Tables of Integrals, Sums, Series and Products
... from Eq (1.232.3) in [8] where Re(w + m) > 0 and Im (m + w) > 0 in order for the sums to converge. We apply Tonelli's theorem for multiple sums, see page 177 in [9] as the summands are of bounded measure over the space ...
... from Eq (1.232.3) in [8] where Re(w + m) > 0 and Im (m + w) > 0 in order for the sums to converge. We apply Tonelli's theorem for multiple sums, see page 177 in [9] as the summands are of bounded measure over the space C × [0, n − 1] × [0, ∞). ...
... from Eq (1.232.3) in [8] where Im(w + m) > 0 in order for the sum to converge. We apply Fubini's theorem for integrals and sums, see page 178 in [9] as the summand is of bounded measure over the space C × [0, ∞). ...
A Prudnikov sum is extended to derive the finite sum of the Hurwitz-Lerch Zeta function in terms of the Hurwitz-Lerch Zeta function. This formula is then used to evaluate a number trigonometric sums and products in terms of other trigonometric functions. These sums and products are taken over positive integers which can be simplified in certain circumstances. The results obtained include generalizations of linear combinations of the Hurwitz-Lerch Zeta functions and involving powers of 2 evaluated in terms of sums of Hurwitz-Lerch Zeta functions. Some of these derivations are in the form of a new recurrence identity and finite products of trigonometric functions.
... From the fundamental theories of the ABC brothers, a series of studies were published soon after. Individual works on this theory developed the general theory of asymmetric elasticity; however, the greatest development was seen in the works [10][11][12][13][14][15][16]. In research works [17][18][19][20][21] examined the present estimation of the mechanics of moment continuums and the potential for their future development. ...
... To determine the images L rr0n and L r 0n in the boundary conditions (23,24) using the addition theorem for Bessel functions [11] and their connection with elementary functions, we find images of the expansion factors in series in terms of the Legendre polynomials of potentials (12) and (13): Here where Γ 1 corresponds to the plane wave, and Γ 2 is the to the spherical wave. ...
This study aims to address the diffraction of non-stationary perturbations with axisymmetric boundaries in a moment elastic framework.
The proposed solution utilizes the Cosserat pseudocontinuum as a model, which represents one of the asymmetric hypotheses of elasticity. The hypothesis posits that a spherical cavity inside an infinite Cosserat pseudocontinuum is subject to either a plane wave or a spherical wave for expansion–compression. The relationship between the non-zero components of the displacement vector and the rotating field is constructed inside a spheroid interrelate system. This system describes the motion of the medium with the extraction taking place at the center of the cavity. In the first stages of existence, the medium exhibits a lack of further disruptions. The initial boundary conditions are represented in terms of dimensionless quantities.
The solution is determined using the expansion of the functions into Legendre and Gegenbauer polynomial series, as well as applying the Laplace transform at each time. The issue at hand is resolved within the domain of Laplace transforms. In the context of linear estimation, the parameters of the original series are obtained by using the Laurent series to analyze images in the vicinity of the period of origin. The findings indicate that the outcomes previously documented in the context of the classical elastic environment align with the solutions obtained via the use of limit techniques.
To facilitate the progress of modern science and technology, it is important to possess a precise comprehension of the deformative processes shown by not only conventional materials, but also those possessing complicated structures. This encompasses materials in which the deformation of the medium may be characterized not only by displacement, but also by rotation. The academic literature generally uses the name "Cosserat medium" to denote the medium characterized by the aforementioned description. Within scholarly discourse, this theory is often referred to as moment, asymmetric, and microstructural elasticity theory. Research has been conducted on the phenomena occurring in pseudo-continuum Cosserat, specifically focusing on the diffraction of waves inside a two-dimensional context, namely by a spherical cavity.
... | | ( )birləşmiş normallanmış Lejandr funksiyasıdır [5]. ...
The properties of (Al2O)6 nanoparticles were investigated by polyempirical Generalized Hukkel method.
Constructed the model of nanoparticle and defined the coordinates of atoms. The calculations were carried out in
valence electronic approximation. For construction of molecular orbitals are used 2s-, 2px -, 2py -, 2pz – valence
atomic orbitals of atoms O and 3s-, 3px -, 3py – and 3pz – valence atomic orbitals of atoms Al. By using the
computer program the orbital energies, ionization potential, the total electronic energy and the effective charge of
atoms of (Al2O)6 nanoparticles were calculated.
... The determination of the subsequent terms does not cause fundamental difficulties, since it is reduced at the chosen value of the reference value 0 xa = to the calculation of tabular integrals of the form [24]: ...
The paper presents a computational-experimental method for the construction of statistical estimates of the error of digital measurements carried out on complex technical objects with metrological support. The algorithm of construction of the estimation of the distribution density function of random error of measurements on the basis of complex application of methods of the theory of characteristic functions, the theory of Lie operator series and methods of statistical modeling in the processing of a limited amount of statistical information in the conditions of small samples is described. The solution of the practical problem on the construction of the estimation of the measurement error distribution density and the problem on the zero mark of the measuring instrument in the probabilistic formulation is given. The presented computational and experimental method can be used to construct statistical estimates for the probability distribution density of the general type, including those given by implicit functions, characteristic functions and operator series.
