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1. Index To Catalogue Of Lattices A catalogue of different types with applications to packing spheres by Neil Sloane and Gabriele Nebe. http://www.research.att.com/~njas/lattices/

A Catalogue of Lattices Keywords : tables, lattices, quadratic forms, lattice packings, lattice coverings, An lattices An* lattices anabasic lattice Barnes-Wall lattices ... E6, E7, E8 lattices and their duals , Eisenstein lattices, Elkies-Shioda lattices face-centered cubic lattice , Hurwitzian lattices, isodual lattices William Jagy: ternary forms that are spinor regular but not regular kissing numbers Kleinian lattices ... Niemeier lattices , Gordon Nipp's tables of quaternary and quinary forms, perfect lattices Quebbemann lattices Rao-Reddy code root lattices ... weight lattices , lattices in , and higher , dimensions, abbreviations change library file in html format to standard format change standard format to GAP format change standard format to MACSYMA format ... change standard format to PARI format , etc. This data-base of lattices is a joint project of Gabriele Nebe , RWTH Aaachen (nebe(AT)math.rwth-aachen.de) and Neil Sloane Our aim is to give information about all the interesting lattices in "low" dimensions (and to provide them with a "home page"!). The data-base now contains about 160,000 lattices! Remarks For the format and for various programs to convert to other formats, see

2. Point Groups And Bravais Lattices Point Groups and Bravais Lattices. The following modules and image archives are made available for educational purposes. high resolution (640x480) animated http://neon.mems.cmu.edu/degraef/pg/index.html

Point Groups and Bravais Lattices The following modules and image archives are made available for educational purposes: high resolution (640x480) animated GIF movies of the 32 point groups, using a helical object as general point; high resolution (120x1200) JPEG images of each point group (essentially the first frame of each of the animated GIF movies); an archive of Rayshade input files used to generate most of the things above. The paragraphs below provide more details on each item. If you plan to use any of this stuff, please read Animated GIF movies The movies can be downloaded as needed from this page , or the entire gzipped and tarred archive can be downloaded from here (48602312 bytes). High resolution JPEG images Another archive...this one contains 32 JPEG files (compressed with quality 100%), one for each point group. The rendering was done at 1200x1200 pixels, and the images in this archive can be printed on a good quality color printer, e.g., on transparencies for classroom use. You can download the archive here (7970148 bytes).

3. Phase Transitions On Lattices Phase transitions and solitons in magnetics, site and continuum percolations with interactive Java applets and comments. http://www.ibiblio.org/e-notes/Perc/contents.htm

Phase transitions on lattices The 2D Ising model Phase transitions in magnetics Magnetics. Magnetization and interaction energy. The Boltzmann distribution. Entropy. The Monte-Carlo method Markov random walks. Importance sampling. The Metropolis algorithm. Detailed balance conditions. Susceptibility and specific heat. The thermostat algorithm. High non-equilibrium spin flip dynamics. Critical slowing down Relaxation time of magnetization. Critical slowing down. Ferromagnetic phase transition Second order phase transition. Critical exponents of the 2D Ising model. Antiferromagnetics Order parameter in antiferromagnetics. Frustrations. Topological exitations in ferromagnetics Vorteces in the XY model Vorteces and antivorteces. Birth and annihilation of the vortex - antivortex pair. The Kosterlitz and Thouless phase transiton. 3D vortex threads and rings. Excitations in 1D spin chain Energy of spin chain. Topological charge. Maps of circle. Continuum limit. Topological solitons in the Heisenberg ferromagnetic Instantons and anti-instantons. Maps of sphere. What does hedgehog hide? Hedgehog zoo Percolation Geometric phase transition 2D site percolation. Cluster labeling. Phase transition. Single 2D and 3D clusters.

4. Lattices Lattices are geometric objects that can be pictorially described as the set of intersection points of a regular (but not necessarily orthogonal) http://www.cs.ucsd.edu/~daniele/lattice/lattice.html

Lattices Lattices are geometric objects that can be pictorially described as the set of intersection points of a regular (but not necessarily orthogonal) n-dimentional infinite grid (See Fig. 1). Figure 1: A Lattice in 2 dimensions Despite their apparent simplicity, lattices are poweful combinatorial objects that can be used to solve many complex problems in mathematics and computer science. In particular they have been widely used in crypotology. Quite peculiarly lattices have been proven useful both in cryptanalysis (i.e., breaking cryptosystems) and cryptography (i.e., designing secure cryptographic functions based on the hardness of certain lattice problems). Lattices are usually specified by a basis (i.e., n linearly indepentend vectors) such that any lattice point can be obtained as an integer linear combination of the basis vectors. For example, the (red) lattice point x in Fig. 1 is a linear combination of the two (blue) basis vectors with integer coefficients 3 and 2. The same lattice (i.e., the same set of intersection points) can be represented by several different bases. For example, the lattice from Figure 1 can also be generated by the following basis: Figure 2: A different basis fot the same lattice Notice that in order to obtain vector x, now the basis vectors must be multiplied by 1 and 1. However, the set of points that can be expressed as an integer linear combination of the basis vectors is exactly the same for the two bases.

5. Bravais Lattices Most solids have periodic arrays of atoms which form what we call a crystal lattice. Amorphous solids and glasses are exceptions. http://hyperphysics.phy-astr.gsu.edu/hbase/solids/bravais.html

The 14 Bravais Lattices Most solids have periodic arrays of atoms which form what we call a crystal lattice . Amorphous solids and glasses are exceptions. The existence of the crystal lattice implies a degree of symmetry in the arrangement of the lattice, and the existing symmetries have been studied extensively. A crystal structure is one of the characteristics of minerals One of the implications of the symmetric lattice of atoms is that it can support resonant lattice vibration modes . These vibrations transport energy and are important in the thermal conductivity of non-metals, and in the heat capacity of all solids. Index References Myers Ch 2 Kittel, Intro to Solid State Ch 1 HyperPhysics Condensed Matter Electricity and Magnetism R Nave Go Back

6. Gallery Of Abrikosov Lattices In Superconductors Vortex lattice in film with artificial periodic pinning, 1996. Lorentz Microscopy Nb film with holes on a square lattice, half matching field http://www.fys.uio.no/super/vortex/

Gallery of Abrikosov Lattices in Superconductors Theoretical Prediction of Vortices in Type-II Superconductors A. A. Abrikosov, Institute of Physical Problems, USSR Soviet Physics JETP 5, 1174 (1957) Nobel prize to A. A. Abrikosov for pioneering contribution to the theory of superconductors First image of Vortex lattice, 1967 Bitter Decoration Pb-4at%In rod, 1.1K, 195G U. Essmann and H. Trauble Max-Planck Institute, Stuttgart Physics Letters 24A, 526 (1967) Vortex lattice in high-Tc superconductor, 1987 Bitter Decoration YBa2Cu3O7 crystal, 4.2K, 52G P. L. Gammel et al. Bell Labs Phys. Rev. Lett. 59, 2592 (1987) STM image of Vortex lattice, 1989 Scanning Tunnel Microscopy H. F. Hess et al. Bell Labs Phys. Rev. Lett. 62, 214 (1989) Vortex chains in tilted field, 1991 Bitter Decoration Bi-Sr-Ca-Cu-O crystal, 35G tilted by 70 degrees C. A. Bolle et al. Bell Labs Phys. Rev. Lett. 66, 112 (1991) Imaging Vortex lattice with Lorentz Microscopy, 1992 Lorentz Microscopy Nb film K. Harada et al. Hitachi Lab Nature 360, 51 (1992)

7. 06: Order, Lattices, Ordered Algebraic Structures Ordered sets, or Lattices, give a uniform structure to, for example, the set of subfields of a field. Various special types of Lattices have particularly http://www.math.niu.edu/~rusin/known-math/index/06-XX.html

Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields POINTERS: Texts Software Web links Selected topics here 06: Order, lattices, ordered algebraic structures Introduction Ordered sets, or lattices, give a uniform structure to, for example, the set of subfields of a field. Various special types of lattices have particularly nice structure and have applications in group theory and algebraic topology, for example. History Applications and related fields A large portion of this field involves simple combinatorial structures on arbitrary sets; see 05: Combinatorics and 03E: Set Theory Linear orderings especially on infinite sets is the study of Ordinals in Set Theory; these are traditionally considered in 03: Mathematical Logic , especially 03G: Algebraic Logic. See 03G05: Boolean algebras and 03G10: Lattices and related structures. Ordered sets may be viewed as topological spaces; see 54: General Topology , especially 54F05: Ordered topological spaces, for more detail. There is significant overlap with 08: General algebraic structures , and orderings (e.g. subgroup lattices) are a natural part of many particular algebraic structures; see

8. Wallpaper Groups: Lattices For any point, the collection of translates of it by translation symmetries of a pattern forms a lattice. In the example at the left, the translates of the http://www.clarku.edu/~djoyce/wallpaper/lattices.html

Lattices Just by considering the translation symmetries of a pattern we can begin to classify patterns. For any point, the collection of translates of it by translation symmetries of a pattern forms a lattice . In the example at the left, the translates of the one point colored red are indicated by dark blue dots. One translation, call it T , is indicated by the green arrows; it translates right and a bit upward. A different translation, call it U , is indicated by the rose colored arrows; it translates up and a bit right. By taking a composition of these two translations and their inverses, you can construct every other translation symmetry of the pattern. Thus, every translation is of the form T m U n where m and n are integers. Furthermore, you can see what composition you need by seeing where the red dot needs to go. So the lattice of dots corresponds to the translation symmetries. Five kinds of lattices We can classify lattices into five different kinds. If a lattice has a square fundamental region, it's called a square lattice hexagonal lattice . That's because in that case, the points in the lattice nearest any one point in the lattice are the vertices of a regular hexagon. (A rhombus is a parallelogram with equal sides.) If a lattice has a rhombus as a fundamental region, it's a

9. ::Fourth International Conference On Concept Lattices And Their Applications:: Concept Lattices and Their Applications. Concept Lattices and Their Applications. Hammamet, Tunisia, October 30November 1st, 2006. Jointly organized by http://www.cck.rnu.tn/cla06/

Home Important dates Call for papers KeyNote Speakers ... Previous CLAs Fourth International Conference On Concept Lattices and Their Applications Hammamet, Tunisia, October 30-November 1st, 2006 Jointly organized by and Sponsored by : :CLA 2006 Last Update : 09/23/2006

10. The Mizar Abstract Of LATTICES (p,q); end; definition let G be non empty /\SemiLattStr, p, q be Element of G; func p /\ q - Element of G equals LATTICESdef 2 (the L_meet of G). http://mizar.uwb.edu.pl/JFM/Vol1/lattices.abs.html

Journal of Formalized Mathematics Volume 1, 1989 University of Bialystok Association of Mizar Users The abstract of the Mizar article: Introduction to Lattice Theory by Stanislaw Zukowski Received April 14, 1989 MML identifier: LATTICES Mizar article MML identifier index environ vocabulary BINOP_1, BOOLE, FINSUB_1, FUNCT_1, SUBSET_1, LATTICES; notation ; constructors ; clusters ; requirements SUBSET BOOLE ; begin definition struct 1-sorted set BinOp of the carrier #) end; definition struct 1-sorted set BinOp of the carrier #) end; definition struct set BinOp of the carrier #) end; definition cluster strict non empty ; cluster strict non empty ; cluster strict non empty LattStr ; end; definition let G be non empty , p, q be Element of G; func Element of G equals :: LATTICES:def 1 (the of G) (p,q); end; definition let G be non empty , p, q be Element of G; func Element of G equals :: LATTICES:def 2 (the of G) (p,q); end; definition let G be non empty , p, q be Element of G; pred

