Russian Journal of Mathematical Physics

Formulas for the asymptotics of some class of integrals of rapidly oscillating functions that generalize the well-known stationary phase method, which were obtained in the previous paper of the author, are applied to integrals arising in the well-known tsunami hydrodynamic piston model in the case of a constant pool bottom. As a result, asymptotic formulas are obtained for the head part of the wave for large values of the time elapsed since the occurrence of the tsunami. These formulas contain some special reference integrals and have different forms depending on combinations of wave and time parameters.

Differential-difference operators are considered on an infinite cylinder. The objective of the paper is to present an index formula for the operators in question. We define the operator symbol as a triple consisting of an internal symbol and conormal symbols on plus and minus infinity. The conormal symbols are families of operators with a parameter and periodic coefficients. Our index formula contains three terms: the contribution of the internal symbol on the base manifold, expressed by an analog of the Atiyah–Singer integral, the contributions of the conormal symbols at infinity, described in terms of the $$\eta$$ -invariant, and also the third term, which also depends on the conormal symbol. The result thus obtained generalizes the Fedosov–Schulze–Tarkhanov formula.

In this paper, we consider the q-analog of the Laplace transform and investigate some properties of the q-Laplace transform. In our investigation, we derive some interesting formulas related to the q-Laplace transform.

This paper addresses the problem of the Chaplygin ball rolling on a horizontal plane which rotates with constant angular velocity. In this case, the equations of motion admit area integrals, an integral of squared angular momentum and the Jacobi integral, which is a generalization of the energy integral, and possess an invariant measure. After reduction the problem reduces to investigating a three-dimensional Poincaré map that preserves phase volume (with density defined by the invariant measure). We show that in the general case the system’s dynamics is chaotic.

In connection with the relationship of the heap paradox with quantum mechanics pointed out by V. P. Maslov, the feasibility and complexity of algorithms for counting very large sets in small time intervals is studied.

The minimality of surfaces defined by certain classical embeddings is proved and properties of sections of the surfaces are studied.

In this paper, we study normal forms of plane curves and knots. We investigate the Euler functional E (the integral of the square of the curvature along the given curve) for closed plane curves, and introduce a closely related functional A, defined for polygonal curves in the plane ℝ2 and its modified version A R , defined for polygonal knots in Euclidean space ℝ3. For closed plane curves, we find the critical points of E and, among them, distinguish the minima of E, which give us the normal forms of plane curves. The minimization of the functional A for plane curves, implemented in a computer animation, gives a very visual approximation of the process of gradient descent along the Euler functional E and, thereby, illustrates the homotopy in the proof of the classical Whitney-Graustein theorem. In ℝ3, the minimization of A R (implemented in a 3D animation) shows how classical knots (or more precisely thin knotted solid tori, which model resilient closed wire curves in space) are isotoped to normal forms.

This paper is devoted to the asymptotic analysis of functions depending on a small parameter characterizing the microinhomogeneous structure of the domain on which the functions are defined. We derive the Friedrichs inequality for these functions and prove the convergence of solutions to corresponding problems posed in a domain perforated aperiodically along the boundary. Moreover, we use numerical simulation to illustrate the results.