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Asymptotics of Hermite–Pade approximants of the first type for Weil functions of discrete Meixner measure and its derivative is studied. We describe a piecewise-analytic curve in the complex plane which attracts part of the zeros of the Meixner multiple orthogonal polynomials. This curve possesses a symmetry property in the equilibrium problem with an external field for logarithmic potential of vector measures with a constraint. Some connection between the considered Hermite–Pade approximants and the theory of Diophantine approximations is presented.

The Newton interpretation of the psi function by rational functions with free poles is studied. A discrete formula for the Rodrigues determinants is obtained and the limit distribution of their zeros is found. The corresponding equilibrium problem of the theory of logarithmic potential is obtained.

Hermite-Padé approximants of the second kind to the Weyl function and its derivatives are investigated. The Weyl function is constructed from the orthogonal Meixner polynomials. The limiting distribution of the zeros of the common denominators of these approximants, which are multiple orthogonal polynomials for a discrete measure, is found. It is proved that the limit measure is the unique solution of the equilibrium problem in the theory of the logarithmic potential with an Angelesco matrix. The effect of pushing some zeros off the real axis to some curve in the complex plane is discovered. An explicit form of the limit measure in terms of algebraic functions is given. Bibliography: 10 titles.

The vocal source and the pulse shape of the glottal flow are determined through the regularized ratio of the speech signal spectra at the intervals of the open and closed vocal slit within each period of the fundamental tone. Three databases were used: Russian numerals for 216 men and 177 women, the base obtained by converting the Russian database by the codec on 9.2 kbps, and the TIMIT database. The pitch period and 7 coefficients for the principal components of the glottal flow provide an average error of recognizing males below 8% for a sequence of 6 vowels. The minimum average recognition error for the initial base of Russian numerals for females makes about 15%, for males in the codec database makes about 15%, and for males in the TIMIT makes about 44%. The minimum average error of males’ recognition in the space of 7 coefficients for the principal components in the Russian database makes about 26%, but about 27% of the speakers have an average error of less than 10%.

The paper considers the inverse problem of finding the shape of a voice-source pulse from a specified segment of a speech signal using a special mathematical model that relates these quantities. A variational method for solving the formulated inverse problem for two new parametric classes of sources is proposed: a piecewise-linear source and an A-source. The error in the obtained approximate solutions of the inverse problem is considered, and a technique to numerically estimate this error is proposed, which is based on the theory of a posteriori estimates of the accuracy in solving ill-posed problems. A computer study of the adequacy of the proposed models of sources, and a study of the a posteriori estimates of the accuracy in solving inverse problems for such sources were performed using various types of voice signals. Numerical experiments for speech signals showed satisfactory properties of such a posteriori estimates, which represent the upper bounds of possible errors in solving the inverse problem. The estimate of the most probable error in determining the source-pulse shapes for the investigated speech material is on average ~7%. It is noted that the a posteriori accuracy estimates can be used as a criterion for the quality of determining the voice-source pulse shape in the speaker-identification problem.

Multiple orthogonal polynomials for three discrete Meixner measures with identical exponential decay at infinity are studied. These polynomials are the denominators of the type II Hermite–Pade approximants to some hypergeometric functions. The limit distribution of zeros of such polynomials scaled in a certain way is described in terms of equilibrium logarithmic potentials and in terms of algebraic curves.

The asymptotic behavior of the Hermite–Pade approximants of the first type for two beta functions are studied. The results are expressed in terms of equilibrium problems of logarithmic potential theory and in terms of meromorphic functions on Riemann surfaces.

The inverse problem with respect to functions proportional to a voice source and volume velocity of the air flow through the glottis is solved as follows: we compute the inverse Fourier transform of the regularized fraction of short-term speech-signal spectra at intervals with an opened (closed) glottis, minimizing the optimality criterion with respect to the regularization parameter and the glottis opening (closing) time. The optimality criterion for solutions includes the values of the volume velocity and its time derivative at the ends of the interval with an opened glottis and the total value of the negative volume velocity. To obtain an empirical error estimate for the solution, experiments using synthesized signals with various parameters, direct measurements of the glottis, and signals synchronously recorded through pairs of microphones of different types are performed. The most probable determination error for the volume velocity is less than 5% for synthetic sources; if the area of the glottis of the source is measured experimentally, then the said error is about 10%. The discrepancy of solutions for the same signal synchronously recorded through a pair of microphones of different types is less than 10%.

