Abstract: Kornel Szlachanyi [28] recently used the term skew-monoidal category for a particular laxi ed version of monoidal category. He showed that bialgebroids H with base ring R could be characterized in terms of skew-monoidal structures on the category of one-sided R-modules for which the lax unit was R itself. We de ne skew monoidales (or skew pseudo-monoids) in any monoidal bicategory M . These are skew-monoidal categories when M is Cat. Our main results are presented at the level of monoidal bicategories. However, a consequence is that quantum categories [10] with base comonoid C in a suitably complete braided monoidal category V are precisely skew monoidales in Comod(V ) with unit coming from the counit of C. Quantum groupoids (in the sense of [6] rather than [10]) are those skew monoidales with invertible associativity constraint. In fact, we provide some very general results connecting opmonoidal monads and skew monoidales. We use a lax version of the concept of warping de ned in [3] to modify monoidal structures.

Abstract: Kornel Szlachanyi recently used the term skew-monoidal category for a particular laxified version of monoidal category. He showed that bialgebroids $H$ with base ring $R$ could be characterized in terms of skew-monoidal structures on the category of one-sided $R$-modules for which the lax unit was $R$ itself. We define skew monoidales (or skew pseudo-monoids) in any monoidal bicategory $\mathscr M$. These are skew-monoidal categories when $\mathscr M$ is $\mathrm{Cat}$. Our main results are presented at the level of monoidal bicategories. However, a consequence is that quantum categories in the sense of Day-Street with base comonoid $C$ in a suitably complete braided monoidal category $\mathscr V$ are precisely skew monoidales in $\mathrm{Comod} (\mathscr V)$ with unit coming from the counit of $C$. Quantum groupoids are those skew monoidales with invertible associativity constraint. In fact, we provide some very general results connecting opmonoidal monads and skew monoidales. We use a lax version of the concept of warping defined recently by Booker-Street to modify monoidal structures.

Abstract: Guba and Sapir asked, in their joint paper [8], if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson's group F. We give an elementary proof for the solution of the latter question. This relies purely on the description of F as the group of piecewise linear orientation-preserving homeomorphisms of the unit. The techniques we develop allow us also to solve the ordinary conjugacy problem as well, and we can compute roots and centralizers. Moreover, these techniques can be generalized to solve the same questions in larger groups of piecewise-linear homeomorphisms.

Abstract: On the universal bundle of unit spinors we study a natural energy functional whose critical points, if dim M \geq 3, are precisely the pairs (g, {\phi}) consisting of a Ricci-flat Riemannian metric g together with a parallel g-spinor {\phi}. We investigate the basic properties of this functional and study its negative gradient flow, the so-called spinor flow. In particular, we prove short-time existence and uniqueness for this flow.

Abstract: A first module divides a string into a number of blocks. A second module associates the blocks with monoid elements in a list of first monoid elements to produce second monoid elements. A third module applies a first function to an initial monoid element and a first of the second monoid elements producing a first calculated monoid element and evaluates an action of the initial monoid element on the first function producing a second function. A fourth module applies the second function to the first calculated monoid element and to a second of the second monoid elements producing a second calculated monoid element and evaluates the action of the first calculated monoid element on the first function producing a third function. Further modules iteratively, corresponding to the number of blocks, apply the produced function to calculated monoid elements and the second monoid elements to produce a hash of the string

Abstract: We give an explicit and character-free construction of a complete set of orthogonal primitive idempotents of a rational group algebra of a finite nilpotent group and a full description of the Wedderburn decomposition of such algebras. An immediate consequence is a well-known result of Roquette on the Schur indices of the simple components of group algebras of finite nilpotent groups. As an application, we obtain that the unit group of the integral group ring ${\mathbb Z} G$ of a finite nilpotent group G has a subgroup of finite index that is generated by three nilpotent groups for which we have an explicit description of their generators. Another application is a new construction of free subgroups in the unit group. In all the constructions dealt with, pairs of subgroups (H, K), called strong Shoda pairs, and explicit constructed central elements e(G, H, K) play a crucial role. For arbitrary finite groups we prove that the primitive central idempotents of the rational group algebras are rational linear combinations of such e(G, H, K), with (H, K) strong Shoda pairs in subgroups of G.

