Related papers: The hanging chain problem with respect to a circle
The effective potential energy of the particles in the field of rotating uniformly magnetized celestial body is investigated. The axis of rotation coincides with the axis of the magnetic field. Electromagnetic field of the body is composed…
A linkage is a finite graph with lengths assigned to each edge. A planar realization is a map to the plane which preserves edge lengths. It can be thought of as a mechanical device formed from stiff rods and rotating joints. We look at the…
The energy spectrum of an electron confined to an arbitrary surface of revolution in an external magnetic field, parallel to the symmetry axis, is studied analitycally and numerically. The problem is reduced via conformal mapping to one on…
Three problems related to the longest chains in an equivalence relation are formulated. Their relevance in estimating the cardinal of quotient spaces, and in particular, of reduced powers and ultrapowers, is indicated.
The displacement and deviation vectors in spaces (manifolds), the tangent bundle of which is endowed with a transport along paths, are introduced. In case these spaces are equipped with a linear connection, the deviation equations (between…
The system of a proton and an electron in an inert and impenetrable spherical cavity is studied by solving Schr\"{o}dinger equation with the correct boundary conditions. The differential equation of a hydrogen atom in a cavity is derived.…
We locate gaps in the spectrum of a Hamiltonian on a periodic cuboidal (and generally hyperrectangular) lattice graph with $\delta$ couplings in the vertices. We formulate sufficient conditions under which the number of gaps is finite. As…
The Cramer-Castillon problem (CCP) consists in finding one or more polygons inscribed in a circle such that their sides pass cyclically through a list of $N$ points. We study this problem where the points are the vertices of a triangle and…
The energy characteristics of a relativistic charged particle in the field of a plane electromagnetic wave of a given amplitude are studied. The dependence of the particle's energy on its phase coordinate is obtained. The maximum value of…
We study solvable spin chains where either fields or couplings vary linearly in space and create a sandwich-like structure of the ground state. We find that the entanglement entropy between two halves of a chain varies logarithmically with…
We examine the behaviour of a charged particle in a two-dimensional confining potential, in the presence of a magnetic field. The confinement serves to remove the otherwise infinite degeneracy, but additional ingredients are required to…
We consider an inverse variational problem for the lines of constant curvature in (pseudo-)Euclidean two-, three-, and four-dimensional spaces. The accumulated results are physically meaningful in the case of relativistic mechanics of…
We consider a quantum particle in a waveguide which consists of an infinite straight Dirichlet strip divided by a thin semitransparent barrier on a line parallel to the walls which is modeled by a $\delta$ potential. We show that if the…
Electron orbits are calculated in solitary two-dimensional axisymmetric electrostatic potential structures, typical of plasma electron holes, in order to establish the conditions for the particles to remain trapped. Analytic calculations of…
In black hole physics, inflationary cosmology, and quantum field theories, it is conjectured that the physical laws are subject to radical changes below the Planck length. Such changes are due to effects of quantum gravity believed to…
We develop a new approach to address some classical questions concerning the size and structure of integer distance sets. Our main result is that any integer distance set in the Euclidean plane is either very sparse or has all but an…
We consider the classical three-body problem with an arbitrary pair potential which depends on the inter-body distance. A general three-body configuration is set by three "radial" and three angular variables, which determine the shape and…
A detailed analysis is given of the angular distribution of an charged particle moving along the arc of a circle. The areas of different angular distribution behavior are highlighted and studied.
We find relations between quantities defining geometry and quantities defining the length of a curve in geometries underlying Electromagnetism and unified model of Electromagnetism and Gravitation. We show that the length of a vector…
We give an algorithm for approximating a given plane curve segment by a planar elastic curve. The method depends on an analytic representation of the space of elastic curve segments, together with a geometric method for obtaining a good…