arXiv subjects

Michael P. Lamoureux

Publications and source records attributed to Michael P. Lamoureux.

9 recordsLinked to original sources

arXiv2025

Calculus: a limitless perspective

We propose a novel foundation for calculus that focuses on the notion of approximations while avoiding the use of limits altogether. Continuity is defined as approximation at a point, while differentiability is defined as approximation with a linear function. The errors in approximation are defined as a class of functions with certain properties; rules for combining error functions lead to all the familiar results in differential calculus. We believe that this approach is more natural for students while still giving a rigourous foundation to differential calculus. We demonstrate its utility by deriving the basic differential rules for trigonometric, hyperbolic and exponential functions, as well as L'Hôpital's Rule, Taylor polynomials, and the Fundamental Theorem of Calculus, all via approximation.

math.HO↗

arXiv2021

Application of an unbalanced optimal transport distance and a mixed L1/Wasserstein distance to full waveform inversion

Full waveform inversion (FWI) is an important and popular technique in subsurface earth property estimation. However, using the least-squares norm in the misfit function often leads to the local minimum solution of the optimization problem, and this phenomenon can be explained with the cycle-skipping artifact. Several methods that apply optimal transport distances to mitigate the cycle-skipping artifact have been proposed recently. The optimal transport distance is designed to compare two probability measures. To overcome the mass equality limit, we introduce an unbalanced optimal transport (UOT) distance with KullbackLeibler divergence to balance the mass difference. Also, a mixed L1/Wasserstein distance is constructed that can preserve the convex properties with respect to shift, dilation, and amplitude change operation. An entropy regularization approach and scaling algorithms are used to compute the distance and the gradient efficiently. Two strategies of normalization methods that transform the seismic signals into non-negative functions are provided. Numerical examples are provided to demonstrate the efficiency and effectiveness of the new method.

math.NA↗

arXiv2015

Constructive solutions to Pólya-Schur problems

We present constructive solutions to the following Pólya-Schur problems concerning linear operators on the space of univariate polynomials: Given subsets $Ω_1$ and $Ω_2$ of the complex plane, determine operators that map all polynomials having no zeros in $Ω_1$ to polynomials having no zeros in $Ω_2$, or to the zero polynomial. We describe an explicit class consisting of rank 1 operators and product-composition operators that solve the stated problems for arbitrary $Ω_1$ and $Ω_2$; and this class is shown to comprise all solutions when $Ω_1$ is bounded and $Ω_2$ has non-empty interior. The latter result encompasses a number of open problems and, moreover, gives explicit solutions in cases of circular domains $Ω_1=Ω_2$ where existing characterizations are non-constructive. The paper also treats problems stemming from digital signal processing that are analogous to Pólya-Schur problems. Specifically, we describe all bounded linear operators on Hardy space that preserve the class of outer functions, as well as those that preserve shifted outer functions.

math.CV↗

arXiv2011

Identification of minimum phase preserving operators on the half line

Minimum phase functions are fundamental in a range of applications, including control theory, communication theory and signal processing. A basic mathematical challenge that arises in the context of geophysical imaging is to understand the structure of linear operators preserving the class of minimum phase functions. The heart of the matter is an inverse problem: to reconstruct an unknown minimum phase preserving operator from its value on a limited set of test functions. This entails, as a preliminary step, ascertaining sets of test functions that determine the operator, as well as the derivation of a corresponding reconstruction scheme. In the present paper we exploit a recent breakthrough in the theory of stable polynomials to solve the stated inverse problem completely. We prove that a minimum phase preserving operator on the half line can be reconstructed from data consisting of its value on precisely two test functions. And we derive an explicit integral representation of the unknown operator in terms of this data. A remarkable corollary of the solution is that if a linear minimum phase preserving operator has rank at least two, then it is necessarily injective.

math.FA↗

arXiv2010

Drawing butterflies from the almost Mathieu operator

Plotting spectra of a range of almost Mathieu operators reveals a beautiful fractal-like image that contains multiple copies of a butterfly image. We demonstrate that plotting the butterflies using a gap-labelling scheme based on K-theory or Chern numbers reveals systematic discontinuities in the gap positioning. A proper image is produced only when we take into account these discontinuities, and close the butterfly wingtips at the points of discontinuity. A conjecture is presented showing a simple formula for locating the discontinuities, and numerical evidence is given to support the conjecture. We also present new renderings of this butterfly.

math.OA↗

arXiv2010

Spectra self-similarity for almost Mathieu operators

We determine numerically the self-similarity maps for spectra of the almost Mathieu operators, a two-dimensional fractal-like structure known as the Hofstadter butterfly. The similarity maps each have a horizontal component determined by certain algebraic maps, and vertical component determined by a Mobius transformation, indexed by a semigroup of the matrix group $GL_2(\Z)$. Based on the numerical evidence, we state and prove a continuity result for the similarity maps. We note a connection between the indexing of the similarity maps and Morita equivalence of rotation algebras $A_θ$, a continuous field of C*-algebras.

math.OA↗

arXiv2006

On the Characteristic Polynomial of the Almost Mathieu Operator

Let theta = p/q with p and q relatively prime and u and v a pair of unitaries such that u v = e^{i theta} v u, where u and v generate the rotation C*-algebra A_theta. Let h_{theta, lambda} = u + u^{-1} + lambda/2(v + v^{-1}) be the almost Mathieu operator. By proving an identity of rational functions we show that for q even, the constant term in the characteristic polynomial of h_{θ, λ} is (-1)^{q/2}(1 + (λ/2)^q) - (z_1^q + z_1^{-q} + (λ/2)^q(z_2^q + z_2^{-q})).

math.OA↗

arXiv1997

Meet irreducible ideals in direct limit algebras

We study the meet irreducible ideals in certain direct limit algebras, namely the strongly maximal triangular subalgebras of AF C*-algebras. These ideals have a description in terms of the coordinates, or spectrum, that is a natural extension of one description of meet irreducible ideals in the upper triangular matrices. Additional information is available if the limit algebra is an analytic subalgebra of its C*-envelope or if the analytic algebra is trivially analytic with an injective 0-cocycle. Completely meet irreducible ideals are considered, and a distance formula is presented.

funct-an↗

arXiv1996

The topology of ideals in some triangular AF algebras

A set of meet-irreducible ideals is described for a class of maximal triangular almost finite algebras. This set forms a topological space under the hull-kernel closure, and there is a one-to-one correspondence between closed sets in this space and ideals in the AF algebra. Some connections with nest representations and nest-primitive ideals are also described.

funct-an↗