... В теоретической литературе, а также в практике полевого электрозондирования получила распространение первая замена, основанная на логарифмических переменных. При этом коэффициенты определяются на основе формулы (8) с использованием интегральных формул 6.611, 6.621 и 6.631 справочника Градштейна-Рыжика [22]. ( ) = ( − 2 ) , ( ) = 0,25 3 (−0,25 2 ). ...
... where lm P are the normalized associated Legendre functions [50,51]. For complex spherical harmonics (SHs), ...
We propose an effective general approach for accurately calculating the electron-electron, nuclear-electron and nuclear-nuclear Coulomb electrostatic interaction energies. Since these interaction energies are fundamental terms in the ab initio, density function and semi-empirical theories, their general examination will make an important contribution to the accurate calculation of the physical and chemical properties of atoms and molecules. It is well known that electron-electron, nuclear-electron and nuclear-nuclear Coulomb electrostatic interaction energies can be reduced to basic two-center Coulomb integrals. The analytical calculation of electrostatic interaction energies with respect to basic two-center Coulomb integrals over Slater type orbitals (STOs) in molecular coordinate systems allows for the routine evaluation of molecular structures and their related properties. In this study, we introduce a new full analytical algorithm for calculating the basic two-center Coulomb integrals over STOs using Guseinov’s auxiliary functions, especially the interactions between electrons. The auxiliary functions are calculated by using the exact recurrence relations developed by Guseinov. Our new approach is successfully tested on data from previously published studies, and can be recommended for the evaluation of related problems in atomic and molecular physics.
... Making use of the known formulas (3.771.2) and (3.771.5) (Gradshtein & Ryzhik, 1971), it can be shown that the application of the Fourier transform to Eqs. (38) in the incompressible case yields ...
AFM (Atomic Force Microscopy)-based stiffness tomography poses novel indentation problems for a heterogeneous elastic sample containing a single or multiple heterogeneities. A relatively stiff infinite elastic fiber buried in an elastic half-space represents a simple model for fibrous organelles inside a living cell. The leading-order asymptotic model for the frictionless unilateral indentation is constructed using the method of matched asymptotic expansions under the assumption that the diameter of the contact area is small compared to the depth of the fiber below the surface subjected to the indentation imaging. The fiber deformation is described in the framework of Euler's theory of bending. The resulting system of integro-differential equations is solved by means of the Fourier transform. The approximate relation between the indenter displacement and the contact force is derived in explicit form. The fiber influence factor is introduced to evaluate the incremental indentation stiffness.
... The above results are formally identical to those corresponding to exact integration of Eq. (38) 1 following Gradshtein and Ryzhik (1971) but with̄, 1 and( ) appearing instead of ( ), 1 ( ) and ( , ) with: ...
... Let us now consider the nonrelativistic limit, in which the thermal kinetic energy of the particles is much less than their rest mass. In this case, we have that ζ ≡ mθ 1, and we can use the large-ζ expansion [21]: ...
In this work we adapt the foundations of relativistic kinetic theory and the Boltzmann equation to particles with Lorentz-violating dispersion relations. The latter are taken to be those associated to two commonly considered sets of coefficients in the minimal Standard-Model Extension. We treat both the cases of classical (Maxwell-Boltzmann) and quantum (Fermi-Dirac and Bose-Einstein) statistics. It is shown that with the appropriate definition of the entropy current, Boltzmann's H-theorem continues to hold. We derive the equilibrium solutions and then identify the Lorentz-violating effects for various thermodynamic variables, as well as for Bose-Einstein condensation. Finally, a scenario with non-elastic collisions between multiple species of particles corresponding to chemical or nuclear reactions is considered.
... The case ∆ < 0 is covered in Appendix B as it is rare. Then, so long as ∆ ≥ 0 the integral I may be expressed as (see Gradshtein and Ryzhik (1971) ...
Void coalescence in columns (or necklace coalescence) is a computationally confirmed and physically observed mechanism of void link-up in metal alloys and polymers that has received little attention in the literature. Here, analytical treatment of the phenomenon proceeds from first principles of limit analysis and homogenization theories. A cylindrical unit cell embedding a cylindrical void of finite height is considered under axially symmetric loading. Two types of trial velocity fields are used in seeking an upper bound to the yield criterion corresponding to the particular regime of coalescence in columns. For each type, exact expressions of the overall yield criterion are obtained, albeit in implicit form when using continuous fields. Upon comparison with other modes of yielding allowing for void growth and coalescence in layers, an actual effective yield domain is obtained so as to ascertain regimes of stress state and microstructural states where void coalescence in columns prevails. The predictions are also assessed against finite element based limit analysis.
... For these 2 cases g B0,0 and g B−ξ 2 ,ξ we have B μ,ξ = R 3 . 2 • ) In a previous paper [4] of ours we proved also that the surface z = a arctan( y x ) + b; a, b ∈ R is a minimal one in the well known 3-dimensional homogeneous space called the Bianchi-Cartan-Vranceanu space which (as a particular d'Atri space denote by C, see [5] and the references therein). We recall briefly and essentially that the following 2-parameters family of homogeneous Riemannian metrics g Cγ,η g Cγ,η = dx 2 + dy 2 (1 + η(x 2 + y 2 )) 2 + (dz + γ 2 ydx − xdy 1 + η(x 2 + y 2 ) ) 2 ; γ, η ∈ R This Riemannian metrics founded by L. Bianchi, E. Cartan and G. Vranceanu. ...
