Semisimple Varieties of Implication Zroupoids

It is a well known fact that Boolean algebras can be defined using only implication and a constant. In 2012, this result was extended to De Morgan algebras in [8] which led Sankappanavar to introduce, and investigate, the variety I of implication zroupoids generalizing De Morgan algebras. His investigations were continued in [3] and [4] in which several new subvarieties of I were introduced and their relationships with each other and with the varieties of [8] were explored. The present paper is a continuation of [8] and [3]. The main purpose of this paper is to determine the simple algebras in I. It is shown that there are exactly five simple algebras in I. From this description we deduce that the semisimple subvarieties of I are precisely the subvarieties of the variety generated by these 5 simple I-zroupoids and are locally finite. It also follows that the lattice of semisimple subvarieties of I is isomorphic to the direct product of a 4-element Boolean lattice and a 4-element chain.Comment: 21 page

Order in Implication Zroupoids

The variety $\mathbf{I}$ of implication zroupoids was defined and investigated by Sankappanavar ([7]) as a generalization of De Morgan algebras. Also, in [7], several new subvarieties of $\mathbf{I}$ were introduced, including the subvariety $\mathbf{I_{2,0}}$, defined by the identity: $x" \approx x$, which plays a crucial role in this paper. Several more new subvarieties of $\mathbf{I}$, including the subvariety $\mathbf{SL}$ of semilattices with a least element $0$, are studied in [3], and an explicit description of semisimple subvarieties of $\mathbf{I}$ is given in [5]. It is well known that the operation $\land$ induces a partial order ($\sqsubseteq$) in the variety $\mathbf{SL}$ and also in the variety $\mathbf{DM}$ of De Morgan algebras. As both $\mathbf{SL}$ and $\mathbf{DM}$ are subvarieties of $\mathbf{I}$ and the definition of partial order can be expressed in terms of the implication and the constant, it is but natural to ask whether the relation $\sqsubseteq$ (now defined) on $\mathbf{I}$ is actually a partial order in some (larger) subvariety of $\mathbf{I}$ that includes $\mathbf{SL}$ and $\mathbf{DM}$. The purpose of the present paper is two-fold: Firstly, a complete answer is given to the above mentioned problem. Indeed, our first main theorem shows that the variety $\mathbf{I_{2,0}}$ is a maximal subvariety of $\mathbf{I}$ with respect to the property that the relation $\sqsubseteq$ is a partial order on its members. In view of this result, one is then naturally led to consider the problem of determining the number of non-isomorphic algebras in $\mathbf{I_{2,0}}$ that can be defined on an $n$-element chain (herein called $\mathbf{I_{2,0}}$-chains), $n$ being a natural number. Secondly, we answer this problem in our second main theorem, which says that, for each $n \in \mathbb{N}$, there are exactly $n$ nonisomorphic $\mathbf{I_{2,0}}$-chains of size $n$.Comment: 35 page

On Implicator Groupoids

In a paper published in 2012, the second author extended the well-known fact that Boolean algebras can be defined using only implication and a constant, to De Morgan algebras-this result led him to introduce, and investigate (in the same paper), the variety I of algebras, there called implication zroupoids (I-zroupoids) and here called implicator gruopids (I- groupoids), that generalize De Morgan algebras. The present paper is a continuation of the paper mentioned above and is devoted to investigating the structure of the lattice of subvarieties of I, and also to making further contributions to the theory of implicator groupoids. Several new subvarieties of I are introduced and their relationship with each other, and with the subvarieties of I which were already investigated in the paper mentioned above, are explored.Comment: This paper, except the appendix, will appear in Algebra Universalis. 25 pages, 4 figures, a revised version with a new titl

Hopanoid lipids may facilitate aerobic nitrogen fixation in the ocean.

Cyanobacterial diazotrophs are considered to be the most important source of fixed N2 in the open ocean. Biological N2 fixation is catalyzed by the extremely O2-sensitive nitrogenase enzyme. In cyanobacteria without specialized N2-fixing cells (heterocysts), mechanisms such as decoupling photosynthesis from N2 fixation in space or time are involved in protecting nitrogenase from the intracellular O2 evolved by photosynthesis. However, it is not known how cyanobacterial cells limit O2 diffusion across their membranes to protect nitrogenase in ambient O2-saturated surface ocean waters. Here, we explored all known genomes of the major marine cyanobacterial lineages for the presence of hopanoid synthesis genes, since hopanoids are a class of lipids that might act as an O2 diffusion barrier. We found that, whereas all non-heterocyst-forming cyanobacterial diazotrophs had hopanoid synthesis genes, none of the marine Synechococcus, Prochlorococcus (non-N2-fixing), and marine heterocyst-forming (N2-fixing) cyanobacteria did. Finally, we conclude that hopanoid-enriched membranes are a conserved trait in non-heterocyst-forming cyanobacterial diazotrophs that might lower the permeability to extracellular O2 This membrane property coupled with high respiration rates to decrease intracellular O2 concentration may therefore explain how non-heterocyst-forming cyanobacterial diazotrophs can fix N2 in the fully oxic surface ocean

