On weyl's identity in the differential geometry of surfaces

Summary At the end of his classical paper on convex surfaces in thelarge,H. Weyl (1916) gave an identity connecting the first and second fundamental forms of a surface. The identity is, however, a purelylocal relation and it holds also when the Gaussian curvature, instead of being positive, is negative. The present paper centers around this identity. Various applications of it are considered and the conditions of validity are specified.     

Strengthening classical results on convergence rates in strong limit theorems

Let X₁, X₂, . . . be i.i.d. random variables, and set S_n=X₁+ . . . +X_n. Several authors proved convergence of series of the type f(ɛ)=∑_nc_nP(|S_n|>ɛa_n),ɛ>α, under necessary and sufficient conditions. We show that under the same conditions, in fact i.e. the finiteness of ∑_nc_nP(|S_n|>ɛa_n),ɛ>α, is equivalent to the convergence of the double sum ∑_k∑_nc_nP(|S_n|>ka_n). Two exceptional series required deriving necessary and sufficient conditions for E[sup_n|S_n|(logn)^η/n]<∞,0≤η≤1.     

Series of tail distributions of finitely inhomogeneous random walks

Let Sn=X₁+?+Xn be a random walk, where the steps Xn are independent random variables having a finite number of possible distributions, and consider general series of the form

(∗)     

4D Equations of motion and fifth force in a 5D general Kaluza–Klein space

In this paper we present a new point of view on space–time–matter (STM) theory. First, some weak points from earlier research papers on STM theory are presented. Then, we obtain in a covariant form the fully general $4D$ 4 D equations of motion for STM theory. This enables us to classify the $5D$ 5 D motions and to give a new definition of the fifth force in $4D$ 4 D physics.     

A new point of view on the fifth force in 4D physics

The purpose of the paper is to define and study, in the presence of the electromagnetic potentials, the fifth force induced by the fifth dimension in the $4D$ physics. A $4D$ tensor calculus and the Riemannian horizontal connection on a $5D$ general Kaluza–Klein space, are the main tools in our study. We prove that the fifth force does not violate the basic concepts in $4D$ physics. Our approach is different from what is known so far in literature.     

Fatty Acid Ethyl Esters (FAEE): A New,Green and Renewable Solvent for the Extraction of Carotenoids from Tomato Waste Products

Currently there is a drive towards the minimisation and reclamation of valuable materials from the waste products of the food and beverage industry. This can be achieved through the extraction of residual nutraceuticals from such materials. Tomato pomace contains carotenoids and other chemicals which can be extracted directly into edible oils to improve the health-giving properties of such oils. We report here a novel green solvent, fatty acid ethyl esters (FAEE), which is significantly more effective than sunflower oil and hexane for the extraction of lycopene and beta-carotene from tomato skin waste. FAEE are a non-toxic renewable resource that is environmentally friendly and to our knowledge has never been used as a vegetal extraction fluid. The efficiency of FAEE extraction was significantly improved relative to both sunflower oil and hexane under ultrasound-assisted extraction (UAE) conditions. In addition, FAEE have the additional and significant advantage that once enriched with the extracted nutraceuticals can be used directly as a food additive.     

Convergence and Precise Asymptotics for Series Involving Self-normalized Sums

Let \(X, X_{1}, X_{2}, \ldots \) be i.i.d. random variables, and set \(S_{n}=X_{1}+\cdots +X_{n}\) and \( V_{n}^{2}=X_{1}^{2}+\cdots +X_{n}^{2}.\) Without any moment conditions on \(X\), assuming that \(\{S_{n}/V_{n}\}\) is tight, we establish convergence of series of the type (*) \(\sum \nolimits _{n}w_{n}P(\left| S_{n}\right| /V_{n}\ge \varepsilon b_{n}),\) \(\varepsilon >0.\) Then, assuming that \(X\) is symmetric and belongs to the domain of attraction of a stable law, and choosing \(w_{n}\) and \(b_{n}\) suitably\(,\) we derive the precise asymptotic behavior of the series (*) as \(\varepsilon \searrow 0. \)