We write and give some minimal surfaces which are solutions of the minimal surfaces equation in the domain B μ,ξ of the Cartesian 3-space R 3 equipped with the 2-parameters family of Riemanian metrics g Bμ,ξ = 1 1 − (μ + ξ 2)(x 2 + y 2) (dx 2 + dy 2 − μω 2 + 2ξωdz + dz 2); μ, ξ ∈ R. where ω = ydx − xdy. g Bμ,ξ are defined on the domain B μ,ξ which is, according to μ, ξ, a region of R 3 or the whole 3-space R 3. The metrics g Bμ,ξ are invariant under rotations about (Oz)-axis and translations along the same axis. They generalyse this one of Heisenberg and Euclidean metrics. The minimal surface equation on B μ,ξ for a graph function z = f (x, y) is f xx (1 + f 2 y − 2ξxf y − μx 2) − 2f xy (f x f y + ξ(yf y − xf x) + μxy) +f yy (1 + f 2 x + 2ξyf x − μy 2) = 0 where the index in f denotes partial derivation. 2782 Zoubir Hanifi et al The affine planes z = f (x, y) = ax + by + c are solutions of previous equation. The euclidean helicoid z = f (x, y) = a tan −1 (y x) + b; a, b , c ∈ R stay as minimal surface in B μ,ξ independently and regardless of μ and ξ. We classify the axially symmetric minimal surfaces in B μ,ξ. In last, we characterise that, the only riemannian metrics which have a constant determinant on B μ,ξ and which admit all the planes as minimal surfaces are Heisenberg's metrics. Mathematics Subject Classification: 49Q05, 53A10, 58B21 (*) The appellation "planal metrics" in the title is given by R. L. Bryant [6] to say that all the metrics in 3-space for which the planes are minimal surfaces.
... Here, the approximation on the right comes from making a change of variables to = 2 , and extending the upper limits on the integrals to ∞. These integrals can be evaluated in terms of special functions (integral 3.462 in [42]) by writing the argument in the exponents in the form: (1 − ) ...
We describe the behavior of an Ising model with orthogonal dynamics, where changes in energy and changes in alignment never occur during the same Monte Carlo (MC) step. This orthogonal Ising model (OIM) allows conservation of energy and conservation of momentum to proceed independently, on their own preferred time scales. MC simulations of the OIM mimic more than twenty distinctive characteristics that are commonly found above and below the glass temperature, Tg. Examples include a specific heat that has hysteresis around Tg, out-of-phase loss that exhibits primary and secondary peaks, super-Arrhenius T dependence for the alpha response time, and fragilities that increase with increasing system size (N). Mean-field theory for energy fluctuations in the OIM yields a novel expression for the super-Arrhenius divergence. Because this divergence is reminiscent of the Vogel-Fulcher-Tammann (VFT) law squared, we call it the VFT2 law. A modified Stickel plot, which linearizes the VFT2 law, gives qualitatively consistent agreement with measurements of primary response (from the literature) on five glass-forming liquids. Such agreement with the OIM suggests that several basic features govern supercooled liquids. The freezing of a liquid into a glass involves an underlying 2nd-order transition that is broadened by finite-size effects. The VFT2 law comes from energy fluctuations that enhance the pathways through an entropy bottleneck, not activation over an energy barrier. Primary response times vary exponentially with inverse N, consistent with the distribution of relaxation times deduced from measurements. System sizes found via the T dependence of the primary response are similar to sizes of independently relaxing regions measured by nuclear magnetic resonance for simple-molecule glass-forming liquids. The OIM provides a broad foundation for more-detailed models of liquid-glass behavior.
... are the normalized wave number and frequency, respectively, as well as the modified dispersion function should be transformed to the following form, (20) where [48]. With the aid of the following integral function [49] ∞ 0 ...
From a theoretical viewpoint, dark matter is a prerequisite entity in the formation processes of large-scale structures on the cosmological (galactic) scales of space and time. The behaviours of Jeans modes for gravitational systems composed of dark and baryonic matters are restudied in the framework of Kaniadakis' statistics and kinetic theory. The results show that the κ parameter and density ratio of dark to baryonic matter ρ∗ have significant effects on the gravitational instabilities of such systems. As a test of the viability of this generalized context, we also prove that the dispersion relations for the Maxwellian case is recovered in the limitation of κ→0. Related results in the present work can provide scientific reference for structure formation on the cosmic Jeans scale.
The weighted variable generalized Hölder spaces Hω(·)(Ω,w)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{\,\omega (\cdot )} (\Omega , w)$$\end{document} defined in terms of the local modulus of continuity are considered. Zygmund-type estimates are obtained for a hypersingular integral operator Dα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^\alpha$$\end{document} defined on a bounded open set Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document} of a metric measure space X=(X,d,μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {X} = (\mathcal {X}, d, \mu )$$\end{document}, assuming the power weight function w(x)=dν(x,a)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(x) = d^{\,\nu }(x, a)$$\end{document}, where a,x∈Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a, x \in \Omega$$\end{document}, 1<Reν<N+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1< \textrm{Re}\,\nu < N + 1$$\end{document}, and N is a parameter that characterizes the measure of balls in X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {X}$$\end{document} with respect to their radius. Based on these estimates, it is proven that, under specific conditions on the characteristic ω(x,h)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (x, h)$$\end{document} of Hω(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{\,\omega (\cdot )}$$\end{document}, Dα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^\alpha$$\end{document} is a bounded operator from Hω(·)(Ω,w)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{\,\omega (\cdot )} (\Omega , w)$$\end{document} to Hω-α(·)(Ω,w)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{\,\omega _{-\alpha } (\cdot )} (\Omega , w)$$\end{document}, where ω-α(x,h):=h-Reαω(x,h)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{-\alpha } (x, h):= h^{\,-\textrm{Re}\,\alpha } \, \omega (x, h)$$\end{document}. This result complements a similar one for the case of 0<Reν≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < \textrm{Re}\,\nu \le 1$$\end{document} that was obtained in a previous study conducted by the authors.
