Hasil untuk "Analytic mechanics"

A. Hasanov, T.G. Ergashev, A.R. Ryskan

Hypergeometric functions are divided into complete and confluent functions. Srivastava and Karlsson were the first to propose a method for constructing the complete set of triple Gaussian hypergeometric series and compiled a table containing definitions and regions of convergence for 205 distinct complete series in three variables. Subsequently, several authors obtained various integral representations and transformation formulas for the functions introduced by Srivastava and Karlsson. More recently, Ergashev identified 395 hypergeometric series of three variables that represent confluent forms of the known 205 complete hypergeometric series. In the present study, new Euler-type integral representations are derived for certain Gaussian hypergeometric functions of three variables. The main results are obtained using properties of the gamma and beta functions. New integral representations are established for 14 functions from the list of confluent hypergeometric functions of three variables. All derived integrals can be regarded as generalized Euler type representations of the classical Gaussian hypergeometric functions of one and two variables. In addition, it is demonstrated how one of these confluent functions, together with its integral representation, can be applied to construct solutions of the three-dimensional singular Helmholtz equation.

Ye.R. Baissalov, J.A. Tussupov

It is shown that the concepts of heir and coheir, introduced by D. Lascar and B. Poizat, play a fundamental role in model theory, particularly in classification theory. The related notions of proper heir and proper coheir are introduced, containing important constructs within themselves. Poizat’s lemma on the existence of a proper heir of any non-definable type over a model is presented as an important fact of existence in unstable theories. The concept of an order trigger in a model is then introduced as the skeleton of an algorithmic device that produces ω-evidence of the order property in it. This evidence is constructed using a method very similar to the “back and forth” method of classical model theory, where at each step two possibilities for choosing elements are alternated. As an example of use, a simplified proof of the characterization theorem of the class of unstable theories using these concepts is explained. It is pointed out that applications of more advanced constructions, such as order triggers, can help in solving problems related to the classification of small, countable and minimal models of unstable theories.

K. Bharatha, R. Rangarajan, C.J. Neethu

The field of nonlinear differential equations have made significant contribution in understanding nonlinear dynamics and its complex phenomenon. One such evolution equation is Kawahara equation, which has gained its importance in plasma physics and allied fields. Many researchers are interested to work on their soliton, multi-solitons solutions and to study other properties such as stability, integrability, conservation laws and so on. The aim of the paper is to study the Coupled Kawahara equation and to deduce its soliton solutions. The coupled equation is treated with the ansatz method and the tanh method to compute soliton solutions. The novelty of this work is to demonstrate the fact, that the derived system efficiently gives two governing equations admitting solitary wave solutions. Further, in the coupled equation, one equation has the nonlinear term vvx addition to the Kawahara equation, while the other is the modified Kawahara equation. Scope for future works is also highlighted.

R.R. Ashurov, U.Kh. Dusanova, N.Sh. Nuraliyeva

This paper investigates a mixed-type partial differential equation involving the Caputo fractional derivative of order ρ ∈ (0,1) for t > 0, and a classical parabolic equation for t < 0. The problem is studied in an arbitrary N-dimensional domain Ω with smooth boundary, subject to Dezin-type non-local boundary and gluing conditions. For the forward problem, existence and uniqueness of the classical solution are established under suitable assumptions on the data, employing the Fourier method. The influence of the parameter λ in the non-local boundary condition on solvability is analyzed. Additionally, an inverse problem is considered, where the source term is separable as F(x,t) = f(x)g(t), with known g(t) and unknown spatial function f(x). Under certain conditions on g(t), the uniqueness and existence of the solution are proven. This work extends previous results on mixed-type equations, highlighting the role of fractional derivatives and nonlocal conditions in both forward and inverse settings. The findings contribute to the theory of mixed-type and fractional differential equations, with potential applications in subdiffusion and related processes.