The extrema of the logarithmic derivative of the mean energy of a voice signal in the frequency range of 1000–3000 Hz are used to determine the instants of opening and closure of the glottis. The inaccuracy of analysis is estimated with the Arctic CMU database, which contains synchronous recordings of speech signals and electro-glottograms. The estimates of the instants of opening and closure of the glottis, found by the developed algorithm, are compared with the instants of the maximum and minimum of the derivative from electro-glottogram signals, which are taken as the “true” instants. The mean square deviation of the glottal opening instant from the extrema of the derivative from the electro-glottogram signals for different speakers is in the range of 1.03–1.64 ms. The error rate of a false estimate of the glottal opening instant is from 0.01 to 0.14%, and the error rate of omission is from 0.42 to 2.38%. An error-detection algorithm is developed. The mean square deviation with an relative—to the period of the fundamental tone—error in detecting the glottal opening instant is in the range of 13–18% for the most probable error from 0 to +5%.

An optimization problem is considered that is formulated in terms of tropical (idempotent) mathematics and consists in the minimization of a nonlinear function in the presence of linear constraints on the domain of admissible values. The objective function is defined on the set of vectors over an idempotent semifield by a matrix with the use of the operation of multiplicative conjugate transposition. The problem considered is a further generalization of several known problems in which the solution involves the calculation of the spectral radius of the matrix. This generalization implies the use of a more complicated objective function compared with that in the above-mentioned problems, and the imposition of additional constraints. To solve the new problem, an auxiliary variable is introduced that describes the minimum value of the objective function. Then the problem reduces to solving an inequality in which the auxiliary variable plays the role of a parameter. Necessary and sufficient conditions for the existence of solutions to the inequality are used to calculate the parameter, and then the general solution of the inequality is taken as a solution to the original optimization problem. Numerical examples of the solution of problems on the set of two-dimensional vectors are presented.

A generalization of orthogonal polynomials of a discrete variable, namely, Charlier polynomials is studied. The limit distribution of their zeros is obtained by the saddle-point method.

The paper addresses the accuracy of an approximate solution to the inverse problem of retrieving the shape of a voice source from a speech signal for a known signal-to-noise ratio (SNR). It is shown that if the source is found as a function of time with the A.N. Tikhonov regularization method, the accuracy of the found approximation is worse than the accuracy of speech signal recording by an order of magnitude. In contrast, adequate parameterization of the source ensures approximate solution accuracy comparable with the accuracy of the problem data. A corresponding algorithm is considered. On the basis of linear (in terms of data errors) estimates of approximate parametric solution accuracy, parametric models with the best accuracy can be chosen. This comparison has been carried out for the known voice source models, i.e., model [17] and the LF model [18]. The advantages of the latter are shown. Thus, for SNR = 40 dB, the relative accuracy of an approximate solution found with this algorithm is about 1% for the LF model and about 2% for model [17] as compared to an accuracy of 7–8% in the regularization method. The role of accuracy estimates found in speaker identification problems is discussed.

The paper studies the asymptotic behavior of the function for the area of the glottis near moments of its opening and closing for two mathematical voice source models. It is shown that in the first model, the asymptotics of the area function obeys a power law with an exponent of no less that 1. Detailed analysis makes it possible to refine these limits depending on the relative sizes of the intervals of a closed and open glottis. This work also studies another parametric model of the area of the glottis, which is based on a simplified physical-geometrical representation of vocal-fold vibration processes. This is a special variant of the well-known two-mass model and contains five parameters: the period of the main tone, equivalent masses on the lower and upper edge of vocal folds, the coefficient of elastic resistance of the lower vocal fold, and the delay time between openings of the upper and lower folds. It is established that the asymptotics of the obtained function for the area of the glottis obey a power law with an exponent of 1 both for opening and closing.

