Coxeter spectral classification of almost TP-critical one-peak posets using symbolic and numeric computations

A survey of the use of a combination of symbolic and numerical calculations is presented. Symbolic calculations primarily refer to the computer processing of procedures from classical algebra, analysis, and calculus. Numerical calculations refer to both numerical mathematics research and scientific computation. This survey is intended to point out a large number of problem areas where a cooperation of symbolic and numerical methods is likely to bear many fruits. These areas include such classical operations as differentiation and integration, such diverse activities as function approximations and qualitative analysis, and such contemporary topics as finite element calculations and computation complexity. It is contended that other less obvious topics such as the fast Fourier transform, linear algebra, nonlinear analysis and error analysis would also benefit from a synergistic approach.

The aim of SNC 2009 is to offer a forum for researchers in symbolic and numeric computation to present their work, interact, exchange ideas and identify important problems in this research area. SNC 2009 continues a successful tradition of previous highly successful workshops in the area of symbolic and numeric computation: SNAP 96, held July 15-17, 1996 in Sophia Antipolis, FranceSNC 2005, held July 19-21, 2005 in Xi'an, ChinaSNC 2007, held July 25-27, 2007 in London Ontario, Canada A warm thank you goes to all those that worked hard to make SNC 2009 happen and in particular the local organizers, the program committee members, the anonymous referees, the invited speakers and the participants. The SNC 2009 Call For Papers solicited submissions in several topics: Hybrid symbolic-numeric algorithmsApproximate polynomial GCD and factorizationSymbolic-numeric methods for solving polynomial systemsResultants and structured matrices for symbolic-numeric computationDifferential equations for symbolic-numeric computationSymbolic-numeric methods for geometric computationSymbolic-numeric algorithms in algebraic geometrySymbolic-numeric algorithms for nonlinear optimizationNumeric computation of characteristic sets and Groebner basesImplementation of symbolic-numeric algorithmsApproximate algebraic algorithmsApplications of symbolic-numeric computation SNC 2009 is sponsored by the University of Tsukuba (http://www.tsukuba.ac.jp/english/). SNC 2009 is in cooperation with ACM SIGSAM (http://www.sigsam.org/) and we wish to thank the Chair of SIGSAM, Dr. Mark Giesbrecht, for his continuous help and support. SNC 2009 is also in cooperation with JSSAC, the Japan Society for Symbolic and Algebraic Computation (http://www.jssac.org/). We hope that the current book of proceedings of SNC 2009 will become a useful resource for researchers in Symbolic-Numeric Computation and other related research areas. We also hope that it will become another testament of the liveliness and vibrancy of this research area and a precursor of the future developments that await us ahead.

Mathematical models of biochemical reaction networks involve nonlinear differential equations with a number of parameters whose values are generally not known a priori. Analysis of such networks includes both analytical and numerical work. The analytical part of such analysis often requires manipulation with massive algebraic expressions that may become challenging due to the inherent nonlinearities and unknown parameters in the model. Computer algebra systems (CASs), which are also capable of performing calculations with symbols and to generate ready‐to‐use computer codes for numerical computations, are excellent tools in such analysis. The present work gives a brief description of how we can benefit from computer algebra to overcome such difficulties and combine symbolic and numeric computation techniques in such analysis with several illustrative biochemical reaction networks. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Algorithms that combine techniques from symbolic and numeric computation have been of increasing importance and interest over the past decade. The necessity to work reliably with imprecise and noisy data, and for speed and accuracy within algebraic and hybrid-numerical problems, has encouraged a new synergy between the numerical and symbolic computing fields. Novel and exciting problems from industrial, mathematical and computational domains are now being explored and solved.

A computational proof of complexity of some restricted counting problems

Maple is a computer algebra system and programming language that can be used for numerical computations, solving equations, manipulating symbolic expressions, plotting, and programming, just to name a few of the basics. APPL is, simply, a set of supplementary Maple commands and procedures that augments the existing computer algebra system. In effect, APPL takes the capabilities of Maple and turns it into a computer algebra system for computational probability. This chapter contains guidelines for using Maple, and discusses the Maple commands that are used in APPL programming. After reading this chapter, an APPL user will have the knowledge necessary to modify the APPL code to meet his or her particular needs. We will start with a discussion of basic numeric computation, then advance to defining variables, symbolic computations, functions, data types, solving equations, calculus and graphing. Then we will discuss the programming features of Maple that facilitate building the APPL language: loops, conditions, and procedures.