Abstract: The problem addressed is the classification up to conjugation of the finite subgroups of the (classical) Morava stabilizer group S_n and the extended Morava stabilizer group G_n(u) associated to a formal group law F of height n over the field F_p of p elements A complete classification in S_n is provided for any positive integer n and prime p Furthermore, we show that the classification in the extended group also depends on F and its associated unit u in the ring of p-adic integers We provide a theoretical framework for the classification in G_n(u), we give necessary and sufficient conditions on n, p and u for the existence in G_n(u) of extensions of maximal finite subgroups of S_n by the Galois group of F_{p^n} over F_p, and whenever such extension exist we enumerate their conjugacy classes We illustrate our methods by providing a complete and explicit classification in the case n=2

Abstract: It is shown that if R is a ring with unit element which is not algebraic over the prime subring of R, then R has a maximal subring. It is shown that whenever R ⊆ T are rings such that there exists a maximal subring V of T, which is integrally closed in T and U(R) ⊈ V, then R has a maximal subring. In particular, it is proved that if R is algebraic over ℤ and there exists a natural number n > 1 with n ∈ U(R), then R has a maximal subring. It is shown that if R is an infinite direct product of certain fields, then the maximal ideals M for which RM (R/M) has maximal subrings are characterized. It is observed that if R is a ring, then either R has a maximal subring or it must be a Hilbert ring. In particular, every reduced ring R with |R|>22ℵ0 or J(R) ≠ 0 has a maximal subring. Finally, the semi-local rings having maximal subrings are fully characterized.

Abstract: Aguiar and Mahajan's bimonoids A in a duoidal category M are studied. Under certain assumptions on M, the Fundamental Theorem of Hopf Modules is shown to hold for A if and only if the unit of A determines an A-Galois extension. Our findings are applied to the particular examples of small groupoids and of Hopf algebroids over a commutative base algebra.

Abstract: For $\lambda$ an $n$-th power of a unit in a finite chain ring we prove that $\lambda$-constacyclic repeated-root codes over some finite chain rings are equivalent to cyclic codes This allows us to simplify the structure of some constacylic codes We also study the $\alpha +p \beta$-constacyclic codes of length $p^s$ over the Galois ring $GR(p^e,r)$

Abstract: We obtain a formula for the density of the free convolution of an arbitrary probability measure on the unit circle of $\mathbb{C}$ with the free multiplicative analogues of the normal distribution on the unit circle. This description relies on a characterization of the image of the unit disc under the subordination function, which also allows us to prove some regularity properties of the measures obtained in this way. As an application, we give a new proof for Biane's classic result on the densities of the free multiplicative analogue of the normal distributions. We obtain analogue results for probability measures on $\mathbb{R}^+$. Finally, we describe the density of the free multiplicative analogue of the normal distributions as an example and prove unimodality and some symmetry properties of these measures.

Abstract: It is shown that any left module A over a ring R can be written as the intersection of a downward directed system of injective submodules of an injective module; equivalently, as an inverse limit of one-to-one homomorphisms of injectives. If R is left Noetherian, A can also be written as the inverse limit of a system of surjective homomorphisms of injectives. Some questions are raised. The flat modules over a ring are precisely the direct limits of projective modules [11] [6] [10, Theorem 2.4.34]. Which modules are, dually, inverse limits of injectives? I sketched the answer in [1], but in view of the limited distribution of that item, it seems worthwhile to make the result more widely available. The construction from [1] is Theorem 2 below; the connecting maps there are inclusions. In Theorem 4, we shall see that the connecting maps can, alternatively, be taken to be onto if R is Noetherian on the appropriate side. In §2 we ask some questions, in §3 we take some steps toward answering one of them, and in §4 we note what the proofs of our results tell us when applied to not necessarily injective modules. Throughout, “ring” means associative ring with unit, and modules are unital. I am indebted to Pace Nielsen for pointing out the need to assume κ regular in Lemma 1, and to the referee for some useful suggestions.

Abstract: A highly concise and stereoselective total synthesis of (5R,7S)-kurzilactone (1) was performed by a convergent approach by means of a Jacobsen's hydrolytic kinetic resolution, a HornerWadsworthEmmons reaction for the construction of the α,β-unsaturated δ-lactone ring system, and a highly diastereoselective Mukaiyama aldol reaction for the introduction of the formal anti-1,3-diol unit (Schemes 2 and 3).

