Lyashko-Looijenga morphisms and submaximal factorisations of a Coxeter element

When W is a finite reflection group, the noncrossing partition lattice NCP_W of type W is a rich combinatorial object, extending the notion of noncrossing partitions of an n-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in NCP_W as a generalised Fuss-Catalan number, depending on the invariant degrees of W. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of NCP_W as fibers of a Lyashko-Looijenga covering (LL), constructed from the geometry of the discriminant hypersurface of W. We study algebraically the map LL, describing the factorisations of its discriminant and its Jacobian. As byproducts, we generalise a formula stated by K. Saito for real reflection groups, and we deduce new enumeration formulas for certain factorisations of a Coxeter element of W.Comment: 18 pages. Version 2 : corrected typos and improved presentation. Version 3 : corrected typos, added illustrated example. To appear in Journal of Algebraic Combinatoric

On the Absence of an Exponential Bound in Four Dimensional Simplicial Gravity

We have studied a model which has been proposed as a regularisation for four dimensional quantum gravity. The partition function is constructed by performing a weighted sum over all triangulations of the four sphere. Using numerical simulation we find that the number of such triangulations containing $V$ simplices grows faster than exponentially with $V$. This property ensures that the model has no thermodynamic limit.Comment: 8 pages, 2 figure

Corrections to Sirlin's Theorem in O(p6)O(p^6) Chiral Perturbation Theory

We present the results of the first two-loop calculation of a form factor in full $SU(3) \times SU(3)$ Chiral Perturbation Theory. We choose a specific linear combination of $\pi^+, K^+, K^0$ and $K\pi$ form factors (the one appearing in Sirlin's theorem) which does not get contributions from order $p^6$ operators with unknown constants. For the charge radii, the correction to the previous one-loop result turns out to be significant, but still there is no agreement with the present data due to large experimental uncertainties in the kaon charge radii.Comment: 6 pages, Latex, 2 LaTeX figure

PT-symmetry and its spontaneous breakdown explained by anti-linearity

The impact of an anti-unitary symmetry on the spectrum of non-Hermitian operators is studied. Wigner's normal form of an anti-unitary operator accounts for the spectral properties of non-Hermitian, PE-symmetric Harniltonians. The occurrence of either single real or complex conjugate pairs of eigenvalues follows from this theory. The corresponding energy eigenstates span either one- or two-dimensional irreducible representations of the symmetry PE. In this framework, the concept of a spontaneously broken PE-symmetry is not needed

Some properties of eigenvalues and eigenfunctions of the cubic oscillator with imaginary coupling constant

Comparison between the exact value of the spectral zeta function, $Z_{H}(1)=5^{-6/5}[3-2\cos(\pi/5)]\Gamma^2(1/5)/\Gamma(3/5)$, and the results of numeric and WKB calculations supports the conjecture by Bessis that all the eigenvalues of this PT-invariant hamiltonian are real. For one-dimensional Schr\"odinger operators with complex potentials having a monotonic imaginary part, the eigenfunctions (and the imaginary parts of their logarithmic derivatives) have no real zeros.Comment: 6 pages, submitted to J. Phys.

Numerical observation of non-axisymmetric vesicles in fluid membranes

By means of Surface Evolver (Exp. Math,1,141 1992), a software package of brute-force energy minimization over a triangulated surface developed by the geometry center of University of Minnesota, we have numerically searched the non-axisymmetric shapes under the Helfrich spontaneous curvature (SC) energy model. We show for the first time there are abundant mechanically stable non-axisymmetric vesicles in SC model, including regular ones with intrinsic geometric symmetry and complex irregular ones. We report in this paper several interesting shapes including a corniculate shape with six corns, a quadri-concave shape, a shape resembling sickle cells, and a shape resembling acanthocytes. As far as we know, these shapes have not been theoretically obtained by any curvature model before. In addition, the role of the spontaneous curvature in the formation of irregular crenated vesicles has been studied. The results shows a positive spontaneous curvature may be a necessary condition to keep an irregular crenated shape being mechanically stable.Comment: RevTex, 14 pages. A hard copy of 8 figures is available on reques

Space of State Vectors in PT Symmetrical Quantum Mechanics

Space of states of PT symmetrical quantum mechanics is examined. Requirement that eigenstates with different eigenvalues must be orthogonal leads to the conclusion that eigenfunctions belong to the space with an indefinite metric. The self consistent expressions for the probability amplitude and average value of operator are suggested. Further specification of space of state vectors yield the superselection rule, redefining notion of the superposition principle. The expression for the probability current density, satisfying equation of continuity and vanishing for the bound state, is proposed.Comment: Revised version, explicit expressions for average values and probability amplitude adde

Compact support probability distributions in random matrix theory

We consider a generalization of the fixed and bounded trace ensembles introduced by Bronk and Rosenzweig up to an arbitrary polynomial potential. In the large-N limit we prove that the two are equivalent and that their eigenvalue distribution coincides with that of the "canonical" ensemble with measure exp[-$n$Tr V(M)]. The mapping of the corresponding phase boundaries is illuminated in an explicit example. In the case of a Gaussian potential we are able to derive exact expressions for the one- and two-point correlator for finite $n$, having finite support

Sums over Graphs and Integration over Discrete Groupoids

We show that sums over graphs such as appear in the theory of Feynman diagrams can be seen as integrals over discrete groupoids. From this point of view, basic combinatorial formulas of the theory of Feynman diagrams can be interpreted as pull-back or push-forward formulas for integrals over suitable groupoids.Comment: 27 pages, 4 eps figures; LaTeX2e; uses Xy-Pic. Some ambiguities fixed, and several proofs simplifie