pinecone/core-2020-05-10-deduplication huggingface dataset:
Computing Orbits of Minimal Parabolic k -subgroups Acting on Symmetric k -varieties

AbstractIn this paper we present an algorithm to compute the orbits of a minimal parabolic k -subgroup acting on a symmetric k -variety and most of the combinatorial structure of the orbit decomposition. This algorithm can be implemented in LiE, GAP4, Magma, Maple or in a separate program. These orbits are essential in the study of symmetric k -varieties and their representations. In a similar way to the special case of a Borel subgroup acting on the symmetric variety, (see A. G. Helminck. Computing B -orbits on G/H. J. Symb. Comput.,21 , 169–209, 1996.) one can use the associated twisted involutions in the restricted Weyl group to describe these orbits (see A. G. Helminck and S. P. Wang. On rationality properties of involutions of reductive groups. Adv. Math., 99, 26–96, 1993). However, the orbit structure in this case is much more complicated than the special case of orbits of a Borel subgroup. We will first modify the characterization of the orbits of minimal parabolic k -subgroups acting on the symmetric k -varieties given in Helminck and Wang (1993), to illuminate the similarity to the one for orbits of a Borel subgroup acting on a symmetric variety in Helminck (1996). Using this characterization we show how the algorithm in Helminck (1996) can be adjusted and extended to compute these twisted involutions as well
Decomposition Plans for Geometric Constraint Systems, Part I: Performance Measures for CAD

AbstractA central issue in dealing with geometric constraint systems for CAD/CAM/CAE is the generation of an optimal decomposition plan that not only aids efficient solution, but also captures design intent and supports conceptual design. Though complex, this issue has evolved and crystallized over the past few years, permitting us to take the next important step: in this paper, we formalize, motivate and explain the decomposition–recombination (DR)-planning problem as well as several performance measures by which DR-planning algorithms can be analyzed and compared. These measures include: generality, validity, completeness, Church–Rosser property, complexity, best- and worst-choice approximation factors, (strict) solvability preservation, ability to deal with underconstrained systems, and ability to incorporate conceptual design decompositions specified by the designer. The problem and several of the performance measures are formally defined here for the first time—they closely reflect specific requirements of CAD/CAM applications. The clear formulation of the problem and performance measures allow us to precisely analyze and compare existing DR-planners that use two well-known types of decomposition methods: SR (constraint shape recognition) and MM (generalized maximum matching) on constraint graphs. This analysis additionally serves to illustrate and provide intuitive substance to the newly formalized measures. In Part II of this article, we use the new performance measures to guide the development of a new DR-planning algorithm which excels with respect to these performance measures
Decomposition Plans for Geometric Constraint Problems, Part II: New Algorithms

AbstractWe systematically design two new decomposition–recombination (DR) planners, geared to perform well with respect to several performance measures. The DR-planning problem and the performance measures were formally defined in Part I of this paper to closely reflect specific requirements of CAD/CAM applications. As expected, in analysis and comparison based on all of these performance measures, one of the new DR-planners, the modified frontier algorithm (MFA), represents a significant improvement over existing planners based on SR (constraint shape recognition) and MM (maximum matching) that were analyzed in Part I. We also present salient heuristics and data structures used in the implementation of MFA
(1 +i) - ary GCD Computation inZ[ i ]as an Analogue to the Binary GCD Algorithm

AbstractWe present a novel algorithm for GCD computation over the ring of Gaussian integersZ[i ], that is similar to the binary GCD algorithm forZ, in which powers of 1 +i are extracted. Our algorithm has a running time of O(n2) bit operations with a small constant hidden in the O -notation if the two input numbers have a length ofO (n) bits. This is noticeably faster than a least remainder version of the Euclidean algorithm inZ[ i ] or the Caviness–Collins GCD algorithm that both have a running time ofO (n·μ(n)) bit operations, whereμ (n) denotes a good upper bound for the multiplication time of n -bit integers. Our new GCD algorithm is also faster by a constant factor than a Lehmer-type GCD algorithm (i.e. in every Euclidean step a small remainder is calculated, but this remainder need not to be a least remainder) inZ[ i ] which achieves a running time of O(n2) bit operations
Detecting Algebraic Curves in Bad Position

AbstractTwo affine algebraic curves over the complex numbers are in bad position if they have at least two distinct common points on some vertical line, i.e. for some α∈C there are distinctβ1 , β2∈C such that (α,β1 ) and (α, β2) are common points of the curves. We describe a method for detecting this situation and use it to develop an algorithm for finding the common points of two algebraic curves along with their multiplicities. We also give a worst case runtime analysis for the two algorithms and include a brief comparison of the second one with an approach using Gröbner bases
Sylvester–Habicht Sequences and Fast Cauchy Index Computation

