On arc index and maximal Thurston-Bennequin number

We discuss the relation between arc index, maximal Thurston--Bennequin number, and Khovanov homology for knots. As a consequence, we calculate the arc index and maximal Thurston--Bennequin number for all knots with at most 11 crossings. For some of these knots, the calculation requires a consideration of cables which also allows us to compute the maximal self-linking number for all knots with at most 11 crossings.Comment: 10 pages, v4: corrected typo

Augmentation of Chinese community relationships (Guanxi) in learning organisations - case study of a private Kunshan (Shanghai) company

The Kunshan factory enjoyed remarkable success over the past 24 months. The quick response, superior teamwork and scouting capability for opportunity and risk on the part of all staff outperformed that of their counterparts in multinational companies. Family ties, community affection, social ‘face’ and personal recommendations co-ordinated company members well. Chinese community relationships had a consolidating and stabilising function in the factory, enabling a convergent corporate movement. This phenomenal practice is explored here to understand quantitatively the roles of Chinese community relationships (guanxi) and learning organisation practice towards corporate performance. In this research, the bricolage method was adopted to cater for the diverse and vibrant nature in the research context. Ethically, this insider research had to ensure that all factory members received fair treatment in the course of the research. Their comments and reflections on learning organisations and Chinese community relationships had to be objectively recorded for examination. An assurance was made that there would be no impact on their job security and access to company caution money remained unchanged during the research, so that they remained in control. Guanxi encourages participation in community activity, at the same time advocating action learning. In research methods, repeated actions, a form of action research, are performed in cycles with input from stakeholders to develop a directed course of further actions. Soft Systems Method, and associated structural tools such as rich pictures and CATWOE, help factory members to dissect the business challenge. Qualitative interviewing was carried out among key managers to devise a master plan for divestment. All employees were recompensed for the disruption caused. New corporate spin-offs were established through the exchange of favours and scouting for opportunities. Later, a focus group was undertaken to evaluate progress in the business units and to examine the contribution of learning organisations and Chinese community relationships. Seven relational outcomes are identified in this research in this Kunshan factory. They are: 1) 360-degree dialogue; 2) group-level exchange of favours; 3) action learning; 4) acculturation; 5) community-prompted foresight capability; 6) personal recommendation; and 7) outside-in mentality. Collectivism on the part of the Chinese community’s participants equalises individual interests through acculturation to establish a common societal and business goal. Sharing a vision and a goal is a pragmatic means of attaining organisational integration and operational excellence. Self-organised vision and mission are attained swiftly through these seven factors to save the effort of initial ice-breaking and team-building activity. A Chinese social system is regarded as a high-context social system – it depends less on precise wording than collective understanding. The Chinese term ‘guanxi’ is used interchangeably in this thesis with ‘Chinese community relationships’ to describe the subtle and delicate interpersonal social interactions perpetuated among ethnic Chinese people in mainland China and overseas in settlements such as Hong Kong, Singapore, Taiwan and elsewhere. Guanxi comprises two aspects. One is a social construct of Chinese society and the other is the value generated by guanxi in action. In combination these position Chinese community relationships as a mediator of organisational transformation such as the implementation of a learning organisation and improved business performance. Historically, guanxi is perceived as having three characteristics: affection (ganqing, in Chinese), reciprocity (renqing) and credit (xinyong). The learning organisation paradigm describes a continuous process of corporate transformation through collective staff learning, aiming to anticipate and cope with the disruptive commercial challenges of today’s business world. When employees are committed to their organisation, it favours business sustainability. Operational and business risks are mitigated through shared vision and mentality. The attributes of individualism and collectivism substantially influence the implementation of a learning organisation. This research uncovers the mediating effect of Chinese community relationships on learning organisation effectiveness for fast business performance through acculturation and personal recommendation. It reveals the social systems in a Chinese community to empower learning organisation practice. The integrational capability of the seven relational outcomes augments guanxi’s mediating effect in empowering learning organisation practice so as to raise corporate performance substantially

Spacetime Foam, Holographic Principle, and Black Hole Quantum Computers

Spacetime foam, also known as quantum foam, has its origin in quantum fluctuations of spacetime. Arguably it is the source of the holographic principle, which severely limits how densely information can be packed in space. Its physics is also intimately linked to that of black holes and computation. In particular, the same underlying physics is shown to govern the computational power of black hole quantum computers.Comment: 8 pages, LaTeX; Talk given by Jack Ng, in celebration of Paul Frampton's 60th birthday, at the Coral Gables Conference (in Fort Lauderdale, Florida on December 17, 2003). To appear in the Proceedings of the 2003 Coral Gables Conferenc

Gravitational Theory with a Dynamical Time

A gravitational theory involving a vector field $\chi^{\mu}$, whose zero component has the properties of a dynamical time, is studied. The variation of the action with respect to $\chi^{\mu}$ gives the covariant conservation of an energy momentum tensor $T^{\mu \nu}_{(\chi)}$. Studying the theory in a background which has killing vectors and killing tensors we find appropriate shift symmetries of the field $\chi^{\mu}$ which lead to conservation laws. The energy momentum that is the source of gravity $T^{\mu \nu}_{(G)}$ is different but related to $T^{\mu \nu}_{(\chi)}$ and the covariant conservation of $T^{\mu \nu}_{(G)}$ determines in general the vector field $\chi^{\mu}$. When $T^{\mu \nu}_{(\chi)}$ is chosen to be proportional to the metric, the theory coincides with the Two Measures Theory, which has been studied before in relation to the Cosmological Constant Problem. When the matter model consists of point particles, or strings, the form of $T^{\mu \nu}_{(G)}$, solutions for $\chi^{\mu}$ are found. For the case of a string gas cosmology, we find that the Milne Universe can be a solution, where the gas of strings does not curve the spacetime since although $T^{\mu \nu}_{(\chi)} \neq 0$, $T^{\mu \nu}_{(G)}= 0$, as a model for the early universe, this solution is also free of the horizon problem. There may be also an application to the "time problem" of quantum cosmology.Comment: 21 pages, discussions extended, some more explicit proofs included, more references include

From computation to black holes and space-time foam

We show that quantum mechanics and general relativity limit the speed $\tilde{\nu}$ of a simple computer (such as a black hole) and its memory space $I$ to \tilde{\nu}^2 I^{-1} \lsim t_P^{-2}, where $t_P$ is the Planck time. We also show that the life-time of a simple clock and its precision are similarly limited. These bounds and the holographic bound originate from the same physics that governs the quantum fluctuations of space-time. We further show that these physical bounds are realized for black holes, yielding the correct Hawking black hole lifetime, and that space-time undergoes much larger quantum fluctuations than conventional wisdom claims -- almost within range of detection with modern gravitational-wave interferometers.Comment: A misidentification of computer speeds is corrected. Our results for black hole computation now agree with those given by S. Lloyd. All other conclusions remain unchange

Twisted Frobenius-Schur indicators for Hopf algebras

The classical Frobenius-Schur indicators for finite groups are character sums defined for any representation and any integer m greater or equal to 2. In the familiar case m=2, the Frobenius-Schur indicator partitions the irreducible representations over the complex numbers into real, complex, and quaternionic representations. In recent years, several generalizations of these invariants have been introduced. Bump and Ginzburg, building on earlier work of Mackey, have defined versions of these indicators which are twisted by an automorphism of the group. In another direction, Linchenko and Montgomery have defined Frobenius-Schur indicators for semisimple Hopf algebras. In this paper, the authors construct twisted Frobenius-Schur indicators for semisimple Hopf algebras; these include all of the above indicators as special cases and have similar properties.Comment: 12 pages. Minor revision