Hasil untuk "Analytic mechanics"

Oğuzhan Kaplan, Manouk Abkarian, Simon Mendez

Airborne transmission has been recognised as an important route of transmission for SARS-CoV-2, the virus responsible for the COVID-19 pandemic. While coughing and sneezing are major aerosol sources, asymptomatic transmission highlights the need to study other exhalation modes in social settings. Gathering around a table, a common scenario for human interactions, may influence airborne transmission by modifying the airflows. Here, we employ high-fidelity large-eddy simulations to investigate the effect of a table for periodic breathing conditions (Reynolds number $Re\approx 10^3$ – $3\times 10^3$ , Froude number $Fr\approx 17$ – $50$ ) as well as during sudden, forceful exhalations at peak values of $Re\approx 1.2\times 10^4$ and $Fr\approx 70$ , mimicking laughter. During downward exhalations, the distance between the source and the table defines a new length scale that constrains the natural spread of buoyant puffs and jets. The table limits forward particle transport but, in doing so, may increase particle concentrations reaching a recipient, raising transmission risks. Simulations of forceful exhalations, such as laughter, further show that the table acts as an inertial filter – intercepting medium-sized particles that would otherwise remain airborne. This introduces a cutoff size dependent on puff inertia, altering the resulting airborne particle size distribution.

Kim Sharp

In statistical mechanics the zeroth law of thermodynamics is taken as a postulate which, as its name indicates, logically precedes the first and second laws. Treating it as a postulate has consequences for how temperature is introduced into statistical mechanics and for the molecular interpretation of temperature. One can, however, derive the zeroth law from first principles starting from a classical Hamiltonian using basic mechanics and a geometric representation of the phase space of kinetic energy configurations - the velocity hypersphere. In this approach there is no difficulty in providing a molecular interpretation of temperature, nor in deriving equality of temperature as the condition of thermal equilibrium. The approach to the macroscopic limit as a function of the number of atoms is easily determined. One also obtains with little difficulty the Boltzmann probability distribution, the statistical mechanical definition of entropy and the configuration partition function. These relations, along with the zeroth law, emerge as straightforward consequences of atoms in random motion.

M.M. Aripov, D. Utebaev, B.D. Utebaev et al.

The issues of stability in solving nonlinear difference equations were considered. Based on a generalized difference analog of the well-known Bihari lemma, stability conditions for a trivial solution based on initial data were obtained, and an a priori estimate of stability under permanent disturbances was determined. The results were used to study the stability of solving explicit and implicit difference schemes approximating nonlinear parabolic equations.

С.З. Джамалов, Ш.Ш. Худойкулов

It is known that V.A. Ilyin and E.I. Moiseev studied generalized nonlocal boundary value problems for the Sturm-Liouville equation, the nonlocal boundary conditions specified at the interior points of the interval under consideration. For such problems, uniqueness and existence theorems for a solution to the problem were proven. There are many difficulties in studying these generalized nonlocal boundary value problems for partial differential equations, especially in obtaining a priori estimates. Therefore, it is necessary to use new methods for solving generalized nonlocal problems (forward problems). As we know, it is not difficult to establish a connection between forward and inverse problems. Therefore, when solving generalized nonlocal boundary value problems for partial differential equations, reducing them to multipoint inverse problems is necessary. The first results in the direction belong to S.Z. Dzhamalov. In his works, he proposed and investigated multipoint inverse problems for some equations of mathematical physics. In this article, the authors studied the correctness of one linear two-point inverse problem for the multidimensional heat conduction equation. Using the methods of a priori estimates, Galerkin’s method, a sequence of approximations and contracting mappings, the unique solvability of the generalized solution of the linear two-point inverse problem for the multidimensional heat equation was proved.

