Algorithms computing O(n, Z)-orbits of P-critical edge-bipartite graphs and P-critical unit forms using Maple and C#

We present combinatorial algorithms constructing loop-free P-critical edge-bipartite (signed) graphs Δ′, with n ≥ 3 vertices, from pairs (Δ, w), with Δ a positive edge-bipartite graph having n-1 vertices and w a sincere root of Δ, up to an action ∗ : UBigrn × O(n, Z) → UBigrn of the orthogonal group O(n, Z) on the set UBigrn of loop-free edge-bipartite graphs, with n ≥ 3 vertices. Here Z is the ring of integers. We also present a package of algorithms for a Coxeter spectral analysis of graphs in UBigrn and for computing the O(n, Z)-orbits of P-critical graphs Δ in UBigrn as well as the positive ones. By applying the package, symbolic computations in Maple and numerical computations in C#, we compute P-critical graphs in UBigrn and connected positive graphs in UBigrn, together with their Coxeter polynomials, reduced Coxeter numbers, and the O(n, Z)-orbits, for n ≤ 10. The computational results are presented in tables of Section 5

Hereditary stable tubes in module categories

The concepts of a self-hereditary stable tube T and a hereditary stable tube T in a module category modA are introduced, where A is a finite dimensional algebra over an algebraically closed field. Characterisations of self-hereditary stable tubes and hereditary stable tubes are given, and illustrative examples of such tubes are presented. Some open problems are presented

Rupture of multiple parallel molecular bonds under dynamic loading

Biological adhesion often involves several pairs of specific receptor-ligand molecules. Using rate equations, we study theoretically the rupture of such multiple parallel bonds under dynamic loading assisted by thermal activation. For a simple generic type of cooperativity, both the rupture time and force exhibit several different scaling regimes. The dependence of the rupture force on the number of bonds is predicted to be either linear, like a square root or logarithmic.Comment: 8 pages, 2 figure

A homological interpretation of the transverse quiver Grassmannians

In recent articles, the investigation of atomic bases in cluster algebras associated to affine quivers led the second-named author to introduce a variety called transverse quiver Grassmannian and the first-named and third-named authors to consider the smooth loci of quiver Grassmannians. In this paper, we prove that, for any affine quiver Q, the transverse quiver Grassmannian of an indecomposable representation M is the set of points N in the quiver Grassmannian of M such that Ext^1(N,M/N)=0. As a corollary we prove that the transverse quiver Grassmannian coincides with the smooth locus of the irreducible components of minimal dimension in the quiver Grassmannian.Comment: final version, 7 pages, corollary 1.2 has been modifie

Dynamic force spectroscopy on multiple bonds: experiments and model

We probe the dynamic strength of multiple biotin-streptavidin adhesion bonds under linear loading using the biomembrane force probe setup for dynamic force spectroscopy. Measured rupture force histograms are compared to results from a master equation model for the stochastic dynamics of bond rupture under load. This allows us to extract the distribution of the number of initially closed bonds. We also extract the molecular parameters of the adhesion bonds, in good agreement with earlier results from single bond experiments. Our analysis shows that the peaks in the measured histograms are not simple multiples of the single bond values, but follow from a superposition procedure which generates different peak positions.Comment: to appear in Europhysics Letter

Descriptions of membrane mechanics from microscopic and effective two-dimensional perspectives

Mechanics of fluid membranes may be described in terms of the concepts of mechanical deformations and stresses, or in terms of mechanical free-energy functions. In this paper, each of the two descriptions is developed by viewing a membrane from two perspectives: a microscopic perspective, in which the membrane appears as a thin layer of finite thickness and with highly inhomogeneous material and force distributions in its transverse direction, and an effective, two-dimensional perspective, in which the membrane is treated as an infinitely thin surface, with effective material and mechanical properties. A connection between these two perspectives is then established. Moreover, the functional dependence of the variation in the mechanical free energy of the membrane on its mechanical deformations is first studied in the microscopic perspective. The result is then used to examine to what extent different, effective mechanical stresses and forces can be derived from a given, effective functional of the mechanical free energy.Comment: 37 pages, 3 figures, minor change

Cycle-finite module categories

We describe the structure of module categories of finite dimensional algebras over an algebraically closed field for which the cycles of nonzero nonisomorphisms between indecomposable finite dimensional modules are finite (do not belong to the infinite Jacobson radical of the module category). Moreover, geometric and homological properties of these module categories are exhibited

Categorical Tinkertoys for N=2 Gauge Theories

In view of classification of the quiver 4d N=2 supersymmetric gauge theories, we discuss the characterization of the quivers with superpotential (Q,W) associated to a N=2 QFT which, in some corner of its parameter space, looks like a gauge theory with gauge group G. The basic idea is that the Abelian category rep(Q,W) of (finite-dimensional) representations of the Jacobian algebra $\mathbb{C} Q/(\partial W)$ should enjoy what we call the Ringel property of type G; in particular, rep(Q,W) should contain a universal `generic' subcategory, which depends only on the gauge group G, capturing the universality of the gauge sector. There is a family of 'light' subcategories $\mathscr{L}_\lambda\subset rep(Q,W)$, indexed by points $\lambda\in N$, where $N$ is a projective variety whose irreducible components are copies of $\mathbb{P}^1$ in one--to--one correspondence with the simple factors of G. In particular, for a Gaiotto theory there is one such family of subcategories, $\mathscr{L}_{\lambda\in N}$, for each maximal degeneration of the corresponding surface $\Sigma$, and the index variety $N$ may be identified with the degenerate Gaiotto surface itself: generic light subcategories correspond to cylinders, while closed-point subcategories to `fixtures' (spheres with three punctures of various kinds) and higher-order generalizations. The rules for `gluing' categories are more general that the geometric gluing of surfaces, allowing for a few additional exceptional N=2 theories which are not of the Gaiotto class.Comment: 142 pages, 8 figures, 5 table

Gorenstein homological algebra and universal coefficient theorems

We study criteria for a ring—or more generally, for a small category—to be Gorenstein and for a module over it to be of finite projective dimension. The goal is to unify the universal coefficient theorems found in the literature and to develop machinery for proving new ones. Among the universal coefficient theorems covered by our methods we find, besides all the classic examples, several exotic examples arising from the KK-theory of C*-algebras and also Neeman’s Brown–Adams representability theorem for compactly generated categories

On Arnold's 14 `exceptional' N=2 superconformal gauge theories

We study the four-dimensional superconformal N=2 gauge theories engineered by the Type IIB superstring on Arnold's 14 exceptional unimodal singularities (a.k.a. Arnold's strange duality list), thus extending the methods of 1006.3435 to singularities which are not the direct sum of minimal ones. In particular, we compute their BPS spectra in several `strongly coupled' chambers. From the TBA side, we construct ten new periodic Y-systems, providing additional evidence for the existence of a periodic Y-system for each isolated quasi-homogeneous singularity with $\hat c<2$ (more generally, for each N=2 superconformal theory with a finite BPS chamber whose chiral primaries have dimensions of the form N/l).Comment: 73 pages, 7 figure