In this paper, evolution properties of a generalized Hermite cosh-Gaussian (GHchG) beam propagating through a chiral medium are investigated. By using the Huygens–Fresnel integral formula, analytical expression of a GHchG beam in a chiral medium is obtained. We numerically simulate the corresponding beam under both chiral and beam parameters. By controlling these parameters, we can exhibit the GHchG beam in many different intensity modes. Results demonstrate that the chiral factor of the medium and the source beam parameters during beam propagation affect the spatial properties of the GHchG beam in the chiral medium. The results of this work could be shown the helpfulness in understanding the interaction between the GHchG beam and a chiral medium.
This paper studies the interference impact on the performance of reconfigurable intelligent surface (RIS)-equipped decode-and-forward (DF) relay networks, where RIS is used as internal part of both the source and relay nodes. For that purpose, approximate closed-form expression is derived for the system outage probability assuming Rayleigh fading channels and opportunistic relaying scheme. Furthermore, to get more insights at the system performance, an approximate but accurate expression is achieved for the outage probability at the high signal-to-noise ratio (SNR) regime, where the system diversity order and coding gain are obtained and analyzed. The findings show that the system can achieve a diversity order of \(G_{d}=\textrm{min}(N_{1},N_{2})K\), where \(N_{1}\) and \(N_{2}\) are the numbers of reflecting elements at the source and relays, respectively, and K is the number of relays. Additionally, results illustrate that for the same diversity order, utilizing one relay with multiple reflecting elements gives better performance than utilizing multiple relays each with a single reflecting element. Moreover, findings show that the number of relays not only affects the system diversity order, but also the coding gain of the first hop when it is dominating the system performance. Furthermore, results illustrate that when \(N_{1}=N_{2}\), the system performance is dominated by the largest interference at either the relay node or the destination. With unequal interference powers, changing number of interferers at the node where interference is larger affects the performance, whereas changing number of interferers at the other node has no effect on the system behavior. On the other hand, when both nodes have the same interference power, changing number of interferers at either node by the same amount results in the same effect on the system performance.
In this paper, a transversely isotropic piezoelectric half-space with the isotropy axis parallel to the z-axis is considered under rotation on a rigid circular disk bonded to the surface of the piezoelectric medium. This is a type of Reissner–Sagoci mixed boundary value problem. By utilizing the Hankel transform, the mixed boundary value problem is simplified into solving a pair of dual integral equations. Full-field analytical expressions for displacement, stresses, and electric displacement inside the half-space are obtained. The shear stresses and electric displacement on the surface are found to be singular at the edge of the rigid circular disk, and the stress intensity factors and electric displacement intensity factor are defined. Numerical results show that material properties and geometric size have significant effects on displacement, shear stresses, and electric displacement.
Aiming at the Bessel higher-order cosh-Gaussian beam and the Bessel higher-order sinh-Gaussian beam, we investigate their propagation properties through turbulent biological tissues. In this respect, the analytical expression of the considered beams is obtained and developed, based on the extended Huygens-Fresnel integral. By numerical simulation, the axial intensity of these beams for biological tissue types including the intestinal epithelium and deep dermis of the mouse in addition the human upper dermis versus the propagation distance as a function of the variations of the laser beam parameters. The obtained results indicate that the resistance of our beams against turbulent biological tissues increases as the source parameter increases counting the decentered parameter, the beam-order of the considered beams and the beam waist width. The findings show that the intensity distribution of the propagation of these beams occurs more quickly when they pass through the deep dermis of the mouse. The results presented in this paper are significant due to their potential application in determining the deterioration or disruption of biological tissue, medical imaging and medical diagnosis.
The purpose of this paper is to explore the evolution behavior of two important laser features: the Bessel higher-order cosh-Gaussian beam and the Bessel higher-order sinh-Gaussian beam propagating through turbulent oceanic environments. Benefiting from the extended Huygens-Fresnel principle, the analytical formulas for the average intensity of the beams passing through oceanic turbulence are derived. The propagation of some laser beams through oceanic turbulence is also deduced as particular cases from the present study. The effects of oceanic turbulence parameters and the source beam parameters are examined to understand their influence on the intensity distribution of the considered beams by using numerical simulations. Our results show that the spreading of these beams depends on their initial parameters and oceanic parameters. Hence, the propagation of the studied beams through oceanic turbulent will be faster with the smaller dissipation rate of the mean square temperature, larger salinity fluctuations, higher rate of dissipation of turbulent kinetic energy per unit mass of fluid and with decreasing the beam width and the parameter Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega$$\end{document}. The outputs of this study have useful applications in optical underwater communication, remote sensing, imaging and others.