A.B. Imanbetova, A.A. Sarsenbi, B. Seilbekov

In this paper, the inverse problem for a fourth-order parabolic equation with a variable complex-valued coefficient is studied by the method of separation of variables. The properties of the eigenvalues of the Dirichlet and Neumann boundary value problems for a non-self-conjugate fourth-order ordinary differential equation with a complex-valued coefficient are established. Known results on the Riesz basis property of eigenfunctions of boundary value problems for ordinary differential equations with strongly regular boundary conditions in the space L2(−1,1) are used. On the basis of the Riesz basis property of eigenfunctions, formal solutions of the problems under study are constructed and theorems on the existence and uniqueness of solutions are proved. When proving theorems on the existence and uniqueness of solutions, the Bessel inequality for the Fourier coefficients of expansions of functions from space L2(−1,1) into a Fourier series in the Riesz basis is widely used. The representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are derived. The convergence of the obtained solutions is discussed.

Zh.A. Balkizov

The paper studies two nonlocal problems with a displacement for the conjugation of two equations of second-order hyperbolic type, with a wave equation in one part of the domain and a degenerate hyperbolic equation of the first kind in the other part. As a non-local boundary condition in the considered problems, a linear system of FDEs is specified with variable coefficients involving the first-order derivative and derivatives of fractional (in the sense of Riemann-Liouville) orders of the desired function on one of the characteristics and on the line of type changing. Using the integral equation method, the first problem is equivalently reduced to a question of the solvability for the Volterra integral equation of the second kind with a weak singularity; and a question of the solvability for the second problem is equivalently reduced to a question of the solvability for the Fredholm integral equation of the second kind with a weak singularity. For the first problem, we prove the uniform convergence of the resolvent kernel for the resulting Volterra integral equation of the second kind and we prove that its solution belongs to the required class. As to the second problem, sufficient conditions are found for the given functions that ensure the existence of a unique solution to the Fredholm integral equation of the second kind with a weak singularity of the required class. In some particular cases, the solutions are written out explicitly.

Carlos R. Argüelles, Santiago Collazo

Galaxy rotation curve (RC) fitting is an important technique which allows the placement of constraints on different kinds of dark matter (DM) halo models. In the case of non-phenomenological DM profiles with no analytic expressions, the art of finding RC best-fits including the full baryonic + DM free parameters can be difficult and time-consuming. In the present work, we use a gradient descent method used in the backpropagation process of training a neural network, to fit the so-called Grand Rotation Curve of the Milky Way (MW) ranging from ∼1 pc all the way to ∼<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mn>5</mn></msup></semantics></math></inline-formula> pc. We model the mass distribution of our Galaxy including a bulge (inner + main), a disk, and a fermionic dark matter (DM) halo known as the Ruffini-Argüelles-Rueda (RAR) model. This is a semi-analytical model built from first-principle physics such as (quantum) statistical mechanics and thermodynamics, whose more general density profile has a <i>dense core</i>–<i>diluted halo</i> morphology with no analytic expression. As shown recently and further verified here, the dark and compact fermion-core can work as an alternative to the central black hole in SgrA* when including data at milliparsec scales from the S-cluster stars. Thus, we show the ability of this state-of-the-art machine learning tool in providing the best-fit parameters to the overall MW RC in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mrow><mo>−</mo><mn>2</mn></mrow></msup></semantics></math></inline-formula>–<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mn>5</mn></msup></semantics></math></inline-formula> pc range, in a few hours of CPU time.

M.I. Qureshi, S.H. Malik, D. Ahmad

The main objective of this article is to provide the analytical solutions (not previously found and not available in the literature) of some problems related with definite integrals integrands of which are the products of the derivatives of Legendre’s polynomials of first kind having different order, with the help of some derivatives of Legendre’s polynomials of first kind P_n(x), Rodrigues formula, Leibnitz’s generalized rule for successive integration by parts and certain values of successive differential coefficients of (x^2 − 1)^r at x = ±1.

M.Zh. Turgumbaev, Z.R. Suleimenova, D.I. Tungushbaeva

In this paper, we consider the questions about the weighted integrability of the sum of series with respect to multiplicative systems with monotone coefficients. Conditions are obtained for weight functions that ensure that the sum of such series belongs to the weighted Lebesgue space. The main theorems are proved without the condition that the generator sequence is bounded; in particular, it can be unbounded. In the case of boundedness of the generator sequence, the proved theorems imply an analogue of the well-known Hardy-Littlewood theorem on trigonometric series with monotone coefficients.