11. QCD And Dense Matter: From Lattices To Stars (INT-04-1) We envision a program that brings together researchers in highdensity QCD, effective theories, lattice QCD, and compact-star physics. http://www.int.washington.edu/PROGRAMS/04-1.html

Organizers: Thomas Schaefer tmschaef@unity.ncsu.edu Gordon Baym gbaym@uiuc.edu Zoltan Fodor fodor@pms2.elte.hu Dam Son son@phys.washington.edu Program Coordinator: Laura Lee lee@phys.washington.edu Talks Online Application form INT Home Page QCD and Dense Matter: From Lattices to Stars March 29 - June 18, 2004 We envision a program that brings together researchers in high-density QCD, effective theories, lattice QCD, and compact-star physics. Until recently researchers in these three areas had little contact with each other. However, in the past few years there have been great advances in understanding dense QCD theoretically, and in simulating the theory using lattice QCD. Simultaneously, there is a wealth of new observations of compact astrophysical objects which could provide evidence for the nature of dense matter. High-density QCD is a new frontier in the investigation of the QCD phase diagram. The theoretical study of the "condensed matter physics" of QCD has led to several surprising discoveries, such as robust color superconducting phases and a new color-flavor-locked phase of quark matter. There are many important problems that remain to be studied. We would like to understand how to combine perturbative QCD and effective field theory techniques in order to perform systematic studies of the phase diagram and the structure of matter at high baryon density. We would like to systematically build in correlations that correspond to the formation of nucleons at moderate density. At even lower density we would like to see whether one can gain additional insights into the traditional nuclear matter problem using methods such as the renormalization group.

12. CaLC 2001 - Cryptography And Lattices Conference The focus of CaLC is on all aspects of Lattices as used in cryptography and complexity theory. We hope that the conference will showcase the current state http://www.math.brown.edu/~jhs/CALC/CALC.html

Cryptography and Lattices Conference 2001 Invited Speakers CaLC Program Directions to Brown Program Committee ... Conference Sponsors Brown University Providence, Rhode Island, USA March 29 and 30, 2001 Click Here For Program Schedule The focus of CaLC is on all aspects of lattices as used in cryptography and complexity theory. We hope that the conference will showcase the current state of lattice theory and will encourage new research in both the theoretical and the practical uses of lattices and lattice reduction in the cryptographic arena. We encourage submission of papers from academia, industry, and other organizations. All submitted papers will be reviewed. Topics of interest include the following, but any paper broadly connected with the use of lattices in cryptography or complexity theory will be given serious consideration: Lattice reduction methods, including theory and practical implementation. Applications of lattice reduction methods in cryptography, cryptanalysis, and related areas of algebra and number theory. Cryptographic constructions such as public key cryptosystems and digital signatures based on lattice problems.

13. Bravais Lattices The last is often described as a centered lattice, a rectangle with an extra point in the middle, to bring out the rectangular nature of the pattern. http://www.uwgb.edu/DutchS/symmetry/bravais.htm

Bravais Lattices Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay First-time Visitors: Please visit . Use "Back" to return here. In two dimensions, periodic unit cells can have one of five basic shapes: general parallelogram, general rectangle, square, 60-120 degree rhombus, and generic rhombus. The last is often described as a "centered" lattice, a rectangle with an extra point in the middle, to bring out the rectangular nature of the pattern. It's not too hard to see the rectangular pattern in a rhombic lattice, but it can be very hard to see the patterns in three dimensional lattices. For this reason, three-dimensional lattices must often be described as unit cells with additional points. There are 14 basic unit cells in three dimensions, called the Bravais Lattices Symbols P - Primitive: simple unit cell F - Face-centered: additional point in the center of each face I - Body-centered: additional point in the center of the cell C - Centered: additional point in the center of each end R - Rhombohedral: Hexagonal class only Isometric Cells The F cell is very important because it is the pattern for cubic closest packing. There is no C cell because such a cell would not have cubic symmetry.

14. DIMACS Workshop On Applications Of Lattices And Ordered Sets To Computer Science DIMACS Workshop, Rutgers University, NJ, USA; 810 July 2003. http://dimacs.rutgers.edu/Workshops/Lattices/

DIMACS Workshop on Applications of Lattices and Ordered Sets to Computer Science July 8 - 10, 2003 DIMACS Center, CoRE Building, Rutgers University Organizers: Jonathan Farley , Massachusetts Institute of Technology Mel Janowitz , DIMACS / Rutgers University, melj@dimacs.rutgers.edu Jimmie Lawson , Louisiana State Univeristy, lawson@math.lsu.edu Michael Mislove , Tulane University, mwm@math.tulane.edu Presented under the auspices of the Special Support: National Science Foundation (NSF) Office of Naval Reasearch (ONR) Workshop Announcement List of Participants Call for Participation Program ... Registration Form (Pre-registration deadline: June 23, 2003) DIMACS Workshop Registration Fees Pre-register before deadline After pre-registration deadline Regular rate $120/day $140/day Academic/nonprofit rate* $60/day $70/day Postdocs $10/day $15/day DIMACS Postdocs $5/day $10/day DIMACS partner institution employees** DIMACS long-term visitors*** Registration fee to be collected on site, cash, check, VISA/Mastercard accepted. Our funding agencies require that we charge a registration fee during the course of the workshop.

15. 8.02T - Electrostatics Visualizations - Lattices 2D. DESCRIPTION Lattices 2D simulates the interaction of charged particles in a two dimensional plane. The particles interact via the classical Coulomb force, http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/31-lattice2

PREVIOUS NEXT SUBJECT: Lattices 2D DESCRIPTION: Lattices 2D simulates the interaction of charged particles in a two dimensional plane. The particles interact via the classical Coulomb force, as well as the repulsive quantum-mechanical Pauli force, which acts at close distances (accounting for the "collisions" between them). Additionally, the motion of the particles is damped by a term proportional to their velocity, allowing them to "settle down" into stable (or meta-stable) states. In this simulation, the proportionality of the Coulomb and Pauli forces has been adjusted to allow for lattice formation, as one might see in a crystal. The "preferred" stable state is a rectangular lattice, although other formations are possible depending on the number of particles and their initial positions. New feature : Selecting a particle and pressing "f" will toggle fieldlines illustrating the local field around that particle. Performance varies depending on the number of particles / fieldlines in the simulation. VISUALIZATION: Fullscreen Version INSTRUCTIONS

16. Lattices In Computer Science (Fall 2004) Lattice Basis Reduction and Integer Programming, by Karen Aardal, nicely explains fixeddimension integer programming using LLL basis reduction http://www.cs.tau.ac.il/~odedr/teaching/lattices_fall_2004/index.html

Computer Science Tel-Aviv University Lattices in Computer Science Fall 2004 Announcements [Dec 30] Extra class on 2004/12/31, Shenkar 114 [Nov 15] Extra class in Shenkar 114 [Nov 12] Extra class on 2004/11/17 [Oct 25] We'll be in Shenkar 104 from now on [Sep 11] The class on 2004/10/18 is cancelled [Sep 11] Try out our discussion board Homeworks Homework 1 Homework 2 Homework 3 Homework 4 Lecture Notes (please leave comments here Introduction (successive minima, Minkowski's theorems, some computational problems) The LLL algorithm Babai's nearest plane algorithm for CVP Attacks on low public exponent RSA using small solutions to small degree polynomials Integer programming in constant dimension ... Some basic complexity results (CVP decision vs search, NP-hardness of exact CVP, SVP not harder than CVP) A simply exponential algorithm for SVP (Ajtai-Kumar-Sivakumar) Dual lattices Introduction to Fourier analysis Banaszczyk's transference theorems ... Worst-case to average-case reduction CVP with preprocessing Administrative Basics Lectures Monday 12:00-15:00 Instructor Oded Regev Textbooks Complexity of Lattice Problems , D. Micciancio and S. Goldwasser

17. Crystal Lattices The following pages contain descriptions and illustrations of some cubic and hexagonal crystal Lattices and their unit cells. http://wb.chem.lsu.edu/htdocs/people/sfwatkins/MERLOT/flattice/00lattice.html

18. Introduction To Lattices And Order - Cambridge University Press This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521784514

19. Spiral Lattices | Phyllotaxis This applet enables the exploration of all possible spiral Lattices and their organization according to their divergence angles, expansion rate, http://maven.smith.edu/~phyllo/Applets/Spiral/Spiral.html

Spiral Lattices Applet Choose the size you want for this applet: Smaller Window (620 by 330) Bigger Window (1000 by 500) Some tips about this applet This applet enables the exploration of all possible spiral lattices and their organization according to their divergence angles, expansion rate, and parastichy numbers. To see the relation between these three entities and visualize the spiral lattices, click the mouse in the Parameter Space. Move the mouse keeping the button pressed. To each point P of the parameter space corresponds a spiral lattice. The divergence angle of the lattice is given by the angle P makes with the x - axis, and expansion rate by the inverse of the distance from P to the center of the parameter space. The option Parastichies shows the spirals that you may have perceived mentally by connecting the dots to their nearest neighbours. The option Closest Points shows the 2 closest points to the blue point on the central circle (which represents potentially the newest primordia on the edge of the meristem).

20. Harmonic Lattice Diagrams, (c) 1998 By Joseph L. Monzo The best way that I have discovered, to grasp as much harmonic information as possible in a justintonation musical tuning system, is the use of lattice http://www.tonalsoft.com/monzo/lattices/lattices.htm

Deutsche Joe Monzo's Harmonic Lattice Diagrams The best way that I have discovered, to grasp as much harmonic information as possible in a just-intonation musical tuning system, is the use of lattice diagrams which portray pitches as points in multi-dimensional space connected by vectors. In a just-intonation tuning system, each note is represented by a ratio which describes that note's relationship to another note, usually one that is used as a reference for the whole system. This reference tone has the ratio 1/1, also described as 1:1 or 1 to 1. Any number can be factored into the series of prime numbers , each of which is a base which has an exponent that is either positive or negative, representing numbers >1 or <1, respectively, unless the exponent=0, which represents 1, the identity in multiplication. Powers of 2 are all harmonically equivalent to this identity 1, thus, powers of 2 represent " octaves ", and thus have no pronounced effect on the harmony, and may be eliminated, unless "octave" registration is specifically under consideration. Therefore the diagrams normally begin with prime-base 3. My lattice diagrams treat each prime base as a unique dimension in space, with all the exponents radiating outward from the central 1/1, which is equivalent to all numbers to the 0th power, or

21. Monadic Distributive Lattices -- Figallo Et Al. 15 (56): 535 -- Logic Journal Of The purpose of this paper is to investigate the variety of algebras, which we call monadic distributive Lattices, as a natural generalization of monadic http://jigpal.oxfordjournals.org/cgi/content/abstract/15/5-6/535