We propose and study a new generalized class of distributions called the Type II Exponentiated Half Logistic-Gompertz-G Power Series (TIIEHL-Gom-GPS) distribution. Some structural properties including expansion of density, ordinary and conditional moments, generating function, order statistics and entropy are derived. We present some special cases of the proposed distribution. The maximum likelihood method is used for estimating the model parameters. The usefulness and importance of the new class of distributions are illustrated by means of two applications to real data sets.
We identify a one-parameter family of inequalities for the Fourier transform whose limiting case is the restriction conjecture for the sphere. Using Stein’s method of complex interpolation, we prove the conjectured inequalities when the target space is $$L^2$$ L 2 , and show that this recovers in the limit the celebrated Tomas-Stein theorem.
It is a brief exposition of results of the investigation of Cauchy problem for some nonlinear equation of pseudoparabolic type that is a generalisation of some model of semiconductor theory. In the paper, the potential theory for the linear part of the equation is elaborated, which demanded quite intricate technique, which can be used in other equations. The properties of the fundamental solution of this linear part are also of interest, because of the singularity of its 1st time derivative. This is not usual for this type of equations. Also, we obtain sufficiant conditions of solvability and of finite-time blow-up.
To study the frequencies and modes of vibrations of a circular plate immersed in a liquid, a new approach has been developed. The technic is based on the use of hypersingular integral equations and the method of prescribed shapes. It is assumed that a round thin elastic plate is immersed in an ideal incompressible fluid, and its motion is considered to be irrotational. Under these conditions, there is a velocity potential that satisfies the Laplace equation everywhere outside the plate, and the no-flow condition is satisfied on the plate surface. The fluid pressure has been determined by using the linearized Cauchy-Lagrange integral. During solving the boundary value problem with regard to the velocity potential, an integral representation in the form of a double layer potential was used. In this case, the potential density is proportional to the fluid pressure drop. The method of given forms made it possible to reduce the problem of determining the added masses of a liquid to solving hypersingular equations on a circular domain. During the research reduction of two-dimensional hypersingular integral equations to one-dimensional ones has been carried out. On condition of this, the inner integrals take the form of elliptic integrals of the second kind, which have no singularities. To calculate the external integral, which exists only in the sense of Hadamard, it is proposed to use the boundary element method. A procedure for calculating the elements of the matrix of a system of linear algebraic equations for finding the unknown density of the double layer potential has been developed. A numerical solution of the hypersingular integral equation has been obtained, and a comparison of the numerical and analytical solutions has been carried out. The right-hand sides of hypersingular integral equations are the forms of vibrations of a rigidly fixed circular plate obtained analytically. A technique for calculating the matrix of added masses has been developed, which made it possible to reduce the problem under consideration to solving the problem of eigenvalues.
We are concerned with boundary value problems for Laplace equation in an unbounded sector $s_\theta$ with vertex at the origin, the boundary conditions being of mixed type and jumping at corner. The boundary conditions are these: Dirichlet datum on one of the radial lines, while on the other the values of an Ventcel boundary condition is prescribed. We are interested in looking for solutions having a prescribed degree of smoothness up to the origin: more precisely we search for solutions of problem having all the derivatives up to the order that are square integrable with a power weight. This problem has a background in physical modeling of electrostatic or thermal imaging. Determining the geometry and the physical nature of an corrosion within a conducting medium from voltage and current measurements on the accessible boundary of the medium can be modeled as an inverse boundary value problem for the Laplace equation subject to appropriate boundary conditions on the corrosion surface. We are interesting in investigation of a regularity properties of solution to the @direct@ problem. Applying Mellin transform we pass to a finite difference equation.We use the methods of V.A.Solonnikov and E.V.Frolova just as in the case of the analogous finite difference equation obtained under the Dirichlet or the Neumann conditions indstead of the Ventcel condition in our case. We obtain the sulution of homogeneous difference equation in the form of infinite product. Then we find asymptotic formulas for this solution.Returning to nonhomogeneous differerence equation we find its solution in the form of contour integral. we define the solution of the starting problem by the help of the inverse Mellin transform. We estimate this solution in the norm of V.Kondratiev spaces $H^k_\mu(s_\theta$ under some conditions on weight $\mu$, higher order of derivatives $k$ and the opening of the angle $\theta$.
This paper deals with the computation of the Lerch transcendent by means of the Gauss-Laguerre formula. An a priori estimate of the quadrature error, that allows to compute the number of quadrature nodes necessary to achieve an arbitrary precision, is derived. Exploiting the properties of the Gauss-Laguerre rule and the error estimate, a truncated approach is also considered. The algorithm used and its Matlab implementation are reported. The numerical examples confirm the reliability of this approach.
The paper presents analytical approach to electrostatic field for dielectric spheroids problem. For both oblate and prolate spheroids fields distributions are derived. The variable separation method is applied. Electrostatic force is evaluated with the help of the Maxwell stress tensor generalized method, material force density, coenergy and equivalent dipole. Levitation forces for both oblate and prolate dielectric spheroids versus axis permittivities and spheroids height are presented.