A.Sh. Akysh (Akishev)

In this paper, based on the splitting method scheme, the existence and uniqueness theorem on the whole time interval t ∈ [0, T), T ≤ ∞ for the full nonlinear Boltzmann equation in the nonequilibrium case is proved where the intermolecular interactions are hard-sphere molecule and central forces. Considering the existence of a bounded solution in the space C, the strict positivity of the solution to the full nonlinear Boltzmann equation is proved when the initial function is positive. On the basis of this some mathematical justification of the H−theorem of Boltzmann is shown.

Gannon E. Lenhart, Andrew B. Royston, Keaton E. Wright

Abstract We present simulations of one magnetic monopole interacting with multiple magnetic singularities. Three-dimensional plots of the energy density are constructed from explicit solutions to the Bogomolny equation obtained by Blair, Cherkis, and Durcan. Animations follow trajectories derived from collective coordinate mechanics on the multi-centered Taub-NUT monopole moduli space. We supplement our numerical results with a complete analytic treatment of the single-defect case.

Alexandra K. Hopkins, Ray Dolan, Katherine S. Button et al.

Positive self-beliefs are important for well-being, and are influenced by how others evaluate us during social interactions. Mechanistic accounts of self-beliefs have mostly relied on associative learning models. These account for choice behaviour but not for the explicit beliefs that trouble socially anxious patients. Neither do they speak to self-schemas, which underpin vulnerability according to psychological research. Here, we compared belief-based and associative computational models of social-evaluation, in individuals that varied in fear of negative evaluation (FNE), a core symptom of social anxiety. We used a novel analytic approach, ‘clinically informed model-fitting’, to determine the influence of FNE symptom scores on model parameters. We found that high-FNE participants learn more easily from negative feedback about themselves, manifesting in greater self-negative learning rates. Crucially, we provide evidence that this bias is underpinned by an overall reduced belief about self-positive attributes. The study population could be characterized equally well by belief-based or associative models, however large individual differences in model likelihood indicated that some individuals relied more on an associative (model-free), while others more on a belief-guided strategy. Our findings have therapeutic importance, as positive belief activation may be used to specifically modulate learning.   Author Summary Understanding how we form and maintain positive self-beliefs is crucial to understanding how things go awry in disorders such as social anxiety. The loss of positive self-belief in social anxiety, especially in inter-personal contexts, is thought to be related to how we integrate evaluative information that we receive from others. We frame this social information integration as a learning problem and ask how people learn whether someone approves of them or not. We thus elucidate why the decrease in positive evaluations manifests only for the self, but not for an unknown other, given the same information. We investigated the mechanics of this learning using a novel computational modelling approach, comparing models that treat the learning process as series of stimulus-response associations with models that treat learning as updating of beliefs about the self (or another). We show that both models characterise the process well and that individuals higher in symptoms of social anxiety learn more from negative information specifically about the self. Crucially, we provide evidence that this originates from a reduction in the amount of positive attributes that are activated when the individual is placed in a social evaluative context.

J. A. Green

D. M. Zuev, K. G. Okhotkin

Modern problems of aerospace industry require consideration of rods experiencing large deflections. The example of such a problem is development of large scale deployable umbrellatype antennas where rods are structural elements. Development of modern analytic methods in the field of solid mechanics allows to model rod bend shapes and to find expressions for maximum deflection. In addition, the analytic methods make it possible to find a full system of solution branches and all possible equilibrium shapes without significant time-consuming for numerical simulations. Wherein relatively simple methods for determining bending shapes in case of large deflections have significant importance for applied use. Namely, they can be used for preliminary design of complex rod constructions. The paper presents the method for obtaining of modified analytic formulas that enable to determine large deflections of a thin elastic cantilever under transverse loading. The method uses a rod’s arc-length saving condition which is important for applied use. The modified formulas allow to achieve accuracy comparable with exact nonlinear solutions given in terms of elliptic integrals and functions. That fact expands the loading range where the linear theory can be used. The authors considered the following cases: concentrated transverse loading on the free end and combined loading (uniformly distributed loading and concentrated transverse loading on the free end). The comparison with experimental data proved accuracy of the proposed method. In addition, the authors obtained approximate formulas based on the modified formulas. The approximate formulas can be use for engineering applications.

