Padé Approximant

We calculate high-temperature graph expansions for the Ising spin glass model with 4 symmetric random distribution functions for its nearest neighbor interaction constants J ij . Series for the Edwards-Anderson susceptibility χ EA are obtained to order 13 in the expansion variable (J/(k B T )) 2 for the general d-dimensional hyper-cubic lattice, where the parameter J determines the width of the distributions. We explain in detail how the expansions are calculated. The analysis, using the Dlog-Padé approximation and the techniques known as M1 and M2, leads to estimates for the critical threshold (J/(k B T c )) 2 and for the critical exponent γ in dimensions 4, 5, 7 and 8 for all the distribution functions. In each dimension the values for γ agree, within their uncertainty margins, with a common value for the different distributions, thus confirming universality.

In Fitzpatrick and Flynn (J. Symbolic Comput. 13 (1992) 133), a Gröbner basis technique for multivariable Padé approximation problems was developed under a rather restrictive hypothesis on the shape of the numerator and denominator in relation to the approximation conditions desired. In this article, we show that their hypotheses can be replaced by other less stringent conditions, and we show how to compute some standard forms of multivariable approximants through several examples.

We show that for a vast class of matrix Lie groups, which includes the orthogonal and the symplectic, diagonal Padé approximants of log((1 + x)/(1x)) are structure preserving. The conditioning of these approximants is analyzed. We also present a new algorithm for the Briggs-Padé method, based on a strategy for reducing the number of square roots in the inverse scaling and squaring procedure.

Based on the homotopy perturbation method (HPM) and Padé approximants (PA), approximate and exact solutions are obtained for cubic Boussinesq and modified Boussinesq equations. The obtained solutions contain solitary waves, rational solutions. HPM is used for analytic treatment to those equations and PA for increasing the convergence region of the HPM analytical solution. The results reveal that the HPM with the enhancement of PA is a very effective, convenient and quite accurate to such types of partial differential equations.

We employ a systematic and model-independent method to extract, from space-and time-like data, the η and η transition form factors (TFFs) obtaining the most precise determination for their low-energy parameters and discuss the Γ η→γγ impact on them. Using TFF data alone, we also extract the ηη mixing parameters, which are compatible to those obtained from more sophisticated and input-demanding procedures.

We consider the generalized anharmonic oscillator p2~_ x2 § fix s. This model is interesting because the coefficients of the perturbation expansions for the energy eigenvalues diverge faster than (2n)!. Evidence is found against the convergence of the Pa44 approximants. The approximation method based on the Borel suminability exhibits the same features found for the fix 4 perturbation. The numerical analysis suggests also the introduction of another summarion method, whose approximants have the useful monotonieity properties of the Pad4 ones.

In this paper, we introduce a novel analytical solution to Tolman-Oppenheimer-Volkoff (TOV) equation, which is ultimately a hydrostatic equilibrium equation derived from the general relativity in the framework of relativistic isothermal spheres. Application of the traditional power series expansions on solving TOV equation results in a limited physical range to the convergent power series solution. To improve the convergence radii of the obtained series solutions, a combination of the two techniques of Euler-Abel transformation and Padé approximation has done. The solutions are given in - and - phase planes taking into account the general relativistic effects = 0.1, 0.2 and 0.3. An Application to a neutron star has done. A Comparison between the results obtained by the suggested approach in the present paper and the numerical one indicates a good agreement with a maximum relative error of order 10 -3 , which establishes the validity and accuracy of the method. The procedure we have applied accelerated the power series solution with about ten times than of traditional one.

The main purpose of the paper is to present some powerful data on the advantage of the rational approximation procedure based on Hermite-Padé polynomials over the Padé approximation procedure. The first part of the paper is devoted to some numerical examples in this direction. The second part will be devoted to some theoretical results. In particular, we demonstrate our ideas about the advantage of rational Hermite-Padé approximants over Padé approximants analyzing the analytical structure of the frequency function ν of the free Van der Pol equation. Bibliography: [47] titles.