Two methods for estimating the vocal-tract length equivalent to the homogeneous acoustic tube length are investigated. One method is based on calculating the tract length from the difference between the frequencies of the adjacent local spectral maxima, which exceed 4 kHz. In the other method, the vocal-tract length is calculated according to the average frequency of the second formant determined by the frequencies of first three formants. In addition, various variants of analysis are discussed irrespective of the context and with allowance for known vowels. The probability that the speaker gender is correctly recognized via two methods is about 13%, and its value is almost independent of the knowledge of the context. The probabilities that male and female voices are correctly recognized according to the difference of higher formants are, respectively, 31 and 25.5% regardless of the context and 37 and 31% with allowance for it. The probabilities of correct recognition of male and female voices reach to 27 and 21.5%, respectively, if context-independent recognition is performed from the average frequency of the second formant and 43 and 35.5% after context-dependent recognition with the known vowel type.

The paper examines two examples of multiple orthogonal polynomials generalizing orthogonal polynomials of a discrete variable, meaning thereby the Meixner polynomials. One example is bound up with a discrete Nikishin system, and the other leads to essentially new effects. The limit distribution of the zeros of polynomials is obtained in terms of logarithmic equilibrium potentials and in terms of algebraic curves. Bibliography: 9 titles.

In the problem of the voice identification, the residual signal that is calculated using the linear prediction is approximated by a mathematical model of the source using the least squares procedure. The approximated results depend on the initial conditions, which must be additionally determined. The problem is solved with the aid of clusterization with respect to multiple data obtained for arbitrary initial conditions. The Euclidian metric appears to be the best one. The centers of clusters for men and women are significantly different. We fail to determine the explicit dependence of the centers of clusters on the fundamental tone frequency.

We construct multipoint Hermite—Pade approximations for two beta functions generating the Nikishin system with infinite discrete measures and unbounded supports. The asymptotic behavior of the approximants is studied. The result is interpreted in terms of the vector equilibrium problem in logarithmic potential theory in the presence of an external field and constraints on measure.

For multiple orthogonal polynomials with respect to two Pollaczek weight functions weak asymptotics are obtained. It is shown that a solution of a vector equilibrium problem of the theory of logarithmic potential in the presence of an external field and with restriction imposed on the measures is given by the limit measure of the distribution of zeros. Bibliography: 17 titles.

Hermite-Pade approximations are constructed for three functions: the square root, logarithm, and arcsine. The Rodrigues formula is presented. An estimate of the measure of linear independence for values of these functions at natural points is obtained.

Efficiency of automatic recognition of male and female voices based on solving the inverse problem for glottis area dynamics and for waveform of the glottal airflow volume velocity pulse is studied. The inverse problem is regularized through the use of analytical models of the voice excitation pulse and of the dynamics of the glottis area, as well as the model of one-dimensional glottal airflow. Parameters of these models and spectral parameters of the volume velocity pulse are considered. The following parameters are found to be most promising: the instant of maximum glottis area, the maximum derivative of the area, the slope of the spectrum of the glottal airflow volume velocity pulse, the amplitude ratios of harmonics of this spectrum, and the pitch. On the plane of the first two main components in the space of these parameters, an almost twofold decrease in the classification error relative to that for the pitch alone is attained. The male voice recognition probability is found to be 94.7%, and the female voice recognition probability is 95.9%.

The Hermite-Pade approximations of the second type for algebra generated by a generalized Nikishin system of Markov functions corresponding to an infinite branching graph are investigated. Arithmetical applications of this construction are given. Namely, lower estimates for polynomials with integer coefficients in logarithms of some rational numbers are obtained. These estimates partially refine some known results obtained earlier by the Siegel method.

We propose a new method for proving the Zudilin-Rivoal theorem stating, in particular, that the sequence of values of the Dirichlet beta function at even natural points contains infinitely many irrational values. For polylogarithms, we use Hermite—Pade approximations of the first type, invariant with respect to the Klein group. Quantitative additions to this theorem are obtained.

Joint rational approximations to a collection of logarithms with a common denominator are considered. In this case each function is approximated at its zero. The asymptotic behavior of approximations is completely described in the real case of two logarithms, convergence of approximations is also investigated. A theoretic-potential interpretation is proposed.

