A Tentative Description of the Crystallography of Amorphous Solids M
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On the Circuit Diameter Conjecture
On the Circuit Diameter Conjecture Steffen Borgwardt1, Tamon Stephen2, and Timothy Yusun2 1 University of Colorado Denver [email protected] 2 Simon Fraser University {tamon,tyusun}@sfu.ca Abstract. From the point of view of optimization, a critical issue is relating the combina- torial diameter of a polyhedron to its number of facets f and dimension d. In the seminal paper of Klee and Walkup [KW67], the Hirsch conjecture of an upper bound of f − d was shown to be equivalent to several seemingly simpler statements, and was disproved for unbounded polyhedra through the construction of a particular 4-dimensional polyhedron U4 with 8 facets. The Hirsch bound for bounded polyhedra was only recently disproved by Santos [San12]. We consider analogous properties for a variant of the combinatorial diameter called the circuit diameter. In this variant, the walks are built from the circuit directions of the poly- hedron, which are the minimal non-trivial solutions to the system defining the polyhedron. We are able to prove that circuit variants of the so-called non-revisiting conjecture and d-step conjecture both imply the circuit analogue of the Hirsch conjecture. For the equiva- lences in [KW67], the wedge construction was a fundamental proof technique. We exhibit why it is not available in the circuit setting, and what are the implications of losing it as a tool. Further, we show the circuit analogue of the non-revisiting conjecture implies a linear bound on the circuit diameter of all unbounded polyhedra – in contrast to what is known for the combinatorial diameter. Finally, we give two proofs of a circuit version of the 4-step conjecture. -
Cons=Ucticn (Process)
DOCUMENT RESUME ED 038 271 SE 007 847 AUTHOR Wenninger, Magnus J. TITLE Polyhedron Models for the Classroom. INSTITUTION National Council of Teachers of Mathematics, Inc., Washington, D.C. PUB DATE 68 NOTE 47p. AVAILABLE FROM National Council of Teachers of Mathematics,1201 16th St., N.V., Washington, D.C. 20036 ED RS PRICE EDRS Pr:ce NF -$0.25 HC Not Available from EDRS. DESCRIPTORS *Cons=ucticn (Process), *Geometric Concepts, *Geometry, *Instructional MateriAls,Mathematical Enrichment, Mathematical Models, Mathematics Materials IDENTIFIERS National Council of Teachers of Mathematics ABSTRACT This booklet explains the historical backgroundand construction techniques for various sets of uniformpolyhedra. The author indicates that the practical sianificanceof the constructions arises in illustrations for the ideas of symmetry,reflection, rotation, translation, group theory and topology.Details for constructing hollow paper models are provided for thefive Platonic solids, miscellaneous irregular polyhedra and somecompounds arising from the stellation process. (RS) PR WON WITHMICROFICHE AND PUBLISHER'SPRICES. MICROFICHEREPRODUCTION f ONLY. '..0.`rag let-7j... ow/A U.S. MOM Of NUM. INCIII01 a WWII WIC Of MAW us num us us ammo taco as mums NON at Ot Widel/A11011 01116111111 IT.P01115 OF VOW 01 OPENS SIAS SO 101 IIKISAMIT IMRE Offlaat WC Of MANN POMO OS POW. OD PROCESS WITH MICROFICHE AND PUBLISHER'S PRICES. reit WeROFICHE REPRODUCTION Pvim ONLY. (%1 00 O O POLYHEDRON MODELS for the Classroom MAGNUS J. WENNINGER St. Augustine's College Nassau, Bahama. rn ErNATIONAL COUNCIL. OF Ka TEACHERS OF MATHEMATICS 1201 Sixteenth Street, N.W., Washington, D. C. 20036 ivetmIssromrrIPRODUCE TmscortnIGMED Al"..Mt IAL BY MICROFICHE ONLY HAS IEEE rano By Mat __Comic _ TeachMar 10 ERIC MID ORGANIZATIONS OPERATING UNSER AGREEMENTS WHIM U. -
Bose-Einstein Condensation Measurements and Superflow in Condensed Helium
