Rational periodic sequences for the Lyness recurrence

Consider the celebrated Lyness recurrence $x_{n+2}=(a+x_{n+1})/x_{n}$ with $a\in\Q$. First we prove that there exist initial conditions and values of $a$ for which it generates periodic sequences of rational numbers with prime periods $1,2,3,5,6,7,8,9,10$ or $12$ and that these are the only periods that rational sequences $\{x_n\}_n$ can have. It is known that if we restrict our attention to positive rational values of $a$ and positive rational initial conditions the only possible periods are $1,5$ and $9$. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for infinitely many positive values of $a,$ positive 9-period rational sequences occur. This last result is our main contribution and answers an open question left in previous works of Bastien \& Rogalski and Zeeman. We also prove that the level sets of the invariant associated to the Lyness map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order $n, n\ge5,$ including $n$ infinity. This fact implies that the Lyness map is a universal normal form for most birrational maps on elliptic curves.

On the bicanonical map of irregular varieties

From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the natural higher-dimensional generalization to this context of curves of genus $2$: varieties birationally equivalent to the theta-divisor of an indecomposable principally polarized abelian variety. The proof is based on the (generalized) Fourier-Mukai transform.

Interaction of Spiral Waves in the Complex Ginzburg-Landau Equation

Solutions of the general cubic complex Ginzburg-Landau equation comprising multiple spiral waves are considered, and laws of motion for the centers are derived. The direction of the motion changes from along the line of centers to perpendicular to the line of centers as the separation increases, with the strength of the interaction algebraic at small separations and exponentially small at large separations. The corresponding asymptotic wave number and frequency are also determined, which evolve slowly as the spirals move

The normal distribution in some constrained sample spaces

Phenomena with a constrained sample space appear frequently in practice. This is the case e.g. with strictly positive data, or with compositional data, like percentages or proportions. If the natural measure of difference is not the absolute one, simple algebraic properties show that it is more convenient to work with a geometry different from the usual Euclidean geometry in real space, and with a measure different from the usual Lebesgue measure, leading to alternative models which better fit the phenomenon under study. The general approach is presented and illustrated using the normal distribution, both on the positive real line and on the D-part simplex. The original ideas of McAlister in his introduction to the lognormal distribution in 1879, are recovered and updated

Bayes linear spaces

Linear spaces consisting of σ-finite probability measures and infinite measures (improper priors and likelihood functions) are defined. The commutative group operation, called perturbation, is the updating given by Bayes theorem; the inverse operation is the Radon-Nikodym derivative. Bayes spaces of measures are sets of classes of proportional measures. In this framework, basic notions of mathematical statistics get a simple algebraic interpretation. For example, exponential families appear as affine subspaces with their sufficient statistics as a basis. Bayesian statistics, in particular some well-known properties of conjugated priors and likelihood functions, are revisited and slightly extended

Exact results for Wilson loops in arbitrary representations

We compute the exact vacuum expectation value of 1/2 BPS circular Wilson loops of TeX = 4 U(N) super Yang-Mills in arbitrary irreducible representations. By localization arguments, the computation reduces to evaluating certain integrals in a Gaussian matrix model, which we do using the method of orthogonal polynomials. Our results are particularly simple for Wilson loops in antisymmetric representations; in this case, we observe that the final answers admit an expansion where the coefficients are positive integers, and can be written in terms of sums over skew Young diagrams. As an application of our results, we use them to discuss the exact Bremsstrahlung functions associated to the corresponding heavy probes.

Site quality curves for birch stands in north-western Spain

Abstract

Bayesian Networks and the Value of the Evidence for the Forensic Two-Trace Transfer Problem

Forensic scientists face increasingly complex inference problems for evaluating likelihood ratios (LRs) for an appropriate pair of propositions. Up to now, scientists and statisticians have derived LR formulae using an algebraic approach. However, this approach reaches its limits when addressing cases with an increasing number of variables and dependence relationships between these variables. In this study, we suggest using a graphical approach, based on the construction of Bayesian networks (BNs). We first construct a BN that captures the problem, and then deduce the expression for calculating the LR from this model to compare it with existing LR formulae. We illustrate this idea by applying it to the evaluation of an activity level LR in the context of the two-trace transfer problem. Our approach allows us to relax assumptions made in previous LR developments, produce a new LR formula for the two-trace transfer problem and generalize this scenario to n traces.

