SIGSAM Bull.最新文献

{"title":"A note on the number of division steps in the Euclidean algorithm","authors":"S. Abramov","doi":"10.1145/377626.377629","DOIUrl":"https://doi.org/10.1145/377626.377629","url":null,"abstract":"Let w be a natural number and let #(w) be the maximal number of divisions that the Euclidean algorithm, ao = qlal+a2 , al ~ q2a2+a3 , ak-2 = qk-lak-l+ak , ak-1 = qkak, (1) needs for a given input (ao, al), where a0 > al = w. Lamd's theorem [2, 1] (this theorem was proved earlier by Finck in 1841 [1]) implies the asymptotic estimate u(~) = O(log w), (2) and log w cannot be replaced by any function h(w) such that h(w) = o(logw), since, if Fo, F1,... is the Fibonacci sequence, for ao = Fk+2, w = at = Fk+x the number of divisions is equal to k. The difference between the latter number and log s w, where ¢ = (1 + z/g)/2, is a bounded value. One of the results related to the average case behavior of the Euclidean algorithm is by Heilbronn [4, 1]: 1 ~ E(v,~) ~ 121n-Aln. ~(v) ~r~ ' l~w~v gcd(v,w)=l where E(v,w) is the number of division steps performed by the Euclidean algorithm on the input (v, w). From this asymptotic equality it follows that for some constant C the inequality 12 In 2 ~(w)> ~ lnw+C (3) holds. Using the standard notation f(n) = O(g(n)), which is defined for functions f(n), g(n) with positive values by f(n) = O(g(n)) if and only if 3cl,c2,no>0, Vn>no, clg(n)~f(n)gc2g(n), we therefore have Theorem 1 #(w) = O(logw). We now prove the following main theorem. Notice that (121n2)/~r 2 < 1/(21n¢), and (4) is stronger than (3) for all large enough w. Additionally, the proof of Theorem 2, which will be given, is elementary and thereby we get an elementary proof of Theorem 1. We start with a lemma on Fibonacci numbers. Lemma 1 For any 0 < d < ~/g, the inequality IF.+1 _¢ < 1 dF~ (5) holds for all large enough n.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2000-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121673018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On computing a separating transcendence basis","authors":"R. Steinwandt","doi":"10.1145/377626.377632","DOIUrl":"https://doi.org/10.1145/377626.377632","url":null,"abstract":"Let <i>k</i>(<i>x</i><inf>1</inf>,&hellip;, <i>x<inf>n</inf></i>)/<i>k</i> be a finitely generated field extension, and <i>g</i><inf>1</inf>,&hellip;,<i>g<inf>r</inf></i> &epsilon; <i>k</i>(x). For <i>k</i>(<i>x</i><inf>1</inf>,&hellip;,<i>x<inf>n</inf></i>)/<i>k</i>(<i>g</i><inf>1</inf>,&hellip;,<i>g<inf>r</inf></i>) being separably generated (which in particular includes the case char(<i>k</i>) = 0) we give a method to compute the transcendence degree and a separating transcendence basis of this extension by means of simple linear algebra techniques.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2000-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130949865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Numerical monsters","authors":"C. Essex, M. Davison, C. Schulzky","doi":"10.1145/377626.377635","DOIUrl":"https://doi.org/10.1145/377626.377635","url":null,"abstract":"When the results of certain computer calculations are shown to be not simply incorrect but dramatically incorrect, we have a powerful reason to be cautious about all computer-based calculations. In this paper we present a \"Rogue's Gallery\" of simple calculations whose correct solutions are obvious to humans but whose numerical solutions are incorrect in pathological ways. We call these calculations, which can be guaranteed to wreak numerical mayhem across both software packages and hardware platforms, \"Numerical Monsters\". Our monsters can be used to provide deep insights into how computer calculations fail, and we use them to engender appreciation for the subject of numerical analysis in our students. Although these monsters are based on well-understood numerical pathologies, even experienced numerical analysts will surprises in their behaviour and can use the lessons they bring to become even better masters of their tools.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"77 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2000-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122849254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Algebraic invariants of graphs; a study based on computer exploration","authors":"Nicolas M. Thiéry","doi":"10.1145/377604.377612","DOIUrl":"https://doi.org/10.1145/377604.377612","url":null,"abstract":"We consider the ring Jn of polynomial invariants overweighted graphs on n vertices. Our primary interest is the use ofthis ring to define and explore algebraic versions of isomorphismproblems of graphs, such as Ulam's reconstruction conjecture. There is a huge body of literature on invariant theory whichprovides both general results and algorithms. However, there is acombinatorial explosion in the computations involved and, to ourknowledge, the ring Jn has only been completelydescribed for n ≤ 4. This led us to study the ring Jn in its own right. Weused intensive computer exploration for small n, and developedPerMuVAR, a library for MuPAD, for computing in invariant rings ofpermutation groups. We present general properties of the ring Jn, as wellas results obtained by computer exploration for small n, includingthe construction of a medium sized generating set forJn. We address several conjectures suggested by thoseresults (low degree system of parameters, unimodality), forJn as well as for more general invariant rings. We alsoshow that some particular sets are not generating, disproving aconjecture of Pouzet related to reconstruction, as well as a lemmaof Grigoriev on the invariant ring over digraphs. We finallyprovide a very simple minimal generating set of the field ofinvariants.