Chern–Simons Invariants of Torus Links

We compute the vacuum expectation values of torus knot operators in Chern–Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus knots and links, and we obtain an analogous formula for Kauffman invariants. We also derive a formula for cable knots. We use our results to test a recently proposed conjecture that relates HOMFLY and Kauffman invariants.     

Design and evaluation of an arbitration-free passive optical crossbar for on-chip interconnection networks

With recent advances in silicon nanophotonics, optical crossbars based on CMOS-compatible microring resonators have emerged as viable on-chip optical interconnection networks to deliver high-bandwidth communication at low power dissipation with a small footprint. This paper describes the design, fabrication and evaluation of an arbitration-free passive crossbar based on a microring resonator matrix that can be used to route wavelength division multiplexing (WDM) signals across the chip. The salient feature of the proposed design is the ability to support multicasting and many-to-one communication efficiently (without arbitration), which makes it suitable for implementing cache coherency protocols and on-chip interconnect in future many-core processors.     

Volterra-Type Equations in S + ′

Necessary and sufficient condition for the unique solvability of Volterra integro-differential equation in $S\prime _ + $ is given.     

Synthesis of novel and uniquely shaped 3-azabicyclo[4.2.0]octan-4-one derivatives by sequential Ugi/[2+2] ene-enone photocycloadditions

[structure: see text]. We report a new methodology for the construction of novel and uniquely shaped 3-azabicyclo[4.2.0]octan-4-one derivatives by combining the Ugi multicomponent reaction with [2+2] enone-olefin photochemical transformations. The overall sequence is capable of creating up to five stereocenters; however, in most cases, only two diastereomers are observed.     

Ultradistribution semigroups

We analyze ultradistribution semigroups with not necessarily densely defined generators and the connections of ultradistribution semigroups with local convoluted semigroups. In such a way, we prove a perturbation type theorem for Gevrey’s ultradistribution semigroups.     

Frames for Weighted Shift-invariant Spaces

In this paper we prove the equivalence of the frame property and the closedness for a weighted shift-invariant space $$ V^p_\mu(\Phi) = \left\{\sum \limits^{r}_{i=1} \sum \limits_{j \in \mathbb{Z}^d} c_{i}(j)\phi_{i}(\cdot-j)\left \vert {\{c_{i}(j)\}}_{j \in \mathbb{Z}^{d}} \in {\ell_{\mu}^{p}}\right.\right\}, \quad p \in [1, \infty], $$ which corresponds to ${{\Phi = \Phi^r = (\phi_1, \phi_2, . . . , \phi_r)^T \in (W^{1}_\omega)^r}}$ . We, also, construct a sequence Φ^2k+1 and the sequence of spaces ${{V^{p}_{\mu} (\Phi^{2k+1})}}$ , ${k \in {\mathbb N}}$ , on ${\mathbb R}$ , with the useful properties in sampling, approximations and stability.     

Resolution of the wavefront set via discrete sets

We introduce discrete wave-front sets of sup type and prove that they coincide with the Hörmander wave-front set of a distribution. To that end we recall the notion of admissible lattice pairs and wave-front sets in Fourier-Lebesgue spaces. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Micro-Local Analysis with Fourier Lebesgue Spaces. Part?I

Let ω,ω ₀ be appropriate weight functions and q∈[1,∞]. We introduce the wave-front set, WF_FL^q(w)(f)\mathrm{WF}_{\mathcal{F}L^{q}_{(\omega)}}(f) of f ? S￠f\in \mathcal{S}' with respect to weighted Fourier Lebesgue space FL^q_(w)\mathcal{F}L^{q}_{(\omega )}. We prove that usual mapping properties for pseudo-differential operators Op (a) with symbols a in S^(w₀)_r,0S^{(\omega _{0})}_{\rho ,0} hold for such wave-front sets. Especially we prove that

Synthesis of fused bicyclic imidazoles by sequential van Leusen/ring-closing metathesis reactions

[reaction: see text]. A new strategy employing the van Leusen three-component reaction and the ring-closing metathesis reaction in a sequential fashion to access fused bicyclic imidazole rings is reported. The two-step sequence generated compounds of significant molecular complexity from simple starting materials in an expedient fashion with excellent yields.