THE ASYMPTOTICS OF A SOLUTION OF THE MULTIDIMENSIONAL HEAT EQUATION WITH UNBOUNDED INITIAL DATA

For the multidimensional heat equation, the long-time asymptotic approximation of the solution of the Cauchy problem is obtained in the case when the initial function grows at infinity and contains logarithms in its asymptotics. In addition to natural applications to processes of heat conduction and diffusion, the investigation of the asymptotic behavior of the solution of the problem under consideration is of interest for the asymptotic analysisof equations of parabolic type. The auxiliary parameter method plays a decisive role in the investigation

EVOLUTION OF A MULTISCALE SINGULARITY OF THE SOLUTION OF THE BURGERS EQUATION IN THE 4-DIMENSIONAL SPACE–TIME

The solution of the Cauchy problem for the vector Burgers equation with a small parameter of dissipation $\varepsilon$ in the $4$-dimensional space-time is studied: $$\mathbf{u}_t + (\mathbf{u}\nabla) \mathbf{u} = \varepsilon \triangle \mathbf{u}, \quad u_{\nu} (\mathbf{x}, -1, \varepsilon) = - x_{\nu} + 4^{-\nu}(\nu + 1) x_{\nu}^{2\nu + 1},$$ With the help of the Cole–Hopf transform $\mathbf{u} = - 2 \varepsilon \nabla \ln H,$ the exact solution and its leading asymptotic approximation, depending on six space-time scales, near a singular point are found. A formula for the growth of partial derivatives of the components of the vector field $\mathbf{u}$ on the time interval from the initial moment to the singular point, called the formula of the gradient catastrophe, is established: $$\frac{\partial u_{\nu} (0, t, \varepsilon)}{\partial x_{\nu}} = \frac{1}{t} \left[ 1 + O \left( \varepsilon |t|^{- 1 - 1/\nu} \right) \right]\!, \quad \frac{t}{\varepsilon^{\nu /(\nu + 1)} } \to -\infty, \quad t \to -0.$$The asymptotics of the solution far from the singular point, involving a multistep reconstruction of the space-time scales, is also obtained: $$ u_{\nu} (\mathbf{x}, t, \varepsilon) \approx - 2 \left( \frac{t}{\nu + 1} \right)^{1/2\nu} \tanh \left[ \frac{x_{\nu}}{\varepsilon} \left( \frac{t}{\nu + 1} \right)^{1/2\nu} \right]\!, \quad \frac{t}{\varepsilon^{\nu /(\nu + 1)} } \to +\infty. $

Numerical simulation of the stress-strain state of the dental system

We present mathematical models, computational algorithms and software, which can be used for prediction of results of prosthetic treatment. More interest issue is biomechanics of the periodontal complex because any prosthesis is accompanied by a risk of overloading the supporting elements. Such risk can be avoided by the proper load distribution and prediction of stresses that occur during the use of dentures. We developed the mathematical model of the periodontal complex and its software implementation. This model is based on linear elasticity theory and allows to calculate the stress and strain fields in periodontal ligament and jawbone. The input parameters for the developed model can be divided into two groups. The first group of parameters describes the mechanical properties of periodontal ligament, teeth and jawbone (for example, elasticity of periodontal ligament etc.). The second group characterized the geometric properties of objects: the size of the teeth, their spatial coordinates, the size of periodontal ligament etc. The mechanical properties are the same for almost all, but the input of geometrical data is complicated because of their individual characteristics. In this connection, we develop algorithms and software for processing of images obtained by computed tomography (CT) scanner and for constructing individual digital model of the tooth-periodontal ligament-jawbone system of the patient. Integration of models and algorithms described allows to carry out biomechanical analysis on three-dimensional digital model and to select prosthesis design.Comment: 19 pages, 9 figure

Joint statistics of amplitudes and phases in Wave Turbulence

Random Phase Approximation (RPA) provides a very convenient tool to study the ensembles of weakly interacting waves, commonly called Wave Turbulence. In its traditional formulation, RPA assumes that phases of interacting waves are random quantities but it usually ignores randomness of their amplitudes. Recently, RPA was generalised in a way that takes into account the amplitude randomness and it was applied to study of the higher momenta and probability densities of wave amplitudes. However, to have a meaningful description of wave turbulence the RPA properties assumed for the initial fields must be proven to survive over the nonlinear evolution time, and such a proof is the main goal of the present paper. We derive an evolution equation for the full probability density function which contains the complete information about the joint statistics of all wave amplitudes and phases. We show that, for any initial statistics of the amplitudes, the phase factors remain statistically independent uniformly distributed variables. If in addition the initial amplitudes are also independent variables (but with arbitrary distributions) they will remain independent when considered in small sets which are much less than the total number of modes. However, if the size of a set is of order of the total number of modes then the joint probability density for this set is not factorisable into the product of one-mode probabilities. In the other words, the modes in such a set are involved in a ``collective'' (correlated) motion. We also study new type of correlators describing the phase statistics.Comment: 27 pages, uses feynmf packag

Influence of CO2-laser pulse parameters on 13.5 nm extreme ultraviolet emission features from irradiated liquid tin target