The spreading of the quantum probability density for the highly-excited states of a single-particle system with an exponential-type potential on the positive semiaxis is quantitatively determined in both position and momentum spaces by means of the Boltzmann–Shannon information entropy. This problem boils down to the calculation of the asymptotics of the entropy-like integrals of the modified Bessel function of the second kind (also called the Mcdonald function or Basset function). The dependence of the two physical entropies on the large quantum number n is given in detail. It is shown that the semiclassical (WKB) position–space entropy grows slower than the corresponding quantity of not only the harmonic oscillator but also the single-particle systems with any power-type potential of the form V(x)=x2k, x∈R and k∈N. The momentum–space entropy, calculated with a method based on the properties of the Mcdonald function, is rigorously found to have a behavior of the form −ln ln n, in strong contrast with the...

1. Studies of the processes of compensation of speech pathology of various types and the artificial disorders of the processes of speech production and perception gave rise to the concept that the system of articulation control was able to solve various inverse problems. The main inverse problem in this system consists in calculation of articulation control commands designed to reproduce acoustic parameters of the speech signal heard. For example, this problem should be solved during language learning. There are grounds to believe that optimum solutions of the main inverse problem and its various variants can be found by the control system using a variety of its own internal criteria of solution selection. Therefore, formulation and analysis of mathematical inverse problems of vocal tract is, on one hand, of considerable interest in terms of elucidation of properties of the articulation control system and, on the other hand, it is a promising system for solving applied problems of automatic recognition, synthesis, and compression of speech. Mathematical simulation based on experimental measurements of vocal tract shape and movement of articulators showed that the variational method is the most effective for solving the inverse problems of speech. This method consists in the search for a conditional optimum of a certain functional (optimality criterion) in the space of parameters of mathematical models of speech production. The necessary condition of adequacy of the inverse problem solution for vocal tract to actual processes of speech production is that the mathematical criteria of optimality during solution of inverse problem are consistent with the internal criteria of optimality of the system of articulation control. It is well known that certain mathematical criteria of optimality (e.g., instantaneous criteria of work and kinetic energy) provide satisfactory solution of both static problems for fricative and vowels [1, 2] and dynamic inverse problems [3, 4]. However, there are grounds to believe that the mathematical criteria of optimality of integral type (i.e., criteria calculated for a certain interval of time) should be used for solving a number of inverse problems of speech. One of such grounds is based on the properties of cross-striated muscles. Even merely to maintain the articulator in a fixed in time position other than neutral, the motor units of crossstriated muscles should be continuously contacted, requiring thereby energy expenditure. Adequate mathematical criteria of optimality during solution of corresponding inverse problems should meet the following requirements: (1) provide adequate reproduction of experimental data; (2) reproduce the effects of compensation in the case of fixation of given articulator; (3) demonstrate the reorganization of control score in the case of modification of the articulation tempo. The goal of this work was to find such adequate criteria of optimality by comparing a number of momentary and integral criteria of optimality. The system of articulators “jaw‐tongue tip” is a convenient test object for detailed studies of various optimality criteria. This system was analyzed in this work. 2. Let x 1 ( t ) be a vertical displacement of the jaw from neutral position; x 2 ( t ) , similar displacement of the tongue tip; and u 1 ( t ) and u 2 ( t ) , the commands of the control systems for these articulators. Each articulator is a system with lumped parameters. Therefore, their movement is described by ordinary differential equations + + = , where k = 1, 2, with zero initial conditions. In these equations, ω k and g k are the proper frequency and damping coefficient of natural oscillations of the k th articulator, respectively; m k is the mass of the articulator. The value u k ( t ) in this case can be interpreted as effective muscular force involved in articulator control. The total displacement of articulators y ( t ) = x 1 ( t ) + x 2 ( t ) , induced by control forces u ( t ) = ( u 1 ( t ), u 2 ( t )) is described by the following equation

The spreading of the position and momentum probability distributions for the stable free oscillations of a circular membrane of radius l is analyzed by means of the associated Boltzmann–Shannon information entropies in the correspondence principle limit (n → ∞, m fixed), where the numbers (n, m), n ∈ ℕ and m ∈ ℤ, uniquely characterize an oscillation of this two-dimensional system. This is done by solving the short-wave asymptotics of the physical entropies in the two complementary spaces, which boils down to the calculation of the asymptotic behavior of certain entropic integrals of Bessel functions. It is rigorously shown that the position and momentum entropies behave as 2 ln (l) + ln (4π) - 2 and ln (n) - 2 ln (l) + ln (2π3) when n → ∞, respectively. So the total entropy sum has a logarithmic dependence on n and it does not depend on the membrane radius. The former indicates that the ordering of short-wavelength oscillations is exactly identical for the entropic sum and the single-particle energy. The latter holds for all oscillations of the membrane because of the uniform scaling invariance of the entropy sum.

