pinecone/core-2020-05-10-deduplication huggingface dataset:
The Product of Like-Indexed Terms in Binary Recurrences

AbstractIn recent work by Hajdu and Szalay, Diophantine equations of the form (ak−1)(bk−1)=x2 were completely solved for a few pairs (a, b). In this paper, a general finiteness theorem for equations of the form ukvk=xn is proved, where uk and vk are terms in certain types of binary recurrence sequences. Also, a unified computational approach for solving equations of the type (ak−1)(bk−1)=x2 is described, and this approach is used to completely solve such equations for almost all (a,b) in the range 1<a<b⩽100. In the final section of this paper, it is shown that the abc conjecture implies much stronger results on these types of Diophantine problems
Finiteness Conjectures for Fl[[T]]-Analytic Extensions of Number Fields

AbstractWe formulate a finiteness conjecture on the image of the absolute Galois group of totally real fields under an even linear representation over a local field of positive characteristic. This is motivated by a recent conjecture of de Jong in the function field case. We discuss the relation to some conjectures of Boston which arise from the conjectures of Fontaine and Mazur and give group-theoretical reformulations of our conjectures. As we will explain, our conjectures have consequences for the structure of universal deformation rings of even residual representations. Finally, we give some evidence for the conjectures themselves and for their consequences
Coverings of Curves with Asymptotically many Rational Points

AbstractIhara defined the quantity A(q), which is the lim sup as g approaches ∞ of the ratio Nq(g)/g, where Nq(g) is the maximum number of rational points a curve of genus g defined over a finite field Fq may have. A(q) is of great relevance for applications to algebraic–geometric codes. It is known that A(q)⩽√q−1 and equality holds when q is a square. By constructing class field towers with good parameters, in this paper we present improvements of lower bounds of A(q) for q an odd power of a prime
Visibility of Shafarevich–Tate Groups of Abelian Varieties

AbstractWe investigate Mazur's notion of visibility of elements of Shafarevich–Tate groups of abelian varieties. We give a proof that every cohomology class is visible in a suitable abelian variety, discuss the visibility dimension, and describe a construction of visible elements of certain Shafarevich–Tate groups. This construction can be used to give some of the first evidence for the Birch and Swinnerton–Dyer conjecture for abelian varieties of large dimension. We then give examples of visible and invisible Shafarevich–Tate groups
On the Bateman–Horn Conjecture

AbstractLet r be a positive integer and f1,…,fr be distinct polynomials in Z[X]. If f1(n),…,fr(n) are all prime for infinitely many n, then it is necessary that the polynomials fi are irreducible in Z[X], have positive leading coefficients, and no prime p divides all values of the product f1(n)···fr(n), as n runs over Z. Assuming these necessary conditions, Bateman and Horn (Math. Comput.16 (1962), 363–367) proposed a conjectural asymptotic estimate on the number of positive integers n⩽x such that f1(n),…,fr(n) are all primes. In the present paper, we apply the Hardy–Littlewood circle method to study the Bateman–Horn conjecture when r⩾2. We consider the Bateman–Horn conjecture for the polynomials in any partition {f1,…,fs}, {fs+1,…,fr} with a linear change of variables. Our main result is as follows: If the Bateman–Horn conjecture on such a partition and change of variables holds true with some conjectural error terms, then the Bateman–Horn conjecture for f1,…,fr is equivalent to a plausible error term conjecture for the minor arcs in the circle method
Value crisis: Affective organization of personal meanings

Contains fulltext :
                  29516.pdf (publisher's version ) (Open Access)The focus of this article is on the specific affective organization of the self in a value crisis. Whereas the self is conceptualized as an organized system of personal meanings, a value crisis is defined as a disorganization of this system. Personal meanings were investigated using a self-confrontation method that is sensitive to the affective properties of personal meanings and their organization into a composite whole. Three groups of students were compared, one high, one medium, and one low in value crisis. Results indicate that the high crisis group had a lower level of intensity of affect referring to self-enhancement, a lower intensity level of affect referring to contact and union with the other, and a lower level of positive affect in comparison with the other groups. In addition, negative meanings were more generalizing in the self of subjects in crisis, whereas positive meanings were more generalizing in the self system of those not in crisis. Moreover, people in crisis showed more discontinuity between their past and future than those not in crisis. Special emphasis was placed on the specific affective organization of subjects in crisis. Two idiographic case studies were presented to illustrate meaningful exceptions to the rule. It was concluded that value crisis is an "in-between state" involving the risk of disorganization of the self, but at the same time including opportunities for innovative self-development
Comparing Curried and Uncurried Rewriting

