Pub Date : 2014-11-03DOI: 10.1112/S1461157015000170
E. Milio
We propose to generalize the work of Regis Dupont for computing modular polynomials in dimension $2$ to new invariants. We describe an algorithm to compute modular polynomials for any invariants derived from theta constants and prove that this algorithm is quasi-linear.Some properties of the modular polynomials with the quotient of theta constants are analyzed.We report on experiments with our implementation.
{"title":"A quasi-linear time algorithm for computing modular polynomials in dimension 2","authors":"E. Milio","doi":"10.1112/S1461157015000170","DOIUrl":"https://doi.org/10.1112/S1461157015000170","url":null,"abstract":"We propose to generalize the work of Regis Dupont for computing modular polynomials in dimension $2$ to new invariants. We describe an algorithm to compute modular polynomials for any invariants derived from theta constants and prove that this algorithm is quasi-linear.Some properties of the modular polynomials with the quotient of theta constants are analyzed.We report on experiments with our implementation.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"18 1","pages":"603-632"},"PeriodicalIF":0.0,"publicationDate":"2014-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157015000170","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411867","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}
Pub Date : 2014-09-29DOI: 10.1112/S1461157014000114
A. Fish, John Taylor
The form of information presented can influence its utility for the conveying of knowledge byaffecting an interpreter’s ability to reason with the information. There are distinct types ofrepresentational systems (e.g. symbolic versus diagrammatic logics), various sub-systems (e.g.propositional versus predicate logics), and even within a single representational system theremay be different means of expressing the same piece of information content. Thus to displayinformation, choices must be made between its different representations, depending upon manyfactors such as: the context, the reasoning tasks to be considered, user preferences or desires (e.g.for short symbolic sentences or minimal clutter within diagrammatic systems). The identificationof all equivalent representations with the same information content is a sensible precursor toattempts to minimize a metric over this class. We posit that defining notions of semantic-redundancy and identifying the syntactic properties that encapsulate redundancy can help inachieving the goal of completely identifying equivalences within a single notational system oracross multiple systems, but that care must be taken when extending systems, since refinementsof redundancy conditions may be necessary even for conservative system extensions. We demonstrate this theory within two diagrammatic systems, which are Euler diagram basednotations. Such notations can be used to represent logical information and have applicationsincluding visualization of database queries, social network visualisation, statistical data visuali-sation, and as the basis of more expressive diagrammatic logics such as constraint languages usedin software specification and reasoning. The development of the new associated machinery andconcepts required is important in its own right since it increases the growing body of knowledgeon diagrammatic logics. In particular, we consider Euler diagrams with shading, and then we conservatively extend the system to include projections, which allow a much greater degree offlexibility of representation. We give syntactic properties that encapsulate semantic equivalencein both systems, whilst observing that the same semantic concept of redundancy is significantlymore difficult to realize as syntactic properties in the extended system with projections.
{"title":"Equivalences in Euler-based diagram systems through normal forms","authors":"A. Fish, John Taylor","doi":"10.1112/S1461157014000114","DOIUrl":"https://doi.org/10.1112/S1461157014000114","url":null,"abstract":"The form of information presented can influence its utility for the conveying of knowledge byaffecting an interpreter’s ability to reason with the information. There are distinct types ofrepresentational systems (e.g. symbolic versus diagrammatic logics), various sub-systems (e.g.propositional versus predicate logics), and even within a single representational system theremay be different means of expressing the same piece of information content. Thus to displayinformation, choices must be made between its different representations, depending upon manyfactors such as: the context, the reasoning tasks to be considered, user preferences or desires (e.g.for short symbolic sentences or minimal clutter within diagrammatic systems). The identificationof all equivalent representations with the same information content is a sensible precursor toattempts to minimize a metric over this class. We posit that defining notions of semantic-redundancy and identifying the syntactic properties that encapsulate redundancy can help inachieving the goal of completely identifying equivalences within a single notational system oracross multiple systems, but that care must be taken when extending systems, since refinementsof redundancy conditions may be necessary even for conservative system extensions. We demonstrate this theory within two diagrammatic systems, which are Euler diagram basednotations. Such notations can be used to represent logical information and have applicationsincluding visualization of database queries, social network visualisation, statistical data visuali-sation, and as the basis of more expressive diagrammatic logics such as constraint languages usedin software specification and reasoning. The development of the new associated machinery andconcepts required is important in its own right since it increases the growing body of knowledgeon diagrammatic logics. In particular, we consider Euler diagrams with shading, and then we conservatively extend the system to include projections, which allow a much greater degree offlexibility of representation. We give syntactic properties that encapsulate semantic equivalencein both systems, whilst observing that the same semantic concept of redundancy is significantlymore difficult to realize as syntactic properties in the extended system with projections.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"431-484"},"PeriodicalIF":0.0,"publicationDate":"2014-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000114","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411188","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}
Pub Date : 2014-08-05DOI: 10.1112/S1461157014000333
Jianwei Li, Phong Q. Nguyen
We present a higher-dimensional generalization of the Gama{Nguyen algorithm (STOC '08) for approximating the shortest vector problem in a lattice. This generalization approximates the densest sublattice by using a subroutine solving the exact problem in low dimension, such as the Dadush{Micciancio algorithm (SODA '13). Our approximation factor corresponds to a natural inequality on Rankin's constant derived from Rankin's inequality.
