The size of triangulations supporting a given link

Let T be a triangulation of S^3 containing a link L in its 1-skeleton. We give an explicit lower bound for the number of tetrahedra of T in terms of the bridge number of L. Our proof is based on the theory of almost normal surfaces.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper13.abs.htm

Schrijver graphs and projective quadrangulations

In a recent paper [J. Combin. Theory Ser. B}, 113 (2015), pp. 1-17], the authors have extended the concept of quadrangulation of a surface to higher dimension, and showed that every quadrangulation of the $n$-dimensional projective space $P^n$ is at least $(n+2)$-chromatic, unless it is bipartite. They conjectured that for any integers $k\geq 1$ and $n\geq 2k+1$, the Schrijver graph $SG(n,k)$ contains a spanning subgraph which is a quadrangulation of $P^{n-2k}$. The purpose of this paper is to prove the conjecture

Converting between quadrilateral and standard solution sets in normal surface theory

The enumeration of normal surfaces is a crucial but very slow operation in algorithmic 3-manifold topology. At the heart of this operation is a polytope vertex enumeration in a high-dimensional space (standard coordinates). Tollefson's Q-theory speeds up this operation by using a much smaller space (quadrilateral coordinates), at the cost of a reduced solution set that might not always be sufficient for our needs. In this paper we present algorithms for converting between solution sets in quadrilateral and standard coordinates. As a consequence we obtain a new algorithm for enumerating all standard vertex normal surfaces, yielding both the speed of quadrilateral coordinates and the wider applicability of standard coordinates. Experimentation with the software package Regina shows this new algorithm to be extremely fast in practice, improving speed for large cases by factors from thousands up to millions.Comment: 55 pages, 10 figures; v2: minor fixes only, plus a reformat for the journal styl

Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables

Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary form. In particular, we can show that for growing $d$ a positive proportion of the binary quadratic forms of discriminant $-d$ is represented by the given quaternary quadratic form.Comment: v5: Some typos correcte

Quadrilateral-octagon coordinates for almost normal surfaces

Normal and almost normal surfaces are essential tools for algorithmic 3-manifold topology, but to use them requires exponentially slow enumeration algorithms in a high-dimensional vector space. The quadrilateral coordinates of Tollefson alleviate this problem considerably for normal surfaces, by reducing the dimension of this vector space from 7n to 3n (where n is the complexity of the underlying triangulation). Here we develop an analogous theory for octagonal almost normal surfaces, using quadrilateral and octagon coordinates to reduce this dimension from 10n to 6n. As an application, we show that quadrilateral-octagon coordinates can be used exclusively in the streamlined 3-sphere recognition algorithm of Jaco, Rubinstein and Thompson, reducing experimental running times by factors of thousands. We also introduce joint coordinates, a system with only 3n dimensions for octagonal almost normal surfaces that has appealing geometric properties.Comment: 34 pages, 20 figures; v2: Simplified the proof of Theorem 4.5 using cohomology, plus other minor changes; v3: Minor housekeepin

Maximal admissible faces and asymptotic bounds for the normal surface solution space

The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but non-linear and non-convex constraint. The main results of this paper are significant improvements upon the best known asymptotic bounds on the number of admissible vertices, using polytopes in both the standard normal surface coordinate system and the streamlined quadrilateral coordinate system. To achieve these results we examine the layout of admissible points within these polytopes. We show that these points correspond to well-behaved substructures of the face lattice, and we study properties of the corresponding "admissible faces". Key lemmata include upper bounds on the number of maximal admissible faces of each dimension, and a bijection between the maximal admissible faces in the two coordinate systems mentioned above.Comment: 31 pages, 10 figures, 2 tables; v2: minor revisions (to appear in Journal of Combinatorial Theory A

Relevance and Recent Developments of Chitosan in Peripheral Nerve Surgery

Developments in tissue engineering yield biomaterials with different supporting strategies to promote nerve regeneration. One promising material is the naturally occurring chitin derivate chitosan. Chitosan has become increasingly important in various tissue engineering approaches for peripheral nerve reconstruction, as it has demonstrated its potential to interact with regeneration associated cells and the neural microenvironment, leading to improved axonal regeneration and less neuroma formation. Moreover, the physiological properties of its polysaccharide structure provide safe biodegradation behavior in the absence of negative side effects or toxic metabolites. Beneficial interactions with Schwann cells (SC), inducing differentiation of mesenchymal stromal cells to SC-like cells or creating supportive conditions during axonal recovery are only a small part of the effects of chitosan. As a result, an extensive body of literature addresses a variety of experimental strategies for the different types of nerve lesions. The different concepts include chitosan nanofibers, hydrogels, hollow nerve tubes, nerve conduits with an inner chitosan layer as well as hybrid architectures containing collagen or polyglycolic acid nerve conduits. Furthermore, various cell seeding concepts have been introduced in the preclinical setting. First translational concepts with hollow tubes following nerve surgery already transferred the promising experimental approach into clinical practice. However, conclusive analyses of the available data and the proposed impact on the recovery process following nerve surgery are currently lacking. This review aims to give an overview on the physiologic properties of chitosan, to evaluate its effect on peripheral nerve regeneration and discuss the future translation into clinical practice

On the critical pair theory in abelian groups : Beyond Chowla's Theorem

We obtain critical pair theorems for subsets S and T of an abelian group such that |S+T| < |S|+|T|+1. We generalize some results of Chowla, Vosper, Kemperman and a more recent result due to Rodseth and one of the authors.Comment: Submitted to Combinatorica, 23 pages, revised versio

Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function

We develop an analog of classical oscillation theory for Sturm-Liouville operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the finiteness of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spectral shift function is established.Comment: 26 page

A multivariable miRNA signature delineates the systemic hemodynamic impact of arteriovenous shunt placement in a pilot study

Arteriovenous (AV) fistulas for hemodialysis can lead to cardiac volume loading and increased serum brain natriuretic peptide (BNP) levels. Whether short-term AV loop placement in patients undergoing microsurgery has an impact on cardiac biomarkers and circulating microRNAs (miRNAs), potentially indicating an increased hemodynamic risk, remains elusive. Fifteen patients underwent AV loop placement with delayed free flap anastomosis for microsurgical reconstructions of lower extremity soft-tissue defects. N-terminal pro-BNP (NT-proBNP), copeptin (CT-proAVP), and miRNA expression profiles were determined in the peripheral blood before and after AV loop placement. MiRNA expression in the blood was correlated with miRNA expression from AV loop vascular tissue. Serum NT-proBNP and copeptin levels exceeded the upper reference limit after AV loop placement, with an especially strong NT-proBNP increase in patients with preexistent cardiac diseases. A miRNA signature of 4 up-regulated (miR-3198, miR-3127-5p, miR-1305, miR-1288-3p) and 2 down-regulated miRNAs (miR30a-5p, miR-145-5p) which are related to cardiovascular physiology, showed a significant systemic deregulation in blood and venous tissue after AV loop placement. AV loop placement causes serum elevations of NT-proBNP, copeptin as well as specific circulating miRNAs, indicating a potentially increased hemodynamic risk for patients with cardiovascular comorbidities, if free flap anastomosis is delayed