pinecone/core-2020-05-10-deduplication huggingface dataset:
An Algorithm for Computing a New Normal Form for Dynamical Systems

AbstractWe propose in this paper a new normal form for dynamical systems or vector fields which improves the classical normal forms in the sense that it is a further reduction of the classical normal forms. We give an algorithm for an effective computation of these normal forms. Our approach is rational in the sense that if the coefficients of the system are in a field K(which, in practice, is Q, R), so is the normal form and all computations are done inK . As a particular case, if the matrix of the linear part is a companion matrix then we reduce the dynamical system to a single differential equation. Our method is applicable for both the nilpotent and the non-nilpotent cases. We have implemented our algorithm in Maple V and obtained many examples of the further reduced normal forms up to some finite order
Efficient Decomposition of Associative Algebras over Finite Fields

AbstractWe present new, efficient algorithms for some fundamental computations with finite-dimensional (but not necessarily commutative) associative algebras over finite fields. For a semisimple algebra A we show how to compute a complete Wedderburn decomposition of A as a direct sum of simple algebras, an isomorphism between each simple component and a full matrix algebra, and a basis for the centre of A. If A is given by a generating set of matrices inFm×m, then our algorithm requires aboutO (m3) operations inF, in addition to the cost of factoring a polynomial inF[ x ] of degree O(m), and the cost of generating a small number of random elements from A. We also show how to compute a complete set of orthogonal primitive idempotents in any associative algebra over a finite field in this same time
Algorithms for Exponentiation in Finite Fields

AbstractGauss periods yield (self-dual) normal bases in finite fields, and these normal bases can be used to implement arithmetic efficiently. It is shown that for a small prime power q and infinitely many integersn , multiplication in a normal basis of Fqn over Fq can be computed with O(n logn loglog n), division with O(n log2n loglog n) operations in Fq, and exponentiation of an arbitrary element in Fqn withO (n2loglog n) operations in Fq. We also prove that using a polynomial basis exponentiation in F 2 n can be done with the same number of operations in F 2 for all n. The previous best estimates were O(n2) for multiplication in a normal basis, andO (n2log n loglog n) for exponentiation in a polynomial basis
Differential Equations of Order Two with One Singular Point

AbstractThe goal of this paper is to describe the set of polynomials r∈C [ x ] such that the linear differential equationy′′=ry has Liouvillian solutions, whereC is an algebraically closed field of characteristic 0. It is known that the differential equation has Liouvillian solutions only if the degree of r is even. Using differential Galois theory we show that the set of such polynomials of degree 2 n can be represented by a countable set of algebraic varieties of dimension n+ 1. Some properties of those algebraic varieties are proved
Two Generator Subgroups ofSL(2,C) and the Hypergeometric, Riemann, and Lamé Equations

AbstractFor the purposes of constructing explicit solutions to second-order linear homogeneous differential equations on the Riemann sphere the Kovacic algorithm partitions the subgroups ofSL (2,C) into four classes and initially determines which class contains the differential Galois group of the input equation. We prove in the case of the hypergeometric and Riemann equations that the relevant class can be determined directly from the coefficients by elementary calculation. We also treat the (non-algebraic form of the) Lamé equation, to which the Kovacic algorithm is not directly applicable. In that instance we combine the Kovacic results with ours to produce an algorithm for determining the class of the associated group. From the group-theoretic viewpoint the problem solved herein is the following: given arbitrary S,T∈SL(2,C), determine which class contains the group 〈S, T〉 generated by S and T
The Schmidt–Kolchin Conjecture

AbstractIn his study of papers by Osgood and Kolchin on rational approximations of algebraic functions, Schmidt observed and proved a theorem which states that the dimension of the vector space of differentially homogeneous differential polynomials is at least 2d. We will show that the dimension is actually equal to 2dand that a basis can be effectively computed. The proof proceeds by developing a term ordering of differential monomials in several variables which ranks monomials relative to a given monomial. This new ordering will be useful in polynomial algorithms in general. The ordering has been implemented in Mathematica
The Differential Ideal [ P ] : M∞