{"title":"Approximating the densest sublattice from Rankin’s inequality","authors":"Jianwei Li, Phong Q. Nguyen","doi":"10.1112/S1461157014000333","DOIUrl":"https://doi.org/10.1112/S1461157014000333","url":null,"abstract":"We present a higher-dimensional generalization of the Gama{Nguyen algorithm (STOC '08) for approximating the shortest vector problem in a lattice. This generalization approximates the densest sublattice by using a subroutine solving the exact problem in low dimension, such as the Dadush{Micciancio algorithm (SODA '13). Our approximation factor corresponds to a natural inequality on Rankin's constant derived from Rankin's inequality.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"92-111"},"PeriodicalIF":0.0,"publicationDate":"2014-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000333","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411898","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}
Pub Date : 2014-08-01DOI: 10.1112/S1461157014000229
Anja Becker, Nicolas Gama, A. Joux
In this paper, we present a heuristic algorithm for solving exact, as well as approximate, shortest vector and closest vector problems on lattices. The algorithm can be seen as a modified sieving algorithm for which the vectors of the intermediate sets lie in overlattices or translated cosets of overlattices. The key idea is hence no longer to work with a single lattice but to move the problems around in a tower of related lattices. We initiate the algorithm by sampling very short vectors in an overlattice of the original lattice that admits a quasi-orthonormal basis and hence an efficient enumeration of vectors of bounded norm. Taking sums of vectors in the sample, we construct short vectors in the next lattice. Finally, we obtain solution vector(s) in the initial lattice as a sum of vectors of an overlattice. The complexity analysis relies on the Gaussian heuristic. This heuristic is backed by experiments in low and high dimensions that closely reflect these estimates when solving hard lattice problems in the average case. This new approach allows us to solve not only shortest vector problems, but also closest vector problems, in lattices of dimension $n$ in time $2^{0.3774n}$ using memory $2^{0.2925n}$. Moreover, the algorithm is straightforward to parallelize on most computer architectures.
{"title":"A sieve algorithm based on overlattices","authors":"Anja Becker, Nicolas Gama, A. Joux","doi":"10.1112/S1461157014000229","DOIUrl":"https://doi.org/10.1112/S1461157014000229","url":null,"abstract":"In this paper, we present a heuristic algorithm for solving exact, as well as approximate, shortest vector and closest vector problems on lattices. The algorithm can be seen as a modified sieving algorithm for which the vectors of the intermediate sets lie in overlattices or translated cosets of overlattices. The key idea is hence no longer to work with a single lattice but to move the problems around in a tower of related lattices. We initiate the algorithm by sampling very short vectors in an overlattice of the original lattice that admits a quasi-orthonormal basis and hence an efficient enumeration of vectors of bounded norm. Taking sums of vectors in the sample, we construct short vectors in the next lattice. Finally, we obtain solution vector(s) in the initial lattice as a sum of vectors of an overlattice. The complexity analysis relies on the Gaussian heuristic. This heuristic is backed by experiments in low and high dimensions that closely reflect these estimates when solving hard lattice problems in the average case. \u0000 \u0000This new approach allows us to solve not only shortest vector problems, but also closest vector problems, in lattices of dimension $n$ in time $2^{0.3774n}$ using memory $2^{0.2925n}$. Moreover, the algorithm is straightforward to parallelize on most computer architectures.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"49-70"},"PeriodicalIF":0.0,"publicationDate":"2014-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000229","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411446","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}
Pub Date : 2014-08-01DOI: 10.1112/S146115701400028X
S. Takemori
Let A(Γ2) denote the ring of scalar valued Siegel modular forms of degree two, level 1 and even weights. In this paper, we prove the determinant of a basis of the module of vector valued Siegel modular forms ⊕ k≡ mod 2 Adetk ⊗Sym(j)(Γ2) over A (Γ2) is equal to a power of the cusp form of degree two and weight 35 up to a constant. Here j = 4, 6 and = 0, 1. The main result in this paper was conjectured by Ibukiyama [7].
