A reason for fusion rules to be even

We show that certain tensor product multiplicities in semisimple braided sovereign tensor categories must be even. The quantity governing this behavior is the Frobenius-Schur indicator. The result applies in particular to the representation categories of large classes of groups, Lie algebras, Hopf algebras and vertex algebras.Comment: 6 pages, LaTe

Infinitesimals without Logic

We introduce the ring of Fermat reals, an extension of the real field containing nilpotent infinitesimals. The construction takes inspiration from Smooth Infinitesimal Analysis (SIA), but provides a powerful theory of actual infinitesimals without any need of a background in mathematical logic. In particular, on the contrary with respect to SIA, which admits models only in intuitionistic logic, the theory of Fermat reals is consistent with classical logic. We face the problem to decide if the product of powers of nilpotent infinitesimals is zero or not, the identity principle for polynomials, the definition and properties of the total order relation. The construction is highly constructive, and every Fermat real admits a clear and order preserving geometrical representation. Using nilpotent infinitesimals, every smooth functions becomes a polynomial because in Taylor's formulas the rest is now zero. Finally, we present several applications to informal classical calculations used in Physics: now all these calculations become rigorous and, at the same time, formally equal to the informal ones. In particular, an interesting rigorous deduction of the wave equation is given, that clarifies how to formalize the approximations tied with Hook's law using this language of nilpotent infinitesimals.Comment: The first part of the preprint is taken directly form arXiv:0907.1872 The second part is new and contains a list of example

The Bloch-Okounkov correlation functions of classical type

Bloch and Okounkov introduced an n-point correlation function on the infinite wedge space and found an elegant closed formula in terms of theta functions. This function has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, etc, and it can also be interpreted as correlation functions on integrable gl_\infty-modules of level one. Such gl_\infty-correlation functions at higher levels were then calculated by Cheng and Wang. In this paper, generalizing the type A results, we formulate and determine the n-point correlation functions in the sense of Bloch-Okounkov on integrable modules over classical Lie subalgebras of gl_\infty of type B,C,D at arbitrary levels. As byproducts, we obtain new q-dimension formulas for integrable modules of type B,C,D and some fermionic type q-identities.Comment: v2, very minor changes, Latex, 41 pages, to appear in Commun. Math. Phy

DBI analysis of generalised permutation branes

We investigate D-branes on the product GxG of two group manifolds described as Wess-Zumino-Novikov-Witten models. When the levels of the two groups coincide, it is well known that there exist permutation D-branes which are twisted by the automorphism exchanging the two factors. When the levels are different, the D-brane charge group demands that there should be generalisations of these permutation D-branes, and a geometric construction for them was proposed in hep-th/0509153. We give further evidence for this proposal by showing that the generalised permutation D-branes satisfy the Dirac-Born-Infeld equations of motion for arbitrary compact, simply connected and simple Lie groups G.Comment: 19 pages, computation in section 3.5.1 corrected, conclusions unchange

Probabilistic theories with purification

We investigate general probabilistic theories in which every mixed state has a purification, unique up to reversible channels on the purifying system. We show that the purification principle is equivalent to the existence of a reversible realization of every physical process, namely that every physical process can be regarded as arising from a reversible interaction of the system with an environment, which is eventually discarded. From the purification principle we also construct an isomorphism between transformations and bipartite states that possesses all structural properties of the Choi-Jamiolkowski isomorphism in quantum mechanics. Such an isomorphism allows one to prove most of the basic features of quantum mechanics, like e.g. existence of pure bipartite states giving perfect correlations in independent experiments, no information without disturbance, no joint discrimination of all pure states, no cloning, teleportation, no programming, no bit commitment, complementarity between correctable channels and deletion channels, characterization of entanglement-breaking channels as measure-and-prepare channels, and others, without resorting to the mathematical framework of Hilbert spaces.Comment: Differing from the journal version, this version includes a table of contents and makes extensive use of boldface type to highlight the contents of the main theorems. It includes a self-contained introduction to the framework of general probabilistic theories and a discussion about the role of causality and local discriminabilit

Resonances in a spring-pendulum: algorithms for equivariant singularity theory

A spring-pendulum in resonance is a time-independent Hamiltonian model system for formal reduction to one degree of freedom, where some symmetry (reversibility) is maintained. The reduction is handled by equivariant singularity theory with a distinguished parameter, yielding an integrable approximation of the Poincaré map. This makes a concise description of certain bifurcations possible. The computation of reparametrizations from normal form to the actual system is performed by Gröbner basis techniques.

