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In this paper, the quasitriangular structures of ω-smash coproduct Hopf algebras Bω ⋈ H as constructed by Caenepeel, Ion, Militaru and Zhu were studied. Necessary and sufficient conditions for ω-smash coproduct Hopf algebras to be quasitriangular Hopf algebras are given in terms of properties of their components. As applications of our results, some special cases are discussed. Especially, The quasitriangular structures for D(H)* and H4ω ⋈ kℤ2 are constructed. [ABSTRACT FROM AUTHOR]

The classical Airy function has been generalized by Kontsevich to a function of a matrix argument, which is an integral over the space of skew-hermitian matrices of a unitary-invariant exponential kernel. In this paper, the Kontsevich integral is further generalized to integrals over the Lie algebra of an arbitrary connected compact Lie group, using exponential kernels invariant under the group. The (real) polynomial defining this kernel is said to have the Airy property if the integral defines a function of moderate growth. A very general sufficient criterion for a polynomial to have the Airy property is given. It is shown that an invariant polynomial on the Lie algebra has the Airy property if its restriction to a Cartan subalgebra has the Airy property. This result is used to evaluate these invariant integrals completely and explicitly on the hermitian matrices, obtaining formulae that contain those of Kontsevich as special cases. [ABSTRACT FROM AUTHOR]

In this paper we consider the Kaluza–Klein field equations in the presence of a generic 5D matter tensor which is governed by a conservation equation due to 5D Bianchi identities. Following a previous work, we provide a consistent approach to matter where the problem of huge massive modes is removed, without relaxing the compactification hypotheses; therefore we perform the dimensional reduction either for metric fields and for matter, thus identifying a pure 4D tensor term, a 4D vector term and a scalar one. Hence we are able to write down a consistent set of equations for the complete dynamics of matter and fields; with respect to the pure Einstein–Maxwell system we now have two additional scalar fields: the usual dilaton one plus a scalar source term. Some significant scenarios involving these terms are discussed and perspectives for cosmological applications are suggested. [ABSTRACT FROM AUTHOR]

In this paper, based on a talk given at the Oberwolfach research centre in May 2008 I will describe how biquandles and their big brother, biracks, can be used to differentiate isotopy classes of virtual (and welded) knots and links. [ABSTRACT FROM AUTHOR]

The present paper is an introduction to a combinatorial theory arising as a natural generalization of classical and virtual knot theory. There is a way to encode links by a class of "realizable" graphs. When passing to generic graphs with the same equivalence relations we get "graph-links". On one hand graph-links generalize the notion of virtual link, on the other hand they do not detect link mutations. We define the Jones polynomial for graph-links and prove its invariance. We also prove some a generalization of the Kauffman–Murasugi–Thistlethwaite theorem on "minimal diagrams" for graph-links. [ABSTRACT FROM AUTHOR]

In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Yd,n(u) and the theory of singular braids. The Yokonuma–Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid SBn into the algebra Yd,n(u). Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relation involving singular crossings, which arises from a quadratic relation in the algebra Yd,n(u). [ABSTRACT FROM AUTHOR]

In this paper we prove the following result: for coprime positive integers p and q with p < q, if r is the least positive integer such that 2p-1 and q + r are coprime, then the minimal degree sequence for a torus knot of type (p, q) is the triple (2p - 1, q + r, d) or (q + r, 2p - 1, d) where q + r + 1 ≤ d ≤ 2q - 1. [ABSTRACT FROM AUTHOR]

Table Algebras (A, B) with distinct non-trivial basis elements a, b ∈ B satisfying either $ab = m \overline{a} + n \overline{b}$ or $ab = m \overline{a} + nb$ for some m, n ∈ ℝ+ are studied in this paper. The case $ab = m \overline{a} + nb$ was partially classified by Arad and Blau, and the case $ab = m \overline{a} + n \overline{b}$ was partially classified by Arad and Cohen. [ABSTRACT FROM AUTHOR]

