Left cells in the weyl group of type 7

By applying an algorithm, we get a representative set of the left cells and all the left cell graphs for the Weyl group W of type E ₇,. From this, we see that the generalized r-invariants characterize all the non-exceptional left cells of W. We give a criterion to check the relation [Ltilde] on the elements in the exceptional left cells of W.     

Efficient preconditioning of linear systems arising from the discretization of hyperbolic conservation laws

In this paper, we describe a novel formulation of a preconditioned BiCGSTAB algorithm for the solution of ill-conditioned linear systems Ax=b. The developed extension enables the control of the residual r _m =b–Ax _m of the approximate solution x _m independent of the specific left, right or two-sided preconditioning technique considered. Thereby, the presented modification does not require any additional computational effort and can be introduced directly into existing computer codes. Furthermore, the proceeding is not restricted to the BiCGSTAB method, hence the strategy can serve as a guideline to extend similar Krylov sub-space methods in the same manner. Based on the presented algorithm, we study the behavior of different preconditioning techniques. We introduce a new physically motivated approach within an implicit finite volume scheme for the system of the Euler equations of gas dynamics which is a typical representative of hyperbolic conservation laws. Thereupon a great variety of realistic flow problems are considered in order to give reliable statements concerning the efficiency and performance of modern preconditioning techniques.     

Loop subgroups of <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">r</Emphasis></Subscript> and the image of their stabilizer subgroups in GL<Subscript><Emphasis Type="Italic">r</Emphasis></Subscript>(ℤ)

For a representative class of subgroups of F _r , the image of their stabilizer subgroup under the action of Aut(F _r ) in GL _r (ℤ) is calculated.     

Force distribution for double-walled carbon nanotubes and gigahertz oscillators

Advances in nanotechnology have led to the creation of many nano-scale devices and carbon nanotubes are representative materials to construct these devices. Double-walled carbon nanotubes with the inner tube oscillating can be used as gigahertz oscillators and form the basis of possible nano-electronic devices that might be instrumental in the micro-computer industry which are predominantly based on electron transport phenomena. There are many experiments and molecular dynamical simulations which show that a wave is generated on the outer cylinder as a result of the oscillation of the inner carbon nanotube and that the frequency of this wave is also in the gigahertz range. As a preliminary to analyze and model such devices, it is necessary to estimate accurately the resultant force distribution due to the inter-atomic interactions. Here we determine some new analytical expressions for the van der Waals force using the Lennard–Jones potential for general lengths of the inner and outer tubes. These expressions are utilized together with Newton’s second law to determine the motion of an oscillating inner tube, assuming that any frictional effects may be neglected. An idealized and much simplified representation of the Lennard–Jones force is used to determine a simple formula for the oscillation frequency resulting from an initial extrusion of the inner tube. This simple formula is entirely consistent with the existing known behavior of the frequency and predicts a maximum oscillation frequency occurring when the extrusion length is (L ₂ – L ₁)/2 where L ₁ and L ₂ are the respective half-lengths of the inner and outer tubes (L ₁ < L ₂).     

Force distribution for double-walled carbon nanotubes and gigahertz oscillators

Advances in nanotechnology have led to the creation of many nano-scale devices and carbon nanotubes are representative materials to construct these devices. Double-walled carbon nanotubes with the inner tube oscillating can be used as gigahertz oscillators and form the basis of possible nano-electronic devices that might be instrumental in the micro-computer industry which are predominantly based on electron transport phenomena. There are many experiments and molecular dynamical simulations which show that a wave is generated on the outer cylinder as a result of the oscillation of the inner carbon nanotube and that the frequency of this wave is also in the gigahertz range. As a preliminary to analyze and model such devices, it is necessary to estimate accurately the resultant force distribution due to the inter-atomic interactions. Here we determine some new analytical expressions for the van der Waals force using the Lennard–Jones potential for general lengths of the inner and outer tubes. These expressions are utilized together with Newton’s second law to determine the motion of an oscillating inner tube, assuming that any frictional effects may be neglected. An idealized and much simplified representation of the Lennard–Jones force is used to determine a simple formula for the oscillation frequency resulting from an initial extrusion of the inner tube. This simple formula is entirely consistent with the existing known behavior of the frequency and predicts a maximum oscillation frequency occurring when the extrusion length is (L ₂ – L ₁)/2 where L ₁ and L ₂ are the respective half-lengths of the inner and outer tubes (L ₁ < L ₂).     