AbstractCurrying is a transformation of term rewrite systems which may contain symbols of arbitrary arity into systems which contain only nullary symbols, together with a single binary symbol called application. We show that for all term rewrite systems (whether orthogonal or not) the following properties are preserved by this transformation: strong normalization, weak normalization, weak Church-Rosser, completeness, semi-completeness, and the non-convertibility of distinct normal forms. Under the condition of left-linearity we show preservation of the properties NF (if a term is reducible to a normal form,then its reducts are all reducible to the same normal form) and UN→ (a term is reducible to at most one normal form).We exhibit counterexamples to the preservation of NF and UN→ for non-left-linear systems.The results extend to partial currying(where some subset of the symbols are curried),and imply some modularity properties for unions of applicative systems
Inductive Theorem Proving for Design Specifications

AbstractWe present a number of new results on inductive theorem provingfor design specifications based on Horn logic with equality.Induction is explicit here because induction orderings aresupposed to be part of the specification. We show how the automatic support for program verification is enhanced if the specification satisfies a bunch of rewrite properties,summarized under the notion of canonicity. The enhancement isdue to inference rules and corresponding strategies whose soundness is implied by the specification's canonicity. The second main result of the paper provides a method for proving canonicity by using the same rules, which are applied in proofs ofconjectures about the specification and the functional-logic programs it contains
Order Stars and Linear Stability Theory

AbstractOrder stars are a powerful modern tool for the development and analysis of numerical methods. They convey important information such as order and stability in a unified framework. A package for rendering order stars becomes part of the standard distribution in the next major release ofMathematica. An introduction to the theory is provided here, set in the context of numerical methods for Ordinary Differential Equations. The implementation is discussed and examples are given to illustrate why a computer algebra system is an ideal environment for the exploration of order stars
Analogs of Gröbner Bases in Polynomial Rings over a Ring

AbstractIn this paper we will define analogs of Gröbner bases forR-subalgebras and their ideals in a polynomial ringR[x1,...,xn] whereRis a noetherian integral domain with multiplicative identity and in which we can determine ideal membership and compute syzygies.The main goal is to present and verify algorithms for constructing these Gröbner basis counterparts.As an application, we will produce a method for computing generators for the first syzygy module of a subset of anR[x1,...,xn] where each coordinate of each syzygy must be an element of the subalgebra
ComputingB-orbits onG/H

AbstractThe orbits of a Borel subgroup acting on a symmetric varietyG/Hoccur in several areas of mathematics. For example, these orbits and their closures are essential in the study of HarishChandra modules (see Vogan, 1983). There are several descriptions of these orbits, but in practice it is actually very difficult and cumbersome to compute the orbits and their closures. Since the characterizations of theseorbits are very combinatorial in nature, this work could conceivably be done by a computer. In this paper we prove a number of additional properties of these orbits and combine these with properties of the various descriptions of these orbits to obtain an efficient algorithm. This algorithm can be implemented on a computer by using existing symbolic manipulation programs or by writing an independent program
Unification in the Union of Disjoint Equational Theories: Combining Decision Procedures

AbstractMost of the work on the combination of unification algorithms for the union of disjoint equational theories has been restricted to algorithms that compute finite complete sets of unifiers.Thus the developed combination methods usually cannot be used to combine decision procedures,i.e., algorithms that just decide solvability of unification problems without computing unifiers.In this paper we describe a combination algorithm for decision procedures that works for arbitrary equational theories, provided that solvability of so–called unification problems with constant restrictions—a slight generalization of unification problems with constants—is decidable for these theories.As a consequence of this new method, we can, for example, show that generalA-unifiability, i.e.,solvability ofA-unification problems with free function symbols, is decidable. HereAstandsfor the equational theory of one associative function symbol.Our method can also be used to combine algorithms that compute finite complete sets of unifiers. Manfred Schmidt–Schauß' combination result, the until now most general result in this direction, can be obtained as a consequence of this fact.We also obtain the new result that unification in the union of disjoint equational theories is finitary, if general unification—i.e., unification of terms with additional free function symbols—is finitary in the single theories
The Method of Multiple Scales: Asymptotic Solutions and Normal Forms for Nonlinear Oscillatory Problems

AbstractThe method of multiple scales is implemented in Maple V Release 2 to generate a uniform asymptotic solutionO(ϵr) for a weakly nonlinear oscillator.In recent work, it has been shown that the method of multiple scales also transforms the differential equations into normal form, so the given algorithm can be used to simplify the equations describing the dynamics of a system near a fixed point.These results are equivalent to those obtained with the traditional method of normal forms which uses a near-identity coordinate transformation to get the system into the “simplest” form.A few Duffing type oscillators are analysed to illustrate the power of the procedure. The algorithm can be modified to take care of systems of ODEs, PDEs and other nonlinear cases
Normalized Rewriting: an Alternative to Rewriting modulo a Set of Equations

