Global asymptotics for polynomials orthogonal with exponential quartic weight

2015 ◽  
pp. 1267-1296
Author(s):  
R. Wong ◽  
L. Zhang
Keyword(s):  
Global Asymptotics

scholarly journals Crystallization of Random Matrix Orbits

2018 ◽  
Vol 2020 (3) ◽  
pp. 883-913 ◽  
Author(s):  
Vadim Gorin ◽  
Adam W Marcus
Keyword(s):  
Random Matrix ◽  
Special Functions ◽  
Free Field ◽  
Gaussian Free Field ◽  
Complex Quaternion ◽  
Cutting Out ◽  
Multiplicative Convolution ◽  
Global Asymptotics ◽  
Associated Special Functions ◽  
Derivatives Of

Abstract Three operations on eigenvalues of real/complex/quaternion (corresponding to $\beta =1,2,4$) matrices, obtained from cutting out principal corners, adding, and multiplying matrices, can be extrapolated to general values of $\beta>0$ through associated special functions. We show that the $\beta \to \infty $ limit for these operations leads to the finite free projection, additive convolution, and multiplicative convolution, respectively. The limit is the most transparent for cutting out the corners, where the joint distribution of the eigenvalues of principal corners of a uniformly-random general $\beta $ self-adjoint matrix with fixed eigenvalues is known as the $\beta $-corners process. We show that as $\beta \to \infty $ these eigenvalues crystallize on an irregular lattice consisting of the roots of derivatives of a single polynomial. In the second order, we observe a version of the discrete Gaussian Free Field put on top of this lattice, which provides a new explanation as to why the (continuous) Gaussian Free Field governs the global asymptotics of random matrix ensembles.

scholarly journals Global asymptotics of particle transport in porous medium

2019 ◽  
Vol 1425 ◽  
pp. 012104
Author(s):  
L I Kuzmina ◽  
Yu V Osipov ◽  
Yu G Zheglova
Keyword(s):  
Porous Medium ◽  
Particle Transport ◽  
Global Asymptotics

Dynamics of Neural Networks as Nonlinear Systems with Several Equilibria

2011 ◽  
pp. 331-357 ◽  
Author(s):  
Daniela Danciu
Keyword(s):  
Neural Network ◽  
Dynamical System ◽  
Neural Networks ◽  
Time Delay ◽  
Dynamical Systems ◽  
Qualitative Properties ◽  
The Neural Network ◽  
The Neural Networks ◽  
Global Asymptotics ◽  
Dynamical Properties

Neural networks—both natural and artificial, are characterized by two kinds of dynamics. The first one is concerned with what we would call “learning dynamics”. The second one is the intrinsic dynamics of the neural network viewed as a dynamical system after the weights have been established via learning. The chapter deals with the second kind of dynamics. More precisely, since the emergent computational capabilities of a recurrent neural network can be achieved provided it has suitable dynamical properties when viewed as a system with several equilibria, the chapter deals with those qualitative properties connected to the achievement of such dynamical properties as global asymptotics and gradient-like behavior. In the case of the neural networks with delays, these aspects are reformulated in accordance with the state of the art of the theory of time delay dynamical systems.

scholarly journals GLOBAL ASYMPTOTICS OF THE FILTRATION PROBLEM IN A POROUS MEDIUM

2019 ◽  
Vol 15 (2) ◽  
pp. 77-85 ◽  
Author(s):  
Lyudmila Kuzmina ◽  
Yuri Osipov ◽  
Yulia Zheglova
Keyword(s):  
Porous Medium ◽  
Asymptotic Solution ◽  
Size Exclusion ◽  
Underground Structures ◽  
Fractional Flow ◽  
Filtration Problem ◽  
Variable Porosity ◽  
Particle Retention ◽  
Global Asymptotics ◽  
Exclusion Mechanism

Filtration of the suspension in a porous medium is important when strengthening the soil and creating watertight partitions for the constructi on of tunnels and underground structures. A model of deep bed filtration with variable porosity and fractional flow, and a size-exclusion mechanism of particle retention are considered. A global asymptotic solution is constructed in the entire domain in which the filtering process takes place. The obtained asymptotics is close to the numerical solution.

scholarly journals GLOBAL ASYMPTOTICS OF STIELTJES–WIGERT POLYNOMIALS

2013 ◽  
Vol 11 (05) ◽  
pp. 1350028 ◽  
Author(s):  
Y. T. LI ◽  
R. WONG
Keyword(s):  
Complex Plane ◽  
Airy Function ◽  
The Other ◽  
Asymptotic Formulas ◽  
Global Asymptotics

Asymptotic formulas are derived for the Stieltjes–Wigert polynomials Sn(z; q) in the complex plane as the degree n grows to infinity. One formula holds in any disc centered at the origin, and the other holds outside any smaller disc centered at the origin; the two regions together cover the whole plane. In each region, the q-Airy function Aq(z) is used as the approximant. For real x > 1/4, a limiting relation is also established between the q-Airy function Aq(x) and the ordinary Airy function Ai (x) as q → 1.

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