A note on E-strings

We study BPS states in type IIA string compactification on a local Calabi-Yau 3-fold which are related to the BPS states of the E-string. Using Picard-Lefshetz transformations of the 3-cycles on the mirror manifold we determine automorphisms of the K-theory of the compact divisor of the Calabi-Yau which maps certain D-brane configurations to a bound state of single D4-brane with multiple D0-branes. This map allows us to write down the generating functions for the multiplicity of these BPS states.Comment: 20 pages, References adde

Membership of the 109th Congress: A Profile

[From Summary] This report presents a profile of the membership of the 109th Congress. Statistical information is included on selected characteristics of Members. This includes data on party affiliation; average age and length of service; occupation; religious affiliation; female and minority Members; foreign-born Members; and military service

Del Pezzo Surfaces and Affine 7-brane Backgrounds

A map between string junctions in the affine 7-brane backgrounds and vector bundles on del Pezzo surfaces is constructed using mirror symmetry. It is shown that the lattice of string junctions with support on an affine 7-brane configuration is isomorphic to the K-theory group of the corresponding del Pezzo surface. This isomorphism allows us to construct a map between the states of the N=2, D=4 theories with E_N global symmetry realized in two different ways in Type IIB and Type IIA string theory. A subgroup of the SL(2,Z) symmetry of the \hat{E}_9 7-brane background appears as the Fourier-Mukai transform acting on the D-brane configurations realizing vector bundles on elliptically fibered B_9.Comment: 19 pages, LaTeX, 2 eps figures. v2: minor changes, version to appear in JHE

M-strings, Elliptic Genera and N=4 String Amplitudes

We study mass-deformed N=2 gauge theories from various points of view. Their partition functions can be computed via three dual approaches: firstly, (p,q)-brane webs in type II string theory using Nekrasov's instanton calculus, secondly, the (refined) topological string using the topological vertex formalism and thirdly, M theory via the elliptic genus of certain M-strings configurations. We argue for a large class of theories that these approaches yield the same gauge theory partition function which we study in detail. To make their modular properties more tangible, we consider a fourth approach by connecting the partition function to the equivariant elliptic genus of R^4 through a (singular) theta-transform. This form appears naturally as a specific class of one-loop scattering amplitudes in type II string theory on T^2, which we calculate explicitly.Comment: 65 pages, a section on calculation of partition function using Nekrasov's instanton calculus is adde

Probabilistic embeddings of the Fr\'echet distance

The Fr\'echet distance is a popular distance measure for curves which naturally lends itself to fundamental computational tasks, such as clustering, nearest-neighbor searching, and spherical range searching in the corresponding metric space. However, its inherent complexity poses considerable computational challenges in practice. To address this problem we study distortion of the probabilistic embedding that results from projecting the curves to a randomly chosen line. Such an embedding could be used in combination with, e.g. locality-sensitive hashing. We show that in the worst case and under reasonable assumptions, the discrete Fr\'echet distance between two polygonal curves of complexity $t$ in $\mathbb{R}^d$, where $d\in\lbrace 2,3,4,5\rbrace$, degrades by a factor linear in $t$ with constant probability. We show upper and lower bounds on the distortion. We also evaluate our findings empirically on a benchmark data set. The preliminary experimental results stand in stark contrast with our lower bounds. They indicate that highly distorted projections happen very rarely in practice, and only for strongly conditioned input curves. Keywords: Fr\'echet distance, metric embeddings, random projectionsComment: 27 pages, 11 figure

Fungal cellulase; production and applications: minireview

Cellulose is the most abundant biomaterial derived from the living organisms on the earth; plant is the major contributor to the cellulose pool present in the biosphere. Cellulose is used in variety of applications ranging from nanomaterials to biofuel production. For biofuel production, cellulose has first to be broken-down into its building blocks; β-D-glucosyl unit which subsequently can be fermented to different product such as ethanol, acetic acids, among others. Cellulase is the enzymatic system, which degrades cellulose chains to glucose monomers. Cellulase is a group of three enzymes endoglucanase, exoglucanases and β-glucosidases which act together to hydrolyze cellulose to glucose units. Cellulases are found in bacteria, fungi, plants, and some animals. Fungi are the preferred source of cellulase for industrial applications since they secrete large quantities of cellulase to culture medium. Despite a remarkable number of fungi found to produce cellulase enzymes, few have been extensively investigated because they produce large quantities of these enzymes extracellularly. In this mini-review, the production of cellulase from fungi and the parameters affecting cellulase production are discussed briefly on light of recent publications. Furthermore, potential applications of cellulase enzymes are highlighted