The aim of the work is the theoretical study in the quasistatic approximation of plasmon oscillations in a metallic nanoparticle having shape of the half of a prolate spheroid. In the introduction the relevance of the work is specified; literature sources that investigate the problem of plasmonic properties of the halves of nanoparticles are considered; problems that impede such researches are noted. In the main part of the work the system of equations was obtained for finding the plasmon permittivity corresponding to the frequencies of plasmon oscillations in metallic prolate hemispheroidal nanoparticle as the eigenvalues. The solution of this system was investigated by the method of truncation, i.e. by taking into account only a finite number of multipoles (spheroidal harmonics). The dependence of the plasmon permittivity of the prolate hemispheroidal nanoparticle on the number of accounted multipoles was numerically investigated. The distribution of the surface electric charge of the plasmon modes excited in the nanoparticle was considered with taking into account the contributions from the finite number of multipoles. New theoretical data obtained in the work can be used for calculations of plasmonic properties of metal hemispheroidal nanoparticles and for analysis and interpretation of experimental data in the optics of such nanoparticles.
“Vesnik of Yanka Kupala State University of Grodno. Series 2. Mathematics. Physics. Informatics, Computer Technology and its Control”. Number 1 (186), 2015, pp. 95–103. [In Russian]
The aim of this article is to show how recent results of the Weinstein equation give new insights for the classical harmonic polynomials. In the first section we present an explicit basis for the vector space \({{\mathcal {H}}}_{n}({\mathbb {R}}^d)\) of homogeneous harmonic polynomials of degree n on \({\mathbb {R}}^d\). In the following two sections we derive an explicit orthonormal basis in case of the dimensions \(d=3\) and \(d=4\).
УДК 517.9 Наведено короткий огляд праць Київської школи математиків, які були опубліковані в радянських журналах 40–70-х років минулого століття. Основні результати подано на мові сучасних методів нескінченновимірного аналізу, що значно спрощує їх доведення. Виведено нелінійні за параметром густини рівняння типу Кірквуда–Зальцбурга для кореляційних функцій канонічного ансамблю. Доведено існування та єдиність їх розв'язків у режимі високої температури та низької густини. Огляд доповнено оригінальним дослідженням одного з авторів [A.~L.~Rebenko, Virial expansions for correlation functions in canonical ensemble, Preprint arXiv:2205.07095 [math-ph], https://doi.org/10.48550/arXiv.2205.07095], в якому побудовано нові розклади кореляційних функцій за параметром густини.
The simple model of the elastic SH-plane wave scattering from the interface crack in a joint of two half-spaces is considered using the Wiener–Hopf technique. Its approximate solution is obtained for the wide crack. The study focuses on the physical features that can be used to recognize interface defects. The far scattering field for arbitrary plane wave illumination angle, including sliding and critical ones, the effects of the mutual edge waves diffraction, and the lateral wave excitation are analyzed. The approximate expressions of the stress intensity factors are obtained and establish their dependencies on the problem parameters. The elastic SH-mode diffraction from the finite interface crack in the joint of the half-space and the layer is considered. The mode transformation properties, as well as the spectral and displacement field characteristics, are analyzed. The problem of the interface crack recognition is discussed.KeywordsHalf-spacesLayerInterface crackSH-wavesWiener–Hopf technique
Modern mechanical engineering sets the tasks of calculating thin-walled structures that simultaneously combine sometimes mutually exclusive properties: lightness and economy on the one hand and high strength and reliability on the other. In this regard, the use of orthotropic materials and plastics seems quite justified.The article demonstrates the complex representation method of the equations of orthotropic shells general theory, which allowed in a complex form to significantly reduce the number of unknowns and the order of the system of diferential equations. A feature of the proposed technique for orthotropic shells is the appearance of complex conjugate unknown functions. Despite this, the proposed technique allows for a more compact representation of the equations, and in some cases it is even possible to calculate a complex conjugate function. In the case of axisymmetric deformation, this function vanishes, and in other cases the influence of the complex conjugate function can be neglected.Verification of the correctness of the proposed technique was demonstrated on a shallow orthotropic spherical shell of rotation under the action of a distributed load. In the limiting case, results were obtained for an isotropic shell as well.
The boundary value problem of transient pressure field development around a horizontal well in a laterally infinite, inhomogeneous, anisotropic reservoir is formulated under assumption of slow spatial variation of the matrix permeability along the well axis. The well is represented as a linear fluid source/sink. The pressure distribution is expressed in the integral form on the basis of the instant point source perturbation function found explicitly. The inverse problem for fluid in/outflow density rates simulation is reduced to solution of the integral equation at a given pressure inside the well. A computational procedure is developed and implemented to predict the in/outflow rates along the well and estimate the impact of the permeability variations on the well performance. Series of calculations for constant, linear, and variable permeability cases are analyzed and compared. The difference of the obtained solution from the so-called “locally-constant” permeability approximation is demonstrated, accuracy and applicability of the latter approach are discussed.
