Binomial Skew Polynomial Rings, Artin-Schelter Regularity, and Binomial Solutions of the Yang-Baxter Equation

2000 Mathematics Subject Classification: Primary 81R50, 16W50, 16S36, 16S37.

Three dimensional Sklyanin algebras and Gröbner bases

We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. this class of algebras has enjoyed much attention. In particular, using tools from algebraic geometry, Feigin, Odesskii \cite{odf}, and Artin, Tate and Van Den Bergh, showed that if at least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter regular. More specifically, $S$ is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates (PHS). It has became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van Den Bergh. Further, we study a wider class of Sklyanin algebras, namely
the situation when all parameters of relations could be different. We call them generalized Sklyanin algebras. We classify up to isomorphism all generalized Sklyanin algebras with the same Hilbert series as commutative polynomials on
3 variables. We show that generalized Sklyanin algebras in general position have a Golod–Shafarevich Hilbert series (with exception of the case of field with two elements).

Class Notes for Math 901/902: Abstract Algebra, Instructor Tom Marley

Topics include: Free groups and presentations; Automorphism groups; Semidirect products; Classification of groups of small order; Normal series: composition, derived, and solvable series; Algebraic field extensions, splitting fields, algebraic closures; Separable algebraic extensions, the Primitive Element Theorem; Inseparability, purely inseparable extensions; Finite fields; Cyclotomic field extensions; Galois theory; Norm and trace maps of an algebraic field extension; Solvability by radicals, Galois' theorem; Transcendence degree; Rings and modules: Examples and basic properties; Exact sequences, split short exact sequences; Free modules, projective modules; Localization of (commutative) rings and modules; The prime spectrum of a ring; Nakayama's lemma; Basic category theory; The Hom functors; Tensor products, adjointness; Left/right Noetherian and Artinian modules; Composition series, the Jordan-Holder Theorem; Semisimple rings; The Artin-Wedderburn Theorem; The Density Theorem; The Jacobson radical; Artinian rings; von Neumann regular rings; Wedderburn's theorem on finite division rings; Group representations, character theory; Integral ring extensions; Burnside's paqb Theorem; Injective modules.

Frame presentations: variants of the reals, rings of functions, their Dedekind completions, and the unit circle

176 p.

Designing folding rings using polynomial continuation

© 2014 by ASME. Two types of foldable rings are designed using polynomial continuation. The first type of ring, when deployed, forms regular polygons with an even number of sides and is designed by specifying a sequence of orientations which each bar must attain at various stages throughout deployment. A design criterion is that these foldable rings must fold with all bars parallel in the stowed position. At first, all three Euler angles are used to specify bar orientations, but elimination is also used to reduce the number of specified Euler angles to two, allowing greater freedom in the design process. The second type of ring, when deployed, forms doubly plane-symmetric (irregular) polygons. The doubly symmetric rings are designed using polynomial continuation, but in this example a series of bar end locations (in the stowed position) is used as the design criterion with focus restricted to those rings possessing eight bars.

On the derived category of a regular toric scheme

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes U s = Spec A s . A quasi-coherent sheaf on X gives rise, by taking sections over the U s , to a diagram of modules over the coordinate rings A s , indexed by the intersection poset S of the covering. If X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category of quasi-coherent sheaves on X can be obtained from a category of Sop-diagrams of chain complexes of modules by inverting maps which induce homology isomorphisms on hyper-derived inverse limits. Moreover, we show that there is a finite set of weak generators, one for each cone in the fan S. The approach taken uses the machinery of Bousfield–Hirschhorn colocalisation of model categories. The first step is to characterise colocal objects; these turn out to be homotopy sheaves in the sense that chain complexes over different open sets U s agree on intersections up to quasi-isomorphism. In a second step it is shown that the homotopy category of homotopy sheaves is equivalent to the derived category of X.