设A(Γ2)表示二阶、一级、偶权的标量值Siegel模形式环。在本文中，我们证明了向量值的Siegel模形式的模的基的行列式⊕k≡mod2 adek⊗Sym(j)(Γ2) / a (Γ2)等于2次和权值为35的尖形的幂直至一个常数。这里j =(4,6)和= (0,1)本文的主要结果是由Ibukiyama b[7]推测出来的。
{"title":"On the computation of the determinant of vector-valued Siegel modular forms","authors":"S. Takemori","doi":"10.1112/S146115701400028X","DOIUrl":"https://doi.org/10.1112/S146115701400028X","url":null,"abstract":"Let A(Γ2) denote the ring of scalar valued Siegel modular forms of degree two, level 1 and even weights. In this paper, we prove the determinant of a basis of the module of vector valued Siegel modular forms ⊕ k≡ mod 2 Adetk ⊗Sym(j)(Γ2) over A (Γ2) is equal to a power of the cusp form of degree two and weight 35 up to a constant. Here j = 4, 6 and = 0, 1. The main result in this paper was conjectured by Ibukiyama [7].","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"247-256"},"PeriodicalIF":0.0,"publicationDate":"2014-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S146115701400028X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411520","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}
Pub Date : 2014-07-29DOI: 10.1112/S1461157016000413
Beth Malmskog, C. Rasmussen
Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over $mathbb{Q}$ with good reduction away from 3, up to $mathbb{Q}$ -isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined and an application to a question of Ihara is discussed.
受N. P. Smart方法的启发，我们描述了一种算法来确定$mathbb{Q}$上的所有Picard曲线，这些曲线从3到$mathbb{Q}$ -同构都很好。建立了此类曲线的同构类与具有有理线性因子的某些五次二元形式的对应关系。给出了一组完整的积分模型，并讨论了其在Ihara问题上的应用。
{"title":"Picard curves over Q with good reduction away from 3","authors":"Beth Malmskog, C. Rasmussen","doi":"10.1112/S1461157016000413","DOIUrl":"https://doi.org/10.1112/S1461157016000413","url":null,"abstract":"Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over $mathbb{Q}$\u0000 with good reduction away from 3, up to $mathbb{Q}$\u0000 -isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined and an application to a question of Ihara is discussed.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"19 1","pages":"382-408"},"PeriodicalIF":0.0,"publicationDate":"2014-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157016000413","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412337","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}
Pub Date : 2014-05-08DOI: 10.1112/S146115701500008X
Brian P. Corr, Tomasz Popiel, C. Praeger
Efficient methods for computing with matrices over finite fields often involve randomised algorithms, where matrices with a certain property are sought via repeated random selection. Complexity analyses for these algorithms require knowledge of the proportion of relevant matrices in the ambient group or algebra. We introduce a method for estimating proportions of families N of elements in the algebra of all d×d matrices over a field of order q, where membership of a matrix in N depends only on its ‘invertible part’. The method is based on estimating proportions of certain subsets of GL(d,q) depending on N, so that existing estimation techniques for nonsingular matrices can be leveraged to deal with families containing singular matrices. As an application we investigate primary cyclic matrices, which are used in the Holt–Rees MEAT-AXE algorithm for testing irreducibility of matrix algebras.
{"title":"Nilpotent-independent sets and estimation in matrix algebras","authors":"Brian P. Corr, Tomasz Popiel, C. Praeger","doi":"10.1112/S146115701500008X","DOIUrl":"https://doi.org/10.1112/S146115701500008X","url":null,"abstract":"Efficient methods for computing with matrices over finite fields often involve randomised algorithms, where matrices with a certain property are sought via repeated random selection. Complexity analyses for these algorithms require knowledge of the proportion of relevant matrices in the ambient group or algebra. We introduce a method for estimating proportions of families N of elements in the algebra of all d×d matrices over a field of order q, where membership of a matrix in N depends only on its ‘invertible part’. The method is based on estimating proportions of certain subsets of GL(d,q) depending on N, so that existing estimation techniques for nonsingular matrices can be leveraged to deal with families containing singular matrices. As an application we investigate primary cyclic matrices, which are used in the Holt–Rees MEAT-AXE algorithm for testing irreducibility of matrix algebras.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"18 1","pages":"404-418"},"PeriodicalIF":0.0,"publicationDate":"2014-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S146115701500008X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411782","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}
Pub Date : 2014-04-01DOI: 10.1112/S1461157014000424
John W. Jones, D. Roberts
We describe an online database of number fields which accompanies this paper. The database centers on complete lists of number fields with prescribed invariants. Our description here focuses on summarizing tables and connections to theoretical issues of current interest.
