New solutions for periodic interfacial gravity waves

Two-dimensional periodic interfacial gravity waves travelling between two homogeneous fluids of finite depth are considered. A boundary-integral-equation method coupled with Fourier expansions of the unknown functions is used to obtain highly accurate solutions. Our numerical results show excellent agreement with those already obtained by Maklakov & Sharipov using a different scheme (J. Fluid Mech., vol. 856, 2018, pp. 673–708). We explore the global bifurcation mechanism of periodic interfacial waves and find three types of limiting wave profiles. The new families of solutions appear either as isolated branches or as secondary branches bifurcating from the primary branch of solutions.

On Solving Problems of Thermal Conductivity on an Anisotropic Plane with a Weakly Permeable Film

The problem of thermal conductivity on an anisotropic plane (x; y) divided into two halfplanes D1(1 < x < 0; y 2 R) and D2(0 < x < 1; y 2 R) by a weakly permeable film x = 0 is considered at given heat sources and a given initial temperature. The anisotropy ellipses are arbitrary (in magnitude and direction) and are the same at all points of the plane. Using the method of convolution of Fourier expansions, the solution of the problem is expressed in single quadratures through the well-known solution of the classical Cauchy problem on an isotropic plane without a film. The results obtained are of practical interest in the problems of heat propagation and conservation in materials with anisotropic properties (crystalline, fibrous materials), in the presence of a thermal insulation film.

About Liquid Filtration Under a Point Dam with a Vertical Weakly Permeable Film

The problem of groundwater filtration under a point dam in a piecewise homogeneous porous medium in the presence of a weakly permeable film under the dam is considered. The filtration area is considered in the form of a vertical half-plane with a horizontal line of water courses. A weakly permeable film divides the filtration area into two quadrants with different constant permeability. By the convolution method of Fourier expansions, the solution of the problem is obtained explicitly. The influence of a weakly permeable film on the filtration process is investigated. It is shown that the presence of a weakly permeable film reduces the filtration rates in the downstream.

Rational functions, cotangent sums and Eichler integrals

With the help of so called pre-weak functions, we formulate a very general transformation law for some holomorphic functions on the upper half plane and motivate the term of a generalized Eisenstein series with real-exponent Fourier expansions. Using the transformation law in the case of negative integers k, we verify a close connection between finite cotangent sums of a specific type and generalized L-functions at integer arguments. Finally, we expand this idea to Eichler integrals and period polynomials for some types of modular forms.

Fourier-Based Automatic Transformation between Mapping Shapes—Cadastral and Land Registry Applications

Sometimes it is necessary to know the transformation to apply to a mapping shape in order to locate its true place. Such an operation can be computed if a corresponding reference object exists and we can identify corresponding points in both shapes. Nevertheless our approach does not need to match any corresponding point beforehand. The method proposed defines a polygon in the frequency domain—two periodic functions are derived from a polygonal or polygon. According to the theory of elliptic Fourier descriptors those two periodic functions can be expressed by Fourier expansions. The transformation can be computed using the coefficients of the harmonics from the corresponding shapes without taking into account where each polygon vertex is placed in the spatial domain. The transformation parameters will be derived by a least squares approach. The geomatics and geosciences applications of this method go from photogrammetry, geographic information system, computer vision, to cadaster and real estates.