Abstract: A new method is presented to describe deformations of an N-membered planar ring (N-ring) molecule in terms of deformation vectors that can be expressed by a set of 2N-3 deformation amplitudes and phase angles. The deformation coordinates are directly derived from the normal vibrational modes of the N-ring and referenced to a regular polygon (N-gon) of unit length. They extend the conceptual approach of the Cremer-Pople puckering coordinates (J. Am. Chem. Soc. 1975, 97, 1354) to the planar ring and make it possible to calculate, e.g., a planar ring of special deformation on a Jahn-Teller surface. It is demonstrated that the 2N-3 deformation parameters are perfectly suited to describe the pseudorotation of a bond through the ring as it is found in cyclic Jahn-Teller systems. In general, an N-membered planar ring can undergo N-2 different bond pseudorotations provided the energetics of such a process is feasible. The Jahn-Teller distortions observed in ring compounds correspond either directly to the basic pseudorotation modes or to linear combinations of them. Any deformed ring molecule can be characterized in terms of the new ring deformation coordinates, which help to identify specific electronic effects. The usefulness of the ring deformation coordinates is demonstrated by calculating the Jahn-Teller surfaces for bond pseudorotation in the case of the cyclopropyl radical cation and cyclobutadiene as well as the ring deformation surfaces of disulfur dinitride and its dianion employing multireference averaged quadratic coupled cluster (MR-AQCC) theory, equation-of-motion coupled cluster theory in form of EOMIP-CCSD, and single determinant coupled cluster theory in form of CCSD(T).

Abstract: A *-ring R is called a *-clean ring if every element of R is the sum of a unit and a projection, and R is called a strongly *-clean ring if every element of R is the sum of a unit and a projection that commute with each other. These concepts were introduced and discussed recently by [L. Vas, *-Clean rings; some clean and almost clean Baer *-rings and von Neumann algebras, J. Algebra324 (2010) 3388–3400]. Here it is proved that a *-ring R is strongly *-clean if and only if R is an abelian, *-clean ring if and only if R is a clean ring such that every idempotent is a projection. As consequences, various examples of strongly *-clean rings are constructed and, in particular, two questions raised in [L. Vas, *-Clean rings; some clean and almost clean Baer *-rings and von Neumann algebras, J. Algebra324 (2010) 3388–3400] are answered.

Abstract: We develop the general theory of RMV-algebras, which are essentially unit intervals in Riesz spaces with strong unit. Since the variety of RMV-algebras is generated by [0, 1], we get an equational characterization of the real product on [0,1] interpreted as scalar multiplication.

Abstract: Let R be a commutative ring with identity. A factorization of a nonunit a ∈ R, a = λa 1…a n (λ a unit, each a i a nonunit) is said to be reduced (resp., μ-reduced) if (resp., for any unit λ′) for i = 1,…, n, and is said to be strongly reduced (resp., strongly μ-reduced) if (resp., for any unit λ′) for any nonempty subset {i 1,…, i s } ⊆ {1,…, n}. In this article, we discuss these various “reduced” factorizations and investigate various factorization properties from integral domains (such as being atomic, a bounded factorization domain, or a UFD) in the context of “reduced” factorizations in commutative rings with zero divisors.

Abstract: Let G be a classical compact Lie group and G μ the associated compact matrix quantum group deformed by a positive parameter μ (or $${\mu\in{\mathbb R}\setminus\{0\}}$$ in the type A case). It is well known that the category of unitary representations of G μ is a braided tensor C*–category. We show that any braided tensor *–functor $${\rho: \text{Rep}(G_\mu)\to\mathcal{M}}$$ to another braided tensor C*–category with irreducible tensor unit is full if |μ| ≠ 1. In particular, the functor of restriction RepG μ → Rep(K) to a proper compact quantum subgroup K cannot be made into a braided functor. Our result also shows that the Temperley–Lieb category $${\mathcal{T}_{\pm d}}$$ for d > 2 can not be embedded properly into a larger category with the same objects as a braided tensor C*–subcategory.

