On homotopy groups of quandle spaces and the quandle homotopy invariant of links

For a quandle X, the quandle space BX is defined, modifying the rack space of Fenn, Rourke and Sanderson (1995) [13], and the quandle homotopy invariant of links is defined in Z[π₂(BX)], modifying the rack homotopy invariant of Fenn, Rourke and Sanderson (1995) [13]. It is known that the cocycle invariants introduced in Carter et al. (2005) [3], Carter et al. (2003) [5], Carter et al. (2001) [6] can be derived from the quandle homotopy invariant.In this paper, we show that, for a finite quandle X, π₂(BX) is finitely generated, and that, for a connected finite quandle X, π₂(BX) is finite. It follows that the space spanned by cocycle invariants for a finite quandle is finitely generated. Further, we calculate π₂(BX) for some concrete quandles. From the calculation, all cocycle invariants for those quandles are concretely presented. Moreover, we show formulas of the quandle homotopy invariant for connected sum of knots and for the mirror image of links.     

Extensional wave motion in homogenous isotropic thermoelastic plate by using asymptotic method

The present investigation is concerned with the study of extensional wave motion in an infinite homogenous isotropic, thermoelastic plate by using asymptotic method. The governing equation for the extensional wave motions have been derived from the system of three-dimensional dynamical equations of linear coupled theory of thermoelasticity. All coefficients of the differential operator are expressed as explicit functions of the material parameters. The velocity dispersion equation for the extensional wave motion is deduced from the three-dimensional analog of Rayleigh–Lamb frequency equation for thermoelastic plate waves. The approximations for long and short waves and expression for group velocity are also derived. The thermoelastic Rayleigh–Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations whose equivalence is established to that of asymptotic method. The dispersion curves for phase velocity and attenuation coefficient are shown graphically for extensional wave motion of the plates.     

An attribute weight based feedback model for multiple attributive group decision analysis problems with group consensus requirements in evidential reasoning context

In an evidential reasoning context, a group consensus (GC) based approach can model multiple attributive group decision analysis problems with GC requirements. The predefined GC is reached through several rounds of group analysis and discussion (GAD) in the approach. However, the GAD with no guidance may not be the most appropriate way to reach the predefined GC because several rounds of GAD will spend a lot of time of all experts and yet cannot help them to effectively emphasize on the assessments which primarily damage the GC. In this paper, an attribute weight based feedback model is constructed to effectively identify the assessments primarily damaging the GC and accelerate the GC convergence. Considering important attributes with the weights more than or at least equal to the mean of the weights of all attributes, the feedback model constructs identification rules to identify the assessments damaging the GC for the experts to renew. In addition, a suggestion rule is introduced to generate appropriate recommendations for the experts to renew their identified assessments. The identification rules are constructed at three levels including the attribute, alternative and global levels. The feedback model is used to solve an engineering project management software selection problem to demonstrate its detailed implementation process, its validity and applicability, and its advantages compared with the GC based approach.     

Dedekind整环上的不可分解扭模

We give a direct proof to the classification of all indecomposable torsion modules over a Dedekind ring. As an application, we classify all indecomposable locally-nilpotent modules over the polynomial algebra with one variable over a field.     

Design of different bandwidth dispersive mirrors

Dispersive mirrors in the wavelength range of 960–1110 nm, 700–900 nm, 950–1150 nm, 600–900 nm, 900–1200 nm, 600–1000 nm and 550–1050 nm are designed, respectively. From the design results, we analyse the choice of material and the approach to dispersion compensation. Furthermore, along with the increase of bandwidth, interaction between group delay dispersion (GDD) oscillation and high reflectivity, and influence of bandwidth on both GDD ripple and reflectivity are discussed.     

On the orthogonality of the phase velocity and its feasibility for plane waves

Phase and group velocities of plane waves may have different directions in a linear and homogeneous medium. This difference may lead to interesting situations such as the phase reversal propagation and the orthogonal phase velocity. Although the former has been analyzed in depth and it has found many applications, the latter has been left as a hypothesis. Actually, due to our best knowledge there is no evidence of such media where the phase velocity is orthogonal to the group velocity. In this paper we discuss this situation thoroughly. We show that a plane wave with orthogonal phase velocity cannot possess linear momentum. Moreover, we show that a plane wave without the linear momentum cannot propagate at all, so the orthogonality of the phase velocity does not take place in reality for electromagnetic plane waves.     

fac-Acetato-bis(pyrazole) complexes: A systematic study on intra- and intermolecular hydrogen bonds

Acetato-bis(pyrazole) complexes [Mo(η³-methallyl)(O₂CMe)(CO)₂(pz^∗H)₂], (methallyl = CH₂C(CH₃)CH₂) and fac-[M(O₂CMe)(CO)₃(pz^∗H)₂], (pz^∗H = pyrazole or 3,5-dimethylpyrazole, dmpzH; M = Mn, Re) are obtained from [Mo(η³-methallyl)Cl(CO)₂(NCMe)₂] or fac-[MBr(CO)₃(NCMe)₂] [M = Mn (synthesized in situ), Re], 2 equiv. of pyrazole, and 1 equiv. of sodium acetate for Mo complexes, or silver acetate for Mn or Re complexes. The chlorido-complexes [Mo(η³-methallyl)Cl(CO)₂L₂] (L = pzH, dmpzH), obtained from the same starting material by substitution of MeCN by pzH or dmpzH, are also described. The crystal structures of the fac-acetato-bis(dimethylpyrazole) complexes present the same pattern of intramolecular hydrogen bonds between the acetate and the dimetylpyrazole ligands, whereas the crystal structures of the fac-acetato-bis(pyrazole) complexes show different hydrogen bonds patterns, with intermolecular interactions. NMR data indicate that these interactions are not maintained in solution.     

On permutation polytopes

In this paper we lay the foundations for the study of permutation polytopes: the convex hull of a group of permutation matrices.We clarify the relevant notions of equivalence, prove a product theorem, and discuss centrally symmetric permutation polytopes. We provide a number of combinatorial properties of (faces of) permutation polytopes. As an application, we classify ?4-dimensional permutation polytopes and the corresponding permutation groups. Classification results and further examples are made available online.We conclude with several questions suggested by a general finiteness result.     

Vertices, sources and Green correspondents of the simple modules for the large Mathieu groups

We investigate the simple modules for the sporadic simple Mathieu groups M₂₂, M₂₃ and M₂₄ as well as those of the automorphism group, the covering groups and the bicyclic extensions of M₂₂ in characteristics 2 and 3. We determine the vertices and sources as well as the Green correspondents of these simple modules. We also find two 3-blocks with elementary abelian defect groups of order 9 in these groups which are Morita equivalent to their Brauer correspondents.     

Valid inequalities for MIPs and group polyhedra from approximate liftings

In this paper, we present an approximate lifting scheme to derive valid inequalities for general mixed integer programs and for the group problem. This scheme uses superadditive functions as the building block of integer and continuous lifting procedures. It yields a simple derivation of new and known families of cuts that correspond to extreme inequalities for group problems. This new approximate lifting approach is constructive and potentially efficient in computation. J.-P. P. Richard was supported by NSF grant DMI-348611.