Seasonality and growth rings in lianas of Bignoniaceae

Lianas are one of the most important components of tropical forest, and yet one of the most poorly known organisms. Therefore, our paper addresses questions on the environmental and developmental aspects that influence the growth of lianas of Bignoniaceae, tribe Bignonieae. In order to better understand their growth, we studied the stem anatomy, seasonality of formation and differentiation of secondary tissues, and the influence of the cambial variant in xylem development on a selected species: Tynanthus cognatus. Afterwards, we compared the results found in T. cognatus with 31 other species of Bignonieae to identify general patterns of growth in lianas of this tribe. We found that cambial activity starts toward the end of the rainy season and onset of the dry season, in contrast to what is known for tropical trees and shrubs. Moreover, their pattern of xylem formation and differentiation is strongly influenced by the presence of massive wedges of phloem produced by a variant cambium. Thus, the variant cambium is the first to commence its activity and only subsequently does cambial activity progress towards the center of the regular region, leading to the formation of confluent growth rings. In summary, we conclude that: the cambium responds to environmental changes; the xylem growth rings are annual and produced in a brief period of about 2 months, something that may explain why lianas possess narrow stems; and furthermore, phloem wedges greatly influence cambial activity.

Aggregates in the strength and gravity regime: Particles sizes in Saturn's rings

Particles in Saturn's main rings range in size from dust to kilometer-sized objects. Their size distribution is thought to be a result of competing accretion and fragmentation processes. While growth is naturally limited in tidal environments, frequent collisions among these objects may contribute to both accretion and fragmentation. As ring particles are primarily made of water ice attractive surface forces like adhesion could significantly influence these processes, finally determining the resulting size distribution. Here, we derive analytic expressions for the specific self-energy Q and related specific break-up energy Q(star) of aggregates. These expressions can be used for any aggregate type composed of monomeric constituents. We compare these expressions to numerical experiments where we create aggregates of various types including: regular packings like the face-centered cubic (fcc), Ballistic Particle Cluster Aggregates (BPCA), and modified BPCAs including e.g. different constituent size distributions. We show that accounting for attractive surface forces such as adhesion a simple approach is able to: (a) generally account for the size dependence of the specific break-up energy for fragmentation to occur reported in the literature, namely the division into "strength" and "gravity" regimes and (b) estimate the maximum aggregate size in a collisional ensemble to be on the order of a few tens of meters, consistent with the maximum particle size observed in Saturn's rings of about 10 m. (c) 2012 Elsevier B.V. All rights reserved.

Magnetic properties and Spin dynamics in magnetic Molecular Rings

Molecular nanomagnets are spin clusters whose topology and magnetic interactions can be modulated at the level of the chemical synthesis. They are formed by a small number of transition metal ions coupled by the Heisenberg's exchange interactions. Each cluster is magnetically isolated from its neighbors by organic ligands, making each unit not interacting with the others. Therefore, we can investigate the magnetic properties of an isolated molecular nanomagnet by bulk measurements. The present thesis has been mostly devoted to the experimental investigation of the magnetic properties and spin dynamics of different classes of antiferromagnetic (AF) molecular rings. This study has been exploiting various techniques of investigations, such as Nuclear Magnetic Resonance (NMR), muon spin relaxation (muSR) and SQUiD magnetometry. We investigate the magnetic properties and the phonon-induced relaxation dynamics of the first regular Cr9 antiferromagnetic (AF) ring, which represents a prototype frustrated AF ring. The magnetically-open AF rings like Cr8Cd are model systems for the study of the microscopic magnetic behaviour of finite AF Heisenberg chains. In this type of system the different magnetic behaviour depends length and on the parity of the chain (odd or even). In order to study the local spin densities on the Cr sites, the Cr-NMR spectra was collected at low temperature. The experimental result confirm the theoretical predictions for the spin configuration. Finally, the study of Dy6, the first rare-earth based ring that has been ever synthesized, has been performed by AC-SQuID and muSR measurements. We found that the dynamics is characterized by more than one characteristic correlation time, whose values depend strongly on the applied field.

