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Consider a random polynomial $P_n$ of degree $n$ whose roots are independent random variables sampled according to some probability distribution $\mu_0$ on the complex plane $\mathbb C$. It is natural to conjecture that, for a fixed $t\in [0,1)$ and as $n\to\infty$, the zeroes of the $[tn]$-th derivative of $P_n$ are distributed according to some measure $\mu_t$ on $\mathbb C$. Assuming either that $\mu_0$ is concentrated on the real line or that it is rotationally invariant, Steinerberger [Proc. AMS, 2019] and O'Rourke and Steinerberger [arXiv:1910.12161] derived nonlocal transport equations for the density of roots. We introduce a different method to treat such problems. In the rotationally invariant case, we obtain a closed formula for $\psi(x,t)$, the asymptotic density of the radial parts of the roots of the $[tn]$-th derivative of $P_n$. Although its derivation is non-rigorous, we provide numerical evidence for its correctness and prove that it solves the PDE of O'Rourke and Steinerberger. Moreover, we present several examples in which the solution is fully explicit (including the special case in which the initial condition $\psi(x,0)$ is an arbitrary convex combination of delta functions) and analyze some properties of the solutions such as the behavior of void annuli and circles of zeroes. As an additional support for the correctness of the method, we show that a similar method, applied to the case when $\mu_0$ is concentrated on the real line, gives a correct result which is known to have an interpretation in terms of free probability.

This paper presents a fast algorithm for computing transport properties of two-dimensional Dirac operators with linear domain walls, which model the macroscopic behavior of the robust and asymmetric transport observed at an interface separating two two-dimensional topological insulators. Our method is based on reformulating the partial differential equation as a corresponding volume integral equation, which we solve via a spectral discretization scheme. We demonstrate the accuracy of our method by confirming the quantization of an appropriate interface conductivity modeling transport asymmetry along the interface, and moreover confirm that this quantity is immune to local perturbations. We also compute the far-field scattering matrix generated by such perturbations and verify that while asymmetric transport is topologically protected the absence of back-scattering is not.

Let $x_1, \dots, x_n$ be $n$ independent and identically distributed random variables with mean zero, unit variance, and finite moments of all remaining orders. We study the random polynomial $p_n$ having roots at $x_1, \dots, x_n$. We prove that for $\ell \in \mathbb{N}$ fixed as $n \rightarrow \infty$, the $(n-\ell)-$th derivative of $p_n^{}$ behaves like a Hermite polynomial: for $x$ in a compact interval,$${n^{\ell/2}} \frac{\ell!}{n!} \cdot p_n^{(n-\ell)}\left( \frac{x}{\sqrt{n}}\right) \rightarrow He_{\ell}(x + \gamma_n),$$ where $He_{\ell}$ is the $\ell-$th probabilists' Hermite polynomial and $\gamma_n$ is a random variable converging to the standard $\mathcal{N}(0,1)$ Gaussian as $n \rightarrow \infty$. Thus, there is a universality phenomenon when differentiating a random polynomial many times: the remaining roots follow a Wigner semicircle distribution.

We consider the quantum electrodynamics of single photons in arrays of one-way waveguides, each containing many atoms. We investigate both chiral and antichiral arrays, in which the group velocities of the waveguides are the same or alternate in sign, respectively. We find that in the continuum limit, the one-photon amplitude obeys a Dirac equation. In the chiral case, the Dirac equation is hyperbolic, while in the antichiral case it is elliptic. This distinction has implications for the nature of photon transport in waveguide arrays. Our results are illustrated by numerical simulations.

We consider the numerical solution of a nonlocal partial differential equation which models the process of collective spontaneous emission in a two-level atomic system containing a single photon. We reformulate the problem as an integro-differential equation for the atomic degrees of freedom, and describe an efficient solver for the case of a Gaussian atomic density. The problem of history dependence arising from the integral formulation is addressed using sum-of-exponentials history compression. We demonstrate the solver on two systems of physical interest: in the first, an initially-excited atom decays into a photon by spontaneous emission, and in the second, a photon pulse is used to an excite an atom, which then decays.

In the present paper we describe a class of algorithms for the solution of Laplace's equation on polygonal domains with Dirichlet and Neumann boundary conditions. It is well known that in such case...

We analyze the convergence and approximation error of the inverse Born series, obtaining results that hold under qualitatively weaker conditions than previously known. Our approach makes use of tools from geometric function theory in Banach spaces. An application to the inverse scattering problem with diffuse waves is described.