Geometric K-Homology of Flat D-Branes

We use the Baum-Douglas construction of K-homology to explicitly describe various aspects of D-branes in Type II superstring theory in the absence of background supergravity form fields. We rigorously derive various stability criteria for states of D-branes and show how standard bound state constructions are naturally realized directly in terms of topological K-cycles. We formulate the mechanism of flux stabilization in terms of the K-homology of non-trivial fibre bundles. Along the way we derive a number of new mathematical results in topological K-homology of independent interest.Comment: 45 pages; v2: References added; v3: Some substantial revision and corrections, main results unchanged but presentation improved, references added; to be published in Communications in Mathematical Physic

Generalized Husimi Functions: Analyticity and Information Content

The analytic properties of a class of generalized Husimi functions are discussed, with particular reference to the problem of state reconstruction. The class consists of the subset of Wodkiewicz's operational probability distributions for which the filter reference state is a squeezed vacuum state. The fact that the function is analytic means that perfectly precise knowledge of its values over any small region of phase space provides enough information to reconstruct the density matrix. If, however, one only has imprecise knowledge of its values, then the amplification of statistical errors which occurs when one attempts to carry out the continuation seriously limits the amount of information which can be extracted. To take account of this fact a distinction is made between explicate, or experimentally accessible information, and information which is only present in implicate, experimentally inaccessible form. It is shown that an explicate description of various aspects of the system can be found localised on various 2 real dimensional surfaces in complexified phase space. In particular, the continuation of the function to the purely imaginary part of complexified phase space provides an explicate description of the Wigner function.Comment: 16 pages, 2 figures, AMS-latex. Replaced with published versio

The carcinogenic potential of tacrolimus ointment beyond immune suppression: a hypothesis creating case report

BACKGROUND: Since tacrolimus ointment was approved by the U.S. Food and Drug Administration (FDA) as a promising treatment for atopic dermatitis, it has been approved in more than 30 additional countries, including numerous European Union member nations. Moreover, in the current clinical routine the use of this drug is no longer restricted to the approved indication, but has been extended to a wide variety of inflammatory skin diseases including some with the potential of malignant transformation. So far, the side-effects reported from the topical use of tacrolimus have been relatively minor (e.g. burning, pruritus, erythema). Recently, however, the FDA reviewed the safety of topical tacrolimus, which resulted in a warning that the use of calcineurin inhibitors may be associated with an increased risk of cancer. CASE PRESENTATION: Oral lichen planus (OLP) was diagnosed in a 56-year-old women in February 1999. After several ineffective local and systemic therapeutic measures an off-label treatment of this recalcitrant condition using Tacrolimus 0.1% ointment was initiated in May 2002. After a few weeks of treatment most of the lesions ameliorated, with the exception of the plaques on the sides of the tongue. Nevertheless, the patient became free of symptoms which, however, reoccurred once tacrolimus was weaned, as a consequence treatment was maintained. In April 2005, the plaques on the left side of the tongue appeared increasingly compact and a biopsy specimen confirmed the suspected diagnosis of an oral squamous cell carcinoma. CONCLUSION: The suspected causal relationship between topical use of tacrolimus and the development of a squamous cell carcinoma prompted us to test the notion that the carcinogenicity of tacrolimus may go beyond mere immune suppression. To this end, tacrolimus has been shown to have an impact on cancer signalling pathways such as the MAPK and the p53 pathway. In the given case, we were able to demonstrate that these pathways had also been altered subsequent to tacrolimus therapy

On the biological relevance of MHC class II and B7 expression by tumour cells in melanoma metastases

A large number of studies have indicated that specific immune reactivity plays a crucial role in the control of malignant melanoma. In this context, expression of MHC I, MHC II and B7 molecules by melanoma cells is seen as relevant for the immune response against the tumour. For a better understanding of the biological relevance of MHC II and B7 expression by tumour cells in metastatic melanoma, we studied the expression of these molecules in melanoma metastases in relation to the inflammatory response, regression of the tumour and survival from 27 patients treated with biochemotherapy (30 mg m−2 Cisplatin and 250 mg m−2 decarbazine (dimethyl-triazene-imidazole-carboxamide, DTIC) on days 1–3 i.v., and 107 IU IFN-α2b 3 days a week s.c., q. 28d). In 19 out of 27 lesions studied, we found expression of MHC II by the tumour cells, while only in one out of 11 tumour biopsies obtained from untreated metastatic melanoma patients, MHC II expression was detected. Expression of B7.1 and B7.2 by tumour cells was found in nine out of 24 and 19 out of 24 lesions, respectively. In all cases where B7.1 expression was found, expression of B7.2 by the tumour cells was also seen. In general, no or only few inflammatory cells positive for B7 were found. Expression of MHC II by tumour cells was positively correlated with the presence of tumour-infiltrating lymphocytes, regression of the lesion, and with time to progression (TTP) and overall survival (OS) of the patient. However, no significant correlation between B7.1 or B7.2 expression and regression of the tumour, TTP or OS was found. In light of other recent findings, these data altogether do support a role as biomarker for MHC II expression by tumour cells; however, its exact immunological pathomechanism(s) remain to be established