Related papers: A strict non-standard inequality .999... < 1

In an attempt to resolve a folklore conjecture of Erd\H{o}s regarding the non-vanishing at $s=1$ of the $L$-series attached to a periodic arithmetical function with period $q$ and values in $\{-1, 1 \}$, Livingston conjectured the…

We prove that the existence of infinitely many $(m_k, n_k) \in \mathbb{N}^2$ such that the difference of harmonic numbers $H_{m_k} - H_{n_k}$ approximates 1 well $$ \lim_{k \rightarrow \infty} \left| \sum_{\ell = n}^{m_k} \frac{1}{\ell} - 1…

We prove in ZFC the existence of a definable, countably saturated elementary extension of the reals. It seems that it has been taken for granted that there is no distinguished, definable nonstandard model of the reals. (This means a…

The notion of topological entropy can be conceptualized in terms of the number of forward trajectories that are distinguishable at resolution $\varepsilon$ within $T$ time units. It can then be formally defined as a limit of a limit…

New exceptional (i.e. non-repeating) prime number multiplets are given and formulated in terms of arithmetic progressions, along with laws governing them. Accompanying repeating prime number multiplets are pointed out. Prime number…

F.: Good morning Hermann, I would like to talk with you about infinitesimals. G.: Tell me Pierre. F.: I'm fed up of all these slanders about my attitude to be non rigorous, so I've started to study nonstandard analysis (NSA) and synthetic…

Building upon previous works of Andr{\'e} and Chudnovsky, we prove a general result concerning the approximations of values at rational points a/b of any G-function F with rational Taylor coefficients by fractions of the form n/(B…

We show that the solid lower bound of about 10^{-44} cm^2 is obtained for the cross section between the supersymmetric dark matter and nucleon in a theory in which the supersymmetric fine-tuning problem is solved without extending the Higgs…

We give a construction of an absolutely normal real number $x$ such that for every integer $b $ greater than or equal to $2$, the discrepancy of the first $N$ terms of the sequence $(b^n x \mod 1)_{n\geq 0}$ is of asymptotic order…

I. J. Good (1941) showed that the set of irrational numbers in $(0,1)$ whose partial quotients $a_n$ tend to infinity is of Hausdorff dimension $1/2$. A number of related results impose restrictions of the type $a_n\in B$ or $a_n\geq f(n)$,…

We study the quantitative unique continuation property of some higher order elliptic operators. In the case of $P=(-\Delta)^m$, where $m$ is a positive integer, we derive lower bounds of decay at infinity for any nontrivial solutions under…

These lecture notes cover classical undecidability results in number theory, Hilbert's 10th problem and recent developments around it, also for rings other than the integers. It also contains a sketch of the authors result that the integers…

There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent $\lambda=n-\alpha$ (that…

In this work, we established symmetric representation of numbers where one can use any of 9 digits giving the same number. The representations of natural numbers from 0 to 1000 are given using only single digit in all the nine cases, i.e.,…

Using an application of Schmidt's Subspace Theorem, this paper gives new transcendence criteria for rapidly converging infinite products of algebraic numbers. The paper also improves existing criteria for irrationality of products and…

This paper presents new lower and upper bounds for the compression rate of binary prefix codes optimized over memoryless sources according to various nonlinear codeword length objectives. Like the most well-known redundancy bounds for…

The Bruss-Robertson inequality gives a bound on the maximal number of elements of a random sample whose sum is less than a specified value, and the extension of that inequality which is given here neither requires the independence of the…

In this paper, first we prove that all finite dimensional special Heisenberg Lie superalgebras with even center have same dimension, say $(2m+1\mid n)$ for some non-negative integers $m,n$ and are isomorphism with them. Further, for a…

We show that the entropy of any black object in any dimension can be understood as the entropy of a highly excited string on the stretched horizon. The string has a gravitationally renormalized tension due to the large redshift near the…

We introduce a linear infinitary $\lambda$-calculus, called $\ell\Lambda_{\infty}$, in which two exponential modalities are available, the first one being the usual, finitary one, the other being the only construct interpreted…