J.A. Otarova

Many problems in mechanics, physics, and geophysics lead to solving partial differential equations that are not included in the known classes of elliptic, parabolic or hyperbolic equations. Such equations, as a rule, began to be called non-classical equations of mathematical physics. The theory of degenerate equations is one of the central branches of the modern theory of partial differential equations. This is primarily due to the identification of a variety of applied problems, the mathematical modeling of which serves the study of various types of degenerate equations. The study of boundary value problems for mixed type’s equations of the fourth-order with power-law degeneration remains relevant. In this work, a boundary value problem in a rectangular domain for a degenerate equation of the fourth-order mixed-type is posed and investigated. Well-posedness of the boundary value problem for a fourth-order partial differential equation is established by proving the existence and uniqueness of the solution. Under sufficient conditions, a solution to the problem under consideration was explicitly found by the variable separation method.

R.R. Ashurov, Yu.E. Fayziev, M.U. Khudoykulova

In recent years, the fractional partial differential equation of the Boussinesq type has attracted much attention from researchers due to its practical importance. In this paper, we study a non-local problem for the Boussinesq type equation Dtαu(t)+A Dtαu(t)+ν2Au(t) =0, 0<t<T, 1<α<3∕2, where Dtα is the Caputo fractional derivative, and A is an abstract operator. In the classical case, i.e., when α = 2, this problem has been studied previously, and an interesting effect has been discovered: the existence and uniqueness of a solution depend significantly on the length of the time interval and the parameter ν. In this note, we show that in the case of a fractional equation, there is no such effect: a solution of the problem exists and is unique for any T and ν.

S. Sivasankar, R. Udhayakumar, V. Muthukumaran et al.

The goal of this study is to propose the existence of mild solutions to delay fractional neutral stochastic differential systems with almost sectorial operators involving the Hilfer fractional (HF) derivative in Hilbert space, which generalized the famous Riemann-Liouville fractional derivative. The main techniques rely on the basic principles and concepts from fractional calculus, semigroup theory, almost sectorial operators, stochastic analysis, and the Mönch fixed point theorem via the measure of noncompactness (MNC). Particularly, the existence result of the equation is obtained under some weakly compactness conditions. An example is given at the end of this article to show the applications of the obtained abstract results.

T.G. Ergashev, A.R. Ryskan, N.N. Yuldashev

Expansion formulas associated with the multidimensional Lauricella hypergeometric functions are wellestablished and extensively utilized. However, the recurrence relations inherit in these formulas add extra complexities to their use. A thorough analysis of the characteristics of these expansion formulas shows that they can be simplified and converted into a more convenient form. This paper presents new recurrence free decomposition formulas, which are employed to solve boundary value problems.

Dominique Levesque, Nicolas Sourlas

One of the important questions in statistical mechanics is how irreversibility (time's arrow) occurs when Newton equations of motion are time reversal invariant. One objection to irreversibility is based on Poincaré's recursion theorem: a classical hamiltonian confined system returns after some time, so-called Poincaré recurrence time (PRT), close to its initial configuration. Boltzmann's reply was that for a $N \sim 10^{23} $ macroscopic number of particles, PRT is very large and exceeds the age of the universe. In this paper we compute for the first time, using molecular dynamics, a typical recurrence time $ T(N)$ for a realistic case of a gas of $N$ particles. We find that $T(N) \sim N^z \exp (y N) $ and determine the exponents $y$ and $z$ for different values of the particle density and temperature. We also compute $y$ analytically using Boltzmann's hypotheses. We find an excellent agreement with the numerical results. This agreement validates Boltzmann's hypotheses which are not yet mathematically proven. We establish that that $T(N) $ exceeds the age of the Universe for a relatively small number of particles, much smaller than $ 10^{23} $.

Zhicheng He, Zhifu Chen, Guilin Liu et al.