In this paper, we prove an L2-concentration result of Zakharov system in space dimension two, with radial initial data $(u_0,n_0,n_1) \in H^s \times L^2 \times H^{-1} (\frac{16}{17} < s < 1)$, when blow up of the solution happens by I-method. In addition to that, we find a blow up character of this system. Furthermore, we improve the global existence result of Bourgain's to the above-mentioned spaces. [ABSTRACT FROM AUTHOR]

We investigate the following existence problem for rational functions: for a given collection Π of partitions of a number n to define whether there exists a rational function f of degree n for which Π is the branch datum. An important particular case when the answer is known is the one when the collection Π contains a partition consisting of a single element (in this case, the corresponding rational function is equivalent to a polynomial). In this paper, we provide a solution in the case when Π contains a partition consisting of two elements. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a classification for the fundamental groups of positively curved n-manifolds which admit isometric torus Tk-actions with $k > \frac{n}{6}$ and n ≥ 25. [ABSTRACT FROM AUTHOR]

In this paper, we give a sufficient (which is also necessary under a compatibility hypothesis) condition on a set of arrows in the quiver of an algebra A so that A is a split extension of A/M, where M is the ideal of A generated by the classes of these arrows. We also compare the notion of split extension with that of semiconvex extension of algebras. [ABSTRACT FROM AUTHOR]

The classification of nilpotent Lie algebras of maximal rank was solved by Santharoubane. In the present paper, we prove that the classification of non-degenerate solvable Lie algebras of maximal rank can be obtained from the work of Santharoubane. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study an analogue of Hilbert's seventeenth problem for functions over a valued field which are integral definite on some definable set; that is, that map the given set into the valuation ring. We use model theory to exhibit a uniform method, on various theories of valued fields, for deriving an algebraic characterization of such functions. As part of this method we refine the concept of a function being integral at a point, and make it dependent on the relevant class of valued fields. We apply our framework to algebraically closed valued fields, model complete theories of difference and differential valued fields, and real closed valued fields. [ABSTRACT FROM AUTHOR]

Modes are idempotent and entropic algebras. Although it had been established many years ago that groupoid modes embed as subreducts of semimodules over commutative semirings, the general embeddability question remained open until Stronkowski and Stanovský's recent constructions of isolated examples of modes without such an embedding. The current paper now presents a broad class of modes that are not embeddable into semimodules, including structural investigations and an analysis of the lattice of varieties. [ABSTRACT FROM AUTHOR]

During the last years nature inspired intelligent techniques have become attractive for analyzing large data sets and solving complex optimization problems. In this paper, one of the most interesting of them, the Ant Colony Optimization (ACO), is used for the construction of a hybrid algorithmic scheme which effectively handles the Pap Smear Cell classification problem. This algorithmic approach is properly combined with a number of nearest neighbor based approaches for performing the requested classification task, through the solution of the so-called optimal feature subset selection problem. The proposed complete algorithmic scheme is tested in two sets of data. The first one consists of 917 images of pap smear cells and the second set consists of 500 images, classified carefully by expert cyto-technicians and doctors. Each cell is described by 20 numerical features, and the cells fall into seven (7) classes, four (4) representing normal cells and three (3) abnormal cases. Nevertheless, from the medical diagnosis viewpoint, a minimum requirement corresponds to the general two-class problem of correct separation between normal from abnormal cells. [ABSTRACT FROM AUTHOR]

Let L be a free restricted Lie algebra and R a restricted ideal of L. Denote by u(L) the restricted enveloping algebra of L and by ω(L) the associative ideal of u(L) generated by L. The purpose of this paper is to identify the subalgebra R ∩ ωn(L)ω(R) in terms of R only. This problem is the analogue of the Fox problem for free groups. [ABSTRACT FROM AUTHOR]

