Related papers: Explicit Multi-Matrix Topological Expansion for Qu…
In this work, we give a rigorous explicit formula for the Lyapunov exponent for some binary infinite products of random $2\times 2$ real matrices. All these products are constructed using only two types of matrices, $A$ and $B$, which are…
We present the generating function approach to the perturbative exponentiation of correlators of a product of Wilson lines and loops. The exponentiated expression is presented in closed form as an algebraic function of correlators of known…
Spectral decomposition of matrices is a recurring and important task in applied mathematics, physics and engineering. Many application problems require the consideration of matrices of size three with spectral decomposition over the real…
We develop a new framework to compute the exact correlators of characteristic polynomials, and their inverses, in random matrix theory. Our results hold for general potentials and incorporate the effects of an external source. In matrix…
We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves…
A combinatorial construction is used to analyze the properties of polyhedral products and generalized moment-angle complexes with respect to certain operations on CW pairs including exponentiation. This allows for the construction of…
In this paper, we propose an incremental algorithm for computing cylindrical algebraic decompositions. The algorithm consists of two parts: computing a complex cylindrical tree and refining this complex tree into a cylindrical tree in real…
We prove that the squared singular values of a fixed matrix multiplied with a truncation of a Haar distributed unitary matrix are distributed by a polynomial ensemble. This result is applied to a multiplication of a truncated unitary matrix…
Borel summable semiclassical expansions in 1D quantum mechanics are considered. These are the Borel summable expansions of fundamental solutions and of quantities constructed with their help. An expansion, called topological,is constructed…
An algorithm for numerically computing the exponential of a matrix is presented. We have derived a polynomial expansion of $e^x$ by computing it as an initial value problem using a symbolic programming language. This algorithm is shown to…
Products and sums of random matrices have seen a rapid development in the past decade due to various analytical techniques available. Two of these are the harmonic analysis approach and the concept of polynomial ensembles. Very recently, it…
The numerical computation of the exponentiation of a real matrix has been intensively studied. The main objective of a good numerical method is to deal with round-off errors and computational cost. The situation is more complicated when…
We assume some standard choices for the branch cuts of a group of functions and consider the problem of then calculating the branch cuts of expressions involving those functions. Typical examples include the addition formulae for inverse…
Tangent Model Composition (TMC) is a method to combine component models independently fine-tuned around a pre-trained point. Component models are tangent vectors to the pre-trained model that can be added, scaled, or subtracted to support…
I study the product of independent identically distributed $D\times D$ random probability matrices. Some exact asymptotic results are obtained. I find that both the left and the right products approach exponentially to a probability…
We study the top Lyapunov exponents of random products of positive $2 \times 2$ matrices and obtain an efficient algorithm for its computation. As in the earlier work of Pollicott, the algorithm is based on the Fredholm theory of…
We provide a recursive method for constructing product formula approximations to exponentials of commutators, giving the first approximations that are accurate to arbitrarily high order. Using these formulas, we show how to approximate…
We make a new attempt at the recently suggested program to express knot polynomials through topological vertices, which can be considered as a possible approach to the tangle calculus: we discuss the Macdonald deformation of the relation…
We develop a class of integrals on a manifold M called exponential iterated integrals, an extension of K. T. Chen's iterated integrals. It is shown that the matrix entries of any upper triangular representation of the fundamental group of M…
Trace and extreme eigenvalues of a product of truncated Toeplitz matrices. The singular case. In a first theorem we give an asymptotic expansion of Tr (T_N (f_1) T_N^{-1}(f_2)) where f1 ({\theta}) = |1 - e^{i {\theta}} | ^{2{{\alpha}1}c1…