Simple Wriggling is Hard unless You Are a Fat Hippo

We prove that it is NP-hard to decide whether two points in a polygonal domain with holes can be connected by a wire. This implies that finding any approximation to the shortest path for a long snake amidst polygonal obstacles is NP-hard. On the positive side, we show that snake's problem is "length-tractable": if the snake is "fat", i.e., its length/width ratio is small, the shortest path can be computed in polynomial time.Comment: A shorter version is to be presented at FUN 201

Cluster Dynamical Mean-field calculations for TiOCl

Based on a combination of cluster dynamical mean field theory (DMFT) and density functional calculations, we calculated the angle-integrated spectral density in the layered $s=1/2$ quantum magnet TiOCl. The agreement with recent photoemission and oxygen K-edge X-ray absorption spectroscopy experiments is found to be good. Th e improvement achieved with this calculation with respect to previous single-site DMFT calculations is an indication of the correlated nature and low-dimensionality of TiOCl.Comment: 9 pages, 3 figures, improved version as publishe

Impact of diagnostic misclassification on estimation of genetic correlations using genome-wide genotypes

Disorders that share genetic risk factors often are placed in closely related diagnostic categories and treated similarly. Until recently, evidence for shared genetic etiology derived from classical research strategies – coaggregation in family and twin studies. Accumulating sufficient numbers of families was often problematic. However, in the era of genome-wide genotyping, we can now directly estimate the degree of sharing of genetic risk factors between disorders. This strategy is practical even for very rare disorders, where it is infeasible to ascertain informative families. Importantly, the estimates of genetic correlations from genome-wide genotypes are derived using such distant relatives that contamination by shared environmental factors seems unlikely. However, any method that seeks to quantify the shared etiology of disorders assumes they can be distinguished diagnostically from one another without error. Here we investigate the impact of misdiagnosis on estimates of genetic correlation both from traditional family data and from genome-wide genotypes of case–control samples from unrelated individuals. Our analyses show similar results for levels of misdiagnosis in both types of data. In both scenarios, genetic variances and heritabilities tend to be slightly underestimated but genetic correlations are overestimated, sometimes substantially so. For example, two genetically distinct but equally heritable disorders each with prevalence 1%, can generate false-positive estimates of genetic correlations of >0.2 in the presence of 10% reciprocal misdiagnosis. Strategies for minimizing the effects of misdiagnosis in cross-disorder genetic studies are discussed

Electronic Structure Calculation by First Principles for Strongly Correlated Electron Systems

Recent trends of ab initio studies and progress in methodologies for electronic structure calculations of strongly correlated electron systems are discussed. The interest for developing efficient methods is motivated by recent discoveries and characterizations of strongly correlated electron materials and by requirements for understanding mechanisms of intriguing phenomena beyond a single-particle picture. A three-stage scheme is developed as renormalized multi-scale solvers (RMS) utilizing the hierarchical electronic structure in the energy space. It provides us with an ab initio downfolding of the global band structure into low-energy effective models followed by low-energy solvers for the models. The RMS method is illustrated with examples of several materials. In particular, we overview cases such as dynamics of semiconductors, transition metals and its compounds including iron-based superconductors and perovskite oxides, as well as organic conductors of kappa-ET type.Comment: 44 pages including 38 figures, to appear in J. Phys. Soc. Jpn. as an invited review pape

Dynamical Mean-Field Theory

The dynamical mean-field theory (DMFT) is a widely applicable approximation scheme for the investigation of correlated quantum many-particle systems on a lattice, e.g., electrons in solids and cold atoms in optical lattices. In particular, the combination of the DMFT with conventional methods for the calculation of electronic band structures has led to a powerful numerical approach which allows one to explore the properties of correlated materials. In this introductory article we discuss the foundations of the DMFT, derive the underlying self-consistency equations, and present several applications which have provided important insights into the properties of correlated matter.Comment: Chapter in "Theoretical Methods for Strongly Correlated Systems", edited by A. Avella and F. Mancini, Springer (2011), 31 pages, 5 figure