Abstract: A monoid S 1 obtained by adjoining a unit element to a 2-testable semigroup S is said to be 2-testable. It is shown that a 2-testable monoid S 1 is either inherently non-finitely based or hereditarily finitely based, depending on whether or not the variety generated by the semigroup S contains the Brandt semigroup of order five. Consequently, it is decidable in quadratic time if a finite 2-testable monoid is finitely based.

Abstract: An element of a ring R is called clean if it is the sum of an idempotent and a unit. A ring R is called clean if each of its element is clean. An element r \in R called regular if r = ryr for some y \in R. The ring R is regular if each of its element is regular. In this paper we define a ring is r-clean if each of its elements is the sum of a regular and an idempotent element. We give some relations between r-clean and clean rings. Finally we investigate some properties of r-clean rings.

Abstract: Let A be a Banach algebra and M be a Banach A- bimodule. We say that a linear mapping : A ! M is a generalized -derivation whenever there exists a -derivation d : A ! M such that (ab) = (a) (b) + (a)d(b), for all a,b 2 A. Giving some facts concerning generalized -derivations, we prove that if A is unital and if : A ! A is a generalized -derivation and there exists an element a 2 A such that d(a) is invertible, then is continuous if and only if d is continuous. We also show that if M is unital, has no zero divisor and : A ! M is a generalized - derivation such that d(1) 6 0, then ker( ) is a bi-ideal of A and ker( ) = ker( ) = ker(d), where 1 denotes the unit element of A.

Abstract: We show that an evolution family of the unit disc is commuting if and only if the associated Herglotz vector field has separated variables. This is the case if and only if the evolution family comes from a semigroup of holomorphic self-maps of the disc.

Abstract: Marciniak and Sehgal showed that if u is a non-trivial bicyclic unit of an integral group ring then there is a bicyclic unit v such that u and v generate a non-abelian free group. A similar result does not hold for Bass cyclic units of infinite order based on non-central elements as some of them have finite order modulo the center. We prove a theorem that suggests that this is the only limitation to obtain a non-abelian free group from a given Bass cyclic unit. More precisely, we prove that if u is a Bass cyclic unit of an integral group ring ℤG of a solvable and finite group G, such that u has infinite order modulo the center of U(ℤG) and it is based on an element of prime order, then there is a non-abelian free group generated by a power of u and a power of a unit in ℤG which is either a Bass cyclic unit or a bicyclic unit.

Abstract: It is shown that the order bidual X~~ of an Archimedean semiprime f -algebra X has a unit element for the Arens multiplication if and only if every positive linear functional on X extends to a positive linear functional on the f -algebra Orth (X) of all orthomorphisms on X.

Abstract: An organic EL device material includes at least a unit including 3,5-biscarbazolylphenyl group, a unit including 4-carbazolylphenyl group, and a compound including a unit including a nitrogen-containing aromatic heterocyclic ring bonding the unit including 3,5-biscarbazolylphenyl group and the unit including 4-carbazolylphenyl group.

Abstract: The title compound, C12H14O3, consists of a chromanone unit with an –OH substituent at the 4-position and methyl substituents on the remaining C atoms of the aromatic ring. The fused pyran­one ring adopts a distorted envelope conformation with the methyl­ene group adjacent to the carbonyl carbon as the flap atom. The crystal structure is stabilized by classical O—H⋯O hydrogen bonds and weak C—H⋯O and C—H⋯π inter­actions, generating a three-dimensional network.

Abstract: Let A be an absolute valued algebra with left unit. We prove that if A contains a nonzero central element, then A is finite dimensional and is isomorphic to \({\mathbb {R}, \mathbb {C}}\) or new classes of four and eight–dimensional absolute valued algebras with left unit. This is more general than those results in [2] and [3].

Abstract: As generalizations of annihilators and associated primes, we introduce the notions of weak annihilators and weak associated primes, respectively. We first study the properties of the weak annihilator of a subset X in a ring R. We next investigate how the weak associated primes of a ring R behave under passage to the skew monoid ring R*M. Let R be a semicommutative ring, and M an ordered monoid and φ: M → Aut(R) a compatible monoid homomorphism. Then we can describe all weak associated primes of the skew monoid ring R*M in terms of the weak associated primes of R in a very straightforward way.

