Multiseries Lie groups and asymptotic modules for characterizing and solving integrable models

A multiseries integrable model (MSIM) is defined as a family of compatible flows on an infinite-dimensional Lie group of N-tuples of formal series around N given poles on the Riemann sphere. Broad classes of solutions to a MSIM are characterized through modules over rings of rational functions, called asymptotic modules. Possible ways for constructing asymptotic modules are Riemann-Hilbert and ∂̄ problems. When MSIM's are written in terms of the group coordinates, some of them can be contracted into standard integrable models involving a small number of scalar functions only. Simple contractible MSIM's corresponding to one pole, yield the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy. Two-pole contractible MSIM's are exhibited, which lead to a hierarchy of solvable systems of nonlinear differential equations consisting of (2 + 1) -dimensional evolution equations and of quite strong differential constraints. © 1989 American Institute of Physics.

NONLINEAR TIME SERIES/SYSTEM ANALYSIS (STATE-SPACE, DYNAMICS, INTERVENTION, RESILIENCE, KALMAN)

A discussion of nonlinear dynamics, demonstrated by the familiar automobile, is followed by the development of a systematic method of analysis of a possibly nonlinear time series using difference equations in the general state-space format. This format allows recursive state-dependent parameter estimation after each observation thereby revealing the dynamics inherent in the system in combination with random external perturbations.^ The one-step ahead prediction errors at each time period, transformed to have constant variance, and the estimated parametric sequences provide the information to (1) formally test whether time series observations y(,t) are some linear function of random errors (ELEM)(,s), for some t and s, or whether the series would more appropriately be described by a nonlinear model such as bilinear, exponential, threshold, etc., (2) formally test whether a statistically significant change has occurred in structure/level either historically or as it occurs, (3) forecast nonlinear system with a new and innovative (but very old numerical) technique utilizing rational functions to extrapolate individual parameters as smooth functions of time which are then combined to obtain the forecast of y and (4) suggest a measure of resilience, i.e. how much perturbation a structure/level can tolerate, whether internal or external to the system, and remain statistically unchanged. Although similar to one-step control, this provides a less rigid way to think about changes affecting social systems.^ Applications consisting of the analysis of some familiar and some simulated series demonstrate the procedure. Empirical results suggest that this state-space or modified augmented Kalman filter may provide interesting ways to identify particular kinds of nonlinearities as they occur in structural change via the state trajectory.^ A computational flow-chart detailing computations and software input and output is provided in the body of the text. IBM Advanced BASIC program listings to accomplish most of the analysis are provided in the appendix. ^

Sobre a integração indefinida de funções racionais complexas: teoria e implementação de algoritmos racionais

We present indefinite integration algorithms for rational functions over subfields of the complex numbers, through an algebraic approach. We study the local algorithm of Bernoulli and rational algorithms for the class of functions in concern, namely, the algorithms of Hermite; Horowitz-Ostrogradsky; Rothstein-Trager and Lazard-Rioboo-Trager. We also study the algorithm of Rioboo for conversion of logarithms involving complex extensions into real arctangent functions, when these logarithms arise from the integration of rational functions with real coefficients. We conclude presenting pseudocodes and codes for implementation in the software Maxima concerning the algorithms studied in this work, as well as to algorithms for polynomial gcd computation; partial fraction decomposition; squarefree factorization; subresultant computation, among other side algorithms for the work. We also present the algorithm of Zeilberger-Almkvist for integration of hyperexpontential functions, as well as its pseudocode and code for Maxima. As an alternative for the algorithms of Rothstein-Trager and Lazard-Rioboo-Trager, we yet present a code for Benoulli’s algorithm for square-free denominators; and another for Czichowski’s algorithm, although this one is not studied in detail in the present work, due to the theoretical basis necessary to understand it, which is beyond this work’s scope. Several examples are provided in order to illustrate the working of the integration algorithms in this text

Guía didáctica para el uso de la calculadora Casio fx-7400GII en la gráfica de funciones racionales y radicales

El presente trabajo fue realizado con el propósito de relacionar los contenidos de funciones racionales y radicales con los recursos tecnológicos, de manera que éstas se conviertan en formas innovadoras de enseñar y aprender. Para ello se ha propuesto elaborar una guía didáctica sobre el uso de la calculadora Casio fx-7400GII en la gráfica y análisis de funciones racionales y radicales. En ésta propuesta, la calculadora es vista como un recurso didáctico, orientado a promover una metodología constructivista en el proceso de enseñanza-aprendizaje en el mencionado tema. La presente propuesta se debe a que en la educación actual no se están utilizando los recursos tecnológicos como es el caso de la calculadora gráfica en el aula de clase, pues no se disponen de guías didácticassencillas adecuados para una determinada asignatura. Es importante usar la calculadora gráfica en el estudio de las funciones racionales y radicales ya que al momento de realizar su estudio y análisis de forma manual se requiere de mucho tiempo y se convierte en una actividad tediosa, en tanto que se puede ahorrar tiempo y esfuerzo utilizando la calculadora Con la presente propuesta se pretende que los estudiantes alcancen un mejor nivel de conocimiento en el tema, a través del uso de la guía en el aula de clases generando un proceso de enseñanza-aprendizaje más práctico y dinámico

Rational Partial Choice Functions and Their Application to Belief Revision.