Актуальность исследования аппроксимации Паде-Фабера (обобщения аппроксимации Паде и аппроксимации Паде-Чебышёва) марковских функций, несомненная с точки зрения математического анализа, дополнительно аргументирована потребностями вычислительной математики. Конструктивно доказана теорема о существовании поддиагональных аппроксимант. Даны разные оценки погрешности аппроксимации. Теоретические утверждения проиллюстрированы результатами численных экспериментов. Библиография: 25 названий. 1 В данной статье речь идет исключительно о так называемой нелинейной аппроксимации Паде-Фабера. 2 Мы используем термин "марковская функция" в слегка расширенном смысле, разрешая носителю порождающей меры быть подмножеством в ]-∞, 0]. Другими словами, мы хотим, чтобы результаты были применимы к импедансным функциям . 3 Как обычно, аппроксиманты порядка [m + l/m] с l -1 могут быть исследованы аналогично: понадобится лишь немного изменить меру, определяющую ортонормированные многочлены. c ⃝ Л. А. Книжнерман, 2009

Квадратура Гаусса-Арнольди для функции (zI -A) -1 ϕ, ϕ и Паде-подобная рациональная аппроксимация функций марковского типа Исследована эффективность применения квадратуры Гаусса-Арнольди для вычисления величины (zI -A) -1 ϕ, ϕ , где A -ограниченный оператор в гильбертовом пространстве, а ϕ -ненулевой вектор из этого пространства. Установлены необходимое и достаточное условия эффективности квадратуры в случае нормального оператора. Приведен пример ненормального оператора, в применении к которому обсуждаемая квадратура неэффективна. Показано, что в определенных случаях квадратура Гаусса-Арнольди связана с Паде-подобной рациональной аппроксимацией (с полюсами в числах Ритца) функций марковского типа и, в частности, может использоваться как средство локализации полюсов рационального возмущения. Даны оценки погрешности, применимые и в тех случаях, когда классическая аппроксимация Паде не работает или ее работоспособность не гарантирована. Теоретические утверждения и гипотезы проиллюстрированы результатами численных экспериментов. Библиография: 44 названия. 2 Для краткости обозначений мы будем использовать матрицы с операторными компонентами и работать с ними по обычным матричным правилам. Например, при h ∈ H символ h * будет обозначать линейный функционал h * : H → C, g → ⟨g, h⟩. Оператор A будет действовать на матрицы покомпонентным образом, скажем, A(z 1 , . . . , zs) = (Az 1 , . . . , Azs). Положим также (g ij ) * = (g * ji ), g ij ∈ H . 3 Этот исторически сложившийся матрично-алгебраический термин не имеет отношения к полю как целостному кольцу с делением.

СЛОБОДАН Ј. ПАВЛОВИЋ* Универзитет у Новом Саду Филозофски факултет Одсек за српски језик и лингвистику ТЕОРИЈСКЕ ОСНОВЕ ДАНИЧИЋЕВЕ ПАДЕЖНЕ СИНТАКСЕ (ПРИЛОГ ИСТОРИЈИ СРПСКЕ ФИЛОЛОГИЈЕ)** У раду се разматрају теоријске и методолошке основе Даничићевог приступа падежу у Србској синтакси (1858). Прва српска синтакса на стаје у време успона компаративне граматике, која у сфери син так сич ких истраживања трага за изворним и општим значењем падежа, да би се већ током прве половине XIX века стабилизовале две супротстављене падежне теорије, и то локалистичка и логичко-граматичка. Да ни чи ћева Србска синтакса настаје као израз компромиса између ових две ју теорија. Кључне речи: српска синтакса, Ђура Даничић, локалистичка падеж на теорија, логичко-граматичка падежна теорија, граматички падежи, просторни падежи.

