Current Applications of Magnetic Nanomaterials for Extraction of Mycotoxins,Pesticides, and Pharmaceuticals in Food Commodities

Environmental pollutants, such as mycotoxins, pesticides, and pharmaceuticals, are a group of contaminates that occur naturally, while others are produced from anthropogenic sources. With increased research on the adverse ecological and human health effects of these pollutants, there is an increasing need to regularly monitor their levels in food and the environment in order to ensure food safety and public health. The application of magnetic nanomaterials in the analyses of these pollutants could be promising and offers numerous advantages relative to conventional techniques. Due to their ability for the selective adsorption, and ease of separation as a result of magnetic susceptibility, surface modification, stability, cost-effectiveness, availability, and biodegradability, these unique magnetic nanomaterials exhibit great achievement in the improvement of the extraction of different analytes in food. On the other hand, conventional methods involve longer extraction procedures and utilize large quantities of environmentally unfriendly organic solvents. This review centers its attention on current applications of magnetic nanomaterials and their modifications in the extraction of pollutants in food commodities.     

On the singular PDE's geometry and boundary value problems

A geometric formulation of singular partial differential equations (PDEs) is considered. Surgery techniques and integral bordism groups are utilized, following previous works by Prástaro on PDEs, in order to build global solutions crossing also singular points and to study their stability properties.     

Estimates on the size of the cycle spectra of Hamiltonian graphs

Given a graph $G$, let $S\left(G\right)$ be the set of all cycle lengths contained in $G$ and let $s\left(G\right)=\left|S\left(G\right)\right|$. Let $?\left(G\right)=\{3,\dots ,n\}?S\left(G\right)$ and let $d$ be the greatest common divisor of $n?2$ and all the positive pairwise differences of elements in $?\left(G\right)$. We prove that if a Hamiltonian graph $G$ of order $n$ has at least $\frac{n(p+2)}{4}+1$ edges, where $p$ is an integer such that $1\le p\le n?2$, then $s\left(G\right)\ge p$ or $G$ is exceptional, by which we mean $d?(??2)$ for some $?\in ?\left(G\right)$. We also discuss cases where $G$ is not exceptional, for example when $n?2$ is prime. Moreover, we show that $s\left(G\right)\ge min\{p,\frac{n?3}{2}\}$, which if $G$ is bipartite implies that $s\left(G\right)\ge min\{?\frac{4(m?1)}{n}?2?,\frac{n?2}{2}\}$, where $m$ is the number of edges in $G$.     

Mathematics teacher training at the State Technical University in Chile

During the fifties and sixties in Chile many mathematics graduates of Pedagogical Institutes became university instructors rather than high school teachers as intended. In the late sixties Masters level programmes began giving more advanced training for university mathematics instructors. A ‘modern’ mathematics curriculum including computer science was introduced in the high schools. The first significant attempt to adapt the training of Chilean high school mathematics teachers to the new academic and social realities was begun in 1976 at the State Technical University with the cooperation of Teachers College, Columbia University, and the support of the Organization of American States.

The four‐year programme leads to Licenciatura (Bachelors) in Mathematics and Computer Education. All courses are taught by faculty members of the Department of Mathematics and Computer Science. The three basic areas of instruction are mathematics, computer science, and mathematics education. Funds provided by the O.A.S. have helped to make possible the development of curricular materials in Spanish, and the acquisition of computing and video tape equipment.     

Renormalization of polygon exchange maps arising from corner percolation

We describe a family {Ψ _α,β } of polygon exchange transformations parameterized by points (α,β) in the square $[0, {\frac{1}{2}}]\times[0, {\frac{1}{2}}]$ . Whenever α and β are irrational, Ψ _α,β has periodic orbits of arbitrarily large period. We show that for almost all parameters, the polygon exchange map has the property that almost every point is periodic. However, there is a dense set of irrational parameters for which this fails. By choosing parameters carefully, the measure of non-periodic points can be made arbitrarily close to full measure. These results are powered by a notion of renormalization which holds in a more general setting. Namely, we consider a renormalization of tilings arising from the Corner Percolation Model.     

T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients

To solve variational indefinite problems, one uses classically the Banach–Ne?as–Babu?ka theory. Here, we study an alternate theory to solve those problems: T-coercivity. Moreover, we prove that one can use this theory to solve the approximate problems, which provides an alternative to the celebrated Fortin lemma. We apply this theory to solve the indefinite problem $\text{ div}\sigma \nabla u=f$ set in $H^1_0$ , with $\sigma $ exhibiting a sign change.     

Heat kernel bounds for the Laplacian on metric graphs of polygonal tilings

We obtain an upper heat kernel bound for the Laplacian on metric graphs arising as one skeletons of certain polygonal tilings of the plane, which reflects the one dimensional as well as the two dimensional nature of these graphs.     

On Fully Idempotent Modules

A submodule N of a module M is idempotent if N = Hom(M, N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands. Counterexamples are given to show that this result is not true in general. It is shown that over commutative Noetherian rings, the fully idempotent modules are precisely the semisimple modules. We also show that the commutative rings over which every module is fully idempotent are exactly the semisimple rings. Idempotent submodules of free modules are characterized.     

On the structure of perfect sets in various topologies associated with tree forcings

We prove that the Ellentuck, Hechler and dual Ellentuck topologies are perfect isomorphic to one another. This shows that the structure of perfect sets in all these spaces is the same. We prove this by finding homeomorphic embeddings of one space into a perfect subset of another. We prove also that the space corresponding to eventually different forcing cannot contain a perfect subset homeomorphic to any of the spaces above.