Necessary and sufficient conditions for choice functions to be rational have been intensively studied in the past. However, in these attempts, a choice function is completely specified. That is, given any subset of options, called an issue, the best option over that issue is always known, whilst in real-world scenarios, it is very often that only a few choices are known instead of all. In this paper, we study partial choice functions and investigate necessary and sufficient rationality conditions for situations where only a few choices are known. We prove that our necessary and sufficient condition for partial choice functions boils down to the necessary and sufficient conditions for complete choice functions proposed in the literature. Choice functions have been instrumental in belief revision theory. That is, in most approaches to belief revision, the problem studied can simply be described as the choice of possible worlds compatible with the input information, given an agent’s prior belief state. The main effort has been to devise strategies in order to infer the agents revised belief state. Our study considers the converse problem: given a collection of input information items and their corresponding revision results (as provided by an agent), does there exist a rational revision operation used by the agent and a consistent belief state that may explain the observed results?

Prime and Semiprime Inner Functions

This paper studies the structure of inner functions under the operation of composition, and in particular the notions or primeness and semiprimeness. Results proved include the density of prime finite Blaschke products in the set of finite Blaschke products, the semiprimeness of finite products of thin Blaschke products and their approximability by prime Blaschke products. An example of a nonsemiprime Blaschke product that is a Frostman Blaschke product is also provided.

Prime and semiprime inner functions

Abstract This paper studies the structure of inner functions under the operation of composition, and in particular the notions or primeness and semiprimeness. Results proved include the density of prime finite Blaschke products in the set of finite Blaschke products, the semiprimeness of finite products of thin Blaschke products and their approximability by prime Blaschke products. An example of a nonsemiprime Blaschke product that is a Frostman Blaschke product is also provided.

Professor Schroeder van der Kolk on the minute structure and functions of the spinal cord and medulla oblongata : and on the proximate cause and rational treatment of epilepsy /

Translation of two memoirs: Anatomisch physiologisch onderzoek over het fijnere zamenstel en de werking van het ruggemerg, published 1854, and Over het fijnere zamenstel en de werking van het verlangde ruggemerg en over de naaste oorzaak van epilepsie en hare rationele behandeling, published 1858.

Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A^(-α/2)b, where A ∈ ℝ^(n×n) is a large, sparse symmetric positive definite matrix and b ∈ ℝ^n is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LL^T is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L^(-T)z, with x = A^(-1/2)z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form ϕn = A^(-α/2)b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t^(-α/2) and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A^(-α/2)b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.

Rational design of artificial cellular niches for tissue engineering

Tissue Engineering is a promising emerging field that studies the intrinsic regenerative potential of the human body and uses it to restore functionality of damaged organs or tissues unable of self-healing due to illness or ageing. In order to achieve regeneration using Tissue Engineering strategies, it is first necessary to study the properties of the native tissue and determine the cause of tissue failure; second, to identify an optimum population of cells capable of restoring its functionality; and third, to design and manufacture a cellular microenvironment in which those specific cells are directed towards the desired cellular functions. The design of the artificial cellular niche has a tremendous importance, because cells will feel and respond to both its biochemical and biophysical properties very differently. In particular, the artificial niche will act as a physical scaffold for the cells, allowing their three-dimensional spatial organization; also, it will provide mechanical stability to the artificial construct; and finally, it will supply biochemical and mechanical cues to control cellular growth, migration, differentiation and synthesis of natural extracellular matrix. During the last decades, many scientists have made great contributions to the field of Tissue Engineering. Even though this research has frequently been accompanied by vast investments during extended periods of time, yet too often these efforts have not been enough to translate the advances into new clinical therapies. More and more scientists in this field are aware of the need of rational experimental designs before carrying out complex, expensive and time-consuming in vitro and in vivo trials. This review highlights the importance of computer modeling and novel biofabrication techniques as critical key players for a rational design of artificial cellular niches in Tissue Engineering.

Karnaugh map analysis and synthesis of threshold functions

A unate function can easily be identified on a Karnaugh map from the well-known property that it cons ist s only ofess en ti al prime implicante which intersect at a common implicant. The additional property that the plot of a unate function F(x, ... XII) on a Karnaugh map should possess in order that F may also be Ivrealizable (n';:; 6) has been found. It has been sh own that the I- realizability of a unate function F corresponds to the ' compac tness' of the plot of F. No resort to tho inequalities is made, and no pre-processing such as positivizing and ordering of the given function is required.

An Algorithmic Characterization of Polynomial Functions over Z(pn)

In this paper we consider polynomial representability of functions defined over , where p is a prime and n is a positive integer. Our aim is to provide an algorithmic characterization that (i) answers the decision problem: to determine whether a given function over is polynomially representable or not, and (ii) finds the polynomial if it is polynomially representable. The previous characterizations given by Kempner (Trans. Am. Math. Soc. 22(2):240-266, 1921) and Carlitz (Acta Arith. 9(1), 67-78, 1964) are existential in nature and only lead to an exhaustive search method, i.e. algorithm with complexity exponential in size of the input. Our characterization leads to an algorithm whose running time is linear in size of input. We also extend our result to the multivariate case.

Simultaneous nonvanishing of Dirichlet L-functions and twists of Hecke-Maass L-functions

We prove that given a Hecke-Maass form f for SL(2, Z) and a sufficiently large prime q, there exists a primitive Dirichlet character chi of conductor q such that the L-values L(1/2, f circle times chi) and L(1/2, chi) do not vanish.