arXiv subjects

Nicholas West

Publications and source records attributed to Nicholas West.

5 recordsLinked to original sources

arXiv2026

LU Factorization of Discrete Random Matrices

We consider the probability that a discrete random matrix $M_n(\xi)$ is \emph{strongly non-singular}, meaning all its leading principal submatrices are non-singular. This property is equivalent to the existence of an LU factorization. We show that for any discrete random variable $\xi$ with finite support and $|\xi|_\infty < 1$, there is a constant probability that $M_n(\xi)$ is strongly non-singular with a growth factor bounded by $n^{5/2+\delta}$. Furthermore, we provide a tight asymptotic lower bound for this probability as $|\xi|_\infty \to 0$. Finally, we provide exact counts for strongly non-singular binary matrices up to $n=9$ and use these to derive improved upper bounds for the Bernoulli case.

math.NA↗

arXiv2026

What is Jackson's constant?

We prove a refinement of Jackson's theorem on the approximation of Lipschitz functions by trigonometric polynomials. Our result precisely characterizes the leading error term associated with Jackson's construction. We do the same for a related construction commonly used in the kernel polynomial method for spectral density estimation, which is slightly better than Jackson's construction in this respect.

math.NA↗

arXiv2026

Spectral density estimation for normal matrices

The spectral density estimation problem asks for an algorithm that, given an $n\times n$ matrix $A$, outputs a probability measure that is a good approximation to the uniform distribution on the eigenvalues of $A$, called the spectral density of $A$. This paper considers the setting where $A$ is a large normal matrix that is accessible only through matrix-vector product queries. We provide an algorithm that makes just $m$ matrix-vector queries to $A$ and returns, with high probability, a measure within earth mover's distance $O(1/m+\log m/{\sqrt n})$ of the true spectral density of $A$. We provide a complementary lower bound that any algorithm producing an $\varepsilon$-approximation to the true spectral density for large matrices must make $\Omega(1/\varepsilon)$ matrix-vector queries. The lower bound holds even for the more restricted case of real symmetric input matrices. In combination with our upper bound, it shows that spectral density estimation is essentially no harder for complex normal matrices than for real symmetric matrices.

math.NA↗

arXiv2019

Human Computer Interaction Design for Mobile Devices Based on a Smart Healthcare Architecture

Smart and IoT-enabled mobile devices have the potential to enhance healthcare services for both patients and healthcare providers. Human computer interaction design is key to realizing a useful and usable connection between the users and these smart healthcare technologies. Appropriate design of such devices enhances the usability, improves effective operation in an integrated healthcare system, and facilitates the collaboration and information sharing between patients, healthcare providers, and institutions. In this paper, the concept of smart healthcare is introduced, including its four-layer information architecture of sensing, communication, data integration, and application. Human Computer Interaction design principles for smart healthcare mobile devices are outlined, based on user-centered design. These include: ensuring safety, providing error-resistant displays and alarms, supporting the unique relationship between patients and healthcare providers, distinguishing end-user groups, accommodating legacy devices, guaranteeing low latency, allowing for personalization, and ensuring patient privacy. Results are synthesized in design suggestions ranging from personas, scenarios, workflow, and information architecture, to prototyping, testing and iterative development. Finally, future developments in smart healthcare and Human Computer Interaction design for mobile health devices are outlined.

cs.HC↗

arXiv2012

Probability of Failure in Hypersonic Engines Using Large Deviations

We consider a reduced order model of an air-breathing hypersonic engine with a time-dependent stochastic inflow that may cause the failure of the engine. The probability of failure is analyzed by the Freidlin-Wentzell theory, the large deviation principle for finite dimensional stochastic differential equations. We compute the asymptotic failure probability by numerically solving the constrained optimization related to the large deviation problem. A large-deviation-based importance sampling suggested by the most probable inflow perturbation is also implemented to compute the probability of failure of the engine. The numerical simulations show that the importance sampling method is much more efficient than the basic Monte Carlo method.

math.PR↗