Zero-Dimensionality and Serre Rings

2000 Mathematics Subject Classification: Primary 13A99; Secondary 13A15, 13B02, 13E05.

Identifying key belief-based targets for promoting regular physical activity among mothers and fathers with young children

We investigated the key beliefs to target in interventions aimed at increasing physical activity (PA) among mothers and fathers of young children. Parents (288 mothers and 292 fathers) completed a Theory of Planned Behaviour belief-based questionnaire and a 1-week follow-up of PA behaviour. We found that a range of behavioural, normative, and control beliefs were significantly correlated with parents’ PA intentions and behaviour, with only a few differences observed in correlations between PA beliefs and intention and behaviour by gender. A range of key beliefs was identified as making independent contributions to parents’ PA intentions; however, the behavioural beliefs about improving parenting practices (β = 0.13), interfering with other commitments (β = −0.29); normative beliefs about people I exercise with (β = 0.20); and control beliefs about lack of time (β = −0.24), inconvenience (β = −0.14), lack of motivation (β = −0.34), were revealed as significant independent predictors of actual PA behaviour. Furthermore, we found that a limited amount of parents already hold these beliefs, suggesting that these key beliefs warrant changing and, therefore, are appropriate targets for subsequent intervention. The current study fills an empirical gap in the PA literature by investigating an at-risk group and using a well established theoretical framework to identify key beliefs that guide parents’ PA decision-making. Overall, we found support for parents being a unique group who hold distinctive behavioural, normative, and control beliefs toward PA. Attention to these key underlying beliefs will assist intervention work aimed at combating inactivity among this at-risk population.

Regular exercise participation mediates the affective response to acute bouts of vigorous exercise

Physical inactivity is a leading factor associated with cardiovascular disease and a major contributor to the global burden of disease in developed countries. Subjective mood states associated with acute exercise are likely to influence future exercise adherence and warrant further investigation. The present study examined the effects of a single bout of vigorous exercise on mood and anxiety between individuals with substantially different exercise participation histories. Mood and anxiety were assessed one day before an exercise test (baseline), 5 minutes before (pre-test) and again 10 and 25 minutes post-exercise. Participants were 31 university students (16 males, 15 females; Age M = 20), with 16 participants reporting a history of regular exercise with the remaining 15 reporting to not exercise regularly. Each participant completed an incremental exercise test on a Monark cycle ergometer to volitional exhaustion. Regular exercisers reported significant post-exercise improvements in mood and reductions in state anxiety. By contrast, non-regular exercisers reported an initial decline in post-exercise mood and increased anxiety, followed by an improvement in mood and reduction in anxiety back to pre-exercise levels. Our findings suggest that previous exercise participation mediates affective responses to acute bouts of vigorous exercise. We suggest that to maximise positive mood changes following exercise, practitioners should carefully consider the individual’s exercise participation history before prescribing new regimes.

A free weekly iron-folic acid supplementation and regular deworming program is associated with improved hemoglobin and iron status indicators in Vietnamese women

Background Anemia due to iron deficiency is recognized as one of the major nutritional deficiencies in women and children in developing countries. Daily iron supplementation for pregnant women is recommended in many countries although there are few reports of these programs working efficiently or effectively. Weekly iron-folic acid supplementation (WIFS) and regular deworming treatment is recommended for non-pregnant women living in areas with high rates of anemia. Following a baseline survey to assess the prevalence of anemia, iron deficiency and soil transmitted helminth infections, we implemented a program to make WIFS and regular deworming treatment freely and universally available for all women of reproductive age in two districts of a province in northern Vietnam over a 12 month period. The impact of the program at the population level was assessed in terms of: i) change in mean hemoglobin and iron status indicators, and ii) change in the prevalence of anemia, iron deficiency and hookworm infections. Method Distribution of WIFS and deworming were integrated with routine health services and made available to 52,000 women. Demographic data and blood and stool samples were collected in baseline, and three and 12-month post-implementation surveys using a population-based, stratified multi-stage cluster sampling design. Results The mean Hb increased by 9.6 g/L (95% CI, 5.7, 13.5, p < 0.001) during the study period. Anemia (Hb<120 g/L) was present in 131/349 (37.5%, 95% CI 31.3, 44.8) subjects at baseline, and in 70/363 (19.3%, 95% CI 14.0, 24.6) after twelve months. Iron deficiency reduced from 75/329 (22.8%, 95% CI 16.9, 28.6) to 33/353 (9.3%, 95% CI 5.7, 13.0) by the 12-mnth survey, and hookworm infection from 279/366 (76.2%,, 95% CI 68.6, 83.8) to 66/287 (23.0%, 95% CI 17.5, 28.5) over the same period. Conclusion A free, universal WIFS program with regular deworming was associated with reduced prevalence and severity of anemia, iron deficiency and ho