Training Needs and Applications for Special Educators with Paraeducator Supervision Responsibilities

This project highlights the need for training for paraeducators working in Special Education settings. Duties performed by paraeducators often overlap with teacher duties and much of the direct instruction provided to students in self-contained classrooms is delivered by paraeducators. The tasks of training and supervising paraeducators often falls on the special education teacher, who may not have received training in supervision. The project includes a brief review of the literature on this topic and two project reports. I designed my project with the objective of improving the quality of instruction being delivered to students in my classroom through training of the paraeducators who deliver much of the instruction. Project reports describe (1) training provided by a teacher to paraeducators who work in a special education post-high classroom; and (2) an application of this training in the form of a curriculum adaptation designed to be implemented by paraeducators to support student IEP goals. Sample lesson plans were designed for a target student and included research-based instructional strategies for supporting a multi-language learning student in a post-high classroom. Report 1 focuses on the preparation and training sessions for paraeducators and Report 2 is a curriculum adaptation designed for paraeducator use with a multi-language learning student. Products from this project include a set of training materials for use with paraeducators and an adapted curriculum and lesson plan ready for paraeducator use. Based on feedback and personal reflections, outcomes of the project include increased preparedness of paraeducators to work with students with disabilities in a post-high setting, in particular students who are culturally and linguistically diverse students with disabilities and multi-language learners with disabilities. The project concludes with reflections on each paraeducator training and on the curriculum adaptation process and implementation

Supersymmetric pairing of kinks for polynomial nonlinearities

We show how one can obtain kink solutions of ordinary differential equations with polynomial nonlinearities by an efficient factorization procedure directly related to the factorization of their nonlinear polynomial part. We focus on reaction-diffusion equations in the travelling frame and damped-anharmonic-oscillator equations. We also report an interesting pairing of the kink solutions, a result obtained by reversing the factorization brackets in the supersymmetric quantum mechanical style. In this way, one gets ordinary differential equations with a different polynomial nonlinearity possessing kink solutions of different width but propagating at the same velocity as the kinks of the original equation. This pairing of kinks could have many applications. We illustrate the mathematical procedure with several important cases, among which the generalized Fisher equation, the FitzHugh-Nagumo equation, and the polymerization fronts of microtubulesComment: 13 pages, 2 figures, revised during the 2nd week of Dec. 200

Extending the applicability of an open-ring trap to perform experiments with a single laser-cooled ion

An open-ring ion trap, also referred to as transparent trap was initially built up to perform $\beta$-$\nu$ correlation experiments with radioactive ions. This trap geometry is also well suited to perform experiments with laser-cooled ions, serving for the development of a new type of Penning trap, in the framework of the project TRAPSENSOR at the University of Granada. The goal of this project is to use a single $^{40}$Ca$^+$ ion as detector for single-ion mass spectrometry. Within this project and without any modification to the initial electrode configuration, it was possible to perform Doppler cooling on $^{40}$Ca$^+$ ions, starting from large clouds and reaching single ion sensitivity. This new feature of the trap might be important also for other experiments with ions produced at Radioactive Ion Beam (RIB) facilities. In this publication, the trap and the laser system will be described, together with their performance with respect to laser cooling applied to large ion clouds down to a single ion.Comment: 9 pages, 13 figure

Influence of a transverse static magnetic field on the magnetic hyperthermia properties and high-frequency hysteresis loops of ferromagnetic FeCo nanoparticles

The influence of a transverse static magnetic field on the magnetic hyperthermia properties is studied on a system of large-losses ferromagnetic FeCo nanoparticles. The simultaneous measurement of the high-frequency hysteresis loops and of the temperature rise provides an interesting insight into the losses and heating mechanisms. A static magnetic field of only 40 mT is enough to cancel the heating properties of the nanoparticles, a result reproduced using numerical simulations of hysteresis loops. These results cast doubt on the possibility to perform someday magnetic hyperthermia inside a magnetic resonance imaging setup.Comment: 6 pages, 3 figure