Integer Case Research Articles

A survey on Büchi’s problem: new presentations and open problems

In a commutative ring with a unit, a Buchi sequence is a sequence such that the second difference of the sequence of its squares is the constant sequence (2). Sequences of elements x n satisfying x n 2 = (x + n)2 for some fixed x are Buchi sequences, which we call trivial. Since we want to study sequences whose elements do not belong to certain subrings (e.g., for fields of rational functions F(z) over a field F, we are interested in sequences that are not over F), the concept of trivial sequences may vary. Buchi’s problem for a ring asks whether there exists a positive integer M such that any Buchi sequence of length M or more is trivial. We survey the current status of knowledge for Buchi’s problem and its analogs for higher-order differences and higher powers. We propose several new and old open problems. We present a few new results and various sketches of proofs of old results (in particular, Vojta’s conditional proof for the case of integers and a rather detailed proof for the case of polynomial rings in characteristic zero), and present a new and short proof of the positive answer to Buchi’s problem over finite fields with p elements (originally proved by Hensley). We discuss applications to logic, which were the initial aim for solving these problems. Bibliography: 30 titles.

Multiparametric programming based algorithms for pure integer and mixed-integer bilevel programming problems

This work introduces two algorithms for the solution of pure integer and mixed-integer bilevel programming problems by multiparametric programming techniques. The first algorithm addresses the integer case of the bilevel programming problem where integer variables of the outer optimization problem appear in linear or polynomial form in the inner problem. The algorithm employs global optimization techniques to convexify nonlinear terms generated by a reformulation linearization technique (RLT). A continuous multiparametric programming algorithm is then used to solve the reformulated convex inner problem. The second algorithm addresses the mixed-integer case of the bilevel programming problem where integer and continuous variables of the outer problem appear in linear or polynomial forms in the inner problem. The algorithm relies on the use of global multiparametric mixed-integer programming techniques at the inner optimization level. In both algorithms, the multiparametric solutions obtained are embedded in the outer problem to form a set of single-level (M)(I)(N)LP problems – which are then solved to global optimality using standard fixed-point (global) optimization methods. Numerical examples drawn from the open literature are presented to illustrate the proposed algorithms.

Ultrastrong coupling between a cavity resonator and the cyclotron transition of a two-dimensional electron gas in the case of an integer filling factor

We investigate theoretically the coupling between a cavity resonator and the cyclotron transition of a two dimensional electron gas under an applied perpendicular magnetic field. We derive and diagonalize an effective quantum Hamiltonian describing the magnetopolariton excitations of the two dimensional electron gas for the case of integer filling factors. The limits of validity of the present approach are critically discussed. The dimensionless vacuum Rabi frequency $\Omega_0/\omega_0$ (i.e., normalized to the cyclotron frequency $\omega_0$) is shown to scale as $\sqrt{\alpha\: n_{QW} \nu}$, where $\alpha$ is the fine structure constant, $n_{QW}$ is the number of quantum wells and $\nu$ is the filling factor in each well. We show that with realistic parameters of a high-mobility semiconductor two dimensional electron gas, the dimensionless coupling $\Omega_0/\omega_0$ can be much larger than 1 in the case of $\nu \gg 1$, the latter condition being typically realized for cyclotron transitions in the microwave range. Implications of such ultrastrong coupling regime are discussed.

Parametric Integer Programming Algorithm for Bilevel Mixed Integer Programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader’s variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case, it yields a “better than fully polynomial time” approximation scheme with running time polynomial in the logarithm of the absolute precision. For the pure integer case where the leader’s variables are integer, and hence optimal solutions are guaranteed to exist, we present an algorithm which runs in polynomial time when the total number of variables is fixed.

Asymptotic normality of b-additive functions on polynomial sequences in number systems

We consider b-additive functions f where b is an algebraic integer over ℤ. In particular, let p be a polynomial, then we show that the asymptotic distribution of f(⌊ p(z)⌋), where ⌊⋅⌋ denotes the integer part with respect to basis b, when z runs through the elements of the ring ℤ[b] is the normal law. This is a generalization of results of Bassily and Kátai (for the integer case) and of Gittenberger and Thuswaldner (for the Gaussian integers).

