structure beams , appropriate decking material movable No design medium falsework Sometimes A latticebridge is a form of truss bridge that uses a large number of small and closely spaced diagonal elements that form a latticework lattice . It was patented by architect Ithiel Town in 1820 and 1835 as Town s lattice truss . Originally a design to allow a substantial bridge to be made from planks ... Roof Construction , 2002. ref Wood lattice truss bridges File Astabula Cty Root Road townsend latice1982.jpg thumb right Root Road Covered Bridge, Astabula, Ohio Bartonsville Covered Bridge Brown Covered Bridge Burt Henry Covered Bridge Cornish Windsor Covered Bridge Kingsley Covered Bridge Pool s Mill Covered Bridge Root Road Covered Bridge Waterford Covered Bridge Windsor Mills Covered Bridge Worrall Covered Bridge Frankenfield Covered Bridge Iron or steel lattice truss bridges File LatticeTrussIowaRiver.jpg ... Carroll built the first completely wrought iron lattice truss bridge. This was built for the New York ... Trail Bridge References Reflist See also Commons category Town Lattice truss bridges Lattice girder ...BridgeTypePix image Guilford vermont bridge covered bridge interior.jpg image title Interior structure of a covered bridge utilizing a plank lattice structure type name Latticebridge sibling names None descendent names None ancestor names Truss bridge carries Pedestrian s, livestock , vehicles span ... of bridge has also been constructed using a large number of relatively light iron or steel members. The individual elements are more easily handled by the construction workers, but the bridge also requires substantial support during construction. A simple lattice truss will transform the applied loads into a thrust, as the bridge will tend to change length under load. This is resisted by pinning the lattice members to the top and bottom chords, which are more substantial than the lattice ... hangar in the Duxford Imperial War Museum The Belfast truss is a cross between the Town s lattice ... more details
wiktionary latticeLattice may refer to In art and design Latticework an ornamental criss crossed framework, an arrangement of crossing laths or other thin strips of material Lattice pastry In engineering A lattice shape truss structure In mathematics Lattice order , a partially ordered set with unique least upper bounds and greatest lower bounds Lattice group , a repeating arrangement of points Lattice discrete subgroup , a discrete subgroup of a topological group with finite covolume Lattice graph , a graph that can be drawn within a repeating arrangement of points Bethe lattice , a regular infinite tree structure Lattice multiplication , a multiplication algorithm suitable for hand calculation Lattice model finance , a method for evaluating stock options that divides time into discrete intervals In science A crystal structure fitting a lattice arrangement Lattice model physics , a model defined not on a continuum, but on a lattice Companies Lattice Semiconductor , a US based integrated circuit manufacturer Lattice, Incorporated, a software company and makers of Lattice C Lattice Group , a former British gas transmission business Other Lattice C , a compiler for the C programming language See also Grid disambiguation Mesh disambiguation Trellis disambiguation In title Lattice disambiguation ar da Gitter de Gitter Begriffskl rung es Ret culo fr Lattice it Reticolo he ka ja Lattice pl Krata pt Ret culo ru sk Mrie ka sv Gitter ... more details
File Darcy Lever geograph.org.uk 93794.jpg right thumb 300px Darcy Lever lattice girder railway bridge, Lancashire Image San Francisco Oakland Bay Bridge Retrofit 3.jpg right thumb 300px This example shows laced struts and ties. Pure tension eyebar s are seen in the background this is the cantilever portion of the eastern span of the San Francisco Oakland Bay Bridge A lattice girder is a girder where the flange s are connected by a Latticework lattice web. ref cite web title Definition of lattice girder publisher Merriam Webster url http webster.com dictionary lattice girder accessdate 2006 12 13 ref This type of design has been supplanted in modern construction with welding welded or bolted joint bolted plate girders, which use more material but have lower fabrication and maintenance costs. The lattice girder was used prior to the development of larger rolled steel plates. The term is also sometimes used to refer to a structural member commonly made using a combination of structural sections connected with diagonal lacing. This member is more correctly referred to as a laced strut ref cite web title Glossary of bridge terminology publisher University of Iowa Lichtenberger Engineering Library url http sdrc.lib.uiowa.edu eng bridges WaddellGlossary GlossL.htm accessdate 2006 12 13 ref or laced tie , as it normally resists axial compression physical compression strut or axial tension mechanics tension tie engineering tie the lattice girder, like any girder, primarily resists bending .... Some structures employing lattice girders Runcorn Railway Bridge Kew Railway Bridge Dowery Dell ... of Liberty The sides of the Cape Fear Memorial Bridge See also Lattice truss bridge an extension of the concept to form a deep truss . Hares Hill Road Bridge References references Category ... using either hot rivets or threaded locator bolts. As with lattice girders, laced struts and ties ... Tower The obsolescent eastern span of the San Francisco Oakland Bay Bridge . The western span has been ... more details