This is a survey of results about the behaviour of Hermite-Pade approximants for graphs of Markov functions, and a survey of interpolation problems leading to Apery's result about the irrationality of the value ζ(3) of the Riemann zeta function. The first example is given of a cyclic graph for which the Hermite-Pade problem leads to Apery's theorem. Explicit formulae for solutions are obtained, namely, Rodrigues' formulae and integral representations. The asymptotic behaviour of the approximants is studied, and recurrence formulae are found.

We give examples of well-posed problems of joint Hermite–Pade approximations of series in two variables. We find Rodrigues formulas and integral representations for solutions. We also study the limit distribution of zeros of the corresponding polynomials. Constructions are based, on the one hand, on the classical Appel polynomials orthogonal in a triangle and, on the other hand, on various ways of proving Apery's theorem about irrationality of the number ζ(3).

The linear independence of the values at certain rational points of generalized polylogarithms, which generate the algebra of all analytic functions with three logarithmic branch points, is established. Hermite-Pade approximations for an Angelesco-Nikishin system defined by a complete binary tree are used.

The inverse problem for the vocal tract is under consideration from the viewpoint of the ill-posed problem theory. The proposed approach, which permits overcoming the difficulties related to ambiguity and instability, is based on the variational regularization with constraints. The work of articulators is used as a functional of regularization and a criterion of optimality for finding an approximate solution. The measured acoustical parameters of the speech signal serve as external constraints while the geometry of the vocal tract, the mechanics of the articulation, and the phonetic properties of the language play the role of internal constraints. An effective numerical implementation of the proposed approach is based on a local piecewise linear approximation of the articulatory-to-acoustics mapping and a polynomial approximation of the discrepancy measure. A heuristic method named the “calibrating curves method” is applied for estimating the accuracy of the obtained approximate solution. It was shown that in some cases the error of the inverse problem solution is weakly dependent on the errors of formant frequency measurements. The vocal tract shapes obtained by virtue of the proposed approach are very close to those measured in X-ray experiments.

A matrix continued fraction is defined and used for the approximation of a function F known as a power series in 1/zwith matrix coefficientsp×q, or equivalently by a matrix of functions holomorphic at infinity. It is a generalization of P-fractions, and the sequence of convergents converges to the given function. These convergents have as denominators a matrix, the columns of which are orthogonal with respect to the linear matrix functional associated to F. The case where the algorithm breaks off is characterized in terms of F.

The solution of the Cauchy problem for semi-infinite chains of ordinary differential equations, studied first by O. I. Bogoyavlenskii in 1987, is obtained in terms of the decomposition in a multidimensional continuous fraction of Markov vector functions (the resolvent functions) related to the chain of a nonsymmetric operator; the decomposition is performed by the Euler-Jacobi-Perron algorithm. The inverse spectral problem method, based on Lax pairs, on the theory of joint Hermite-Pade approximations, and on the Sturm-Liouville method for finite difference equations is used.

An algebraic theory of orthogonality for vector polynomials with respect to a matrix of linear forms is presented including recurrence relations, extension of the Shohat?Favard theorem, of the Christoffel?Darboux formula, and its converse. The connection with orthogonal matrix polynomials is described.

The Hermite-Pade approximants are studied for systems of Markov functions (introduced in this paper) with structure described by a graph. Results of an asymptotic nature are stated in terms of certain equilibrium problems of potential theory concerning vector potentials.

A new proof of the fact that π2 is transcendental is proposed. A modification of Hermite's method for an expressly constructed Nikishin system is used. The Beukers integral, which was previously used to prove Apery's theorem on the irrationality of ζ(2) and ζ(3) is a special case of this construction.

We study the stationary Schrodinger equation with the degenerate potential U(x) = x2r, r e Z+, which describes phase transitions in quantum systems. The symmetry of the problem is revealed and an analytical procedure is developed for finding the eigenvalues of such potentials. The eigenvalues and the lowest energy levels are found numerically for r=2, 3, ..., 18.