JLowTempPhys DOI 10.1007/s10909-013-0855-0 Bose-Einstein Condensation Measurements and Superflow in Condensed Helium H.R. Glyde Received: 9 November 2012 / Accepted: 18 January 2013 © Springer Science+Business Media New York 2013 Abstract We review the formulation and measurement of Bose-Einstein condensa- tion (BEC) in liquid and solid helium. BEC is defined for a Bose gas and subsequently for interacting systems via the one-body density matrix (OBDM) valid for both uni- form and non-uniform systems. The connection between the phase coherence created by BEC and superflow is made. Recent measurements show that the condensate frac- tion in liquid 4He drops from 7.25 ± 0.75 % at saturated vapor pressure (p ≈ 0) to 2.8 ± 0.2 % at pressure p = 24 bars near the solidification pressure (p = 25.3 bar). Extrapolation to solid densities suggests a condensate fraction in the solid of 1 % or less, assuming a frozen liquid structure such as an amorphous solid. Measurements in the crystalline solid have not been able to detect a condensate with an upper limit 4 set at n0 ≤ 0.3 %. Opportunities to observe BEC directly in liquid He confined in porous media, where BEC is localized to patches by disorder, and in amorphous solid helium is discussed. Keywords Solid · Liquid · Helium · BEC · OBDM · Neutron scattering 1 Introduction This article is part of a series motivated by reports of possible superfluidity in solid helium, initially in 2004 [1, 2]. Other articles in this series survey extensively the experiments and theory in this new field and the current status of the field. -
Drug-Rich Phases Induced by Amorphous Solid Dispersion: Arbitrary Or Intentional Goal in Oral Drug Delivery?
pharmaceutics Review Drug-Rich Phases Induced by Amorphous Solid Dispersion: Arbitrary or Intentional Goal in Oral Drug Delivery? Kaijie Qian 1 , Lorenzo Stella 2,3 , David S. Jones 1, Gavin P. Andrews 1,4, Huachuan Du 5,6,* and Yiwei Tian 1,* 1 Pharmaceutical Engineering Group, School of Pharmacy, Queen’s University Belfast, 97 Lisburn Road, Belfast BT9 7BL, UK; [email protected] (K.Q.); [email protected] (D.S.J.); [email protected] (G.P.A.) 2 Atomistic Simulation Centre, School of Mathematics and Physics, Queen’s University Belfast, 7–9 College Park E, Belfast BT7 1PS, UK; [email protected] 3 David Keir Building, School of Chemistry and Chemical Engineering, Queen’s University Belfast, Stranmillis Road, Belfast BT9 5AG, UK 4 School of Pharmacy, China Medical University, No.77 Puhe Road, Shenyang North New Area, Shenyang 110122, China 5 Laboratory of Applied Mechanobiology, Department of Health Sciences and Technology, ETH Zurich, Vladimir-Prelog-Weg 4, 8093 Zurich, Switzerland 6 Simpson Querrey Institute, Northwestern University, 303 East Superior Street, 11th Floor, Chicago, IL 60611, USA * Correspondence: [email protected] (H.D.); [email protected] (Y.T.); Tel.: +41-446339049 (H.D.); +44-2890972689 (Y.T.) Abstract: Among many methods to mitigate the solubility limitations of drug compounds, amor- Citation: Qian, K.; Stella, L.; Jones, phous solid dispersion (ASD) is considered to be one of the most promising strategies to enhance D.S.; Andrews, G.P.; Du, H.; Tian, Y. the dissolution and bioavailability of poorly water-soluble drugs. -
Decomposing Deltahedra
Decomposing Deltahedra Eva Knoll EK Design ([email protected]) Abstract Deltahedra are polyhedra with all equilateral triangular faces of the same size. We consider a class of we will call ‘regular’ deltahedra which possess the icosahedral rotational symmetry group and have either six or five triangles meeting at each vertex. Some, but not all of this class can be generated using operations of subdivision, stellation and truncation on the platonic solids. We develop a method of generating and classifying all deltahedra in this class using the idea of a generating vector on a triangular grid that is made into the net of the deltahedron. We observed and proved a geometric property of the length of these generating vectors and the surface area of the corresponding deltahedra. A consequence of this is that all deltahedra in our class have an integer multiple of 20 faces, starting with the icosahedron which has the minimum of 20 faces. Introduction The Japanese art of paper folding traditionally uses square or sometimes rectangular paper. The geometric styles such as modular Origami [4] reflect that paradigm in that the folds are determined by the geometry of the paper (the diagonals and bisectors of existing angles and lines). Using circular paper creates a completely different design structure. The fact that chords of radial length subdivide the circumference exactly 6 times allows the use of a 60 degree grid system [5]. This makes circular Origami a great tool to experiment with deltahedra (Deltahedra are polyhedra bound by equilateral triangles exclusively [3], [8]). After the barn-raising of an endo-pentakis icosi-dodecahedron (an 80 faced regular deltahedron) [Knoll & Morgan, 1999], an investigation of related deltahedra ensued. -