The combination of symbolic computation and numerical calculation will undoubtedly increase the power of control systems design. Although several control design toolboxes exist, many based on the simulation software MATLAB, they are primarily geared for matrix operations. General nonlinear controller design certainly needs the combination of both symbolic and numerical computations. People may argue that one can use the MATLAB symbolic toolbox which is essentially a small subset of Maple-V. However, this symbolic toolbox can only be used to solve small scale nonlinear control design problems. It is not suitable for general nonlinear computer aided control systems design without further C code for interface, as one cannot dynamically allocate memory for an indexed function of time t of arbitrary dimension, for example. This interface will use no less effort than using Mathlink as an interface between Mathematica and MATLAB. Dynamic sliding mode control using state feedback for minimum phase MIMO systems and for MIMO uncertain systems has been developed. These methods have been implemented with Mathematica and MATLAB using Mathlink as an interface. These packages have been used for nonlinear control design.

In the January 2002 edition of SIAM News, Nick Trefethen announced the '$100, 100-Digit Challenge'. In this note he presented ten easy-to-state but hard-to-solve problems of numerical analysis, and challenged readers to find each answer to ten-digit accuracy. Trefethen closed with the enticing comment: 'Hint: They're hard! If anyone gets 50 digits in total, I will be impressed.' This challenge obviously struck a chord in hundreds of numerical mathematicians worldwide, as 94 teams from 25 nations later submitted entries. Many of these submissions exceeded the target of 50 correct digits; in fact, 20 teams achieved a perfect score of 100 correct digits. Trefethen had offered $100 for the best submission. Given the overwhelming response, a generous donor (William Browning, founder of Applied Mathematics, Inc.) provided additional funds to provide a $100 award to each of the 20 winning teams. Soon after the results were out, four participants, each from a winning team, got together and agreed to write a book about the problems and their solutions. The team is truly international: Bornemann is from Germany, Laurie is from South Africa, Wagon is from the USA, and Waldvogel is from Switzerland. This book provides some mathematical background for each problem, and then shows in detail how each of them can be solved. In fact, multiple solution techniques are mentioned in each case. The book describes how to extend these solutions to much larger problems and much higher numeric precision (hundreds or thousands of digit accuracy). The authors also show how to compute error bounds for the results, so that one can say with confidence that one's results are accurate to the level stated. Numerous numerical software tools are demonstrated in the process, including the commercial products Mathematica, Maple and Matlab. Computer programs that perform many of the algorithms mentioned in the book are provided, both in an appendix to the book and on a website. In the process, the authors take the reader on a wide-ranging tour of modern numerical mathematics, with enough background material so that even readers with little or no training in numerical analysis can follow. Here is a list of just a few of the topics visited: numerical quadrature (i.e., numerical integration), series summation, sequence extrapolation, contour integration, Fourier integrals, high-precision arithmetic, interval arithmetic, symbolic computing, numerical linear algebra, perturbation theory, Euler-Maclaurin summation, global minimization, eigenvalue methods, evolutionary algorithms, matrix preconditioning, random walks, special functions, elliptic functions, Monte-Carlo methods, and numerical differentiation.

One can reasonably say that we always prove geometric theorems using deductive method.The deductive method is too often used such that we get the impression that there is no other alternative appropriate method.In this paper we will use inductive method.We need many special cases to prove so that we will use the computer algebra system to assist us.We use symbolic computation (using CAS and Fortran) to compute with rational number.First we prove geometric theorems about triangles and conic sections.In those cases we only need linear algebra so that using rational numbers will eliminate rounding error.Every theorem can be proved with special cases using rational numbers.Then we move on to noneuclidean geometry which is one of the most important topics in general relativity.We need calculus so that we use real number with high precision to prove special cases.The main goal of this research is to make mathematics closer to physics and by doing that we get a deeper understanding of general relativity.

A method for control employing rule-based search is reviewed, and a Rule-Based Controller achieving economical real-time performance is described. Code optimization, in the form of LISP-to-Pascal knowledge base translation, provides real-time search execution speed and a processing environment enabling highly integrated symbolic and numeric computation. With a multiprocessor software architecture specifying rule-based protocol for control task communication, and a hardware architecture providing concurrent implementation within a multi-microprocessor system, the controller realizes a set of cooperating real-time expert systems. Based on experience gained through the design and implementation of a Rule-Based Flight Control System, the proposed approach appears applicable to a large class of complex control problems.

