scholarly journals Non-linear PDEs for gap probabilities in random matrices and KP theory

2012 ◽  
Vol 241 (23-24) ◽  
pp. 2265-2284 ◽  
Author(s):  
M. Adler ◽  
M. Cafasso ◽  
P. van Moerbeke
Keyword(s):  
Random Matrices ◽  
Non Linear ◽  
Gap Probabilities ◽  
Kp Theory ◽  
Linear Pdes

scholarly journals Singular values and evenness symmetry in random matrix theory

2016 ◽  
Vol 28 (5) ◽  
pp. 873-891 ◽  
Author(s):  
Folkmar Bornemann ◽  
Peter J. Forrester
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Functional Form ◽  
Matrix Theory ◽  
Stereographic Projection ◽  
Singular Values ◽  
Unitary Symmetry ◽  
Matrix Ensembles ◽  
Gap Probabilities ◽  
Orthogonal Ensembles

AbstractComplex Hermitian random matrices with a unitary symmetry can be distinguished by a weight function. When this is even, it is a known result that the distribution of the singular values can be decomposed as the superposition of two independent eigenvalue sequences distributed according to particular matrix ensembles with chiral unitary symmetry. We give decompositions of the distribution of singular values, and the decimation of the singular values – whereby only even, or odd, labels are observed – for real symmetric random matrices with an orthogonal symmetry, and even weight. This requires further specifying the functional form of the weight to one of three types – Gauss, symmetric Jacobi or Cauchy. Inter-relations between gap probabilities with orthogonal and unitary symmetry follow as a corollary. The Gauss case has appeared in a recent work of Bornemann and La Croix. The Cauchy case, when appropriately specialised and upon stereographic projection, gives decompositions for the analogue of the singular values for the circular unitary and circular orthogonal ensembles.

scholarly journals Thinning and conditioning of the circular unitary ensemble

2017 ◽  
Vol 06 (02) ◽  
pp. 1750007 ◽  
Author(s):  
Christophe Charlier ◽  
Tom Claeys
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Random Matrices ◽  
Unitary Matrices ◽  
Toeplitz Determinants ◽  
Random Unitary Matrices ◽  
Gap Probabilities ◽  
Unitary Ensemble

We apply the operation of random independent thinning on the eigenvalues of [Formula: see text] Haar distributed unitary random matrices. We study gap probabilities for the thinned eigenvalues, and we study the statistics of the eigenvalues of random unitary matrices which are conditioned such that there are no thinned eigenvalues on a given arc of the unit circle. Various probabilistic quantities can be expressed in terms of Toeplitz determinants and orthogonal polynomials on the unit circle, and we use these expressions to obtain asymptotics as [Formula: see text].

scholarly journals Mean Field Limit of Interacting Filaments and Vector Valued Non-linear PDEs

2017 ◽  
Vol 166 (5) ◽  
pp. 1276-1309 ◽  
Author(s):  
Hakima Bessaih ◽  
Michele Coghi ◽  
Franco Flandoli
Keyword(s):  
Mean Field ◽  
Field Limit ◽  
Mean Field Limit ◽  
Non Linear ◽  
Vector Valued ◽  
Linear Pdes

scholarly journals Free Convective Flow of Water/Ethylene Glycol Based Micropolar Nanofluid Over a Shrinking Sheet

2021 ◽  
Vol 11 (5) ◽  
pp. 12596-12605
Keyword(s):  
Ethylene Glycol ◽  
Shrinking Sheet ◽  
Heat Equations ◽  
Shooting Technique ◽  
Similarity Transformations ◽  
Micropolar Nanofluid ◽  
Non Linear ◽  
Linear Differential ◽  
The Impact ◽  
Linear Pdes

In this article, the free convective micropolar nanofluid was investigated over a shrinking sheet in the presence of a heat source by tacking into the water/ethylene glycol-based nanofluid account. The physical problem is first modeled. Under the assumptions of Boussinesq's approximation, the governing equations are reduced into non-linear PDEs. The combined non-linear PDEs representing momentum and non-homogeneous heat equations were reduced to a series of regular non-linear differential equations with appropriate similarity transformations. By applying the Runge-Kutta procedure followed by the Shooting technique, the transformed equations are then solved. Via the diagrams, the impact of related parameters characterizing the flow was presented and then addressed. It is observed that volumetric fraction has a substantial influence on the velocity profile and also induces a decrease in the boundary layer because water-based nanofluid has high thermal conductivity relative to Ferro nanofluid based on ethylene glycol.

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