Characteristics of the Solid State and Differences Between Crystalline and Amorphous Solids
IJRECE VOL. 7 ISSUE 2 Apr.-June 2019 ISSN: 2393-9028 (PRINT) | ISSN: 2348-2281 (ONLINE) Characteristics of the Solid State and Differences between Crystalline and Amorphous Solids Anu Assistant Professor, Department of Physics, Indus Degree college, Kinana Abstract: “A crystal is a solid composed of atoms (ions or possess the unique property of being rigid. Such solids are molecules) arranged in an orderly repetitive array.” Most of known as true solids e.g., NaCl, KCl, Sugar, Ag, Cu etc. the naturally occurring solids are found to have definite On the other hand the solid which loses shape on long crystalline shapes which can be recognized easily. These are standing, flows under its own weight and is easily distorted by in large size because these are formed very slowly, thus even mild distortion force, is called pseudo solid e.g., glass, particles get sufficient time to get proper position in the pith etc. Some solids such as NaCl, sugar, sulphur etc. have crystal structure. Some crystalline solids are so small that properties not only of rigidity and incompressibility but also appear to be amorphous. But on examination under a powerful of having typical geometrical forms. These solids are called microscope they are also found to have a definite crystalline crystalline solids. In such solids there is definite arrangement shape. Such solids are known as micro-crystalline solids. of particles (atoms, ions or molecules) throughout the entire three dimensional network of a crystal. This is named as long- Key Words: Crystal, Goniometer, Amorphous solids range order. This three dimensional arrangement is called crystal lattice or space lattice. -
Upper Bound on the Packing Density of Regular Tetrahedra and Octahedra
Upper bound on the packing density of regular tetrahedra and octahedra Simon Gravel Veit Elser and Yoav Kallus Department of Genetics Laboratory of Atomic and Solid State Physics Stanford University School of Medicine Cornell University Stanford, California, 94305-5120 Ithaca, NY 14853-2501 Abstract We obtain an upper bound to the packing density of regular tetrahedra. The bound is obtained by showing the existence, in any packing of regular tetrahedra, of a set of disjoint spheres centered on tetrahedron edges, so that each sphere is not fully covered by the packing. The bound on the amount of space that is not covered in each sphere is obtained in a recursive way by building on the observation that non-overlapping regular tetrahedra cannot subtend a solid angle of 4π around a point if this point lies on a tetrahedron edge. The proof can be readily modified to apply to other polyhedra with the same property. The resulting lower bound on the fraction of empty space in a packing of regular tetrahedra is 2:6 ::: × 10−25 and reaches 1:4 ::: × 10−12 for regular octahedra. 1 Introduction The problem of finding dense arrangements of non-overlapping objects, also known as the packing prob- lem, holds a long and eventful history and holds fundamental interest in mathematics, physics, and computer science. Some instances of the packing problem rank among the longest-standing open problems in mathe- matics. The archetypal difficult packing problem is to find the arrangements of non-overlapping, identical balls that fill up the greatest volume fraction of space. The face-centered cubic lattice was conjectured to realize the highest packing fraction by Kepler, in 1611, but it was not until 1998 that this conjecture was established arXiv:1008.2830v1 [math.MG] 17 Aug 2010 using a computer-assisted proof [1] (as of March 2009, work was still in progress to “provide a greater level of certification of the correctness of the computer code and other details of the proof” [2]) . -
Triangulations and Simplex Tilings of Polyhedra
