Mean eigenvalues for simple, simply connected, compact Lie groups

We determine for each of the simple, simply connected, compact and complex Lie groups SU(n), Spin$(4n+2)$ and $E_6$ that particular region inside the unit disk in the complex plane which is filled by their mean eigenvalues. We give analytical parameterizations for the boundary curves of these so-called trace figures. The area enclosed by a trace figure turns out to be a rational multiple of $\pi$ in each case. We calculate also the length of the boundary curve and determine the radius of the largest circle that is contained in a trace figure. The discrete center of the corresponding compact complex Lie group shows up prominently in the form of cusp points of the trace figure placed symmetrically on the unit circle. For the exceptional Lie groups $G_2$, $F_4$ and $E_8$ with trivial center we determine the (negative) lower bound on their mean eigenvalues lying within the real interval $[-1,1]$. We find the rational boundary values -2/7, -3/13 and -1/31 for $G_2$, $F_4$ and $E_8$, respectively.Comment: 12 pages, 8 figure

Inner metric properties of 2-dimensional semi-algebraic sets

Sin resume

Pigmentation, Melanocyte Colonization, and p53 Status in Basal Cell Carcinoma

Basal cell carcinoma (BCC) is the most common neoplasm in the Caucasian population. Only a fraction of BCC exhibits pigmentation. Lack of melanocyte colonization has been suggested to be due to p53-inactivating mutations in the BCC cells interfering with the p53-proopiomelanocortin pathway and the production of alpha melanocyte-stimulating hormone in the tumor. To evaluate this, we determined tumor pigmentation as well as expression of melan-A and of p53 in 49 BCC tissues by means of immunohistochemistry. As expected, we observed a positive relation between tumor pigmentation and melan-A positive intra-tumoral melanocytes. Melanocyte colonization and, to a lesser extent, p53 overexpression showed intraindividual heterogeneity in larger tumors. p53 overexpression, which is indicative of p53 mutations, was not correlated to melanocyte colonization of BCC. Sequencing of exon 5–8 of the p53 gene in selected BCC cases revealed that colonization by melanocytes and BCC pigmentation is neither ablated by p53 mutations nor generally present in BCCs with wild-type p53

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

Nonnegatively curved homogeneous metrics obtained by scaling fibers of submersions

We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists, so that the homogeneous space G/H is the total space of a Riemannian submersion. We study the question as to whether enlarging the fibers of the submersion by a constant scaling factor retains the nonnegative curvature in the case that the deformation starts at a normal homogeneous metric. We classify triples of groups (H,K,G) where nonnegative curvature is maintained for small deformations, using a criterion proved by Schwachh\"ofer and Tapp. We obtain a complete classification in case the subgroup H has full rank and an almost complete classification in the case of regular subgroups.Comment: 23 pages; minor revisions, to appear in Geometriae Dedicat

In vitro evidence for senescent multinucleated melanocytes as a source for tumor-initiating cells

Oncogenic signaling in melanocytes results in oncogene-induced senescence (OIS), a stable cell-cycle arrest frequently characterized by a bi- or multinuclear phenotype that is considered as a barrier to cancer progression. However, the long-sustained conviction that senescence is a truly irreversible process has recently been challenged. Still, it is not known whether cells driven into OIS can progress to cancer and thereby pose a potential threat. Here, we show that prolonged expression of the melanoma oncogene N-RAS61K in pigment cells overcomes OIS by triggering the emergence of tumor-initiating mononucleated stem-like cells from senescent cells. This progeny is dedifferentiated, highly proliferative, anoikis-resistant and induces fast growing, metastatic tumors. Our data describe that differentiated cells, which are driven into senescence by an oncogene, use this senescence state as trigger for tumor transformation, giving rise to highly aggressive tumor-initiating cells. These observations provide the first experimental in vitro evidence for the evasion of OIS on the cellular level and ensuing transformation

On the total curvatures of a tame function

Given a definable function f, enough differentiable, we study the continuity of the total curvature function t --> K(t), total curvature of the level {f=t}, and the total absolute curvature function t-->|K| (t), total absolute curvature of the level {f=t}. We show they admits at most finitely many discontinuities

Loop Quantum Cosmology II: Volume Operators

Volume operators measuring the total volume of space in a loop quantum theory of cosmological models are constructed. In the case of models with rotational symmetry an investigation of the Higgs constraint imposed on the reduced connection variables is necessary, a complete solution of which is given for isotropic models; in this case the volume spectrum can be calculated explicitly. It is observed that the stronger the symmetry conditions are the smaller is the volume spectrum, which can be interpreted as level splitting due to broken symmetries. Some implications for quantum cosmology are presented.Comment: 21 page