Tomás Oliveira e Silva

Modelling and Identification with Rational Orthogonal Basis Functions

Abstract Decomposing dynamical systems in terms of orthogonal expansions enables the modelling/approximation of a system with a finite length expansion. By flexibly tuning the basis functions to underlying system characteristics, the rate of convergence of these expansions can be drastically increased, leading to highly accurate models (small bias) being represented by few parameters (small variance). Additionally algorithmic and numerical aspects are favourable. A recently developed general theory for basis construction will be presented, that is a generalization of the classical Laguerre theory. The basis functions are applied in problems of identification, approximation, realization, uncertainty modelling, and adaptive filtering, particularly exploiting the property that basis function models are linearly parametrized. Besides powerful algorithms, they also provide useful analysis tools for understanding the underlying identification/approximation algorithms.

Optimality conditions for truncated Laguerre networks

Presents optimality conditions for truncated Laguerre networks both for continuous time and for discrete time. Contrary to the current approach, no assumption is made about the whiteness of the power spectrum of the input signal. Curiously, the results obtained have the same form as those published for the more restrictive case where the input signal is an impulse at the time origin (or white noise). >

Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅10¹⁸

This paper describes how the even Goldbach conjecture was confirmed to be true for all even numbers not larger than 4 · 1018. Using a result of Ramaré and Saouter, it follows that the odd Goldbach conjecture is true up to 8.37 · 1026. The empirical data collected during this extensive verification effort, namely, counts and first occurrences of so-called minimal Goldbach partitions with a given smallest prime and of gaps between consecutive primes with a given even gap, are used to test several conjectured formulas related to prime numbers. In particular, the counts of minimal Goldbach partitions and of prime gaps are in excellent accord with the predictions made using the prime k-tuple conjecture of Hardy and Littlewood (with an error that appears to be O( √ t log log t), where t is the true value of the quantity being estimated). Prime gap moments also show excellent agreement with a generalization of a conjecture made in 1982 by Heath-Brown. The Goldbach conjecture [13] is a famous mathematical problem whose proof, or disproof, has so far resisted the passage of time [20, Problem C1]. (According to [1], Waring and, possibly, Descartes also formulated similar conjectures.) It states, in its modern even form, that every even number larger than four is the sum of two odd prime numbers, i.e., that n = p + q. Here, and in what follows, n will always be an even integer larger than four, and p and q will always be odd prime numbers. The additive decomposition n = p + q is called a Goldbach partition of n. The one with the smallest p will be called the minimal Goldbach partition of n; the corresponding p will be denoted by p(n) and the corresponding q by q(n). It is known that up to a given number x at most O(x) even integers do not have a Goldbach partition [30], and that every large enough even number is the sum of a prime and the product of at most two primes [24]. Furthermore, according to [48], every odd number greater that one is the sum of at most five primes. As described in Table 1, over a time span of more than a century the even Goldbach conjecture was confirmed to be true up to ever-increasing upper limits. Section 1 describes the methods that were used by the first author, with computational help from the second and third authors, and others, to set the limit of verification of the Goldbach conjecture at 4 · 10. Section 2 presents a small subset of the empirical data that was gathered during the verification, namely, counts and first occurrences of primes in minimal Goldbach partitions, and counts and first occurrences of prime gaps, and compares it with the predictions made by Received by the editor May 21, 2012 and, in revised form, December 6, 2012. 2010 Mathematics Subject Classification. Primary 11A41, 11P32, 11N35; Secondary 11N05, 11Y55.

Optimality conditions for truncated Kautz networks with two periodically repeating complex conjugate poles

Presents optimality conditions for the approximation of a SISO system by a truncated Kautz network with two repeating complex conjugate poles. Both the continuous and the discrete time cases are discussed. The author approaches the problem in a system approximation framework, and he does not assume that the input signal is white (or an impulse). The results obtained generalize the results already known for truncated Laguerre networks. >

Maximum excursion and stopping time record-holders for the 3x + 1 problem: computational results

This paper presents some results concerning the search for initial values to the so-called 3x+1 problem which give rise either to function iterates that attain a maximum value higher than all function iterates for all smaller initial values, or which have a stopping time higher than those of all smaller initial values. Our computational results suggest that for an initial value of n, the maximum value of the function iterates is bounded from above by n 2 f(n), with f(n) either a constant or a very slowly increasing function of n. As a byproduct of this (exhaustive) search, which was performed up to n. = 3. 2 53 2.702. 10 16 , the 3x + 1 conjecture was verified up to that same number.

On the construction of orthonormal basis functions for system identification

Abstract In this paper we present an alternative derivation of orthonormal rational families in H 2 , based on previous work by Roberts and Mullis. The construction is based on balanced state space realizations of all-pass transfer functions, and generalizes recent work of Heuberger et al. The emphasis is put on the simplicity of this approach compared with traditional methods. We also present some historical remarks concerning earlier approaches to this problem, as well as completeness ami uniform boundedness conditions for this family of orthonormal functions.

A N-Width Result for the Generalized Orthonormal Basis Function Model

Abstract In this paper we apply the Kolmogorov n-width concept to a model set based on the so-called generalized orthonormal basis functions (GOBFs), recently developed by Heuberger and co-workers. The main result of this paper is the characterization of the subset of H2(C+) for which the GOBFs are optimal in the Kolmogorov n-width sense. It turns out that each function of this subset is analytic outside a region of the complex plane that depends only on the poles of the GOBFs and on a parameter that defi1ws the size of the region of non-analyticity.

Stationarity conditions for the L 2 error surface of the generalized orthonormal basis functions lattice filter

Abstract Recently, a novel model to perform system identification, based on the so-called generalized orthonormal basis functions, appeared in the automatic control literature. This model generalizes the Laguerre and “two-parameter” Kautz models (which are IIR models with a restricted structure), and has some remarkable properties. Among them is the fact that under ideal conditions the correlation matrix of its internal signals has a block Toeplitz structure. In this paper we explore this property of the correlation matrix, with the result that a lattice version of the aforementioned model is uncovered. This lattice model is then used to determine under which conditions the models mean-squared-error has a stationary point with respect to the position of its poles, assuming that there is no external feedback between the models output and input. The results of this study generalize known similar results for Laguerre and “two-parameter” Kautz filters. As a by-product of this study the Stationarity conditions for the error surface of ARMA( m , m − 1) filters with respect to their pole positions, in an output error configuration and for an arbitrary input signal, are obtained.

Optimal Pole Conditions for Laguerre and Two-Parameter Kautz Models: A Survey of Known Results

Abstract In this paper we present the stationarity conditions for Laguerre and two-parameter Kautz models under a plethora of different conditions, viz., any combination of i) continuous-time or discrete-time models ii) impulsive (or white-noise) input signals or arbitrary input signals iii) a time- or frequency-domain performance criterion based on an Lp, lp or an Hp norm, 1 iv) existence, or not, of interpolation constraints in the response of the model to known signals, in the time- and/or frequency domains v) utilization, or not, of a predefined set of fixed poles in the model. For Laguerre models, in all possible cases the optimality conditions take the same form, namely, either the last optimal weight of the model vanishes or the last optimal weight of the model of the next higher order vanishes. For two-parameter Kautz models similar conditions hold, with the sole difference that the last two weights of the model vanish, instead of only a single weight. Unfortunately, these conditions are also satisfied in other stationarity points of the performance criterion.

A proof of the conjecture of Cohen and Mullen on sums of primitive roots

We prove that for all q > 61, every non-zero element in the nite eld Fq can be written as a linear combination of two primitive roots of Fq. This resolves a conjecture posed by Cohen and Mullen.