@import "/resource/css/hw.css"; @import "/resource/css/igpl.css"; Skip Navigation Oxford Journals Contact Us My Basket ... Volume 15, Number 5-6 Pp. 535-551 Logic Journal of IGPL Advance Access originally published online on September 26, 2007 Logic Journal of IGPL 2007 15(5-6):535-551; doi:10.1093/jigpal/jzm039 This Article Full Text (PDF) All Versions of this Article: most recent References Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive ... Request Permissions Google Scholar Articles by Figallo, A. V. Articles by Ziliani, A. Monadic Distributive Lattices Aldo V. Figallo and Alicia Ziliani Abstract The purpose of this paper is to investigate the variety of algebras, which we call monadic distributive lattices, as a natural generalization of monadic Heyting algebras [16]. It is worth mentioning that the latter is a proper subvariety of the first one, as it is

22. Geometry Of Lattices And Algorithms papers, tables and software for computational geometry of numbers. http://fma2.math.uni-magdeburg.de/~latgeo/

Geometry of Lattices and Algorithms News February - April 2008 we take part in the HIM Junior Trimester Program on Computational Mathematics , focusing on Extreme Geometric Structures . In two workshops on '' Linear and semidefinite programming bounds '' (February 26 - February 29) and '' Experimentation with, construction of, and and enumeration of optimal geometric structures '' (March 25 - March 28) we will address the basic tasks of finding constructions of extreme geometric structures and finding obstructions for their existence. Older News With Mathieu Dutour Sikiric the classification of perfect forms (lattices) in dimension 8 was completed on 4th October 2005. There are no more than the 10916 forms found before (see Jacques Martinet's homepage ). The file perfect-forms-dim8.txt contains a complete list in human readable format, together with their contiguities. We will report on this in our paper Classification of eight dimensional perfect forms . In dimension 9 we found more than 500000 perfect forms (see the compressed file perfect-forms-dim9.txt.gz

23. Quantum Gases In Optical Lattices - Physicsworld.com The newfound ability to confine ultracold quantum gases in optical Lattices is already having a major impact in fields as diverse as condensed-matter http://physicsworld.com/cws/article/print/19273

@import url(/cws/css/screen.css); @import url(/cws/css/themes/phw.css); @import url(/cws/css/datePicker.css); Skip to the content A community website from IOP Publishing Sign in Forgotten your password? Sign up physicsworld.com ... Advertising Whole site Print edition News In depth Jobs Events Companies Products Search Related Stories Quantum criticality in metals Making atoms cooler A Fermi gas of atoms Waves and turbulence cause a stir in superfluids ... Fermionic first for condensates Related Links DIGITAL EDITION DECEMBER ISSUE NOW AVAILABLE Members of the Institute of Physics can access a full digital version of Physics World magazine. Simply login here and follow the Physics World link. Corporate Partners For maximum exposure, become a Corporate Partner. Contact our sales team Corporate Partners Features Apr 10, 2004 Quantum gases in optical lattices Arrays of ultracold atoms trapped by artificial crystals of light can be used in a wide variety of experiments in quantum physics Mountains of potential Imagine having an artificial substance in which you can control almost all aspects of the underlying periodic structure and the interactions between the atoms that make up this dream material. Such a substance would allow us to explore a whole range of fundamental phenomena that are extremely difficult - or impossible - to study in real materials. It may sound too good to be true, but over the last two years physicists have come extremely close to achieving this goal. This breakthrough has been made possible by the convergence of two related but previously distinct realms of research in atomic physics: quantum gases and optical lattices. The new-found ability to confine ultracold quantum gases in optical lattices is already having a major impact in fields as diverse as condensed-matter physics and quantum information processing.

24. Private Page Martinet Perfect Lattices in Euclidean Spaces, SpringerVerlag, Grundleheren der Preface, Introduction and Contents of the book Perfect Lattices in Euclidean http://www.math.u-bordeaux1.fr/~martinet/

Jacques Martinet phone: +33-5-40-00-60-96 (From France : 05-56-84-60- 96) fax : +33-5-56-84-69-50 (From France : 05-40-00-69-50) martinet a` math point u-bordeaux1.fr 33405, Talence cedex France J. Th. Nombres Bordeaux U.F.R. Math-info GDR Th. Nombres bibli ... Orgue Talence Links to WEB pages of some mathematicians working on lattices and coding theory. Christine Bachoc Eva Bayer Richard Borcherds Henry Cohn ... Nebe-Sloane's catalogue Other links Karim Belabas Henri Cohen Denis Simon Colloque "Georges Gras" ... SMF Recent publications BOOKS dvi postscript pdf Perfect Lattices in Euclidean Spaces, Springer-Verlag, Grundleheren der Mathematischen Wissenschaften 327, Heidelberg, 2003. Preface, Introduction and Contents of the book "Perfect Lattices in Euclidean Spaces". postscript pdf Corrected and extended reference list of the book "Perfect Lattices in Euclidean Spaces". dvi postscript pdf Comparison between the French and English editions. dvi postscript pdf Erratum to the book "Perfect Lattices in Euclidean Spaces". dvi postscript pdf Complements to the book "Perfect Lattices in Euclidean Spaces".

25. Optical Lattices Because the periodic arrangement of trapping sites resembles a crystalline lattice, we call this system an optical lattice. http://physics.nist.gov/Divisions/Div842/Gp4/lattices.html

Laser Cooling and Trapping Group Bose Einstein Condensation Cold Collisions Optical Lattices Optical Tweezers (postdoctoral positions) Optical Lattices In the figure below we show the optical potential the atoms experience as they move in a standing wave light field formed at the intersection of four laser beams. Laser cooling causes the atoms to lose enough energy that they become trapped in the egg-carton like potential wells. Because the periodic arrangement of trapping sites resembles a crystalline lattice, we call this system an optical lattice To establish the existence of this optical lattice we have performed a Bragg scattering experiment in which we observe the reflection of a probe laser beam as it coherently diffracts from our sample. Because the spacing in our lattice is on the micron scale (the wavelength of the trapping light) instead of the Angstrom scale, large-angle Bragg scattering occurs for visible light instead of x-rays. et al . Phys. Rev. Lett. Current areas of research involve using Bragg scattering to explore the dynamics of atoms trapped in optical lattices, including approach to equilibrium, parametric driving, and the creation of breathing-mode wave packets. Contact: Trey Porto National Institute of Standards and Technology Gaithersburg, MD 20899

26. Point Lattices We call this set L the point lattice (or just lattice) described by the basis B. Point Lattices are pervasive structures in mathematics, and have been http://www.farcaster.com/papers/sm-thesis/node5.html

Next: Reduced Lattice Bases Up: Introduction Previous: Introduction Point Lattices Let B be a set of vectors in . If these vectors are independent, then they form a basis of and any point in n -space may be written as a linear combination of vectors in B Consider the set of points which may be written as the sum of integer multiples of the basis vectors: We call this set L the point lattice (or just lattice ) described by the basis B . Point lattices are pervasive structures in mathematics, and have been studied extensively. See [ ], for example, for a survey of the field. In the area of combinatorial mathematics alone it is possible to phrase many different problems as questions about lattices. Integer programming [ ], factoring polynomials with rational coefficients [ ], integer relation finding [ ], integer factoring [ ], and diophantine approximation [ ] are just a few of the areas where lattice problems arise. In some cases, such as integer programming existence problems, it is necessary to determine whether a convex body in contains a lattice point (for some specific lattice). In other cases the items of interest are short vectors in the lattice. As we shall see below, for certain cryptographic applications, we would like to be able to quickly determine the Euclidean-norm shortest nonzero vector in a lattice. It is important to note that the difficulty of finding the Euclidean-norm shortest nonzero vector in a lattice is an open question. If

27. Lattices, Universal Algebra And Applications This conference was organized by the Project Lattices, Universal Algebra and Algebraic Logic of the Centro de Álgebra da Universidade de Lisboa (CAUL). http://ptmat.fc.ul.pt/~uaconf03/

Lattices, Universal Algebra and Applications Lisbon May 28-30, 2003 CAUL The conference included six invited lectures and about thirty contributed talks. Over 60 participants came from all over Europe, plus Japan, Canada and the United States. Larger picture Main Speakers Program and Abstracts Conference fee ... Organizing Committee Conference location The conference took place at the Complexo Interdisciplinar da Universidade de Lisboa , Avenida Professor Gama Pinto, 2, 1609-003 LISBOA. Maps and directions can be obtained from CAUL's website Invited speakers W. Blok (University of Illinois at Chicago) M. Gehrke (New Mexico State University) M. Haviar (Matej Bel University) K. Kearnes (University of Colorado) B. Monjardet (University of Paris I) D. Mundici (University of Florence) Organizing Committee Gabriela Bordalo (CAUL, University of Lisbon) Isabel Ferreirim (CAUL, University of Lisbon) Program and Abstracts The Conference Program is available, in

28. The Math Forum - Math Library - Order/Lattices The Math Forum s Internet Math Library is a comprehensive catalog of Web sites and Web pages relating to the study of mathematics. This page contains sites http://mathforum.org/library/topics/lattices/

Browse and Search the Library Home Math Topics Discrete Math : Order/Lattices Library Home Search Full Table of Contents Suggest a Link ... Library Help Selected Sites (see also All Sites in this category Lattice Theory and Geometry of Numbers (The Geometry Junkyard) - David Eppstein, Theory Group, ICS, UC Irvine An extensive annotated list of links to material on lattices. A lattice is an infinite arrangement of points spaced with sufficient regularity that one can shift any point onto any other point by some symmetry of the arrangement. More formally, a lattice can be defined as a discrete subgroup of a finite-dimensional vector space (the subgroup is often required not to lie within any subspace of the vector space, which can be expressed formally by saying that the quotient of the space by the lattice is compact). more>> Order, Lattices, Ordered Algebraic Structures - Dave Rusin; The Mathematical Atlas A short article designed to provide an introduction to ordered sets. Ordered sets, or lattices, give a uniform structure to, for example, the set of subfields of a field. Various special types of lattices have particularly nice structure and have applications in group theory and algebraic topology, for example. History; applications and related fields and subfields; textbooks, reference works, and tutorials; software and tables; other web sites with this focus. more>> All Sites - 22 items found, showing 1 to 22

29. Phong NGUYEN -- Publications Rankin s Constant and Blockwise Lattice Reduction (CRYPTO 06) We survey the main examples of the two faces of Lattices in cryptology. http://www.di.ens.fr/~pnguyen/pub.html

Surveys and Tutorials Journal Papers Conference Papers Workshop Papers Technical Reports Click here for my DBLP list of publications , which is almost complete. Papers for which submission is in progress are not available. For my Ph.D. , see there . If you can't read postscript, use this Surveys La cryptologie : Enjeux et perspectives New Trends in Cryptology (STORK Project, 2003) The Two Faces of Lattices in Cryptology (Calc '01) Lattice Reduction in Cryptology: An Update (ANTS-IV, 2000) Journal Papers Testing Set Proportionality and the Adam Isomorphism of Circulant Graphs (2005) *Journal of Discrete Algorithms*, 2005 Hidden number problem with hidden multipliers, timed-release crypto and noisy exponentiation (2003) *Mathematics of computation*, Vol 72, Number 243, Pages 1473-1485 The Insecurity of the Elliptic Curve Digital Signature Algorithm with Partially Known Nonces (2003) *Design, Codes and Cryptography*, Vol 30, Number 2, Pages 201-217 The Insecurity of the Digital Signature Algorithm with Partially Known Nonces (2002) *J. of Cryptology*, Vol. 15, Number 3, Pages 151-176 International Conference Papers Full Key-Recovery Attacks on HMAC/NMAC-MD4 and NMAC-MD5 (CRYPTO '07) New Chosen-Ciphertext Attacks on NTRU (PKC '07) Rankin's Constant and Blockwise Lattice Reduction (CRYPTO '06) Learning a Parallelepiped: Cryptanalysis of GGH and NTRU Signatures (EUROCRYPT '06, Best Paper Award)

