Abstract:
Let $Z_n(s; a_1,..., a_n)$ be the Epstein zeta function defined as the meromorphic continuation of the function \sum_{k\in\Z^n\setminus\{0\}}(\sum_{i=1}^n [a_i k_i]^2)^{-s}, \text{Re} s>\frac{n}{2} to the complex plane. We show that for fixed $s\neq n/2$, the function $Z_n(s; a_1,..., a_n)$, as a function of $(a_1,..., a_n)\in (\R^+)^n$ with fixed $\prod_{i=1}^n a_i$, has a unique minimum at the point $a_1=...=a_n$. When $\sum_{i=1}^n c_i$ is fixed, the function $$(c_1,..., c_n)\mapsto Z_n(s; e^{c_1},..., e^{c_n})$$ can be shown to be a convex function of any $(n-1)$ of the variables $\{c_1,...,c_n\}$. These results are then applied to the study of the sign of $Z_n(s; a_1,..., a_n)$ when $s$ is in the critical range $(0, n/2)$. It is shown that when $1\leq n\leq 9$, $Z_n(s; a_1,..., a_n)$ as a function of $(a_1,..., a_n)\in (\R^+)^n$, can be both positive and negative for every $s\in (0,n/2)$. When $n\geq 10$, there are some open subsets $I_{n,+}$ of $s\in(0,n/2)$, where $Z_{n}(s; a_1,..., a_n)$ is positive for all $(a_1,..., a_n)\in(\R^+)^n$. By regarding $Z_n(s; a_1,..., a_n)$ as a function of $s$, we find that when $n\geq 10$, the generalized Riemann hypothesis is false for all $(a_1,...,a_n)$.

Abstract:
This paper deals with the distribution of $\alpha \zeta^{n} \bmod 1$, where $\alpha\neq 0,\zeta>1$ are fixed real numbers and $n$ runs through the positive integers. Denote by $\Vert.\Vert$ the distance to the nearest integer. We investigate the case of $\alpha\zeta^{n}$ all lying in prescribed small intervals modulo $1$ for all large $n$, with focus on the case $\Vert\alpha \zeta^{n}\Vert \leq \epsilon$ for small $\epsilon>0$. We are particularly interested in what we call cardinality gap phenomena. For example for fixed $\zeta>1$ and small $\epsilon>0$ there are at most countably many values of $\alpha$ such that $\Vert\alpha \zeta^{n}\Vert \leq \epsilon$ for all large $n$, whereas larger $\epsilon$ induces an uncountable set. We investigate the value of $\epsilon$ at which the gap occurs. We will pay particular attention to the case of algebraic and, more specific, rational $\zeta>1$. Results concerning Pisot and Salem numbers and the reduced length of a polynomial such as some contribution to Mahler's $3/2$-problem are implicitly deduced. We study similar questions for fixed $\alpha\neq 0$.

Abstract:
In many cases the convexity of the image of a linear map with range is $R^n$ is automatic because of the facial structure of the domain of the map. We develop a four step procedure for proving this kind of ``automatic convexity''. To make this procedure more efficient, we prove two new theorems that identify the facial structure of the intersection of a convex set with a subspace in terms of the facial structure of the original set. Let $K$ be a convex set in a real linear space $X$ and let $H$ be a subspace of X that meets $K$. In Part I we show that the faces of $K\cap H$ have the form $F\cap H$ for a face $F$ of $K$. Then we extend our intersection theorem to the case where $X$ is a locally convex linear topological space, $K$ and $H$ are closed, and $H$ has finite codimension in $X$. In Part II we use our procedure to ``explain'' the convexity of the numerical range (and some of its generalizations) of a complex matrix. In Part III we use the topological version of our intersection theorem to prove a version of Lyapunov's theorem with finitely many linear constraints. We also extend Samet's continuous lifting theorem to the same constrained siuation.

Abstract:
The purpose of this paper is finding the essential attributes underlying the convexity theorems for momentum maps. It is shown that they are of topological nature; more specifically, we show that convexity follows if the map is open onto its image and has the so called local convexity data property. These conditions are satisfied in all the classical convexity theorems and hence they can, in principle, be obtained as corollaries of a more general theorem that has only these two hypotheses. We also prove a generalization of the "Lokal-global-Prinzip" that only requires the map to be closed and to have a normal topological space as domain, instead of using a properness condition. This allows us to generalize the Flaschka-Ratiu convexity theorem to non-compact manifolds.

Abstract:
A stable stratified night during the experimental SABLES-98 campaign was studied. It was found that the night could be divided into three distinct parts: an initial phase of transition, with significant changes in all the variables until a quasi-steady regime was reached; a second part dominated by the surface radiative cooling; and a final stage that revealed no significant changes. Each of these intervals was studied separately through the analysis of the vertical profiles, turbulence, spectra and the probability density functions.

Abstract:
In this paper, approximate convexity and approximate midconvexity properties, called $\varphi$-convexity and $\varphi$-midconvexity, of real valued function are investigated. Various characterizations of $\varphi$-convex and $\varphi$-midconvex functions are obtained. Furthermore, the relationship between $\varphi$-midconvexity and $\varphi$-convexity is established.

Abstract:
abstract this article deals with the analysis of wittgenstein？s proposals made in his notes the big typescript. in this work, he approaches the problem of colours from four different perspectives. first, he points out the importance of the language of colours in orden to understanding colours on their nature. second, he criticizes the logical position in tractratus logicus philosophicus for being too vague. third, he reconstructs the difference between phenomenological and physical language. he considers these two aspects as the most relevant peculiarities within the analysis of colours and at the same time criticizes both edmund husserl？s and ernst mach？s proposals. finally, he sets a frame work for a theory of colours that he calls ？minima visibilia？

Abstract:
We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well as functions related to them, using transformations under which the zeros remain unchanged. We give upper as well as lower bounds for the distance between consecutive zeros in several cases.

Abstract:
We give bounds on the successive minima of an $o$-symmetric convex body under the restriction that the lattice points realizing the successive minima are not contained in a collection of forbidden sublattices. Our investigations extend former results to forbidden full-dimensional lattices, to all successive minima and complement former results in the lower dimensional case.

Abstract:
An experimental phenomenon is reported about crosswell physical model.From the waveforms received in receiver borehole,we discover the amplitude of tube wave obviously attenuated in the interface between the layered medium,but the amplitude would strengthen after the receiver depart from the interface.