AbstractLet F be a finite subset of the differential polynomial algebra k { y 1,⋯ , yn } . In order to determine membership in the radical differential ideal { F }, one is led to express { F } as the intersection of differential ideals of the form [ P ] : M∞ for suitable subsets P and M of k { y 1,⋯ , yn } . One criterion for “suitability" is that the ideal [ P ] : M∞ should be radical; another is that the question of membership in this ideal should be reducible to the question of membership in its algebraic counterpart (P) : M∞. Lazard’s lemma provides sufficient conditions for the first criterion to hold; Rosenfeld’s lemma provides sufficient conditions for the second criterion to hold. In this paper, we prove substantially strengthened versions of both of these results, and apply them to Mansfield’s algorithms (Mansfield, 1993)for solving systems of PDEs
Computation of Hilbert Polynomials in Two Variables

AbstractLet R be a ring of polynomials in m+n indeterminatesx1 , . . . , xm, y1, . . . ,yn over a field K and let M be a finitely generated R -module. Furthermore, let (Rrs)r, s∈Nbe the natural double filtration of the ring R and let (Mrs)r,s∈Nbe the corresponding double filtration of the module M associated with the given system of generators. We introduce a special type of reduction in a free R -module and develop the appropriate technique of characteristic sets that allows us to prove the existence and find methods and algorithms of computation of a numerical polynomial in two variables φ(t1, t2) such thatφ (r, s) =dimKMrsfor all sufficienly large r, s∈N. The results obtained are applied in differential algebra where the classical theorems on differential dimension polynomials and methods of computation of such polynomials are generalized to the case of differential structures with two basic sets of derivation operators
Multibasic and Mixed Hypergeometric Gosper-Type Algorithms

AbstractGosper’s summation algorithm finds a hypergeometric closed form of an indefinite sum of hypergeometric terms, if such a closed form exists. We extend his algorithm to the case when the terms are simultaneously hypergeometric and multibasic hypergeometric. We also provide algorithms for finding polynomial as well as hypergeometric solutions of recurrences in the mixed case. We do not require the bases to be transcendental, but only that q1k1⋯qmkm≠1 unless k1=⋯=km= 0. Finally, we generalize the concept of greatest factorial factorization to the mixed hypergeometric case
New Structure Theorem for Subresultants

AbstractWe give a new structure theorem for subresultants precising their gap structure and derive from it a new algorithm for computing them. If d is a bound on the degrees and τ a bound on the bit size of the minors extracted from Sylvester matrix, our algorithm has O(d2) arithmetic operations and size of intermediate computations 2 τ. The key idea is to precise the relations between the successive Sylvester matrix ofA and B on one hand and of A and XB on the other hand, using the notion of G-remainder that we introduce. We also compare our new algorithm with another algorithm with the same characteristics which appeared inDucos (2000)
Solving Systems of Strict Polynomial Inequalities

AbstractWe present an algorithm for finding an explicit description of solution sets of systems of strict polynomial inequalities, correct up to lower dimensional algebraic sets. Such a description is sufficient for many practical purposes, such as volume integration, graphical representation of solution sets, or global optimization over open sets given by polynomial inequality constraints. Our algorithm is based on the cylindrical algebraic decomposition algorithm. It uses a simplified projection operator, and constructs only rational sample points
Algorithmic Computation of de Rham Cohomology of Complements of Complex Affine Varieties