In this paper we produce a finite algebra which generates a variety with a PSPACE-complete membership problem. We produce another finite algebra with a γ function that grows exponentially. The results are obtained via a modification of a construction of the algebra A(T) that was introduced by McKenzie in 1996. [ABSTRACT FROM AUTHOR]

In this paper, we define the notion of dimension filtration of an Artinian module and study a class of Artinian modules, called sequentially co-Cohen–Macaulay modules, which contains strictly all co-Cohen–Macaulay modules. Some characterizations of co-Cohen–Macaulayness in terms of the Matlis duality and of local homology are also given. [ABSTRACT FROM AUTHOR]

Non-commutative Poisson algebras are the algebras having both an associative algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on $so_n\widetilde{({\Bbb C}_Q)}$ are determined. [ABSTRACT FROM AUTHOR]

In this paper, Kaplansky-type theorems are given to characterize GCD-monoids and valuation monoids. Also, (unique) r-factorable monoids are defined and it is shown that S is a Krull monoid if and only if S is a unique t-factorable (resp., w-factorable) monoid if and only if S is a t-factorable (resp., w-factorable) t-Prüfer monoid. [ABSTRACT FROM AUTHOR]

A Weyl type algebra is defined in the book [4]. A Weyl type non-associative algebra $\overline{WP_{m,n,s}}$ and its restricted subalgebra $\overline{WP_{m,n,s}}_r$ are defined in various papers (see [1, 3, 11, 12]). Several authors find all the derivations of an associative (a Lie, a non-associative) algebra (see [1, 2, 4, 6, 11, 12]). We define the non-associative simple algebra $\overline{WP_{A_{\cdot n}g_n,A_{\cdot m},A_{\cdot s}}}_B$ and the semi-Lie algebra $\overline{WP_{A_{\cdot n}g_n,A_{\cdot m},A_{\cdot s}}}_{B_{[\,,\,]}}$, where $B=\{1,\partial_1,\partial_2,\partial_{12},\partial_{1}^2,\partial_{2}^2\}$. We prove that the algebra is simple and find all its non-associative algebra derivations. [ABSTRACT FROM AUTHOR]

For a quantum Markov semigroup ${\mathcal T}$ on the algebra ${\mathcal B} ({\mathsf h})$ with a faithful invariant state ρ, we can define an adjoint $\widetilde T$ with respect to the scalar product determined by ρ. In this paper, we solve the open problems of characterizing adjoints $\widetilde T$ that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators H, Lk in the Gorini–Kossakowski–Sudarshan–Lindblad representation ${\mathcal L} (x) =i [H, x] -\frac{1}{2} \sum_k (L^*_k L_k x -2L^*_k x L_k +xL^*_k L_k)$ of the generator of ${\mathcal T}$. We study the adjoint semigroup with respect to both scalar products 〈a, b〉 =tr (ρa*b) and 〈a, b〉 =tr (ρ1/2a*ρ1/2 b). [ABSTRACT FROM AUTHOR]

A new evolving network based on the scale-free network of Barabási and Albert (BA) is studied, and the accelerated attachment of new edges is considered in its evolving process. The accelerated attachment is different from the previous accelerated growth of edges and has two particular meanings in this paper. One is that a new vertex with the edges is inserted into the network with acceleration at each time step; the other is that, with a given probability, some additional edges are linked with the vertices in proportion to the number of their obtained edges in the latest evolving periods. The new model describes the cases of those complex networks with a few exceptional vertices. The attachment mechanism of the new adding edges for these vertices does not follow the preferential attachment rule. Comparing with the linear edge growth model, the characteristics of the accelerated growth model are studied theoretically and numerically. We show that the degree distributions of these models have a power law decay and the exponents are larger than that of the BA model. We point out that the characteristics of the exceptional vertices and the aging vertices in an aging network are not identical. The reasons for neglecting this attachment in most of evolving networks are also summarized. [ABSTRACT FROM AUTHOR]

