Kato square root problem with unbounded leading coefficients

We prove the Kato conjecture for elliptic operators, $L=-\nabla\cdot\left((\mathbf A+\mathbf D)\nabla\ \right)$, with $\mathbf A$ a complex measurable bounded coercive matrix and $\mathbf D$ a measurable real-valued skew-symmetric matrix in $\mathbb{R}^n$ with entries in $BMO(\mathbb{R}^n)$;\, i.e., the domain of $\sqrt{L}\,$ is the Sobolev space $\dot H^1(\mathbb{R}^n)$ in any dimension, with the estimate $\|\sqrt{L}\, f\|_2\lesssim \| \nabla f\|_2$

On the Independent Domination Number of Regular Graphs

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. In this paper, we consider questions about independent domination in regular graphs

Cerebellum Transcriptome of Mice Bred for High Voluntary Activity Offers Insights into Locomotor Control and Reward-Dependent Behaviors.

The role of the cerebellum in motivation and addictive behaviors is less understood than that in control and coordination of movements. High running can be a self-rewarding behavior exhibiting addictive properties. Changes in the cerebellum transcriptional networks of mice from a line selectively bred for High voluntary running (H) were profiled relative to an unselected Control (C) line. The environmental modulation of these changes was assessed both in activity environments corresponding to 7 days of Free (F) access to running wheel and to Blocked (B) access on day 7. Overall, 457 genes exhibited a significant (FDR-adjusted P-value < 0.05) genotype-by-environment interaction effect, indicating that activity genotype differences in gene expression depend on environmental access to running. Among these genes, network analysis highlighted 6 genes (Nrgn, Drd2, Rxrg, Gda, Adora2a, and Rab40b) connected by their products that displayed opposite expression patterns in the activity genotype contrast within the B and F environments. The comparison of network expression topologies suggests that selection for high voluntary running is linked to a predominant dysregulation of hub genes in the F environment that enables running whereas a dysregulation of ancillary genes is favored in the B environment that blocks running. Genes associated with locomotor regulation, signaling pathways, reward-processing, goal-focused, and reward-dependent behaviors exhibited significant genotype-by-environment interaction (e.g. Pak6, Adora2a, Drd2, and Arhgap8). Neuropeptide genes including Adcyap1, Cck, Sst, Vgf, Npy, Nts, Penk, and Tac2 and related receptor genes also exhibited significant genotype-by-environment interaction. The majority of the 183 differentially expressed genes between activity genotypes (e.g. Drd1) were under-expressed in C relative to H genotypes and were also under-expressed in B relative to F environments. Our findings indicate that the high voluntary running mouse line studied is a helpful model for understanding the molecular mechanisms in the cerebellum that influence locomotor control and reward-dependent behaviors

Voluntary Wheel Running Reverses Age-Induced Changes in Hippocampal Gene Expression

Normal aging alters expression of numerous genes within the brain. Some of these transcription changes likely contribute to age-associated cognitive decline, reduced neural plasticity, and the higher incidence of neuropathology. Identifying factors that modulate brain aging is crucial for improving quality of life. One promising intervention to counteract negative effects of aging is aerobic exercise. Aged subjects that exercise show enhanced cognitive performance and increased hippocampal neurogenesis and synaptic plasticity. Currently, the mechanisms behind the anti-aging effects of exercise are not understood. The present study conducted a microarray on whole hippocampal samples from adult (3.5-month-old) and aged (18-month-old) male BALB/c mice that were individually housed with or without running wheels for 8 weeks. Results showed that aging altered genes related to chromatin remodeling, cell growth, immune activity, and synapse organization compared to adult mice. Exercise was found to modulate many of the genes altered by aging, but in the opposite direction. For example, wheel running increased expression of genes related to cell growth and attenuated expression of genes involved in immune function and chromatin remodeling. Collectively, findings show that even late-onset exercise may attenuate age-related changes in gene expression and identifies possible pathways through which exercise may exert its beneficial effects

Multiple novel prostate cancer susceptibility signals identified by fine-mapping of known risk loci among Europeans

Genome-wide association studies (GWAS) have identified numerous common prostate cancer (PrCa) susceptibility loci. We have fine-mapped 64 GWAS regions known at the conclusion of the iCOGS study using large-scale genotyping and imputation in 25 723 PrCa cases and 26 274 controls of European ancestry. We detected evidence for multiple independent signals at 16 regions, 12 of which contained additional newly identified significant associations. A single signal comprising a spectrum of correlated variation was observed at 39 regions; 35 of which are now described by a novel more significantly associated lead SNP, while the originally reported variant remained as the lead SNP only in 4 regions. We also confirmed two association signals in Europeans that had been previously reported only in East-Asian GWAS. Based on statistical evidence and linkage disequilibrium (LD) structure, we have curated and narrowed down the list of the most likely candidate causal variants for each region. Functional annotation using data from ENCODE filtered for PrCa cell lines and eQTL analysis demonstrated significant enrichment for overlap with bio-features within this set. By incorporating the novel risk variants identified here alongside the refined data for existing association signals, we estimate that these loci now explain ∼38.9% of the familial relative risk of PrCa, an 8.9% improvement over the previously reported GWAS tag SNPs. This suggests that a significant fraction of the heritability of PrCa may have been hidden during the discovery phase of GWAS, in particular due to the presence of multiple independent signals within the same regio

Meta-analysis of genome-wide association studies discovers multiple loci for chronic lymphocytic leukemia.

