In this paper, we study the probability distribution of observable $s = (1/N)\sum_{i=N-N'+1}^N x_i$, with $1 \leq N' N$ and $x_1<x_2<\cdots< x_N$ representing ordered positions $N$ particles in a $1d$ one-component plasma, i.e., harmonically confined charges on line, pairwise repulsive Coulomb interaction $|x_i-x_j|$. This represents an example truncated linear statistics -- here center mass $N...

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In this article we present a method to determine the band spectrum, band gaps, and discrete energy levels, of a one-dimensional photonic crystal with localized impurities. For one-dimensional crystals with piecewise constant refractive indices we develop an algorithm to recover the refractive index distribution from the period map. Finally, we derive the relationship between the period map and ...

We give the first explicit computations of rational homotopy groups of spaces of “long knots” in Euclidean spaces. We define a spectral sequence which converges to these rational homotopy groups whose E term is defined in terms of braid Lie algebras. For odd k we establish a vanishing line for this spectral sequence, show the Euler characteristic of the rows of this E term is zero, and make cal...

We study one dimensional intersections of M5 branes with M5 and M2 branes. On the worldvolume of the M5-brane, such an intersection appears as a string soliton. We study this worldvolume theory in two different regimes: 1) Where the worldvolume theory is formulated in flat space and 2) where the worldvolume theory is studied in the supergravity background produced by a stack of M5 (or M2) brane...