{"title":"A database of number fields","authors":"John W. Jones, D. Roberts","doi":"10.1112/S1461157014000424","DOIUrl":"https://doi.org/10.1112/S1461157014000424","url":null,"abstract":"We describe an online database of number fields which accompanies this paper. The database centers on complete lists of number fields with prescribed invariants. Our description here focuses on summarizing tables and connections to theoretical issues of current interest.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"7 1","pages":"595-618"},"PeriodicalIF":0.0,"publicationDate":"2014-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000424","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411550","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}
Pub Date : 2014-04-01DOI: 10.1112/S1461157013000260
A. Jooste, K. Jordaan
The zeros of certain different sequences of orthogonal polynomials interlace in a well-defined way. The study of this phenomenon and the conditions under which it holds lead to a set of points that can be applied as bounds for the extreme zeros of the polynomials. We consider different sequences of the discrete orthogonal Meixner and Kravchuk polynomials and use mixed three-term recurrence relations, satisfied by the polynomials under consideration, to identify bounds for the extreme zeros of Meixner and Kravchuk polynomials.
{"title":"Bounds for zeros of Meixner and Kravchuk polynomials","authors":"A. Jooste, K. Jordaan","doi":"10.1112/S1461157013000260","DOIUrl":"https://doi.org/10.1112/S1461157013000260","url":null,"abstract":"The zeros of certain different sequences of orthogonal polynomials interlace in a well-defined way. The study of this phenomenon and the conditions under which it holds lead to a set of points that can be applied as bounds for the extreme zeros of the polynomials. We consider different sequences of the discrete orthogonal Meixner and Kravchuk polynomials and use mixed three-term recurrence relations, satisfied by the polynomials under consideration, to identify bounds for the extreme zeros of Meixner and Kravchuk polynomials.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"47-57"},"PeriodicalIF":0.0,"publicationDate":"2014-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157013000260","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411272","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}
Pub Date : 2014-03-27DOI: 10.1112/S1461157014000461
R. Broker, Everett W. Howe, K. Lauter, P. Stevenhagen
We study the problem of efficiently constructing a curve C of genus 2 over a finite field F for which either the curve C itself or its Jacobian has a prescribed number N of F-rational points. In the case of the Jacobian, we show that any `CM-construction' to produce the required genus-2 curves necessarily takes time exponential in the size of its input. On the other hand, we provide an algorithm for producing a genus-2 curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus-2 curve having exactly 10^2014 + 9703 (prime) points, and two genus-2 curves each having exactly 10^2013 points. In an appendix we provide a complete parametrization, over an arbitrary base field k of characteristic neither 2 nor 3, of the family of genus-2 curves over k that have k-rational degree-3 maps to elliptic curves, including formulas for the genus-2 curves, the associated elliptic curves, and the degree-3 maps.
{"title":"Genus-2 curves and Jacobians with a given number of points","authors":"R. Broker, Everett W. Howe, K. Lauter, P. Stevenhagen","doi":"10.1112/S1461157014000461","DOIUrl":"https://doi.org/10.1112/S1461157014000461","url":null,"abstract":"We study the problem of efficiently constructing a curve C of genus 2 over a finite field F for which either the curve C itself or its Jacobian has a prescribed number N of F-rational points. \u0000In the case of the Jacobian, we show that any `CM-construction' to produce the required genus-2 curves necessarily takes time exponential in the size of its input. \u0000On the other hand, we provide an algorithm for producing a genus-2 curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus-2 curve having exactly 10^2014 + 9703 (prime) points, and two genus-2 curves each having exactly 10^2013 points. \u0000In an appendix we provide a complete parametrization, over an arbitrary base field k of characteristic neither 2 nor 3, of the family of genus-2 curves over k that have k-rational degree-3 maps to elliptic curves, including formulas for the genus-2 curves, the associated elliptic curves, and the degree-3 maps.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"32 1","pages":"170-197"},"PeriodicalIF":0.0,"publicationDate":"2014-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000461","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411619","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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