Abstract: This paper introduces two new Burnside rings for a finite group $G$, called the slice Burnside ring and the section Burnside ring. They are built as Grothendieck rings of the category of morphisms of $G$-sets, and of Galois morphisms of $G$-sets, respectively. The well known results on the usual Burnside ring, concerning ghost maps, primitive idempotents, and description of the prime spectrum, are extended to these rings. It is also shown that these two rings have a natural structure of Green biset functor. The functorial structure of unit groups of these rings is also discussed.

Abstract: The invention discloses a path computing method and unit in an optical network. The method comprises the following steps: releasing ring protection information of each link in the optical network by a routing protocol message, wherein the ring protection information comprises the identification of each link and the identification of a ring to which each link belongs; receiving the ring protectioninformation by the routing protocol message, and according to the ring protection information, combining the links with the same ring identification into a ring so as to obtain ring topology information; and according to the ring topology information, carrying out path computing. Compared with the prior art, an optimal path can be computed by using the method and unit provided by the invention.

Abstract: PROBLEM TO BE SOLVED: To provide an organic semiconductor material having improved solubility to a solventSOLUTION: The organic semiconductor material has a sulfur-containing condensed ring compound The sulfur-containing condensed ring compound has a condensed ring skeleton with 4-10 condensed rings combining m unit A represented by general formula (1)((point) indicates a condensation position), and n unit B represented by general formula (2)((point) indicates a condensation position) so as to satisfy the relations 4≤3 m+n≤10, m≥1, n≥0 A specific functional group is bonded to each benzene ring configuring a condensed ring skeleton so as to be asymmetric on the condensed ring skeleton

Abstract: We investigate Buchbaum and Eisenbud's construction of the second symmetric power S 2(X) of a chain complex X of modules over a commutative ring R. We state and prove a num- ber of results from the folklore of the subject for which we know of no good direct references. We also provide several explicit com- putations and examples. We use this construction to prove the following version of a result of Avramov, Buchweitz, and S ega: let R! S be a module-nite ring homomorphism such that R is noe- therian and local, and such that 2 is a unit in R. Let X be a complex of nite rank free S-modules such that Xn = 0 for each n < 0. If(n AssR(Hn(X S X)) Ass(R) and if Xp' Sp for each p2 Ass(R), then X' S.

Abstract: Note that there are five mutually non-isomorphic groups of order 12: two decomposable Abelian groups G1 ∼ C3 × C4 and G2 ∼ C3×K4; three indecomposable non-Abelian groups A4, Q12 and D12. For a finite field F, the structure of U (FA4) was determined by R.K. Sharma, J.B. Srivastava and M. Khan in 2007. In this paper, we determine the structure of the unit group U (FG) of the group ring FG, where F is a finite field, and G= G1, G2, Q12 or D12.

Abstract: Let $G$ be a minimal locally compact groupoid with compact metrizable unit space and let $E$ be a continuous $G$-Hilbert bundle. We show that a bounded continuous cocycle $c: G\ra r^*E$ is necessarily a continuous coboundary. This is a groupoid version of a classical theorem of Gottschalk and Hedlund.

Abstract: In this paper we consider the times-q map on the unit inter- val as a subshift of finite type by identifying each number with its base q expansion, and we study certain non-dense orbits of this system where no element of the orbit is smaller than some fixed parameter c. The Hausdorff dimension of these orbits can be calculated using the spec- tral radius of the transition matrix of the corresponding subshift, and using simple methods based on Euclidean division in the integers, we completely characterize the characteristic polynomials of these matrices as well as give the value of the spectral radius for certain values of c. It is known through work of Urbanski and Nilsson that the Hausdorff dimension of the orbits mentioned above as a map of c is continuous and constant almost every- where, and as a new result we give some asymptotic results on how this map behaves as q ! 1.