Computing the qq-Multiplicity of the Positive Roots of slr+1(C)\mathfrak{sl}_{r+1}(\mathbb{C}) and Products of Fibonacci Numbers

Using Kostant's weight multiplicity formula, we describe and enumerate the terms contributing a nonzero value to the multiplicity of a positive root $\mu$ in the adjoint representation of $\mathfrak{sl}_{r+1}(\mathbb{C})$, which we denote $L(\tilde{\alpha})$, where $\tilde{\alpha}$ is the highest root of $\mathfrak{sl}_{r+1}(\mathbb{C})$. We prove that the number of terms contributing a nonzero value in the multiplicity of the positive root $\mu=\alpha_i+\alpha_{i+1}+\cdots+\alpha_j$ with $1\leq i\leq j\leq r$ in $L(\tilde{\alpha})$ is given by the product $F_{i}\cdot F_{r-j+1}$, where $F_n$ is the $n^{\text{th}}$ Fibonacci number. Using this result, we show that the $q$-multiplicity of the positive root $\mu=\alpha_i+\alpha_{i+1}+\cdots+\alpha_j$ with $1\leq i\leq j\leq r$ in the representation $L(\tilde{\alpha})$ is precisely $q^{r-h(\mu)}$, where $h(\mu)=j-i+1$ is the height of the positive root $\mu$. Setting $q=1$ recovers the known result that the multiplicity of a positive root in the adjoint representation of $\mathfrak{sl}_{r+1}(\mathbb{C})$ is one.Comment: 16 pages, 0 figure

Gross\u27s Proof of Local Existence for the Coupled Maxwell-Dirac Equations

The Maxwell-Dirac equations are a model for the interaction of a relativistic electron with an electromagnetic field. It is to be expected that the initial value problem will have an unique solution which exists for all time t \u3e0, for all appropriate initial conditions. This is not yet known, but in 1966, Leonard Gross proved a local existence theorem. In this thesis, we will present an overview of Leonard Gross\u27s proof

Flattened Stirling Permutations

Recall that a Stirling permutation is a permutation on the multiset $\{1,1,2,2,\ldots,n,n\}$ such that any numbers appearing between repeated values of $i$ must be greater than $i$. We call a Stirling permutation ``flattened'' if the leading terms of maximal chains of ascents (called runs) are in weakly increasing order. Our main result establishes a bijection between flattened Stirling permutations and type $B$ set partitions of $\{0,\pm1,\pm2,\ldots,\pm (n-1)\}$, which are known to be enumerated by the Dowling numbers, and we give an independent proof of this fact. We also determine the maximal number of runs for any flattened Stirling permutation, and we enumerate flattened Stirling permutations with a small number of runs or with two runs of equal length. We conclude with some conjectures and generalizations worthy of future investigation.Comment: 15 pages, 1 figure, 2 tabl

They Be Taggin, Don\u27t They?: The Acquisition of Invariant Be

Does African-American English (AAE) have a separate grammar from Standard American English (SAE)? AAE has a number of distinctive features, several of which have been discussed extensively in the sociolinguistic literature: the variable use of third-person singular -s; the absence of plura -s; the variable use of copula; and others. (See Labov, 1966, 1969a, b, and Fasold, 1972 for discussion.) Perhaps the most distinctive feature of AAE is habitual or invariant be

Direct activation of NADPH oxidase 2 by 2-deoxyribose-1-phosphate triggers nuclear factor kappa B-dependent angiogenesis.

AbstractAims: Deoxyribose-1-phosphate (dRP) is a proangiogenic paracrine stimulus released by cancer cells, platelets, and macrophages and acting on endothelial cells. The objective of this study was to clarify how dRP stimulates angiogenic responses in human endothelial cells.Results: Live cell imaging, electron paramagnetic resonance, pull-down of dRP-interacting proteins, followed by immunoblotting, gene silencing of different NADPH oxidases (NOXs), and their regulatory cosubunits by small interfering RNA (siRNA) transfection, and experiments with inhibitors of the sugar transporter glucose transporter 1 (GLUT1) were utilized to demonstrate that dRP acts intracellularly by directly activating the endothelial NOX2 complex, but not NOX4. Increased reactive oxygen species generation in response to NOX2 activity leads to redox-dependent activation of the transcription factor nuclear factor kappa B (NF-κB), which, in turn, induces vascular endothelial growth factor receptor 2 (VEGFR2) upregulation. Using endothelial tube formation assays, gene silencing by siRNA, and antibody-based receptor inhibition, we demonstrate that the activation of NF-κB and VEGFR2 is necessary for the angiogenic responses elicited by dRP. The upregulation of VEGFR2 and NOX2-dependent stimulation of angiogenesis by dRP were confirmed in excisional wound and Matrigel plug vascularization assays in vivo using NOX2−/− mice.Innovation: For the first time, we demonstrate that dRP acts intracellularly and stimulates superoxide anion generation by direct binding and activation of the NOX2 enzymatic complex.Conclusions: This study describes a novel molecular mechanism underlying the proangiogenic activity of dRP, which involves the sequential activation of NOX2 and NF-κB and upregulation of VEGFR2. Antioxid. Redox Signal. 28, 110–130