30. 3D Crystal Lattices Bodycentered Face-centered Crystal Lattice Hexagonal crystal lattice Hexagonal close-packing. In hexagonal close-packing, the atoms in the third http://www.tg.rim.or.jp/~kanai/chemistry/cry3d-e.htm

3D Crystal Lattices Unit cell Hexagonal close-packing The crystal structure of most metalic elements is very simple and beautiful. The arrangement of the atoms is that which one obtains when packing spheres of equal size in the most efficient manner. This arrangement is called close-packing. There are two of these, one called hexagonal close-packing and the other cubic close-packing. In close-packing, each sphere(atom) has twelve nearest-neighbors, all equidistant. Hexagonal close-packing In hexagonal close-packing, the atoms in the third layer lie directly over those of the first. Cubic close-packing Cubic close-packing Ionic Crystal The crystal structure of cesium chloride is the same as that of a body-centered type, if one overlooks the difference between cation and anion. The number of anions surrounding each cation in this structure is eight. In the structure of sodium chloride, each positive ion is surrounded by six negative ions arranged in an oxtahedron, and vice versa. Note that the arrangement of the negative ion is the same of a face-centered type. Na (blue), Cs

31. Fields, Lattices And Condensed Matter Fields, Lattices and Condensed Matter. A symposium in honour of John Cardy s 60th birthday. The Rudolf Peierls Centre for Theoretical Physics http://www-thphys.physics.ox.ac.uk/user/FabianEssler/Fields.html

Fields, Lattices and Condensed Matter A symposium in honour of John Cardy's 60th birthday The Rudolf Peierls Centre for Theoretical Physics Oxford University December 14-15, 2007 This symposium will celebrate the outstanding scientific career of John Cardy on the occasion of his 60th birthday. To this end, the symposium will gather a number of leading researchers from the international community in Theoretical Physics to discuss recent developments in the fields to which John has contributed so greatly. Invited speakers: I. Affleck (UBC) R. Dijkgraaf (Amsterdam) P. Dorey (Durham) B. Duplantier (Saclay) K. Gawedzki (Lyon) F.D.M. Haldane (Princeton) A. Lamacraft (Virginia) G. Mussardo (Trieste) B. Nienhuis (Amsterdam) N. Read (Yale) H. Saleur (Saclay) A. Tsvelik (Brookhaven) W. Werner (Orsay) J.B. Zuber (Paris) Organising committee: John Chalker (Oxford) Fabian Essler (Oxford) Paul Fendley (All Souls) Martin Howard (Norwich) Jesper Jacobsen (Orsay) Mike Teper (All Souls) John Wheater (Oxford) Participation: There is a registration fee of 20 pounds, to be paid upon arrival. Due to the limited space in the lecture theatre, participants are encouraged to register at their earliest convenience, and no later than November 15, 2007. To register, please send an email to "jesper dot jacobsen at u-psud dot fr", with the text "Cardy symposium" in the subject line.

32. SPHERICON HOMEPAGE: Lattices 24 Sphericons will lattice into a truncated octahedra. You can put an infinite amount of these truncated octahedra together to fill space. http://homepage.ntlworld.com/paul.roberts99/Lattices.htm

Lattices Contents Two Sphericons Strings Four Sphericons Nine Sphericons ... Main Menu Two Sphericons If properly aligned, two Sphericons will roll around each other ad infinitum. This forms the basis of most other Sphericon lattices. Back to top Strings In a configuration whereby Sphericons are joined and poised to roll, they could be in a straight line, in closed rings of 4, 6, etc., spirals, and more. These strings could fold up into sheets, archimedean solids, etc. by only turning one a small amount. Back to top 4 Sphericons Four Sphericons will turn together (Click here or here to see them rolling) ad-infinitem with two surfaces of each rolling around others so that eight surfaces touch at all times. Let's say one revolution of the block is where a point of one returns to the same spot. For half of this revolution, in two separated periods each of a quarter revolution, the 4 are in perfect symmetry about the centre (four fold). Then, (without actually separating) they split into pairs (that is to say they have only 2 fold symmetry). Whether they split horizontally or vertically seems to be a random decision. In the animation, we have alternated it. This randomness makes it difficult to animate in some computer packages. Every half turn, two tangentially rolling pairs go out of synchronization with the other two pairs by 1/18

33. Fields Institute - Lattices And Trajectories Lattices and Trajectories A Symposium of Mathematical Chemistry in honour of Ray Kapral and Stu Whittington Fields Institute, Toronto http://www.fields.utoronto.ca/programs/scientific/06-07/lattices/

Home About Us NPCDS/PNSDC Mathematics Education ... Search SCIENTIFIC PROGRAMS AND ACTIVITIES December 24, 2007 May 31 - June 2, 2007 Lattices and Trajectories: A Symposium of Mathematical Chemistry in honour of Ray Kapral and Stu Whittington Fields Institute, Toronto Organizing Committee: Jeremy Schofield (Chemistry, UToronto) John Valleau (Chemistry, UToronto) Robbie Grunwald (Chemistry, UToronto) Gary Iliev (Chemistry, UToronto) Maria Sabaye Moghaddam (Biochemistry, UToronto) Chemistry The Connaught Fund Audio and slides of talks Registered Participants Symposium Schedule Visitor Resources ... Housing and Hotels This symposium will celebrate the outstanding scientific work and stellar academic careers of Ray Kapral and Stu Whittington at the University of Toronto on the occasion of their 65th birthdays. To this end, the symposium will draw together a number of leading researchers from an international community in Chemical Physics to discuss recent developments in the fields to which the honorees have contributed so meaningfully. Sessions Knots and Polymers Random Spatial Processes Mixed Quantum-Classical Dynamics Nonlinear Systems and Pattern Formation Current List of Invited Speakers

34. Home Page Of Gerald Teschl Jacobi operators and completely integrable nonlinear Lattices. The Toda lattice; Lax pairs, the Toda hierarchy, and hyperelliptic curves http://www.mat.univie.ac.at/~gerald/ftp/book-jac/index.html

@import url("../../gerald.css"); Books Gerald Teschl Faculty of Mathematics University of Vienna Math. Dep. Uni. Vienna ESI Home ... Links Book Mathematical Surveys and Monographs , Volume 72. Jacobi Operators and Completely Integrable Nonlinear Lattices Gerald Teschl Abstract This book is intended to serve both as an introduction and a reference to spectral and inverse spectral theory of Jacobi operators (i.e., second order symmetric difference operators) and applications of these theories to the Toda and Kac-van Moerbeke hierarchy. Starting from second order difference equations we move on to self-adjoint operators and develop discrete Weyl-Titchmarsh-Kodaira theory, covering all classical aspects like Weyl m -functions, spectral functions, the moment problem, inverse spectral theory, and uniqueness results. Next, we investigate some more advanced topics like locating the essential, absolutely continuous, and discrete spectrum, oscillation theory, trace formulas, random operators, almost periodic operators, (quasi-)periodic operators, scattering theory, and spectral deformations. Then, the Lax approach is used to introduce the Toda hierarchy and its modified counterpart, the Kac-van Moerbeke hierarchy. Uniqueness and existence theorems for the initial value problem

35. Residuated Lattices An Algebraic Glimpse At Substructural Logics This is also where we begin investigating Lattices of logics and varieties, rather than particular examples. We continue in this vein by presenting a number http://www.elsevier.com/wps/product/cws_home/711437

Home Site map Elsevier websites Alerts ... Residuated Lattices: An Algebraic Glimpse at Substructural Logics, 151 Book information Product description Audience Author information and services Ordering information Bibliographic and ordering information Conditions of sale Book-related information Submit your book proposal Other books in same subject area About Elsevier Select your view RESIDUATED LATTICES: AN ALGEBRAIC GLIMPSE AT SUBSTRUCTURAL LOGICS, 151 To order this title, and for more information, click here By Nikolaos Galatos , School of Information Science, Japan Advanced Institute of Science and Technology Peter Jipsen , Chapman University, Orange, USA Tomasz Kowalski , Australian National University, Canberra, Australia Hiroakira Ono , Japan Advanced Institute of Science and Technology, Ishikawa, Japan Included in series Studies in Logic and the Foundations of Mathematics, Description Audience This book is intended for: Research mathematicians and graduate students and: Computer scientists Contents Contents List of Figures List of Tables Introduction Chapter 1. Getting started Chapter 2. Substructural logics and residuated lattices Chapter 3. Residuation and structure theory Chapter 4. Decidability Chapter 5. Logical and algebraic properties Chapter 6. completions and finite embeddability Chapter 7. Algebraic aspects of cut elimination Chapter 8. Glivenko theorems Chapter 9. Lattices of logics and varieties Chapter 10. Splittings Chapter 11. Semisimplicity Bibliography Index Hardbound, 532 pages, publication date: APR-2007

36. Molecular Model Kits: Organic Molecule Structures, Lattices Molecular model kits for chemistry, biochemistry, crystal Lattices; build ice, diamond, graphite, solid state, rutile, wurtzite, lithium niobate, http://www.indigo.com/models/orbit-molecular-model-kits.html

Comments? Email us ezAd.Shown = false; Orbit Molecular Model Kits Home rare earth magnets, molecular models, credit card magnifiers, school science supplies Search Molecular ... Order Orbit Molecular Model Kits Orbit molecular model kits provide an economical way to build a wide variety of specific models such as diamond, sodium choride, ice , etc. If you don't see what you want, we can quote you a custom kit. Note: If you are taking organic chemistry, you may really be looking for one of our student molecular model sets All Orbit Kits are supplied with our "wobbly" bonds which are made from special plastic tubing and have the following advantages: lattice structures such as diamond are stiff layered compounds such as graphite are flexible ("wobbly") very durable ; can be reused in lab/class settings make virtually indestructible permanent models clear bonds make overall molecular geometry more visible can accomodate non-VSEPR geometries Note: Models come unassembled with instructions unless noted otherwise. Models can be assembled for triple the price ; be sure to indicate this in the "Comments" section of the order form.

37. Identical Dual Lattices Ans Subdivision Of Space This phenomenon of the two identical subspaces, the surface in between, and the two selfdual Lattices representing them makes the exhaustive search of any http://www.mi.sanu.ac.yu/vismath/koren/index.html

Identical Dual Lattices and Subdivision of Space By Ami Korren Name Ami Korren, Architect, (b. Tiberias, Israel, 1950). Address: E-mail: Korren-a@inter.net.il Fields of interest: Architecture, Morphology, geometry, architectural crystallography, and Minimal and consequential forms. Publication and exhibition: A.Korren Periodic 2-manifold surfaces that divide the space into two identical subspaces , Master thesis, Technion, Israel, 1993. ,Periodic Hyperbolic Surfaces and Subdivision of 3-Space, Katachi U Symmetry, Springer-Verlag, Tokyo, , Self-Dual Space Lattices, and Periodic Hyperbolic Surfaces, Symmetry Natural and Artificial, Symmetrion Budapest Science in the Arts –Art in the Sciences , Ernst Museum, Budapest, 1999. Introduction The morphological dealing with space subdivision by continuous 2-manifold surfaces is one of the important issues for understanding organized space and its order. The phenomenon of periodic surfaces that divide the space into two identical subspaces was dealt with over the years by various researchers from different scientific fields, such as mathematicians, architects, crystallographers, physicians, etc. These periodic 2-manifold surfaces are continuous and divide the space into two identical subspaces, which are graphically characterized by two dual tunneled space networks. These surfaces have the shape of a sponge structure.