Orbital state and magnetic properties of LiV_2 O_4

LiV_2 O_4 is one of the most puzzling compounds among transition metal oxides because of its heavy fermion like behavior at low temperatures. In this paper we present results for the orbital state and magnetic properties of LiV_2 O_4 obtained from a combination of density functional theory within the local density approximation and dynamical mean-field theory (DMFT). The DMFT equations are solved by quantum Monte Carlo simulations. The trigonal crystal field splits the V 3d orbitals such that the a_{1g} and e_{g}^{pi} orbitals cross the Fermi level, with the former being slightly lower in energy and narrower in bandwidth. In this situation, the d-d Coulomb interaction leads to an almost localization of one electron per V ion in the a_{1g} orbital, while the e_{g}^{pi} orbitals form relatively broad bands with 1/8 filling. 2The theoretical high-temperature paramagnetic susceptibility chi(T) follows a Curie-Weiss law with an effective paramagnetic moment p_{eff}=1.65 in agreement with the experimental results.Comment: 11 pages, 10 figures, 2 table

Static perfect fluids with Pant-Sah equations of state

We analyze the 3-parameter family of exact, regular, static, spherically symmetric perfect fluid solutions of Einstein's equations (corresponding to a 2-parameter family of equations of state) due to Pant and Sah and "rediscovered" by Rosquist and the present author. Except for the Buchdahl solutions which are contained as a limiting case, the fluids have finite radius and are physically realistic for suitable parameter ranges. The equations of state can be characterized geometrically by the property that the 3-metric on the static slices, rescaled conformally with the fourth power of any linear function of the norm of the static Killing vector, has constant scalar curvature. This local property does not require spherical symmetry; in fact it simplifies the the proof of spherical symmetry of asymptotically flat solutions which we recall here for the Pant-Sah equations of state. We also consider a model in Newtonian theory with analogous geometric and physical properties, together with a proof of spherical symmetry of the asymptotically flat solutions.Comment: 32 p., Latex, minor changes and correction

A Genetic Investigation of Sex Bias in the Prevalence of Attention-Deficit/Hyperactivity Disorder

Background Attention-deficit/hyperactivity disorder (ADHD) shows substantial heritability and is two to seven times more common in male individuals than in female individuals. We examined two putative genetic mechanisms underlying this sex bias: sex-specific heterogeneity and higher burden of risk in female cases. Methods We analyzed genome-wide autosomal common variants from the Psychiatric Genomics Consortium and iPSYCH Project (n = 20,183 cases, n = 35,191 controls) and Swedish population register data (n = 77,905 cases, n = 1,874,637 population controls). Results Genetic correlation analyses using two methods suggested near complete sharing of common variant effects across sexes, with rg estimates close to 1. Analyses of population data, however, indicated that female individuals with ADHD may be at especially high risk for certain comorbid developmental conditions (i.e., autism spectrum disorder and congenital malformations), potentially indicating some clinical and etiological heterogeneity. Polygenic risk score analysis did not support a higher burden of ADHD common risk variants in female cases (odds ratio [confidence interval] = 1.02 [0.98–1.06], p = .28). In contrast, epidemiological sibling analyses revealed that the siblings of female individuals with ADHD are at higher familial risk for ADHD than the siblings of affected male individuals (odds ratio [confidence interval] = 1.14 [1.11–1.18], p = 1.5E-15). Conclusions Overall, this study supports a greater familial burden of risk in female individuals with ADHD and some clinical and etiological heterogeneity, based on epidemiological analyses. However, molecular genetic analyses suggest that autosomal common variants largely do not explain the sex bias in ADHD prevalence

Functional approach to 2+1 dimensional gravity coupled to particles

The quantum gravity problem of N point particles interacting with the gravitational field in 2+1 dimensions is approached working out the phase-space functional integral. The maximally slicing gauge is adopted for a non compact open universe with the topology of the plane. The conjugate momenta to the gravitational field are related to a class of meromorphic quadratic differentials. The boundary term for the non compact space is worked out in detail. In the extraction of the physical degrees of freedom functional determinants related to the puncture formulation of string theory occur and cancel out in the final reduction. Finally the ordering problem in the definition of the functional integral is discussed.Comment: 29 pages LaTe