Abstract: We study the recently discovered isomorphisms between hyperbolic Weyl groups and modular groups over integer domains in normed division algebras. We show how to realize the group action via fractional linear transformations on generalized upper half-planes over the division algebras, focussing on the cases involving quaternions and octonions. For these we construct automorphic forms, whose explicit expressions depend crucially on the underlying arithmetic properties of the integer domains. Another main new result is the explicit octavian realization of W + (E10), which contains as a special case a new realization of W + (E8) in terms of unit octavians and their automorphism group.

Abstract: Let R be an associative ring with identity. An element x∈R is said to be weakly exchange if there exists an idempotent e∈R such that e∈xR and 1−e∈(1−x)R or 1−e∈(1+x)R. The ring R is said to be weakly exchange if all of its elements are weakly exchange. In this paper an element-wise characterization is given, and it is shown that weakly-Abel weakly exchange rings are weakly clean. Moreover, a relation between unit regular rings and weakly clean rings is also obtained.

Abstract: The invention relates to a design method of a low-profile dual-polarized tile antenna unit adopting T-shaped microstrip feeding. The design method is characterized in that an antenna patch unit adopts a T-shaped microstrip line for coupled feeding, a feeding network and the patch unit are located in one layer, and two polarizing and feeding networks are arranged; and a parasitic unit is located just above an excitation unit. The antenna has capability of suppressing antenna system low-frequency signal interference, and has standing wave ratio smaller than 1.8 and cross polarization level smaller than -17dB in 20% relative bandwidth; and the T-shaped microstrip line is adopted for coupled feeding as well as is located in one plane with an antenna radiation unit, thus the thickness of the antenna unit is significantly reduced. The obtained antenna unit not only can be used as a unit of a tile type array antenna, but also can be independently used as a transceiver antenna of a polarization diversity system.

Abstract: Let D E be an extension of integral domains, 0 a numerical semigroup with 0 ( N0, 0 D 0nf0g and RD DC ET0 U. In this paper, we completely characterize when R is a weakly Krull domain, an AWFD or a GWFD. We also prove that R is never a WFD. Let T.D/ be the abelian group of t-invertible fractional t-ideals of D under the t- multiplication I JD.I J/t , and let Inv.D/ and Prin.D/ be the subgroups of T.D/ consisting respectively of invertible fractional ideals of D and nonzero principal fractional ideals of D. Then it is clear that Prin.D/ Inv.D/ T.D/. The t-class group of D is an abelian group Cl.D/D T.D/= Prin.D/ and the Picard group Pic.D/D Inv.D/= Prin.D/ is a subgroup of Cl.D/. The local t-class group G.D/ of D is defined by G.D/D Cl.D/= Pic.D/. Let X 1 .D/ stand for the set of height-one prime ideals of D. We say that D is a weakly Krull domain if DD T P2X 1 .D/ DP and this intersection has finite character, i.e., each nonzero element d2 D is a unit in DP for all but a finite number of P's in X 1 .D/; D is a weakly factorial domain (WFD) if every nonzero nonunit element of D is a product of primary elements; D is an almost weakly factorial domain

Abstract: Techniques described in the disclosure are generally related to generating points of a domain. A tessellation unit may determine outer ring point coordinates for a point of an outer ring of the domain, and inner ring point coordinates for a point of an inner ring of the domain. The inner ring is inner to the outer ring within the domain. The tessellation unit may enqueue the inner ring point coordinates at a location of a queue, read the inner ring point coordinates from the queue, and read the outer ring point coordinates from the queue when the outer ring is not an outermost ring, where the outer ring point coordinates were previously enqueued in the queue when the outer ring was a previous inner ring. The tessellation unit may connect the inner ring coordinates and the outer ring coordinates each of which being read from the queue.

Abstract: Let R be the ring of linear transformations of a right vector space over a division ring D. Three results are proved: (1) if |D|>4, then for any a∈R there exists a unit u of R such that a+u,a−u and a−u−1 are units of R; (2) if |D|>3 , then for any a∈R there exists a unit u of R such that both a+u and a−u−1 are units of R; (3) if |D|>2 , then for any a∈R there exists a unit u of R such that both a−u and a−u−1 are units of R. The second result extends the main result in H. Chen, [‘Decompositions of countable linear transformations’, Glasg. Math. J. (2010), doi:10.1017/S0017089510000121] and the third gives an affirmative answer to the question raised in the same paper.