AbstractIn the first part of this paper, we introducenormalized rewriting, a new rewrite relation. It generalizes former notions of rewriting modulo a set of equationsE, dropping some conditions onE. For example,Ecan now be the theory of identity, idempotence, the theory of Abelian groups or the theory of commutative rings. We give a new completion algorithm for normalized rewriting. It contains as an instance the usual AC completion algorithm, but also the well-known Buchberger algorithm for computing Gröbner bases of polynomial ideals. In the second part, we investigate the particular case of completion of ground equations. In this case we prove by a uniform method that completion moduloEterminates, for some interesting theoriesE. As a consequence, we obtain the decidability of the word problem for some classes of equational theories, including the AC-ground case (a result known since 1991), the ACUI-ground case (a new result to our knowledge), and the cases of ground equations modulo the theory of Abelian groups and commutative rings, which is already known when the signature contains only constants, but is new otherwise. Finally, we give implementation results which show the efficiency of normalized completion with respect to completion modulo AC
Memory Mangement in the PoSSo Solver

AbstractA uniform general purpose garbage collector may not always provide optimal performance. Sometimes an algorithm exhibits a predictable pattern of memory usage that could be exploited, delaying as much as possible the intervention of the collector. This requires a collector whose strategy can be customized to the need of an algorithm. We present a dynamic memory management framework which allows such customization, while preserving the convenience of automatic collection in the normal case. The Customizable Memory Management (CMM) organizes memory in multiple heaps, each one encapsulating a particular storage discipline. The default heap for collectable objects uses the technique of mostly copying garbage collection, providing good performance and memory compaction. Customization of the collector is achieved through object orientation by specialising the collector methods for each heap class. We describe how the CMM has been exploited in the implementation of the Buchberger algorithm, by using a special heap for temporary objects created during polynomial reduction. The solution drastically reduces the overall cost of memory allocation in the algorithm
Improved Techniques for Factoring Univariate Polynomials

AbstractThe paper describes improved techniques for factoring univariate polynomials over the integers. The authors modify the usual linear method for lifting modular polynomial factorizations so that efficient early factor detection can be performed. The new lifting method is universally faster than the classical quadratic method, and is faster than a linear method due to Wang, provided we lift sufficiently high. Early factor detection is made more effective by also testing combinations of modular factors, rather than just single modular factors. Various heuristics are presented that reduce the cost of the factor testing or that increase the chance of successful testing. Both theoretical and empirical computing times are presented
Implicitization of Rational Parametric Surfaces

AbstractA generalized projective implicitization theorem is presented that can be used to solve the implicitization of rational parametric curves and surfaces in an affine space. The Groebner bases technique is used to implement the algorithm. The algorithm has the advantages that it can handle base points in a parametrization, and no extra factors will be introduced into an implicit equation. The complexity of the algorithm in terms of the degrees of the polynomials in the Groebner basis is better than the existing metho
Construction of Finitely Presented Lie Algebras and Superalgebras

AbstractWe consider the following problem: what is the most general Lie algebra or superalgebra satisfying a given set of Lie polynomial equations? The presentation of Lie (super)algebras by a finite set of generators and defining relations is one of the most general mathematical and algorithmic schemes of their analysis. That problem is of great practical importance covering applications ranging from mathematical physics to combinatorial algebra. Some particular applications are construction of prolongation algebras in the Wahlquist-Estabrook method for integrability analysis of nonlinear partial differential equations and investigation of Lie (super)algebras arising in different (super)symmetrical physical models. The finite presentations also indicate a way toq-quantize Lie (super)algebras. To solve this problem, one should perform a large volume of algebraic transformations which is sharply increased with growth of the number of generators and relations. For this reason one needs, in practice, to use a computer algebra tool. We describe here an algorithm for constructing the basis of a finitely presented Lie (super)algebra and its commutator table, and its implementation inC. Some computer results illustrating our algorithm and its actual implementation are also presented
Modular Algorithm for Sparse Multivariate Polynomial Interpolationand its Parallel Implementation

