Abstract: This is a survey of recently published results. We introduce and study a wide class of algebras associated to directed graphs and related to factorizations of noncommutative polynomials. In particular, we show that for many well-known graphs such algebras are Koszul and compute their Hilbert series. Let R be an associative ring with unit and P(t) = a0t n +a1t n−1 +� � � +an be a polynomial over R. Here t is an independent central variable. We consider factorizations of P(t) into a product

Abstract: It is shown that if A is a compact operator on a Hilbert space with its numerical range W(A) contained in the closed unit disc D and with W(A) intersecting the unit circle at infinitely many points, then W(A) is equal to D. This is an infinite-dimensional analogue of a result of Anderson for finite matrices.

Abstract: We argue that a selfconsistent spatial coarse-graining, which involves interacting (anti)calorons of unit topological charge modulus, implies that real-time loop expansions of thermodynamical quantities in the deconfining phase of SU(2) and SU(3) Yang-Mills thermodynamics are, modulo 1PI resummations, determined by a finite number of connected bubble diagrams.

Abstract: If C = C(R) denotes the center of a ring R and g(x) is a polynomial in C(x), Camillo and Sim´ on called a ring g(x)-clean if every element is the sum of a unit and a root of g(x). If V is a vector space of countable dimension over a division ring D, they showed that end DV is g(x)-clean provided that g(x) has two roots in C(D). If g(x) = x − x 2 this shows that end DV is clean, a result of Nicholson and Varadarajan. In this paper we remove the countable condition, and in fact prove that end RM is g(x)-clean for any semisimple module M over an arbitrary ring R provided that g(x) ∈ (x − a)(x − b)C(x) where a, b ∈ C and both b and b − a are units in R.

Abstract: Let G be a Lie group which is the union of an ascending sequence of Lie groups G_n (all of which may be infinite-dimensional). We study the question when G is the direct limit of the G_n's in the category of Lie groups, topological groups, smooth manifolds, resp., topological spaces. Full answers are obtained for G the group Diff_c(M) of compactly supported smooth diffeomorphisms of a sigma-compact smooth manifold M, and for test function groups C^infty_c(M,H) of compactly supported smooth maps with values in a finite-dimensional Lie group H. We also discuss the cases where G is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie group-valued analytic maps, or a weak direct product of Lie groups.

Abstract: The subset multiplicatively generated by any given set of quantum minors and the unit element in the quantum matrix bialgebra satisfies the left and right Ore conditions.

Abstract: We prove that an ideal I of a regular ring R is separative if and only if each a ∊ R satisfying Rr(a)aR = Ral(a)R = RaR(1 − a)R ⊆ I is unit-regular. If I is a separative ideal of a regular ring R, then each a ∊ R satisfying Rar(a2) = l(a2)aR = R(a − a2) R ⊆ I is clean. Some applications are also obtained.

Abstract: Disclosed is a luminescent or charge-transporting polymer compound having a main chain containing a divalent heterocyclic group, a divalent condensed polycyclic hydrocarbon group including no five-membered ring, a group represented by the formula (1) below or a divalent aromatic amine group as a repeating unit, and a functional side chain containing at least one functional group selected from the group consisting of hole injecting/transporting groups, electron injecting/transporting groups, and light-emitting groups. This polymer compound is characterized in that the functional group is directly bonded to a saturated carbon atom in the repeating unit or bonded to the repeating unit via X in an -RJ-X- group (wherein RJ represents an optionally substituted alkylene group, X represents a direct bond, an oxygen atom, a sulfur atom, C=O, C(=O)-O, S=O, SiR8R9, NR10, BR11, PR12 or P(=O)R13. (In the formula, the ring A and the ring B independently represent an optionally substituted aromatic hydrocarbon ring, and the aromatic hydrocarbon ring in the ring A and the aromatic hydrocarbon ring in the ring B have different ring structures from each other; two bonding hands are respectively present on the ring A and/or the ring B; Rw and Rx independently represent a hydrogen atom or a substituent, and Rw and Rx may combine together to form a ring.)

Abstract: The bearing arrangement has a bearing (1), which has three or more rings. The bearing has an inner ring (3) which rotates during use as intended, and a central ring (4) and an outer ring (5). A sensor and an exciter element (12) are arranged on the inner ring or on a component (13) which is connected to the inner ring. A detecting unit (14) is arranged on the central ring or on the outer ring for detecting and forwarding a signal which is measured by a sensor (7,8,9,10,11). An independent claim is also included for a method for control of a bearing arrangement.

Abstract: The present invention is a method of database searching. First, a language is selected and elements received. The system is searched to identify a unit number associated with each element, which is linked to a data unit containing morphological variants of the element. If none are identified, the element is broken into sub-textual units that may contain a prefix, compound-prefix, and/or suffix along with a primary element. A unit number is then obtained for the primary element. If this does not result in a match, the elements may be saved in a database for further linguistic development. A unit number associated with each matched element is then chosen, and the elements contained in the data units linked to the unit numbers are compared to a database index. If an element is associated with multiple unit numbers, this process is repeated until all data units have been compared to the database.

Abstract: We classify groups G such that the unit group U(ZG) is hypercentral. In the second part we classify groups G whose modular group algebra has hyperbolic unit group V(KG).

