We present a 17-dimensional family of multiplication schemes for 3×3 matrices with 23 multiplications applicable to arbitrary coefficient rings.
我们给出了适用于任意系数环的具有23次乘法的3×3矩阵的17维乘法方案族。
{"title":"A family of schemes for multiplying 3 × 3 matrices with 23 coefficient multiplications","authors":"Marijn J. H. Heule, Manuel Kauers, M. Seidl","doi":"10.1145/3377006.3377015","DOIUrl":"https://doi.org/10.1145/3377006.3377015","url":null,"abstract":"We present a 17-dimensional family of multiplication schemes for 3×3 matrices with 23 multiplications applicable to arbitrary coefficient rings.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"53 1","pages":"118 - 121"},"PeriodicalIF":0.1,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3377006.3377015","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41437218","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}
Beke (1894) gave an algorithm that factors any differential operator and that algorithm can be used for difference operators as well. Bronstein improved Beke’s algorithm (see [2]). To find an order-m right-hand factor of an order-n operator using Beke-Bronstein’s algorithm, we need to create a difference system of order N = ( n m ) and solve that system. Experiments show that this is practical for n ≤ 8, or n = 9 if m ≤ 3, but beyond that N becomes too large. Our goal is a new method to find factors without increasing the order.
{"title":"Fast algorithm for factoring difference operators","authors":"Yi Zhou, M. V. Hoeij","doi":"10.1145/3377006.3377023","DOIUrl":"https://doi.org/10.1145/3377006.3377023","url":null,"abstract":"Beke (1894) gave an algorithm that factors any differential operator and that algorithm can be used for difference operators as well. Bronstein improved Beke’s algorithm (see [2]). To find an order-m right-hand factor of an order-n operator using Beke-Bronstein’s algorithm, we need to create a difference system of order N = ( n m ) and solve that system. Experiments show that this is practical for n ≤ 8, or n = 9 if m ≤ 3, but beyond that N becomes too large. Our goal is a new method to find factors without increasing the order.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"53 1","pages":"150 - 152"},"PeriodicalIF":0.1,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3377006.3377023","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43330177","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}
Let C be the field of complex numbers and C(x) be the field of rational functions in the variables x = x1, . . . , xn over C. Let Si be the shift operator with respect to xi on C(x) defined as Si(f(x1, . . . , xn)) = f(x1, . . . , xi−1, xi + 1, xi+1, . . . , xn) for any f ∈ C(x). Definition 1 (Hypergeometric and hyperarithmetic terms). A nonzero term H(x) : Nn → C is said to be hypergeometric over C(x) if there exist rational functions f1, . . . , fn ∈ C(x) such that
{"title":"Additive Ore-Sato theorem","authors":"Shaoshi Chen, Jing Guo","doi":"10.1145/3377006.3377009","DOIUrl":"https://doi.org/10.1145/3377006.3377009","url":null,"abstract":"Let C be the field of complex numbers and C(x) be the field of rational functions in the variables x = x1, . . . , xn over C. Let Si be the shift operator with respect to xi on C(x) defined as Si(f(x1, . . . , xn)) = f(x1, . . . , xi−1, xi + 1, xi+1, . . . , xn) for any f ∈ C(x). Definition 1 (Hypergeometric and hyperarithmetic terms). A nonzero term H(x) : Nn → C is said to be hypergeometric over C(x) if there exist rational functions f1, . . . , fn ∈ C(x) such that","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"53 1","pages":"96 - 98"},"PeriodicalIF":0.1,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3377006.3377009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46509116","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}
Patricia Pascual-Ortigosa, E. Sáenz-de-Cabezón, H. Wynn
We develop algorithms for the analysis of multi-state k-out-of-n systems and their reliability based on commutative algebra.
基于交换代数，提出了多态k-out- n系统及其可靠性分析算法。
{"title":"Algebraic algorithms for the reliability analysis of multi-state k-out-of-n systems","authors":"Patricia Pascual-Ortigosa, E. Sáenz-de-Cabezón, H. Wynn","doi":"10.1145/3377006.3377022","DOIUrl":"https://doi.org/10.1145/3377006.3377022","url":null,"abstract":"We develop algorithms for the analysis of multi-state k-out-of-n systems and their reliability based on commutative algebra.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"53 1","pages":"146 - 149"},"PeriodicalIF":0.1,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3377006.3377022","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48831344","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}
In this poster we present the results of [1]. We consider the problem of detecting chaotic behaviors in discrete dynamical systems. We propose an algebraic criterion for determining whether all the zeros of a given polynomial are outside the unit circle in the complex plane. This criterion is used to deduce critical algebraic conditions for the occurrence of chaos in multi-dimensional discrete systems based on Marotto's theorem. Using these algebraic conditions we reduce the problem of analyzing chaos induced by snapback repeller to an algebraic problem, and introduce an algorithmic approach to solve this problem by means of symbolic computation. The proposed approach is effective as shown by several examples and can be used to determine the possibility that all the fixed points are snapback repellers.