AbstractA new algorithm for sparse multivariate polynomial interpolation is presented. It is a multi-modular extension of the Ben-Or and Tiwari algorithm, and is designed to be a practical method to construct symbolic formulas from numeric data produced by vector or massively-parallel processors. The main idea in our algorithm comes from the well-known technique for primality test based on Fermat's theorem, and is the application of the generalized Chinese remainder theorem to the monomial exponents. We regard the exponent vector of each multivariate monomial as a mixed-radix representation of the corresponding exponent value obtained after the transformation by Kronecker's technique. It is shown by complexity comparison and experimental results that the step for univariate polynomial factorization is most expensive in our algorithm, and its parallelization is considered. Also reported are some empirical results of the parallelization on KLIC, a portable system of a concurrent logic programming language KL1
Strategy-Accurate Parallel Buchberger Algorithms

AbstractWe describe two parallel versions of the Buchberger algorithm for computing Gröbner bases, one for the general case and one for homogeneous ideals, which exploit coarse grain parallelism. For the general case, to avoid the growth in number and complexity of the polynomials to reduce, the algorithm adheres strictly to the same strategies as the best sequential implementation. A suitable communication procotol has been designed to ensure proper synchronization of the various processes and to limit their idle time. We provide a detailed analysis the maximum potential degree of parallelism that is achievable with such architecture. The analysis corresponds to the results of our experimental implementation and also explains similar results obtained by other authors
A Distributed and Cooperative Environment for Computer Algebra

AbstractIn this paper, we describe theCentral Control, a software component that enables several symbolic systems to cooperate and exchange data. The Central Control has been designed to be the kernel of an environment for scientific computations which can offer a common and concurrent access to several tools needed by the scientist and the engineer: general purpose and specialized computer algebra systems, visualization tools, links with numerical libraries and tools to manipulate numerical programsetc. The user can interact with the Central Control through one or more (graphical) user interfaces. The Central Control achieves its goals by requiring as little as possible from the tools and by using a particular programming language to provide a unified view for the objects and the operations performed by the connected tools. The Central Control will be used as the basis of theComprehensive Solverthat will provide common access to all the programs developed within thePOSSO ESPRIT/BRAproject. We give a simple example of an actual use of the Central Control for computing primary decompositions of ideals
Bidirectional Exact Integer Division

AbstractDivision of integers is calledexactif the remainder is zero. We show that the high-order part and the low-order part of the exact quotient can be computed independently from each other. A sequential implementation of this algorithm is up to twice as fast as ordinary exact division and four times as fast as the general classical division algorithm if the dividend is twice as long as the divisor. A shared-memory parallel implementation on two processors gains another factor of two in speed
Performance Analysis of the Parallel Karatsuba Multiplication Algorithm for Distributed Memory Architectures

AbstractWe present three parallel implementations of the Karatsuba algorithm for long integer multiplication on a distributed memory architecture and discuss the experimental results obtained on a Paragon computer. The first two implementations have both time complexityO(n) onnlog23processors, but present different behavior for inputs of practical size. The third algorithm has complexityO(nlog2nonnprocessors, offering therefore better asymptotic efficiency. A refinement of the asymptotic analysis for the important case of a constant number of processors takes into account sequential parts of the algorithm and communications overhead. It is shown that the theoretically best speed-up and efficiency can be obtained with two of the algorithms for sufficient problem size. The experimental results confirm the analysis
Strategy Compliant Multi-Threaded Term Completion

AbstractWe report on the design, implementation, and performance,of the parallel term-rewriting system PaReDuX.We discuss the parallelization of three term completion procedures: Knuth-Bendix completion, completion modulo AC, and unfailing completion. Our parallelization is strategy-compliant, i.e., the parallel code performs exactly the same work as the sequential code, but the work load is shared by many processors. PaReDuX is designed for shared memory parallel architectures, such as multi-processor workstations, where it shows good performance on a variety of examples
On the Reconstruction of Proofs in Distributed Theorem Proving: a Modified Clause-Diffusion Method

AbstractProof reconstruction is the operation of extracting the computed prooffrom the trace of a theorem-proving run. We study the problem of proof reconstruction indistributed theorem provingbecause of the distributed nature of the derivation and especially because of deletions of clauses bycontraction,it may happen that a deductive process generates the empty clause, but does not have all the necessary information to reconstruct the proof. We analyse this problem and we present a method for distributed theorem proving, calledModified Clause-Diffusion, which guarantees that the deductive process that generates the empty clause will be able to reconstruct the distributed proof. This result is obtained without imposing a centralized control on the deductive processes or resorting to a round of post-processing with ad hoc communication. We prove that Modified Clause-Diffusion is fair (hence complete) and guarantees proof reconstruction. First we define a set of conditions, next we prove that they are sufficient for proof reconstruction, then we show that Modified Clause-Diffusion satisfies them. Fairness is proved in the same way, which has the advantage that the sufficient conditions provide a treatment of the problem relevant for distributed theorem proving in general