М. Мирсабуров, А.С. Бердышев, C.Б. Эргашева
et al.
The work is devoted to the formulation and study of the solvability for a problem with missing conditions on the characteristic boundary of the domain and an analogue of the Frankl condition on the segment of the degeneracy for a hyperbolic equation. The difference between this problem and known local and nonlocal problems is that, firstly, a hyperbolic equation is taken with arbitrary positive power degeneracy and singular coefficients on the part of the boundary, and secondly, the characteristic boundary of the domain is arbitrarily divided into two pieces and the value of the desired function is set on the first piece, and the second piece is freed from the boundary condition and this missing Goursat condition is replaced by an analogue of the Frankl condition on the degeneracy segment, and the value of an unknown function on another characteristic boundary of the domain is also considered to be known. The conditions for the coefficients of the equation and the data of the formulated problem, ensuring the validity of the uniqueness theorem are found. The theorem of the existence of a solution to the problem is proved by reducing to the problem of solving a non-standard singular integral equation with a non-Fredholm integral operator in the non-characteristic part of the equation, the kernel of which has an isolated first-order singularity. Applying the Carleman regularization method to the received equation, the Wiener-Hopf integral equation is obtained. It is proved that the index of the Wiener-Hopf equation is zero, therefore it is uniquely reduced to the Fredholm integral equation of the second kind, the solvability of which follows from the uniqueness of the problem’s solution.
The classical inequalities of Bochkarev play a very important role in harmonic analysis. The meaning of these inequalities lies in the connection between the metric characteristics of functions and the summability of their Fourier coefficients. One of the most important directions of harmonic analysis is the theory of Fourier series. His interest in this direction is explained by his applications in various departments of modern mathematics and applied sciences, as well as the availability of many unsolved problems. One of these problems is the study of the interrelationships of the integral properties of functions and the properties of the sum of its coefficients. The solution of these problems was dedicated to the efforts of many mathematicians. And further research in this area are important and interesting problems and can give new, unexpected effects. In the article we receive a two-dimensional analog of the Bochkarev type theorem for the Fourier transform.
Fish often swim in crystallized group formations (schooling) and orient themselves against the incoming flow (rheotaxis). At the intersection of these two phenomena, we investigate the emergence of unique schooling patterns through passive hydrodynamic mechanisms in a fish pair, the simplest subsystem of a school. First, we develop a fluid dynamics-based mathematical model for the positions and orientations of two fish swimming against a flow in an infinite channel, modelling the effect of the self-propelling motion of each fish as a point-dipole. The resulting system of equations is studied to gain an understanding of the properties of the dynamical system, its equilibria and their stability. The system is found to have five types of equilibria, out of which only upstream swimming in in-line and staggered formations can be stable. A stable near-wall configuration is observed only in limiting cases. It is shown that the stability of these equilibria depends on the flow curvature and streamwise interfish distance, below critical values of which, the system may not have a stable equilibrium. The study reveals that simply through passive fluid dynamics, in the absence of any other feedback/sensing, we can justify rheotaxis and the existence of stable in-line and staggered schooling configurations.
Pietro Salizzoni, Cosimo Peruzzi, Massimo Marro
et al.
We investigate the ventilation conditions required to control the propagation of smoke, produced by a tunnel fire, in the presence of two inertial forcings: a transverse extraction system and a longitudinal flow. For that purpose, we performed a series of experiments in a reduced-scale tunnel, using a mixture of air and helium to simulate the release of hot smoke during a fire. Experiments were designed to focus on the ventilation flows that allow the buoyant release to be confined between two adjacent extraction vents. Different source conditions, in terms of density and velocity of the buoyant release, were analysed along with different vent configurations. Experiments allowed us to quantify the increase of the extraction velocity needed to confine the buoyant smoke, overcoming the effect of an imposed longitudinal velocity. Vents with a rectangular shape, and spanning over the whole tunnel width, provide the best performance. Finally, we studied the stratification conditions of the flow, individuating four regimes. Interestingly, when the stratification conditions fade out, as both the longitudinal flow and vertical extraction flows increase, the flow dynamics becomes almost independent of the forcing induced by the presence of buoyant smoke, which eventually acts as a passive scalar transported by the flow.
The statistical convergence is defined for sequences with the asymptotic density on the natural numbers, in general. In this paper, we introduce the statistical convergence in vector lattices by using the finite additive measures on directed sets. Moreover, we give some relations between the statistical convergence and the lattice properties such as the order convergence and lattice operators.
