A DEIM Induced CUR Factorization

We derive a CUR matrix factorization based on the Discrete Empirical Interpolation Method (DEIM). For a given matrix $A$, such a factorization provides a low rank approximate decomposition of the form $A \approx C U R$, where $C$ and $R$ are subsets of the columns and rows of $A$, and $U$ is constructed to make $CUR$ a good approximation. Given a low-rank singular value decomposition $A \approx V S W^T$, the DEIM procedure uses $V$ and $W$ to select the columns and rows of $A$ that form $C$ and $R$. Through an error analysis applicable to a general class of CUR factorizations, we show that the accuracy tracks the optimal approximation error within a factor that depends on the conditioning of submatrices of $V$ and $W$. For large-scale problems, $V$ and $W$ can be approximated using an incremental QR algorithm that makes one pass through $A$. Numerical examples illustrate the favorable performance of the DEIM-CUR method, compared to CUR approximations based on leverage scores

Classifying the 2022 status of Tsuga Canadensis (Eastern Hemlock) along the Kentucky portion of the Pine Mountain Wildlands Corridor.

The invasion of the Hemlock Woolly Adelgid (HWA) (Adelges tsugae) has posed a continual threat in the United States to the Eastern Hemlock (Tsuga canadensis) trees since the 1950s. HWA feed on eastern hemlock needles, reducing the amount of healthy photosynthesizing vegetative area. The use of satellite imagery has been instrumental in identifying areas of eastern hemlock presence. Satellite platforms like Landsat and AVIRIS are commonly used for identification, classification, and mapping of eastern hemlock. Sentinel-2 imagery was released in 2015 for free access. It has a finer spatial grain of with the majority of the bands at 10 and 20 m compared to the 30m resolution of Landsat, for example, and has multiple NIR and SWIR bands where previously used satellites have only one of each, making it ideal for the classification of eastern hemlock trees in the eastern United States. The study will use summer and winter Sentinel-2 imagery in an attempt to answer three questions: 1) What is the current extent of eastern hemlock along the portion of the Pine Mountain Wildlands Corridor within Kentucky? 2) Can various stages of hemlock decline be identified within areas of known hemlock presence? Using a Random Forest classification method in the ArcGIS Pro Environment, hemlock presence was predicted with a 94% accuracy. The variation in spectral signature of eastern hemlock due to decline led to the inability to predict health stages, however, hemlock canopy coverage was predicted with an 83.6% accuracy. Mapping eastern hemlock trees can inform land management of the status of hemlock death, implications on forest health for areas of death, and identify areas in which treatment is needed on their lands

Symbolic Powers of Monomial Ideals

We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal $I$ in $k[x_0, \ldots, x_n]$ we show $I^{t(m+e-1)-e+r)}$ is a subset of $M^{(t-1)(e-1)+r-1}(I^{(m)})^t$ for all positive integers $m$, $t$ and $r$, where $e$ is the big-height of $I$ and $M = (x_0, \ldots, x_n)$. This captures two conjectures ($r=1$ and $r=e$): one of Harbourne-Huneke and one of Bocci-Cooper-Harbourne. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals.Comment: 15 pages. Fixed typ

Spectral approximation of banded Laurent matrices with localized random perturbations

This paper explores the relationship between the spectra of perturbed infinite banded Laurent matrices $L(a)+K$ and their approximations by perturbed circulant matrices $C_{n}(a)+P_{n}KP_{n}$ for large $n$. The entries $K_{jk}$ of the perturbation matrices assume values in prescribed sets $\Omega_{jk}$ at the sites $(j,k)$ of a fixed set $E$, and are zero at the sites $(j,k)$ outside $E$. With ${\cal K}_{\Omega}^{E}$ denoting the ensemble of these perturbation matrices, it is shown that \ud \displaystyle\lim_{n\to\infty} \ud \displaystyle\bigcup_{K\in{\cal K}_{\Omega}^{E}}\ud sp(C_{n}(a)+P_{n}KP_{n})=\ud \displaystyle\bigcup_{K\in{\cal K}_{\Omega}^{E}}\ud sp(L(a)=K)\ud under several fairly general assumptions on $E$ and $\Omega$

Growth rate for the expected value of a generalized random Fibonacci sequence

A random Fibonacci sequence is defined by the relation g_n = | g_{n-1} +/- g_{n-2} |, where the +/- sign is chosen by tossing a balanced coin for each n. We generalize these sequences to the case when the coin is unbalanced (denoting by p the probability of a +), and the recurrence relation is of the form g_n = |\lambda g_{n-1} +/- g_{n-2} |. When \lambda >=2 and 0 < p <= 1, we prove that the expected value of g_n grows exponentially fast. When \lambda = \lambda_k = 2 cos(\pi/k) for some fixed integer k>2, we show that the expected value of g_n grows exponentially fast for p>(2-\lambda_k)/4 and give an algebraic expression for the growth rate. The involved methods extend (and correct) those introduced in a previous paper by the second author

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as $\lambda \to \infty$, $\dim (\sigma(H_\lambda)) \cdot \log \lambda$ converges to an explicit constant ($\approx 0.88137$). We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schr\"odinger dynamics generated by the Fibonacci Hamiltonian.Comment: 23 page

Feminist phenomenology and the woman in the running body

Modern phenomenology, with its roots in Husserlian philosophy, has been taken up and utilised in a myriad of ways within different disciplines, but until recently has remained relatively under-used within sports studies. A corpus of sociological-phenomenological work is now beginning to develop in this domain, alongside a longer standing literature in feminist phenomenology. These specific social-phenomenological forms explore the situatedness of lived-body experience within a particular social structure. After providing a brief overview of key strands of phenomenology, this article considers some of the ways in which sociological, and particularly feminist phenomenology, might be used to analyse female sporting embodiment. For illustrative purposes, data from an autophenomenographic project on female distance running are also included, in order briefly to demonstrate the application of phenomenology within sociology, as both theoretical framework and methodological approach

Sporting embodiment: sports studies and the (continuing) promise of phenomenology

Whilst in recent years sports studies have addressed the calls ‘to bring the body back in’ to theorisations of sport and physical activity, the ‘promise of phenomenology’ remains largely under-realised with regard to sporting embodiment. Relatively few accounts are grounded in the ‘flesh’ of the lived sporting body, and phenomenology offers a powerful framework for such analysis. A wide-ranging, multi-stranded, and interpretatively contested perspective, phenomenology in general has been taken up and utilised in very different ways within different disciplinary fields. The purpose of this article is to consider some selected phenomenological threads, key qualities of the phenomenological method, and the potential for existentialist phenomenology in particular to contribute fresh perspectives to the sociological study of embodiment in sport and exercise. It offers one way to convey the ‘essences’, corporeal immediacy and textured sensuosity of the lived sporting body. The use of Interpretative Phenomenological Analysis (IPA) is also critically addressed. Key words: phenomenology; existentialist phenomenology; interpretative phenomenological analysis (IPA); sporting embodiment; the lived-body; Merleau-Pont