Pasquale del Pezzo, Duke of Caianello, Neapolitan mathematician

This article is dedicated to a reconstruction of some events and achievements, both personal and scientific, in the life of the Neapolitan mathematician Pasquale del Pezzo, Duke of Caianello.

A Petri Nets-based Scheduling Methodology forMultipurpose Batch Plants.

This article presents an optimization methodology of batch production processes assembled by shared resources which rely on a mapping of state-events into time-events allowing in this way the straightforward use of a well consolidated scheduling policies developed for manufacturing systems. A technique to generate the timed Petri net representation from a continuous dynamic representation (Differential-Algebraic Equations systems (DAEs)) of the production system is presented together with the main characteristics of a Petri nets-based tool implemented for optimization purposes. This paper describes also how the implemented tool generates the coverability tree and how it can be pruned by a general purpose heuristic. An example of a distillation process with two shared batch resources is used to illustrate the optimization methodology proposed.

Mekanismien dynamiikan simuloinnissa sovellettavia numeerisia- ja mallinnusmenetelmiä

Koneet voidaan usein jakaa osajärjestelmiin, joita ovat ohjaus- ja säätöjärjestelmät, voimaa tuottavat toimilaitteet ja voiman välittävät mekanismit. Eri osajärjestelmiä on simuloitu tietokoneavusteisesti jo usean vuosikymmenen ajan. Osajärjestelmien yhdistäminen on kuitenkin uudempi ilmiö. Usein esimerkiksi mekanismien mallinnuksessa toimilaitteen tuottama voimaon kuvattu vakiona, tai ajan funktiona muuttuvana voimana. Vastaavasti toimilaitteiden analysoinnissa mekanismin toimilaitteeseen välittämä kuormitus on kuvattu vakiovoimana, tai ajan funktiona työkiertoa kuvaavana kuormituksena. Kun osajärjestelmät on erotettu toisistaan, on niiden välistenvuorovaikutuksien tarkastelu erittäin epätarkkaa. Samoin osajärjestelmän vaikutuksen huomioiminen koko järjestelmän käyttäytymissä on hankalaa. Mekanismien dynamiikan mallinnukseen on kehitetty erityisesti tietokoneille soveltuvia numeerisia mallinnusmenetelmiä. Useimmat menetelmistä perustuvat Lagrangen menetelmään, joka mahdollistaa vapaasti valittaviin koordinaattimuuttujiin perustuvan mallinnuksen. Numeerista ratkaisun mahdollistamiseksi menetelmän avulla muodostettua differentiaali-algebraaliyhtälöryhmää joudutaan muokkaamaan esim. derivoimalla rajoiteyhtälöitä kahteen kertaan. Menetelmän alkuperäisessä numeerisissa ratkaisuissa kaikki mekanismia kuvaavat yleistetyt koordinaatit integroidaan jokaisella aika-askeleella. Tästä perusmenetelmästä johdetuissa menetelmissä riippumattomat yleistetyt koordinaatit joko integroidaan ja riippuvat koordinaatit ratkaistaan rajoiteyhtälöiden perusteella tai yhtälöryhmän kokoa pienennetään esim. käyttämällä nopeus- ja kiihtyvyysanalyyseissä eri kiertymäkoordinaatteja kuin asema-analyysissä. Useimmat integrointimenetelmät on alun perin tarkoitettu differentiaaliyhtälöiden (ODE) ratkaisuunjolloin yhtälöryhmään liitetyt niveliä kuvaavat algebraaliset rajoiteyhtälöt saattavat aiheuttaa ongelmia. Nivelrajoitteiden virheiden korjaus, stabilointi, on erittäin tärkeää mekanismien dynamiikan simuloinnin onnistumisen ja tulosten oikeellisuuden kannalta. Mallinnusmenetelmien johtamisessa käytetyn virtuaalisen työn periaatteen oletuksena nimittäin on, etteivät rajoitevoimat tee työtä, eli rajoitteiden vastaista siirtymää ei tapahdu. Varsinkaan monimutkaisten järjestelmien pidemmissä analyyseissä nivelrajoitteet eivät toteudu tarkasti. Tällöin järjestelmän energiatasapainoei toteudu ja järjestelmään muodostuu virtuaalista energiaa, joka rikkoo virtuaalisen työn periaatetta, Tästä syystä tulokset eivät enää pidäpaikkaansa. Tässä raportissa tarkastellaan erityyppisiä mallinnus- ja ratkaisumenetelmiä, ja vertaillaan niiden toimivuutta yksinkertaisten mekanismien numeerisessa ratkaisussa. Menetelmien toimivuutta tarkastellaan ratkaisun tehokkuuden, nivelrajoitteiden toteutumisen ja energiatasapainon säilymisen kannalta.

