The term photonic crystals appears because of the analogy between electron waves in crystals and the light waves in artificial periodic dielectric structures. During the recent years the investigation of one-, two-and three-dimensional periodic structures has attracted a widespread attention of the world optics community because of great potentiality of such structures in advanced applied optical fields. The interest in periodic structures has been stimulated by the fast development of semiconductor technology that now allows the fabrication of artificial structures, whose period is comparable with the wavelength of light in the visible and infrared ranges.

http://ieeexplore.ieee.org/iel5/6354/16969/00781867.pdf

Photonic crystals

The presence of tiny holes in an opaque metal film, with sizes smaller than the wavelength of incident light, leads to a wide variety of unexpected optical properties such as strongly enhanced transmission of light through the holes and wavelength filtering. These intriguing effects are now known to be due to the interaction of the light with electronic resonances in the surface of the metal film, and they can be controlled by adjusting the size and geometry of the holes. This knowledge is opening up exciting new opportunities in applications ranging from subwavelength optics and optoelectronics to chemical sensing and biophysics.

Light in tiny holes

Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

Bound states in the continuum (BICs) are waves that remain localized even though they coexist with a continuous spectrum of radiating waves that can carry energy away. Their very existence defies conventional wisdom. Although BICs were first proposed in quantum mechanics, they are a general wave phenomenon and have since been identified in electromagnetic waves, acoustic waves in air, water waves and elastic waves in solids. These states have been studied in a wide range of material systems, such as piezoelectric materials, dielectric photonic crystals, optical waveguides and fibres, quantum dots, graphene and topological insulators. In this Review, we describe recent developments in this field with an emphasis on the physical mechanisms that lead to BICs across seemingly very different materials and types of waves. We also discuss experimental realizations, existing applications and directions for future work. The fascinating wave phenomenon of ‘bound states in the continuum’ spans different material and wave systems, including electron, electromagnetic and mechanical waves. In this Review, we focus on the common physical mechanisms underlying these bound states, whilst also discussing recent experimental realizations, current applications and future opportunities for research.

Bound states in the continuum

The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

/pdf/an-introduction-to-random-matrices-1gerxro6r6.pdf

An Introduction to Random Matrices

We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e−nV(x)dx on the line as n ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel-Rotach-type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]).



The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [19, 20]. The Riemann-Hilbert problem is analyzed in turn using the steepest-descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμV for V as analyzed in [8]. © 1999 John Wiley & Sons, Inc.

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx= e Q(x) dx on the real line, where Q(x)=∑ 2m k=0 qkx k , q2m> 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [22, 23]. We employ the steepest-descent-type method introduced in [18] and further developed in [17, 19] in order to obtain uniform Plancherel-Rotach-type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the orthogonal polynomials. c 1999 John Wiley & Sons, Inc.

/pdf/strong-asymptotics-of-orthogonal-polynomials-with-respect-to-4s7xvhewsy.pdf

Strong asymptotics of orthogonal polynomials with respect to exponential weights

We investigated the forces that connect the genetic program of development to morphogenesis in Drosophila. We focused on dorsal closure, a powerful model system for development and wound healing. We found that the bulk of progress toward closure is driven by contractility in supracellular "purse strings" and in the amnioserosa, whereas adhesion-mediated zipping coordinates the forces produced by the purse strings and is essential only for the end stages. We applied quantitative modeling to show that these forces, generated in distinct cells, are coordinated in space and synchronized in time. Modeling of wild-type and mutant phenotypes is predictive; although closure in myospheroid mutants ultimately fails when the cell sheets rip themselves apart, our analysis indicates that beta(PS) integrin has an earlier, important role in zipping.

https://science.sciencemag.org/content/sci/300/5616/145.full.pdf

Forces for Morphogenesis Investigated with Laser Microsurgery and Quantitative Modeling

New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems

There are many physical systems which display shocks i.e. regions in space where the solution develops extremely large slopes. In general, such systems are too complicated to be treated by exact calculation and their properties are best studied through the proof of general theorems. A model of the formation and propagation of dispersive shocks in one space dimension, in which explicit calculation is possible, is given by the initial value problem for the Korteweg-de Vries equation: 

$$ {u_{t}} - 6u{u_{x}} + {\varepsilon ^{2}}u{u_{{xxx}}} = 0 $$

(1.1a)



$$ u(x,o,\varepsilon ) = - v(x)$$

(1.1b)

in the limit e → 0.

Stephanos Venakides

Papers

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

Strong asymptotics of orthogonal polynomials with respect to exponential weights

Forces for Morphogenesis Investigated with Laser Microsurgery and Quantitative Modeling

New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems

The Small Dispersion Limit of the Korteweg-De Vries Equation