AbstractLet X=Cn. In this paper we present an algorithm that computes the de Rham cohomology groups HdRi(U,C ) where U is the complement of an arbitrary Zariski-closed set Y in X. Our algorithm is a merger of the algorithm given inOaku and Takayama (1999), who considered the case where Y is a hypersurface, and our methods from Walther (1999) for the computation of local cohomology. We further extend the algorithm to compute de Rham cohomology groups with supports HdR, Zi(U,C ) where again U is an arbitrary Zariski-open subset of X and Z is an arbitrary Zariski-closed subset of U. Our main tool is a generalization of the restriction process from Oaku and Takayama (in press) to complexes of modules over the Weyl algebra. The restriction rests on an existence theorem onVd -strict resolutions of complexes that we prove by means of an explicit construction via Cartan–Eilenberg resolutions. All presented algorithms are based on Gröbner basis computations in the Weyl algebra and the examples are carried out using the computer system Kan by Takayama (1999)
Applications of Some Properties of the Canonical Module in Computational Projective Algebraic Geometry

AbstractWe here develop some new algorithms for computing several invariants attached to a projective scheme (dimension, Hilbert polynomial, unmixed part,... ) that are based on liaison theory and therefore connected to properties of the canonical module. The main features of these algorithms are their simplicity and the fact that their complexity is controlled by the complexity of the output, when it exceeds a linear function on the degrees of the input. We also give bounds for the Castelnuovo-Mumford regularity (which controls the complexity of the output) in low dimension, and give a reasonable algorithm to check smoothness of the unmixed part (one case where good bounds are known for complexity)
On Rational Geometry of Conic Sections

AbstractSimple geometric objects and transformations appear in representations and algorithms of geometric facilities in computer applications such as modelling, robotics, or graphics. Usually, these applications only support objects and transformations fully describable by rational parameters, and a computer display of points of the objects at least implicitly requires points with rational coordinates. In this setting we investigate some basic questions of the geometry of rational conic sections, when the geometry is defined by the group of rational projective transformations, the group of rational affine transformations, or the group of rational rigid transformations. Some results follow classical results, while others turn out to be quite different. In particular, we obtain a complete classification scheme for nondegenerate rational conics for rational affine geometry and a constructive method for production of a minimal set of representatives of all equivalence classes
Invariants, Patterns and Weights for Ordering Terms

AbstractWe prove that any simplification order over arbitrary terms is an extension of an order by weight, by considering a related monadic term algebra called the spine. We show that any total ground-stable simplification order on the spine lifts to an order on the full term algebra. Conversely, under certain restrictions, a simplification ordering on the term algebra defines a weight function on the spine, which in turn can be lifted to a weight order on the original ground terms which contains the original order. We investigate the Knuth–Bendix and polynomial orders in this light. We provide a general framework for ordering terms by counting embedded patterns, which gives rise to many new orderings. We examine the recursive path order in this context
Toward Symbolic Integration of Elliptic Integrals

AbstractA method is proposed by which elliptic integrals can be integrated symbolically without information regarding limits of integration and branch points of the integrand that is required in integral tables using Legendre’s integrals. However, it is assumed that when all polynomials in the integrand have been factored symbolically into linear factors, the exponents of all distinct linear factors are known. The recurrence relations are one-parameter relations, all formulas are given explicitly, and the integral is eventually expressed in terms of canonicalR-functions, with no increase in their number if neither limit of integration is a branch point of the integrand. It is the use ofR-functions rather than Legendre’s integrals that makes it possible to carry out the whole process symbolically. If (possibly complex) numerical values of the symbols are known, there are published algorithms for numerical computation of theR-functions
Decomposition of Polynomials with Respect to the Cyclic Group of Orderm

AbstractGiven a polynomial solution of a differential equation, its m -ary decomposition, i.e. its decomposition as a sum of m polynomials P[ j ](x) =∑kαj,kxλj, kcontaining only exponentsλj, k with λj,k+1−λj,k=m, is considered. A general algorithm is proposed in order to build holonomic equations for the m -ary parts P[ j ](x) starting from the initial one, which, in addition, provides a factorized form of them. Moreover, these differential equations are used to compute expansions of the m -ary parts of a given polynomial in terms of classical orthogonal polynomials. As illustration, binary and ternary decomposition of these classical families are worked out in detail
Partial Differential Equations and Bivariate Orthogonal Polynomials