The quasi -relativistic harmonic oscillator bound -states constructed by Znojil ( 1996 J. Phys. A29 2905) are investigated via a new methodical proposal. Compared to those obtained by an anonymous referee ( from a direct numerical integration method) of Znojil's paper [3], our results appeared to be more favorable than those obtained by Znojil via quasi -perturbative, variational, Hill -determinant and Riccati -Padé methods. Bound -states with larger angular momenta l are also constructed.

Analytical solutions for the normal stress differences in large-amplitude oscillatory shear flow (LAOS), for continuum or molecular models, normally take the form of the first few terms of a series expansion in the shear rate amplitude. Here, we improve the accuracy of these truncated expansions by replacing them with rational functions called Padé approximants. Specifically, we examine replacing the truncated expansion for the corotational Jeffreys fluid with its Padé approximants for the normal stress differences. We begin by extending the Goddard integral expansion (GIE) from its third [J Non-Newt Fluid Mech 166, 1081 (2011)] to its fifth order term in the shear rate amplitude. We then explore the Padé approximants for this extension. We uncover the best approximant, the approximant, and compare this with both the GIE and the exact solution [Macromol Theory Simul 24, 352 (2015)]. We use Ewoldt grids to show the stunning accuracy of our approximant in LAOS. We quantify this accuracy with an objective function and then map its onto Pipkin space. We find the approximant to be a simple accurate expression for the normal stress differences in LAOS. Our applications illustrate how to use our new approximant reliably. For this, we use the Spriggs relations to generalize our best approximant to multimode, and then, we compare with measurements on molten high-density polyethylene, and on dissolved polyisobutylene in isobutylene oligomer.

The nonlocal boundary conditions (NLBCs) for high-order finite-difference parabolic equations (PEs) are obtained by Z transformation of the discrete PE in a homogeneous medium. The considered NLBCs include the free-space radiation condition, possibly with a density jump at the NLBC interface, the NLBC at an arbitrary impedance interface, and the NLBCs for sources and the starting field beyond the NLBC interface. The derivation is presented for the multiterm Padé PE model OWWE (one-way wave equation), but the developed technique is applicable to a broad class of finite-difference PEs. The obtained NLBCs are exact for the given finite-difference scheme. They are not limited by the order of Padé approximation or by the PE steps in range and depth. The NLBC convolution coefficients are calculated by the numerical inverse-Z transformation. The accuracy and performance of the algorithm are analyzed for several benchmark problems. The solution is robust for the range steps over 25 waveleng...

The photoabsorption cross-section of an ion immersed in a plasma is studied on the basis of the Thomas–Fermi approximation for the equilibrium electron distribution and Bloch's classical hydrodynamic model for collective motion of the electrons. The frequency-dependent cross-section scales with the nuclear charge, and depends strongly on the plasma density and temperature. An approximation of the frequency dependence is constructed with the aid of sum rules and Padé approximants.

The fact that physical phenomena are modelled, mostly, by nonlinear differential equations underlines the importance of having reliable methods to solve them. In this work, we present a comparison of homotopy perturbation method (HPM), nonlinearities distribution homotopy perturbation method (NDHPM), Picard, and Picard-Pade´methods to solve Michaelis-Menten equation. The results show that NDHPM possesses the smallest average absolute relative error 1.51(À2) of all tested methods, in the range of r 2 ½0; 5. Also, we introduce the combination of Picard's iterative method and Pade´approximants as an alternative to reduce complexity of Picard's solutions and increase accuracy.

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This work is devoted to the thermodynamics of high-temperature dense hydrogen plasmas in the pressure region between 10 −1 and 10 2 Mbar. In particular we present for this region results of extensive calculations based on a recently developed path integral Monte Carlo scheme (direct PIMC). This method allows for a correct treatment of the thermodynamic properties of hot dense Coulomb systems. Calculations were performed in a broad region of the nonideality parameter Γ 3 and degeneracy parameter n e Λ 3 10. We give a comparison with a few available results from other path integral calculations (restricted PIMC) and with analytical calculations based on Padé approximations for strongly ionized plasmas. Good agreement between the results obtained from the three independent methods is found.