AbstractIn this paper we show how Schönhage’s strategy for computing continued fractions (Schönhage 1971) can be combined with the theory of sub-resultants (Habicht, 1948; Collins, 1967; Brown, 1971; Brown and Traub, 1971; Loos, 1982; Gonzalez et al., 1990, 1994; Ducos, 1996; Ho and Yap, 1996; Lazard, 1998; Quitté, 1998) in order to compute the Cauchy index of a rational function or the signature of a non-singular Hankel matrix in a fast and also storage efficient way. Over the integers our algorithms have bit complexityO (M(d, σ) ·log(d)) withσ=O(dτ) where M(d,σ ) =O(dσ·log(dσ) ·loglog(dσ)) is Schönhage’s bound for multiplication of integer polynomials of degrees bounded by d and bit size bounded by σ in the multi-tape Turing machine model (Schönhage, 1982). Thus our bound isO(d2τ·log(dτ) ·loglog(dτ) ·log(d)).As a byproduct of the necessary analysis we obtain a refinement of the Sub-resultant Theorem. We present a new exact divisibility for sub-resultants in the defective case which extends the formulæ for the non-defective situation in a natural way. We also prove that the size of coefficients in the ordinary remainder sequence is quadratic in d
Implicitization of a General Union of Parametric Varieties

AbstractWe reconduct the computation of the equations defining a union V of parametric varieties, up to a given degree d, to the computation of the equations, of degree ≤d, vanishing on a finite set of points of V. If V is general this gives a new algorithm for implicitizing V. We have compared the implementation of this algorithm with the Hilbert-driven elimination algorithm included in the software packages CoCoA 3.7 and Singular 1.2, obtaining significant time savings. Moreover, the implementation of the algorithm leads to two intriguing conjectures on the natural number of generators of the union of general parametric curves or surfaces
The Control Layer in Open Mechanized Reasoning Systems: Annotations and Tactics

AbstractWe are interested in developing a methodology for integrating mechanized reasoning systems such as Theorem Provers, Computer Algebra Systems, and Model Checkers. Our approach is to provide a framework for specifying mechanized reasoning systems and to use specifications as a starting point for integration. We build on the work presented by Giunchigliaet al. (1994) which introduces the notion of Open Mechanized Reasoning Systems (OMRS) as a specification framework for integrating reasoning systems. An OMRS specification consists of three components: the logic component, the control component, and the interaction component. In this paper we focus on the control level. We propose to specify the control component by first adding control knowledge to the data structures representing the logic by means of annotations and then by specifying proof strategies via tactics. To show the adequacy of the approach we present and discuss a structured specification of constraint contextual rewriting as a set of cooperating specialized reasoning modules
Automatic Derivation of the Irrationality of e

AbstractAs part of a project on automatic generation of proofs involving both logic and computation, we have automatically generated a proof of the irrationality of e. The proof involves inequalities, bounds on infinite series, type distinctions (between real numbers and natural numbers), a subproof by mathematical induction, and significant mathematical steps, including correct simplification of expressions involving factorials and summing an infinite geometrical series. Metavariables are instantiated by inference rules embodying mathematical knowledge, rather than only by unification. The proof is generated completely automatically, without any interactive component
On the Role of OpenMath in Interactive Mathematical Documents

AbstractThe standard OpenMath is an enabling technology for creating an integrated computer environment in which software packages for computer algebra and for proof checking can be combined. Here we demonstrate how OpenMath can be employed for generating interactive mathematical documents containing primality proofs. Our case study takes place within a browser; once a prime number is specified, a document appears summarizing the proof in a number of assertions. By clicking an assertion regarding the truth of an arithmetic equality, a computer algebra calculation is invoked verifying the equality. By clicking an assertion regarding a specific mathematical lemma called Pocklington’s Criterion, a verification of the corresponding formal proof is carried out by a proof checker. Moreover, the whole document is structured in such a way that it can be easily translated to a formal proof object. OpenMath supports the interaction between the document as it appears in the browser and the mathematical software packages. This paper begins with an introduction to OpenMath and a brief comparison with MathML
MBase: Representing Knowledge and Context for the Integration of Mathematical Software Systems

AbstractIn this article we describe the data model of the MB ase system, a web-based, distributed mathematical knowledge base. This system is a mathematical service in MathWeb that offers a universal repository of formalized mathematics where the formal representation allows semantics-based retrieval of distributed mathematical facts. We classify the data necessary to represent mathematical knowledge and analyze its structure. For the logical formulation of mathematical concepts, we propose a methodology for developing representation formalisms for mathematical knowledge bases. We propose to concretely equip knowledge bases with a hierarchy of logical systems that are linked by logic morphisms. These mappings relativize formulae and proofs and thus support translation of the knowledge to the various formats currently in use in deduction systems. On the other hand they define higher language features from simpler ones and ultimately serve as a means to found the whole knowledge base in axiomatic set theory. The viability of this approach is proven by developing a sorted record-λ -calculus with dependent sorts and labeled abstraction that is well-suited both for formalizing mathematical practice and supporting efficient inference services. This “mathematical vernacular" is an extension of a sortedλ -calculus by records, dependent record sorts and selection sorts
A General Framework to Build Contextual Cover Set Induction Provers

