The almost-sure asymptotic behavior of the solution to the stochastic heat equation with L\'evy noise

We examine the almost-sure asymptotics of the solution to the stochastic heat equation driven by a L\'evy space-time white noise. When a spatial point is fixed and time tends to infinity, we show that the solution develops unusually high peaks over short time intervals, even in the case of additive noise, which leads to a breakdown of an intuitively expected strong law of large numbers. More precisely, if we normalize the solution by an increasing nonnegative function, we either obtain convergence to $0$, or the limit superior and/or inferior will be infinite. A detailed analysis of the jumps further reveals that the strong law of large numbers can be recovered on discrete sequences of time points increasing to infinity. This leads to a necessary and sufficient condition that depends on the L\'evy measure of the noise and the growth and concentration properties of the sequence at the same time. Finally, we show that our results generalize to the stochastic heat equation with a multiplicative nonlinearity that is bounded away from zero and infinity.Comment: Forthcoming in The Annals of Probabilit

Parameter Estimation of Sigmoid Superpositions: Dynamical System Approach

Superposition of sigmoid function over a finite time interval is shown to be equivalent to the linear combination of the solutions of a linearly parameterized system of logistic differential equations. Due to the linearity with respect to the parameters of the system, it is possible to design an effective procedure for parameter adjustment. Stability properties of this procedure are analyzed. Strategies shown in earlier studies to facilitate learning such as randomization of a learning sequence and adding specially designed disturbances during the learning phase are requirements for guaranteeing convergence in the learning scheme proposed.Comment: 30 pages, 7 figure

Chaotic Field Theory - a Sketch

Spatio-temporally chaotic dynamics of a classical field can be described by means of an infinite hierarchy of its unstable spatio-temporally periodic solutions. The periodic orbit theory yields the global averages characterizing the chaotic dynamics, as well as the starting semiclassical approximation to the quantum theory. New methods for computing corrections to the semiclassical approximation are developed; in particular, a nonlinear field transformation yields the perturbative corrections in a form more compact than the Feynman diagram expansions.Comment: 22 pp, 24 figs, uses elsart.cl

Spectral density of generalized Wishart matrices and free multiplicative convolution

We investigate the level density for several ensembles of positive random matrices of a Wishart--like structure, $W=XX^{\dagger}$, where $X$ stands for a nonhermitian random matrix. In particular, making use of the Cauchy transform, we study free multiplicative powers of the Marchenko-Pastur (MP) distribution, ${\rm MP}^{\boxtimes s}$, which for an integer $s$ yield Fuss-Catalan distributions corresponding to a product of $s$ independent square random matrices, $X=X_1\cdots X_s$. New formulae for the level densities are derived for $s=3$ and $s=1/3$. Moreover, the level density corresponding to the generalized Bures distribution, given by the free convolution of arcsine and MP distributions is obtained. We also explain the reason of such a curious convolution. The technique proposed here allows for the derivation of the level densities for several other cases.Comment: 10 latex pages including 4 figures, Ver 4, minor improvements and references updat

The Emergence of Gravitational Wave Science: 100 Years of Development of Mathematical Theory, Detectors, Numerical Algorithms, and Data Analysis Tools

On September 14, 2015, the newly upgraded Laser Interferometer Gravitational-wave Observatory (LIGO) recorded a loud gravitational-wave (GW) signal, emitted a billion light-years away by a coalescing binary of two stellar-mass black holes. The detection was announced in February 2016, in time for the hundredth anniversary of Einstein's prediction of GWs within the theory of general relativity (GR). The signal represents the first direct detection of GWs, the first observation of a black-hole binary, and the first test of GR in its strong-field, high-velocity, nonlinear regime. In the remainder of its first observing run, LIGO observed two more signals from black-hole binaries, one moderately loud, another at the boundary of statistical significance. The detections mark the end of a decades-long quest, and the beginning of GW astronomy: finally, we are able to probe the unseen, electromagnetically dark Universe by listening to it. In this article, we present a short historical overview of GW science: this young discipline combines GR, arguably the crowning achievement of classical physics, with record-setting, ultra-low-noise laser interferometry, and with some of the most powerful developments in the theory of differential geometry, partial differential equations, high-performance computation, numerical analysis, signal processing, statistical inference, and data science. Our emphasis is on the synergy between these disciplines, and how mathematics, broadly understood, has historically played, and continues to play, a crucial role in the development of GW science. We focus on black holes, which are very pure mathematical solutions of Einstein's gravitational-field equations that are nevertheless realized in Nature, and that provided the first observed signals.Comment: 41 pages, 5 figures. To appear in Bulletin of the American Mathematical Societ

Theory and applications of free-electron vortex states

Both classical and quantum waves can form vortices: with helical phase fronts and azimuthal current densities. These features determine the intrinsic orbital angular momentum carried by localized vortex states. In the past 25 years, optical vortex beams have become an inherent part of modern optics, with many remarkable achievements and applications. In the past decade, it has been realized and demonstrated that such vortex beams or wavepackets can also appear in free electron waves, in particular, in electron microscopy. Interest in free-electron vortex states quickly spread over different areas of physics: from basic aspects of quantum mechanics, via applications for fine probing of matter (including individual atoms), to high-energy particle collision and radiation processes. Here we provide a comprehensive review of theoretical and experimental studies in this emerging field of research. We describe the main properties of electron vortex states, experimental achievements and possible applications within transmission electron microscopy, as well as the possible role of vortex electrons in relativistic and high-energy processes. We aim to provide a balanced description including a pedagogical introduction, solid theoretical basis, and a wide range of practical details. Special attention is paid to translate theoretical insights into suggestions for future experiments, in electron microscopy and beyond, in any situation where free electrons occur.Comment: 87 pages, 34 figure