We propose a method to reconstruct the optical properties of a highly scattering medium from acousto-optic measurements. The method is based on solving an inverse problem with internal data for a system of diffusion equations.

In the present paper we describe a class of algorithms for the solution of Laplace's equation on polygonal domains with Neumann boundary conditions. It is well known that in such cases the solutions have singularities near the corners which poses a challenge for many existing methods. If the boundary data is smooth on each edge of the polygon, then in the vicinity of each corner the solution to the corresponding boundary integral equation has an expansion in terms of certain (analytically available) singular powers. Using the known behavior of the solution, universal discretizations have been constructed for the solution of the Dirichlet problem. However, the leading order behavior of solutions to the Neumann problem is $O(t^{\mu})$ for $\mu \in (-1/2,0)$ depending on the angle at the corner (compared to $O(C+t^{\mu})$ with $\mu>1/2$ for the Dirichlet problem); this presents a significant challenge in the design of universal discretizations. Our approach is based on using the discretization for the Dirichlet problem in order to compute a solution in the "weak sense" by solving an adjoint linear system; namely, it can be used to compute inner products with smooth functions accurately, but it cannot be interpolated. Furthermore we present a procedure to obtain accurate solutions arbitrarily close to the corner, by solving a sequence of small local subproblems in the vicinity of that corner. The results are illustrated with several numerical examples., Comment: 25 pages, 6 figures

There are several hybrid inverse problems for equations of the form $$\nabla \cdot D(x) \nabla u - \sigma (x) u = 0$$ in which we want to obtain the coefficients D and \(\sigma \) on a domain \(\varOmega \) when the solutions u are known. One approach is to use two solutions \(u_1\) and \(u_2\) to obtain a transport equation for the coefficient D, and then solve this equation inward from the boundary along the integral curves of a vector field X defined by \(u_1\) and \(u_2\). Bal and Ren have shown that for any nontrivial choices of \(u_1\) and \(u_2\), this method suffices to recover the coefficients almost everywhere on a dense set in \(\varOmega \) Bal and Ren in (Inv Prob 075003 [3]). This article presents an alternate proof of the same result from a dynamical systems point of view.

Let $$\Omega \subset \mathbb {R}^2$$ be a bounded, convex domain and let u be the solution of $$-\Delta u = 1$$ vanishing on the boundary $$\partial \Omega $$ . The estimate $$\begin{aligned} \Vert \nabla u\Vert _{L^{\infty }(\Omega )} \le c |\Omega |^{1/2} \end{aligned}$$ is classical. We use the P-functional, the stability theory of the torsion function and Brownian motion to establish the estimate for a universal $$c < (2\pi )^{-1/2}$$ . We also give a numerical construction showing that the optimal constant satisfies $$c \ge 0.358$$ . The problem is important in different settings: (1) as the maximum shear stress in Saint Venant Elasticity Theory, (2) as an optimal control problem for the constrained maximization of the lifetime of Brownian motion started close to the boundary, and (3) optimal Hermite–Hadamard inequalities for subharmonic functions on convex domains.

In this paper we characterize the behavior of solutions to systems of boundary integral equations associated with Laplace transmission problems in composite media consisting of regions with polygonal boundaries. In particular we consider triple junctions, i.e. points at which three distinct media meet. We show that, under suitable conditions, solutions to the boundary integral equations in the vicinity of a triple junction are well-approximated by linear combinations of functions of the form $t^\beta,$ where $t$ is the distance of the point from the junction and the powers $\beta$ depend only on the material properties of the media and the angles at which their boundaries meet. Moreover, we use this analysis to design efficient discretizations of boundary integral equations for Laplace transmission problems in regions with triple junctions and demonstrate the accuracy and efficiency of this algorithm with a number of examples., Comment: 32 pages, 11 figures

Symmetry analysis and conservation laws are widely used in analyzing and solving differential equations. Conservation laws are also called first integrals when dealing with ordinary differential equations (ODEs). We explore the complementary nature of symmetry analysis and conservation laws; specifically, the use of symmetries to find integrating factors and, conversely, the use of conservation laws to seek new symmetries. In particular, we show two new fundamental results. Firstly, we show that a higher-order symmetry of an ODE induces a Lie point symmetry of the corresponding integrating factor determining equations (IFDEs). Moreover, we obtain an explicit expression for the infinitesimal generator of this induced point symmetry, enabling one to find new integrating factors of existing ODEs and thus possibly to solve a wider range of ODEs. Secondly, we show that the converse also holds, i.e. a point symmetry of the IFDEs for a given ODE induces a higher-order symmetry of the ODE.

