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Designer spin systems via inverse statistical mechanics

Physical Review B 2013年第13期88卷 134104-134104页

作者： Robert A. DiStasio, Jr. Étienne Marcotte Roberto Car Frank H. Stillinger Salvatore Torquato Department of Chemistry Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

In this work, we extend recent inverse statistical-mechanical methods developed for many-particle systems to the case of spin systems. For simplicity, we focus in this initial study on the two-state Ising model with radial spin-spin interactions of finite range (i.e., extending beyond nearest-neighbor sites) on the square lattice under periodic boundary conditions. Our interest herein is to find the optimal set of shortest-range pair interactions within this family of Hamiltonians, whose corresponding ground state is a targeted spin configuration such that the difference in energies between the energetically closest competitor and the target is maximized. For an exhaustive list of competitors, this optimization problem is solved exactly using linear programming. The possible outcomes for a given target configuration can be organized into the following three solution classes: unique (nondegenerate) ground state (class I), degenerate ground states (class II), and solutions not contained in the previous two classes (class III). We have chosen to study a general family of striped-phase spin configurations comprised of alternating parallel bands of up and down spins of varying thicknesses and a general family of rectangular block checkerboard spin configurations with variable block size, which is a generalization of the classic antiferromagnetic Ising model. Our findings demonstrate that the structurally anisotropic striped phases, in which the thicknesses of up- and down-spin bands are equal, are unique ground states for isotropic short-ranged interactions. By contrast, virtually all of the block checkerboard targets are either degenerate or fall within class III solutions. The degenerate class II spin configurations are identified up to a certain block size. We also consider other target spin configurations with different degrees of global symmetries and order. Our investigation reveals that the solution class to which a target belongs depends sensitively on the nature

关键词： INVERSE functions LATTICE dynamics STRUCTURAL optimization ANTIFERROMAGNETIC materials GROUND state (Quantum mechanics) ANISOTROPIC crystals

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11th German Conference on Chemoinformatics (GCC 2015) Abstracts

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JOURNAL OF CHEMINFORMATICS 2016年第SUPPL 1期8卷 1-27页

作者： [Anonymous] GDCh-CIC Division Associated Board Member Beilstein-Institut zur Förderung der Chemischen Wissenschaften Trakehner Str. 7-9 60487 Frankfurt Germany. ufechner@beilstein-institut.de. Division Medicinal Chemistry Amsterdam Institute for Molecules Medicines and Systems (AIMMS) VU University Amsterdam The Netherlands. Centre for Bioinformatics Uni Hamburg Bundesstr. 43 20146 Hamburg Germany. Sanofi-Aventis Deutschland GmbH 65926 Frankfurt am Main Germany. stefan.guessregen@***. Sanofi-Aventis Deutschland GmbH 65926 Frankfurt am Main Germany. GlaxoSmithKline Stevenage SG1 2NY UK. nicola.j.richmond@***. Discngine Paris 75011 France. Organisch-Chemisches Institut Westfälische Wilhelms-Universität Münster Germany. marwin.segler@wwu.de. Organisch-Chemisches Institut Westfälische Wilhelms-Universität Münster Germany. Bundeskriminalamt Wiesbaden Central Analytics II 65173 Wiesbaden Germany. Department of Chemistry and Biochemistry University of California Santa Barbara CA 93111 USA. shea@chem.ucsb.edu. Department of Chemistry and Biochemistry University of California Santa Barbara CA 93111 USA. Chemistry Department Ludwig-Maximilians-Universität München Butenandtstr. 7 81377 Munich Germany. Inorganic Chemistry and Center for Nanointegration University of Duisburg-Essen Essen Germany. CAM-D Technologies Essen Germany. Institute for Bioinformatics and Chemoinformatics Westphalian University of Applied Sciences Recklinghausen Germany. Institute for Bioinformatics and Chemoinformatics Westphalian University of Applied Sciences Recklinghausen Germany. achim.zielesny@w-hs.de. Leiden University Leiden Netherlands. j.fraaije@chem.leidenuniv.nl. Culgi BV Berlin Germany. Physikalische Chemie III TU Dortmund 44227 Dortmund Germany. stefan.kast@tu-dortmund.de. Centre for Molecular Informatics Department of Chemistry University of Cambridge Lensfield Road Cambridge CB2 1EW United Kingdom. kcb27@cam.ac.uk. Centre for Molecular Informatics Department of Chemistry

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Jammed lattice sphere packings

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Physical Review E 2013年第6期88卷 062151-062151页

作者： Yoav Kallus Étienne Marcotte Salvatore Torquato Princeton Center for Theoretical Science Princeton University Princeton New Jersey 08544 USA Department of Physics Princeton University Princeton New Jersey 08544 USA Department of Chemistry Princeton University Princeton New Jersey 08544 USA Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA Princeton Institute of the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA

We generate and study an ensemble of isostatic jammed hard-sphere lattices. These lattices are obtained by compression of a periodic system with an adaptive unit cell containing a single sphere until the point of mechanical stability. We present detailed numerical data about the densities, pair correlations, force distributions, and structure factors of such lattices. We show that this model retains many of the crucial structural features of the classical hard-sphere model and propose it as a model for the jamming and glass transitions that enables exploration of much higher dimensions than are usually accessible.

