated kinematic system when dealing with nonholonomic systems with controls. If the system is fully actuated, both systems are equivalent. However, if it is underactuated an extra condition must be imposed to ensure that a weak equivalence holds. Furthermore, the notion of decoupling vector fields is generalized to a vector field distribution. This point of view may be used to obtain better solutions when a cost function is introduced to the controllability problem. The results are applied to some particular examples and to mechanical systems with symmetries.      

2DT-FP: A parallel functional programming language on two-dimensional data

We propose a new paradigm for programming multiprocessor systems, 2DT (two-dimensional transformations). 2DT programs are composed of local computations on linear data (columns) and global transformations on 2-dimensional combinations of columns (2D-arrays). Local computations can be expressed in a functional or imperative base language; a typed variant of Backus' FP is chosen for 2DT-FP. The level of abstraction of 2DT makes it suitable for programming a relevant set of algorithms, in general any algorithms, whose data can be easily mapped to 2-dimensional representations. A set of examples tries to prove this claim. An interleaving semantics for 2DT-FP is given, exposing the potential for parallel execution of 2DT-FP programs. The claim is proved that any sequential and thus any parallel execution will deliver the same result. The implementation realized on the Intel Hypercube is described. Supported by the Deutsche Forschungsgemeinschaft, DFG