Abstract: . An element of a ring is called strongly nil clean provided thatit can be written as the sum of an idempotent and a nilpotent elementthat commute. We characterize, in this article, the strongly nil cleannessof 2 2 and 3 3 matrices over R [ x ] = ( x 2 1)where R is a commutativelocal ring with characteristic 2. Matrix decompositions over elds arederived as special cases. 1. IntroductionLet R be an associative ring with identity. An element a 2 R is said to bestrongly clean provided that there exist an idempotent e 2 R and a unit u 2 R such that a = e + u and eu = ue . Strongly clean matrices over commutativelocal rings were extensively studied by many authors from very fft viewpoints (cf. [1-2], [4-7] and [9-12]). In [5], Diesl introduced the concept ofstrongly nil cleanness. An element a 2 R is strongly nil clean provided thatthere exist an idempotent e 2 R and a nilpotent element u 2 R such that a = e + u and eu = ue . Every strongly nil clean element is strongly clean (cf.[5, Proposition 3.1.3]). But the converse is not true, e.g., 2

Abstract: Objective: The aim of the study was to describe patterns of injuries sustained by nurses during the restraining of aggressive patients and to identify factors in the restraining process that can be modified to improve the safety of nurses during restraining. Design: Within‑method triangulation was used in this study and involved two quantitative data collection methods. Setting: An adult acute psychiatric unit in Victoria, Australia. Subjects: Seven male and twenty‑six female nurses. Main outcome measures: The outcome measures are patterns of injuries and ways of reducing injuries. Results: Incident reports showed more than half of all injuries occurred in the afternoon shift and during the holding stage of restraining. Eighty percent of the injured nurses sustained multiple injuries. Questionnaire results showed that restraining was associated with an estimated increased risk of being injured of 25% (RR = 1.25, 95% CI= 0.97 to 1.61, p > 0.05). The proportion of injuries was higher among female nurses (52.38%) compared with male nurses (28.57%), (RR=0.51, 95% CI = 0.15 to 1.74, p > 0.05). Lack of group co‑ordination was perceived as the main contributor to injury. Introducing easier restraining techniques and increasing the training period were identified as ways that might improve the safety of nurses. Conclusion: Most injuries occurred at the holding stage of restraining and in the afternoon shift. Many participants sustained multiple injuries and most of the injuries were caused by physical assaults. There is need for improving group coordination during restraining to increase the safety of nurses.

Abstract: It is proved that if $D$ is a $UFD$ and $R$ is a $D$-algebra, such that $U(R)\cap D eq U(D)$, then $R$ has a maximal subring. In particular, if $R$ is a ring which either contains a unit $x$ which is not algebraic over the prime subring of $R$, or $R$ has zero characteristic and there exists a natural number $n>1$ such that $\frac{1}{n}\in R$, then $R$ has a maximal subring. It is shown that if $R$ is a reduced ring with $|R|>2^{2^{\aleph_0}}$ or $J(R) eq 0$, then any $R$-algebra has a maximal subring. Residually finite rings without maximal subrings are fully characterized. It is observed that every uncountable $UFD$ has a maximal subring. The existence of maximal subrings in a noetherian integral domain $R$, in relation to either the cardinality of the set of divisors of some of its elements or the height of its maximal ideals, is also investigated.

Abstract: Let G be an 1-connected simple Lie group, and let x\inG be a group element. We determine the isomorphism type of the centralizer C_{x} in term of a minimal geodesic joinning the group unit e\inG to x. This result is applied to classify the isomorphism types of maximal subgroups of maximal rank of G, and the isomorphism types of parabolic subgroups of G.

Abstract: Let K be a number field with unit rank at least four, containing a subfield M such that K/M is Galois of degree at least four. We show that the ring of integers of K is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann Hypothesis for Dedekind zeta functions. We prove this unconditionally.

Abstract: A ring R with identity is called "clean" if for every element a ∈ R, there exist an idempotent e and a unit u in R such that a = u +e. Let C(R) denote the center of a ring R and g(x) be a polynomial in C(R)(x). An element r ∈ R is called "g(x)-clean" if r = u + s where g(s )=0a ndu is a unit of R and R is g(x)-clean if every element is g(x)-clean. In this paper we define a ring to be weakly g(x)-clean if each element of R can be written as either the sum or difference of a unit and ar oot ofg(x).

Abstract: We take account of the stability of higher ring derivation in intuitionistic fuzzy Banach algebra associated to the Jensen type functional equation. In addition, we deal with the superstability of higher ring derivation in intuitionistic fuzzy Banach algebra with unit.