Chronic lymphocytic leukemia (CLL) is a common lymphoid malignancy with strong heritability. To further understand the genetic susceptibility for CLL and identify common loci associated with risk, we conducted a meta-analysis of four genome-wide association studies (GWAS) composed of 3,100 cases and 7,667 controls with follow-up replication in 1,958 cases and 5,530 controls. Here we report three new loci at 3p24.1 (rs9880772, EOMES, P=2.55 × 10(-11)), 6p25.2 (rs73718779, SERPINB6, P=1.97 × 10(-8)) and 3q28 (rs9815073, LPP, P=3.62 × 10(-8)), as well as a new independent SNP at the known 2q13 locus (rs9308731, BCL2L11, P=1.00 × 10(-11)) in the combined analysis. We find suggestive evidence (P<5 × 10(-7)) for two additional new loci at 4q24 (rs10028805, BANK1, P=7.19 × 10(-8)) and 3p22.2 (rs1274963, CSRNP1, P=2.12 × 10(-7)). Pathway analyses of new and known CLL loci consistently show a strong role for apoptosis, providing further evidence for the importance of this biological pathway in CLL susceptibility

Domination results: vertex partitions and edge weight functions

D.Phil.Domination in graphs is now well studied in graph theory and the literature on this subject has been surveyed and detailed in the two books by Haynes, Hedetniemi, and Slater [45, 46]. In this thesis, we continue the study of domination, by adding to the theory; improving a number of known bounds and solving two previously published conjectures. With the exception of the introduction, each chapter in this thesis corresponds to a single paper already published or submitted as a journal article. Despite the seeming disparity in the content of some of these articles, there are two overarching goals achieved in this thesis. The rst is an attempt to partition the vertex set of a graph into two sets, each holding a speci c domination-type property. The second is simply to improve known bounds for various domination parameters. In particular, an edge weighting function is presented which has been useful in providing some of these bounds. Although the research began as two separate areas of focus, there has been a fair degree of overlap and a number of the results contained in this thesis bridge the gap quite pleasingly. Specially, Chapter 11 uses the edge weighting function to prove a bound on one of the sets in our most fundamental partitions, while the improvement on a known bound presented in Chapter 7 was inspired by considering the possible existence of another partition. This latter proof relies implicitly on the `almost' existence of such a partition

A characterization of graphs with disjoint dominating and total dominating sets

A dominating set of a graph is a set of vertices such that every vertex not in the set is adjacent to a vertex in the set, while a total dominating set of a graph is a set of vertices such that every vertex is adjacent to a vertex in the set. In this paper, we provide a constructive characterization of graphs whose vertex set can be partitioned into a dominating set and a total dominating set.Keywords: Domination; total domination; vertex partitionQuaestiones Mathematicae 32(2009), 119–12

Characterization of the prohormone complement in Amphiprion and related fish species integrating genome and transcriptome assemblies.

The Amphiprion (anemonefish or clownfish) family of teleost fish, which is not a common model species, exhibits multiple unique characteristics, including social control of body size and protandrous sex change. The social changes in sex and body size are modulated by neuropeptide signaling pathways. These neuropeptides are formed from complex processing from larger prohormone proteins; understanding the neuropeptide complement requires information on complete prohormones sequences. Genome and transcriptome information within and across 22 teleost fish species, including 11 Amphiprion species, were assembled and integrated to achieve the first comprehensive survey of their prohormone genes. This information enabled the identification of 175 prohormone isoforms from 159 prohormone proteins across all species. This included identification of 9 CART prepropeptide genes and the loss of insulin-like 5B and tachykinin precursor 1B genes in Pomacentridae species. Transcriptome assemblies generally detected most prohormone genes but provided fewer prohormone genes than genome assemblies due to the lack of expression of prohormone genes or specific isoforms and tissue sampled. Comparisons between duplicate genes indicated that subfunctionalization, degradation, and neofunctionalization may be occurring between all copies. Characterization of the prohormone complement lays the foundation for future peptidomic investigation of the molecular basis of social physiology and behavior in the teleost fish