For time integration in finite element analysis, a higher order counterpart of the widely used Newmark method is formulated by applying the three step fourth order Adams-Moulton (AM) method to lightly damped systems with accelerations. A linear system arises for which the solution effort is exactly the same as in Newmark. Using Pade approximations, step-wise and cumulative errors in both methods are assessed for both free and forced response. For comparable accuracy Newmark requires much smaller time steps than AM, even in low frequency modes. Numerical damping at fourth order is introduced into AM. Newmark is numerically stable for all time steps. However, owing to extraneous eigenvalues AM exhibits a critical ratio of time step to period, above which numerical instability occurs. This is problematic in high frequency modes with small time periods. It is shown that this stability limit is not obviated by damping, whether viscous or 'numerical'. Instead, a discussion is given of removal of higher order modes using filters based on the Wavelet Transform.

It is shown that the non-relativistic ground state energy of helium-like and lithium-like ions with static nuclei can be interpolated in full physics range of nuclear charges Z with accuracy of not less than 6 decimal digits (d.d.) or 7-8 significant digits (s.d.) using a meromorphic function in appropriate variable with a few free parameters. It is demonstrated that finite nuclear mass effects do not change 4-5 s.d. for Z ∈ [1, 50] for 2-,3-electron systems and the leading relativistic and QED corrections leave unchanged 3-4 s.d. for Z ∈ [1, 12] in the ground state energy for 2-electron system, thus, the interpolation reproduces definitely those figures. A meaning of proposed interpolation is in a construction of unified, two-point Pade approximant (for both small and large Z expansions) with fitting some parameters at intermediate Z.

Concerns about the future energy needs challenge energy experts to present advanced power grids making them smarter. Smart grids employ digital technology and are based on highly collaborative and responsive decision-making strategies. Current trends to control and manage their operation lead to the use of multi-agent system (MAS) technology. This paper aims to present the design, specification, and application of the Python Agent DEvelopment (PADE) framework, an open source platform implemented in Python language and conceived for the implementation of multi-agent systems on power systems. PADE is compliant with specifications of the Foundation for Intelligent Physical Agents and eases the development of solutions to power systems based on MAS. For demonstration of the effectiveness of PADE, an automatic restoration system MAS-based is presented. A test bed that encompasses agents embedded on computation devices is used to demonstrate the practical implementation of a PADE-based multi-agent systems in a power distribution network.

This paper describes the integration process between two tools in order to perform co-simulation for representation and analysis of dynamic environments in the context of smart grids. The integrated tools are Mosaik, a software to co-simulation management, and PADE, a software to multi-agent systems development. As a study case for demonstrate the integration, a scenary was utilized composed of a low voltage electricity distribution grid with 37 load bus, 20 photo-voltaic distributed generations, randomly connected to load bus, as well as, 20 PADE agents associated to distributed generation, modeling the behavior of electricity storage systems. The simulation results show the integration happening and demonstrate how useful is to model the dynamics of distributed electric resources with multi-agent systems.

In this work, the Daftardar-Jafari method has been submitted to deal with the approximated solutions for the nonlinear damped generalized regularized long-wave (DGRLW) equation with some given variable coefficient. by using this iterative technique we have got rid of any additional assumptions in order to get the solution of the nonlinear DGRLW equation where this solution has been evaluated in a series form with several computable components. Three numerical examples have been tested to show the easy-applicable scheme for the method where the third example has been solved by using the modified Daftardar-Jafari method in order to deal with an inhomogeneous case for the DGRLW equation. Also, our approximate solutions have been numerically reported and figured with the exact solution.

This paper investigates the problem of boundary layer stagnation point flow and heat transfer of couple stress fluid containing nanoparticles and flowing over an exponentially stretching surface in a porous medium. The governing equations of the couple stress fluid model for velocity, temperature and nanoparticle profiles are given as part of the boundary layer approach. The nonlinear partial differential equations are simplified by using similar transformations. The analytical solutions of simplified equations are found using the homotopy analysis method. The convergence of the HAM solutions is discussed by plotting -curves and also through homotopy padé approximation. The physical features of pertinent parameters are discussed through graphs.