Shape-preferred orientations in metamorphic tectonites are commonly defined by 3D fabric elements (Chap. 2). To understand the development of such fabrics, it is necessary to study the behavior of 3D fabric elements in given flow fields. The classic works of Jeffery (Proc R Soc Lond Ser A- 102(715):161–179, 1922) and Eshelby (Proc R Soc Lond Ser A- 241(1226):376–396, 1957, Proc R Soc Lond Ser A- 252(1271):561–569, 1959) and the extension of Eshelby’s theory to viscous deformation (Bilby et al., Tectonophysics 28(4):265–274, 1975; Jiang, J Struct Geol 29(2):189–200, 2007; Jiang, J Struct Geol 50:22–34, 2013; Jiang, J Struct Geol 68:247–272, 2014) have been used for the motion of rigid and deformable elements. In this Chapter, we focus on the rotation of rigid elements in slow viscous flows. We first develop the governing equation for rotation. Jeffery’s result for the angular velocity of a rigid ellipsoid in slow viscous flows is combined with the rotation equation to form an initial value problem for the motion of a rigid ellipsoid in viscous flow. The problem is solved analytically for a spheroid in monoclinic flows. The problem is solved numerically for more general situations. Mathcad and MATLAB programs for numerical solutions are provided.
Closed-form solution for the plane strain problem of static deformation of a homogeneous, isotropic, elastic layer of uniform thickness resting over a rough rigid base caused by a vertical dip-slip fault located in the elastic layer has been studied. The stresses and displacements have been obtained using the Airy stress function approach in the integral form by applying suitable boundary conditions at the free surface. To evaluate the integrals analytically, the denominator term has been replaced with a finite sum of exponential terms using the Least Square approximation. The integral expressions for the displacements have been evaluated analytically. The variation of horizontal and vertical displacement has been computed numerically for different source locations for two mechanically different crusts: continental crust and oceanic crust.
The paper considers a rectangular orthotropic insulated slab on an elastic foundation, modeled by an elastic homogeneous isotropic layer rigidly connected to a non-deformable foundation. Elastic and nonlinear calculations of this plate opening. The calculation of a flexible orthotropic slab on an elastic foundation in a nonlinear formulation is carried out iteratively by the method of B. N. Zhemochkin. To determine the coefficients of the canonical equations and free terms, a mixed method of structural mechanics was used. The deflections of a slab with a pinched normal in the main system of the mixed method due to the action of a concentrated force are determined by the Ritz method when the deflections are represented as a power polynomial in a new original expression, which is proposed by the author for the first time in the studies. This expression satisfies not only the boundary conditions of the pinched slab in terms of displacements, but also the biharmonic equation. In nonlinear calculations, when finding the variable (secant) stiffness for the Zhemochkin section, at each iteration, the “stiffness – curvature” dependence is used for each of the X and Y directions, approximated by a nonlinear function, the nature of the dependence of which graphically indicates the nonlinear-elastic operation of the orthotropic plate and its deformation taking into account crack formation and crack opening. The algorithm for the above solution is implemented using the Wolfram Mathematica 11.3 computer program.
Modern mechanical engineering sets the tasks of calculating thin-walled structures that simultaneously combine sometimes mutually exclusive properties: lightness and economy on the one hand and high strength and reliability on the other. In this regard, the use of orthotropic materials and plastics seems quite justified.The article demonstrates the complex representation method of the equations of the orthotropic shells general theory, which allowed in a complex form to significantly reduce the number of unknowns and the order of the system of differential equations. A feature of the proposed technique for orthotropic shells is the appearance of complex conjugate unknown functions. Despite this, the proposed technique allows for a more compact representation of the equations, and in some cases it is even possible to calculate a complex conjugate function. In the case of axisymmetric deformation, this function vanishes, and in other cases the influence of the complex conjugate function can be neglected. Verification of the correctness of the proposed technique was demonstrated on a shallow orthotropic spherical shell of rotation under the action of a distributed load. In the limiting case, results were obtained for an isotropic shell as well.
In various sections of the classical Fourier Analysis, the problem of the convergence of the integrals ∫0∞f(λt)g(t)dtas λ→+0 and λ→+∞ under various assumptions on the functions f and g are considered. In this paper, we study some analogues of such problems for weight integrals of the form ∫0∞f(λt)g(t)t2α+1dt,α>−1/2,for functions f and g from some weighted functional classes that are connected with Fourier–Bessel harmonic analysis.
In the present section based on the principles of the linear theory of elasticity the plane strain state of piecewise homogeneous planes containing rectilinear or arch-shaped interfacial absolutely rigid thin inclusions under different conditions of contact between the inclusion and the matrix is considered. We assume that the short sides of inclusion are not contacted with the matrix. For some cases the exact solutions of general problems are constructed, when the piecewise homogeneous planes contain interfacial cracks and the mixed conditions are given on the banks, the special cases of which can be considered as a presence of absolutely rigid inclusions.KeywordsPiecewise homogeneous planeInterfacial absolutely rigid thin inclusionCrackArc-shaped interfacial absolutely rigid thinMixed boundary value problem
Most of the estimators of parameters of rare and large events, among which we distinguish the extreme value index (EVI) for maxima, one of the primary parameters in statistical extreme value theory (EVT), are averages of statistics, based on the k upper observations. They can thus be regarded as the logarithm of the geometric mean, i.e. the logarithm of the power mean of order p = 0 of a certain set of statistics. Only for heavy tails, i.e. a positive EVI, quite common in many areas of application, and trying to improve the performance of the classical Hill EVI-estimators, instead of the aforementioned geometric mean, we can more generally consider the power mean of order-p (MOp), and build associated MOp EVI–estimators. The normal asymptotic behaviour of MOp EVI- estimators has already been obtained for p < 1/(2ξ), with consistency achieved for p < 1/ξ, where ξ denotes the EVI. We shall now consider the non-regular case, p ≥ 1/(2ξ), a situation in which either normal or non-normal sum-stable laws can be obtained, together with the possibility of an ‘almost degenerate’ EVI-estimation.
