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arXiv2012

Evaluation of Multi-Sums for Large Scale Problems

A big class of Feynman integrals, in particular, the coefficients of their Laurent series expansion w.r.t.\ the dimension parameter $\ep$ can be transformed to multi-sums over hypergeometric terms and harmonic sums. In this article, we present a general summation method based on difference fields that simplifies these multi--sums by transforming them from inside to outside to representations in terms of indefinite nested sums and products. In particular, we present techniques that assist in the task to simplify huge expressions of such multi-sums in a completely automatic fashion. The ideas are illustrated on new calculations coming from 3-loop topologies of gluonic massive operator matrix elements containing two fermion lines, which contribute to the transition matrix elements in the variable flavor scheme.

math-ph↗

arXiv2012

Computable Hilbert Schemes

In this PhD thesis we propose an algorithmic approach to the study of the Hilbert scheme. Developing algorithmic methods, we also obtain general results about Hilbert schemes. In Chapter 1 we discuss the equations defining the Hilbert scheme as subscheme of a suitable Grassmannian and in Chapter 5 we determine a new set of equations of degree lower than the degree of equations known so far. In Chapter 2 we study the most important objects used to project algorithmic techniques, namely Borel-fixed ideals. We determine an algorithm computing all the saturated Borel-fixed ideals with Hilbert polynomial assigned and we investigate their combinatorial properties. In Chapter 3 we show a new type of flat deformations of Borel-fixed ideals which lead us to give a new proof of the connectedness of the Hilbert scheme. In Chapter 4 we construct families of ideals that generalize the notion of family of ideals sharing the same initial ideal with respect to a fixed term ordering. Some of these families correspond to open subsets of the Hilbert scheme and can be used to a local study of the Hilbert scheme. In Chapter 6 we deal with the problem of the connectedness of the Hilbert scheme of locally Cohen-Macaulay curves in the projective 3-space. We show that one of the Hilbert scheme considered a "good" candidate to be non-connected, is instead connected. Moreover there are three appendices that present and explain how to use the implementations of the algorithms proposed.

math.AG↗

arXiv2013

Effective Differential Lüroth's Theorem

This paper focuses on effectivity aspects of the Lüroth's theorem in differential fields. Let $\mathcal{F}$ be an ordinary differential field of characteristic 0 and $\mathcal{F} $ be the field of differential rational functions generated by a single indeterminate $u$. Let be given non constant rational functions $v_1,...,v_n\in \mathcal{F} $ generating a differential subfield $\mathcal{G}\subseteq \mathcal{F} $. The differential Lüroth's theorem proved by Ritt in 1932 states that there exists $v\in \mathcal G$ such that $\mathcal{G}= \mathcal{F} $. Here we prove that the total order and degree of a generator $v$ are bounded by $\min_j \textrm{ord} (v_j)$ and $(nd(e+1)+1)^{2e+1}$, respectively, where $e:=\max_j \textrm{ord} (v_j)$ and $d:=\max_j \textrm{deg} (v_j)$. As a byproduct, our techniques enable us to compute a Lüroth generator by dealing with a polynomial ideal in a polynomial ring in finitely many variables.

math.AC↗

arXiv2012

Can the Eureqa symbolic regression program, computer algebra and numerical analysis help each other?

The Eureqa symbolic regression program has recently received extensive press praise. A representative quote is "There are very clever 'thinking machines' in existence today, such as Watson, the IBM computer that conquered Jeopardy! last year. But next to Eureqa, Watson is merely a glorified search engine." The program was designed to work with noisy experimental data. However, if the data is generated from an expression for which there exists more concise equivalent expressions, sometimes some of the Eureqa results are one or more of those more concise equivalents. If not, perhaps one or more of the returned Eureqa results might be a sufficiently accurate approximation that is more concise than the given expression. Moreover, when there is no known closed form expression, the data points can be generated by numerical methods, enabling Eureqa to find expressions that concisely fit those data points with sufficient accuracy. In contrast to typical regression software, the user does not have to explicitly or implicitly provide a specific expression or class of expressions containiing unknown constants for the software to determine. Is Eureqa useful enough in these regards to provide an additional tool for experimental mathematics, computer algebra users and numerical analysis? Yes if used carefully. Can computer algebra and numerical methods help Eureqa? Definitely.

cs.MS↗

arXiv2012

Subtotal ordering -- a pedagogically advantageous algorithm for computing total degree reverse lexicographic order

