相关论文: On Vojta's $1+\epsilon$ Conjecture
In 1930s Paul Erdos conjectured that for any positive integer C in any infinite +1 -1 sequence (x_n) there exists a subsequence x_d, x_{2d}, ... , x_{kd} for some positive integers k and d, such that |x_d + x_{2d} + ... + x_{kd}|> C. The…
The Annals of Applied Probability 16 (2006) 984--1033 [URL: http://projecteuclid.org/euclid.aoap/1151592257]
In the abstract of [1] we read: "We obtain so far unproved properties of a ratio involving a classof Hermite and parabolic cylinder functions." However, we explain how some of the main results in that paper were already proved in [2],…
The document tries to put focus on sequences with certain properties and periods leading to the first value smaller than the starting value in the Collatz problem. With the idea that, if all starting numbers lead ultimately to a smaller…
The aim of this work is to expose some asymptotic series associated to some expressions involving the volume of the n-dimensional unit ball. All proofs and the methods used for improving the classical inequalities announced in the final…
We clarify and extend insights from Lavrentiev's seminal paper. We examine the original theorem dealing with the absence of the Lavrentiev phenomenon, a cornerstone issue in the calculus of variations. We point out some inconsistencies in…
Comment on "Liquids on Topologically Nanopatterned Surfaces" by O. Gang et al, Phys. Rev. Lett. 95, 217801 (2005). See also an erratum published by O. Gang et al (Phys Rev Lett, to appear)
The study of the diameter of the graph of polyhedra is a classical problem in the theory of linear programming. While transportation polytopes are at the core of operations research and statistics it is still open whether the Hirsch…
We will prove the Brannan conjecture for particular values of the parameter. The basic tool of the study is an integral representation published in a recent work [3].
We prove that the proof of existence of weighted local median order of weighted tournaments is wrong and that the proof of the correct statement which asserts that every digraph obtained from a tournament by deleting a set of arcs incident…
We prove that at least $\left( \dfrac{(1+\epsilon)2m}{N-1}+1+\epsilon \right)^N$, where $0\leqslant \epsilon <1$, many general points, satisfy Demailly's conjecture. Previously, it was known to be true for at least $(2m+2)^N$ many general…
We give a combinatorial upper bound for the gonality of a curve that is defined by a bivariate Laurent polynomial with given Newton polygon. We conjecture that this bound is generically attained, and provide proofs in a considerable number…
Exploring the Collatz Conjecture and changing the expression from 3n + 1 to 5n + 1, we found patterns in different sets of numbers. Some numbers reduce to one (as stated in the Collatz Conjecture), some might escape to infinity, and some…
In a recent work [JNT \textbf{129}, 2154 (2009)], Gun and co-workers have claimed that the number $\,\log{\Gamma(x)} + \log{\Gamma(1-x)}\,$, $x$ being a rational number between $0$ and $1$, is transcendental with at most \emph{one} possible…
WITHDRAWN: The proof contains an uncorrectable gap in the proof of theorem 7 on page 11. A proof of the Krzyz conjecture is presented, based on the application of the variational method, as well as on the use of two classical results and…
In [arXiv:2305.05022], Cohen proved a higher dimensional fractal uncertainty principle for line porous sets. The purpose of this expository note is to provide a different point of view on some parts of Cohen's proof, particularly suited to…
This is an elementary note. It corrects a mistake in the reformulation of the Riemann Hypothesis in J. Havil's book Gamma: Exploring Euler's Constant.
Several arguments concerning the "vexatae quaestiones" of the gravity field of a point mass and of the wavy gravity field.
In this comment, we propose possible errors in constructing the continuum contribution in the sum rule studied by M. C. Birse and B. Krippa, Phys. Rev. C. {\bf 54} (1996) 3240.
The aim of this short note is to clarify some of the claims made in the comparison made in [S. Lloyd, On the uncomputability of the spectral gap, arXiv:1602.05924] between our recent result [T.S. Cubitt, D. Perez-Garcia, M.M. Wolf,…