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"1418 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2000-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"113994947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Resistance scaling and random walk dimensions for finitely ramified Sierpinski carpets","authors":"C. Schulzky, A. Franz, K. Hoffmann","doi":"10.1145/377604.377608","DOIUrl":"https://doi.org/10.1145/377604.377608","url":null,"abstract":"We present a new algorithm to calculate the random walk dimensionof finitely ramified Sierpinski carpets. The fractal structure isinterpreted as a resistor network for which the resistance scalingexponent is calculated using Mathematica. A fractal form of theEinstein relation, which connects diffusion with conductivity, isused to give a numerical value for the random walk dimension.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"154 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2000-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134263628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"East Coast Computer Algebra Day","authors":"M. Giesbrecht","doi":"10.1145/377604.569765","DOIUrl":"https://doi.org/10.1145/377604.569765","url":null,"abstract":"s of Invited Talks Subresultants revisited Joachim yon zur Gathen Fachbereich 17 Mathematik-Informatik Universit~t Paderborn Paderborn, Germany Starting in the late 1960s, Collins and Brown & Tranb invented polynomial remainder sequences (PRS) in order to apply the Euclidean algorithm to integer polynomials. Subresultants play a major role in this theory. We compare the various notions of subresultants, give a general and precise definition of PRS, and clean up some loose ends: • prove a 1971 conjecture of Brown that all results in the subresultant PRS are integer polynomials, • show an exponential lower bound on the pseudo PRS. Lastly, we show how Kronecker had, already in the 1870s, discovered many of the fundamental properties of Euclid's algorithm for polynomials. Some problems in general purpose computer algebra systems design Michael Monagan Center for Experimental and Computational Mathematics Simon Fraser University In this talk I will present three problems of interest to the computer algebra community. The first is the problem of implementing modular algorithms efficiently. Application of the Chinese remainder theorem to solve the GCD and Groebner bases problems leads to a big loss of efficiency because the data structure overhead overwhelms the cost of the modular arithmetic. The second problem is how to build a system so that all the components interact well. I will take as an example a problem of automatic differentiation from astrophysics where the function to be differentiated involves the solution of a non-linear equation. Can the CAS differentiate commands like f so lve (f--0,x--a); in a program? The third problem is a problem of trying to implement generic algorithms, efficiently. I will take as an example a linear p-adic Newton iteration. A generic version of this algorithm would work over Z mod p~ and over Fix] mod x ~ for example. Iterative solution of algebraic problems with polynomials Hans J. Stetter Technical University of Vienna Vienna, Austria In Numerical Analysis, it is standard to use an iterative solution procedure for a nonlinear problem. In Computer Algebra, one prefers exact finite manipulations which preserve the algebraic structure (like in Groebner basis computation); but often, in the end, an iterative numerical procedure can not be avoided (e.g. for zeros of a polynomial system). Furthermore, algebraic problems from Scientific Computing generally contain some \"empiric\" data so that their results are only defined to a limited accuracy. In this situation, an iterative approach may reduce to a few (or just one) step(s). We will at tempt to demonstrate how iterative procedures can be built upon the algebraic structure of a variety of problems for which such an approach has not been considered so far: After some discussion of zero clusters of univariate and systems of multivariate polynomials, we will mainly consider overdetermined problems like greatest common divisors, multivariate factorization, etc.; here the","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2000-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132713125","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Southern Ontario Numerical Analysis Day","authors":"Robert M Corless","doi":"10.1145/377604.569766","DOIUrl":"https://doi.org/10.1145/377604.569766","url":null,"abstract":"s of Invited Talks C o m p u t e r Algebra for Differential-Algebraic Equat ions (DAEs) Gilles ViUard LMC-IMA G / Equipe Caleul Formel Grenoble, l, ranee This talk will survey the activities of a five year project (95-99) of the computer algebra team of Grenoble on DAEs. The ACTE project has been supported by \"Electricit~ de France\" (EDF) in relation with the development of the Eurostag software for power systems simulation. Answering to questions of the company EDF, ACTE's main concerns have been to investigate how computer algebra could be involved for the study and the numerical integration of DAEs. Three main topics have been distinguished and will be presented: • The validation of numerical solutions of low index DAEs using global error estimators. Convergence analysis and numerical tests are proposed. • A fully symbolic index reduction algorithm is designed as an effective implementation of Rabier and Rheinbold's method for quasi-linear DAEs. • A cheap hybrid numeric/symbolic is proposed for the numerical integration of high index quasi-linear DAEs. It is based on fast exact linear algebra for a preliminary reduction of the index. Differential-Algebraic Systems in Industr ia l P lant Simulat ion: S o m e Personal Experiences Grant Stephenson Honeywell Hi-Spec Solutions London, Canada In the field of industrial plant simulation, mathematical models of industrial processes are solved to predict plant operation under varying conditions. These models c~a be used for many purposes including plant design, analysis