Triangulations and Simplex Tilings of Polyhedra by Braxton Carrigan A dissertation submitted to the Graduate Faculty of Auburn University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Auburn, Alabama August 4, 2012 Keywords: triangulation, tiling, tetrahedralization Copyright 2012 by Braxton Carrigan Approved by Andras Bezdek, Chair, Professor of Mathematics Krystyna Kuperberg, Professor of Mathematics Wlodzimierz Kuperberg, Professor of Mathematics Chris Rodger, Don Logan Endowed Chair of Mathematics Abstract This dissertation summarizes my research in the area of Discrete Geometry. The par- ticular problems of Discrete Geometry discussed in this dissertation are concerned with partitioning three dimensional polyhedra into tetrahedra. The most widely used partition of a polyhedra is triangulation, where a polyhedron is broken into a set of convex polyhedra all with four vertices, called tetrahedra, joined together in a face-to-face manner. If one does not require that the tetrahedra to meet along common faces, then we say that the partition is a tiling. Many of the algorithmic implementations in the field of Computational Geometry are dependent on the results of triangulation. For example computing the volume of a polyhedron is done by adding volumes of tetrahedra of a triangulation. In Chapter 2 we will provide a brief history of triangulation and present a number of known non-triangulable polyhedra. In this dissertation we will particularly address non-triangulable polyhedra. Our research was motivated by a recent paper of J. Rambau [20], who showed that a nonconvex twisted prisms cannot be triangulated. As in algebra when proving a number is not divisible by 2012 one may show it is not divisible by 2, we will revisit Rambau's results and show a new shorter proof that the twisted prism is non-triangulable by proving it is non-tilable. -
Irreversible Transition of Amorphous and Polycrystalline Colloidal Solids Under Cyclic Deformation
PHYSICAL REVIEW E 98, 062607 (2018) Irreversible transition of amorphous and polycrystalline colloidal solids under cyclic deformation Pritam Kumar Jana,1,2,* Mikko J. Alava,1 and Stefano Zapperi1,3,4 1COMP Centre of Excellence, Department of Applied Physics, Aalto University, P.O. Box 11100, FI-00076 Aalto, Espoo, Finland 2Université Libre de Bruxelles (ULB), Interdisciplinary Center for Nonlinear Phenomena and Complex Systems, Campus Plaine, CP 231, Blvd. du Triomphe, B-1050 Brussels, Belgium 3Center for Complexity and Biosystems, Department of Physics, University of Milano, via Celoria 16, 20133 Milano, Italy 4CNR-ICMATE, Via R. Cozzi 53, 20125, Milano, Italy (Received 26 June 2018; published 14 December 2018) Cyclic loading on granular packings and amorphous media leads to a transition from reversible elastic behavior to an irreversible plasticity. In the present study, we investigate the effect of oscillatory shear on polycrystalline and amorphous colloidal solids by performing molecular dynamics simulations. Our results show that close to the transition, both systems exhibit enhanced particle mobility, hysteresis, and rheological loss of rigidity. However, the rheological response shows a sharper transition in the case of the polycrystalline sample as compared to the amorphous solid. In the polycrystalline system, we see the disappearance of disclinations, which leads to the formation of a monocrystalline system, whereas the amorphous system hardly shows any ordering. After the threshold strain amplitude, as we increase the strain amplitude both systems get fluid. In addition to that, particle displacements are more homogeneous in the case of polycrystalline systems as compared to the amorphous solid, mainly when the strain amplitude is larger than the threshold value. -
1 Mar 2014 Polyhedra, Complexes, Nets and Symmetry