38. CLA 2007 Fifth International Conference on Concept Lattices and Their Applications. Montpellier, France, October 2426, 2007. Jointly organized by LIRMM University http://www.lirmm.fr/cla07/

CLA 2007 Home Technical Content Call for papers Invited Speakers ... Sponsors CLA 2007 Fifth International Conference on Concept Lattices and Their Applications Montpellier, France, October 24-26, 2007 Jointly organized by LIRMM University of Montpellier II News Selected papers accepted to CLA 2007 will be invited to further revise and extend their work in a special issue of International Journal of General Systems on Concept Lattices and Applications . Selected papers will be competitively chosen by the Program Chairs and the CLA steering committee based on the quality of their content and presentation. Online Paper Submission System is ON. Abstract submission deadline extended to 20 June Paper submission deadline extended to 29 June Online Registration System is ON. Online Final Paper Submission System is ON. The upload option of the submission interface is open. So you can upload your final version safely. Program schedule online. Early registration until September 30. Proceedings Online.

39. The Bravais Lattices Song The Bravais Lattices Song. by Walter Fox Smith. Tune I Am the Very Model of a Modern Major General , from The Pirates of Penzance , by William Gilbert http://www.haverford.edu/physics/songs/bravais.htm

The Bravais Lattices Song by Walter Fox Smith Tune: "I Am the Very Model of a Modern Major General", from "The Pirates of Penzance" Lyrics: Web Page Word Format PDF recording Real Audio Piano: Bruce Morrison .... Chorus: Marian McKenzie, Michael K. McCutchan, Faith H. McKenzie RealAudio format is faster for modem downloads, mp3 format is slightly higher audio quality. Background courtesy of Free Backgrounds.com

40. Bravais Lattices 3D visualization of the 14 Bravais Lattices, amino acids and more. http://phycomp.technion.ac.il/~sshaharr/intro.html

The fourteen Bravais lattices There are fourteen distinct space groups that a Bravais lattice can have. Thus, from the point of view of symmetry, there are fourteen different kinds of Bravais lattices. Auguste Bravais (1811-1863) was the first to count the categories correctly. The seven crystal systems I list below the seven crystal systems and the Bravais lattices belonging to each. Cubic (3 lattices) The cubic system contains those Bravias lattices whose point group is just the symmetry group of a cube. Three Bravais lattices with nonequivalent space groups all have the cubic point group. They are the simple cube body-centered cubic , and face-centered cubic Tetragonal (2 lattices) The simple tetragonal is made by pulling on two opposite faces of the simple cubic and stretching it into a rectangular prism with a square base, but a height not equal to the sides of the square. By similarly stretching the body-centered cubic one more Bravais lattice of the tetragonal system is constructed, the centered tetragonal Orthorhombic (4 lattices) The simple orthorhombic is made by deforming the square bases of the tetragonal into rectangles, producing an object with mutually perpendicular sides of three unequal lengths. The

41. Lattices Central to the lattice machinery in Magma is a highly optimized LLL algorithm. The LLL algorithm takes a basis of a lattice and returns a new basis of the http://magma.maths.usyd.edu.au/magma/Features/node166.html

Next: Lattices: Construction and Operations Up: Lattices and Quadratic Forms Previous: Lattices and Quadratic Forms Lattices A lattice in Magma is a -module contained in or , together with a positive definite inner product. The information specifying a lattice is a basis, given by a sequence of elements in or , and a bilinear product , given by v w v M w tr for a positive definite matrix M . Central to the lattice machinery in Magma is a highly optimized LLL algorithm. The LLL algorithm takes a basis of a lattice and returns a new basis of the lattice which is LLL-reduced which usually means that the vectors of the new basis have small norms. The Magma LLL algorithm is based on the FP-LLL algorithm of Schnorr and Euchner and the de Weger integral algorithm but includes various optimizations, with particular attention to different kinds of input matrices. Lattices: Construction and Operations Lattices: Properties Lattices: Reduction Lattices: Automorphisms ... -Lattices Next: Lattices: Construction and Operations Up: Lattices and Quadratic Forms Previous: Lattices and Quadratic Forms

42. Diamant: Lattices The lecture will start by recalling how one can use a lattice basis reduction algorithm for solving systems of linear equations over the ring of integers. http://homepages.cwi.nl/~aardal/diamant-lattice/

DIAMANT Intercity Event: Lattice Day When and where When: November 10, 2006. 11:00-17:00. Where: Gorlaeus Laboratorium, Universiteit Leiden, Einsteinweg 55. English version 11:00 - 12:00: zaal 3, Gorlaeus 13:30 - 17:00: zaal 1, Gorlaeus Program Hendrik W. Lenstra (Universiteit Leiden) A new type of lattices. Abstract. The lecture will start by recalling how one can use a lattice basis reduction algorithm for solving systems of linear equations over the ring of integers. An analysis of this application suggests that one can more appropriately handle it by means of a new notion of lattice, for which the length function takes values in an ordered vector space of dimension greater than one. The full theory of these generalized lattices, as well as the corresponding basis reduction algorithms, remain to be developed. No previous knowledge of lattices is necessary for following the lecture. Lunch Phong Nguyen Hermite's constant and lattice reduction algorithms. Abstract. Lattice reduction is a computationally hard problem of interest to both public-key cryptography and public-key cryptanalysis. Despite its importance, extremely few algorithms are known. In this talk, we will survey all lattice reduction algorithms known, and we will try to speculate on future developments. In doing so, we will emphasize a connection between those algorithms and the historical mathematical problem of bounding Hermite's constant. Friedrich Eisenbrand Integer programming: Results in fixed dimension

43. Monomer Adsorption On Equilateral Triangular Lattices With Attractive First-neig We have recently studied a model of monomer adsorption on infinitely long equilateral triangular Lattices with terraces of finite width M and nonperiodic http://pubs.acs.org/cgi-bin/abstract.cgi/langd5/asap/abs/la702128f.html

[Journal Home Page] [Search the Journals] [Table of Contents] [PDF version of this article] ... [Purchase Article] Langmuir, ASAP Article Web Release Date: November 30, Monomer Adsorption on Equilateral Triangular Lattices with Attractive First-neighbor Interactions Alain J. Phares,* David W. Grumbine, Jr., and Francis J. Wunderlich Department of Physics, Villanova University, Mendel Hall, Villanova, PA 19085-1699, and Department of Physics, Saint Vincent College, Latrobe, Pennsylvania 15650-4580 Received July 16, 2007 In Final Form: September 28, 2007 Abstract: We have recently studied a model of monomer adsorption on infinitely long equilateral triangular lattices with terraces of finite width M and nonperiodic boundaries. This study was restricted to the case of repulsive adsorbate-adsorbate first-neighbor interactions but included attractive, repulsive, and negligible second-neighbor interactions. The present work extends this study to the case of attractive first-neighbors, and the phases are determined, as before, with a confidence exceeding 10 significant figures. Phase diagrams are included for terrace widths M 11. Most of the occupational characteristics of the phases fit exact analytic expressions in

44. [0709.3112] Lie Point Symmetries Of Difference Equations And Lattices Abstract A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and Lattices, while leaving http://arxiv.org/abs/0709.3112

arXiv.org math-ph Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages Full-text links: Download: PostScript PDF Other formats Citations CiteBase p revious n ... ext Mathematical Physics Title: Lie point symmetries of difference equations and lattices Authors: Decio Levi Pavel Winternitz (Submitted on 19 Sep 2007) Abstract: A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant. The method is applied to several examples. The found symmetry groups are used to obtain particular solutions of differential-difference equations. Subjects: Mathematical Physics (math-ph) Journal reference: J. Phys. A: Math. Gen., 33, No 47 (2000), 8507-8523 Cite as: arXiv:0709.3112v1 [math-ph] Submission history view email Wed, 19 Sep 2007 21:48:31 GMT (22kb) Which authors of this paper are endorsers? Link back to: arXiv form interface contact

45. Crystals And Lattices The optical properties of crystals are macroscopic ones, since the wavelengths of 500 nm or so average over many lattice points, and depend on indices of http://mysite.du.edu/~jcalvert/phys/lattice.htm

Crystals and Lattices A different approach to understanding the reciprocal lattice Contents Introduction The Direct Lattice The Reciprocal Lattice X-Ray Diffraction ... References Introduction The word "crystal" is from the Greek krustallo s, "ice," and was first applied to quartz, which seemed to be a kind of permanent ice. After the late middle ages, it came to mean any substance, usually a mineral, showing plane faces in symmetrical relations. Until X-rays allowed us to examine these bodies in microscopic (actually, sub-microscopic) detail after 1912, the faces were the only obvious clue to internal structure. Crystal faces, produced naturally, vary widely in shapes and sizes, even in the same crystal. Niels Stensen (Nicolaus Steno, 1638-1686) recognized in 1669 that the angles between the normals to faces were more fundamental than any accidents of shape, and were the same in all crystals of a given substance. This allowed the classification of the symmetry of crystals, and established goniometry as the basis for mathematical crystallography. Many crystalline substances do not occur as crystals with well-defined faces (as euhedral crystals), and the individual crystals may be tiny, so their natures remained obscure. All metals, for example, are crystalline. When we say "microscopic," we generally mean "on an atomic scale," not literally microscopic. "Macroscopic" is used when the atomic constitution is not apparent.