Abstract: PROBLEM TO BE SOLVED: To provide a reproducing unit and a reproducing method capable of displaying warning information on content having a time limit according to user's intent. SOLUTION: The reproducing unit comprises: a communication section for receiving the content having a time limit from an external device via a network; a storage section for storing the content having a time limit in a storage region via the communication section; an input section inputting warning conditions; and a management section for generating a warning image or warning sound for output when the time of the content having the time limit stored in the storage section is compared with current time and time from the current time to the time limit meets the warning conditions from the input section. COPYRIGHT: (C)2008,JPO&INPIT

Abstract: A ring R is called (left principally) quasi-Baer if the left annihilator of every (principal) left ideal of R is generated by an idempotent. Let R be a ring, G be an ordered monoid acting on R by fl and R be G-compatible. It is shown that R is (left principally) quasi-Baer if and only if skew monoid ring Rfl(G) is (left principally) quasi-Baer. If G is an abelian monoid, then R is (left principally) quasi-Baer if and only if the Cohn-Jordan exten- sion A(R;fl) is (left principally) quasi-Baer if and only if left Ore quotient ring G i1 Rfl(G) is (left principally) quasi-Baer.

Abstract: We consider the quantization of inversion in commutative p-normed quasi-Banach algebras with unit. The standard questions considered for such an algebra A with unit e and Gelfand transform x bar right arrow (x) over cap are: (i) Is K-nu = sup{parallel to(e - x)(-1)parallel to(p) : x is an element of A, parallel to x parallel to(p) = delta} bounded? Both questions are related to a "uniform spectral radius" of the algebra, r(infinity)(A), introduced by Bjork. Question (i) has an affirmative answer if and only if r(infinity)(A) < 1, and this result is extended to more general nonlinear extremal problems of this type. Question (ii) is more difficult, but it can also be related to the uniform spectral radius. For algebras with compact Gelfand transform we prove that the answer is "yes" for all delta is an element of (0, 1) if and only if r(infinity)(A) = 0. Finally, we specialize to semisimple Beurling type algebras l(w)(p)(D), where 0 < p < 1 and D = N or D = Z. We show that the number r(infinity)(l(w)(p)(D)) can be effectively computed in terms of the underlying weight. In particular, this solves questions (i) and (ii) for many of these algebras. We also construct weights such that the corresponding Beurling algebra has a compact Gelfand transform, but the uniform spectral radius equals an arbitrary given number in (0, 1]. (Less)

Abstract: We study one-parameter families of S-unit equations of the form f(t)u+g(t)v=h(t), where f, g, and h are univariate polynomials over a number field, t is an S-integer, and u and v are S-units. For many possible choices of f, g, and h, we are able to determine all but finitely many solutions to the corresponding one-parameter family of S-unit equations. The results are obtained as consequences of some recent results on integral points on surfaces.

Abstract: Using the computational algebra system GAP (this http URL) and the GAP package LAGUNA (this http URL), we checked that all 2-groups of order not greater than 32 are determined by normalized unit groups of their modular group algebras over the field of two elements.

Abstract: It is shown that every element in a one-sided unit-regular ring is the sum of an idempotent and a one-sided unit. Also we prove that every element in a regular ring satisfying related comparability is the sum of an idempotent and the product of two one-sided units.

Abstract: The jewellery e.g. watch (5) includes an inner ring (2), intermediate ring (3) and outer ring (4) nested inside each other. The pivoting zones (9) vary the degree of nesting of the rings, by pivoting the ring relating to each other about a common axis. An indexing unit has pin secured to the inner ring for cooperating with the edge of an elongated aperture in the outer ring, so as to immobilize the rings in several relative positions such as nested position and tilted position.

Abstract: The device has an inertia unit (3) movable in a section and held in a force-free condition in a neutral position. The inertia unit acts on a locking unit (6) during occurrence of acceleration forces, so that the locking unit locks a bolting device in a locked position. The inertia unit is movable in all three directions from its neutral position. The locking unit is designed as a locking pin (10).

Abstract: We study one-parameter families of S-unit equations of the form f(t)u+g(t)v=h(t), where f, g, and h are univariate polynomials over a number field, t is an S-integer, and u and v are S-units. For many possible choices of f, g, and h, we are able to determine all but finitely many solutions to the corresponding one-parameter family of S-unit equations. The results are obtained as consequences of some recent results on integral points on surfaces.

Abstract: For Jason Luther, the I-Search paper alone did not seem to do enough to help students think critically. In this article, he shows how he supplements the I-Search paper with a unit to develop habits and methods of inquiry utilizing documentary films, creative nonfiction, and feature articles discussed through Socratic seminars.

Abstract: PROBLEM TO BE SOLVED: To solve the problem, wherein a pairing arithmetic speed is slow, since the amount of operation required for pairing arithmetic is much larger than that of a regular ellipse arithmetic operation. SOLUTION: In the invention, high speed is achieved by using a characteristic of a finite field. Assuming r as an element on the finite field GF(p k/2 ) and f as an element on the finite field GF(p k ) which is calculated by Miller algorithm, calculation is performed using the algorithm in which a calculating amount is less than Miller algorithm for imaging to an element f' on the finite field GF(p k ) which satisfies f'=rf. Instead of calculating an inverse element, p to the k-th power of a multiplying element is calculated as a quasi inverse element. The polynomial expansion of multiplications of elements of GF(p k ) in which the calculation amount is large, is performed by using the quasi inverse elements. High speed is achieved by calculating terms beforehand, in which calculation amount is large, as a common term of a repeated calculation, by performing the polynomial expansion. COPYRIGHT: (C)2007,JPO&INPIT

Abstract: Let R be an Artinian ring (not necessarily with unit element), let Z(R) be its center, and let R° be the group of invertible elements of the ring R with respect to the operation a ∘ b = a + b + ab. We prove that the adjoint group R° is nilpotent and the set Z(R) + R° generates R as a ring if and only if R is the direct sum of finitely many ideals each of which is either a nilpotent ring or a local ring with nilpotent multiplicative group.