Studies on integrating kinematic design method with mechanical systems simulation techniques

Theultimate goal of any research in the mechanism/kinematic/design area may be called predictive design, ie the optimisation of mechanism proportions in the design stage without requiring extensive life and wear testing. This is an ambitious goal and can be realised through development and refinement of numerical (computational) technology in order to facilitate the design analysis and optimisation of complex mechanisms, mechanical components and systems. As a part of the systematic design methodology this thesis concentrates on kinematic synthesis (kinematic design and analysis) methods in the mechanism synthesis process. The main task of kinematic design is to find all possible solutions in the form of structural parameters to accomplish the desired requirements of motion. Main formulations of kinematic design can be broadly divided to exact synthesis and approximate synthesis formulations. The exact synthesis formulation is based in solving n linear or nonlinear equations in n variables and the solutions for the problem areget by adopting closed form classical or modern algebraic solution methods or using numerical solution methods based on the polynomial continuation or homotopy. The approximate synthesis formulations is based on minimising the approximation error by direct optimisation The main drawbacks of exact synthesis formulationare: (ia) limitations of number of design specifications and (iia) failure in handling design constraints- especially inequality constraints. The main drawbacks of approximate synthesis formulations are: (ib) it is difficult to choose a proper initial linkage and (iib) it is hard to find more than one solution. Recentformulations in solving the approximate synthesis problem adopts polynomial continuation providing several solutions, but it can not handle inequality const-raints. Based on the practical design needs the mixed exact-approximate position synthesis with two exact and an unlimited number of approximate positions has also been developed. The solutions space is presented as a ground pivot map but thepole between the exact positions cannot be selected as a ground pivot. In this thesis the exact synthesis problem of planar mechanism is solved by generating all possible solutions for the optimisation process ¿ including solutions in positive dimensional solution sets - within inequality constraints of structural parameters. Through the literature research it is first shown that the algebraic and numerical solution methods ¿ used in the research area of computational kinematics ¿ are capable of solving non-parametric algebraic systems of n equations inn variables and cannot handle the singularities associated with positive-dimensional solution sets. In this thesis the problem of positive-dimensional solutionsets is solved adopting the main principles from mathematical research area of algebraic geometry in solving parametric ( in the mathematical sense that all parameter values are considered ¿ including the degenerate cases ¿ for which the system is solvable ) algebraic systems of n equations and at least n+1 variables.Adopting the developed solution method in solving the dyadic equations in direct polynomial form in two- to three-precision-points it has been algebraically proved and numerically demonstrated that the map of the ground pivots is ambiguousand that the singularities associated with positive-dimensional solution sets can be solved. The positive-dimensional solution sets associated with the poles might contain physically meaningful solutions in the form of optimal defectfree mechanisms. Traditionally the mechanism optimisation of hydraulically driven boommechanisms is done at early state of the design process. This will result in optimal component design rather than optimal system level design. Modern mechanismoptimisation at system level demands integration of kinematic design methods with mechanical system simulation techniques. In this thesis a new kinematic design method for hydraulically driven boom mechanism is developed and integrated in mechanical system simulation techniques. The developed kinematic design method is based on the combinations of two-precision-point formulation and on optimisation ( with mathematical programming techniques or adopting optimisation methods based on probability and statistics ) of substructures using calculated criteria from the system level response of multidegree-of-freedom mechanisms. Eg. by adopting the mixed exact-approximate position synthesis in direct optimisation (using mathematical programming techniques) with two exact positions and an unlimitednumber of approximate positions the drawbacks of (ia)-(iib) has been cancelled.The design principles of the developed method are based on the design-tree -approach of the mechanical systems and the design method ¿ in principle ¿ is capable of capturing the interrelationship between kinematic and dynamic synthesis simultaneously when the developed kinematic design method is integrated with the mechanical system simulation techniques.