AbstractIn 1929, S. Bochner identified the families of polynomials which are eigenfunctions of a second-order linear differential operator. What is the appropriate generalization of this result to bivariate polynomials? One approach, due to Krall and Sheffer in 1967 and pursued by others, is to determine which linear partial differential operators have orthogonal polynomial solutions with all the polynomials in the family of the same degree sharing the same eigenvalue. In fact, such an operator only determines a multi-dimensional eigenspace associated with each eigenvalue; it does not determine the individual polynomials, even up to a multiplicative constant. In contrast, our approach is to seek pairs of linear differential operators which have joint eigenfunctions that comprise a family of bivariate orthogonal polynomials. This approach entails the addition of some “normalizing" or “regularity" conditions which allow determination of a unique family of orthogonal polynomials. In this article we formulate and solve such a problem and show with the help of Mathematica that the only solutions are disk polynomials. Applications are given to product formulas and hypergroup measure algebras
Program Development Schemata as Derived Rules

AbstractWe show how the formalization and application of schemata for program development can be reduced to the formalization and application of derived rules of inference. We formalize and derive schemata as rules in theories that axiomatize program data and programs themselves. We reduce schema-based program development to ordinary theorem proving, where higher-order unification is used to apply rules. Conceptually, our formalization is simple and unifies divergent views of schemata, program synthesis, and program transformation. Practically, our formalization yields a simple methodology for carrying out development using existing logical frameworks; we illustrate this in the domain of logic program synthesis and transformation using the Isabelle logical framework
A Calculus for Schemas in Z

AbstractThe popularity and flexibility of the Z notation can largely be attributed to its notion of schemas. We describe these schemas and illustrate their various common uses in Z. We also present a collection of logical laws for manipulating these schemas. These laws are capable of supporting reasoning about the Z schema calculus in its full generality. This is demonstrated by presenting some theorems about the removability of schemas from Z specifications, together with outline proofs. We survey briefly models against which this logical system may be proven sound, and other related logics for Z
An Abstract Formalization of Correct Schemas for Program Synthesis

AbstractProgram schemas should capture not only structured program design principles, but also domain knowledge, both of which are of crucial importance for hierarchical program synthesis. However, most researchers represent schemas as purely syntactic constructs, which can provide only a program template, but not the domain knowledge. In this paper, we take a semantic approach and show that a schema S consists of a syntactic part, namely a template T, and a semantic part. Template T is formalized as an open (first-order) logic program in the context of the problem domain, characterized as a first-order axiomatization, called a specification frameworkF, which is the semantic part. F endows the schema S with a formal semantics, and enables us to define and reason about its correctness. Naturally, correct schemas can be used to guide the synthesis of correct programs
Proofs as Schemas and Their Heuristic Use

AbstractAutomated theorem proving essentially amounts to solving search problems. Despite significant progress in recent years theorem provers still have many shortcomings. The use of machine-learning techniques such as schemas is acknowledged as promising, but difficult to apply in the area of theorem proving. We propose a simple form of schemas, and to make use of a schema heuristically by integrating it with a search-guiding heuristic. Experiments have demonstrated that the approach allows a theorem prover to find proofs significantly faster and to prove hard problems that were out of reach before
A Formula for the General Solution of a Constant-coefficient Difference Equation

AbstractWe give a formula for the general solution of a d th-order linear difference equation with constant coefficients in terms of one of the solutions of its associated homogeneous equation. The formula neither uses the roots of the characteristic equation nor their multiplicities. It can be readily generalized to the case where the domain of the difference equation is the real numbers, and the initial values are given by a function defined on the interval [0,d ). In both cases, we express the general solution of the difference equation in terms of a single solution of its associated homogeneous equation at integer arguments
Quantifier Elimination for Trigonometric Polynomials by Cylindrical Trigonometric Decomposition

