Geometric properties of boundary sections of solutions to the Monge--Amp\`ere equation and applications

In this paper, we establish several geometric properties of boundary sections of convex solutions to the Monge-Amp\`ere equations: the engulfing and separating properties and volume estimates. As applications, we prove a covering lemma of Besicovitch type, a covering theorem and a strong type $p-p$ estimate for the maximal function corresponding to boundary sections. Moreover, we show that the Monge-Amp\`ere setting forms a space of homogeneous type.Comment: 24 page

The 3-fold vertex via stable pairs

The theory of stable pairs in the derived category yields an enumerative geometry of curves in 3-folds. We evaluate the equivariant vertex for stable pairs on toric 3-folds in terms of weighted box counting. In the toric Calabi-Yau case, the result simplifies to a new form of pure box counting. The conjectural equivalence with the DT vertex predicts remarkable identities. The equivariant vertex governs primary insertions in the theory of stable pairs for toric varieties. We consider also the descendent vertex and conjecture the complete rationality of the descendent theory for stable pairs.Comment: Typos fixed. 40 pages, 8 figure

From Triumph to Tragedy: African American Soldiers Fight For Citizenship and Manhood in the Spanish-American-Cuban-Filipino War

The purpose of this study is to provide a re-examination of the black soldier in the Spanish-American-Cuban-Filipino War. Specifically, by adding a gender analysis, this study will demonstrate that black soldiers fought in the war for two principle reasons: first, it was a means of exercising their citizenship; and secondly, it was a means of demonstrating that they were real men. Reflecting on an era when proving one\u27s manhood was a national obsession--this thesis provides a critical window through which we can reconstruct their motivations for fighting in America\u27s first overseas war

Global W2,pW^{2,p} estimates for solutions to the linearized Monge--Amp\`ere equations

In this paper, we establish global $W^{2,p}$ estimates for solutions to the linearized Monge-Amp\`ere equations under natural assumptions on the domain, Monge-Amp\`ere measures and boundary data. Our estimates are affine invariant analogues of the global $W^{2,p}$ estimates of Winter for fully nonlinear, uniformly elliptic equations, and also linearized counterparts of Savin's global $W^{2,p}$ estimates for the Monge-Amp\`ere equations.Comment: v2: presentation improve

Restriction theorems for μ\mu-(semi)stable framed sheaves

We provide a generalization of Mehta-Ramanathan theorems to framed sheaves: we prove that the restriction of a $\mu$-semistable framed sheaf on a nonsingular projective irreducible variety, of dimension greater or equal than two, to a general hypersurface of sufficiently high degree is again $\mu$-semistable. The same holds for $\mu$-stability under some additional assumptions.Comment: v2: incorporates changes suggested by the referee. v3: some intermediate results on the (relative) Harder-Narasimhan filtration and the Jordan-Holder filtrations have been extended to the case of framed sheaves on schemes. The main results are unchanged. To appear in Journal of Pure and Applied Algebr

The structure of volcanic cristobalite in relation to its toxicity; relevance for the variable crystalline silica hazard