46. CCD Lattices In Presheaf Categories In this paper we give a characterization of constructively completely distributive (CCD) Lattices in presheaves on C, for C a small category with pullbacks. http://www.tac.mta.ca/tac/volumes/18/6/18-06abs.html

CCD lattices in presheaf categories G. S. H. Cruttwell, F. Marmolejo and R. J. Wood In this paper we give a characterization of constructively completely distributive (CCD) lattices in presheaves on C, for C a small category with pullbacks. Keywords: Constructive complete distributivity, presheaves, Beck-Chevalley condition, Frobenius reciprocity, change of base 2000 MSC: 18B35, 06D10, 06B23 Theory and Applications of Categories, Vol. 18, 2007, No. 6, pp 157-171. http://www.tac.mta.ca/tac/volumes/18/6/18-06.dvi http://www.tac.mta.ca/tac/volumes/18/6/18-06.ps http://www.tac.mta.ca/tac/volumes/18/6/18-06.pdf ftp://ftp.tac.mta.ca/pub/tac/html/volumes/18/6/18-06.dvi ... TAC Home

47. Problem-set/algebra/lattices problemset/algebra/Lattices. Click here to see the number of accesses to this problem-set/algebra/Lattices created 07/15/86 revised 07/20/88 http://www.netlib.org/problem-set/algebra/lattices/

problem-set/algebra/lattices Click here to see the number of accesses to this library. # problem-set/algebra/lattices # created : 07/15/86 # revised : 07/20/88 # file problems

48. Module Lattices In AIPS++, we have used the ability to call many things by one generic name (Lattice) to create a number of classes which have different storage techniques http://aips2.nrao.edu/docs/lattices/implement/Lattices.html

Module Lattices Changes made in the current development cycle can be found in the changelog Description classes Regular N-dimensional data structures. Prerequisite Programmers of new Lattice classes should understand Inheritance Users of the Lattice classes should understand Polymorphism. class IPosition class Array Review Status Reviewed By: Peter Barnes Date Reviewed: Etymology Lattice: "A regular, periodic configuration of points, particles, or objects, throughout an area of a space..." (American Heritage Directory) This definition matches our own: an N-dimensional arrangement of data on regular orthogonal axes. In AIPS++, we have used the ability to call many things by one generic name (Lattice) to create a number of classes which have different storage techniques (e.g. core memory, disk, etc...). The name Lattice should make the user think of a class interface (or member functions) which all Lattice objects have in common. If functions require a Lattice argument, the classes described here may be used interchangeably, even though their actual internal workings are very different. Synopsis The Lattice module may be broken up into a few areas: Lattices - the actual holders of lattice-like data which all share a common interface . The following items are all Lattices and may be used polymorphically wherever a Lattice is called for. The ArrayLattice class adds the interface requirements of a Lattice to an AIPS++ Array . The data inside an ArrayLattice are not stored on disk. This n-dimensional array class is the simplest of the Lattices. Users construct the ArrayLattice with an argument which is either an IPosition which describes the array shape or a previously instantiated Array object that may already contain data. In the former case, some Lattice operation must be done to fill the data. The ArrayLattice, like all Lattices, may be iterated through with a

49. Science News Online - This Week - Feature Article - 7/5/97 A lattice is a regular array of points, each one specified by a set of coordinates Lattices may put security codes on a firmer footing. Science 273(Aug. http://www.sciencenews.org/sn_arc97/7_5_97/bob1.htm

July 5, 1997 Hiding in Lattices An improved mathematical strategy for encrypting data By IVARS PETERSON S ending a message over the Internet is like mailing a postcard. It can be read by anyone. Protecting sensitive information whether a credit card number, a password, or other data requires encrypting the message so that no eavesdropper or thief can read the contents in transit. Indeed, many online retailers now use systems that routinely encrypt any information a customer enters to order a product. In general, the hiding is accomplished in such a way that breaking the code involves solving an extraordinarily difficult mathematical problem one so hard that even a thief with access to the world's most powerful supercomputers would fail. For instance, it's easy to multiply two numbers. It's considerably more difficult, given that product, to work out what numbers were multiplied together to generate it. Determining that 57,814,193 is the product of the two prime numbers 7,079 and 8,167 requires much more computer time than multiplying the two primes. The belief that factoring huge numbers is intrinsically difficult underlies one widely used cryptosystem. Cracking such codes typically requires factoring numbers that are 200 or more digits long (SN: 5/14/94, p. 308).

50. Dynamics Of Coupled Map Lattices cml2004, cml 2004, coupled map Lattices at ihp, coupled map Lattices 2004, Institut Henri Poincare, dynamics of coupled map Lattices. http://www.cpht.polytechnique.fr/cpth/cml2004/

School-forum C OUPLED M AP L ATTICES 2004 I nstitut H enri ... P Paris, June 21 - July 2, 2004 Supporting institutions PROGRAMME: Mornings Lectures Afternoons Working, posters and Young scientists sessions LECTURERS V. Afraimovich P. Ashwin C. Baesens L. Bunimovich R. Coutinho B. Ermentrout M. Floria P. Glendinning H. de Jong W. Just G. Keller R. Lima Y. Maistrenko R. MacKay TOPICS Statistical and topological properties : physical measures, phase transitions, spectral theory, ergodic properties, spatial chaos, attractors, bifurcations Synchronisation and globally coupled systems : riddled basins, Lyapunov vectors and exponents, clustering. Monotone extended systems : Frenkel-Kontorova models, continuous time, discrete time. Related spatially extended systems models : integro-differential equations. Dynamics of genetic regulation networks : continuous time, discrete time, piecewise affine models. Lecture Notes List of participants ORGANISERS J.-R. Chazottes , CPHT, CNRS-Ecole Polytechnique, Palaiseau, France, jeanrene@cpht.polytechnique.fr B. Fernandez

51. ARCC Workshop: Sphere Packings, Lattices, And Infinite Dimensional Algebra The AIM Research Conference Center (ARCC) will host a focused workshop on Sphere Packings, Lattices, and Infinite Dimensional Algebra, August 16 to August http://www.aimath.org/ARCC/workshops/spherepacking.html

Sphere Packings, Lattices, and Infinite Dimensional Algebra August 16 to August 20, 2004 at the American Institute of Mathematics , Palo Alto, California organized by Lisa Carbone Noam Elkies , and Jim Lepowsky This workshop, sponsored by AIM and the NSF , will focus on sphere packings and lattice packings, with particular attention to dimensions 8 and 24 and the connection with automorphic forms and Moonshine. The problem of packing identical spheres as densely as possible in Euclidean space has a 400 year history, having been initiated by Johannes Kepler in 1611. Though the problem is unsolved in general today, attempts to solve it have led to the discovery of a wealth of mathematics. The Leech lattice appears in several places in `Moonshine' which is a term first coined by J. Conway and S. Norton in 1979 to describe the mysterious connections between finite sporadic simple groups and modular functions. The solution of the Monstrous moonshine conjectures by Frenkel, Lepowsky, Meurman and Borcherds expanded the Moonshine horizon to include interrelationships between lattices and hyperbolic reflection groups, generalized Kac-Moody Lie algebras, vertex (operator) algebras, automorphic forms and conformal field theory. Finding optimal sphere and lattice packings is an active area of current research. Many of the intricate connections between lattices, packings groups, automorphic forms and problems in Moonshine are not fully understood and give rise to substantial open problems. The proposed workshop will bring together specialists from these diverse but related areas, providing a unique opportunity for them to interact.

52. Optical Lattices - Quantiki An optical lattice is simply a set of standing wave lasers. The electric field of these lasers can interact with atoms the atoms see a potential and http://www.quantiki.org/wiki/index.php/Optical_Lattices

@import "/wiki/skins/quantiki/mediawiki.css"; @import "/wiki/skins/quantiki/style.css"; Wiki CMS Article Discussion ... History Optical Lattices An optical lattice is simply a set of standing wave lasers. The electric field of these lasers can interact with atoms - the atoms see a potential and therefore congregate in the potential minima. In the case of a typical one-dimensional setup, the wavelength of the opposing lasers is chosen so that the light shift is negative. This means that the potential minima occur at the intensity maxima of the standing wave. Furthermore, the natural beam width constrains the system to being one-dimensional. For quantum computation, we would like to be able to initialise the system so that there is an atom (generally, we think about alkali atoms, such as Rubidium) in every lattice site. This is accomplished by making use of the Mott insulator phase transition . This is done by starting with a Bose-Einstein condensate (BEC) and turning on the standing wave laser. At some critical value of the intensity, the atoms change from being in the BEC phase to the Mott insulator phase, which means that there are integer numbers of atoms in each lattice site. Purification procedures and suitable choice of the density of the BEC enable us to limit this to a single atom per lattice site. Usefully, we can assume that the atoms do not interact provided the intensity of the lasers is large enough. Some of the electronic levels of Rubidium.

53. Introduction To Concept Lattices In this paper we give Mizar formalization of concept Lattices. Concept Lattices stem from the socalled formal concept analysis - a part of applied http://mizar.org/JFM/Vol10/conlat_1.html

Journal of Formalized Mathematics Volume 10, 1998 University of Bialystok Association of Mizar Users Introduction to Concept Lattices Christoph Schwarzweller Summary. In this paper we give Mizar formalization of concept lattices. Concept lattices stem from the so-called formal concept analysis - a part of applied mathematics that brings mathematical methods into the field of data analysis and knowledge processing. Our approach follows the one given in [ MML Identifier: The terminology and notation used in this paper have been introduced in the following articles [ Contents (PDF format) Formal Contexts Derivation Operators Formal Concepts Concept Lattices Bibliography 1] Grzegorz Bancerek. Complete lattices Journal of Formalized Mathematics 2] Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps Journal of Formalized Mathematics 3] Czeslaw Bylinski. Binary operations Journal of Formalized Mathematics 4] Czeslaw Bylinski. Functions and their basic properties Journal of Formalized Mathematics 5] Czeslaw Bylinski. Functions from a set to a set Journal of Formalized Mathematics 6] Czeslaw Bylinski.

54. Bravais Lattices NanoLanguage has builtin support for all 14 three-dimensional Bravais Lattices. In fact, in order to perform proper calculations for bulk systems, http://www.atomistix.com/manuals/ATK_2.3/html/ref.bravaislattices.html

Atomistix ToolKit 2.3 Core NanoLanguage elements twoCenterIntegralParameters   addToSample Name Bravais lattices The 14 Bravais lattices used to describe crystal structures. Synopsis Namespace ATK.KohnSham or ATK.TwoProbe Object SimpleCubic a Object BodyCenteredCubic a Object FaceCenteredCubic a Object Rhombohedral a alpha Object Hexagonal a c Object SimpleTetragonal a c Object BodyCenteredTetragonal a c Object SimpleOrthorhombic a b c Object BodyCenteredOrthorhombic a b c Object FaceCenteredOrthorhombic a b c Object BaseCenteredOrthorhombic a b c Object SimpleMonoclinic a b c beta Object BaseCenteredMonoclinic a b c beta Object Triclinic a b c alpha beta gamma Description NanoLanguage has built-in support for all 14 three-dimensional Bravais lattices. In fact, in order to perform proper calculations for bulk systems, it is necessary to use these classes, instead of defining your own lattice vectors. This will (at least in the future) allow ATK to take advantage of the symmetries of the lattice, and the user does not have to worry about the exact definition of the lattice vectors, symmetry points in the Brillouin zone, and so on. None of the parameters have any default value, and they must all be specified with units (see the examples below).

55. Focus On Cold Atoms In Optical Lattices The varying light intensity creates a periodic potential landscape – the optical lattice – for the atoms via an ACStark shift interaction. http://www.iop.org/EJ/abstract/1367-2630/8/8/E02

@import url(http://ej.iop.org/style/nu/NJP4.css); Athens login IOP login: Password: Create account Alerts Contact us IOP Journals Home ... Content finder EDITORIAL Focus on Cold Atoms in Optical Lattices New J. Phys. doi:10.1088/1367-2630/8/8/E02 Imagine you could produce an artificial crystal for quantum matter, defect free and with complete control over the periodic crystal potential. The shape of the periodic potential, its depth and the interactions between the underlying particles could be changed at will and the particles could be moved around in a highly controlled way, at essentially zero temperature. This sounds almost too good to be true, but it is in fact what optical lattices have made possible for cold and ultracold atoms. T c superconductivity, but so far even the exact phase diagram of this basic Hamiltonian is still under discussion after decades of theoretical research on this problem. Another research focus in this respect has been to use ultracold atoms in optical lattices for the investigation of quantum magnetism. Here the spin state of an atom does not have to be restricted to two possible values, as for electrons, but can cover several possible magnetic substates, yielding rich and novel quantum phases. In the latest proposals, it has also been shown how quantum spin models could be robustly and efficiently simulated with polar molecules trapped in optical lattices. Next to exploring the fundamental behaviour of quantum matter in periodic potentials, optical lattices have also shown to be very useful for the generation of ultracold molecules. Imagine for example having two atoms at each lattice site of an optical lattice. Then photoassociation or Feshbach resonances can be used to convert these atom pairs into stable molecules in a defined rotational-vibrational quantum state. One might consider such a coherent conversion of atoms into molecules to be the ultimate quantum limit that we can reach in the control over a 'chemical reaction'.