Subterranean Termites

8 pp., 9 figures, 5 color photosSubterranean termites are the most destructive pests of wood in the United States. This publication explains the distribution, identification, biology and habits of subterranean termites. It also describes the damage they cause and suggests preventive measures

Integration of electrocatalysts with silicon microcone arrays for minimization of optical and overpotential losses during sunlight-driven hydrogen evolution

Microstructured photoelectrode morphologies can advantageously facilitate integration of optically absorbing electrocatalysts with semiconducting light absorbers, to maintain low overpotentials for fuel production without producing a substantial loss in photocurrent density. We report herein the use of arrays of antireflective, high-aspect-ratio Si microcones (μ-cones), coupled with light-blocking Pt and Co–P catalysts, as photocathodes for H_2 evolution. Thick (∼16 nm) layers of Pt or Co–P deposited onto Si μ-cone arrays yielded absolute light-limited photocurrent densities of ∼32 mA cm^(−2), representing a reduction in light-limited photocurrent density of 6% relative to bare Si μ-cone-array photocathodes, while maintaining high fill factors and low overpotentials for H_2 production from 0.50 M H_2SO_4(aq). The Si μ-cone arrays were embedded in a flexible polymeric membrane and removed from the Si substrate, to yield flexible photocathodes consisting of polymer-embedded arrays of free-standing μ-cones that evolved hydrogen from 0.50 M H_2SO_4(aq)

Induction of Group IVC Phospholipase A2 in Allergic Asthma: Transcriptional Regulation by TNF-α in Bronchoepithelial Cells

Airway inflammation in allergen-induced asthma is associated with eicosanoid release. These bioactive lipids exhibit anti- and pro-inflammatory activities with relevance to pulmonary pathophysiology. We hypothesized that sensitization/challenge using an extract from the ubiquitous fungus, Aspergillus fumigatus (Af), in a mouse model of allergic asthma would result in altered phospholipase gene expression, thus modulating the downstream eicosanoid pathway. We observed the most significant induction in the group IVC phospholipase A2 (cPLA2γ or PLA2G4C). Our results infer that Af extract can induce cPLA2γ levels directly in eosinophils while induction in lung epithelial cells is most likely a consequence of TNF-α secretion by Af-activated macrophages. The mechanism of TNF-α-dependent induction of cPLA2γ gene expression was elucidated through a combination of promoter deletions, ChIP and overexpression studies in human bronchoepithelial cells, leading to the identification of functionally relevant CRE, NF-κB and E-box promoter elements. ChIP analysis demonstrated that RNA polymerase II, c-Jun/ATF-2, p65/p65 and USF1/USF2 complexes are recruited to the cPLA2γ enhancer/promoter in response to TNF-α with overexpression and dominant negative studies implying a strong level of cooperation and interplay between these factors. Overall, our data link cytokine-mediated alterations in cPLA2γ gene expression with allergic asthma and outline a complex regulatory mechanism

A Super-Oxidized Radical Cationic Icosahedral Boron Cluster

While the icosahedral closo-[B₁₂H₁₂]²⁻ cluster does not display reversible electrochemical behavior, perfunctionalization of this species via substitution of all 12 B–H vertices with alkoxy or benzyloxy (OR) substituents engenders reversible redox chemistry, providing access to clusters in the dianionic, monoanionic, and neutral forms. Here, we evaluated the electrochemical behavior of the electron-rich B₁₂(O-3-methylbutyl)₁₂ (1) cluster and discovered that a new reversible redox event that gives rise to a fourth electronic state is accessible through one-electron oxidation of the neutral species. Chemical oxidation of 1 with [N(2,4-Br₂C₆H₃)₃]·⁺ afforded the isolable [1]·⁺ cluster, which is the first example of an open-shell cationic B₁₂ cluster in which the unpaired electron is proposed to be delocalized throughout the boron cluster core. The oxidation of 1 is also chemically reversible, where treatment of [1]·⁺ with ferrocene resulted in its reduction back to 1. The identity of [1]·⁺ is supported by EPR, UV–vis, multinuclear NMR (¹H, ¹¹B), and X-ray photoelectron spectroscopic characterization