关键词： LATTICE theory SPHERE packings ISOSTATIC pressing STABILITY (Mechanics) MATHEMATICAL models GLASS transitions

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Optimal frames and Newton's method

Optimal frames and Newton's method

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作者： Fickus, Matthew Mixon, Dustin G. Department of Mathematics Air Force Institute of Technology Wright-Patterson AFB OH 45433 United States Program in Applied and Computational Mathematics Princeton University Princeton NJ United States

Given a parametrized family of finite frames, we consider the optimization problem of finding the member of this family whose coefficient space most closely contains a given data vector. This nonlinear least squares problem arises naturally in the context of a certain type of radar system. We derive analytic expressions for the first and second partial derivatives of the objective function in question, permitting this optimization problem to be efficiently solved using Newton's method. We also consider how sensitive the location of this minimizer is to noise in the data vector. We further provide conditions under which one should expect the minimizer of this objective function to be unique. We conclude by discussing a related variational-calculus- based approach for solving this frame optimization problem over an interval of time. © 2012 Copyright.

关键词： Newton-Raphson method

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Approximating algebraic space curves by circular arcs 1

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7th International Conference on Curves and Surfaces, Curves and Surfaces 2010

作者： Béla, Szilvia Jüttler, Bert Doctoral Program in Computational Mathematics Johannes Kepler University 4040 Linz Austria Institute of Applied Geometry Johannes Kepler University Altenberger Str. 69 4040 Linz Austria

ISBN: (数字)9783642274138

ISBN: (纸本)9783642274121

We introduce a new method to approximate algebraic space curves. The algorithm combines a subdivision technique with local approximation of piecewise regular algebraic curve segments. The local technique computes pairs of polynomials with modified Taylor expansions and generates approximating circular arcs. We analyze the connection between the generated approximating arcs and the osculating circles of the algebraic curve. © 2012 Springer-Verlag.

关键词： Approximation algorithms

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Landauʼs necessary density conditions for the Hankel transform

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Journal of Functional Analysis 2012年第4期262卷 1845-1866页

作者： Luís Daniel Abreu Afonso S. Bandeira Department of Mathematics of University of Coimbra 3001-454 Coimbra Portugal Program in Applied and Computational Mathematics Princeton University NJ 08544 USA

We will prove an analogue of Landauʼs necessary conditions [H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967) 37–52] for spaces of functions whose Hankel transform is supported in a measurable subset S of the positive semi-axis. As a special case, necessary density conditions for the existence of Fourier–Bessel frames are obtained.

关键词： Sampling and interpolation Beurling–Landau density Hankel transform Bessel functions Fourier–Bessel frames

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Using a level set to model multiple myeloma induced bone loss

Using a level set to model multiple myeloma induced bone los...

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6th European Congress on computational Methods in applied Sciences and Engineering, ECCOMAS 2012

作者： Ayati, Bruce P. Graham, Jason M. Holstein, Sarah A. Department of Mathematics and Program in Applied Mathematical and Computational Sciences University of Iowa Iowa City IA 52242-1419 United States Department of Internal Medicine University of Iowa Iowa City IA 52242-1081 United States

ISBN: (纸本)9783950353709

Multiple myeloma is a hematological malignancy characterized by proliferation of malignant plasma cells and derangement of bone homeostasis. Myeloma bone disease results in significant morbidity as a result of bone pain, hypercalcemia, diffuse osteopenia, and pathologic fractures. We present a spatially explicit mathematical model of multiple myeloma and bone remodeling, synthesizing the existing model of local "microenvironment" interactions in Ayati et al. 2010 [1] with a level set approach for representing the sharp interface been bone and marrow introduced in [6]. computational results show the feasibility of using a level set to capture the spatial structure in the context of a geometrically straightforward interface, but one that nonetheless captures the essence of the rich geometries seen in bone marrow biopsy slides. In particular, we are able to model the formation of an osteolytic lesion in the case of multiple myeloma dysregulated bone remodeling, but not, using the same remodeling parameter set, in the case of normal bone remodeling.

关键词： Bone

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Computation of sparse low degree interpolating polynomials and their application to derivative-free optimization

Computation of sparse low degree interpolating polynomials a...

