Our main task of this lecture is to show a polynomial time algorithm which approximately solves the Shortest Vector Problem (SVP) within a factor of 2 for lattices of dimension n. It may seem that such an algorithm with exponential error bound is either obvious or useless. However, the algorithm of Lenstra, Lenstra and Lovász (LLL) is widely regarded as one of the most beautiful algorithms and ...

The present study was carried out to show the potential of root exudates mobilize potentially toxic elements (PTE) in rhizospheric carbonated soils. Five different contaminated soils were collected from five former mining districts northern Tunisia (Jebel Hallouf (H3), Sidi-Bouaouane (B1), Jebel Ghozlane (G7), Hammam Zriba (Z2) and Jalta (J2)). abundant minerals these are quartz, calcite clays....

The aim of this paper is to present an answer to Problem 1 of [M1], [M2]. We do this by proving in particular that if μ is a strong limit singular cardinal, θ = (2), 2 = μ then there are Boolean algebras B1,B2 such that c(B1) = μ, c(B2) < θ but c(B1 ∗ B2) = μ. Further we improve this result, deal with the method and the neccessity of the assumptions. The research partially supported by “The Isr...

We investigate when an algebraic function of the form φ(λ) = −B(λ)+ √ R(λ) A(λ) , where R(λ) is a polynomial of odd degree N = 2g + 1 with coefficients in C, can be written as a periodic α-fraction of the form φ(λ) = [b0; b1, b2, . . . , bN ]α = b0 + λ− α1 b1 + λ−α2 b2+ . . .bN−1+ λ−αN bN+ λ−α1 b1+ λ−α2 b2+ . . . , for some fixed sequence αi. We show that this problem has a natural answer given...

Aflatoxin B1 (AFB1), one of the most potent naturally occurring mutagens and carcinogens, causes significant threats to the food industry and animal production. In this study, 25 bacteria isolates were collected from grain kernels and soils displaying AFB1 reduction activity. Based on its degradation effectiveness, isolate N17-1 was selected for further characterization and identified as Pseudo...

Classical probability theory considers probability distributions that assign probabilities to all events (at least in the finite case). However, there are natural situations where only part of the process is controlled by some probability distribution while for the other part we know only the set of possibilities without any probabilities assigned. We adapt the notions of algorithmic informatio...

We show that the first betti number b1 (0) = d im H 1(0, !R) of a compact Riemannian orbifold 0 with Ricci curvature Ric(O ) ~ -(n 1)k and d iameter diam(O) :5 D is bounded above by a constant r (n, kD2 ) ~ 0 , depending only on dimension , curvature and diameter. In the case when t he orbi fold has nonnegative Ricci curvature, we show that the b1 (0) is bounded above by the dimension dim 0 , a...

Introduction Roughly 30 peer reviewed papers have been published on B1+ mapping in the last 2 years from 2010 to 2011. This shows the vitality of a research area that has started back in the 80’s [1, 2]. The RF field behavior in biological tissue and its spatial variations was assessed very early [3]. The transmit RF field can be described by a time independent, spatially varying B1+ map times ...

Introduction: Parallel Transmission (PTx) offers more control over B1+ than single channel systems by independently controlling elements of transmit arrays, each with spatially distinct sensitivity patterns. The majority of PTx methods require accurate knowledge of these sensitivities, the measurement of which is often time consuming for the whole array. In this work, we exploit the observation...