{"title":"Algebraic approach to chaos induced by snapback repeller","authors":"Bo Huang, W. Niu","doi":"10.1145/3377006.3377016","DOIUrl":"https://doi.org/10.1145/3377006.3377016","url":null,"abstract":"In this poster we present the results of [1]. We consider the problem of detecting chaotic behaviors in discrete dynamical systems. We propose an algebraic criterion for determining whether all the zeros of a given polynomial are outside the unit circle in the complex plane. This criterion is used to deduce critical algebraic conditions for the occurrence of chaos in multi-dimensional discrete systems based on Marotto's theorem. Using these algebraic conditions we reduce the problem of analyzing chaos induced by snapback repeller to an algebraic problem, and introduce an algorithmic approach to solve this problem by means of symbolic computation. The proposed approach is effective as shown by several examples and can be used to determine the possibility that all the fixed points are snapback repellers.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"53 1","pages":"122 - 125"},"PeriodicalIF":0.1,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3377006.3377016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46838300","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}
With the progress of algebraic computations on polynomials and matrices, we are paying more attention to approximate algebraic algorithms. Among approximate algebraic algorithms, those for calculating approximate greatest common divisor (GCD) consider a pair of given polynomials f and g that are relatively prime in general, and find f and g which are close to f and g, respectively, in the sense of polynomial norm, and have the GCD of certain degree. The algorithms can be classified into two categories: 1) for a given tolerance (magnitude) of ||f - f|| and ||g - g||, make the degree of approximate GCD as large as possible, and 2) for a given degree d, minimize the magnitude of ||f - f|| and ||g - g||.
{"title":"The GPGCD algorithm with the Bézout matrix","authors":"Boming Chi, Akira Terui","doi":"10.1145/3377006.3377010","DOIUrl":"https://doi.org/10.1145/3377006.3377010","url":null,"abstract":"With the progress of algebraic computations on polynomials and matrices, we are paying more attention to approximate algebraic algorithms. Among approximate algebraic algorithms, those for calculating approximate greatest common divisor (GCD) consider a pair of given polynomials f and g that are relatively prime in general, and find f and g which are close to f and g, respectively, in the sense of polynomial norm, and have the GCD of certain degree. The algorithms can be classified into two categories: 1) for a given tolerance (magnitude) of ||f - f|| and ||g - g||, make the degree of approximate GCD as large as possible, and 2) for a given degree d, minimize the magnitude of ||f - f|| and ||g - g||.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"53 1","pages":"99 - 102"},"PeriodicalIF":0.1,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3377006.3377010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46894505","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}
Consider the polynomial mapping defined by the projection to the first coordinate on a real, smooth and compact hypersurface. The critical points of this mapping in generic coordinates have several applications in real algebraic geometry. We provide bit complexity estimates for computing them. Generic coordinates are obtained by applying a randomly chosen linear change of variables to the polynomial defining the hypersurface. The coordinates are sufficiently generic when the Jacobian matrix of the system under study has full rank at the critical points and when the number of critical points is finite. We have proven a new quantitative extension of Thom's weak transversality theorem [1]. By applying this extension, we are able to choose sufficiently generic changes of variables with arbitrarily high probability.
{"title":"Bit complexity for critical point computation in smooth and compact real hypersurfaces","authors":"J. Elliott, É. Schost","doi":"10.1145/3377006.3377014","DOIUrl":"https://doi.org/10.1145/3377006.3377014","url":null,"abstract":"Consider the polynomial mapping defined by the projection to the first coordinate on a real, smooth and compact hypersurface. The critical points of this mapping in generic coordinates have several applications in real algebraic geometry. We provide bit complexity estimates for computing them. Generic coordinates are obtained by applying a randomly chosen linear change of variables to the polynomial defining the hypersurface. The coordinates are sufficiently generic when the Jacobian matrix of the system under study has full rank at the critical points and when the number of critical points is finite. We have proven a new quantitative extension of Thom's weak transversality theorem [1]. By applying this extension, we are able to choose sufficiently generic changes of variables with arbitrarily high probability.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"53 1","pages":"114 - 117"},"PeriodicalIF":0.1,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3377006.3377014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41852757","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}
A common advice in automated guessing is that when the input is a sequence like 2, 0, 0, 7, 0, 0,17,..., one should remove the zeros before passing the data to the guesser. On this poster, we explain why this approach is sound, and which problem may arise if one does not take this step.
{"title":"Why you should remove zeros from data before guessing","authors":"Manuel Kauers, Thibaut Verron","doi":"10.1145/3377006.3377017","DOIUrl":"https://doi.org/10.1145/3377006.3377017","url":null,"abstract":"A common advice in automated guessing is that when the input is a sequence like 2, 0, 0, 7, 0, 0,17,..., one should remove the zeros before passing the data to the guesser. On this poster, we explain why this approach is sound, and which problem may arise if one does not take this step.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"53 1","pages":"126 - 129"},"PeriodicalIF":0.1,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3377006.3377017","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46244201","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}
For the problem of sparse interpolation of multivariate polynomials, we propose robust computation methods based on the modified numerical Ben-Or/Tiwari algorithm by M. Giesbrecht, G. Labahn, and W.-s. Lee.
{"title":"Robust computation methods for sparse interpolation of multivariate polynomials","authors":"Kazuki Kondo, Hiroshi Sekigawa","doi":"10.1145/3377006.3377018","DOIUrl":"https://doi.org/10.1145/3377006.3377018","url":null,"abstract":"For the problem of sparse interpolation of multivariate polynomials, we propose robust computation methods based on the modified numerical Ben-Or/Tiwari algorithm by M. Giesbrecht, G. Labahn, and W.-s. Lee.","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"53 1","pages":"130 - 133"},"PeriodicalIF":0.1,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3377006.3377018","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42448726","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":"Methods for simplifying differential equations","authors":"Shayea Aldossari, M. V. Hoeij","doi":"10.1145/3377006.3377008","DOIUrl":"https://doi.org/10.1145/3377006.3377008","url":null,"abstract":"","PeriodicalId":41965,"journal":{"name":"ACM Communications in Computer Algebra","volume":"53 1","pages":"93 - 95"},"PeriodicalIF":0.1,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3377006.3377008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47316900","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}
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