Galactic-wide outflows driven by active galactic nuclei (AGNs) is a routinely invoked feedback mechanism in galaxy evolution models. Hitherto, the interplay among the interstellar gas on galactic scales, the propagation of AGN outflows and the fundamental AGN parameters during evolution remains elusive. Powerful nuclear outflows are found to favorably exist at early AGN stages usually associated with high accretion rates and weak narrow emission lines. In a sample of quasars emitting Mg II narrow absorption lines (NALs) from the Sloan Digital Sky Survey, we discover an unprecedented phenomenon where galaxy-scale inflow-dominated transforming into outflow-dominated gas accompanied by an increasing strength of the narrow [O III] line, at a confidence level of 6.7σ. The fact that nuclear outflows diminish while galaxy-wide outflows intensifies as AGNs evolve implies that early-stage outflows interact with interstellar medium on galactic scales and trigger the gradual transformation into galaxy-wide outflows, providing observational links to the hypothetical multi-stage propagation of AGN outflows that globally regulates galaxy evolution.

S. Katsyv, V. Kukharchuk, N. Kondratenko et al.

The paper considers the problems that arise when using the theory of fuzzy sets to solve applied problems. Unlike stochastic methods, which are based on statistical data, fuzzy set theory methods make sense to apply when statistical data are not available. In these cases, algorithms should be based on membership functions formed by experts who are specialists in this field of knowledge. Ideally, complete information about membership functions is required, but this is an impractical procedure. More often than not, even the most experienced expert can determine only their carriers or separate sets of the α-cuts for unknown fuzzy parameters of the system. Building complete membership functions of unknown fuzzy parameters on this basis is risky and unreliable. Therefore, the paper proposes an extension of the fuzzy sets theory axiomatics in order to introduce non-traditional (less demanding on the completeness of data on membership functions) extension principles and operations on fuzzy sets. The so-called α-weak operations on fuzzy sets are proposed, which are based on the use of separate sets of the α-cuts. It is also shown that all classical theorems of Cantor sets theory apply in the extended axiomatic theory. New extension principles of generalization have been introduced, which allow solving problems in conditions of significant uncertainty of information.

Jeremy Canfield, Anna Galler, James K. Freericks

Quantum mechanics has about a dozen exactly solvable potentials. Normally, the time-independent Schrödinger equation for them is solved by using a generalized series solution for the bound states (using the Fröbenius method) and then an analytic continuation for the continuum states (if present). In this work, we present an alternative way to solve these problems, based on the Laplace method. This technique uses a similar procedure for the bound states and for the continuum states. It was originally used by Schrödinger when he solved the wave functions of hydrogen. Dirac advocated using this method too. We discuss why it is a powerful approach to solve all problems whose wave functions are represented in terms of confluent hypergeometric functions, especially for the continuum solutions, which can be determined by an easy-to-program contour integral.

F.N. Dekhkonov

In this paper, we consider a boundary control problem for a parabolic equation in a segment. In the part of the domain’s bound it is a given value of the solution and it is required to find controls to get the average value of the solution. The given control problem is reduced to a system of Volterra integral equations of the first kind. By the mathematical-physics methods it is proved that like this control functions exist over some domain, the necessary estimates were found and obtained.

Midas Segers, Aderik Voorspoels, Takahiro Sakaue et al.

Proteins often regulate their activities via allostery - or action at a distance - in which the binding of a ligand at one binding site influences the affinity for another ligand at a distal site. Although less studied than in proteins, allosteric effects have been observed in experiments with DNA as well. In these experiments two or more proteins bind at distinct DNA sites and interact indirectly with each other, via a mechanism mediated by the linker DNA molecule. We develop a mechanical model of DNA/protein interactions which predicts three distinct mechanisms of allostery. Two of these involve an enthalpy-mediated allostery, while a third mechanism is entropy driven. We analyze experiments of DNA allostery and highlight the distinctive signatures allowing one to identify which of the proposed mechanisms best fits the data.