Total degree reverse lexicographic order is currently generally regarded as most often fastest for computing Groebner bases. This article describes an alternate less mysterious algorithm for computing this order using exponent subtotals and describes why it should be very nearly the same speed the traditional algorithm, all other things being equal. However, experimental evidence suggests that subtotal order is actually slightly faster for the Mathematica Groebner basis implementation more often than not. This is probably because the weight vectors associated with the natural subtotal weight matrix and with the usual total degree reverse lexicographic weight matrix are different, and Mathematica also uses those the corresponding weight vectors to help select successive S polynomials and divisor polynomials: Those selection heuristics appear to work slightly better more often with subtotal weight vectors. However, the most important advantage of exponent subtotals is pedagogical. It is easier to understand than the total degree reverse lexicographic algorithm, and it is more evident why the resulting order is often the fastest known order for computing Groebner bases. Keywords: Term order, Total degree reverse lexicographic, tdeg, grevlex, Groebner basis

cs.SC↗

arXiv2012

Simplifying products of fractional powers of powers

Most computer algebra systems incorrectly simplify (z - z)/(sqrt(w^2)/w^3 - 1/(w*sqrt(w^2))) to 0 rather than to 0/0. The reasons for this are: 1. The default simplification doesn't succeed in simplifying the denominator to 0. 2. There is a rule that 0 is the result of 0 divided by anything that doesn't simplify to either 0 or 0/0. Try it on your computer algebra systems! This article describes how to simplify products of the form w^a*(w^b1)^g1 ... (w^bn)^gn correctly and well, where w is any real or complex expression and the exponents are rational numbers. It might seem that correct good simplification of such a restrictive expression class must already be published and/or built into at least one widely used computer-algebra system, but apparently this issue has been overlooked. Default and relevant optional simplification was tested with 86 examples for n=1 on Derive, Maple, Mathematica, Maxima and TI-CAS. Totaled over all five systems, 11% of the results were not equivalent to the input everywhere, 50% of the results did not simplify to 0 a result that was equivalent to 0, and at least 16% of the results exhibited one or more of four additional flaw types. There was substantial room for improvement in all five systems, including the two for which I was a co-author. The good news is: These flaws are easy to fix.

cs.SC↗

arXiv2012

Series misdemeanors

cs.SC↗

arXiv2012

Residues and Telescopers for Rational Functions

We give necessary and sufficient conditions for the existence of telescopers for rational functions of two variables in the continuous, discrete and q-discrete settings and characterize which operators can occur as telescopers. Using this latter characterization, we reprove results of Furstenberg and Zeilberger concerning diagonals of power series representing rational functions. The key concept behind these considerations is a generalization of the notion of residue in the continuous case to an analogous concept in the discrete and q-discrete cases.

math.CO↗

arXiv2012

Series Crimes

Puiseux series are power series in which the exponents can be fractional and/or negative rational numbers. Several computer algebra systems have one or more built-in or loadable functions for computing truncated Puiseux series. Some are generalized to allow coefficients containing functions of the series variable that are dominated by any power of that variable, such as logarithms and nested logarithms of the series variable. Some computer algebra systems also have built-in or loadable functions that compute infinite Puiseux series. Unfortunately, there are some little-known pitfalls in computing Puiseux series. The most serious of these is expansions within branch cuts or at branch points that are incorrect for some directions in the complex plane. For example with each series implementation accessible to you: Compare the value of (z^2 + z^3)^(3/2) with that of its truncated series expansion about z = 0, approximated at z = -0.01. Does the series converge to a value that is the negative of the correct value? Compare the value of ln(z^2 + z^3) with its truncated series expansion about z = 0, approximated at z = -0.01 + 0.1i. Does the series converge to a value that is incorrect by 2pi i? Compare arctanh(-2 + ln(z)z) with its truncated series expansion about z = 0, approximated at z = -0.01. Does the series converge to a value that is incorrect by about pi i? At the time of this writing, most implementations that accommodate such series exhibit such errors. This article describes how to avoid these errors both for manual derivation of series and when implementing series packages.

cs.SC↗

arXiv2012

A "Hybrid" Approach for Synthesizing Optimal Controllers of Hybrid Systems: A Case Study of the Oil Pump Industrial Example

In this paper, we propose an approach to reduce the optimal controller synthesis problem of hybrid systems to quantifier elimination; furthermore, we also show how to combine quantifier elimination with numerical computation in order to make it more scalable but at the same time, keep arising errors due to discretization manageable and within bounds. A major advantage of our approach is not only that it avoids errors due to numerical computation, but it also gives a better optimal controller. In order to illustrate our approach, we use the real industrial example of an oil pump provided by the German company HYDAC within the European project Quasimodo as a case study throughout this paper, and show that our method improves (up to 7.5%) the results reported in [3] based on game theory and model checking.

eess.SY↗

arXiv2012

State Space Exploration of RT Systems in the Cloud

The growing availability of distributed and cloud computing frameworks make it possible to face complex computational problems in a more effective and convenient way. A notable example is state-space exploration of discrete-event systems specified in a formal way. The exponential complexity of this task is a major limitation to the usage of consolidated analysis techniques and tools. We present and compare two different approaches to state-space explosion, relying on distributed and cloud frameworks, respectively. These approaches were designed and implemented following the same computational schema, a sort of map & fold. They are applied on symbolic state-space exploration of real-time systems specified by (a timed extension of) Petri Nets, by readapting a sequential algorithm implemented as a command-line Java tool. The outcome of several tests performed on a benchmarking specification are presented, thus showing the convenience of cloud approaches.

