On the Factor Refinement Principle and its Implementation on Multicore Architectures

The factor refinement principle turns a partial factorization of integers (or polynomi­ als) into a more complete factorization represented by basis elements and exponents, with basis elements that are pairwise coprime. There are lots of applications of this refinement technique such as simplifying systems of polynomial inequations and, more generally, speeding up certain algebraic algorithms by eliminating redundant expressions that may occur during intermediate computations. Successive GCD computations and divisions are used to accomplish this task until all the basis elements are pairwise coprime. Moreover, square-free factorization (which is the first step of many factorization algorithms) is used to remove the repeated patterns from each input element. Differentiation, division and GCD calculation op­ erations are required to complete this pre-processing step. Both factor refinement and square-free factorization often rely on plain (quadratic) algorithms for multipli­ cation but can be substantially improved with asymptotically fast multiplication on sufficiently large input. In this work, we review the working principles and complexity estimates of the factor refinement, in case of plain arithmetic, as well as asymptotically fast arithmetic. Following this review process, we design, analyze and implement parallel adaptations of these factor refinement algorithms. We consider several algorithm optimization techniques such as data locality analysis, balancing subproblems, etc. to fully exploit modern multicore architectures. The Cilk++ implementation of our parallel algorithm based on the augment refinement principle of Bach, Driscoll and Shallit achieves linear speedup for input data of sufficiently large size

Oodatavate vastuste ja arvutialgebra süsteemide vastuste erinevused koolimatemaatika võrrandite puhul

Arvutialgebra süsteemidega saab lahendada erinevat tüüpi matemaatikaülesandeid, sealhulgas koolimatemaatika võrrandeid. Sageli langevad arvutialgebra süsteemide vastused kokku koolikontekstis oodatavate vastustega (koolivastustega), vahel aga mitte. Sellised ootamatud arvutialgebra süsteemide vastused on tihti küll matemaatiliselt korrektsed, aga mõne teise standardi järgi, näiteks kompleksarvude vallas. Arvutialgebra süsteemide vastuste ja koolivastuste erinevuste süstemaatiline ülevaade on kasulik arvutialgebra süsteemide arendamisel ning õppetöö planeerimisel. Käesolev dissertatsioon annab ülevaate arvutialgebra süsteemide vastuste ja koolivastuste erinevustest ning nende põhjustest koolimatemaatika võrrandite puhul. Erinevuste spektrit selgitatakse kahe võimaliku klassifikatsiooni abil. Esimese klassifikatsiooni aluseks on see, kas arvutialgebra süsteemi vastus sisaldab rohkem või vähem lahendeid kui oodatav vastus. Teine klassifikatsioon on sisupõhisem ja toob esile vastuste kuju, täielikkuse, arvuvallast sõltuvuse, harunemise ja automaatse lihtsustamise teemad. Arvuvalla ja harunemisega seotud erinevusi käsitletakse dissertatsioonis põhjalikumalt eraldi peatükkides. Koolivastuste ja arvutialgebra süsteemide vastuste erinevusi saab kasutada õpetamisel ja õppimisel. Käesolev dissertatsioon pakubki välja pedagoogilise lähenemise, mis põhineb arvutialgebra süsteemide vastuste ja õppijate endi vastuste võrdlemisel paaristööna. Lisaks õpetamisele ja õppimisele saab selle formaadiga koguda andmeid õppijate teemamõistmise kohta. Väljapakutud lähenemist kasutati tunnisituatsioonis trigonomeetriliste võrrandite käsitlemisel. Põhjalikumalt analüüsiti, kui adekvaatselt õppijad tuvastasid enda vastuse ja arvutialgebra vastuse ekvivalentsust/mitteekvivalentsust ja korrektsust. Leiti, et isegi kui õppijate lahendus paistab korrektne, võib siiski olla lünki arusaamises.It is possible to solve most mathematical problems, including equations of school mathematics, with the help of Computer Algebra Systems (CAS). The answers offered by CAS (CAS answers) often coincide with the answers that are expected in the school context (school answers), but sometimes not. Such unexpected CAS answers are often correct, but based on different standards, for example in complex domain. A systematic review of the differences between CAS answers and school answers is useful for development of CAS and organizing the teaching process. A review of the differences between CAS answers and school answers and their reasons in case of school mathematics equations is provided in this dissertation. The spectrum of differences is explained by using two possible classifications. A key criterion of the first classification is comparing whether the CAS answer includes a larger or a smaller number of solutions than the expected answer. The other classification is more content-oriented, highlighting the issues of the form, completeness, dependence on the number domain, branching and automatic simplification of answers. The differences caused by number domain and branching are discussed separately in greater depth in separate chapters. The differences between school answers and CAS answers can be used in teaching and learning. This dissertation proposes a pedagogical approach that is based on comparative discussions on students' answers and CAS answers in pairs. In addition to teaching and learning, the format is also suitable for collecting data on students' understandings and misunderstandings. The proposed approach was used in lessons on trigonometric equations. The focus was on analyzing whether students can adequately identify the equivalence/non-equivalence and correctness of their answer and CAS answer. It is found that even if a student's solution looks to be correct, students can have misunderstandings and knowledge gaps

