We attempt a broad exploration of properties and connections between the symmetry function of a convex set S ⊂ IRn and other arenas of convexity including convex functions, convex geometry, probability theory on convex sets, and computational complexity. Given a point x ∈ S, let sym(x, S) denote the symmetry value of x in S: sym(x, S) := max{α ≥ 0 : x + α(x− y) ∈ S for every y ∈ S} , which esse...

1. Macdonald polynomials Macdonald polynomials P λ (x; q, t) are orthogonal symmetric polynomials which are the natural multivariable generalisation of the continuous q-ultraspherical polyno-mials C n (x; β|q) [2] which, in their turn, constitute an important class of hyper-geometric orthogonal polynomials in one variable. Polynomials C n (x; β|q) can be obtained from the general Askey-Wilson p...

Knop and Sahi simultaneously introduced a family of non-homogeneous, non-symmetric polynomials, Gα(x; q, t). The top homogeneous components of these polynomials are the non-symmetric Macdonald polynomials, Eα(x; q, t). An appropriate Hecke algebra symmetrization of Eα yields the Macdonald polynomials, Pλ(x; q, t). A search for explicit formulas for the polynomials Gα(x; q, t) led to the main re...

Within the framework of PT symmetric quantum mechanics (which works with parity-pseudo-Hermitian Hamiltonians H = PHP and real spectra) we describe the double-well effects of tunneling by a pair of δ functions with mutually complex conjugate strengths. The model is exactly solvable and exhibits several interesting features. We observe an amazingly robust absence of any PT symmetry breaking. A q...

Let D1 and D2 be two irreducible bounded symmetric domains in the complex spaces V1 and V2 respectively. Let E be the Euclidean metric on V2 and h the Bergman metric on V1. The Bloch constant b(D1,D2) is defined to be the supremum of E(f ′(z)x, f ′(z)x) 1 2 /hz(x, x)1/2, taken over all the holomorphic functions f : D1 → D2 and z ∈ D1, and nonzero vectors x ∈ V1. We find the constants for all th...

| Let X be a CAT ( 1) space which is spherically symmetric around some point x0 2 X and whose boundary has nite positive s dimensional Hausdor measure. Let = ( x)x2X be a conformal density of dimension d > s=2 on @X. We prove that x0 is a weak limit of measures supported on spheres centered at x0. These measures are expressed in terms of the total mass function of and of the d dimensional spher...

X-linked adrenoleukodystrophy (X-ALD) is an inherited disease caused by peroxisomal dysfunction. X-ALD usually involves the cerebral white matter in an approximately symmetric way. We report a 10-year-old boy with the cerebral form of X-ALD who presented with cognitive impairment and left spastic hemiparesis. Brain magnetic resonance imaging (MRI) showed asymmetric lesions, and the lesions in t...

The performance of a polarizing beam splitter based on the one-dimensional photonic crystals (1D-PCs), is theoretically investigated. The polarizing beam splitter consists of a symmetric stack of the low-index quarter-wave plates and the high-index half-wave plates with a central defect layer of air. The linear transmission properties of the polarizing beam splitter are numerically simulat...

We study the “flat” directions of non-BPS extremal black hole attractors for N = 2, d = 4 supergravities whose vector multiplets’ scalar manifold is endowed with homogeneous symmetric special Kähler geometry. The non-BPS attractors with non-vanishing central charge have a moduli space described by real special geometry (and thus related to the d = 5 parent theory), whereas the moduli spaces of ...

Abstract Using umbral calculus, we establish a symmetric identity for any sequence of polynomials satisfying A ′ n +1 ( x ) = + 1) with 0 constant polynomial. This allows us to obtain in simple way some known relations involving Apostol-Bernoulli polynomials, Apostol-Euler and generalized Bernoulli attached primitive Dirichlet character.