Stability of Soft Quasicrystals in a Coupled-Mode Swift-Hohenberg Model for Three-Component Systems

In this article, we discuss the stability of soft quasicrystalline phases in a coupled-mode Swift-Hohenberg model for three-component systems, where the characteristic length scales are governed by the positive-definite gradient terms. Classic two-mode approximation method and direct numerical minimization are applied to the model. In the latter approach, we apply the projection method to deal with the potentially quasiperiodic ground states. A variable cell method of optimizing the shape and size of higher-dimensional periodic cell is developed to minimize the free energy with respect to the order parameters. Based on the developed numerical methods, we rediscover decagonal and dodecagonal quasicrystalline phases, and find diverse periodic phases and complex modulated phases. Furthermore, phase diagrams are obtained in various phase spaces by comparing the free energies of different candidate structures. It does show not only the important roles of system parameters, but also the effect of optimizing computational domain. In particular, the optimization of computational cell allows us to capture the ground states and phase behavior with higher fidelity. We also make some discussions on our results and show the potential of applying our numerical methods to a larger class of mean-field free energy functionals.Comment: 26 pages, 13 figures; accepted by Communications in Computational Physic

Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth

We investigate crossing minimization for 1-page and 2-page book drawings. We show that computing the 1-page crossing number is fixed-parameter tractable with respect to the number of crossings, that testing 2-page planarity is fixed-parameter tractable with respect to treewidth, and that computing the 2-page crossing number is fixed-parameter tractable with respect to the sum of the number of crossings and the treewidth of the input graph. We prove these results via Courcelle's theorem on the fixed-parameter tractability of properties expressible in monadic second order logic for graphs of bounded treewidth.Comment: Graph Drawing 201

Frequency-Weighted Model Reduction with Applications to Structured Models

In this paper, a frequency-weighted extension of a recently proposed model reduction method for linear systems is presented. The method uses convex optimization and can be used both with sample data and exact models. We also obtain bounds on the frequency-weighted error. The method is combined with a rank-minimization heuristic to approximate multiinput– multi-output systems.We also present two applications— environment compensation and simplification of interconnected models — where we argue the proposed methods are useful

The F-Landscape: Dynamically Determining the Multiverse

We evolve our Multiverse Blueprints to characterize our local neighborhood of the String Landscape and the Multiverse of plausible string, M- and F-theory vacua. Building upon the tripodal foundations of i) the Flipped SU(5) Grand Unified Theory (GUT), ii) extra TeV-Scale vector-like multiplets derived out of F-theory, and iii) the dynamics of No-Scale Supergravity, together dubbed No-Scale F-SU(5), we demonstrate the existence of a continuous family of solutions which might adeptly describe the dynamics of distinctive universes. This Multiverse landscape of F-SU(5) solutions, which we shall refer to as the F-Landscape, accommodates a subset of universes compatible with the presently known experimental uncertainties of our own universe. We show that by secondarily minimizing the minimum of the scalar Higgs potential of each solution within the F-Landscape, a continuous hypervolume of distinct minimum minimorum can be engineered which comprise a regional dominion of universes, with our own universe cast as the bellwether. We conjecture that an experimental signal at the LHC of the No-Scale F-SU(5) framework's applicability to our own universe might sensibly be extrapolated as corroborating evidence for the role of string, M- and F-theory as a master theory of the Multiverse, with No-Scale supergravity as a crucial and pervasive reinforcing structure.Comment: 15 Pages, 7 Figures, 1 Tabl