The use of data-mining for the automatic formation of tactics

This paper discusses the usse of data-mining for the automatic formation of tactics. It was presented at the Workshop on Computer-Supported Mathematical Theory Development held at IJCAR in 2004. The aim of this project is to evaluate the applicability of data-mining techniques to the automatic formation of tactics from large corpuses of proofs. We data-mine information from large proof corpuses to find commonly occurring patterns. These patterns are then evolved into tactics using genetic programming techniques

Grupos de investigación escuela de ingeniería

This report presents the advances in the process of building and consolidating its research capacity based on the following research groups of the School of Engineering: Marine Science Group, Environmental Geology and Seismic Engineering, GIPAB – Research Group in Environmental and biotechnological Processes, Research Group in Production Engineering, Computers in Education Research and Development Group, Virtual Reality, Software Engineering, GIRSD – Research Group in Networks and Distributed Systems, GEMI – Research Group in Industrial Maintenance, Mechatronics and Machine Design, Laboratory of CAD/CAM/CAE, and Research Group in Bioengineering EAFIT -CES.Este cuaderno presenta los avances en el proceso de construcción y fortalecimiento de la capacidad investigativa alrededor de los siguientes grupos de investigación actualmente constituidos en la Escuela de Ingeniería: Área de Ciencias del Mar, Geología Ambiental e Ingeniería Sísmica, GIPAB – Grupo de Investigación en Procesos Ambientales y Biotecnológicos, Grupo de Investigación en Ingeniería de Producción, Informática Educativa, Realidad Virtual, Ingeniería de Software, GIRSD – Grupo de Investigación en Redes y Sistemas Distribuidos, GEMI – Grupo de Estudios de Mantenimiento Industrial, Mecatrónica y Diseño de Máquinas, Laboratorio de CAD/CAM/CAE y Grupo de Investigación en Bioingeniería EAFIT -CES

Homomorphic Encryption and the Approximate GCD Problem

With the advent of cloud computing, everyone from Fortune 500 businesses to personal consumers to the US government is storing massive amounts of sensitive data in service centers that may not be trustworthy. It is of vital importance to leverage the benefits of storing data in the cloud while simultaneously ensuring the privacy of the data. Homomorphic encryption allows one to securely delegate the processing of private data. As such, it has managed to hit the sweet spot of academic interest and industry demand. Though the concept was proposed in the 1970s, no cryptosystem realizing this goal existed until Craig Gentry published his PhD thesis in 2009. In this thesis, we conduct a study of the two main methods for construction of homomorphic encryption schemes along with functional encryption and the hard problems upon which their security is based. These hard problems include the Approximate GCD problem (A-GCD), the Learning With Errors problem (LWE), and various lattice problems. In addition, we discuss many of the proposed and in some cases implemented practical applications of these cryptosystems. Finally, we focus on the Approximate GCD problem (A-GCD). This problem forms the basis for the security of Gentry\u27s original cryptosystem but has not yet been linked to more standard cryptographic primitives. After presenting several algorithms in the literature that attempt to solve the problem, we introduce some new algorithms to attack the problem

