Uniform asymptotics of the coefficients of unitary moment polynomials

Keating and Snaith showed that the $2k^{th}$ absolute moment of the characteristic polynomial of a random unitary matrix evaluated on the unit circle is given by a polynomial of degree $k^2$. In this article, uniform asymptotics for the coefficients of that polynomial are derived, and a maximal coefficient is located. Some of the asymptotics are given in explicit form. Numerical data to support these calculations are presented. Some apparent connections between random matrix theory and the Riemann zeta function are discussed.Comment: 31 pages, 1 figure, 2 tables. A few minor misprints fixe

Preparation of a Resorbable Osteoinductive Tricalcium Phosphate Ceramic

Over the past decade we have demonstrated numerous times that calcium phosphates can be rendered with osteoinductive properties by introducing specific surface microstructures1. Since most of these calcium phosphates contained hydroxyapatite, they are either slowly or not resorbable2. Resorbability is an often sought after characteristic of calcium phosphates so that they can be gradually replaced by newly formed bone. The objective of this study was to prepare a resorbable surface microstructured tricalcium phosphate (TCP) ceramic and evaluate its osteoinductive property and resorption rate after intramuscular implantation in dogs. This material was then compared to the established and slowly resorbable osteoinductive biphasic calcium phosphate ceramic (BCP)

Photonic quasicrystals for general purpose nonlinear optical frequency conversion

We present a general method for the design of 2-dimensional nonlinear photonic quasicrystals that can be utilized for the simultaneous phase-matching of arbitrary optical frequency-conversion processes. The proposed scheme--based on the generalized dual-grid method that is used for constructing tiling models of quasicrystals--gives complete design flexibility, removing any constraints imposed by previous approaches. As an example we demonstrate the design of a color fan--a nonlinear photonic quasicrystal whose input is a single wave at frequency $\omega$ and whose output consists of the second, third, and fourth harmonics of $\omega$, each in a different spatial direction

What is a crystal?

Almost 25 years have passed since Shechtman discovered quasicrystals, and 15 years since the Commission on Aperiodic Crystals of the International Union of Crystallography put forth a provisional definition of the term crystal to mean ``any solid having an essentially discrete diffraction diagram.'' Have we learned enough about crystallinity in the last 25 years, or do we need more time to explore additional physical systems? There is much confusion and contradiction in the literature in using the term crystal. Are we ready now to propose a permanent definition for crystal to be used by all? I argue that time has come to put a sense of order in all the confusion.Comment: Submitted to Zeitschrift fuer Kristallographi

Theory of high harmonic generation in relativistic laser interaction with overdense plasma

High harmonic generation due to the interaction of a short ultra relativistic laser pulse with overdense plasma is studied analytically and numerically. On the basis of the ultra relativistic similarity theory we show that the high harmonic spectrum is universal, i.e. it does not depend on the interaction details. The spectrum includes the power law part $I_n\propto n^{-8/3}$ for $n<\sqrt{8\alpha}\gamma_{\max}^3$, followed by exponential decay. Here $\gamma_{\max}$ is the largest relativistic $\gamma$-factor of the plasma surface and $\alpha$ is the second derivative of the surface velocity at this moment. The high harmonic cutoff at $\propto \gamma_{\max}^3$ is parametrically larger than the $4 \gamma_{\max}^2$ predicted by the ``oscillating mirror'' model based on the Doppler effect. The cornerstone of our theory is the new physical phenomenon: spikes in the relativistic $\gamma$-factor of the plasma surface. These spikes define the high harmonic spectrum and lead to attosecond pulses in the reflected radiation.Comment: 12 pages, 9 figure

Univalent Foundations and the UniMath Library

We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (section 3). On the way our odyssey from the foundations to the "horizon" of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander