Related papers: Some Problems in Number Theory I: The Circle Probl…
There exists "a square problem": in a unit square is there a point with four rational distances to the vertices? This problem is still regarded as unproved. Yang showed proofs for several special cases of the square problem. By the…
A lattice-type regularization of the supersymmetric field theories on a supersphere is constructed by approximating the ring of scalar superfields by an integer-valued sequence of finite dimensional rings of supermatrices and by using the…
The fourth chapter of the collection of problems in cosmology, devoted to black holes. Consists of basic introduction, sections on Schwarzschild and Kerr black holes, a section on particles' motion and collisions in general black hole…
In this paper, we study a point-hyper plane incidence theorem in matrix rings, which generalizes all previous works in literature of this direction.
We announce numerous new results in the theory of orthogonal polynomials on the unit circle.
We find tight estimates for the minimum number of proper subspaces needed to cover all lattice points in an n-dimensional convex body symmetric about the origin. We also find the order of magnitude of the number of (n-1)-dimensional…
We describe the linear and nonlinear stability and instability of certain symmetric configurations of point vortices on the sphere forming relative equilibria. These configurations consist of one or two rings, and a ring with one or two…
Lefschetz thimbles regularisation of (lattice) field theories was put forward as a possible solution to the sign problem. Despite elegant and conceptually simple, it has many subtleties, a major one boiling down to a plain question: how…
Let $L$ be any integral lattice in the 2-dimensional Euclidean space. Generalizing the earlier works of Hiroshi Maehara and others, we prove that for every integer $n>0$, there is a circle in the plane $\mathbb{R}^{2}$ that passes through…
This is an expository paper. A $1$-cycle in a graph is a set $C$ of edges such that every vertex is contained in an even number of edges from $C$. E.g., a cycle in the sense of graph theory is a $1$-cycle, but not vice versa. It is easy to…
Recent development in numerical simulations of supersymmetric Yang-Mills (SYM) theories on the lattice is reviewed.
In this paper, we introduce and develop the circle embedding method. This method hinges essentially on a combinatorial-geometric structure which we choose to call circles of partition. We provide applications in the context of problems that…
An equivalence relation in the symmetric group, where is a positive integer has been considered. An algorithm for calculation of the number of the equivalence classes by this relation for arbitrary integer has been described.
We examine the metrics that arise when a finite set of points is embedded in the real line, in such a way that the distance between each pair of points is at least 1. These metrics are closely related to some other known metrics in the…
The First Hilbert problem is studied in this paper by applying two instruments: a new methodology distinguishing between mathematical objects and mathematical languages used to describe these objects; and a new numeral system allowing one…
For a given family of similar shapes, what we call a "unit shape" strongly analogizes the role of the unit circle within the family of all circles. Within many such families of similar shapes, we present what we believe is naturally and…
Plateau's problem is not a single conjecture or theorem, but rather an abstract framework, encompassing a number of different problems in several related areas of mathematics. In its most general form, Plateau's problem is to find an…
We investigate the definability (reducts) lattice of the order of integers and describe a sublattice generated by relations 'between', 'cycle', 'separation', 'neighbor', '1-codirection', 'order' and equality'. Some open questions are…
In this paper we discuss about properties of lattices and its application in theoretical and algorithmic number theory. This result of Minkowski regarding the lattices initiated the subject of Geometry of Numbers, which uses geometry to…
Among seven problems, proposed for XXI century by Clay Mathematical Institute, there are two stemming from physics. One of them is called "Yang-Mills Existence and Mass Gap". The detailed statement of the problem, written by A. Jaffe and E.…