56. OLAQUi: Optical Lattices And Quantum Information A quantum phase transition can be used to prepare exactly one atom per lattice site, where each atom can be considered as quantum bit. http://olaqui.df.unipi.it/

O ptical La ttices and Qu antum I nformation Community Research Specific Targeted Research Projects Sixth Framework Programme Home Partners News / Events Restricted Documents ... Links For suggestions contact: Francesca Usala usala@df.unipi.it Last update: 10/07/2007 OLAQUi: Optical Lattices and Quantum Information Specific Targeted Research Project Contract No 013501 A system of neutral atoms stored in an optical lattice is a promising candidate for implementing scalable quantum computing. A quantum phase transition can be used to prepare exactly one atom per lattice site, where each atom can be considered as quantum bit. Based on the so-called Mott-Insulator state several schemes for quantum computation have been proposed, including proposals for the creation of entanglement, computation with cluster states and quantum simulations. It is planned to use a Mott insulator as a quantum register, in which one can encode qu-bits in the single atoms on each lattice site and quantum gates can be implemented acting on different atoms of the lattice. Crucial advantages are: the simple quantum-level structures of atoms; the insulation from the environment which leads to a strong suppression of decoherence, and the ability to trap and act on a very large ensemble of identical atoms.

57. Lattice (Sphinx-4) Lattices describe all theories considered by the Recognizer that have not been pruned out. Lattices are a directed graph containing Nodes and Edges . http://cmusphinx.sourceforge.net/sphinx4/javadoc/edu/cmu/sphinx/result/Lattice.h

Overview Package Class Tree Deprecated Index Help ... METHOD edu.cmu.sphinx.result Class Lattice java.lang.Object edu.cmu.sphinx.result.Lattice public class Lattice extends java.lang.Object Provides recognition lattice results. Lattices are created from Results which can be partial or final. Lattices describe all theories considered by the Recognizer that have not been pruned out. Lattices are a directed graph containing Nodes and Edges . A Node that correponds to a theory that a word was spoken over a particular period of time. An Edge that corresponds to the score of one word following another. The usual result transcript is the sequence of Nodes though the Lattice with the best scoring path. Lattices are a useful tool for analyzing "alternate results". A Lattice can be created from a Result that has a full token tree (with its corresponding AlternativeHypothesisManager). Currently, only the WordPruningBreadthFirstSearchManager has an AlternativeHypothesisManager. Furthermore, the lattice construction code currently only works for linguists where the WordSearchState returns false on the isWordStart method, i.e., where the word states appear at the end of the word in the linguist.

58. Bravais Crystal Lattices There are 14 distinct Bravais crystal Lattices. The Lattices can be used to describe the geometrical symmetry of a crystal. The Bravais Lattices are http://chemistry.about.com/od/crystallography/ig/Bravais-Crystal-Lattices/

zOBT=" Ads" zGCID=" test1" zGCID=" test1 test13" zJs=10 zJs=11 zJs=12 zJs=13 zc(5,'jsc',zJs,9999999,'') z160=zpreC(160,600);z336=zpreC(336,280);z728=zpreC(728,90);z133=zpreC(336,133);zItw=160 if(thin)gEI('globalwrapper').className='thin'; Chemistry var h2=document.getElementsByTagName("h2")[0];if(h2.getElementsByTagName("a")[0].firstChild.nodeValue.length>29)h2.className="long"; Home Education Chemistry Search over 1.4 million articles by over 600 experts Chemistry About.com Search Chemistry Basics Science Projects How Things Work ... on the Chemistry Forum Bravais Crystal Lattices h1 = document.getElementById("title").getElementsByTagName("h1")[0];h1.innerHTML = widont(h1.innerHTML); From Anne Marie Helmenstine, Ph.D. , About.com More About: bravais lattices crystallography crystals There are 14 distinct Bravais crystal lattices. The Bravais lattices are sometimes called space lattices. The lattices can be used to describe the geometrical symmetry of a crystal. All crystals can be described by one of these arrangements. Enter Gallery Images of Triclinic Bravais Crystal Lattice Simple Monoclinic Bravais Crystal Lattice Base Centered Monoclinic Bravais Crystal Lattice Simple Orthorhombic Bravais Crystal Lattice ... Simple Cubic or Isometric Bravais Crystal Lattice Prev Next Graphic Index Text Index More about Crystals Crystal Photo Gallery Types of Crystals Crystal Science Fair Projects Grow Crystals Big Alum Crystal Silver Crystals Charcoal Crystal Garden zSB(2,5);

59. Project-Lande:A Coq Library Of Lattices We have developed a library of Coq modules for implementing Lattices, the fundamental data structure of most static analysers. http://ralyx.inria.fr/2006/Raweb/lande/uid20.html

Team Lande Members Overall Objectives Project overview View by sections View by topics Scientific Foundations Static program analysis Program testing Reachability analysis over term rewriting systems Software A Coq library of lattices Timbuk: a tree automata library Protocol Animator Automatic functional test case generation New Results Access control model for interactive devices From certified abstract interpretation to proof-carrying code Certificates checkers for arithmetics invariants Resource-oriented analysis ... Dynamic and static information flow analysis Contracts and Grants with Industry The CASTLES RNTL project The RNTL CAT project Static analysis of Java Midlets with rewriting and reachability analysis European FET-Integrated project MOBIUS Other Grants and Activities The ACI Sécurité Informatique project DISPO The ACI Sécurité Informatique project V3F The ACI Sécurité Informatique project SATIN Dissemination Conferences: program committees, organization, invitations PhD theses defended PhD and Habilitation committees Teaching: university courses and summer schools ... Administrative responsibilities Bibliography Major publications Year Publications References in notes Inria ... Project: Lande Project : lande Section: Software Keywords Constructive logic Coq modules Lattices A Coq library of lattices Participant David Pichardie . Using the abstract interpretation methodology, static analyses are specified as least solution of system of equations (inequations) on lattice structures. The library of Coq modules allows to construct complex and efficient lattices by combination of functors and base lattices. The lattice signature possesses a parameter which ensure termination of a generic fixed-point solver. The delicate problem of termination of fixpoint iterations is hence dealt with once and for all when building a lattice as a combination of the different lattice functors.

60. On The Characterization Of Modular And Distributive Lattices The last part treats of intervallike subLattices of any lattice. Properties of relational structures, posets, Lattices and maps. http://www.cs.ualberta.ca/~piotr/Mizar/mirror/htdocs/JFM/Vol10/yellow11.html

Journal of Formalized Mathematics Volume 10, 1998 University of Bialystok Association of Mizar Users On the Characterization of Modular and Distributive Lattices Adam Naumowicz University of Bialystok Summary. This article contains definitions of the ``pentagon'' lattice $N_5$ and the ``diamond'' lattice $M_3$. It is followed by the characterization of modular and distributive lattices depending on the possible shape of substructures. The last part treats of interval-like sublattices of any lattice. This work has been supported by KBN Grant 8 T11C 018 12. MML Identifier: The terminology and notation used in this paper have been introduced in the following articles [ Contents (PDF format) Preliminaries Main Part Diamond and Pentagon Intervals of a Lattice Bibliography 1] Grzegorz Bancerek. The ordinal numbers Journal of Formalized Mathematics 2] Grzegorz Bancerek. Complete lattices Journal of Formalized Mathematics 3] Grzegorz Bancerek. Bounds in posets and relational substructures Journal of Formalized Mathematics 4] Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps

61. Perfect Lattices In Euclidean Spaces - Zahlentheorie Journals, Books & Online Me Perfect Lattices in Euclidean Spaces Zahlentheorie Kombinatorik. Lattices are discrete subgroups of maximal rank in a Euclidean space. http://www.springer.com/dal/home/math/numbers?SGWID=1-10048-22-1476196-0

62. International Focus Workshop On Mobile Fermions And Bosons On Frustrated Lattice Mobile Fermions and Bosons on Frustrated Lattices. International focus workshop – January 11 13, 2007. Scientific Coordinators http://www.mpipks-dresden.mpg.de/~itfrus07/

63. Lattice (order) - Wikipedia, The Free Encyclopedia In mathematics, a lattice is a partially ordered set (or poset) in which every pair of elements has a unique supremum (the elements least upper bound; http://en.wikipedia.org/wiki/Lattice_(order)

var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia"; Lattice (order) From Wikipedia, the free encyclopedia Jump to: navigation search The name "lattice" is suggested by the form of the Hasse diagram depicting it. In mathematics , a lattice is a partially ordered set (or poset ) in which every pair of elements has a unique supremum (the elements' least upper bound; called their join ) and an infimum (greatest lower bound; called their meet ). Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities . Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra Semilattices include lattices, which in turn include Heyting and Boolean algebras . These "lattice-like" structures all admit order-theoretic as well as algebraic descriptions. Contents Lattices as posets Lattices as algebraic structures Connection between the two definitions Examples ... edit Lattices as posets Consider a poset L L is a lattice if For any two elements x and y of L x y join , or supremum ) and a greatest lower bound ( meet , or infimum The join and meet of x and y are denoted by and , respectively. Because joins and meets are assumed to exist in a lattice

64. Crystal Lattice Structures A list of the possible crystal lattice structures and coordinates of atoms in such crystals. http://cst-www.nrl.navy.mil/lattice/

Navigation: Crystal Sites Home Strukturbericht Designation Pearson Symbol ... Other Sites Questions? FAQ Submitting additions to the database NRL Sites NRL Home MSCT 6000 MSTD 6300 CCMS 6390 Navy Sites DoN ONR Recruiting FOIA Crystal Lattice Structures Indexed by Strukturbericht Designation Pearson Symbol Space Group ... Structures of Intermetallic Alloy Phases This page offers a concise index of common crystal lattice structures. A graphical representation as well as useful information about the lattices can be obtained by clicking on the desired structure below. This page currently contains links to 273 structures in 94 of the 230 space groups. Newest Structures: Proposed CdPt CaC (11.5 K superconductor) LiBC ... HoCoGa Simple Cubic and related structures Cubic Close Packed and related structures Body Centered Cubic and related structures Hexagonal Close Packed and related structures Carbon and Related Structures Manganese Structures The Laves Phases Perovskite and Related Structures Quartz (SiO and Related Structures Sulfur and its Compounds Lanthanides, Actinides

65. Lattice -- From Wolfram MathWorld Note that this type of lattice is distinct from the regular array of points known as a point lattice (or informally as a mesh or grid). http://mathworld.wolfram.com/Lattice.html

Search Site Algebra Applied Mathematics Calculus and Analysis ... Insall Lattice An algebra is called a lattice if is a nonempty set and are binary operations on , both and are idempotent commutative , and associative , and they satisfy the absorption law . The study of lattices is called lattice theory Note that this type of lattice is distinct from the regular array of points known as a point lattice (or informally as a mesh or grid). While every point lattice is a lattice under the ordering inherited from the plane, many lattices are not point lattices. Lattices offer a natural way to formalize and study the ordering of objects using a general concept known as the poset (partially ordered set). A lattice as an algebra is equivalent to a lattice as a poset 1. Let the poset be a lattice. Set and . Then the algebra is a lattice. 2. Let the algebra be a lattice. Set iff . Then is a poset , and the poset is a lattice. 3. Let the poset be a lattice. Then 4. Let the algebra be a lattice. Then The following inequalities hold for any lattice: A lattice can be obtained from a lattice-ordered poset by defining and for any . Also, from a lattice

66. Lattice Theory This lattice is infinte and subdirectly irreducible but the variety it generates contains only finitely many other varieties, refuting an old conjecture. http://www.math.hawaii.edu/LatThy/

The Lattice Theory Homepage J. B. Nation's counter example to the finite height conjecture This lattice is infinte and subdirectly irreducible but the variety it generates contains only finitely many other varieties, refuting an old conjecture. Think of this lattice as wrapped around a cylinder so that the elements at the sides with the same labels are identified. Note the middle part of this diagrams moves up when going from left to right so that when it is wrapped around the cylinder it is helical. This page will have link to various subbranches of lattice theory containing general information and problems. A Java program indicating how the automatic lattice drawing works General Algebra and Lattice Theory Homepage Universal Algebra and Lattice Theory Bibliographic Database You can search for papers by a particular author or on a particular subject. You can choose to have the output in either BibTeX or AmSTeX format. You can even add or correct entries in the database. Free and Finitely Presented Lattices A description of the subject and a list of open problems.