A minimum length covering subgraph of a network

This paper concerns the problem of locating a central facility on a connected networkN. The network,N, could be representative of a transport system, while the central facility takes the form of a connected subgraph ofN. The problem is to locate a central facility of minimum length so that each of several demand points onN is covered by the central facility: a demand point atv _i inN is covered by the central facility if the shortest path distance betweenv _i and the closest point in the central facility does not exceed a parameterr _i . This location problem is NP-hard, but for certain special cases, efficient solution methods are available.     

(Z2)k-Actions with fixed point set of constant codimension 2k + 81

The ideal J _*,k/2 ^k+8 of cobordism classes in the unoriented cobordism ring MO* containing a representative admitting a (Z ₂) ^k -action with fixed point set of constant codimension 2 ^k + 8 is determined for k ≥ 4.     

Markov classes in certain finite quotients of artin’s braid group

This paper studies three finite quotients of the sequence of braid groups {B _n;n = 1,2,…}. Each has the property that Markov classes in {ie160-1} = ∐B _n pass to well-defined equivalence classes in the quotient. We are able to solve the Markov problem in two of the quotients, obtaining canonical representatives for Markov classes and giving a procedure for reducing an arbitrary representative to the canonical one. The results are interpreted geometrically, and related to link invariants of the associated links and the value of the Jones polynomial on the corresponding classes. This material is based upon work partially supported by the National Science Foundation under Grant No. DMS-8503758.     

Commuting Involutions with Fixed Point Set of Variable Codimension

Special generators of the unoriented cobordism ring MO _* are constructed to determine some groups of cobordism classes in MO _n containing a representative M ⁿ admitting a (Z ₂) ^k -action with the fixed point set of (n−l _i )-dimensional submanifolds of M ⁿ . This work is supported by HNSF and BSHJJ     

Robust Stability and Stabilization of a Class of Nonlinear Switched Discrete-Time Systems with Time-Varying Delays

In this paper, we investigate the problems of robust delay-dependent ℒ₂ gain analysis and feedback control synthesis for a class of nominally-linear switched discrete-time systems with time-varying delays, bounded nonlinearities and real convex bounded parametric uncertainties in all system matrices under arbitrary switching sequences. We develop new criteria for such class of switched systems based on the constructive use of an appropriate switched Lyapunov-Krasovskii functional coupled with Finsler’s Lemma and a free-weighting parameter matrix. We establish an LMI characterization of delay-dependent conditions under which the nonlinear switched delay system is robustly asymptotically stable with an ℒ₂-gain smaller than a prescribed constant level. Switched feedback schemes, based on state measurements, output measurements or by using dynamic output feedback, are designed to guarantee that the corresponding switched closed-loop system enjoys the delay-dependent asymptotic stability with an ℒ₂ gain smaller than a prescribed constant level. All the developed results are expressed in terms of convex optimization over LMIs and tested on representative examples.     

Decompositions of graphs into trees

Let θ be a family of graphs. By a θ-decomposition of a graph G we mean a partition λ of the edge set of G such that every F ? π spans in G a subgraph isomorphic to a graph in θ. In this paper we state the following conjecture: If T₁ and T₂ are two trees having relatively prime sizes then there exists c = c(T₁ T₂) such that every graph G satisfying the condition δ(G) ? c has a {T₁, T₂}-decom-position. We prove this conjecture for some special pairs of trees. In particular, we prove it in the following cases: (i) T₁ and T₂ are stars having relatively prime sizes; (ii) T₁ and T₂ are paths having relatively prime sizes; and. (iii) T₁ = T₂ - {v}, where v is a terminal vertex in T ₂.     

Necessary conditions for Schur-positivity

In recent years, there has been considerable interest in showing that certain conditions on skew shapes A and B are sufficient for the difference s _A −s _B of their skew Schur functions to be Schur-positive. We determine necessary conditions for the difference to be Schur-positive. Specifically, we prove that if s _A −s _B is Schur-positive, then certain row overlap partitions for A are dominated by those for B. In fact, our necessary conditions require a weaker condition than the Schur-positivity of s _A −s _B ; we require only that, when expanded in terms of Schur functions, the support of s _A contains that of s _B . In addition, we show that the row overlap condition is equivalent to a column overlap condition and to a condition on counts of rectangles fitting inside A and B. Our necessary conditions are motivated by those of Reiner, Shaw and van Willigenburg that are necessary for s _A =s _B , and we deduce a strengthening of their result as a special case.     