In this paper, we construct special radicals using class pairs of near-rings. We establish necessary conditions for a class pair to be a special radical class. We then define Jacobson-type near-rings and show that in most cases the class of all near-rings of this type is a special radical class. Subsequently, we investigate the relationship between each Jacobson-type near-ring and the corresponding matrix near-ring. [ABSTRACT FROM AUTHOR]

In this paper, we classify the fine representations and R-groups of quasi-split simple groups, and describe the set of good roots with the help of Yan's method. Part of the results are not new, but the method used here is new and efficient. [ABSTRACT FROM AUTHOR]

A semigroup S is called an eventually regular semigroup if for every a ∈ S, there exists a positive integer n such that an is regular. In this paper, the R-unipotent, inverse semigroup and group congruences on an eventually regular semigroup S are described by means of certain congruence pairs (ξ, K), where ξ is a normal congruence on the subsemigroup 〈E(S)〉 generated by E(S), and K is a normal subsemigroup of S. [ABSTRACT FROM AUTHOR]

In this paper, δ-lifting and δ-supplemented modules are defined as generalizations of lifting and supplemented modules. Several properties of these modules are proved. New characterizations of δ-semiperect and δ-perfect rings studied in [9] are obtained using δ-lifting and δ-supplemented modules. [ABSTRACT FROM AUTHOR]

The notion of xst-rings was introduced by García and Marín in 1999. In this paper, we characterize Morita-like equivalences for right xst-rings, obtain the universal theory of Morita equivalences, and prove that two right xst-rings R and T are Morita-like equivalent if and only if there is a Morita context between R and T. We also prove that Morita-like equivalences can be realized by the covariant functors Hom and ⊗ for these rings. [ABSTRACT FROM AUTHOR]

The Held group He discovered by Held [10] is a sporadic simple group of order 4030387200 = 210.33.52.73.17. The group He has 11 conjugacy classes of maximal subgroups as determined by Butler [5] and listed in the 픸핋핃픸핊. Held himself determined much of the local structure of He as well as the conjugacy classes of its elements. Thompson calculated the character table of He. In the present paper, we determine the Fischer–Clifford matrices and hence compute the character table of the non-split extension 3·S7, which is a maximal subgroups of He of index 226560 using the technique of Fischer–Clifford matrices. Most of the computations were carried out with the aid of the computer algebra system 픾픸ℙ. [ABSTRACT FROM AUTHOR]

A temporal flexible planning problem that involves contingent and requirement events can be formulated as a simple temporal network with uncertainty (STNU). An STNU is controllable when there is a strategy for executing the requirement events (or actions) in such a way that all the conditions involving contingent events can be satisfied in all situations. The most interesting and useful controllability property is dynamic controllability in which the remaining actions in an STNU can always be scheduled under all possible feasible durations of future contingent events when all the past contingent events are known. In this paper, we propose and study a novel problem of assigning bounds on the duration of each requirement link in order for the resulting STNU to be dynamically controllable and to minimize the total cost over the allowed durations of all requirement links. We first prove the NP hardness of the problem with a linear cost function. We then formulate the dynamic controllability of an STNU as the constraints in a nonlinear optimization problem. Finally, we present methods for reducing the number of constraints in order to make the problem tractable and to demonstrate the computational performance of our methods. [ABSTRACT FROM AUTHOR]

This paper is devoted to a study on the way generalized substitutions - a multi-dimensional extension of substitutions - act on multi-dimensional Sturmian sequences. We give a sufficient condition under which these multi-dimensional Sturmian sequences are obtained by iterated compositions of generalized substitutions. This condition relies on Brun expansions - a multi-dimensional extension of continued fraction expansions. [ABSTRACT FROM AUTHOR]

Conformal field theory in two dimensions has, over the years, been an extremely useful tool for a variety of physical and mathematical problems. In this paper, perhaps one of the most profound and foundational aspects of the subject is reviewed in detail, namely that of an operator algebra of the theory. This aspect is then demonstrated in some modern applications, first stochastic Loewner evolution and then logarithmic conformal field theory. [ABSTRACT FROM AUTHOR]