67. Introduction To Cubic Crystal Lattice Structures A site introducing the properties of crystals with a cubic unit cell. http://www.okstate.edu/jgelder/solstate.html

Introduction to Cubic Crystal Lattice Structures The outstanding macroscopic properties of crystalline solids are rigidity, incompressibility and characteristic shape. All crystalline solids are composed of orderly arrangements of atoms, ions, or molecules. The macroscopic result of the microscopic arrangements of the atoms, ions or molecules is exhibited in the symmetrical shapes of the crystalline solids Solids are either amorphous, without form, or crystalline. In crystalline solid s the array of particles are well ordered. Crystalline solids have definite, rigid shapes with clearly defined faces. The arrangement of the atoms, ions or molecules are very ordered and repeat in 3-dimensions. Small, 3-dimensional, repeating units called unit cells are responsible for the order found in crystalline solids. The unit cell can be thought of as a box which when stacked together in 3-dimensions produces the crystal lattice. There are a limited number of unit cells which can be repeated in an orderly pattern in three dimensions. We will explore the cubic system in detail to understand the structure of most metals and a wide range of ionic compounds. In the cubic crystal system three types of arrangements are found;

68. Crystal Atomic Lattice Viewer Crystal Atomic Lattice Viewer. If your browser supports Java you should be able to rotate these structures using your mouse!! Diamond. Buckminsterfullerine http://www.le.ac.uk/eg/spg3/atomic.html

Crystal Atomic Lattice Viewer If your browser supports Java you should be able to rotate these structures using your mouse!! Diamond Buckminsterfullerine Sodium Chloride Magnesium Tungsten Carbide The above Java applets have been developed by Simon Gill from Sun Microsystems Molecule Viewer freely available applet are in the source code LatticeViewer.java which also requires Matrix3D.java . These applets are best viewed on a high performance computer using the JDK 'appletviewer'. If you do use this applet I would be pleased if you let me know. The atomic positions of the following crystal structures were obtained from the Complex Systems Theory branch of the US Navy.

69. Reciprocal Lattice In particular we will concentrate on the reciprocal lattice and its relationship with the real lattice. The reciprocal lattice can be observed if we shine http://www.chembio.uoguelph.ca/educmat/chm729/recip/vlad.htm

"... The nature is created so that all the simple in it is true, and all the complicated is false." Grigory Skovoroda.(Ukrainian philosopher, XVIII century) As the title figure of this page shows, you will find below a lecture by an imaginary professor (in the middle) about the peculiar ways in which scientists extract atomic level information about the structure of crystals. In particular we will concentrate on the reciprocal lattice and its relationship with the real lattice. The reciprocal lattice can be observed if we shine X-rays or other short wave radiation onto the real lattice. Unlike the real lattice, the reciprocal lattice can be confusing and needs definite knowledge to be interpreted. However even at this point we can state that the things which are larger in real space are smaller in reciprocal space by definition. :Before we go any further, let me introduce you to the student P: Reciprocal space is also called Fourier space, k- space, or momentum space in contrast to real space or direct space. The concept of the reciprocal lattice was devised to tabulate two important properties of crystal planes: their slopes and their interplanar distances.

70. Defied 3D Teaching Explorer 3D - Bravais lattice. Cubic simple. Live3D Applet move to rotate s for stereo ? Shift - move for zoom. Body Centered Cubic http://feynman.phy.ulaval.ca/marleau/bravais3D_1.htm

Prof. Luc Marleau, Ph. D. Full professor Department of physics, engineering physics and optics Research Personal information Research activities Researchers-students Recent publications ... How to contact Teaching Teaching Courses Lecture notes Projects PHY-10518 ... Particle Physics: Reactions and Conservation Laws Links Theoretical physical group Department Université Laval Québec City Languages Select interface language: Teaching - Explorer 3D - Bravais lattice Cubic simple Live3D Applet: move to rotate " s " for stereo Shift - move for zoom Body Centered Cubic Live3D Applet: move to rotate " s " for stereo Shift - move for zoom Face Centered Cubic Live3D Applet: move to rotate " s " for stereo Shift - move for zoom Simple Tetragonal Live3D Applet: move to rotate " s " for stereo Shift - move for zoom Centered Tetragonal Live3D Applet: move to rotate " s " for stereo Shift - move for zoom Simple Orthorhombic Live3D Applet: move to rotate " s " for stereo Shift - move for zoom C Face (or Base) Centered Orthorhombic Live3D Applet: move to rotate " s " for stereo Shift - move for zoom Body Centered Orthorhombic Live3D Applet: move to rotate " s " for stereo Shift - move for zoom Face Centered Orthorhombic Live3D Applet: move to rotate " s " for stereo Shift - move for zoom Simple Monoclinic Live3D Applet: move to rotate " s " for stereo Shift - move for zoom C Face Centered Monoclinic Live3D Applet: move to rotate " s " for stereo Shift - move for zoom Triclinic Live3D Applet: move to rotate " s " for stereo

71. The Reciprocal Lattice To give a firm mathematical understanding of the reciprocal lattice, of the relationships between real and reciprocal space and of their implications for http://www.iucr.org/comm/cteach/pamphlets/4/index.html

Next: 1. Introduction The Reciprocal Lattice A. Authier Download in PDF format 1. Introduction 2. Crystallographic Definition 2.1 Definition ... 5. Diffraction Condition in the Reciprocal Lattice Teaching Aims To give a firm mathematical understanding of the reciprocal lattice, of the relationships between real and reciprocal space and of their implications for X-ray diffraction. Level This approach would be suitable for final year undergraduates in physics and mathematics or for initial post-graduate students in other disciplines provided that their mathematical background is adequate. Background A familiarity with vector manipulation is needed and, for certain sections, an understanding of tensor calculus. Practical Resources No specific practical resources are required. Time Required for Teaching If the mathematical background is already adequate this could be taught in 3 or 4 lectures. More would be required, however, if time has to be spent on mathematical equations and derivations as in places the treatment given is very concise. IUCr Webmaster

72. 200 Lattice Themed Brushes For Photoshop (R) And Photoshop (R) Elements, Mixed, 200 Lattice brushes for Photoshop (R). Brushes for Photoshop (R) CS3 CS2 CS 7 and Photoshop (R) Elements 5 4 3 2. The brushes set includes lattice brushes, http://www.graphicxtras.com/products/psbrush_more10.htm

catalog home mail cart ... up DOWNLOAD PAYPAL, Visa, Mastercard, Amex, Wire transfer, Check, maestro and many others Free sampler/demo Manual Gallery graphicxtras.com ... Mail any questions Related products 3D/Spheres brushes Spirals1 brushes Starry1 brushes Flames1 brushes ... 10000 Brushes Includes maze brushes, grid brushes, lattice brushes, connection brushes, circuit brushes, distorted brushes, thin lattice brushes, patchy brushes ,node brushes, thin brushes, square brushes, circular brushes and many more ... Brushes are easy to install (in presets) Brushes are easy to use, use with most brush tools. Modify via brushes palette adding spacing + size. Combine brushes, apply with blending modes + apply to layers + apply to channels. Use brushes to create designs + frames + backgrounds + drawings + distressed effects + grunge brush effects + overlays + much more. Use with effects. Use brushes to create web designs, leaflets, images, books, packaging, scrap books and more... Royalty-free, commercial and hobby work Notes Native format ABR Brushes.

73. Concept Lattice-based Knowledge Discovery Among those communities, there are several subareas of research that can be unified by their common interest in concept lattice http://www.ksl.stanford.edu/iccs2001/clkdd2001/

ICCS 2001 Workshop 2001 International Workshop on Concept Lattice-based theory, methods and tools for Knowledge Discovery in Databases Stanford University, California, USA July 30, 2001 Workshop of the 9th International Conference on Conceptual Structures ( ICCS-2001 WORKSHOP CONTEXT Throughout the last decade, Knowledge Discovery in Databases (KDD) has become an increasingly important topic in research as well as in industrial applications, up to being now a well-established interdisciplinary research area, benefiting yet from diverse influences such as: Databases, Data Analysis and Machine Learning Technology. Among those communities, there are several sub-areas of research that can be unified by their common interest in "concept lattice structures" which, as a consequence, start to play an important role in Data Mining. Over the last two decades, several trends of works have demonstrated how concept lattices formalize conceptual structures by coding any kind of dualities, and can be used to address a variety of problems in Databases, Data Analysis and Machine Learning. These include : - association rules or data dependencies in Databases, searching frequent item sets, indexing documents for information retrieval...;

74. Lattice Command for style none scale is not specified (nor any optional args) for all other styles scale = reduced density rho* (for LJ units) scale = lattice constant in http://lammps.sandia.gov/doc/lattice.html

LAMMPS WWW Site LAMMPS Documentation LAMMPS Commands lattice command Syntax: lattice style scale keyword values ... style = none or sc or bcc or fcc or diamond or sq or or hex or custom scale = scale factor between lattice and simulation box for style none : scale is not specified (nor any optional args) for all other styles: scale = reduced density rho* (for LJ units) scale = lattice constant in Angstroms (for real or metal units) zero or more keyword/value pairs may be appended keyword = origin or orient or spacing or or or or basis origin values = x y z x,y,z = fractions of a unit cell (0 <= x,y,z orient values = dim i j k dim = x or y or z i,j,k = integer lattice directions spacing values = dx dy dz dx,dy,dz = lattice spacings in the x,y,z box directions values = x y z x,y,z = primitive vector components that define unit cell basis values = x y z x,y,z = fractional coords of a basis atom (0 <= x,y,z Examples: Description: Define a lattice for use by other commands. In LAMMPS, a lattice is simply a set of points in space, determined by a unit cell with basis atoms, that is replicated infinitely in all dimensions. The arguments of the lattice command can be used to define a wide variety of crystallographic lattices. A lattice is used by LAMMPS in two ways. First, the