AbstractCover set induction is known as a proof method that keeps the advantages of explicit induction and proof by consistency. Most implicit induction proof procedures are defined in a cover set induction framework. Contextual cover set (CCS) is a new concept that fully characterizes explicit induction schemes, such as the cover sets, and many simplification techniques as those specific to the “proof by consistency" approach. Firstly, we present an abstract inference system uniformly defined in terms of contextual cover sets as our general framework to build implicit induction provers. Then, we show that it generalizes existing cover set induction procedures. This paper also contributes to the general problem of assembling reasoning systems in a sound manner. Elementary CCSs are generated by reasoning modules that implement various simplification techniques defined for a large class of deduction mechanisms such as rewriting, conditional rewriting and resolution-based methods for clauses. We present a generic and sound integration schema of reasoning modules inside our procedure together with a simple methodology for improvements and incremental sound extensions of the concrete proof procedures. As a case study, the inference system of the SPIKE theorem prover has been shown to be an instance of the abstract inference system integrating reasoning modules based on rewriting techniques defined for conditional theories. Our framework allows formodular and incremental sound extensions of SPIKE when new reasoning techniques are proposed. An extension of the prover, incorporating inductive semantic subsumption techniques, has proved the correctness of the MJRTY algorithm by performing a combination of arithmetic and inductive reasoning
Fast Evaluation of Holonomic Functions Near and in Regular Singularities

AbstractA holonomic function is an analytic function, which satisfies a linear differential equationLf= 0 with polynomial coefficients. In particular, the elementary functions exp,log,sin, etc., and many special functions such as erf, Si, Bessel functions, etc., are holonomic functions. In a previous paper, we have given an asymptotically fast algorithm to evaluate a holonomic function f at a non-singular point z′ on the Riemann surface of f, up to any number of decimal digits while estimating the error. However, this algorithm becomes inefficient, when z′ approaches a singularity of f. In this paper, we obtain efficient algorithms for the evaluation of holonomic functions near and in singular points where the differential operator L is regular (or, slightly more generally, where L is quasi-regular—a concept to be introduced below)
Predicting Zero Coefficients in Formal Power Series Computations

AbstractWe consider the problem of predicting long sequences of zero coefficients in a power series obtained by multiplication, division or reversion (where all coefficients are integers). We describe efficient randomized algorithms whose probability of error can be controlled by the user. A runtime analysis is given and some experimental results are also presented that compare our algorithms with classical ones for formal power series computations. We envisage the algorithms given here as being of greatest use in situations where several processors are available so that the possibility of a long sequence of zeros can be tested in parallel to the normal computation of coefficients
Irreducible Decomposition of Curves

AbstractIn this paper, we propose a fast semi-numerical algorithm for computing all irreducible branches of a curve in Cτdefined by polynomials with rational coefficients, we also treat the case of a non-reduced curve. Our algorithm does not requires a prior projection procedure which, in many cases, is difficult to achieve. It relies on a fine analysis of a generic “fat" section of the curve. Our approach could be applied to more general situations, it generalizes our previous study on absolute factorization of polynomials
On Computations in Kummer Extensions

AbstractLet k be an algebraic number field containing a primitive m th root of unity. An extension K=k([m ] μ) of k with μ∈k is called a Kummer extension. These extensions have been studied extensively in the past and they play an important role in class field theory. Recently many new algorithms dealing with Kummer extensions emerged. In this paper we will give algorithms to solve two problems, which are of particular interest; the computation of the relative discriminant dK/k and the computation of Hilbert norm symbols
A New Approach to Primary Decomposition

AbstractThe first purpose of this paper is to describe a new mathematical approach for the computation of an irredundant primary decomposition of a given polynomial ideal I. This presentation will be formed of three parts: a decomposition of the associated radical ideal I to an intersection of prime ideals Pi, then the determination of ideals Iiwhose radical is prime (equal to Pi), and finally, the extraction of the possible embedded components included in Ii. The second is to give an implementation of this algorithm via a new software component, called The Central Control22Developed inside the PoSSo European project., in which we implemented distributed algorithms performing the basic operations of algebraic geometry
Solvability of Context Equations with Two Context Variables is Decidable

