Legendrian contact homology in R3\mathbb{R}^3

This is an introduction to Legendrian contact homology and the Chekanov-Eliashberg differential graded algebra, with a focus on the setting of Legendrian knots in $\mathbb{R}^3$.Comment: v3: 59 pages, 27 figures; introduction rewritten, sections 5 and 6 switched, many small revision

Legendrian and transverse twist knots

In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the $m(5_2)$ knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least $n$ different Legendrian representatives with maximal Thurston--Bennequin number of the twist knot $K_{-2n}$ with crossing number $2n+1$. In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that $K_{-2n}$ has exactly $\lceil\frac{n^2}2\rceil$ Legendrian representatives with maximal Thurston--Bennequin number, and $\lceil\frac{n}{2}\rceil$ transverse representatives with maximal self-linking number. Our techniques include convex surface theory, Legendrian ruling invariants, and Heegaard Floer homology.Comment: 27 pages, v3: added figure, other minor changes, to appear in JEM

Critique of proposed limit to space--time measurement, based on Wigner's clocks and mirrors

Based on a relation between inertial time intervals and the Riemannian curvature, we show that space--time uncertainty derived by Ng and van Dam implies absurd uncertainties of the Riemannian curvature.Comment: 5 pages, LaTex, field "Author:" correcte

Turbo Detection of Space-time Trellis-Coded Constant Bit Rate Vector-Quantised Videophone System using Reversible Variable-Length Codes, Convolutional Codes and Turbo Codes

In this treatise we characterise the achievable performance of a proprietary video transmission system, which employs a Constant Bit Rate (CBR) video codec that is concatenated with one of three error correction codecs, namely a Reversible Variable-Length Code (RVLC), a Convolutional Code (CC) or a convolutional-based Turbo Code (TC). In our investigations, the CBR video codec was invoked in conjunction with Space-Time Trellis Coding (STTC) designed for transmission over a dispersive Rayleigh fading channel. At the receiver, the channel equaliser, the STTC decoder and the RVLC, CC or TC decoder, as appropriate, employ the Max-Log Maximum A-Posteriori (MAP) algorithm and their operations are performed in an iterative 'turbo-detection' fashion. The systems were designed for maintaining similar error-free video reconstruction qualities, which were found to be subjectively pleasing at a Peak Signal to Noise Ratio (PSNR) of 30.6~dB, at a similar decoding complexity per decoding iteration. These design criteria were achieved by employing differing transmission rates, with the CC- and TC-based systems having a 22% higher bandwidth requirement. The results demonstrated that the TC-, RVLC- and CC-based systems achieve acceptable subjective reconstructed video quality associated with an average PSNR in excess of 30~dB for $E_b/N_0$ values above 4.6~dB, 6.4~dB and 7.7~dB, respectively. The design choice between the TC- and RVLC-based systems constitutes a trade-off between the increased error resilience of the TC-based scheme and the reduced bandwidth requirement of the RVLC-based scheme

Invariants of Legendrian Knots and Coherent Orientations

We provide a translation between Chekanov's combinatorial theory for invariants of Legendrian knots in the standard contact R^3 and a relative version of Eliashberg and Hofer's Contact Homology. We use this translation to transport the idea of ``coherent orientations'' from the Contact Homology world to Chekanov's combinatorial setting. As a result, we obtain a lifting of Chekanov's differential graded algebra invariant to an algebra over Z[t,t^{-1}] with a full Z grading.Comment: 32 pages, 17 figures; small technical corrections to proof of Thm 3.7 and example 4.

Critical Fidelity

Using a Wigner Lorentzian Random Matrix ensemble, we study the fidelity, $F(t)$, of systems at the Anderson metal-insulator transition, subject to small perturbations that preserve the criticality. We find that there are three decay regimes as perturbation strength increases: the first two are associated with a gaussian and an exponential decay respectively and can be described using Linear Response Theory. For stronger perturbations $F(t)$ decays algebraically as $F(t)\sim t^{-D_2}$, where $D_2$ is the correlation dimension of the critical eigenstates.Comment: 4 pages, 3 figures. Revised and published in Phys. Rev. Let

Hegemonic conceptualizations of empowerment in entrepreneurship and their suitability for collective contexts

The relationship between empowerment and entrepreneurship in collective societies is, in our view, insufficiently examined. Accepted definitions of empowerment and the assumptions underlying programs and research designs based on them result in outcomes that self-fulfil and, as a result, disappoint. Several issues are prevalent: the empowerment potential of programs is overestimated and the dominant view of what constitutes an ‘empowered self’ does not go deep enough to explore, and reframe, the self and its relationship to agency—two issues at the core of empowerment definitions and formulations. In this conceptual article, we examine the entrepreneurship and empowerment literature to suggest ways forward for the future health and relevance of the subject area. We highlight a serious methodological and perceptual issue within the literature, which offers many opportunities for theory development in the field

Charged black holes in Vaidya backgrounds: Hawking's Radiation

In this paper we propose a class of embedded solutions of Einstein's field equations describing non-rotating Reissner-Nordstrom-Vaidya and rotating Kerr-Newman-Vaidya black holes.Comment: 30 pages, latex file, no figure

Disjoint difference families and their applications

Difference sets and their generalisations to difference families arise from the study of designs and many other applications. Here we give a brief survey of some of these applications, noting in particular the diverse definitions of difference families and the variations in priorities in constructions. We propose a definition of disjoint difference families that encompasses these variations and allows a comparison of the similarities and disparities. We then focus on two constructions of disjoint difference families arising from frequency hopping sequences and showed that they are in fact the same. We conclude with a discussion of the notion of equivalence for frequency hopping sequences and for disjoint difference families