Laser-produced plasma (LPP) induced during irradiation of a liquid tin droplet with diameter of 150 um and 180 um by CO2 laser pulse with various pulse durations and energies is considered. The two-dimensional radiative magnetohydrodynamic (RMHD) plasma code is used to simulate the emission and plasma dynamics of multicharged ion tin LPP. Results of simulations for various laser pulse durations and 75-600 mJ pulse energies with Gaussian and experimentally taken temporal profiles are discussed. It is found that if the mass of the target is big enough to provide the plasma flux required (the considered case) a kind of dynamic quasi-stationary plasma flux is formed. In this dynamic quasi-stationary plasma flux, an interlayer of relatively cold tin vapor with mass density of 1-2 g/cm3 is formed between the liquid tin droplet and low density plasma of the critical layer. Expanding of the tin vapor from the droplet provides the plasma flux to the critical layer. In critical layer the plasma is heated up and expands faster. In the simulation results with spherical liquid tin target, the CE into 2${\pi}$ is of 4% for 30 ns FWHM and just slightly lower - of 3.67% for 240 ns FWHM for equal laser intensities of 14 GW/cm2. This slight decay of the in-band EUV yield with laser pulse duration is conditioned by an increasing of radiation re-absorption by expanding plasma from the target, as more cold plasma is produced with longer pulse. The calculated direction diagrams of in-band EUV emission permit to optimize a collector configuration

Energy cascades and spectra in turbulent Bose-Einstein condensates

We present a numerical study of turbulence in Bose-Einstein condensates within the 3D Gross-Pitaevskii equation. We concentrate on the direct energy cascade in forced-dissipated systems. We show that behavior of the system is very sensitive to the properties of the model at the scales greater than the forcing scale, and we identify three different universal regimes: (1) a non-stationary regime with condensation and transition from a four-wave to a three-wave interaction process when the largest scales are not dissipated, (2) a steady weak wave turbulence regime when largest scales are dissipated with a friction-type dissipation, (3) a state with a scale-by-scale balance of the linear and the nonlinear timescales when the large-scale dissipation is a hypo-viscosity.Comment: 10 pages, 5 figure

Interaction of Kelvin waves and nonlocality of energy transfer in superfluids

We argue that the physics of interacting Kelvin Waves (KWs) is highly nontrivial and cannot be understood on the basis of pure dimensional reasoning. A consistent theory of KW turbulence in superfluids should be based upon explicit knowledge of their interactions. To achieve this, we present a detailed calculation and comprehensive analysis of the interaction coefficients for KW turbuelence, thereby, resolving previous mistakes stemming from unaccounted contributions. As a first application of this analysis, we derive a local nonlinear (partial differential) equation. This equation is much simpler for analysis and numerical simulations of KWs than the Biot-Savart equation, and in contrast to the completely integrable local induction approximation (in which the energy exchange between KWs is absent), describes the nonlinear dynamics of KWs. Second, we show that the previously suggested Kozik-Svistunov energy spectrum for KWs, which has often been used in the analysis of experimental and numerical data in superfluid turbulence, is irrelevant, because it is based upon an erroneous assumption of the locality of the energy transfer through scales. Moreover, we demonstrate the weak nonlocality of the inverse cascade spectrum with a constant particle-number flux and find resulting logarithmic corrections to this spectrum

Gravity wave turbulence in a laboratory flume

We present an experimental study of the statistics of surface gravity wave turbulence in a flume of a horizontal size 12×6  m. For a wide range of amplitudes the wave energy spectrum was found to scale as Eω∼ω-ν in a frequency range of up to one decade. However, ν appears to be nonuniversal: it depends on the wave intensity and ranges from about 6 to 4. We discuss our results in the context of existing theories and argue that at low wave amplitudes the wave statistics is affected by the flume finite size, and at high amplitudes the wave breaking effect dominates

DISPERSIVE RAREFACTION WAVE WITH A LARGE INITIAL GRADIENT

Consider the Cauchy problem for the Korteweg-de Vries equation with a small parameter at the highest derivative and a large gradient of the initial function. Numerical and analytical methods show that the obtained using renormalization formal asymptotics, corresponding to rarefaction waves, is an asymptotic solution of the KdV equation. The graphs of the asymptotic solutions are represented, including the case of non-monotonic initial data

New possibilities and tools for corporate strategic management for supporting its high competitiveness and economic effectiveness

The purpose of the article is to determine new possibilities and tools for corporate strategic management and to substantiate the necessity for their usage for supporting a company’s competitiveness and economic effectiveness in modern Russia. The information and analytical basis of the research consists of the materials of the Report on global competitiveness for 2016-2017, prepared within the World economic forum. For verification of the offered hypothesis, the methods of deduction, induction, synthesis, systemic, problem, and structural & functional analysis, as well as analysis of causal connections, and the methods of modeling and formalization were used. Because of the research, a conclusion is made that imperfection of the process of strategic management is a reason for low competitiveness and economic effectiveness of modern Russian companies. To solve this problem, the authors substantiate the necessity for applying the principle of interactivity, which supposes consideration of new possibilities and usage of leading tools in the process of corporate strategic management and develop the recommendations and offer an interactive model for strategic management of a modern company in modern Russia for the purpose of supporting high competitiveness and economic effectiveness of domestic entrepreneurship.peer-reviewe