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作者： Bandeira, A.S. Scheinberg, K. Vicente, L.N. Program in Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States Department of Industrial and Systems Engineering Lehigh University Harold S. Mohler Laboratory 200 West Packer Avenue Bethlehem PA 18015-1582 United States CMUC Department of Mathematics University of Coimbra 3001-501 Coimbra Portugal

Interpolation-based trust-region methods are an important class of algorithms forDerivative-Free Optimizationwhich rely on locally approximating an objective function by quadratic polynomial interpolation models, frequently built from less points than there are basis components. Often, in practical applications, the contribution of the problem variables to the objective function is such that many pairwise correlations between variables are negligible, implying, in the smooth case, a sparse structure in the Hessian matrix. To be able to exploit Hessian sparsity, existing optimization approaches require the knowledge of the sparsity structure. The goal of this paper is to develop and analyze a method where the sparse models are constructed automatically. The sparse recovery theory developed recently in the field of compressed sensing characterizes conditions under which a sparse vector can be accurately recovered from few random measurements. Such a recovery is achieved by minimizing the l1-norm of a vector subject to the measurements constraints. We suggest an approach for building sparse quadratic polynomial interpolationmodels byminimizing the l1-norm of the entries of themodel Hessian subject to the interpolation conditions. We show that this procedure recovers accurate models when the function Hessian is sparse, using relatively few randomly selected sample points. Motivated by this result, we developed a practical interpolation-based trust-region method using deterministic sample sets and minimum l1-norm quadratic models. Our computational results show that the new approach exhibits a promising numerical performance both in the general case and in the sparse one. © Springer and Mathematical Optimization Society 2012.

关键词： Interpolation

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Organizing principles for dense packings of nonspherical hard particles: Not all shapes are created equal

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Physical Review E 2012年第1期86卷 011102-011102页

作者： Salvatore Torquato Yang Jiao []Department of Chemistry Department of Physics Princeton Center for Theoretical Science Program of Applied and Computational Mathematics Princeton Institute of the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA

We have recently devised organizing principles to obtain maximally dense packings of the Platonic and Archimedean solids and certain smoothly shaped convex nonspherical particles [Torquato and Jiao, Phys. Rev. E 81, 041310 (2010)]. Here we generalize them in order to guide one to ascertain the densest packings of other convex nonspherical particles as well as concave shapes. Our generalized organizing principles are explicitly stated as four distinct propositions. All of our organizing principles are applied to and tested against the most comprehensive set of both convex and concave particle shapes examined to date, including Catalan solids, prisms, antiprisms, cylinders, dimers of spheres, and various concave polyhedra. We demonstrate that all of the densest known packings associated with this wide spectrum of nonspherical particles are consistent with our propositions. Among other applications, our general organizing principles enable us to construct analytically the densest known packings of certain convex nonspherical particles, including spherocylinders, “lens-shaped” particles, square pyramids, and rhombic pyramids. Moreover, we show how to apply these principles to infer the high-density equilibrium crystalline phases of hard convex and concave particles. We also discuss the unique packing attributes of maximally random jammed packings of nonspherical particles.

关键词： PARTICLES (Nuclear physics) PACKINGS (Chromatography) CONCAVE functions CATALANS POLYHEDRA

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Densest binary sphere packings

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Physical Review E 2012年第2期85卷 021130-021130页

作者： Adam B. Hopkins Frank H. Stillinger Salvatore Torquato Department of Chemistry Department of Physics Princeton Center for Theoretical Science Princeton Institute for the Science and Technology of Materials Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

The densest binary sphere packings in the α-x plane of small to large sphere radius ratio α and small sphere relative concentration x have historically been very difficult to determine. Previous research had led to the prediction that these packings were composed of a few known “alloy” phases including, for example, the AlB2 (hexagonal ω), HgBr2, and AuTe2 structures, and to XYn structures composed of close-packed large spheres with small spheres (in a number ratio of n to 1) in the interstices, e.g., the NaCl packing for n=1. However, utilizing an implementation of the Torquato-Jiao sphere-packing algorithm [Torquato and Jiao, Phys. Rev. E 82, 061302 (2010)], we have discovered that many more structures appear in the densest packings. For example, while all previously known densest structures were composed of spheres in small to large number ratios of one to one, two to one, and very recently three to one, we have identified densest structures with number ratios of seven to three and five to two. In a recent work [Hopkins et al., Phys. Rev. Lett. 107, 125501 (2011)], we summarized these findings. In this work, we present the structures of the densest-known packings and provide details about their characteristics. Our findings demonstrate that a broad array of different densest mechanically stable structures consisting of only two types of components can form without any consideration of attractive or anisotropic interactions. In addition, the structures that we have identified may correspond to currently unidentified stable phases of certain binary atomic and molecular systems, particularly at high temperatures and pressures.

关键词： SPHERE packings RADIUS (Geometry) PHASE transformations (Physics) ALGORITHMS TEMPERATURE effect ANISOTROPY PRESSURE

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