cs.SE↗

arXiv2012

New techniques for computing the ideal class group and a system of fundamental units in number fields

We describe a new algorithm for computing the ideal class group, the regulator and a system of fundamental units in number fields under the generalized Riemann hypothesis. We use sieving techniques adapted from the number field sieve algorithm to derive relations between elements of the ideal class group, and $p$-adic approximations to manage the loss of precision during the computation of units. This new algorithm is particularily efficient for number fields of small degree for which a speed-up of an order of magnitude is achieved with respect to the standard methods.

math.NT↗

arXiv2012

A polynomial time algorithm for computing the HNF of a module over the integers of a number field

We present a variation of the modular algorithm for computing the Hermite Normal Form of an $\OK$-module presented by Cohen, where $\OK$ is the ring of integers of a number field K. The modular strategy was conjectured to run in polynomial time by Cohen, but so far, no such proof was available in the literature. In this paper, we provide a new method to prevent the coefficient explosion and we rigorously assess its complexity with respect to the size of the input and the invariants of the field K.

cs.SC↗

arXiv2012

Improvements in the computation of ideal class groups of imaginary quadratic number fields

We investigate improvements to the algorithm for the computation of ideal class groups described by Jacobson in the imaginary quadratic case. These improvements rely on the large prime strategy and a new method for performing the linear algebra phase. We achieve a significant speed-up and are able to compute ideal class groups with discriminants of 110 decimal digits in less than a week.

math.NT↗

arXiv2012

Synthesis of Minimal Error Control Software

Software implementations of controllers for physical systems are at the core of many embedded systems. The design of controllers uses the theory of dynamical systems to construct a mathematical control law that ensures that the controlled system has certain properties, such as asymptotic convergence to an equilibrium point, while optimizing some performance criteria. However, owing to quantization errors arising from the use of fixed-point arithmetic, the implementation of this control law can only guarantee practical stability: under the actions of the implementation, the trajectories of the controlled system converge to a bounded set around the equilibrium point, and the size of the bounded set is proportional to the error in the implementation. The problem of verifying whether a controller implementation achieves practical stability for a given bounded set has been studied before. In this paper, we change the emphasis from verification to automatic synthesis. Using synthesis, the need for formal verification can be considerably reduced thereby reducing the design time as well as design cost of embedded control software. We give a methodology and a tool to synthesize embedded control software that is Pareto optimal w.r.t. both performance criteria and practical stability regions. Our technique is a combination of static analysis to estimate quantization errors for specific controller implementations and stochastic local search over the space of possible controllers using particle swarm optimization. The effectiveness of our technique is illustrated using examples of various standard control systems: in most examples, we achieve controllers with close LQR-LQG performance but with implementation errors, hence regions of practical stability, several times as small.

eess.SY↗

arXiv2012

Delta-Complete Decision Procedures for Satisfiability over the Reals

We introduce the notion of "δ-complete decision procedures" for solving SMT problems over the real numbers, with the aim of handling a wide range of nonlinear functions including transcendental functions and solutions of Lipschitz-continuous ODEs. Given an SMT problem φand a positive rational number δ, a δ-complete decision procedure determines either that φis unsatisfiable, or that the "δ-weakening" of φis satisfiable. Here, the δ-weakening of φis a variant of φthat allows δ-bounded numerical perturbations on φ. We prove the existence of δ-complete decision procedures for bounded SMT over reals with functions mentioned above. For functions in Type 2 complexity class C, under mild assumptions, the bounded δ-SMT problem is in NP^C. δ-Complete decision procedures can exploit scalable numerical methods for handling nonlinearity, and we propose to use this notion as an ideal requirement for numerically-driven decision procedures. As a concrete example, we formally analyze the DPLL framework, which integrates Interval Constraint Propagation (ICP) in DPLL(T), and establish necessary and sufficient conditions for its δ-completeness. We discuss practical applications of δ-complete decision procedures for correctness-critical applications including formal verification and theorem proving.

cs.LO↗

arXiv2012

Computational linear algebra over finite fields

We present here algorithms for efficient computation of linear algebra problems over finite fields.

cs.SC↗

arXiv2012

Matrix Formula of Differential Resultant for First Order Generic Ordinary Differential Polynomials

In this paper, a matrix representation for the differential resultant of two generic ordinary differential polynomials $f_1$ and $f_2$ in the differential indeterminate $y$ with order one and arbitrary degree is given. That is, a non-singular matrix is constructed such that its determinant contains the differential resultant as a factor. Furthermore, the algebraic sparse resultant of $f_1, f_2, δf_1, δf_2$ treated as polynomials in $y, y', y"$ is shown to be a non-zero multiple of the differential resultant of $f_1, f_2$. Although very special, this seems to be the first matrix representation for a class of nonlinear generic differential polynomials.

cs.SC↗