Compressible Matrix Algebras and the Distance from Projections to Nilpotents

In this thesis we address two problems from the fields of operator algebras and operator theory. In our first problem, we seek to obtain a description of the unital subalgebras $\mathcal{A}$ of $\mathbb{M}_n(\mathbb{C})$ with the property that $E\mathcal{A}E$ is an algebra for all idempotents $E\in\mathbb{M}_n(\mathbb{C})$. Algebras with this property are said to be \textit{idempotent compressible}. Likewise, we wish to determine which unital subalgebras of $\mathbb{M}_n(\mathbb{C})$ satisfy the analogous property for projections (i.e., self-adjoint idempotents). Such algebras are said to be \textit{projection compressible}. We begin by constructing various examples of idempotent compressible subalgebras of $\mathbb{M}_n(\mathbb{C})$ for each integer $n\geq 3$. Using a case-by-case analysis based on reduced block upper triangular forms, we prove that our list includes all unital projection compressible subalgebras of $\mathbb{M}_3(\mathbb{C})$ up to similarity and transposition. A similar examination indicates that the same phenomenon occurs in the case of unital subalgebras of $\mathbb{M}_n(\mathbb{C})$, $n\geq 4$. We therefore demonstrate that the notions of projection compressibility and idempotent compressibility coincide for unital subalgebras of $\mathbb{M}_n(\mathbb{C})$, and obtain a complete classification of the unital algebras admitting these properties up to similarity and transposition. In our second problem, we address the question of computing the distance from a non-zero projection to the set of nilpotent operators acting on $\mathbb{C}^n$. Building on MacDonald's results in the rank-one case, we prove that the distance from a rank $n-1$ projection to the set of nilpotents in $\mathbb{M}_n(\mathbb{C})$ is $\frac{1}{2}\sec\left(\frac{\pi}{\frac{n}{n-1}+2}\right)$. For each $n\geq 2$, we construct examples of pairs $(Q,T)$ where $Q$ is a projection of rank $n-1$ and $T\in\mathbb{M}_n(\mathbb{C})$ is a nilpotent of minimal distance to $Q$. Moreover, it is shown that any two such pairs are unitarily equivalent. We end by discussing possible extensions of these results in the case of projections of intermediate ranks

Matrix Algebras with a Certain Compression Property I

An algebra $\mathcal{A}$ of $n\times n$ complex matrices is said to be projection compressible if $P\mathcal{A}P$ is an algebra for all orthogonal projections $P\in\mathbb{M}_n(\mathbb{C})$. Analogously, $\mathcal{A}$ is said to be idempotent compressible if $E\mathcal{A}E$ is an algebra for all idempotents $E\in\mathbb{M}_n(\mathbb{C})$. In this paper we construct several examples of unital algebras that admit these properties. In addition, a complete classification of the unital idempotent compressible subalgebras of $\mathbb{M}_3(\mathbb{C})$ is obtained up to similarity and transposition. It is shown that in this setting, the two notions of compressibility agree: a unital subalgebra of $\mathbb{M}_3(\mathbb{C})$ is projection compressible if and only if it is idempotent compressible. Our findings are extended to algebras of arbitrary size in the sequel to this paper.Comment: 23 page

I (Don't) Feel Pretty: Weight Perception, Self-Esteem, and Gender in Adolescence and Young Adulthood

Life course scholars have established that early life experiences are crucial factors in shaping developmental trajectories. Early adolescent health has been one of the most widely studied of these factors, with increased attention being paid to the enduring effects of obesity across the life course. While obesity is shown to be associated with several negative social and psychological outcomes, less is known regarding how self-perceived weight affects individuals’ mental health trajectories through adolescence and young-adulthood. To address this gap, I use nationally representative longitudinal panel data to analyze how self-perceived weight affects self-esteem trajectories during the transition to adulthood. Furthermore, I test the extent to which these processes work differently for young men and women. Consistent with previous literature, I find that, on average, self-esteem increases linearly through adolescence and young adulthood. While weight perception is significantly and negatively associated with self-esteem trajectories for both men and women, the effect is greater for women than men. These findings are relevant for scholars and health practitioners and, in the conclusion of this article, the implications of these findings are discussed

Endogenous versus exogenous carbohydrate oxidation measured by stable isotopes in pre-pubescent children plus 13C abundances in foods consumed three days prior

