Detecting signals of weakly first-order phase transitions in two-dimensional Potts models

We investigate the first-order phase transitions of the $q$-state Potts models with $q = 5, 6, 7$, and $8$ on the two-dimensional square lattice, using Monte Carlo simulations. At the very weakly first-order transition of the $q=5$ system, the standard data-collapse procedure for the order parameter, carried out with results for a broad range of system sizes, works deceptively well and produces non-trivial critical exponents different from the trivial values expected for first-order transitions. However, a more systematic study reveals significant drifts in the `pseudo-critical' exponents as a function of the system size. For this purpose, we employ two methods of analysis: the data-collapse procedure with narrow range of the system size, and the Binder-cumulant crossing technique for pairs of system sizes. In both methods, the estimates start to drift toward the trivial values as the system size used in the analysis exceeds a certain `cross-over' length scale. This length scale is remarkably smaller than the correlation length at the transition point for weakly first-order transitions, e.g., less than one tenth for $q=5$, in contrast to the naive expectation that the system size has to be comparable to or larger than the correlation length to observe the correct behavior. The results overall show that proper care is indispensable to diagnose the nature of a phase transition with limited system sizes.Comment: 10 pages, 7 figures. One figure has been replaced to make our claim cleare

Mapping molar shapes on signaling pathways

A major challenge in evolutionary developmental biology is to understand how genetic mutations underlie phenotypic changes. In principle, selective pressures on the phenotype screen the gene pool of the population. Teeth are an excellent model for understanding evolutionary changes in the genotype-phenotype relationship since they exist throughout vertebrates. Genetically modified mice (mutants) with abnormalities in teeth have been used to explore tooth development. The relationship between signaling pathways and molar shape, however, remains elusive due to the high intrinsic complexity of tooth crowns. This hampers our understanding of the extent to which developmental factors explored in mutants explain developmental and phenotypic variation in natural species that represent the consequence of natural selection. Here we combine a novel morphometric method with two kinds of data mining techniques to extract data sets from the three-dimensional surface models of lower first molars: i) machine learning to maximize classification accuracy of 22 mutants, and ii) phylogenetic signal for 31 Murinae species. Major shape variation among mutants is explained by the number of cusps and cusp distribution on a tooth crown. The distribution of mutant mice in morphospace suggests a nonlinear relationship between the signaling pathways and molar shape variation. Comparative analysis of mutants and wild murines reveals that mutant variation overlaps naturally occurring diversity, including more ancestral and derived morphologies. However, taxa with transverse lophs are not fully covered by mutant variation, suggesting experimentally unexplored developmental factors in the evolutionary radiation of Murines. Author summary Teeth are found in almost all vertebrates, and they show many different morphologies. In mammals, especially the cheek teeth or molars are highly diverse in shape, reflecting a vast range of dietary habits and efficiency of occlusion. As teeth are the most durable part of the body, they preserve well in the fossil record. The diversity of molar fossils has been useful in reconstructing the diet and phylogeny of extinct mammals. Genetically modified mice (mutants) show diverse modifications of their molar morphology, but we lack computational tools to test to what extent mutant morphologies account for the natural diversity found in the wild. We developed data mining using machine learning and phylogeny-based methods to analyze three-dimensional molar shapes in mouse mutants and natural species. Although many mutants and species have comparable features, most of the mutant molar variation covers the more evolutionarily ancestral than the more evolutionary derived shapes. Yet to be explored developmental factors may underly the more extreme shapes.Peer reviewe

Higher-continuity s-version of finite element method with B-spline functions

This paper proposes a strategy to solve the problems of the conventional s-version of finite element method (SFEM) fundamentally. Because SFEM can reasonably model an analytical domain by superimposing meshes with different spatial resolutions, it has intrinsic advantages of local high accuracy, low computation time, and simple meshing procedure. However, it has disadvantages such as accuracy of numerical integration and matrix singularity. Although several additional techniques have been proposed to mitigate these limitations, they are computationally expensive or ad-hoc, and detract from its strengths. To solve these issues, we propose a novel strategy called B-spline based SFEM. To improve the accuracy of numerical integration, we employed cubic B-spline basis functions with $C^2$-continuity across element boundaries as the global basis functions. To avoid matrix singularity, we applied different basis functions to different meshes. Specifically, we employed the Lagrange basis functions as local basis functions. The numerical results indicate that using the proposed method, numerical integration can be calculated with sufficient accuracy without any additional techniques used in conventional SFEM. Furthermore, the proposed method avoids matrix singularity and is superior to conventional methods in terms of convergence for solving linear equations. Therefore, the proposed method has the potential to reduce computation time while maintaining a comparable accuracy to conventional SFEM.Comment: 40 pages, 15 figures and 2 table