Geometric fit of a point set by generalized circles

In our paper we approximate a set of given points by a general circle. More precisely, given two norms k ₁ and k ₂ and a set of points in the plane, we consider the problem of locating and scaling the unit circle of norm k ₁ such that the sum of weighted distances between the circumference of the circle and the given points is minimized, where the distance is measured by a norm k ₂. We present results for the general case. In the case that k ₁ and k ₂ are both polyhedral norms, we are able to solve the problem by investigating a finite candidate set.     

Decentralized Reliable Control of Interconnected Time-Delay Systems Against Sensor Failures

In this paper, we study the problem of designing decentralized reliable feedback control methods under a class of control failures for a class of linear interconnected continuous-time systems having internal subsystem time-delays and additional time-delay couplings. These failures are described by a model that takes into consideration possible outages or partial failures in every single actuator of each decentralized controller. The decentralized control design is performed through two steps. First, a decentralized stabilizing reliable feedback control set is derived at the subsystem level through the construction of appropriate Lyapunov-Krasovskii functional and, second, a feasible linear matrix inequalities procedure is then established for the effective construction of the control set under different feedback schemes. Two schemes are considered: the first is based on state-measurement and the second utilizes static output-feedback. The decentralized feedback gains in both schemes are determined by convex optimization over linear matrix inequalities. We characterize decentralized linear matrix inequality-based feasibility conditions such that every local closed-loop subsystem of the linear interconnected delay system is delay-dependent robustly asymptotically stable with an γ-level ℒ₂-gain. The developed results are tested on a representative example.     

Construct irreducible representations of quantum groups U q (ƒ m (K))

In this paper, we construct families of irreducible representations for a class of quantum groups U _q (ƒ _m (K)). First, we give a natural construction of irreducible weight representations for U _q (ƒ _m (K)) using methods in spectral theory developed by Rosenberg. Second, we study the Whittaker model for the center of U _q (ƒ _m (K)). As a result, the structure of Whittaker representations is determined, and all irreducible Whittaker representations are explicitly constructed. Finally, we prove that the annihilator of a Whittaker representation is centrally generated.      

F_2+vF_2上的循环码

本文研究了环F_2+vF_2上的循环码.利用标准形生成元集刻画了环F_2+vF_2上的循环码的代数结构,证明了环F_2+vF_2上的每一个非零的循环码均有唯一的标准形生成元集,进而得到了每一个循环码均是由一个多项式生成的.     

On the existence of unitarily invariant norm under some conditions

Let r ₁, …, r _m be positive real numbers and A ₁, …, A _m be n × n matrices with complex entries. In this article, we present a necessary and sufficient condition for the existence of a unitarily invariant norm ‖·‖, such that ‖A _i ‖ = r _i , for i = 1, …, m. Then we identify the greatest unitarily invariant norm which satisfies this condition. Using this, we get an approximation of unitarily invariant norms. Although the minimum unitarily invariant norm which satisfies this condition does not exist in general, we find conditions over A _i s and r _i s which are sufficient for the existence of such a norm. Finally, we get a characterization of unitarily invariant norms.     

The complexity of the Specht modules corresponding to hook partitions

We show that the complexity of the Specht module corresponding to any hook partition is the p-weight of the partition. We calculate the variety and the complexity of the signed permutation modules. Let E _s be a representative of the conjugacy class containing an elementary abelian p-subgroup of a symmetric group generated by s disjoint p-cycles. We give formulae for the generic Jordan types of signed permutation modules restricted to E _s and of Specht modules corresponding to hook partitions μ restricted to E _s where s is the p-weight of μ.      

A lattice-theoretic approach to arbitrary real functions on frames

In this paper we model discontinuous extended real functions in pointfree topology following a lattice-theoretic approach, in such a way that, if L is a subfit frame, arbitrary extended real functions on L are the elements of the Dedekind-MacNeille completion of the poset of all extended semicontinuous functions on L. This approach mimicks the situation one has with a T₁-space X, where the lattice F?(X) of arbitrary extended real functions on X is the smallest complete lattice containing both extended upper and lower semicontinuous functions on X. Then, we identify real-valued functions by lattice-theoretic means. By construction, we obtain definitions of discontinuous functions that are conservative for T₁-spaces. We also analyze semicontinuity and introduce definitions which are conservative for T₀-spaces.