We perform a dimension analysis for colliding viscoelastic spheres to show that the coefficient of normal restitution ⑀ depends on the impact velocity g as ⑀ϭ1Ϫ␥ 1 g 1/5 ϩ␥ 2 g 2/5 ϯ•••, in accordance with recent findings. We develop a simple theory to find explicit expressions for coefficients ␥ 1 and ␥ 2. Using these and few next expansion coefficients for ⑀(g) we construct a Padé approximation for this function which may be used for a wide range of impact velocities where the concept of the viscoelastic collision is valid. The obtained expression reproduces quite accurately the existing experimental dependence ⑀(g) for ice particles.

This paper presents a numerical comparison of third order o L stable − numerical schemes for the two-dimensional parabolic partial differential equations subject to nonlocal boundary conditions. These numerical schemes are based on Pade` approximations to the matrix exponentials arising from the use of the method of lines and are tested on a model problem. The numerical comparison shows that the (0, 3)-Padé scheme is efficient and provide very accurate results as compared to (1, 2)-Padé scheme.

Asthma is one of the most prevalent and costly chronic conditions in the United States, which cannot be cured. However, accurate and timely surveillance data could allow for timely and targeted interventions at the community or individual level. Current national asthma disease surveillance systems can have data availability lags of up to two weeks. Rapid progress has been made in gathering nontraditional, digital information to perform disease surveillance. We introduce a novel method of using multiple data sources for predicting the number of asthma-related emergency department (ED) visits in a specific area. Twitter data, Google search interests, and environmental sensor data were collected for this purpose. Our preliminary findings show that our model can predict the number of asthma ED visits based on near-real-time environmental and social media data with approximately 70% precision. The results can be helpful for public health surveillance, ED preparedness, and targeted patient interventions.

This paper presents a numerical comparison of third order o L stable − numerical schemes for the two-dimensional parabolic partial differential equations subject to nonlocal boundary conditions. These numerical schemes are based on Pade` approximations to the matrix exponentials arising from the use of the method of lines and are tested on a model problem. The numerical comparison shows that the (0, 3)-Padé scheme is efficient and provide very accurate results as compared to (1, 2)-Padé scheme.

This paper presents a numerical comparison of third order o L stable − numerical schemes for the two-dimensional parabolic partial differential equations subject to nonlocal boundary conditions. These numerical schemes are based on Pade` approximations to the matrix exponentials arising from the use of the method of lines and are tested on a model problem. The numerical comparison shows that the (0, 3)-Padé scheme is efficient and provide very accurate results as compared to (1, 2)-Padé scheme.

Behavioral models are used for top-down design and for bottom-up verification. The efficient simulation of large-scale dynamical system needs a systematic procedure for order reduction of the original circuit. A method based on the moment matching for behavioral model generation of analog integrated circuits and by comparison some approaches to obtain an accurate approximation of the circuit function are presented. It is shown that we can handle the MIMO system using the semi-state method and Padé approximation. An illustrative example is given and some conclusions are pointed out.

In this investigation, the series solutions of mixed convection boundary layer flow over a vertical permeable cylinder are constructed. Two types of series as well numerical solutions are presented by choosing exponential and rational bases. The resulting differential system are solved by employing homotopy analysis method and Pade technique which have been proven to be successful in tackling non-linear problems. We offer various verifications of the solutions by comparing to existing, documented results and also mathematically, through reduction of sundry parameters. The convergence of the series solutions have been discussed explicitly. Comparison with existing results reveal that the series solutions are not only valid for large (aiding flow) but also for small values (opposing flow) of l and the dual solutions do not obtain in both cases.