Purpose: The purposes of the present study were to (a) examine resting metabolism, substrate utilization, and endogenous versus exogenous carbohydrate (CHO) oxidation before and after 30-g rapidly-digesting carbohydrate (RDC) ingestion using indirect calorimetry and breath test analysis of stable isotope concentrations in pre-pubescent children and (b) report the 13C abundances in foods consumed for three days prior. Methods: Nineteen children (n 1⁄4 10 boys, n 1⁄4 9 girls) at Tanner stage I or II participated (mean age ± 95% CI 1⁄4 9.84 ± 0.77 y) in this study. Food was administered to the children for three days preceding their scheduled breath tests. Breath tests and indirect calorimetry were performed after an 8-h fast before and 60 min following consumption of a 30-g simple RDC drink consisting of maltodextrin and sucrose. Open circuit spirometry and indirect calorimetry monitored resting metabolism and CHO oxidation. Separate breath samples were taken every 15 min. Samples of all foods and breath samples were analyzed for 13C and 12C abundances with a stable-isotope mass spectrometer. Results: 13C in expired breath samples were 23.81 + 1.64‰ at baseline and increased every 15 min after consumption of the CHO drink (p \u3c 0.001e0.009). Cumulative total, endogenous, and exogenous CHO utilization increased during the post-prandial period (p \u3c 0.001). Endogenous CHO oxidation was consistently greater than exogenous CHO oxidation (p \u3c 0.001e0.002). Blood glucose was elevated from baseline at 30- and 60-min post-prandial (p \u3c 0.001). Insulin did not change over time (p 1⁄4 0.184). Conclusions: The foods provided during the 3-day controlled diet effectively minimized 13C variation prior to metabolic testing. The 13C abundances of foods reported herein should serve as practical recommendations to reduce 13C intake before breath tests. While endogenous CHO oxidation remained greater in proportion to exogenous CHO oxidation, these findings suggest that even a relatively small amount of RDC can increase exogenous CHO oxidation and blood glucose in normal-weight children. To further examine shifts in endogenous versus exogenous CHO utilization, we recommend that future studies take steps to minimize 13C variation before breath tests and examine changes in substrate metabolism at rest and during exercise in normal weight and overweight pre-pubescent children. Clinical trial registration number: NCT03185884

Basic Understanding of Condensed Phases of Matter via Packing Models

Packing problems have been a source of fascination for millenia and their study has produced a rich literature that spans numerous disciplines. Investigations of hard-particle packing models have provided basic insights into the structure and bulk properties of condensed phases of matter, including low-temperature states (e.g., molecular and colloidal liquids, crystals and glasses), multiphase heterogeneous media, granular media, and biological systems. The densest packings are of great interest in pure mathematics, including discrete geometry and number theory. This perspective reviews pertinent theoretical and computational literature concerning the equilibrium, metastable and nonequilibrium packings of hard-particle packings in various Euclidean space dimensions. In the case of jammed packings, emphasis will be placed on the "geometric-structure" approach, which provides a powerful and unified means to quantitatively characterize individual packings via jamming categories and "order" maps. It incorporates extremal jammed states, including the densest packings, maximally random jammed states, and lowest-density jammed structures. Packings of identical spheres, spheres with a size distribution, and nonspherical particles are also surveyed. We close this review by identifying challenges and open questions for future research.Comment: 33 pages, 20 figures, Invited "Perspective" submitted to the Journal of Chemical Physics. arXiv admin note: text overlap with arXiv:1008.298

Pan-Cancer Analysis of lncRNA Regulation Supports Their Targeting of Cancer Genes in Each Tumor Context

Long noncoding RNAs (lncRNAs) are commonly dys-regulated in tumors, but only a handful are known toplay pathophysiological roles in cancer. We inferredlncRNAs that dysregulate cancer pathways, onco-genes, and tumor suppressors (cancer genes) bymodeling their effects on the activity of transcriptionfactors, RNA-binding proteins, and microRNAs in5,185 TCGA tumors and 1,019 ENCODE assays.Our predictions included hundreds of candidateonco- and tumor-suppressor lncRNAs (cancerlncRNAs) whose somatic alterations account for thedysregulation of dozens of cancer genes and path-ways in each of 14 tumor contexts. To demonstrateproof of concept, we showed that perturbations tar-geting OIP5-AS1 (an inferred tumor suppressor) andTUG1 and WT1-AS (inferred onco-lncRNAs) dysre-gulated cancer genes and altered proliferation ofbreast and gynecologic cancer cells. Our analysis in-dicates that, although most lncRNAs are dysregu-lated in a tumor-specific manner, some, includingOIP5-AS1, TUG1, NEAT1, MEG3, and TSIX, synergis-tically dysregulate cancer pathways in multiple tumorcontexts

Pan-cancer Alterations of the MYC Oncogene and Its Proximal Network across the Cancer Genome Atlas

Although theMYConcogene has been implicated incancer, a systematic assessment of alterations ofMYC, related transcription factors, and co-regulatoryproteins, forming the proximal MYC network (PMN),across human cancers is lacking. Using computa-tional approaches, we define genomic and proteo-mic features associated with MYC and the PMNacross the 33 cancers of The Cancer Genome Atlas.Pan-cancer, 28% of all samples had at least one ofthe MYC paralogs amplified. In contrast, the MYCantagonists MGA and MNT were the most frequentlymutated or deleted members, proposing a roleas tumor suppressors.MYCalterations were mutu-ally exclusive withPIK3CA,PTEN,APC,orBRAFalterations, suggesting that MYC is a distinct onco-genic driver. Expression analysis revealed MYC-associated pathways in tumor subtypes, such asimmune response and growth factor signaling; chro-matin, translation, and DNA replication/repair wereconserved pan-cancer. This analysis reveals insightsinto MYC biology and is a reference for biomarkersand therapeutics for cancers with alterations ofMYC or the PMN

Genomic, Pathway Network, and Immunologic Features Distinguishing Squamous Carcinomas

This integrated, multiplatform PanCancer Atlas study co-mapped and identified distinguishing molecular features of squamous cell carcinomas (SCCs) from five sites associated with smokin