The paper considers the space of generalized fractional-maximal function, constructed on the basis of a rearrangement-invariant space. Two types of cones generated by a nonincreasing rearrangement of a generalized fractional-maximal function and equipped with positive homogeneous functionals are constructed. The question of embedding the space of generalized fractional-maximal function in a rearrangementinvariant space is investigated. This question reduces to the embedding of the considered cone in the corresponding rearrangement-invariant spaces. In addition, conditions for covering a cone generated by generalized fractional-maximal function by the cone generated by generalized Riesz potentials are given. Cones from non-increasing rearrangements of generalized potentials were previously considered in the works of M. Goldman, E. Bakhtigareeva, G. Karshygina and others.
We simulate the head-on collision between vortex rings with circulation Reynolds numbers of 4000 using an adaptive, multiresolution solver based on the lattice Green's function. The simulation fidelity is established with integral metrics representing symmetries and discretization errors. Using the velocity gradient tensor and structural features of local streamlines, we characterize the evolution of the flow with a particular focus on its transition and turbulent decay. Transition is excited by the development of the elliptic instability, which grows during the mutual interaction of the rings as they expand radially at the collision plane. The development of antiparallel secondary vortex filaments along the circumference mediates the proliferation of small-scale turbulence. During turbulent decay, the partitioning of the velocity gradients approaches an equilibrium that is dominated by shearing and agrees well with previous results for forced isotropic turbulence. We also introduce new phase spaces for the velocity gradients that reflect the interplay between shearing and rigid rotation and highlight geometric features of local streamlines. In conjunction with our other analyses, these phase spaces suggest that, while the elliptic instability is the predominant mechanism driving the initial transition, its interplay with other mechanisms, e.g. the Crow instability, becomes more important during turbulent decay. Our analysis also suggests that the geometry-based phase space may be promising for identifying the effects of the elliptic instability and other mechanisms using the structure of local streamlines. Moving forward, characterizing the organization of these mechanisms within vortices and universal features of velocity gradients may aid in modelling turbulent flows.
A boundary value problem for a fractionally loaded heat equation is considered in the first quadrant. The loaded term has the form of the Riemann-Liouville’s fractional derivative with respect to the time variable, and the order of the derivative in the loaded term is less than the order of the differential part. The study is based on reducing the boundary value problem to a Volterra integral equation. The kernel of the obtained integral equation contains a special function, namely, the Wright function. The kernel is estimated, and the conditions for the unique solvability of the integral equation are obtained.
The study considers the solvability of a mixed problem for a Hilfer type partial differential equation of the even order with initial value conditions and small positive parameters in mixed derivatives in threedimensional domain. It studies the solution to this fractional differential equation of higher order in the class of regular functions. The case, when the order of fractional operator is 1 < α < 2, is examined. During this study the authors use the Fourier series method and obtain a countable system of ordinary differential equations. The initial value problem is integrated as an ordinary differential equation and the integrated constants find by the aid of given initial value conditions. Using the Cauchy–Schwarz inequality and the Bessel inequality, it is proved the absolute and uniform convergence of the obtained Fourier series. The stability of the solution to the mixed problem on the given functions is studied.
M.T. Jenaliyev, M.G. Yergaliyev, A.A. Assetov
et al.
We consider some initial boundary value problems for the Burgers equation in a rectangular domain, which in a sense can be taken as a model one. The fact is that such a problem often arises when studying the Burgers equation in domains with moving boundaries. Using the methods of functional analysis, priori estimates, and Faedo-Galerkin in Sobolev spaces and in a rectangular domain, we show the correctness of the initial boundary value problem for the Burgers equation with nonlinear boundary conditions of the Neumann type.
Periodic water waves generate Stokes drift as manifest from the orbits of Lagrangian particles not fully closing. Stokes drift can contribute to the transport of floating marine litter, including plastic. Previously, marine litter objects have been considered to be perfect Lagrangian tracers, travelling with the Stokes drift of the waves. However, floating marine litter objects have large ranges of sizes and densities, which potentially result in different rates of transport by waves due to the non-Lagrangian behaviour of the objects. Through a combination of theory and experiments for idealised spherical objects in deep-water waves, we show that different objects are transported at different rates depending on their size and density, and that larger buoyant objects can have increased drift compared with Lagrangian tracers. We show that the mechanism for the increased drift observed in our experiments comprises the variable submergence and the corresponding dynamic buoyancy force components in a direction perpendicular to the local water surface. This leads to an amplification of the drift of these objects compared to the Stokes drift when averaged over the wave cycle. Using an expansion in wave steepness, we derive a closed-form approximation for this increased drift, which can be included in ocean-scale models of marine litter transport.
