ERG and Schwinger-Dyson Equations - Comparison in formulations and applications --

The advantageous points of ERG in applications to non-perturbative analyses of quantum field theories are discussed in comparison with the Schwinger-Dyson equations. First we consider the relation between these two formulations specially by examining the large N field theories. In the second part we study the phase structure of dynamical symmetry breaking in three dimensional QED as a typical example of the practical applications.Comment: 13 pages, 6 figures, references are added. Invited talk presented at Second Conference on the Exact RG, Sept 18 - 22, 2000, Rome, Ital

Critical points and resonance of hyperplane arrangements

If F is a master function corresponding to a hyperplane arrangement A and a collection of weights y, we investigate the relationship between the critical set of F, the variety defined by the vanishing of the one-form w = d log F, and the resonance of y. For arrangements satisfying certain conditions, we show that if y is resonant in dimension p, then the critical set of F has codimension at most p. These include all free arrangements and all rank 3 arrangements.Comment: revised version, Canadian Journal of Mathematics, to appea

A Case of Generalized Acanthosis Nigricans with Positive Lupus Erythematosus-Related Autoantibodies and Antimicrosomal Antibody: Autoimmune Acanthosis Nigricans?

Acanthosis nigricans (AN) is a hyperpigmented keratotic skin lesion known to be associated with malignant disease and endocrinopathy. We report a very rare case of generalized AN with Sjögren's syndrome- and systemic lupus erythematosus-like features but without type B insulin resistance. Neither internal malignancy nor other endocrinological disorders, including glucose intolerance, were detected during a 10-year clinical course with benign diffuse papillomatosis extending from the mucosa of the larynx to the esophagogastric junction. The case was complicated with chronic thyroiditis and interstitial pneumonia, which were not treated with any medication. AN skin lesions and mucosal papillomatosis regressed with oral cyclosporine A, accompanied by the lowering of autoantibody titers. This is the first report of generalized AN involving an area from the mucosa of the larynx to the esophagogastric junction accompanied by autoimmune manifestations which responded to systemic immunosuppressive therapy

Spin Driven Jahn-Teller Distortion in a Pyrochlore system

The ground-state properties of the spin-1 antiferromagnetic Heisenberg model on the corner-sharing tetrahedra, pyrochlore lattice, is investigated. By breaking up each spin into a pair of 1/2-spins, the problem is reduced to the equivalent one of the spin-1/2 tetrahedral network in analogy with the valence bond solid state in one dimension. The twofold degeneracy of the spin-singlets of a tetrahedron is lifted by a Jahn-Teller mechanism, leading to a cubic to tetragonal structural transition. It is proposed that the present mechanism is responsible for the phase transition observed in the spin-1 spinel compounds ZnV$_2$O$_4$ and MgV$_2$O$_4$.Comment: 4 pages, 3 eps figures, REVTeX, to appear in Phys. Rev. Let

Higher homotopy groups of complements of complex hyperplane arrangements

We generalize results of Hattori on the topology of complements of hyperplane arrangements, from the class of generic arrangements, to the much broader class of hypersolvable arrangements. We show that the higher homotopy groups of the complement vanish in a certain combinatorially determined range, and we give an explicit Z\pi_1-module presentation of \pi_p, the first non-vanishing higher homotopy group. We also give a combinatorial formula for the \pi_1-coinvariants of \pi_p. For affine line arrangements whose cones are hypersolvable, we provide a minimal resolution of \pi_2, and study some of the properties of this module. For graphic arrangements associated to graphs with no 3-cycles, we obtain information on \pi_2, directly from the graph. The \pi_1-coinvariants of \pi_2 may distinguish the homotopy 2-types of arrangement complements with the same \pi_1, and the same Betti numbers in low degrees.Comment: 24 pages, 3 figure