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New method for finding exact solutions of nonlinear differential equations is presented. It is based on constructing the polygon corresponding to the equation studied. The algorithms of power geometry are used. The method is applied for…
Rotation curves of spiral galaxies are known with reasonable precision for a large number of galaxies with similar morphologies. The data implies that non-Keplerian fall--off is seen. This implies that (i) large amounts of dark matter must…
Scintillometer measurements of the turbulence inner-scale length $l_o$ and refractive index structure function $C_n^2$ allow for the retrieval of large-scale area-averaged turbulent fluxes in the atmospheric surface layer. This retrieval…
We give direct and inverse theorems for the weighted approximation of functions with endpoint singularities by combinations of Bernstein operators.
We prove finite time blowup of the Burgers-Hilbert equation. We construct smooth initial data with finite $H^5$-norm such that the $L^\infty$-norm of the spacial derivative of the solution blows up. The blowup is an asymptotic self-similar…
In this note, we consider some Burgers-like equations involving Laguerre derivatives and demonstrate that it is possible to construct specific exact solutions using separation of variables. We prove that a general scheme exists for…
The Fisher and Burgers' equations with finite memory transport, describing reaction-diffusion and convection-diffusion processes, respectively have recently attracted a lot of attention in the context of chemical kinetics, mathematical…
We prove some combinatorial results required for the proof of the following conjecture of Nekrasov: The generating function of closed string invariants in local Calabi-Yau geometries obtained by appropriate fibrations of $A_N$ singularities…
The generating functionals (GF) method in Bogolyubov's formulation and its application for particle physics is considered. Effectiveness of the method is illustrated by two examples. So, GF method can be used as the technical trick solving…
Motivated by the classical ideas of generating functions for orthogonal polynomials, we initiate a new line of investigation on "generating operators" for a family of differential operators between two manifolds. We prove a novel formula of…
The formation of singularities in finite time in non-local Burgers' equations, with time-fractional derivative, is studied in detail. The occurrence of finite time singularity is proved, revealing the underlying mechanism, and precise…
The functional RG (FRG) approach to pinning of $d$-dimensional manifolds is reexamined at any temperature $T$. A simple relation between the coupling function $R(u)$ and a physical observable is shown in any $d$. In $d=0$ its beta function…
This research presents a new turbulence generation method based on stochastic wavelets and tests its various properties in both homogeneous and inhomogeneous turbulence. Turbulence field can be generated with less basis compared to previous…
Our previous work on the one-dimensional KPZ equation with sharp wedge initial data is extended to the case of the joint height statistics at n spatial points for some common fixed time. Assuming a particular factorization, we compute an…
By considering the partition function of the topological 2D gravity, a conformal field theory on the Airy curve emerges as the mirror theory of Gromov-Witten theory of a point. In particular, a formula for bosonic n-point functions in terms…
Self-gravitating isothermal supersonic turbulence is analyzed in the asymptotic limit of large Reynolds numbers. Based on the inviscid invariance of total energy, an exact relation is derived for homogeneous, (not necessarily isotropic)…
Nelkin scaling, the scaling of moments of velocity gradients in terms of the Reynolds number, is an alternative way of obtaining inertial-range information. It is shown numerically and theoretically for the Burgers equation that this…
We investigate a way to approximate the maximum of a polynomial over a polytopal region by using Handelman's polynomial decomposition and continuous multivariate generating functions. The maximization problem is NP-hard, but our…
Ouroboros functions have shown some interesting properties when subjected to conventional operations. The aim of this paper is to continue our investigation and prove some additional properties of these functions. Using algebraic methods,…
We investigate correlation functions in a periodic box-ball system. For the second and the third nearest neighbor correlation functions, we give explicit formulae obtained by combinatorial methods. A recursion formula for a specific…