Let H be a coalgebra. In this paper, we show that H is a dual quasi-bialgebra if and only if the category ${\cal M}^{H}$ of comodules is a tensor category; and H is a braided dual quasi-bialgebra if and only if ${\cal M}^{H}$ is a braided tensor category. If H is a braided dual quasi-Hopf algebra, it is shown that the antipode of H is inner, i.e., s2(h) = ∑ τ (h1)h2τ-1(h3). [ABSTRACT FROM AUTHOR]

Let K be a complete field with respect to a real non-trivial valuation v, and $\bar{v}$ be the extension of v to an algebraic closure $\overline{K}$ of K. A well-known result of Ostrowski asserts that the limit of a Cauchy sequence of elements of $\overline{K}$ does not always belong to $\overline{K}$ unless $\overline{K}$ is a finite extension of K. In this paper, it is shown that when a Cauchy sequence { bn } of elements of $\overline{K}$ is such that the sequence { [K(bn) : K] } of degrees of the extensions K(bn) / K does not tend to infinity as n approaches infinity, then { bn } has a limit in $\overline{K}$. We also give a characterization of those Cauchy sequences { bn } of elements of $\overline{K}$ whose limit is not in $\overline{K}$, which generalizes a result of Alexandru, Popescu and Zaharescu. [ABSTRACT FROM AUTHOR]

This paper surveys, and in some cases generalizes, many of the recent results on homomorphisms and the higher Ext groups for q-Schur algebras and for the Hecke algebra of type A. We review various results giving isomorphisms between Ext groups in the two categories, and discuss those cases where explicit results have been determined. [ABSTRACT FROM AUTHOR]

In this paper a *-algebra of regular functions on the Shilov boundary S(픻) of bounded symmetric domain 픻 is constructed. The algebras of regular functions on S(픻) are described in terms of generators and relations for two particular series of bounded symmetric domains. Also, the degenerate principal series of quantum Harish–Chandra modules related to S(픻) = Un is investigated. [ABSTRACT FROM AUTHOR]

In this paper we find a description, up to equivalence, of all irreducible representations for a class of algebras generated by four linearly related idempotents. [ABSTRACT FROM AUTHOR]

Lie bialgebras occur as the principal objects in the infinitesimalization of the theory of quantum groups — the semi-classical theory. Their relationship with the quantum theory has made available some new tools that we can apply to classical questions. In this paper, we study the simple complex Lie algebras using the double-bosonization construction of Majid. This construction expresses algebraically the induction process given by adding and removing nodes in Dynkin diagrams, which we call Lie induction. We first analyze the deletion of nodes, corresponding to the restriction of adjoint representations to subalgebras. This uses a natural grading associated to each node. We give explicit calculations of the module and algebra structures in the case of the deletion of a single node from the Dynkin diagram for a simple Lie (bi-)algebra. We next consider the inverse process, namely that of adding nodes, and give some necessary conditions for the simplicity of the induced algebra. Finally, we apply these to the exceptional series of simple Lie algebras, in the context of finding obstructions to the existence of finite-dimensional simple complex algebras of types E9, F5 and G3. In particular, our methods give a new point of view on why there cannot exist such an algebra of type E9. [ABSTRACT FROM AUTHOR]

