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Differential Equations on Polytopes: Laplacians and Lagrangian Manifolds, Corresponding to Semiclassical Motion
The aim of this work is to describe certain constructions and results concerning differential operators on polyhedral surfaces. In particular, we...
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Extensions of Symmetric Operators and Evolution Equations on Singular Spaces
Differential operators on varieties with singularities were studied in a great number of papers. General theory of such operators is rather...
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A note on lattice points in model domains of finite type in \({\mathbb{R}^d}\)
We study the Fourier transforms of indicator functions of some special high-dimensional finite type domains and obtain estimates of the associated...
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The Laplace Transform, Mirror Symmetry, and the Topological Recursion of Eynard–Orantin
This paper is based on the author’s talk at the 2012 Workshop on Geometric Methods in Physics held in Białowieża, Poland. The aim of the talk is to...
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On a Diophantine inequality involving prime powers
Suppose that N is a sufficiently large real number. In this paper it is proved that if 1 < c < 108/53, c ≠ 2, then the Diophantine inequality
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Weighted divisor sums and Bessel function series, IV
One fragment (p. 335) published with Ramanujan’s Lost Notebook contains two formulas, each involving a finite trigonometric sum and a doubly infinite...
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The sphere problem and the L-functions
We improve the upper bound for the lattice point discrepancy of large spheres under conjectural properties of the real L -functions. In connection...
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Randomness of the square root of 2 and the giant leap, part 2
We prove that the “quadratic irrational rotation” exhibits a central limit theorem. More precisely, let α be an arbitrary real root of a quadratic...
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Symmetrically constrained compositions
Given integers a 1 , a 2 ,…, a n , with a 1 + a 2 +⋅⋅⋅+ a n ≥1, a symmetrically constrained composition λ 1 + λ 2 +⋅⋅⋅+ λ n = M of M into n nonnegative parts is...
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On the distribution of reducible polynomials
Let ℛ n ( t ) denote the set of all reducible polynomials p ( X ) over ℤ with degree n ≥ 2 and height ≤ t . We determine the true order of magnitude of the...
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Randomness of the square root of 2 and the Giant Leap, Part 1
We prove that the “quadratic irrational rotation” exhibits a central limit theorem. More precisely, let α be an arbitrary real root of a quadratic...
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On Differences of Two kth Powers of Integers
The arithmetic function rk−(n) counts the number of ways to write a natural number n as the difference of two kth powers (k ≥ 3 fixed). The...
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On neighbouring matrices with quadratic elementary divisors
This paper presents new results on matrices with quadratic elementary divisors neighbouring a given matrix A and relates them to those given by Kakan....