Algorithms for Colourful Simplicial Depth and Medians in the Plane

The colourful simplicial depth of a point x in the plane relative to a configuration of n points in k colour classes is exactly the number of closed simplices (triangles) with vertices from 3 different colour classes that contain x in their convex hull. We consider the problems of efficiently computing the colourful simplicial depth of a point x, and of finding a point, called a median, that maximizes colourful simplicial depth. For computing the colourful simplicial depth of x, our algorithm runs in time O(n log(n) + k n) in general, and O(kn) if the points are sorted around x. For finding the colourful median, we get a time of O(n^4). For comparison, the running times of the best known algorithm for the monochrome version of these problems are O(n log(n)) in general, improving to O(n) if the points are sorted around x for monochrome depth, and O(n^4) for finding a monochrome median.Comment: 17 pages, 8 figure

Stabilizing the Complex Structure in Heterotic Calabi-Yau Vacua

In this paper, we show that the presence of gauge fields in heterotic Calabi-Yau compacitifications causes the stabilisation of some, or all, of the complex structure moduli of the Calabi-Yau manifold while maintaining a Minkowski vacuum. Certain deformations of the Calabi-Yau complex structure, with all other moduli held fixed, can lead to the gauge bundle becoming non-holomorphic and, hence, non-supersymmetric. This leads to an F-term potential which stabilizes the corresponding complex structure moduli. We use 10- and 4-dimensional field theory arguments as well as a derivation based purely on algebraic geometry to show that this picture is indeed correct. An explicit example is presented in which a large subset of complex structure moduli is fixed. We demonstrate that this type of theory can serve as the hidden sector in heterotic vacua and can co-exist with realistic particle physics.Comment: 17 pages, Late

Numerical Hermitian Yang-Mills Connections and Vector Bundle Stability in Heterotic Theories

A numerical algorithm is presented for explicitly computing the gauge connection on slope-stable holomorphic vector bundles on Calabi-Yau manifolds. To illustrate this algorithm, we calculate the connections on stable monad bundles defined on the K3 twofold and Quintic threefold. An error measure is introduced to determine how closely our algorithmic connection approximates a solution to the Hermitian Yang-Mills equations. We then extend our results by investigating the behavior of non slope-stable bundles. In a variety of examples, it is shown that the failure of these bundles to satisfy the Hermitian Yang-Mills equations, including field-strength singularities, can be accurately reproduced numerically. These results make it possible to numerically determine whether or not a vector bundle is slope-stable, thus providing an important new tool in the exploration of heterotic vacua.Comment: 52 pages, 15 figures. LaTex formatting of figures corrected in version 2

Centerpoints: a link between optimization and convex geometry

We introduce a concept that generalizes several different notions of a “centerpoint” in the literature. We develop an oracle-based algorithm for convex mixed-integer optimization based on centerpoints. Further, we show that algorithms based on centerpoints are “best possible” in a certain sense. Motivated by this, we establish several structural results about this concept and provide efficient algorithms for computing these points

B-L Cosmic Strings in Heterotic Standard Models

E_{8} X E_{8} heterotic string and M-theory, when compactified on smooth Calabi-Yau manifolds with SU(4) vector bundles, can give rise to softly broken N=1 supersymmetric theories with the exact matter spectrum of the MSSM, including three right-handed neutrinos and one Higgs-Higgs conjugate pair of supermultiplets. These vacua have the SU(3)_{C} X SU(2)_{L} X U(1)_{Y} gauge group of the standard model augmented by an additional gauged U(1)_{B-L}. Their minimal content requires that the B-L symmetry be spontaneously broken by a vacuum expectation value of at least one right-handed sneutrino. The soft supersymmetry breaking operators can induce radiative breaking of the B-L gauge symmetry with an acceptable B-L/electroweak hierarchy. In this paper, it is shown that U(1)_{B-L} cosmic strings occur in this context, potentially with both bosonic and fermionic superconductivity. We present a numerical analysis that demonstrates that boson condensates can, in principle, form for theories of this type. However, the weak Yukawa and gauge couplings of the right-handed sneutrino suggests that bosonic superconductivity will not occur in the simplest vacua in this context. The electroweak phase transition also disallows fermion superconductivity, although substantial bound state fermion currents can exist.Comment: 41 pages, 5 figure

U(n) Spectral Covers from Decomposition

We construct decomposed spectral covers for bundles on elliptically fibered Calabi-Yau threefolds whose structure groups are S(U(1) x U(4)), S(U(2) x U(3)) and S(U(1) x U(1) x U(3)) in heterotic string compactifications. The decomposition requires not only the tuning of the SU(5) spectral covers but also the tuning of the complex structure moduli of the Calabi-Yau threefolds. This configuration is translated to geometric data on F-theory side. We find that the monodromy locus for two-cycles in K3 fibered Calabi-Yau fourfolds in a stable degeneration limit is globally factorized with squared factors under the decomposition conditions. This signals that the monodromy group is reduced and there is a U(1) symmetry in a low energy effective field theory. To support that, we explicitly check the reduction of a monodromy group in an appreciable region of the moduli space for an $E_6$ gauge theory with (1+2) decomposition. This may provide a systematic way for constructing F-theory models with U(1) symmetries.Comment: 41 pages, 14 figures; v2: minor improvements and a reference adde