Let E be a row finite directed graph with no sinks and (XE, σE) the one-sided edge shift space. Then the graph C*-algebra C*(E) contains the commutative algebra C0(XE). Moreover if E is locally finite so that the canonical completely positive map ϕE on C*(E) is well-defined, ϕE|C0(XE) coincides with the *-homomorphism $\sigma_E^*$. In this paper we first show that if two edge shift spaces (XE, σE) and (XF, σF) are topologically conjugate, there is an isomorphism of C*(E) onto C*(F), and if the graphs are locally finite the isomorphism transforms ϕE|C0(XE) onto ϕF|C0(XF), which has been known for Cuntz–Krieger algebras. Let ht(ϕE) be Voiculescu–Brown topological entropy of ϕE. In case E is finite, it is well-known that the values ht(ϕE), $ht(\phi_E|_{\mathcal A_E})$, hl(E) and hb(E) all coincide, where $\mathcal A_E$ is the AF core of C*(E) and hl(E), hb(E) are the loop, block entropies of E respectively. If E is irreducible and infinite, $h_l(E)\leq ht(\phi_E|_{\mathcal A_E})$ has been known recently, and here we show that $ht(\phi_E|_{\mathcal A_E})\leq h_b (E^t)$, where Et is the transposed graph of E. Also some dynamical systems related with AF subalgebras $\mathcal A_E(v)$ of $\mathcal A_E$ are examined to prove that $h_l(E)\leq ht(\phi_E|_{\mathcal A_E(v)})$ for each vertex v. [ABSTRACT FROM AUTHOR]

We consider a discrete-time multi-server queue for which the moments of the stationary queue length can be expressed in terms of series over the zeros in the closed unit disk of a queue-specific characteristic function. In many important cases these zeros can be considered to be located on a queue-specific curve, called generalized Szegö curve. By adopting a special parametrization of these Szegö curves, the relevant zeros occur as equidistant samples of a 2π-periodic function whose Fourier coefficients can be determined analytically. Thus the series occurring in the expressions for the moments can be written as Fourier aliasing series with terms given in analytic form. This gives rise to formulas for e.g. the mean and variance of the queue length that are reminiscent of Spitzer's identity for the moment generating function of the steady-state waiting time for a G/G/1 queue. Indeed, by considering the queue under investigation as a G/G/1 queue, the new formulas for the mean and variance also follow from Spitzer's identity. The approach in this paper can also be used to compute the probability distribution function of the queue length in analytic form. [ABSTRACT FROM AUTHOR]

The period-1 motion of a piecewise-linear system under a periodic excitation is predicted analytically through the Poincaré mapping and the corresponding mapping sections formed by the switch planes pertaining to the two constraints. The mapping relationship generates a set of nonlinear algebraic equations from which the period-1 motion is determined analytically. The stability and bifurcation of the period-1 motion are determined, and numerical simulations are carried out for confirmation of the analytical prediction of period-1 motion. An unsymmetrical stable period-1 motion is observed. This investigation helps us understand the dynamical behavior of period-1 motion in the piecewise-linear system and more efficiently obtain other periodic motions and chaos through numerical simulations. The similar methodology presented in this paper can be used for other nonsmooth dynamical systems. [ABSTRACT FROM AUTHOR]

The main result of this paper is the following: Let A be a self-injective Nakayama K-algebra, which is basic and connected. Suppose that A is a right Ae-module of τAe-period 1. (1) If A is an algebra whose Jacobson radical square is zero, then ${\Bbb A}(\tau_{A^{e}};A)\cong K[Z]$. (2) If A is an algebra whose Jacobson radical square is not zero, then ${\Bbb A}(\tau_{A^{e}};A)\cong K[X,Z]/(X^{n})$ for some positive integer n. [ABSTRACT FROM AUTHOR]

This paper presents a fast algorithm for solving the n-queens problem. The basic idea of this algorithm is to use pre-computed solutions in 75% of the cases, while the remaining cases are solved by calling the Sosic's algorithm. The novelty of this algorithm is in the observation that these pre-computable cases exhibit a modular nature. In addition, the pre-computed solutions run 100 times faster than Sosic's algorithm in most cases. [ABSTRACT FROM AUTHOR]

In this paper, we study triples of the form (E, θ, ϕ) over a compact Riemann surface, where (E, θ) is a Higgs bundle and ϕ is a global holomorphic section of the Higgs bundle. Our main result is an description of a birational equivalence which relates geometrically the moduli space of Higgs bundles of rank r and degree d to the moduli space of Higgs bundles of rank r-1 and degree d. [ABSTRACT FROM AUTHOR]