We consider the theory of radiative transport in quasi-homogeneous random media. We derive the radiative transport equation that governs the propagation of light in such media. This result provides conditions under which it is justified to apply radiative transport theory to spatially inhomogeneous media.

t-distributed stochastic neighbor embedding (t-SNE) is widely used for visualizing single-cell RNA-sequencing (scRNA-seq) data, but it scales poorly to large datasets. We dramatically accelerate t-SNE, obviating the need for data downsampling, and hence allowing visualization of rare cell populations. Furthermore, we implement a heatmap-style visualization for scRNA-seq based on one-dimensional t-SNE for simultaneously visualizing the expression patterns of thousands of genes. Software is available at https://github.com/KlugerLab/FIt-SNE and https://github.com/KlugerLab/t-SNE-Heatmaps .

We examine the propagation of single photons in periodic and disordered dimer chains coupled to one-dimensional chiral and bidirectional waveguides. Each dimer is composed of two dipole-coupled atoms. In the disordered setting, we separately treat two types of position disorder, namely in dimer length and in dimer separation. The focus of this study is to understand in what ways the interplay between dipole-dipole interactions and directionality of photon emission can impact the transport of single photons. Cold atoms trapped near optical fibers can serve as an experimentally realizable platform for the models that we consider.

Architectures based on waveguide quantum electrodynamics have emerged as promising candidates for quantum networks. In this paper, we analyze the propagation of single-photons in disordered many-atom waveguides. We pay special attention to the influence of chirality (directionality of photon transport) on the formation of localized photonic states, considering separately the cases of the disorder in the atomic positions and in the atomic transition frequencies.

We consider the acousto-optic effect in a random medium. We derive the radiative transport equations that describe the propagation of multiply-scattered light in a medium whose dielectric permittivity is modulated by an acoustic wave. Using this result, we present an analysis of the sensitivity of an acousto-optic measurement to the presence of a small absorbing inhomogeneity., 14 pages, 3 figures

We present an algorithm for solving inverse problems on graphs analogous to those arising in diffuse optical tomography for continuous media. In particular, we formulate and analyze a discrete version of the inverse Born series, proving estimates characterizing the domain of convergence, approximation errors, and stability of our approach. We also present a modification which allows additional information on the structure of the potential to be incorporated, facilitating recovery for a broader class of problems.

The couplons of the cardiomyocyte form nanospaces within the cell that place the L-type calcium channel (Ca(v)1.2), situated on the plasmalemma, in opposition to the type 2 ryanodine receptor (RyR2), situated on the sarcoplasmic reticulum. These two molecules, which form the basis of excitation-contraction coupling, are separated by a very limited space, which allows a few Ca(2+) ions passing through Ca(v)1.2 to activate the RyR2 at concentration levels that would be deleterious to the whole cell. The limited space also allows Ca(2+) inactivation of Ca(v)1.2. We have found that not all couplons are the same and that their properties are likely determined by their molecular partners which, in turn, determine their excitability. In particular, there are a class of couplons that lie outside the RyR2-Ca(v)1.2 dyad; in this case, the RyR2 is close to caveolin-3 rather than Ca(v)1.2. These extra-dyadic couplons are probably controlled by the multitude of molecules associated with caveolin-3 and may modulate contractile force under situations such as stress. It has long been assumed that like the skeletal muscle, the RyR2 in the couplon are arranged in a structured array with the RyR2 interacting with each other via domain 6 of the RyR2 molecule. This arrangement was thought to provide local control of RyR2 excitability. Using 3D electron tomography of the couplon, we show that the RyR2 in the couplon do not form an ordered pattern, but are scattered throughout it. Relatively few are in a checkerboard pattern--many RyR2 sit edge-to-edge, a configuration which might preclude their controlling each other's excitability. The discovery of this structure makes many models of cardiac couplon function moot and is a current avenue of further research.