In the context of Pad~ approximation theory some methods are studied for the calculation of the resonances and the phase shifts in potential scattering. They are extensions of a method already used for the determination of the bound states and are based on the variational content of the Padd approximation. Essentially these methods amount to work with off-shell amplitudes taking the off-shell momenta as variational parameters. Numerical examples are reported which always show a definite improvement over the results of the conventional on-shell Padd approximation.

In the context of Pad~ approximation theory some methods are studied for the calculation of the resonances and the phase shifts in potential scattering. They are extensions of a method already used for the determination of the bound states and are based on the variational content of the Padd approximation. Essentially these methods amount to work with off-shell amplitudes taking the off-shell momenta as variational parameters. Numerical examples are reported which always show a definite improvement over the results of the conventional on-shell Padd approximation.

Resumo: This paper describes the integration process between two tools in order to perform co-simulation for representation and analysis of dynamic environments in the context of smart grids. The integrated tools are Mosaik, a software to co-simulation management, and PADE, a software to multi-agent systems development. As a study case for demonstrate the integration, a scenary was utilized composed of a low voltage electricity distribution grid with 37 load bus, 20 photo-voltaic distributed generations, randomly connected to load bus, as well as, 20 PADE agents associated to distributed generation, modeling the behavior of electricity storage systems. The simulation results show the integration happening and demonstrate how useful is to model the dynamics of distributed electric resources with multi-agent systems.

The number of visits is decreasing because in addition to the impact of the pandemic, but visitors' interest in staying at the Pade Hotel Aceh has also decreased. Friendly and responsive service can increase guest satisfaction. Location is a key factor influencing stay decisions, while hotel facilities also play an important role in determining customer comfort and satisfaction. This study aims to identify the effect of service quality, facilities, and location on stay decisions at the Pade Hotel Aceh. There were 100 respondents in this study where participants included all visitors who stayed at Pade Hotel Aceh. The results showed a significant influence of Service Quality and Location on visitors' Staying Decisions. Meanwhile, the Facility factor proved to have no significant effect on the Staying Decision. The measurement of the representation of this research variable is 66.2% so it is necessary to consider other additional factors related that can influence visitors' decisions to stay.

Pasional MR 164 najstariji je hagiografski, rukopisni latinski kodeks čuvan u Knjižnici Zagrebačke nadbiskupije Metropolitani. Ovaj obiman hagiografski zbornik per circulum anni pisan karolinom napisan je najvjerojatnije u drugoj polovici 10. stoljeća za potrebe ravenske crkve Sant'Apollinare Nuovo što se dokazuje sanktoralom i tekstualno-umjetničkim analizama. Hagiografski tekstovi koji su u njemu zapisani osobito su starog postanja i napisani su odmah nakon vremena persekucija. Inačica latinskog jezika najstarijih hagiografskih tekstova jest biblijski latinitet čija je osnova vulgarni latinitet koji u sebi sadrži priličan broj hebrejizama i grecizama, naslijeđenih iz svetopisamskih prijevoda. Rad analizira osobito česte hebrejizme, grecizme, vulgarizme u biblijskom jeziku ne-biblijskih tekstova, kao i elemente sintakse padeža klasičnog latiniteta koji su zbog svoje specifične i česte upotrebe postali sasvim osobiti za ovu inačicu latinskog jezika. Ključne riječi. -Pasional MR 164, vulgarni latinitet, biblijski latinitet, hagiografski tekstovi.

This paper is concerned with a thorough investigation in achieving exact analytical solution for the Van der Pol (VDP) nonlinear oscillators models via Adomian decomposition method (ADM). The models are nonlinear time dependent second order ordinary differential equations. ADM has already been applied, in existing literatures, to obtain approximate results. But, we adapt the method by adjusting the source term; a procedure that is base on the asymptotic Taylor's series expansion on the term that would have resulted to proliferation of terms during the invertible process. Then, the rational Pade Approximant is applied to clarify and get a better understanding of the uniqueness and convergence of our findings. Two models were used as illustrations and their result pictured to indicate their behaviour in the given domains. And, we found that the adaptation on the models yielded exact results which were further displayed in constructed tables.