Identifying quantitative operation principles in metabolic pathways: a systematic method for searching feasible enzyme activity patterns leading to cellular adaptive responses

Background: Optimization methods allow designing changes in a system so that specific goals are attained. These techniques are fundamental for metabolic engineering. However, they are not directly applicable for investigating the evolution of metabolic adaptation to environmental changes. Although biological systems have evolved by natural selection and result in well-adapted systems, we can hardly expect that actual metabolic processes are at the theoretical optimum that could result from an optimization analysis. More likely, natural systems are to be found in a feasible region compatible with global physiological requirements. Results: We first present a new method for globally optimizing nonlinear models of metabolic pathways that are based on the Generalized Mass Action (GMA) representation. The optimization task is posed as a nonconvex nonlinear programming (NLP) problem that is solved by an outer- approximation algorithm. This method relies on solving iteratively reduced NLP slave subproblems and mixed-integer linear programming (MILP) master problems that provide valid upper and lower bounds, respectively, on the global solution to the original NLP. The capabilities of this method are illustrated through its application to the anaerobic fermentation pathway in Saccharomyces cerevisiae. We next introduce a method to identify the feasibility parametric regions that allow a system to meet a set of physiological constraints that can be represented in mathematical terms through algebraic equations. This technique is based on applying the outer-approximation based algorithm iteratively over a reduced search space in order to identify regions that contain feasible solutions to the problem and discard others in which no feasible solution exists. As an example, we characterize the feasible enzyme activity changes that are compatible with an appropriate adaptive response of yeast Saccharomyces cerevisiae to heat shock Conclusion: Our results show the utility of the suggested approach for investigating the evolution of adaptive responses to environmental changes. The proposed method can be used in other important applications such as the evaluation of parameter changes that are compatible with health and disease states.

Theta-duality on Prym varieties and a Torelli Theorem

Let $\pi : \widetilde C \to C$ be an unramified double covering of irreducible smooth curves and let $P$ be the attached Prym variety. We prove the scheme-theoretic theta-dual equalities in the Prym variety $T(\widetilde C)=V^2$ and $T(V^2)=\widetilde C$, where $V^2$ is the Brill-Noether locus of $P$ associated to $\pi$ considered by Welters. As an application we prove a Torelli theorem analogous to the fact that the symmetric product $D^{(g)}$ of a curve $D$ of genus $g$ determines the curve.

Wavelet Filter Parameterization for Compression of Spectral Images

Vaatimus kuvatiedon tiivistämisestä on tullut entistä ilmeisemmäksi viimeisen kymmenen vuoden aikana kuvatietoon perustuvien sovellutusten myötä. Nykyisin kiinnitetään erityistä huomiota spektrikuviin, joiden tallettaminen ja siirto vaativat runsaasti levytilaa ja kaistaa. Aallokemuunnos on osoittautunut hyväksi ratkaisuksi häviöllisessä tiedontiivistämisessä. Sen toteutus alikaistakoodauksessa perustuu aallokesuodattimiin ja ongelmana on sopivan aallokesuodattimen valinta erilaisille tiivistettäville kuville. Tässä työssä esitetään katsaus tiivistysmenetelmiin, jotka perustuvat aallokemuunnokseen. Ortogonaalisten suodattimien määritys parametrisoimalla on työn painopisteenä. Työssä todetaan myös kahden erilaisen lähestymistavan samanlaisuus algebrallisten yhtälöiden avulla. Kokeellinen osa sisältää joukon testejä, joilla perustellaan parametrisoinnin tarvetta. Erilaisille kuville tarvitaan erilaisia suodattimia sekä erilaiset tiivistyskertoimet saavutetaan eri suodattimilla. Lopuksi toteutetaan spektrikuvien tiivistys aallokemuunnoksen avulla.