Related papers: A variation on the theme of Nicomachus
We discuss issues relating to the topology of Euclidean de Sitter space. We show that in (2+1) dimensions, the Euclidean continuation of the`causal diamond', i.e the region of spacetime accessible to a timelike observer, is a…
During a first St. Petersburg period Leonhard Euler, in his early twenties, became interested in the Basel problem: summing the series of inverse squares (posed by Pietro Mengoli in mid 17th century). In the words of Andre Weil (1989) "as…
In this paper, we study the gaps between primes in Beatty sequences following the methods in the recent breakthrough of J. Maynard.
In this work we investigate Cauchy problem and initial boundary value problem for time-fractional Airy equation on the graphs with infinite and finite bonds. We studied properties of potentials for this equation and using these properties…
We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period…
In this paper we adopt a geometric point of view regarding a famous conjecture due to Littlewood in diophantine approximation of real numbers. Following the spirit of the geometric theory of continued fractions, we give a sufficient…
We establish a version of Kn\"{o}rrer's Periodicity Theorem in the context of noncommutative invariant theory. Namely, let $A$ be a left noetherian AS-regular algebra, let $f$ be a normal and regular element of $A$ of positive degree, and…
In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck.…
Three-body and n-body problems in celestial mechanics are age-old and challenging puzzles. In recent years, several breakthroughs are made in finding periodic orbits for three-body problem. And Bohua Sun proposed a conjecture on Kepler's…
Solutions to the Einstein equations for Bianchi IX cosmologies are examined through the use of Ellis MacCallum Wainwright (expansion-normalized) variables. Using an iterative map derived from the Einstein equations one can construct an…
We provide several simple recursive formulae for the moment sequence of infinite Bernoulli convolution. We relate moments of one infinite Bernoulli convolution with others having different but related parameters. We give examples relating…
The Belinkskii, Khalatnikov and Lifshitz conjecture says that as one approaches space-like singularities in general relativity, 'time derivatives dominate over spatial derivatives' so that the dynamics at any spatial point is well captured…
We put a new conjecture on primes from the point of view of its binary expansions and make a step towards justification.
Here are two problems. First, understand the dynamics of a tiling billiard in a cyclic quadrilateral periodic tiling. Second, describe the topology of connected components of plane sections of a centrally symmetric subsurface $S \subset…
We are concerned with the invariants of Steiner chains consisting of four circles. In particular, we compute the invariant moments of curvatures in Steiner 4-chains and give two applications of the obtained formulas. Specifically, we…
Difficulties around the idea of spontaneous breaking of time translation symmetry in a closed quantum mechanical system are identified, and then overcome in a simple model. The possibility of ordering in imaginary time is also discussed.
We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes,…
Lambert's theorem (1761) on the elapsed time along a Keplerian arc drew the attention of several prestigious mathematicians. In particular, they tried to give simple and transparent proofs of it (see our timeline \S 9). We give two new…
We consider the Boltzmann system corresponding to the motion of a billiard with a linear boundary under the influence of a gravitational field. We derive analytic conditions of Cayley's type for periodicity of its trajectories and provide…
We introduce nine further local facets of the classical Problem of Time, and underlying Background Independence aspects, the previous two Articles having covered one further facet-and-aspect each for a total of eleven. I.e. 1) Constraint…