ABS methods for continuous and integer linear equations and optimization

ABS methods are a large class of algorithms for solving continuous and integer linear algebraic equations, and nonlinear continuous algebraic equations, with applications to optimization. Recent work by Chinese researchers led by Zunquan Xia has extended these methods also to stochastic, fuzzy and infinite systems, extensions not considered here. The work on ABS methods began almost thirty years. It involved an international collaboration of mathematicians especially from Hungary, England, China and Iran, coordinated by the university of Bergamo. The ABS method are based on the rank reducing matrix update due to Egervary and can be considered as the most fruitful extension of such technique. They have led to unification of classes of methods for several problems. Moreover they have produced some special algorithms with better complexity than the standard methods. For the linear integer case they have provided the most general polynomial time class of algorithms so far known; such algorithms have been extended to other integer problems, as linear inequalities and LP problems, in over a dozen papers written by Iranian mathematicians led by Nezam Mahdavi-Amiri. ABS methods can be implemented generally in a stable way, techniques existing to enhance their accuracy. Extensive numerical experiments have shown that they can outperform standard methods in several problems. Here we provide a review of their main properties, for linear systems and optimization. We also give the results of numerical experiments on some linear systems. This paper is dedicated to Professor Egervary, developer of the rank reducing matrix update, that led to ABS methods.

Detecting changes from short to long memory

This paper studies well-known tests by Kim et al. (J Econom 109:389–392, 2002) and Busetti and Taylor (J Econom 123:33–66, 2004) for the null hypothesis of short memory against a change to nonstationarity, I (1). The potential break point is not assumed to be known but estimated from the data. First, we show that the tests are also applicable for a change from I (0) to a fractional order of integration I (d) with d > 0 (long memory) in that the tests are consistent. The rates of divergence of the test statistics are derived as functions of the sample size T and d. Second, we compare their finite sample power experimentally. Third, we consider break point estimation for a change from I (0) to I (d) for finite samples in computer simulations. It turns out that the estimators proposed for the integer case (d = 1) are practically reliable only if d is close enough to 1.

Area and power efficient mismatched filters based on sidelobe inversion

A computationally efficient mismatched filter comprised of a matched filter in cascade with a multi-stage filter based on sidelobe inversion is proposed. For this approach to work, the given code has to satisfy certain conditions as derived in this work. For the proposed filter to be of practical significance, the given autocorrelation must have a computationally efficient representation including the case of sparse and/or small integer valued sidelobes such as in Barker codes. When implemented in VLSI, significantly smaller chip area and less power are required compared to the length-optimal filters achieving comparable sidelobe suppression.

Hypergeometric functions with rational arguments

We elaborate on the expansion of hypergeometric functions about rational parameters, where we focus mainly on the integer and half-integer case. The strategy and the basic steps of a recently developed algorithm for the expansion about half-integer parameters are described. The algorithm is implemented in the Mathematica package HypExp, by means of which we derive (partially new) results of selected multi-loop Feynman diagrams. Moreover, we give a new formulation of a conjecture in the context of a three-loop master integral in HQET.

Nonconservative Systems within Fractional Generalized Derivatives

A fractional derivative generalizes an ordinary derivative, and therefore the derivative of the product of two functions differs from that for the classical (integer) case; the integration by parts for Riemann-Liouville fractional derivatives involves both the left and right fractional derivatives. Despite these restrictions, fractional calculus models are good candidates for description of nonconservative systems. In this article, nonconservative Lagrangian mechanics are investigated within the fractional generalized derivative approach. The fractional Euler-Lagrange equations based on the Riemann-Liouville fractional derivatives are briefly presented. Using generalized fractional derivatives, we give a meaning for the term which appears in fractional Euler-Lagrange equations and contains the second order fractional derivative. The fractional Lagrangians and Hamiltonians of two illustrative nonconservative mechanical systems are investigated in detail.

New Results on the Controllability and Observability of Fractional Dynamical Systems

In this article, we describe some new results on the controllability and observability of linear dynamical systems involving a fractional derivative of order α. We show that the input control energy required to drive the state in a given direction and the energy available at the output are related to the observability and the controllability Gramians, respectively. Our results can be considered as a generalization of the known corresponding results for the integer order case to the fractional order one, since for α 1 1, the results for the integer case are recovered.

Knuth relations for the hyperoctahedral groups

C. Bonnafé, M. Geck, L. Iancu, and T. Lam have conjectured a description of Kazhdan-Lusztig cells in unequal parameter Hecke algebras of type B which is based on domino tableaux of arbitrary rank. In the integer case, this generalizes the work of D. Garfinkle. We adapt her methods and construct a family of operators which generate the equivalence classes on pairs of arbitrary rank domino tableaux described in the above conjecture.