Polyhedra, Complexes, Nets and Symmetry Egon Schulte∗ Northeastern University Department of Mathematics Boston, MA 02115, USA March 4, 2014 Abstract Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial, and algebraic properties. They can be viewed as finite or infinite 3-periodic graphs (nets) equipped with additional structure imposed by the faces, allowed to be skew, zig- zag, or helical. A polyhedron or complex is regular if its geometric symmetry group is transitive on the flags (incident vertex-edge-face triples). There are 48 regular polyhedra (18 finite polyhedra and 30 infinite apeirohedra), as well as 25 regular polygonal complexes, all infinite, which are not polyhedra. Their edge graphs are nets well-known to crystallographers, and we identify them explicitly. There also are 6 infinite families of chiral apeirohedra, which have two orbits on the flags such that adjacent flags lie in different orbits. 1 Introduction arXiv:1403.0045v1 [math.MG] 1 Mar 2014 Polyhedra and polyhedra-like structures in ordinary euclidean 3-space E3 have been stud- ied since the early days of geometry (Coxeter, 1973). However, with the passage of time, the underlying mathematical concepts and treatments have undergone fundamental changes. Over the past 100 years we can observe a shift from the classical approach viewing a polyhedron as a solid in space, to topological approaches focussing on the underlying maps on surfaces (Coxeter & Moser, 1980), to combinatorial approaches highlighting the basic incidence structure but deliberately suppressing the membranes that customarily span the faces to give a surface. -
Evidence for a Superglass State in Solid 4He
Evidence for a Superglass State in Solid 4He B. Hunt,1§ E. Pratt,1§ V. Gadagkar,1 M. Yamashita,1,2 A. V. Balatsky3 and J.C. Davis1,4 1 Laboratory of Atomic and Solid State Physics, Department of Physics, Cornell University, Ithaca, NY 14853, USA. 2 Department of Physics, Kyoto University, Kyoto 606-8502, Japan. 3 T-Division, Center for Integrated Nanotechnologies, MS B 262, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. 4 Scottish Universities Physics Alliance, School of Physics and Astronomy, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland, UK. §These authors contributed equally to this work. Science 324, 632 (1 May 2009) Although solid helium-4 (4He) may be a supersolid it also exhibits many phenomena unexpected in that context. We studied relaxation dynamics in the resonance frequency f(T) and dissipation D(T) of a torsional oscillator containing solid 4He. With the appearance of the “supersolid” state, the relaxation times within f(T) and D(T) began to increase rapidly together. More importantly, the relaxation processes in both D(T) and a component of f(T) exhibited a complex synchronized ultra-slow evolution towards equilibrium. Analysis using a generalized rotational susceptibility revealed that, while exhibiting these apparently glassy dynamics, the phenomena were quantitatively inconsistent with a simple excitation freeze-out transition because the variation in f was far too large. One possibility is that amorphous solid 4He represents a new form of supersolid in which dynamical excitations within the solid control the superfluid phase stiffness. A ‘classic’ supersolid (1-5) is a bosonic crystal with an interpenetrating superfluid component. -
SOLID STATE PHYSICS PART II Optical Properties of Solids
SOLID STATE PHYSICS PART II Optical Properties of Solids M. S. Dresselhaus 1 Contents 1 Review of Fundamental Relations for Optical Phenomena 1 1.1 Introductory Remarks on Optical Probes . 1 1.2 The Complex dielectric function and the complex optical conductivity . 2 1.3 Relation of Complex Dielectric Function to Observables . 4 1.4 Units for Frequency Measurements . 7 2 Drude Theory{Free Carrier Contribution to the Optical Properties 8 2.1 The Free Carrier Contribution . 8 2.2 Low Frequency Response: !¿ 1 . 10 ¿ 2.3 High Frequency Response; !¿ 1 . 11 À 2.4 The Plasma Frequency . 11 3 Interband Transitions 15 3.1 The Interband Transition Process . 15 3.1.1 Insulators . 19 3.1.2 Semiconductors . 19 3.1.3 Metals . 19 3.2 Form of the Hamiltonian in an Electromagnetic Field . 20 3.3 Relation between Momentum Matrix Elements and the E®ective Mass . 21 3.4 Spin-Orbit Interaction in Solids . 23 4 The Joint Density of States and Critical Points 27 4.1 The Joint Density of States . 27 4.2 Critical Points . 30 5 Absorption of Light in Solids 36 5.1 The Absorption Coe±cient . 36 5.2 Free Carrier Absorption in Semiconductors . 37 5.3 Free Carrier Absorption in Metals . 38 5.4 Direct Interband Transitions . 41 5.4.1 Temperature Dependence of Eg . 46 5.4.2 Dependence of Absorption Edge on Fermi Energy . 46 5.4.3 Dependence of Absorption Edge on Applied Electric Field . 47 5.5 Conservation of Crystal Momentum in Direct Optical Transitions . 47 5.6 Indirect Interband Transitions .