We consider the Dirac equation with position-dependent mass for a q-deformed Pöschl–Teller potential plus a Coulomb-like tensor interaction in the limits of spin and pseudospin symmetries. Under the condition of spin symmetry, the bound state energy eigenvalue equation and the two radial components of the Dirac wave function in terms of the Jacobi polynomials are obtained approximately for an arbitrary shifted spin-orbit quantum number λκ = κ + H and for any value of the deformation parameter q ≥ 1. In the case of the pseudospin symmetry limit, there are no bound states. The Dirac equation describes a free physical system and the two components of the wave function are expressed in terms of the spherical Bessel functions.
Previously, the author obtained an analytical solution for determining the vertical displacements of the homogeneous isotropic quarter-space face, which is affected by a vertical concentrated force. This expression gave an exact solution for determining the vertical displacements of a quarter-space face made of incompressible material and an approximate solution for a Poisson ratio different from 0.5. Later, in the published paper by S. V. Bosakov and P. D. Skachek “Action of Concentrated Force on 1/8 of Homogeneous Isotropic Space”, it was shown that by combining solutions for determining
the vertical displacements of a quarter-space and a half-space from the action of concentrated forces, one can find vertical displacements for one-eighth face of a homogeneous isotropic space. The resulting expressions allow solving contact problems for non-classical domains in the form of a quarter of a space and one eighth of a space. Below, the author obtains the first approximation for displacements of a quarter-space face from the action of a vertical concentrated force. In this paper, the author gives the first approximation for determining the vertical displacements of a quarter-space face from the action of a vertical concentrated force, widely using the special approximation method developed in the works of V. M. Alexandrov and allowing to calculate successfully improper integrals. The constructed graphs show good results when determining displacements with a Poisson’s ratio different from 0.5 and equal to 0.5. It should be noted that the approach indicated in the paper by S. V. Bosakov and P. D. Skachek can be successfully used in determining all displacements of the faces of one-eighth face of a homogeneous isotropic sharpened forces tangent to the edge of a quarter-space and this will allow to solve contact problems taking into account friction forces in the contact zone of the beam or plate.
In this paper, we investigate theoretically the properties of a vortex cosine-hyperbolic-Gaussian beam (vChGB) propagating in uniaxial crystals orthogonal to the optical axis. Analytical expressions for the vChGB propagating through a uniaxial crystal orthogonal to the optical axis are derived. From the obtained formulas the evolution properties of the intensity and phase distributions of the output beam are analyzed with numerical examples. It is shown that a vChGB propagating in the uniaxial crystal keeps its initial profile nearly invariant for a small propagation distance, whereas, during the beam propagation the anisotropy of the uniaxial crystal influences strongly the beam properties. The output beam becomes astigmatic, and the intensity and phase distributions are dependent on the ratio of refractive indices of the crystal and the initial beam parameters. The present study may provide a convenient way to generate astigmatic vortex hollow beams.
The aim of this paper is to discuss the problem of wave diffraction from the finite wedge on a rigorous level using the method of analytical regularization. We apply the Kontorovich–Lebedev integrals that are considered in principal value sense and the eigenfunctions series for this purpose. The problem is reduced to a couple of the independent infinite systems of linear algebraic equations (ISLAE) of the first kind. The convolution type operators are singled out from them and the inverse operators are represented in analytical form. These two couples of operators are called the regularizing operators. They are used to reduce the initial ISLAE of the first kind to the ISLAE of the second kind. The numerical examples of wave scattering from the wedge are analysed.
In the present work, we study problem related to the approximation of continuous $2\pi$-periodic functions by linear means of their Fourier series. The simplest example of a linear approximation of periodic function is the approximation of this function by partial sums of the Fourier series. However, as well known, the sequence of partial Fourier sums is not uniformly convergent over the class of continuous $2\pi$-periodic functions. Therefore, a significant number of papers is devoted to the research of the approximative properties of different approximation methods, which are generated by some transformations of the partial sums of the Fourier series. The methods allow us to construct sequence of trigonometrical polynomials that would be uniformly convergent for all functions $f \in C$. Particularly, Ceśaro means and Fejer sums have been widely studied in past decades.One of the important problems in this field is the study of the exact constant in an inequality for upper bounds of linear means deviations of the Fourier sums on fixed classes of periodic functions. Methods of investigation of integral representations for trigonometric polynomial deviations are generated by linear methods of summation of the Fourier series. They were developed in papers of Nikolsky, Stechkin, Nagy and others. The paper presents known results related to the approximation of classes of continuous functions by linear means of the Fourier sums and new facts obtained for some particular cases.In the paper, it is studied the approximation by the Ceśaro means of Fourier sums in Lipschitz class. In certain cases, the exact inequalities are found for upper bounds of deviations in the uniform metric of the second order rectangular Ceśaro means on the Lipschitz class of periodic functions in two variables.
The entropies of some solid metals have been analytically calculated by the use of n-dimensional Debye approximation in this study. The obtained formula is valid for all temperature ranges from low to melting temperature. We have compared our results with available numerical and experimental data for room temperature ( ). As can be seen that our results are in good agreement with literature.
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