A Polynomial Analogue of the 3n + 1 Problem

viewed as mapping the integers to itself. The famous 3n + I conjecture postulates that after a finite number of iterations one always arrives at the value 1. This has been shown to be true for all n < 240. Whether it converges to 1 for every positive integer n is an open problem, see Lagarias [3]. The intricate relationship between the ring Z of integers and the polynomial ring Fq[x] of polynomials in a single variable x over the finite field Fq has been studied extensively. This discussion has included many topics such as comparisons between prime integers and irreducible polynomials, Goldbach type problems in both the in teger and polynomial settings (Effinger and Hayes [1]), and the study of twin primes and twin irreducibles and their densities, see [2]. In [2], the authors allude to the fact that in many (but not all) cases an analogous problem in the polynomial setting may be easier to resolve than the original problem. The result presented here indeed provides an example of that phenomenon. We start with the binary field F2 since this is the case most analogous to the integer version, see [2]. In the integer case, the first two primes are of course 2 and 3. In the F2[x] setting, the first two irreducible polynomials are x and x + 1. The analogue to the 3n + 1 problem for polynomials is the map Ci(f(x)) (x + l)f(x) + l if /(0)^0 fix) JK } if/(0) = 0 x

Effects of interaction-induced second Landau level mixing in theν=1quantum Hall effect

The work by Mandal and Jain [Solid State Commun. 118, 503 (2001)] suggests that interaction-induced mixing with the second composite fermion Landau level can lead to renormalization of the electron correlation function exponent in the fractional quantum Hall effect. In the work reported here, a similar mixing with the second electronic Landau level is studied in the $\ensuremath{\nu}=1$ integer case. The ground state is calculated by use of the Hartree\char21{}Fock algorithm, and the electron density and electron correlation function on the edge are calculated. It is shown that the interaction gives rise to oscillations in the density profile. In particular, a short range interaction gives a profile qualitatively similar to the results reported by Mandal and Jain. On the other hand, no renormalization of the correlation function exponent is found.

Critical exponents in a quantum phase transition of an anisotropic 2D antiferromagnet

I use the two-step density-matrix renormalization group method to extract the critical exponents $\ensuremath{\beta}$ and $\ensuremath{\nu}$ in the transition from a N\'eel $Q=(\ensuremath{\pi},\ensuremath{\pi})$ phase to a magnetically disordered phase with a spin gap. I find that the exponent $\ensuremath{\beta}$ computed from the magnetic side of the transition is consistent with that of the classical Heisenberg model, but not the exponent $z\ensuremath{\nu}$ computed from the disordered side. This suggests a possible discrepancy with the classical behavior. I also show the contrast between integer and half-integer spin cases.

Chebyshev expansion for the component functions of the almost‐Mathieu operator

AbstractThe component functions {Ψn (ε)} (n 2 Z+) from difference Schrödinger operators, can be formulated in a second order linear difference equation. Then the Harper equation, associated to almost‐Mathieu operator, is a prototypical example. Its spectral behavior is amazing. Here, due the cosine coefficient in Harper equation, the component functions are expanded in a Chebyshev series of first kind, Tn (cos 2πθ). It permits us a particular method for the θ variable separation. Thus, component functions can be expressed as an inner product, equation image (cos 2πθ) · equation image . A matrix block transference method is applied for the calculation of the vector equation image . When θ is integer, {Ψn (ε) is the sum of component from equation image . The complete family of Chebyshev Polynomials can be generated, with fit initial conditions. The continuous spectrum is one band with Lebesgue measure equal to 4. When θ is not integer the inner product Ψn can be seen as a perturbation of vector equation image on the sum of components from the vector equation image . When equation image , with p and q coprime, periodic perturbation appears, the connected band from the integer case degenerates in q sub‐bands. When θ is irrational, ergodic perturbation produces that one band spectrum from integer case degenerates to a Cantor set. Lebesgue measure is Lσ = 4(1 – |λ|), 0 &lt; |λ| ≤ 1. In this situation, the series solution becomes critical. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Gaussian Laws for the Main Parameters of the Euclid Algorithms