of operating problems, assessment of control strategies, optimization of plant operation and training of plant operators. A plant simulation is typically comprised of a large number of mathematical models each representing some aspect of the processing performed by individual pieces of equipment or the control strategies implemented in the distributed control system. These mathematical models are logically connected to form an integrated model of the plant just as the process piping, electrical wiring and pneumatic tubing connect the equipment in the physical plant. Each equipment model is either a system of nonlinear algebraic equations or differential-algebraic equations. This presentation describes some of the presenter's experiences building and solving mathematical models for piping networks, cryogenic heat exchangers used in natural gas liquefaction and distillation columns in the context of training operators of industrial plants. Solving D D E s in Mat lab Lawrence F. Sharnpine Southern Methodist University, T e x a s With the goal of making it as easy as possible to solve a large class of delay differential equations (DDEs), the speaker and S. Thompson of l:tadford University developed a program, dde23, to solve DDEs with constant delays in Matlab. The speaker will discuss briefly important differences between DDEs and ODEs and then how numerical methods for ODEs can be extended to solve DDEs. Some examples will show that restr","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"196 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2000-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116514404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"OpenMath: compositionality achieved at last","authors":"Andreas Strotmann, L. Kohout","doi":"10.1145/362001.362024","DOIUrl":"https://doi.org/10.1145/362001.362024","url":null,"abstract":"As a language for exchanging computer-\"understandable\" representations of mathematical formulas and concepts, OpenMath has a fairly standard syntactic structure for most of the mathematical notions that have so far been formalized in its \"Content Dictionaries.\" The syntax for most operators is closely related to the well-known and decades-old LISP prefix notation for function application.For syntactic representations of a particular large class of mathematical operators, however, OpenMath takes a very unusual approach when compared with any of the existing Computer Algebra systems, say. This class contains integration and differentiation operators, sums and products, and other generalized quantifiers.In our paper we argue that OpenMath's novel approach to representing generalized quantifiers is superior to the classic representations. In particular, we show that this unusual feature of OpenMath has been designed to adhere to the Compositionality Principle[10], a design principle that classic representations, including older versions of OpenMath, have been violating[12].Having thus shown the importance of compositionality for the design of modern OpenMath, we then proceed to show that the Compositionality Principle is a fundamental research instrument in the Formal Semantics branch of linguistics[9], and argue that, like the use of this particular guiding principle for improvements in the design of a representation for generalized quantifiers, the study of the underlying structure of natural language as discovered by various branches of linguistics can provide many more suggestions for further improvements to OpenMath and to its sibling, MathML. We give some examples to substantiate our claim.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2000-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129922578","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"OMDoc: an infrastructure for OpenMath content dictionary information","authors":"M. Kohlhase","doi":"10.1145/362001.362021","DOIUrl":"https://doi.org/10.1145/362001.362021","url":null,"abstract":"The OPENMATH framework for transmitting mathematical objects over the Internet relies on the concept of Content Dictionaries (CDs) to define the semantics of mathematical objects. This is an essential measure for establishing a meaningful communication amongst mathematical software systems (and humans).Currently, the infrastructure for conceiving, administering, viewing CDs is limited to a file-based almost flat repository. In this paper, we propose to use the OMDoc extension of the OPENMATH XML encoding as an infrastructure to express and manipulate content dictionary information. OMDoc extends OPENMATH by adding support for document markup (making the CDs more readable to the human user) and structured specification (making them more explicit, formal, and allow the user to reuse, and inherit CD information in a flexible, but well-defined way).","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2000-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124155545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"OpenMath support under CSL-hosted REDUCE","authors":"A. Triulzi","doi":"10.1145/362001.362017","DOIUrl":"https://doi.org/10.1145/362001.362017","url":null,"abstract":"OpenMath is a standard for representing mathematical objects. This paper describes an attempt to include OpenMath support into the computer algebra system REDUCE. This support is limited to the version based upon Cambridge Standard Lisp and is based upon the OpenMath C Library developed under Esprit project 24.969. The rationale for this effort was to avoid having to rewrite substantial amounts of code due to changes in the OpenMath standard but instead relying on the underlying C Library to provide the necessary modifications.","PeriodicalId":314801,"journal":{"name":"SIGSAM Bull.","volume":"426 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2000-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115645799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}