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We study some problems in metric Diophantine approximation over local fields of positive characteristic.
Orthogonalization is one of few mathematical methods conforming to mathematical standards for approximation. Finding a consistent PC matrix of a given an inconsistent PC matrix is the main goal of a pairwise comparisons method. We introduce…
We introduce a diophantine property of a log canonical algebra, and use it to describe the restriction of a log canonical algebra of general type to a log canonical center of codimension one.
A Diophantine $m$-tuple is a set of $m$ distinct integers such that the product of any two distinct elements plus one is a perfect square. It was recently proven that there is no Diophantine quintuple in positive integers. We study the same…
It is shown how strictly four-dimensional integration by parts combined with differential renormalization and its infrared analogue can be applied for calculation of Feynman diagrams.
Two program packages are presented for evaluating one-loop amplitudes. They can work either in dimensional regularization or in constrained differential renormalization. The latter method is found at the one-loop level to be equivalent to…
In this paper we present a new method of solving the classical diophantine equation $A^4+B^4=C^4+D^4$. Two methods of solving this equation, given by Euler, yield parametric solutions given by polynomials of degrees 7 and 13. Several other…
This paper develops the metric theory of simultaneous inhomogeneous Diophantine approximation on a planar curve with respect to multiple approximating functions. Our results naturally generalize the homogeneous Lebesgue measure and Hausdor?…
A lens limited by diffraction and having two parabolic surfaces is presented. The knowledge of the following parameters: object distance, relative refractive index, lens thickness, and image distance, enables to analytically calculate the…
The Diophantine equation $A^4+hB^4=C^4+hD^4$, where $h$ is a fixed arbitrary positive integer, has been investigated by some authors. Currently, by computer search, the integer solutions of this equation are known for all positive integer…
Some diophantine problems are stated for the Omega constant and, more generally, the values of Lambert $W$-function and their $p$-adic extensions.
This paper investigates a generalized version of Diophantine tuples in finite fields. Applying Shparlinski's method, we obtain power-saving results on the number of such tuples.
We consider multiparameter dynamics on the space of unimolular lattices. Along with quantitative nondivergence we prove that multiplicative Diophantine exponents of hyperplanes are inherited by their nondegenerate submanifolds.
Let $p$ and $q$ be distinct primes such that $q+1 | p-1$. In this paper we find all integer solutions $a$, $b$ to the equation $1/a + 1/b = (q+1)/pq$ using only elementary methods.
In a small simply-connected closed 4-manifold, we construct infinitely many pairs of exotic codimension-$1$ submanifolds with diffeomorphic complements that remain exotic after any number of stabilizations by $ S^2 \times S^2$. We also give…
In this paper, convergent, multi-infinite, series solutions are derived for the homoclinic orbits of a canonical fourth-order ODE system, in both reversible and non-reversible cases. This ODE includes traveling-wave reductions of many…
We propose a method to monochromatize multiple orders of high harmonics by using a proper designed multilayer mirror. Multilayer mirrors designed by our concept realize the perfect extraction of a single-order harmonic from multiple-order…
We discuss a problem initially thought for the Mathematical Olympiad but which has several interpretations. The recurrence sequences involved in this problem may be generalized to recurrence sequences related to a much larger set of…
A basic question of Diophantine approximation, which is the first issue we discuss, is to investigate the rational approximations to a single real number. Next, we consider the algebraic or polynomial approximations to a single complex…
We give the general solution of three Diophantine equations in the ring of integer of the algebraic number field ${\bf Q}[{\sqr 5}]$. These equations are related to the problem of determination of the minimum distance in quasicrystals with…