Analytical properties of the Lambert W function

This research studies analytical properties of one of the special functions, the Lambert W function. W function was re-discovered and included into the library of the computer-algebra system Maple in 1980’s. Interest to the function nowadays is due to the fact that it has many applications in a wide variety of fields of science and engineering. The project can be broken into four parts. In the first part we scrutinize a convergence of some previously known asymptotic series for the Lambert W function using an experimental approach followed by analytic investigation. Particularly, we have established the domain of convergence in real and complex cases, given a comparative analysis of the series properties and found asymptotic estimates for the expansion coefficients. The main analytical tools used herein are Implicit Function Theorem, Lagrange Inversion Theorem and Darboux’s Theorem. In the second part we consider an opportunity to improve convergence prop­ erties of the series under study in terms of the domain of,convergence and rate of convergence. For this purpose we have studied a new invariant transformation defined by parameter p, which retains the basic series structure. An effect of parameter p on a size of the domain of convergence and rate of convergence of the series has been studied theoretically and numerically using M a p l e . We have found that an increase in parameter p results in an extension of the domain of convergence while the rate of convergence can be either raised or lowered. We also considered an expansion of W(x) in powers of Inx. For this series we found three new forms for a representation of the expansion coefficients in terms of different special numbers and accordingly have obtained different ways ito\u27compute the expansion coefficients. As an extra consequence we have obtained some combinatorial relations including the Carlitz-Riordan identities. In the third part we study the properties of the polynomials appearing in the expressions for the higher derivatives of the Lambert W function. It is shown that the polynomial coefficients form a positive sequence that is log-concave and unimodal, which implies that the positive real branch of the Lambert W function is Bernstein and its derivative is a Stieltjes function. In the fourth part we show that many functions containing Ware Stieltjes functions. In terms of the result obtained in the third part, we, in fact, obtain one more way to establish that the derivative of W function is a Stieltjes function. We have extended the properties of the set of Stieltjes functions and also proved a generalization of a conjecture of Jackson, Procacci &; Sokal. In addition, we have considered a relation of W to the class of completely monotonic functions and shown that W is a complete Bernstein function. . We give explicit Stieltjes representations of functions of W, We also present integral representations of W which are associated with the properties of its being a Bernstein and Pick function. Representations based on Poisson and Burniston- Siewert integrals are given as well. The results are obtained relying on the fact that the all of the above mentioned classes are characterized by their own integral forms and using Cauchy Integral Formula, Stieltjes-Perron Inversion Formula and properties of W itself

Probabilistic Inference with Generating Functions for Population Dynamics of Unmarked Individuals

Modeling the interactions of different population dynamics (e.g. reproduction, migration) within a population is a challenging problem that underlies numerous ecological research questions. Powerful, interpretable models for population dynamics are key to developing intervention tactics, allocating limited conservation resources, and predicting the impact of uncertain environmental forces on a population. Fortunately, probabilistic graphical models provide a robust mechanistic framework for these kinds of problems. However, in the relatively common case where individuals in the population are unmarked (i.e. indistinguishable from one another), models of the population dynamics naturally contain a deceptively challenging statistical feature: discrete latent variables with unbounded/countably infinite support. Unfortunately, existing inference algorithms for discrete distributions are applicable only for finite distributions and while approximate inference algorithms exist for countably infinite discrete distributions, they are generally unreliable and inefficient. In this work, we develop the first known general-purpose polynomial-time exact inference algorithms for this class of models using a novel representation based on probability generating functions. These methods are flexibe, easy to use, and significantly faster than existing approximate solutions. We also introduce a novel approximation scheme based on this technique that allows it to gracefully scale to populations well beyond the computational limits of any previously known exact or approximate general-purpose inference algorithm for population dynamics. Finally, we conduct an ecological case study on historical data demonstrating the downstream impact of these advances to a large scale population monitoring setting

Analyzing Satisfiability and Refutability in Selected Constraint Systems

This dissertation is concerned with the satisfiability and refutability problems for several constraint systems. We examine both Boolean constraint systems, in which each variable is limited to the values true and false, and polyhedral constraint systems, in which each variable is limited to the set of real numbers R in the case of linear polyhedral systems or the set of integers Z in the case of integer polyhedral systems. An important aspect of our research is that we focus on providing certificates. That is, we provide satisfying assignments or easily checkable proofs of infeasibility depending on whether the instance is feasible or not. Providing easily checkable certificates has become a much sought after feature in algorithms, especially in light of spectacular failures in the implementations of some well-known algorithms. There exist a number of problems in the constraint-solving domain for which efficient algorithms have been proposed, but which lack a certifying counterpart. When examining Boolean constraint systems, we specifically look at systems of 2-CNF clauses and systems of Horn clauses. When examining polyhedral constraint systems, we specifically look at systems of difference constraints, systems of UTVPI constraints, and systems of Horn constraints. For each examined system, we determine several properties of general refutations and determine the complexity of finding restricted refutations. These restricted forms of refutation include read-once refutations, in which each constraint can be used at most once; literal-once refutations, in which for each literal at most one constraint containing that literal can be used; and unit refutations, in which each step of the refutation must use a constraint containing exactly one literal. The advantage of read-once refutations is that they are guaranteed to be short. Thus, while not every constraint system has a read-once refutation, the small size of the refutation guarantees easy checkability