Remarks on "Resolving isospectral `drums' by counting nodal domains"

In [3] the authors studied the 4-parameter family of isospectral flat 4-tori T^\pm(a,b,c,d) discovered by Conway and Sloane. With a particular method of counting nodal domains they were able to distinguish these tori (numerically) by computing the corresponding nodal sequences relative to a few explicit tuples (a,b,c,d). In this note we confirm the expectation expressed in [3] by proving analytically that their nodal count distinguishes any 4-tuple of distinct positive real numbers.Comment: 5 page

BDDC and FETI-DP under Minimalist Assumptions

The FETI-DP, BDDC and P-FETI-DP preconditioners are derived in a particulary simple abstract form. It is shown that their properties can be obtained from only on a very small set of algebraic assumptions. The presentation is purely algebraic and it does not use any particular definition of method components, such as substructures and coarse degrees of freedom. It is then shown that P-FETI-DP and BDDC are in fact the same. The FETI-DP and the BDDC preconditioned operators are of the same algebraic form, and the standard condition number bound carries over to arbitrary abstract operators of this form. The equality of eigenvalues of BDDC and FETI-DP also holds in the minimalist abstract setting. The abstract framework is explained on a standard substructuring example.Comment: 11 pages, 1 figure, also available at http://www-math.cudenver.edu/ccm/reports

MyChemise: A 2D drawing program that uses morphing for visualisation purposes

MyChemise (My Chemical Structure Editor) is a new 2D structure editor. It is designed as a Java applet that enables the direct creation of structures in the Internet using a web browser. MyChemise saves files in a digital format (.cse) and the import and export of .mol files using the appropriate connection tables is also possible

On the Nodal Count Statistics for Separable Systems in any Dimension

We consider the statistics of the number of nodal domains aka nodal counts for eigenfunctions of separable wave equations in arbitrary dimension. We give an explicit expression for the limiting distribution of normalised nodal counts and analyse some of its universal properties. Our results are illustrated by detailed discussion of simple examples and numerical nodal count distributions.Comment: 21 pages, 4 figure

Planktonic Aggregates as Hotspots for Heterotrophic Diazotrophy: The Plot Thickens

Biological dinitrogen (N-2) fixation is performed solely by specialized bacteria and archaea termed diazotrophs, introducing new reactive nitrogen into aquatic environments. Conventionally, phototrophic cyanobacteria are considered the major diazotrophs in aquatic environments. However, accumulating evidence indicates that diverse non-cyanobacterial diazotrophs (NCDs) inhabit a wide range of aquatic ecosystems, including temperate and polar latitudes, coastal environments and the deep ocean. NCDs are thus suspected to impact global nitrogen cycling decisively, yet their ecological and quantitative importance remain unknown. Here we review recent molecular and biogeochemical evidence demonstrating that pelagic NCDs inhabit and thrive especially on aggregates in diverse aquatic ecosystems. Aggregates are characterized by reduced-oxygen microzones, high C:N ratio (above Redfield) and high availability of labile carbon as compared to the ambient water. We argue that planktonic aggregates are important loci for energetically-expensive N-2 fixation by NCDs and propose a conceptual framework for aggregate-associated N-2 fixation. Future studies on aggregate-associated diazotrophy, using novel methodological approaches, are encouraged to address the ecological relevance of NCDs for nitrogen cycling in aquatic environments

Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count

The nodal domains of eigenvectors of the discrete Schrodinger operator on simple, finite and connected graphs are considered. Courant's well known nodal domain theorem applies in the present case, and sets an upper bound to the number of nodal domains of eigenvectors: Arranging the spectrum as a non decreasing sequence, and denoting by $\nu_n$ the number of nodal domains of the $n$'th eigenvector, Courant's theorem guarantees that the nodal deficiency $n-\nu_n$ is non negative. (The above applies for generic eigenvectors. Special care should be exercised for eigenvectors with vanishing components.) The main result of the present work is that the nodal deficiency for generic eigenvectors equals to a Morse index of an energy functional whose value at its relevant critical points coincides with the eigenvalue. The association of the nodal deficiency to the stability of an energy functional at its critical points was recently discussed in the context of quantum graphs [arXiv:1103.1423] and Dirichlet Laplacian in bounded domains in $R^d$ [arXiv:1107.3489]. The present work adapts this result to the discrete case. The definition of the energy functional in the discrete case requires a special setting, substantially different from the one used in [arXiv:1103.1423,arXiv:1107.3489] and it is presented here in detail.Comment: 15 pages, 1 figur

Fuzzy Implications: Some Recently Solved Problems

In this chapter we discuss some open problems related to fuzzy implications, which have either been completely solved or those for which partial answers are known. In fact, this chapter also contains the answer for one of the open problems, which is hitherto unpublished. The recently solved problems are so chosen to reflect the importance of the problem or the significance of the solution. Finally, some other problems that still remain unsolved are stated for quick reference

A publicly accessible database for Clostridioides difficile genome sequences supports tracing of transmission chains and epidemics

Clostridioides difficile is the primary infectious cause of antibiotic-associated diarrhea. Local transmissions and international outbreaks of this pathogen have been previously elucidated by bacterial whole-genome sequencing, but comparative genomic analyses at the global scale were hampered by the lack of specific bioinformatic tools. Here we introduce a publicly accessible database within EnteroBase (http://enterobase.warwick.ac.uk) that automatically retrieves and assembles C. difficile short-reads from the public domain, and calls alleles for core-genome multilocus sequence typing (cgMLST). We demonstrate that comparable levels of resolution and precision are attained by EnteroBase cgMLST and single-nucleotide polymorphism analysis. EnteroBase currently contains 18 254 quality-controlled C. difficile genomes, which have been assigned to hierarchical sets of single-linkage clusters by cgMLST distances. This hierarchical clustering is used to identify and name populations of C. difficile at all epidemiological levels, from recent transmission chains through to epidemic and endemic strains. Moreover, it puts newly collected isolates into phylogenetic and epidemiological context by identifying related strains among all previously published genome data. For example, HC2 clusters (i.e. chains of genomes with pairwise distances of up to two cgMLST alleles) were statistically associated with specific hospitals (P<10−4) or single wards (P=0.01) within hospitals, indicating they represented local transmission clusters. We also detected several HC2 clusters spanning more than one hospital that by retrospective epidemiological analysis were confirmed to be associated with inter-hospital patient transfers. In contrast, clustering at level HC150 correlated with k-mer-based classification and was largely compatible with PCR ribotyping, thus enabling comparisons to earlier surveillance data. EnteroBase enables contextual interpretation of a growing collection of assembled, quality-controlled C. difficile genome sequences and their associated metadata. Hierarchical clustering rapidly identifies database entries that are related at multiple levels of genetic distance, facilitating communication among researchers, clinicians and public-health officials who are combatting disease caused by C. difficile