An induced real quaternion spherical ensemble of random matrices

We study the induced spherical ensemble of non-Hermitian matrices with real quaternion entries (considering each quaternion as a [Formula: see text] complex matrix). We define the ensemble by the matrix probability distribution function that is proportional to [Formula: see text] These matrices can also be constructed via a procedure called ‘inducing’, using a product of a Wishart matrix (with parameters [Formula: see text]) and a rectangular Ginibre matrix of size [Formula: see text]. The inducing procedure imposes a repulsion of eigenvalues from [Formula: see text] and [Formula: see text] in the complex plane with the effect that in the limit of large matrix dimension, they lie in an annulus whose inner and outer radii depend on the relative size of [Formula: see text], [Formula: see text] and [Formula: see text]. By using functional differentiation of a generalized partition function, we make use of skew-orthogonal polynomials to find expressions for the eigenvalue [Formula: see text]-point correlation functions, and in particular the eigenvalue density (given by [Formula: see text]). We find the scaled limits of the density in the bulk (away from the real line) as well as near the inner and outer annular radii, in the four regimes corresponding to large or small values of [Formula: see text] and [Formula: see text]. After a stereographic projection, the density is uniform on a spherical annulus, except for a depletion of eigenvalues on a great circle corresponding to the real axis (as expected for a real quaternion ensemble). We also form a conjecture for the behavior of the density near the real line based on analogous results in the [Formula: see text] and [Formula: see text] ensembles; we support our conjecture with data from Monte Carlo simulations of a large number of matrices drawn from the [Formula: see text] induced spherical ensemble. This ensemble is a quaternionic analog of a model of a one-component charged plasma on a sphere, with soft wall boundary conditions.

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The matrix analogue of the Blackwell renewal theorem on the real line

Sbornik Mathematics ◽

10.1070/sm2006v197n03abeh003762 ◽

2006 ◽

Vol 197 (3) ◽

pp. 369-386 ◽

Cited By ~ 8

Author(s):

M S Sgibnev

Keyword(s):

Real Line ◽

Renewal Theorem ◽

The Real ◽

The Matrix

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Recurrence for Markov processes on N lines

Journal of Applied Probability ◽

10.1017/s0021900200114627 ◽

1971 ◽

Vol 8 (04) ◽

pp. 724-730

Author(s):

Mark Pinsky

Keyword(s):

Markov Process ◽

Characteristic Function ◽

Markov Processes ◽

Real Line ◽

Transition Function ◽

Characteristic Functions ◽

The Real ◽

Matrix Logarithm ◽

The Matrix ◽

Continuous Markov Process

Let Λ = R 1 × {1, 2, ···, N} denote N copies of the real line and ξ(t) = (X(t), α(t))be a right-continuous Markov process taking values in A having transition function of the form P(t, (x, α), A × {β}) = Fαβ (t, A – x). Fukushima and Hitsuda [2] have found the most general such transition function; the (matrix) logarithm of its characteristic function is decomposed into a Lévy-Khintchine integral on the diagonal and multiples of characteristic functions off the diagonal.

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Vector Interpretation of the Matrix Orthogonality on the Real Line

Acta Applicandae Mathematicae ◽

10.1007/s10440-010-9577-3 ◽

2010 ◽

Vol 112 (3) ◽

pp. 357-383 ◽

Cited By ~ 2

Author(s):

A. Branquinho ◽

F. Marcellán ◽

A. Mendes

Keyword(s):

Real Line ◽

The Real ◽

The Matrix

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On the Schur product ofH-matrices and non-negative matrices, and related inequalities

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100037932 ◽

1964 ◽

Vol 60 (3) ◽

pp. 425-431 ◽

Cited By ~ 26

Author(s):

M. S. Lynn

Keyword(s):

Real Line ◽

Positive Definite ◽

Positive Definite Matrices ◽

The Real ◽

Schur Product ◽

Symmetric Positive Definite ◽

The Matrix ◽

Image Position ◽

Symmetric Positive Definite Matrices

1.Introduction. Let ℛndenote the set of alln×nmatrices with real elements, and letdenote the subset of ℛnconsisting of all real,n×n, symmetric positive-definite matrices. We shall use the notationto denote that minor of the matrixA= (aij) ∈ ℛnwhich is the determinant of the matrixTheSchur Product(Schur (14)) of two matricesA, B∈ ℛnis denned bywhereA= (aij),B= (bij),C= (cij) andLet ϕ be the mapping of ℛninto the real line defined byfor allA∈ ℛn, where, as in the sequel,.

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Recurrence for Markov processes on N lines

Journal of Applied Probability ◽

10.2307/3212236 ◽

1971 ◽

Vol 8 (4) ◽

pp. 724-730

Author(s):

Mark Pinsky

Keyword(s):

Markov Process ◽

Characteristic Function ◽

Markov Processes ◽

Real Line ◽

Transition Function ◽

Characteristic Functions ◽

The Real ◽

Matrix Logarithm ◽

The Matrix ◽

Continuous Markov Process

Let Λ = R1 × {1, 2, ···, N} denote N copies of the real line and ξ(t) = (X(t), α(t))be a right-continuous Markov process taking values in A having transition function of the form P(t, (x, α), A × {β}) = Fαβ(t, A – x). Fukushima and Hitsuda [2] have found the most general such transition function; the (matrix) logarithm of its characteristic function is decomposed into a Lévy-Khintchine integral on the diagonal and multiples of characteristic functions off the diagonal.

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Vector Semi-Fredholm Toeplitz Operators and Mean Winding Numbers

Nagoya Mathematical Journal ◽

10.1017/s0027763000009703 ◽

2009 ◽

Vol 195 ◽

pp. 57-75

Author(s):

Dmitry Yakubovich

Keyword(s):

Toeplitz Operator ◽

Real Line ◽

Matrix Function ◽

Toeplitz Operators ◽

Necessary Conditions ◽

Winding Number ◽

The Real ◽

The Matrix ◽

Complex Valued

AbstractFor a continuous nonvanishing complex-valued function g on the real line, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with matrix-valued symbol G to be semi-Fredholm in terms of mean winding numbers of det G. The matrix function G is assumed to be continuous on the real line, and no other apriori assumptions on it are made.

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The properties of Bessel-type fractional derivatives and integrals

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v11i8.203 ◽

2016 ◽

pp. 3973-3982

Author(s):

V. R. Lakshmi Gorty

Keyword(s):

Fractional Order ◽

Real Line ◽

Computer Algebra System ◽

Fractional Derivatives ◽

Graphical Representation ◽

Fractional Integrals ◽

The Real ◽

Fractional Derivatives And Integrals ◽

Half Axis ◽

Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

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