V.I. Senashov, I.A. Paraschuk

We consider the problem of recognizing a group by its bottom layer. This problem is solved in the class of layer-finite groups. A group is layer-finite if it has a finite number of elements of every order. This concept was first introduced by S. N. Chernikov. It appeared in connection with the study of infinite locally finite p-groups in the case when the center of the group has a finite index. S. N. Chernikov described the structure of an arbitrary group in which there are only finite elements of each order and introduced the concept of layer-finite groups in 1948. Bottom layer of the group G is a set of its elements of prime order. If have information about the bottom layer of a group we can receive results about its recognizability by bottom layer. The paper presents the examples of groups that are recognizable, almost recognizable and unrecognizable by its bottom layer under additional conditions.

M.B. Muratbekov, M.M. Muratbekov

The Schrödinger operator L = -∆ + q ( x ) , x ∈ Rn, is one of the main operators of modern quantum mechanics and theoretical physics. It is known that many fundamental results have been obtained for the Schrödinger operator L . Among them, for example, are questions about the existence of a resolvent, separability (coercive estimate), various weight estimates, estimates of intermediate derivatives of functions from the domain of definition of an operator, estimates of eigenvalues and singular numbers ( s -numbers). At present, there are various generalizations of the above results for elliptic operators. For general differential operators, the solution of such problem as a whole is far from complete. In particular, as far as we know, there was no result until now showing the existence of the resolvent and coercivity, as well as the discreteness of the spectrum of a hyperbolic type operator in an infinite domain with increasing and oscillating coefficients. It is easy to see that the study of some classes of differential operators of hyperbolic type defined in the space L 2( Rn+1), using the Fourier method, can be reduced to the study of the Schrödinger operator with a negative parameter : Lt =-∆+(-t2+itb(x)+q(x)), where t is a parameter (-∞ < t < ∞) , i 2 = -1. Hence, it is easy to see that we get - t 2 → -∞ when | t | → ∞ for the operator Lt . Consequently, a completely different situation arises here compared to the Schrödinger operator L = -∆ + q ( x ), and in particular, the methods worked out for the Schrödinger operator L turn out to be little adapted when studying the Schrödinger operator Lt with a negative parameter. All these questions indicate the relevance and novelty of this work. In the paper we study the problems of the existence of the resolvent and the coercivity of the Schrödinger operator with a negative parameter.

R. Parker, S. Vijayakar, T. Imajo

Xiaomin Zhao, Ye-hwa Chen, H. Zhao et al.

There are many achievements in the field of analytical mechanics, such as Lagrange Equation, Hamilton’s Principle, Kane’s Equation. Compared to Newton–Euler mechanics, analytical mechanics have a wider range of applications and the formulation procedures are more mathematical. However, all existing methods of analytical mechanics were proposed based on some auxiliary variables. In this review, a novel analytical mechanics approach without the aid of Lagrange’s multiplier, projection, or any quasi or auxiliary variables is introduced for the central problem of mechanical systems. Since this approach was firstly proposed by Udwadia and Kalaba, it was called Udwadia–Kalaba Equation. It is a representation for the explicit expression of the equations of motion for constrained mechanical systems. It can be derived via the Gauss’s principle, d’Alembert’s principle or extended d’Alembert’s principle. It is applicable to both holonomic and nonholonomic equality constraints, as long as they are linear with respect to the accelerations or reducible to be that form. As a result, the Udwadia–Kalaba Equation can be applied to a very broad class of mechanical systems. This review starts with introducing the background by a brief review of the history of mechanics. After that, the formulation procedure of Udwadia–Kalaba Equation is given. Furthermore, the comparisons of Udwadia–Kalaba Equation with Newton–Euler Equation, Lagrange Equation and Kane’s Equation are made, respectively. At last, three different types of examples are given for demonstrations.

M. Prime

V.А. Nastasenko, M.V. Babiy, V.O. Protsenko

The article presents the results of experimental research of the cutting process with cutoff tools with laterally mounted multifaceted unresharpenable plates (MUP), which allowed to confirm their efficiency and progressiveness. As a result of research and experimental data processing, for the first time, the mathematical models were obtained that adequately describe force parameters (Pz and Py) of cutting process by the proposedcutoff tools. The rational values of rake and relief angles are determined.