AbstractContext unification is a natural variant of second-order unification that represents a generalization of word unification at the same time. While second-order unification is well known to be undecidable and word unification is decidable it is currently an open question if the solvability of context equations is decidable. We show that the solvability of systems of context equations with two context variables is decidable. The context variables may have an arbitrary number of occurrences, and the equations may also contain an arbitrary number of individual variables. The result holds under the assumption that the first-order background signature is finite
Gröbner Bases over Galois Rings with an Application to Decoding Alternant Codes

AbstractWe develop a theory of Gröbner bases over Galois rings, following the usual formulation for Gröbner bases over finite fields. Our treatment includes a division algorithm, a characterization of Gröbner bases, and an extension of Buchberger’s algorithm. One application is towards the problem of decoding alternant codes over Galois rings. To this end we consider the module M= {(a, b) :aS≡b modxr} of all solutions to the so-called key equation for alternant codes, where S is a syndrome polynomial. In decoding, a particular solution (Σ, Ω) ∈M is sought satisfying certain conditions, and such a solution can be found in a Gröbner basis of M. Applying techniques introduced in the first part of this paper, we give an algorithm which returns the required solution
A Higher-order Interpretation of Deductive Tableau

AbstractThe Deductive Tableau of Manna and Waldinger is a formal system with an associated methodology for synthesizing functional programs by existence proofs in classical first-order theories. We reinterpret the formal system in a setting that is higher-order in two respects: higher-order logic is used to formalize a theory of functional programs and higher-order resolution is used to synthesize programs during proof. Their synthesis methodology can be applied in our setting as well as new methodologies that take advantage of these higher-order features. The reinterpretation gives us a framework for directly formalizing and implementing the Deductive Tableau system in standard theorem provers that support the kinds of higher-order reasoning listed above. We demonstrate this, as well as a new development methodology, within a conservative extension of higher-order logic in the Isabelle system. We report too on a case-study in synthesizing sorting programs
Computing the Radical of an Ideal in Positive Characteristic

AbstractWe propose a method for computing the radical of an arbitrary ideal in the polynomial ring in n variables over a perfect field of characteristic p> 0. In our method Buchberger’s algorithm is performed once in n variables and a Gröbner basis conversion algorithm is performed at most ⌈nlogpd⌉ times in 2 n variables, where d is the maximum of total degrees of generators of the ideal and 3. Next we explain how to compute radicals over a finitely generated coefficient field over a field K, when we have a radical computation method over the field K. Thus we can compute radicals over any finitely generated field over a perfect field
On Efficient Sparse Integer Matrix Smith Normal Form Computations

AbstractWe present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo powers of word-size primes. Consequently, the algorithm does not suffer from coefficient growth. We have implemented several variants of this algorithm (elimination and/or black box techniques) since practical performance depends strongly on the memory available. Our method has proven useful in algebraic topology for the computation of the homology of some large simplicial complexes
Two Computational Techniques for Singularity Resolution

AbstractThis paper presents two efficient computational techniques in algebraic geometry. The first one allows the elimination of redundancies in the representation of quasi-projective varieties by atlases of affine charts. The second simplifies the computations with exponentiated ideals by attaching rational weights to the generators, applying Hironaka’s theory of idealistic exponents. As the main application, we used these techniques to speed up Villamayor’s algorithm for resolving hypersurface singularities in any dimension
The Parametrization of Canal Surfaces and the Decomposition of Polynomials into a Sum of Two Squares

AbstractA canal surface in R3, generated by a parametrized curveC=m(t), is the Zariski closure of the envelope of the set of spheres with radius r(t) centered at m(t). This concept is a generalization of the classical notion of an offsets of a plane curve: first, the canal surface is a surface in 3-space rather than a curve inR2 and second, the radius function r(t) is allowed to vary with the parametert . In case r(t) =const, the resulting envelope is called a pipe surface. In this paper we develop an elementary symbolic method for generating rational parametrizations of canal surfaces generated by rational curves m(t) with rational radius variation r(t). This method leads to the problem of decomposing a polynomial into a sum of two squares over R. We discuss decomposition algorithms which give symbolic and numerical answers to this problem
Constructing Homomorphism Spaces and Endomorphism Rings

AbstractWe present a new deterministic algorithm for constructing homomorphism spaces and endomorphism rings of finite-dimensional modules. The modules are given via vertex projective presentations over path algebras and finite-dimensional quotients of path algebras. We use the theory of right Gröbner bases to encode modules and to construct appropriate systems of equations for computing homomorphism spaces and endomorphism rings. The algorithm is implemented in the computer algebra system GAP and is included in Hopf, a computational package for noncommutative algebra. The performance of our implementation for computing endomorphism rings is experimentally compared with the implementation in Magma for the same class of modules. These experiments show that our implementation has a better time complexity