The polygenic nature of telomere length and the anti-ageing properties of lithium

Telomere length is a promising biomarker for age-related disease and a potential anti-ageing drug target. Here, we study the genetic architecture of telomere length and the repositioning potential of lithium as an anti-ageing medication. LD score regression applied to the largest telomere length genome-wide association study to-date, revealed SNP-chip heritability estimates of 7.29%, with polygenic risk scoring capturing 4.4% of the variance in telomere length in an independent cohort (p = 6.17 × 10-5). Gene-enrichment analysis identified 13 genes associated with telomere length, with the most significant being the leucine rich repeat gene, LRRC34 (p = 3.69 × 10-18). In the context of lithium, we confirm that chronic use in a sample of 384 bipolar disorder patients is associated with longer telomeres (p = 0.03). As complementary evidence, we studied three orthologs of telomere length regulators in a Caenorhabditis elegans model of lithium-induced extended longevity and found all transcripts to be affected post-treatment (p  0.05). Consequently, this suggests that lithium may be catalysing the activity of endogenous mechanisms that promote telomere lengthening, whereby its efficacy eventually becomes limited by each individual's inherent telomere maintenance capabilities. Our work indicates a potential use of polygenic risk scoring for the prediction of adult telomere length and consequently lithium's anti-ageing efficacy

Calmodulin-like proteins localized to the conoid regulate motility and cell invasion by Toxoplasma gondii

Toxoplasma gondii contains an expanded number of calmodulin (CaM)-like proteins whose functions are poorly understood. Using a combination of CRISPR/Cas9-mediated gene editing and a plant-like auxin-induced degron (AID) system, we examined the roles of three apically localized CaMs. CaM1 and CaM2 were individually dispensable, but loss of both resulted in a synthetic lethal phenotype. CaM3 was refractory to deletion, suggesting it is essential. Consistent with this prediction auxin-induced degradation of CaM3 blocked growth. Phenotypic analysis revealed that all three CaMs contribute to parasite motility, invasion, and egress from host cells, and that they act downstream of microneme and rhoptry secretion. Super-resolution microscopy localized all three CaMs to the conoid where they overlap with myosin H (MyoH), a motor protein that is required for invasion. Biotinylation using BirA fusions with the CaMs labeled a number of apical proteins including MyoH and its light chain MLC7, suggesting they may interact. Consistent with this hypothesis, disruption of MyoH led to degradation of CaM3, or redistribution of CaM1 and CaM2. Collectively, our findings suggest these CaMs may interact with MyoH to control motility and cell invasion

Thoracic and Lumbar Vertebral Bone Mineral Density Changes in a Natural Occurring Dog Model of Diffuse Idiopathic Skeletal Hyperostosis

Ankylosing spinal disorders can be associated with alterations in vertebral bone mineral density (BMD). There is however controversy about vertebral BMD in patients wuse idiopathic skeletal hyperostosis (DISH). DISH in Boxer dogs has been considered a natural occurring disease model for DISH in people. The purpose of this study was to compare vertebral BMD between Boxers with and without DISH. Fifty-nine Boxers with (n=30) or without (n=29) DISH that underwent computed tomography were included. Vertebral BMD was calculated for each thoracic and lumbar vertebra by using an earlier reported and validated protocol. For each vertebral body, a region of interest was drawn on the axial computed tomographic images at three separate locations: immediately inferior to the superior end plate, in the middle of the vertebral body, and superior to the inferior end plate. Values from the three axial slices were averaged to give a mean Hounsfield Unit value for each vertebral body. Univariate statistical analysis was performed to identify factors to be included in a multivariate model. The multivariate model including all dogs demonstrated that vertebral DISH status (Coefficient 24.63; 95% CI 16.07 to 33.19; p <0.001), lumbar vertebrae (Coefficient -17.25; 95% CI -23.42 to -11.09; p < 0.01), and to a lesser extent higher age (Coefficient -0.56; 95% CI -1.07 to -0.05; p = 0.03) were significant predictors for vertebral BMD. When the multivariate model was repeated using only dogs with DISH, vertebral DISH status (Coefficient 20.67; 95% CI, 10.98 to 30.37; p < 0.001) and lumbar anatomical region (Coefficient -38.24; 95% CI, -47.75 to -28.73; p < 0.001) were again predictors for vertebral BMD but age was not. The results of this study indicate that DISH can be associated with decreased vertebral BMD. Further studies are necessary to evaluate the clinical importance and pathophysiology of this finding