A generalization of the Krylov-Eckhoff method is investigated for removing of the classical Gibbs phenomenon. Convergence acceleration scheme for Fourier expansions of piecewise smooth functions is derived. Numerical results are presented and discussed.

In this paper, we present a new ANM continuation algorithm with a predictor based on a new Padé approximant and without the use of a correction process. The ANM is a numerical method to obtain the solution of a nonlinear problem as a succession of branches . Each branch is represented by a vectorial series which is obtained by inverting only one tangent stiffness matrix. The series representation can be replaced by a rational representation which reduces the number of branches necessary to obtain the entire branch of desired solution. In this work, we discuss the use of a new vectorial Padé approximants in the ANM algorithm. In previous works [1, 2, 4, 5], the Padé approximants have been introduced after an orthonormalization of the terms of the vectorial series. In a recent article [8], we have demonstrated that the coefficients b M i of the Padé approximants can be chosen in an arbitrary manner. For this purpose, we propose a new choice of vectorial Padé approximants {U M p } at order M which minimizes the relative error between two consecutive vectorial Padé approximants {U M p } at order M and {U M-1 p } at order M -1. This minimization has been made by a judicious choice of the coefficients b M i of the Padé {U M p } at order M as a function of coefficients b M-1 i of the Padé approximants {U M-1 p } at order M -1 obtained by the classical process of Gram-Schmidt orthonormalization. A comparison of the obtained results with those computed by the use of classical Padé approximants is presented.

In this paper, we propose a new analytical formula to define the next branch in the Asymptotic Numerical Method (ANM) using the Padé approximants. The proposed formula is based on the computation of the relative error of two consecutive Padé approximants. This formula is obtained by developing the relative error with respect to the path parameter. An appropriate matrix formulation is adopted for the computation of this relative error. A comparison between the analytical formula proposed in this paper and the classical continuation Padé approximants using the step length computed numerically using dichotomy method is presented for examples of buckling structures.

A new semi-empirical method for constructing the dipole moments of diatomic molecules as functions of the internuclear distance R∈[0,∞) is suggested. The dipole moment is described by a piecewise continuous function specified by a power-law polynomial with the asymptotic µ(R) → R 3 for small R, the dipole moment and its derivatives for equilibrium positions of nuclei in the molecule, and the results of ab initio calculations of the dipole moment for large R. The method is hydrogen halide molecules.

A.P. Ivanov, Physical Foundations of Hydrooptics [in Russian], Nauka i Tekhnika, Minsk (1975). 10. V.E. Zuev, Propagation of Visible and Infrared Waves in the Atmosphere [in Russian], Soy. Radio, Moscow (1970). 11. S.B. Mogil'nitskii, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 11, 117 (1979). PADE FORM OF EFFECTIVE ROTATIONAL MOLECULAR HAMILTONIANS V. N. Bryukhanov, Yu. S. Makushkin, V1. G. Ty~aterev, and V. N. CherepanoV UDC 539.194 Effective rotational molecular Hamiltenians (ERH) are constructed which differ from the traditional representation of the ERH as a power series in the total angular momentum. A Padeform of the ERH is systematically obtained and the nonuniqueness of this form is emphasized. Rapid progress in high and superhigh resolution spectroscopic techniques has led to several questions in the theory of molecular spectra. For example, recent studies of infrared and microwave molecular spectra have shown a very strong dependence of the rotational states on quantum number, and this is difficul~ to interpret with the traditional representation of the effective rotational Hamiltonian as a polynomial in powers of projections of the total angular momentum Jc~. Derivations of other forms of the ERH different from the traditional ones have not appeared in the literature. In the present paper, we use methods previously developed by us to obtain systematically the Pade-form of the ERH and also formulas for some of the spectroscopic constants which appear in one of the possible rational function representations of the ERH.