A.R. Yeshkeyev, I.O. Tungushbayeva, S.M. Amanbekov

This article is devoted to the study of Jonsson quasivarieties in a signature enriched with new predicate and constant symbols. New concepts of semantic Jonsson quasivariety and fragment-conservativeness of the center of the Jonsson theory are introduced. The cosemanticness classes of the Jonsson spectrum constructed for a semantic Jonsson quasvariety are considered. In this case, the Kaiser hull of the semantic Jonsson quasivariety is assumed to be existentially prime. By constructing a central type for classes of theories from the Jonsson spectrum, the following results are formulated and proved. In the first main result, the necessary and sufficient condition is given for the center of the cosemanticness class of an existentially prime semantic Jonsson quasivariety to be λ-stable. The second result is the criterion for the center of the class of theories to be ω-categorical in the enriched language. The obtained theorems can be useful in continuing studies of various Jonsson algebras, in particular, Jonsson quasivarieties.

R.Ye. Uteshova, Ye.V. Kokotova

This paper deals with a problem of finding a bounded solution of a system of nonhomogeneous linear differential equations with an unbounded matrix of coefficients on a finite interval. The right-hand side of the equation belongs to a space of continuous functions bounded with some weight; the weight function is chosen taking into account the behavior of the coefficient matrix. The problem is studied using a modified version of the parameterization method with non-uniform partitioning. Necessary and sufficient conditions of well-posedness of the problem are obtained in terms of a bilaterally infinite matrix of special structure.

Marie-Sophie Hartig, Sönke Schuster, Gudrun Wanner

Tilt-to-length coupling is a technical term for the cross-coupling of angular or lateral jitter into an interferometric phase signal. It is an important noise source in precision interferometers and originates either from changes in the optical path lengths or from wavefront and clipping effects. Within this paper, we focus on geometric TTL coupling and categorize it into a number of different mechanisms for which we give analytic expressions. We then show that this geometric description is not always sufficient to predict the TTL coupling noise within an interferometer. We, therefore, discuss how understanding the geometric effects allows TTL noise reduction already by smart design choices. Additionally, they can be used to counteract the total measured TTL noise in a system. The presented content applies to a large variety of precision interferometers, including space gravitational wave detectors like LISA.

K.H.F. Jwamer, Sh.Sh. Ahmed, D.Kh. Abdullah

In this paper, we suggest two new methods for approximating the solution to the Volterra integro-fractional differential equation (VIFDEs), based on the normal quadratic spline function and the second method used the Richardson Extrapolation technique the usage of discrete collocation points. The fractional derivatives are regarded in the Caputo perception. A new theorem for the Richardson Extrapolation points for using the finite difference approximation of Caputo derivative is introduced with their proof. New techniques using the first derivative at the initial point such that obtained by follow two cases the first using trapezoidal rule and the second using the first step of linear spline function using the Richardson Extrapolation method. Specifically, the program is given in examples analysis in Matlab (R2018b). Numerical examples are available to illuminate the productivity and trustworthiness of the methods, as well as, follow the Clenshaw Curtis rule for calculating the required integrals for those equations.

М.B. Muratbekov, A.O. Suleimbekova

Partial differential equations of the third order are the basis of mathematical models of many phenomena and processes, such as the phenomenon of energy transfer of hydrolysis of adenosine triphosphate molecules along protein molecules in the form of solitary waves, i.e. solitons, the process of transferring soil moisture in the aeration zone, taking into account its movement against the moisture potential. In particular, this class includes the nonlinear Korteweg-de Vries equation, which is the main equation of modern mathematical physics. It is known that various problems have been studied for the Korteweg-de Vries equation and many fundamental results obtained. In this paper, issues about the existence of a resolvent and separability (maximum smoothness of solutions) of a class of linear singular operators of the Korteweg-de Vries type in the case of an unbounded domain with strongly increasing coefficients are investigated.

A. Ashyralyev, M. Urun

In this study the source identification problem for the one-dimensional Schr¨odinger equation with non-local boundary conditions is considered. A second order of accuracy Crank-Nicolson difference scheme for the numerical solution of the differential problem is presented. Stability estimates are proved for the solution of this difference scheme. Numerical results are given.