AbstractGiven a formula Φ in r variables, some of them quantified and/or occurring as arguments in trigonometric functions, we consider in this paper the problem of finding a quantifier-free formula equivalent toΦ . We present an algorithm that first computes a decomposition of the space so that the polynomials occurring inΦ are sign-invariant over each cell of this decomposition. Then the cells over which Φ is true are collected: their description gives a quantifier-free formula equivalent to Φ. The algorithm is an adaption of a well-known algorithm for the same problem with polynomial functions
Deciding Hopf Bifurcations by Quantifier Elimination in a Software-component Architecture

AbstractIn this paper we give a semi-algebraic description of Hopf bifurcation fixed points for a given parameterized polynomial vector field. The description is carried out by use of the Hurwitz determinants, and produces a first-order formula which is transformed into a quantifier-free formula by the use of usual-quantifier elimination algorithms. We apply techniques from the theory of sub-resultant sequences and of Gröbner bases to come up with efficient reductions, which lead to quantifier elimination questions that can often be handled by existing quantifier elimination packages. We could implement the algorithms for the conditions on Hopf bifurcations by combining the computer algebra system Maple with packages for quantifier elimination using a Java-based component architecture recently developed by the second author. In addition to some textbook examples we applied our software system to an example discussed in a recent research paper
Minimally Generating Ideals of Rational Parametric Curves in Polynomial Time

AbstractWe present an algorithm for computing a minimal set of generators for the ideal of a rational parametric projective curve in polynomial time. The method exploits the availability of polynomial algorithms for the computation of minimal generators of an ideal of points and is an alternative to the existing Gröbner bases techniques for the implicitization of curves. The termination criterion is based on the Castelnuovo–Mumford regularity of a curve. The described computation also yields the Hilbert function and, hence, the Hilbert polynomial and the Poincaré series of the curves. Moreover, it can be applied to unions of rational curves. We have compared the implementation of our algorithm with the Hilbert driven elimination algorithm included in CoCoA 3.6 and Singular 1.2, obtaining, in general, significant improvements in timings
Solving Index Form Equations in Fields of Degree 9 with Cubic Subfields

AbstractWe describe an efficient algorithm for solving index form equations in number fields of degree 9 which are composites of cubic fields with coprime discriminants. We develop the algorithm in detail for the case of complex cubic fields, but the main steps of the procedure are also applicable for other cases. Our most important tool is the main theorem of a recent paper of Gaál (1998a). In view of this result the index form equation in the ninth degree field implies relative index form equations over the subfields. In our case these equations are cubic relative Thue equations over cubic fields. The main purpose of the paper is to show that this approach is much more efficient than the direct method, which consists of reducing the index form equation to unit equations over the normal closure of the original field. At the end of the paper we describe our computational experience. Many ideas of the paper can be applied to develop fast algorithms for solving index form equations in other types of higher degree fields which are composites of subfields
Theorem Proving Techniques for View Deletion in Databases

AbstractIn this paper, we show how techniques from first-order theorem proving can be used for efficient deductive database updates. The key idea is to transform the given database, together with the update request, into a (disjunctive) logic program and to apply the hyper-tableau calculus (Baumgartner et al. 1996) to solve the original update problem. The resulting algorithm has the following properties: it works goal-directed (i.e. the search is driven by the update request), it is rational in the sense that it satisfies certain rationality postulates stemming from philosophical works on belief dynamics, and, unlike comparable approaches, it is of polynomial space complexity. To obtain soundness and completeness results, the hyper-tableau calculus is slightly modified for minimal model reasoning. Besides a direct proof we give an alternate proof which gives insights into the relation to previous approaches. As a by-product we thereby derive a soundness and completeness result of hyper-tableaux for computing minimal abductive explanations