Colored Non-Crossing Euclidean Steiner Forest

Given a set of $k$-colored points in the plane, we consider the problem of finding $k$ trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For $k=1$, this is the well-known Euclidean Steiner tree problem. For general $k$, a $k\rho$-approximation algorithm is known, where $\rho \le 1.21$ is the Steiner ratio. We present a PTAS for $k=2$, a $(5/3+\varepsilon)$-approximation algorithm for $k=3$, and two approximation algorithms for general~$k$, with ratios $O(\sqrt n \log k)$ and $k+\varepsilon$

Quarter-Filled Honeycomb Lattice with a Quantized Hall Conductance

We study a generic two-dimensional hopping model on a honeycomb lattice with strong spin-orbit coupling, without the requirement that the half-filled lattice be a Topological Insulator. For quarter-(or three-quarter) filling, we show that a state with a quantized Hall conductance generically arises in the presence of a Zeeman field of sufficient strength. We discuss the influence of Hubbard interactions and argue that spontaneous ferromagnetism (which breaks time-reversal) will occur, leading to a quantized anomalous Hall effect.Comment: 4 pages, 3 figure

A local families index formula for d-bar operators on punctured Riemann surfaces

Using heat kernel methods developed by Vaillant, a local index formula is obtained for families of d-bar operators on the Teichmuller universal curve of Riemann surfaces of genus g with n punctures. The formula also holds on the moduli space M{g,n} in the sense of orbifolds where it can be written in terms of Mumford-Morita-Miller classes. The degree two part of the formula gives the curvature of the corresponding determinant line bundle equipped with the Quillen connection, a result originally obtained by Takhtajan and Zograf.Comment: 47 page

Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints

We introduce a new structure for a set of points in the plane and an angle $\alpha$, which is similar in flavor to a bounded-degree MST. We name this structure $\alpha$-MST. Let $P$ be a set of points in the plane and let $0 < \alpha \le 2\pi$ be an angle. An $\alpha$-ST of $P$ is a spanning tree of the complete Euclidean graph induced by $P$, with the additional property that for each point $p \in P$, the smallest angle around $p$ containing all the edges adjacent to $p$ is at most $\alpha$. An $\alpha$-MST of $P$ is then an $\alpha$-ST of $P$ of minimum weight. For $\alpha < \pi/3$, an $\alpha$-ST does not always exist, and, for $\alpha \ge \pi/3$, it always exists. In this paper, we study the problem of computing an $\alpha$-MST for several common values of $\alpha$. Motivated by wireless networks, we formulate the problem in terms of directional antennas. With each point $p \in P$, we associate a wedge $W_p$ of angle $\alpha$ and apex $p$. The goal is to assign an orientation and a radius $r_p$ to each wedge $W_p$, such that the resulting graph is connected and its MST is an $\alpha$-MST. (We draw an edge between $p$ and $q$ if $p \in W_q$, $q \in W_p$, and $|pq| \le r_p, r_q$.) Unsurprisingly, the problem of computing an $\alpha$-MST is NP-hard, at least for $\alpha=\pi$ and $\alpha=2\pi/3$. We present constant-factor approximation algorithms for $\alpha = \pi/2, 2\pi/3, \pi$. One of our major results is a surprising theorem for $\alpha = 2\pi/3$, which, besides being interesting from a geometric point of view, has important applications. For example, the theorem guarantees that given any set $P$ of $3n$ points in the plane and any partitioning of the points into $n$ triplets, one can orient the wedges of each triplet {\em independently}, such that the graph induced by $P$ is connected. We apply the theorem to the {\em antenna conversion} problem

Enhanced chondrogenic phenotype of primary bovine articular chondrocytes in Fibrin-Hyaluronan hydrogel by multi-axial mechanical loading and FGF18

