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

Relative CC"-Numerical Ranges for Applications in Quantum Control and Quantum Information

Motivated by applications in quantum information and quantum control, a new type of $C$"-numerical range, the relative $C$"-numerical range denoted $W_K(C,A)$, is introduced. It arises upon replacing the unitary group U(N) in the definition of the classical $C$"-numerical range by any of its compact and connected subgroups $K \subset U(N)$. The geometric properties of the relative $C$"-numerical range are analysed in detail. Counterexamples prove its geometry is more intricate than in the classical case: e.g. $W_K(C,A)$ is neither star-shaped nor simply-connected. Yet, a well-known result on the rotational symmetry of the classical $C$"-numerical range extends to $W_K(C,A)$, as shown by a new approach based on Lie theory. Furthermore, we concentrate on the subgroup $SU_{\rm loc}(2^n) := SU(2)\otimes ... \otimes SU(2)$, i.e. the $n$-fold tensor product of SU(2), which is of particular interest in applications. In this case, sufficient conditions are derived for $W_{K}(C,A)$ being a circular disc centered at origin of the complex plane. Finally, the previous results are illustrated in detail for $SU(2) \otimes SU(2)$.Comment: accompanying paper to math-ph/070103

Localized Exotic Smoothness

Gompf's end-sum techniques are used to establish the existence of an infinity of non-diffeomorphic manifolds, all having the same trivial ${\bf R^4}$ topology, but for which the exotic differentiable structure is confined to a region which is spatially limited. Thus, the smoothness is standard outside of a region which is topologically (but not smoothly) ${\bf B^3}\times {\bf R^1}$, where ${\bf B^3}$ is the compact three ball. The exterior of this region is diffeomorphic to standard ${\bf R^1}\times {\bf S^2}\times{\bf R^1}$. In a space-time diagram, the confined exoticness sweeps out a world tube which, it is conjectured, might act as a source for certain non-standard solutions to the Einstein equations. It is shown that smooth Lorentz signature metrics can be globally continued from ones given on appropriately defined regions, including the exterior (standard) region. Similar constructs are provided for the topology, ${\bf S^2}\times {\bf R^2}$ of the Kruskal form of the Schwarzschild solution. This leads to conjectures on the existence of Einstein metrics which are externally identical to standard black hole ones, but none of which can be globally diffeomorphic to such standard objects. Certain aspects of the Cauchy problem are also discussed in terms of ${\bf R^4_\Theta}$\models which are ``half-standard'', say for all $t<0,$ but for which $t$ cannot be globally smooth.Comment: 8 pages plus 6 figures, available on request, IASSNS-HEP-94/2

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.

Tamoxifen as a single agent for advanced melanoma in postmenopausal women. A phase II study of the EORTC malignant melanoma cooperative group

Between 1983 and 1989 a phase II study was carried out by the EORTC Malignant Melanoma Cooperative Group in which postmenopausal women with advanced melanoma but a good performance status received tamoxifen, 40 mg per day, as a single agent. From 12 centres, 114 patients were registered of whom 107 appeared to be eligible and 102 evaluable. Seven died of progressive disease within 4 weeks, eight others had early progressive disease (of whom seven died within 7 weeks). The response rate was 4.9%, the complete response rate 1%. Without prior chemotherapy three of 58 responded, with prior chemotherapy two of 44. Except for one of 46 patients with lung metastases who experienced a PR of these metastases, all responders were patients with slowly growing small soft tissue metastases. We conclude that, because of the few side effects, tamoxifen can be recommended for (postmenopausal) patients who have only a small number of slowly growing metastases and who are not yet candidates for treatment with toxic drugs. © 1992 Rapid Communications of Oxford Ltd.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Identification of HNPCC by molecular analysis of colorectal and endometrial tumors

Abstract. Hereditary nonpolyposis colorectal cancer (HNPCC, Lynch syndrome) is a dominantly inherited syndrome characterized by the development of colorectal cancer, endometrial cancer and other cancers and the presence of microsatellite instability (MSI) in tumors. The Bethesda guidelines have been proposed for the identification of families suspected of HNPCC that require further molecular analysis. We have evaluated the yield of MSI-analysis in a large series of Dutch families suspected of HNPCC. We also analysed whether the loss of mismatch repair (MMR) protein detected by immunohistochemistry (IHC) of colorectal cancer (CRC) and endometrial cancer correlated with the presence of MSI and/or a MMR gene mutation. The results showed that the Bethesda criteria with a few modifications are appropriate to identify families eligible for genetic testing. In addition, we found that MSI and IHC-analysis of CRC using antibodies against MLH1, MSH2, MSH6 and PMS2 proteins are equally effective for identifying carriers of the known MMR gene defects. However, as long as the role of other putative MMR genes in hereditary CRC has not been elucidated, IHC-analysis cannot completely replace MSI. For this reason, we prefer MSI-analysis as first step in families suspected of HNPCC. On the other hand, in families fulfilling the revised Amsterdam criteria in which the probability of detecting a mutation is relatively high, we would recommend IHC as first diagnostic step because the result might predict the specific underlying MMR gene mutation. MSI or IHC-analysis of endometrial cancer alone was found to be less sensitive compared with these tests performed in colorectal cancer. Therefore, probably the best approach in the analysis of this cancer is to perform both techniques. The identification of HNPCC is important as it makes it possible to target effective preventative measures. Our studies showed that MSI and IHC analysis of colorectal and endometrial cancer, are reliable cost-effective tools that can be used to identify patients with HNPCC