We provide sharp estimates for the probabilistic behaviour of the main parameters of the Euclid Algorithms, both on polynomials and on integer numbers. We study in particular the distribution of the bit-complexity which involves two main parameters: digit-costs and length of remainders. We first show here that an asymptotic Gaussian law holds for the length of remainders at a fraction of the execution, which exhibits a deep regularity phenomenon. Then, we study in each framework—polynomials (P) and integer numbers (I)—two gcd algorithms, the standard one (S) which only computes the gcd, and the extended one (E) which also computes the Bezout pair, and is widely used for computing modular inverses. The extended algorithm is more regular than the standard one, and this explains that our results are more precise for the Extended algorithm: we exhibit an asymptotic Gaussian law for the bit-complexity of the extended algorithm, in both cases (P) and (I). We also prove that an asymptotic Gaussian law for the bit-complexity of the standard gcd in case (P), but we do not succeed obtaining a similar result in case (I). The integer study is more involved than the polynomial study, as it is usually the case. In the polynomial case, we deal with the central tools of the distributional analysis of algorithms, namely bivariate generating functions. In the integer case, we are led to dynamical methods, which heavily use the dynamical system underlying the number Euclidean algorithm, and its transfer operator. Baladi and Vallee (J. Number Theory 110(2):331–386, 2005) have recently designed a general framework for “distributional dynamical analysis”, where they have exhibited asymptotic Gaussian laws for a large family of parameters. However, this family does not contain neither the bit-complexity cost nor the size of remainders, and we have to extend their methods for obtaining our results. Even if these dynamical methods are not necessary in case (P), we explain how the polynomial dynamical system can be also used for proving our results. This provides a common framework for both analyses, which well explains the similarities and the differences between the two cases (P) and (I), for the algorithms themselves, and also for their analysis. An extended abstract of this paper can be found in Lhote and Vallee (Proceedings of LATIN’06, Lecture Notes in Computer Science, vol. 3887, pp. 689–702, 2006).

Multiple objective minimum cost flow problems: A review

In this paper, theory and algorithms for solving the multiple objective minimum cost flow problem are reviewed. For both the continuous and integer case exact and approximation algorithms are presented. In addition, a section on compromise solutions summarizes corresponding results. The reference list consists of all papers known to the authors which deal with the multiple objective minimum cost flow problem.

LOCALIZED WAVEFUNCTIONS AND MAGNETIC BAND STRUCTURE FOR LATERAL SEMICONDUCTOR SUPERLATTICES

In this paper we present calculations on the electronic band structure of a two-dimensional lateral superlattice subject to a perpendicular magnetic field by employing a projection operator technique based on the ray-group of magnetotranslation operators. We construct a new basis of appropriately symmetrized Bloch-like wavefunctions as linear combination of well-localized magnetic Wannier functions. The magnetic field was consistently included in the Wannier functions defined in terms of free-electron eigenfunctions in the presence of external magnetic field in the symmetric gauge. Using the above basis, we calculate the magnetic energy spectrum of electrons in a lateral superlattice with bi-directional weak electrostatic modulation. Both a square lattice and a triangular one are considered as special cases. Our approach based on group theory handles the cases of integer and rational magnetic fluxes in a uniform way and the provided basis could be convenient for further both analytic and numerical calculations.

Optimizing over the split closure

The polyhedron defined by all the split cuts obtainable directly (i.e. without iterated cut generation) from the LP-relaxation P of a mixed integer program (MIP) is termed the (elementary, or rank 1) split closure of P. This paper deals with the problem of optimizing over the elementary split closure. This is accomplished by repeatedly solving the following separation problem: given a fractional point, say x, find a rank-1 split cut violated by x or show that none exists. Following Caprara and Letchford [17], we formulate this separation problem as a nonlinear mixed integer program that can be treated as a parametric mixed integer linear program (PMILP) with a single parameter in the objective function and the right hand side. We develop an algorithmic framework to deal with the resulting PMILP by creating and maintaining a dynamically updated grid of parameter values, and use the corresponding mixed integer programs to generate rank 1 split cuts. Our approach was implemented in the COIN-OR framework using CPLEX 9.0 as a general purpose MIP solver. We report our computational results on well-known benchmark instances from MIPLIB 3.0 and several classes of structured integer and mixed integer problems. Our computational results show that rank-1 split cuts close more than 98% of the duality gap on 15 out of 41 mixed integer instances from MIPLIB 3.0. More than 75% of the duality gap can be closed on an additional 10 instances. The average gap closed over all 41 instances is 72.78%. In the pure integer case, rank-1 split cuts close more than 75% of the duality gap on 13 out of 24 instances from MIPLIB 3.0. On average, rank 1 split cuts close about 72% of the duality gap on these 24 instances. We also report results on several classes of structured problems: capacitated versions of warehouse location, single-source facility location, p-median, fixed charge network flow, multi-commodity network design with splittable and unsplittable flows, and lot sizing. The fraction of the integrality gap closed varies for these problem classes between 100 and 67%. We also gathered statistics on the average coefficient size (absolute value) of the disjunctions generated. They turn out to be surprisingly small.