Possible Counterexample of the Riemann Hypothesis

Under the assumption that the Riemann hypothesis is true, von Koch deduced the improved asymptotic formula $\theta(x) = x + O(\sqrt{x} \times \log^{2} x)$, where $\theta(x)$ is the Chebyshev function. A precise version of this was given by Schoenfeld: He found under the assumption that the Riemann hypothesis is true that $\theta(x) < x + \frac{1}{8 \times \pi} \times \sqrt{x} \times \log^{2} x$ for every $x \geq 599$. On the contrary, we prove if there exists some real number $x \geq 2$ such that $\theta(x) > x + \frac{1}{\log \log \log x} \times \sqrt{x} \times \log^{2} x$, then the Riemann hypothesis should be false. In this way, we show that under the assumption that the Riemann hypothesis is true, then $\theta(x) < x + \frac{1}{\log \log \log x} \times \sqrt{x} \times \log^{2} x$.

Possible Counterexample of the Riemann Hypothesis

Under the assumption that the Riemann hypothesis is true, von Koch deduced the improved asymptotic formula $\theta(x) = x + O(\sqrt{x} \times \log^{2} x)$, where $\theta(x)$ is the Chebyshev function. A precise version of this was given by Schoenfeld: He found under the assumption that the Riemann hypothesis is true that $\theta(x) < x + \frac{1}{8 \times \pi} \times \sqrt{x} \times \log^{2} x$ for every $x \geq 599$. On the contrary, we prove if there exists some real number $x \geq 2$ such that $\theta(x) > x + \frac{1}{\log \log x} \times \sqrt{x} \times \log^{2} x$, then the Riemann hypothesis should be false. In this way, we show that under the assumption that the Riemann hypothesis is true, then $\theta(x) < x + \frac{1}{\log \log x} \times \sqrt{x} \times \log^{2} x$.

Limits to Aggregation and Uncertain Rescues

Limited aggregation holds that we are only sometimes, not always, permitted to aggregate. Aggregation is permissible only when the harms and benefits are relevant to one another. But how should limited aggregation be extended to cases in which we are uncertain about what will happen? In this article, I provide a challenge to ex post limited aggregation. I reconstruct a precise version of ex post limited aggregation that relies on the notion of ex post claims. However, building a theory of limited aggregation based on ex post claims leads to a dilemma. This shows that ex post limited aggregation is currently far away from being a well-defined alternative, strengthening the case for ex ante limited aggregation.

Genizah Fragment’s Version of the Amoraic Statements in Bavli Eruvin 103a

The Genizah fragment Cambridge U-L T-S F2 (2) 23, numbered C98948 in the Friedberg Jewish Manuscript Society, includes among other things the amoraic controversy between R. Eleazar and R. Jose son of R. Hanina, as well as the give and take between R. Safra and Abaye in Tractate Eruvin 103a. Some of the researchers are divided concerning the initial formation of the sugya. The controversy between R. Eleazar and R. Jose son of R. Hanina as presented in the fragment’s version poses difficulties and interferes with the ordered understanding of the methods utilized by these amoraim to solve the contradiction between the Mishna in Eruvin and the Mishna in Pesaḥim. The purpose of the article is to present the difficulties in the fragment’s version with regard to the abovementioned amoraic controversy and reach conclusions regarding the precise original version of the fragment. Thus too in the matter of the fragment’s version of the give and take between R. Safra and Abaye, which differs from other versions. The purpose of the article is to examine the fragment’s version of this give and take in comparison to other versions and reach conclusions regarding the clearest and most precise version compared to the other versions.

Variational Approaches to P(X)-Laplacian-Like Problems with Neumann Condition Originated from a Capillary Phenomena

This article presents several sufficient conditions for the existence of at least one weak solution and infinitely many weak solutions for the following Neumann problem, originated from a capillary phenomena, $$\begin{equation*} \left\{\begin{array}{ll} -{\rm div}\bigg(\bigg(1+\frac{|\nabla u|^{p(x)}}{\sqrt{1+|\nabla u|^{2p(x)}}}\bigg) |\nabla u|^{p(x)-2}\nabla u\bigg)+\alpha(x)|u|^{p(x)-2}u\\=\lambda f(x,u) \mbox{in}\,\,\Omega,\\ \frac{\partial u}{\partial \nu}=0\mbox{on}\,\,\partial \Omega \end{array}\right. \end{equation*}$$where $\Omega \subset \mathbb{R}^N$$(N\geq 2)$ is a bounded domain with boundary of class $C^1,$$\nu$ is the outer unit normal to $\partial \Omega,$$\lambda>0$, $\alpha\in L^{\infty}(\Omega),$$f:\Omega\times\mathbb{R}\to\mathbb{R}$ is an $L^1$-Carathéodory function and $p\in C^0(\overline{\Omega})$. Our technical approach is based on variational methods and we use a more precise version of Ricceri’s Variational Principle due to Bonanno and Molica Bisci. Some recent results are extended and improved. Some examples are presented to illustrate the application of our main results.

ENRIQUES’ CLASSIFICATION IN CHARACTERISTIC 0$" height="12pt">: THE -THEOREM

The main goal of this paper is to show that Castelnuovo–Enriques’ $P_{12}$ - theorem (a precise version of the rough classification of algebraic surfaces) also holds for algebraic surfaces $S$ defined over an algebraically closed field $k$ of positive characteristic ( $\text{char}(k)=p>0$ ). The result relies on a main theorem describing the growth of the plurigenera for properly elliptic or properly quasielliptic surfaces (surfaces with Kodaira dimension equal to 1). We also discuss the limit cases, i.e., the families of surfaces which show that the result of the main theorem is sharp.

Text as Sliding Signifier

In the 1980s Ricoeur conceptualized metaphoricity and narrativity as twin ends of a discursive field governed by the productive imagination. A decade earlier Ricoeur was working at a significantly different proposition. He wanted to establish a parallel, in fact a strong homology, between metaphor and text. In both cases Ricoeur articulated a complex criteriology to establish the parallelism between the terms. Should we regard the earlier parallel as a first and less precise version of the later? a distinct thesis? a reconcilable claim? a category error? Was the rejection and return an evolution, a clarification, or simply a different topic? The thesis of this close reading of Ricoeur’s middle period writings on metaphor is that the shift of analogical relations is actually a migration from the first to the second homology, and that the second homology is the correct one.