General algorithm of computation of c-table and detection of valleys

We present a review of all interesting results concerning the c-table obtained by the authors for the last two decades. These results are not widely known because they were presented in publications of limited circulation. We discuss different computational aspects of software producing the c-tables in the presence of blocs and their evolution following the evolution of the computer environment: effects of the use of 32-bit arithmetic .≈8 digits), 64-bit arithmetic (double precision, ≈16 digits), and Bailey’s Fortran multiprecision package .32 or 64 digits), competition between the ascending and descending algorithms, relationship between the complexity of computation and precision, overflow and underflow problems, competition between different formulas allowing one to overcome the blocs in the c-table, practical simple criterion of detecting numerical zeros in the c-table allowing to identify the blocs, and automatic detection of valleys.Наведено огляд усіх цікавих результатів щодо c-таблиць, одержаних авторами протягом двох останніх десятиліть, які маловідомі з причини публікації у виданнях обмеженого поширення. Розглянуто різні обчислювальні аспекти програм, що продукують с-таблиці з наявністю блоків, а також їх еволюцію, обумовлену еволюцією комп'ютерного середовища, а саме: наслідки використання 32-бітової арифметики (кз 8 розрядів), 64-бітової арифметики (подвійна точність, ≈ 16 розрядів) та високоточного пакету Фортрана Бейлі (32 або 64 розряди), порівняння зростаючих та спадних алгоритмів, зв'язок між складністю обчислень і точністю, проблеми надпотоків та недостатніх потоків, порівняння різних формул, шо дозволяють уникнути блоків у c-таблицях, практичний простий критерій для визначення числових нулів у c-таблицях, що дозволяють ідентифікувати блоки, автоматичне визначення точок мінімуму

On the relation between measures defining the Stieltjes and the inverted Stieltjes functions

A compact formula is found for the measure of the inverted Stieltjes function expressed by the measure of the original Stieltjes function.Встановлено формулу для міри оберненої функції Стільтьєса, що виражена через міру початкової функції Стільтьєса

Universal analytic properties of noise. Introducing the J-Matrix formalism

We propose a new method in the spectral analysis of noisy time-series data for damped oscillators. From the Jacobi three terms recursive relation for the denominators of the Pad\'e Approximations built on the well-known Z-transform of an infinite time-series, we build an Hilbert space operator, a J-Operator, where each bound state (inside the unit circle in the complex plane) is simply associated to one damped oscillator while the continuous spectrum of the J-Operator, which lies on the unit circle itself, is shown to represent the noise. Signal and noise are thus clearly separated in the complex plane. For a finite time series of length 2N, the J-operator is replaced by a finite order J-Matrix J_N, having N eigenvalues which are time reversal covariant. Different classes of input noise, such as blank (white and uniform), Gaussian and pink, are discussed in detail, the J-Matrix formalism allowing us to efficiently calculate hundreds of poles of the Z-transform. Evidence of a universal behaviour in the final statistical distribution of the associated poles and zeros of the Z-transform is shown. In particular the poles and zeros tend, when the length of the time series goes to infinity, to a uniform angular distribution on the unit circle. Therefore at finite order, the roots of unity in the complex plane appear to be noise attractors. We show that the Z-transform presents the exceptional feature of allowing lossless undersampling and how to make use of this property. A few basic examples are given to suggest the power of the proposed method.Comment: 14 pages, 8 figure

Strong-Coupling Expansion for the Hubbard Model

A strong-coupling expansion for models of correlated electrons in any dimension is presented. The method is applied to the Hubbard model in $d$ dimensions and compared with numerical results in $d=1$. Third order expansion of the Green function suffices to exhibit both the Mott metal-insulator transition and a low-temperature regime where antiferromagnetic correlations are strong. It is predicted that some of the weak photoemission signals observed in one-dimensional systems such as $SrCuO_2$ should become stronger as temperature increases away from the spin-charge separated state.Comment: 4 pages, RevTex, 3 epsf figures include

Asymptotic Improvement of Resummation and Perturbative Predictions in Quantum Field Theory

The improvement of resummation algorithms for divergent perturbative expansions in quantum field theory by asymptotic information about perturbative coefficients is investigated. Various asymptotically optimized resummation prescriptions are considered. The improvement of perturbative predictions beyond the reexpansion of rational approximants is discussed.Comment: 21 pages, LaTeX, 3 tables; title shortened; typographical errors corrected; minor changes of style; 2 references adde

General algorithm of computation of c-table and detection of valleys

International audienc

On the relation between measures defining the Stieltjes and the inverted Stieltjes functions

International audienc