We formulate and analyze difference equations on graphs analogous to time-independent diffusion equations arising in the study of diffuse scattering in continuous media. Moreover, we show how to construct solutions in the presence of weak scatterers from the solution to the homogeneous (background problem) using Born series, providing necessary conditions for convergence and demonstrating the process through numerous examples. In addition, we outline a method for finding Green's functions for Cayley graphs for both abelian and non-abelian groups. Finally, we conclude with a discussion of the effects of sparsity on our method and results, outlining the simplifications that can be made provided that the scatterers are weak and well-separated.

Agonist-stimulated smooth muscle Ca2+ waves regulate blood vessel tone and vasomotion. Previous studies employing cytoplasmic Ca2+ indicators revealed that these Ca2+ waves were stimulated by a combination of inositol 1,4,5-trisphosphate- and Ca2+ -induced Ca2+ release from the endo/sarcoplasmic reticulum. Herein, we present the first report of endothelin-1 stimulated waves of Ca2+ depletion from the sarcoplasmic reticulum of vascular smooth muscle cells using a calsequestrin-targeted Ca2+ indicator. Our findings confirm that these waves are due to regenerative Ca2+ -induced Ca2+ release by the receptors for inositol 1,4,5-trisphosphate. Our main new finding is a transient elevation in SR luminal Ca2+ concentration ([Ca2+](SR)) both at the site of wave initiation, just before regenerative Ca2+ release commences, and at the advancing wave front, during propagation. This strongly suggests a role for [Ca2+](SR) in the activation of inositol 1,4,5-trisphosphate receptors during agonist-induced calcium waves. In addition, quantitative analysis of the gradual decrease in the velocity of the depletion wave, observed in the absence of external Ca2+, indicates continuity of the lumen of the sarcoplasmic reticulum network. Finally, our observation that the depletion wave was arrested by the nuclear envelope may have implications for selective Ca2+ signalling.

Agonist-stimulated waves of elevated cytoplasmic Ca2+ concentration ([Ca2+]i ) regulate blood vessel tone and vasomotion in vascular smooth muscle. Previous studies employing cytoplasmic Ca2+ indicators revealed that these Ca2+ waves were generated by a combination of inositol 1,4,5-trisphosphate (IP3) and Ca2+ induced Ca2+ release (CICR) from the sarcoplasmic reticulum (SR); although, some of the mechanistic details remain uncertain. However, these findings were derived indirectly from observing agonist-induced [Ca2+]i fluctuations in the cytoplasm.Here, for the first time, we have recorded Endothelin-1 (ET-1) induced waves of Ca2+ depletion from the SR lumen in vascular smooth muscle cells (VSMCs) using a calsequestrin-targeted Ca2+ indicator. Our findings show that these waves: (1) are due to regenerative CICR by the receptors for IP3 (IP3R), (2) have a marked latency period, (3) are characterized by a transient increase in SR Ca2+ ([Ca2+]SR ) both at the point of origin and at the wave front, (4) proceed with diminishing velocity, and (5) are arrested by the nuclear envelope. Our quantitative model indicates that the gradual decrease in the velocity of the SR depletion wave, in the absence of external Ca2+, results from continuity of the SR luminal network.

We present results from a novel study of Ca2+ waves in vascular smooth muscle cells using a sarcoplasmic reticulum (SR)-targeted Ca2+ indicator that specifically binds to the luminal protein calsequestrin.Agonist-stimulated waves of elevated cytoplasmic calcium concentration regulate blood vessel tone, and vasomotion in vascular smooth muscle. Previous studies employing cytoplasmic calcium indicators revealed that these calcium waves are generated by a combination of inositol 1,4,5-trisphosphate (IP3) and calcium-induced calcium release (CICR) from the SR. Our findings confirm that these waves are due to regenerative CICR by the receptors for IP3 (IP3R).The main new finding from our bservations is a transient elevation in luminal SR Ca2+ concentration ([Ca2+]_SR) both at the site of wave initiation, just before regenerative Ca2+ release commences, and at the advancing wave front, during wave propagation. This strongly suggests a role of [Ca2+]_SR in activation of IP3R.In addition, we find that these depletion waves are characterized by a decreasing velocity as they progress. We developed a quantitative diffusional model to analyze this finding and conclude that the gradual decrease in the velocity of the SR Ca2+ depletion wave, observed in the absence of external calcium, indicates continuity of the lumen of the sarcoplasmicreticulum network.Finally, our observation that the depletion wave was arrested by the nuclear envelope may have implications for selective Ca2+ signalling.