Current treatments for cartilage lesions are often associated with fibrocartilage formation and donor site morbidity. Mechanical and biochemical stimuli play an important role in hyaline cartilage formation. Biocompatible scaffolds capable of transducing mechanical loads and delivering bioactive instructive factors may better support cartilage regeneration. In this study we aimed to test the interplay between mechanical and FGF-18 mediated biochemical signals on the proliferation and differentiation of primary bovine articular chondrocytes embedded in a chondro-conductive Fibrin-Hyaluronan (FB/HA) based hydrogel. Chondrocytes seeded in a Fibrin-HA hydrogel, with or without a chondro-inductive, FGFR3 selective FGF18 variant (FGF-18v) were loaded into a joint-mimicking bioreactor applying controlled, multi-axial movements, simulating the natural movements of articular joints. Samples were evaluated for DNA content, sulphated glycosaminoglycan (sGAG) accumulation, key chondrogenic gene expression markers and histology. Under moderate loading, samples produced particularly significant amounts of sGAG/DNA compared to unloaded controls. Interestingly there was no significant effect of FGF-18v on cartilage gene expression at rest. Following moderate multi-axial loading, FGF-18v upregulated the expression of Aggrecan (ACAN), Cartilage Oligomeric Matrix Protein (COMP), type II collagen (COL2) and Lubricin (PRG4). Moreover, the combination of load and FGF-18v, significantly downregulated Matrix Metalloproteinase-9 (MMP-9) and Matrix Metaloproteinase-13 (MMP-13), two of the most important factors contributing to joint destruction in OA. Biomimetic mechanical signals and FGF-18 may work in concert to support hyaline cartilage regeneration and repair. Statement of significance: Articular cartilage has very limited repair potential and focal cartilage lesions constitute a challenge for current standard clinical procedures. The aim of the present research was to explore novel procedures and constructs, based on biomaterials and biomechanical algorithms that can better mimic joints mechanical and biochemical stimulation to promote regeneration of damaged cartilage. Using a hydrogel-based platform for chondrocyte 3D culture revealed a synergy between mechanical forces and growth factors. Exploring the mechanisms underlying this mechano-biochemical interplay may enhance our understanding of cartilage remodeling and the development of new strategies for cartilage repair and regeneration

Harris criterion on hierarchical lattices: Rigorous inequalities and counterexamples in Ising systems

Random bond Ising systems on a general hierarchical lattice are considered. The inequality between the specific heat exponent of the pure system, $\alpha_p$, and the crossover exponent $\phi$, $\alpha_p<=\phi$, gives rise to a possibility of a negative $\alpha_p$ along with a positive $\phi$, leading to random criticality in disagreement with the Harris criterion. An explicit example where this really happens for an Ising system is presented and discussed. In addition to that, it is shown that in presence of full long-range correlations, the crossover exponent is larger than in the uncorrelated case.Comment: 8 pages, 3 figure

Quantum Fields in Hyperbolic Space-Times with Finite Spatial Volume

The one-loop effective action for a massive self-interacting scalar field is investigated in $4$-dimensional ultrastatic space-time $R \times H^3/\Gamma$, $H^3/\Gamma$ being a non-compact hyperbolic manifold with finite volume. Making use of the Selberg trace formula, the $\zeta$-function related to the small disturbance operator is constructed. For an arbitrary gravitational coupling, it is found that $\zeta(s)$ has a simple pole at $s=0$. The one-loop effective action is analysed by means of proper-time regularisations and the one-loop divergences are explicitly found. It is pointed out that, in this special case, also $\zeta$-function regularisation requires a divergent counterterm, which however is not necessary in the free massless conformal invariant coupling case. Finite temperature effects are studied and the high-temperature expansion is presented. A possible application to the problem of the divergences of the entanglement entropy for a free massless scalar field in a Rindler-like space-time is briefly discussed.Comment: 13 pages, LaTex. The contribution of hyperbolic elements has been added. Other minor corrections and reference

Engineering Art Galleries

The Art Gallery Problem is one of the most well-known problems in Computational Geometry, with a rich history in the study of algorithms, complexity, and variants. Recently there has been a surge in experimental work on the problem. In this survey, we describe this work, show the chronology of developments, and compare current algorithms, including two unpublished versions, in an exhaustive experiment. Furthermore, we show what core algorithmic ingredients have led to recent successes