According to Committee on Programs for Advanced Study of Mathematics and Science in American High Schools [1] (Gollub et al. (eds.) in Learning and Understanding. Improving Advanced Study of Mathematics and Science in U.S. Hight Schools. National Academic Press, Washington, 2002), “the primary goal of advanced study in any discipline should be for students to achieve a deep conceptual understanding of the disciplines content”. It is undoubted that abstraction is one of the skills that teachers wish to improve in their students, but, how can teachers take advantage of technological resources, such as CAS or DGS, as help in their classes in undergraduate courses? One concept, whose importance is both theoretical and practical, corresponds to the coordinate transformation, in particular orthogonal coordinate systems. We can use trigonometric constructions to find the transformation equations, namely, the algorithm for transforming a Cartesian system into other coordinate system, as cylindrical or spherical coordinates. Not only, if we add the knowledge and some techniques from Linear Algebra, we can motivate new mathematical properties, but also we will increase considerably the abstract reasoning and symbolic calculation. We know that visualization helps intuitive understanding. Therefore, we propose using CAS and DGS to show how a triad, of basis vectors, is continuously changing direction, keeping the norm vector without change, and how this match visualization with the reasoning from theories of linear algebra.

Unification of calculus on Time Scales with mathematica

We propose a novel computational approach to the approximate input/state feedback linearization problem by interpolating a finite number of local linear coordinate transforms and static state feedback designs. For a class of single-input nonlinear systems, the approximate approach allows the main assumptions underlying exact input/state feedback linearization (involutivity, smoothness and controllability everywhere) to be relaxed. Moreover, the present approach relies only on simple numeric linear algebraic computations, in strong contrast to the exact input/state feedback linearization approach that relies on the solution of a partial differential equation and other symbolic computations. In contrast to related approaches to approximation feedback linearization, the feedback design need not be restricted to a neighborhood of the equilibrium manifold. It is shown that the approximation error goes to zero uniformly as the resolution of the state space partitioning increases. Explicit expressions for the approximation error allows the accuracy and robustness of the design to be assessed.

Mathematics is a ubiquitous foundation of science, technology, and engineering. Specific areas of mathematics, such as numeric and symbolic computation or logics, enjoy considerable software support. Working mathematicians have recently started to adopt Web 2.0 environments, such as blogs and wikis, but these systems lack machine support for knowledge organization and reuse, and they are disconnected from tools such as computer algebra systems or interactive proof assistants. We argue that such scenarios will benefit from Semantic Web technology.Conversely, mathematics is still underrepresented on the Web of [Linked] Data. There are mathematics-related Linked Data, for example statistical government data or scientific publication databases, but their mathematical semantics has not yet been modeled. We argue that the services for the Web of Data will benefit from a deeper representation of mathematical knowledge.Mathematical knowledge comprises structures given in a logical language --formulae, statements e.g. axioms, and theo-ries --, a mixture of rigorous natural language and symbolic notation in documents, application-specific metadata, and discussions about conceptualizations, formalizations, proofs, and counter-examples. Our review of vocabularies for representing these structures covers ontologies for mathematical problems, proofs, interlinked scientific publications, scientific discourse, as well as mathematical metadata vocabularies and domain knowledge from pure and applied mathematics.Many fields of mathematics have not yet been implemented as proper Semantic Web ontologies; however, we show that MathML and OpenMath, the standard XML-based exchange languages for mathematical knowledge, can be fully integrated with RDF representations in order to contribute existing mathematical knowledge to the Web of Data.We conclude with a roadmap for getting the mathematical Web of Data started: what datasets to publish, how to interlink them, and how to take advantage of these new connections.

We apply perturbation theory to the hydrogen atom in a magnetic field by means of a recurrence relation for properly chosen moments of the wave function. The method is suitable for both numerical and symbolic computation and allows the treatment of classes of states with common symmetry properties. We derive analytic expressions in terms of the principal quantum number for the perturbation corrections to the energy of the unperturbed states with quantum numbers n-1=\ensuremath{\Vert}m\ensuremath{\Vert}=l, n-2=\ensuremath{\Vert}m\ensuremath{\Vert}=l-1, n-2=\ensuremath{\Vert}m\ensuremath{\Vert}=l, n-3=\ensuremath{\Vert}m\ensuremath{\Vert}=l-1, and (n-3=\ensuremath{\Vert}m\ensuremath{\Vert}=l,n-3=\ensuremath{\Vert}m\ensuremath{\Vert}=l-2) some of which have not been reported before. Disagreements with some previous results are pointed out.