In the differential geometry of curves and surfaces, the curvatures of curves and surfaces are often calculated and results are given. In particular, the results given by using the apparatus of the curve - surface pair are important in terms of what kind of surface the surface indicates. In this study, some relationships between curvatures of the parallel surface pair ( X,Xr ) via structure functions of non - developable ruled surface X ( u,v ) = a ( u ) + vb ( u ) are established such that a ( u ) is striction curve of non - developable surface and b ( u ) is a unit spherical curve in E 3. Especially, it is examined whether the non - developable surface Xr is minimal surface, linear Weingarten surface and Weingarten surface. X and its parallel Xr are expressed on the Helicoid surface sample. It is indicated on the figure with the help of S W P. Moreover, curvatures of curve - surface pairs ( X,a ) and ( Xr,β ) are investigated and some conclusions are obtained.
It is known that the eigenvalues λn(n = 1, 2, ...) numbered in decreasing order and taking the multiplicity of the self-adjoint Sturm-Liouville operator with a completely continuous inverse operator L−1 have the following property (∗) λn → 0, when n → ∞, moreover, than the faster convergence to zero so the operator L−1 is best approximated by finite rank operators.
The following question:
- Is it possible for a given nonlinear operator to indicate a decreasing numerical sequence characterized by the property (∗)?
naturally arises for nonlinear operators. In this paper, we study the above question for the nonlinear Sturm-Liouville operator. To solve the above problem the theorem on the maximum regularity of the solutions of the nonlinear Sturm-Liouville equation with greatly growing and rapidly oscillating potential in the space L2(R) (R = (−∞, ∞)) is proved. Twosided estimates of the Kolmogorov widths of the sets associated with solutions of the nonlinear SturmLiouville equation are also obtained. As is known, the obtained estimates of Kolmogorov widths give the opportunity to choose approximation apparatus that guarantees the minimum possible error.
Analytic first and second derivatives of the energy are developed for the fragment molecular orbital method interfaced with molecular mechanics in the electrostatic embedding scheme at the level of Hartree-Fock and density functional theory. The importance of the orbital response terms is demonstrated. The role of the electrostatic embedding upon molecular vibrations is analyzed, comparing force field and quantum-mechanical treatments for an ionic liquid and a solvated protein. The method is applied for 100 protein conformations sampled in MD to take into account the complexity of a flexible protein structure in solution, and a good agreement to experimental data is obtained: frequencies from an experimental IR spectrum are reproduced within 17 cm$^{-1}$.
This contribution aims to shed light on mathematical epidemic dynamics modelling from the viewpoint of analytical mechanics. To set the stage, it recasts the basic SIR model of mathematical epidemic dynamics in an analytical mechanics setting. Thereby, it considers two possible re-parameterizations of the basic SIR model. On the one hand, it is proposed to re-scale time, while on the other hand, to transform the coordinates, i.e.\ the independent variables. In both cases, Hamilton's equations in terms of a suited Hamiltonian as well as Hamilton's principle in terms of a suited Lagrangian are considered in minimal and extended phase and state space coordinates, respectively. The corresponding Legendre transformations relating the various options for the Hamiltonians and Lagrangians are detailed. Ultimately, this contribution expands on a multitude of novel vistas on mathematical epidemic dynamics modelling that emerge from the analytical mechanics viewpoint. As result, it is believed that interesting and relevant new research avenues open up when exploiting in depth the analogies between analytical mechanics and mathematical epidemic dynamics modelling.
Stephan Sponar, Rene I. P. Sedmik, Mario Pitschmann
et al.
Among the known particles, the neutron takes a special position, as it provides experimental access to all four fundamental forces and a wide range of hypothetical interactions. Despite being unstable, free neutrons live long enough to be used as test particles in interferometric, spectroscopic, and scattering experiments probing low-energy scales. As was already recognized in the 1970s, fundamental concepts of quantum mechanics can be tested in neutron interferometry using silicon perfect-single-crystals. Besides allowing for tests of uncertainty relations, Bell inequalities and alike, neutrons offer the opportunity to observe the effects of gravity and hypothetical dark forces acting on extended matter wave functions. Such tests gain importance in the light of recent discoveries of inconsistencies in our understanding of cosmology as well as the incompatibility between quantum mechanics and general relativity. Experiments with low-energy neutrons are thus indispensable tools for probing fundamental physics and represent a complementary approach to colliders. In this review we discuss the history and experimental methods used at this low-energy frontier of physics and collect bounds and limits on quantum mechanical relations and dark energy interactions.
In given work are considered model - theoretical properties of companions of (n1; n2) - Jonsson theory. Also were considered a communications between center and (n1; n2) - Jonsson theory. Herewith considered theories is perfect in the sense of the existence of appropriate model companion. In given article introduced new concepts: (n1; n2) - Jonsson theory, D - model companion. New results are shown with respect to model companions of - Jonsson theory and 1 - perfect 1 - Jonsson theory.