Let R be a ring, M a right R-module and a hereditary torsion theory in Mod-R with associated torsion functor τ for the ring R. Then M is called τ-supplemented when for every submodule N of M there exists a direct summand K of M such that K ≤ N and N/K is τ-torsion module. In [4], M is called almost τ-torsion if every proper submodule of M is τ-torsion. We present here some properties of these classes of modules and look for answers to the following questions posed by the referee of the paper [4]: (1) Let a module M = M′ ⊕ M″ be a direct sum of a semisimple module M′ and τ-supplemented module M″. Is M τ-supplemented? (2) Can one find a non-stable hereditary torsion theory τ and τ-supplemented modules M′ and M″ such that M′ ⊕ M″ is not τ-supplemented? (3) Can one find a stable hereditary torsion theory τ and a τ-supplemented module M such that M/N is not τ-supplemented for some submodule N of M? (4) Let τ be a non-stable hereditary torsion theory and the module M be a finite direct sum of almost τ-torsion submodules. Is M τ-supplemented? (5) Do you know an example of a torsion theory τ and a τ-supplemented module M with τ-torsion submodule τ(M) such that M/τ(M) is not semisimple? [ABSTRACT FROM AUTHOR]

Majid and Bespalov obtained a braided interpretation of Radford's theorem about Hopf algebras with projection. In this paper, we extend these results and study relations between the antipodes of the Hopf algebras that appear in this situation. [ABSTRACT FROM AUTHOR]

In this paper we present duality theory for compact groups in the case when the C&ast;-algebra A, the fixed point algebra of the corresponding Hilbert C&ast;-system (F; G), has a nontrivial center Z ⊃ C1 and the relative commutant satisfies the minimality condition A′ ∩ F = Z, as well as a technical condition called regularity. The abstract characterization of the mentioned Hilbert C&ast;-system is expressed by means of an inclusion of C&ast;-categories TC < T , where TC is a suitable DR-category and T a full subcategory of the category of endomorphisms of A. Both categories have the same objects and the arrows of T can be generated from the arrows of TC and the center Z. A crucial new element that appears in the present analysis is an abelian group C(G), which we call the chain group of G, and that can be constructed from certain equivalence relation defined on Ĝ, the dual object of G. The chain group, which is isomorphic to the character group of the center of G, determines the action of irreducible endomorphisms of A when restricted to Z. Moreover, C(G) encodes the possibility of defining a symmetry ∈ also for the larger category T of the previous inclusion. [ABSTRACT FROM AUTHOR]

Most of the work in the field of topology optimization is concentrated on using sensitivity analysis and optimality criteria methods that need explicit formulation. The design systems are often hard-coded for a specific problem with specialized optimization and FEM routines. This paper presents a work that uses a system approach to solid free form design. It attempts to develop a general topology optimization system that has a wide range of applicability by making use of sophisticated optimization and FEM packages available. A computer design system is implemented with an integration of commercial codes CFSQP and NASTRAN. A pre-processor and a post-processor are developed to assist the optimal design process. The system is tested with benchmark cases for minimum mean compliance and minimum weight designs. The results for the cases are presented, demonstrating the ability of the system to handle complex cases with practical feasibility. The implementation is evaluated with a parametric study of its performance. The key factors for the common problems of topology optimization are examined, including the mesh dependency and numerical instability. The computational efficiency is further studied to indicate the direction for further improvement of the system. [ABSTRACT FROM AUTHOR]

For a given maximal subgroup M of a finite group G, a subgroup C of G is said to be a completion for M in G if C is not contained in M while every G-invariant proper subgroup of C is contained in M. A completion C of M is called maximal if M has no any completion which properly contains C. In this paper, the authors weaken the condition of maximal completions by defining s-completions. Using s-completions, the authors obtain some new characterizations of the solvable and supersolvable groups. [ABSTRACT FROM AUTHOR]