BACKGROUND: Respirable crystalline silica (RCS) continues to pose a risk to human health worldwide. Its variable toxicity depends on inherent characteristics and external factors which influence surface chemistry. Significant population exposure to RCS occurs during volcanic eruptions, where ashfall may cover hundreds of square km and exposure may last years. Occupational exposure also occurs through mining of volcanic deposits. The primary source of RCS from volcanoes is through collapse and fragmentation of lava domes within which cristobalite is mass produced. After 30 years of research, it is still not clear if volcanic ash is a chronic respiratory health hazard. Toxicological assays have shown that cristobalite-rich ash is less toxic than expected. We investigate the reasons for this by determining the physicochemical/structural characteristics which may modify the pathogenicity of volcanic RCS. Four theories are considered: 1) the reactivity of particle surfaces is reduced due to co-substitutions of Al and Na for Si in the cristobalite structure; 2) particles consist of aggregates of cristobalite and other phases, restricting the surface area of cristobalite available for reactions in the lung; 3) the cristobalite surface is occluded by an annealed rim; 4) dissolution of other volcanic particles affects the surfaces of RCS in the lung. METHODS: The composition of volcanic cristobalite crystals was quantified by electron microprobe and differences in composition assessed by Welch's two sample t-test. Sections of dome-rock and ash particles were imaged by scanning and transmission electron microscopy, and elemental compositions of rims determined by energy dispersive X-ray spectroscopy. RESULTS: Volcanic cristobalite contains up to 4 wt. % combined Al(2)O(3) and Na(2)O. Most cristobalite-bearing ash particles contain adhered materials such as feldspar and glass. No annealed rims were observed. CONCLUSIONS: The composition of volcanic cristobalite particles gives insight into previously-unconsidered inherent characteristics of silica mineralogy which may affect toxicity. The structural features identified may also influence the hazard of other environmentally and occupationally produced silica dusts. Current exposure regulations do not take into account the characteristics that might render the silica surface less harmful. Further research would facilitate refinement of the existing simple, mass-based silica standard by taking into account composition, allowing higher standards to be set in industries where the silica surface is modified.Natural Environment Research Council (NERC)Moyes Foundation - studentshi

Inflammatory thresholds and the species-specific effects of colonising bacteria in stable chronic obstructive pulmonary disease

There has been increasing interest in the use of newer, culture-independent techniques to study the airway microbiome of COPD patients. We investigated the relationships between the three common potentially pathogenic microorganisms (PPMs) Haemophilus influenzae, Streptococcus pneumoniae and Moraxella catarrhalis, as detected by quantitative PCR (qPCR), and inflammation and health status in stable patients in the London COPD cohort

Stability and BPS branes

We define the concept of Pi-stability, a generalization of mu-stability of vector bundles, and argue that it characterizes N=1 supersymmetric brane configurations and BPS states in very general string theory compactifications with N=2 supersymmetry in four dimensions.Comment: harvmac, 18 p

Curve counting via stable pairs in the derived category

For a nonsingular projective 3-fold $X$, we define integer invariants virtually enumerating pairs $(C,D)$ where $C\subset X$ is an embedded curve and $D\subset C$ is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of $X$. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of $X$. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We prove that our integrality predictions for Gromov-Witten invariants agree with the BPS integrality. Conversely, the BPS geometry imposes strong conditions on the enumeration of pairs.Comment: Corrected typos and duality error in Proposition 4.6. 47 page

Loss of MeCP2 in adult 5-HT neurons induces 5-HT1A autoreceptors, with opposite sex-dependent anxiety and depression phenotypes

The 5-HT1A autoreceptor mediates feedback inhibition of serotonin (5-HT) neurons, and is implicated in major depression. The human 5-HT1A gene (HTR1A) rs6295 risk allele prevents Deaf1 binding to HTR1A, resulting in increased 5-HT1A autoreceptor transcription. Since chronic stress alters HTR1A methylation and expression, we addressed whether recruitment of methyl-binding protein MeCP2 may alter Deaf1 regulation at the HTR1A locus. We show that MeCP2 enhances Deaf1 binding to its HTR1A site and co-immunoprecipitates with Deaf1 in cells and brain tissue. Chromatin immunoprecipitation assays showed Deaf1-dependent recruitment of MeCP2 to the mouse HTR1A promoter, and MeCP2 modulated human and mouse HTR1A gene transcription in a Deaf1-dependent fashion, enhancing Deaf1-induced repression at the Deaf1 site. To address the role of MeCP2 in HTR1A regulation in vivo, mice with conditional knockout of MeCP2 in adult 5-HT neurons (MeCP2 cKO) were generated. These mice exhibited increased 5-HT1A autoreceptor levels and function, consistent with MeCP2 enhancement of Deaf1 repression in 5-HT neurons. Interestingly, female MeCP2-cKO mice displayed reduced anxiety, while males showed increased anxiety and reduced depression-like behaviors. These data uncover a novel role for MeCP2 in 5-HT neurons to repress HTR1A expression and drive adult anxiety- and depression-like behaviors in a sex-specific manner