In mathematics , the term lattice group is used for two distinct notions a lattice group lattice , a discrete subgroup of R sup n sup and its generalizations. a lattice ordered group , a group that with a partial ordering that is a lattice order lattice order. mathdab ... more details
In mineralogy , atomic lattice refers to the arrangement of atoms into a crystal structure . In order theory , a lattice order lattice is called an atomic lattice if the underlying partial order is atomic order theory atomic . disambig ... more details
In crystallography , a lattice plane of a given Bravais lattice is a plane or family of parallel planes whose intersections with the lattice or any crystalline structure of that lattice are periodic functions periodic i.e. are described by 2d Bravais lattices and intersect the Bravais lattice equivalently, a lattice plane is any plane containing at least three noncollinear Bravais lattice points. ref name Ash76 Neil W. Ashcroft and N. David Mermin, Solid State Physics Harcourt New York, 1976 . ref All lattice planes can be described by a set of integer Miller indices , and vice versa all integer Miller indices define lattice planes . ref name Ash76 Conversely, planes that are not lattice planes have aperiodic intersections with the lattice called quasicrystal s this is known as a cut and project construction of a quasicrystal and is typically also generalized to higher dimensions . ref J. B. Suck, M. Schreiber, and P. H ussler, eds., Quasicrystals An Introduction to Structure, Physical Properties, and Applications Springer Berlin, 2004 . ref References references Category Crystallography Category Geometry geometry stub de Gitterebene it Piani reticolari ... more details
Unreferenced date December 2009 In mathematics , the n dimensional integer lattice or cubic lattice , denoted Z sup n sup , is the lattice group lattice in the Euclidean space R sup n sup whose lattice points are n tuples n tuples of integer s. The two dimensional integer lattice is also called the square lattice , or grid lattice. Z sup n sup is the simplest example of a root lattice . The integer lattice is an odd unimodular lattice . Automorphism group The automorphism group or group of congruence relation congruence s of the integer lattice consists of all permutation s and sign changes of the coordinates, and is of order 2 sup n sup n nowiki nowiki . As a matrix group it is given by the set of all n × n signed permutation matrices . This group is isomorphic to the semidirect product math mathbb Z 2 n rtimes S n math where the symmetric group S sub n sub acts on Z sub 2 sub sup n sup by permutation this is a classic example of a wreath product . For the square lattice, this is the group of the square, or the dihedral group of order 8 for the three dimensional cubic lattice, we get the group of the cube, or octahedral group , of order 48. Diophantine geometry In the study of Diophantine geometry , the square lattice of points with integer coordinates is often referred to as the Diophantine plane . In mathematical terms, the Diophantine plane is the Cartesian product math scriptstyle mathbb Z times mathbb Z math of the ring of all integers math scriptstyle mathbb Z math . The study of Erd s Diophantine graph Diophantine figures focuses on the selection of nodes in the Diophantine plane such that all pairwise distances are integer. Coarse geometry In coarse structure coarse geometry , the integer lattice is coarsely equivalent to Euclidean space . See also Regular grid DEFAULTSORT Integer Lattice Category Euclidean geometry Category Lattice points Category Diophantine geometry ... more details
Unreferenced date December 2009 Image Square Lattice.svg thumb 300px An upright square lattice left and a diagonal square lattice right . Image Square Lattice Tiling.svg thumb Upright square tiling . The vertices of all squares together with their centers form an upright square lattice. For each color the centers of the squares of that color form a diagonal square lattice which is in linear scale &radic 2 times as large as the upright square lattice. In mathematics , the square lattice is a type of Lattice group lattice in a two dimensional Euclidean space . It is the two dimensional version of the integer lattice . It is one of the five types of two dimensional lattices as classified by their symmetry ... of an image of the lattice are by far the most common. They can conveniently be referred to as upright square lattice and diagonal square lattice . They differ by an angle of 45 . This is related to the fact that a square lattice can be partitioned into two square sub lattices, as is evident in the colouring of a checkerboard . Symmetry The square lattice s symmetry category is wallpaper group p4m. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. An upright square lattice can be viewed as a diagonal square lattice ..., after adding the centers of the squares of an upright square lattice we have a diagonal square lattice with a mesh size that is 2 times as small as that of the original lattice. A pattern with 4 fold rotational symmetry has a square lattice of 4 fold rotocenters that is a factor 2 finer and diagonally oriented relative to the lattice of translational symmetry . With respect to reflection axes ... lattice of 4 fold rotocenters, with these rotocenters at the centers of the squares formed by the reflection ... directions they are linearly a factor 2 denser. See also square tiling hexagonal lattice centered square number Gaussian integer DEFAULTSORT Square Lattice Category Euclidean geometry Category Lattice ... more details
notability Products date April 2012 primary sources date April 2012 The Lattice Project combines computing resource s, Grid middleware, specialized scientific application software and web services into a comprehensive Grid computing system for scientific analysis. A major aspect of the project makes use of the Berkeley Open Infrastructure for Network Computing BOINC platform. The Lattice Project maintains a separate http boinc.umiacs.umd.edu BOINC web site . External links http lattice.umiacs.umd.edu The Lattice Project web site Category Berkeley Open Infrastructure for Network Computing projects Lattice Project, The BOINC topics DEFAULTSORT Lattice Project science software stub pt The Lattice Project es The Lattice Project ... more details
In Lattice order lattice theory , a Lattice order bounded lattice L is called a 0,1 simple lattice if nonconstant lattice homomorphisms of L preserve the identity of its top and bottom elements. That is, if L is 0,1 simple and is a function from L to some other lattice that preserves joins and meets and does not map every element of L to a single element of the image, then it must be the case that sup 1 sup 0 0 and sup 1 sup 1 1 . For instance, let L sub n sub be a lattice with n Atom order theory atoms a sub 1 sub , a sub 2 sub , ..., a sub n sub , top and bottom elements 1 and 0, and no other elements. Then for n 3, L sub n sub is 0,1 simple. However, for n 2, the function that maps 0 and a sub 1 sub to 0 and that maps a sub 2 sub and 1 to 1 is a homomorphism, showing that L sub 2 sub is not 0,1 simple. External links mathworld urlname 01 SimpleLattice title 0,1 Simple Lattice author Matt Insall Category Lattice theory algebra stub pt 0,1 simples lattice ... more details
, c and angles between the sides given by , , ref The lattice constant or lattice parameter refers to the constant distance between unit cell s in a crystal lattice . Lattices in three dimensions generally have three lattice constants, referred to as a , b , and c . However, in the special case .... A group of lattice constants could be referred to as lattice parameters . However, the full set of lattice parameters consist of the three lattice constants and the three angles between them. For example the lattice constant for a common carbon diamond is a 3.57 at 300 K . The structure is equilateral although its actual shape can not be determined from only the lattice constant. Furthermore, in real applications, typically the average lattice constant is given. As lattice constants have the dimension of length, their SI unit is the meter . Lattice constants are typically on the order of several angstrom s i.e. tenths of a nanometre . Lattice constants can be determined using techniques such as X ray diffraction or with an atomic force microscope . In epitaxy epitaxial growth , the lattice constant is a measure of the structural compatibility between different materials. Lattice ... epitaxy epitaxial growth of thicker layers without defects. Lattice matching Matching of lattice structures ... , and aluminium arsenide have almost equal lattice constants, making it possible to grow almost arbitrarily thick layers of one on the other one. Lattice grading Typically, films of different materials grown on the previous film or substrate are chosen to match the lattice constant of the prior layer to minimize film stress. An alternative method is to grade the lattice constant from one value ... layer will have a ratio to match the underlying lattice and the alloy at the end of the layer growth will match the desired final lattice for the following layer to be deposited. The rate of change ... more details
Lattice model may refer to Lattice model physics , a physical model that is defined on a periodic function periodic structure with a repeating elemental unit pattern, as opposed to the continuum theory continuum of space or spacetime. Lattice model finance , a discrete time model of the varying price over time of the underlying financial instrument, during the life of the instrument. Lattice model mathematics , a regular tiling of a space by a primitive cell. Hidden Markov Models Lattice model computational biology , equivalent to Markov chains formulated e.g. with the help of Hidden Markov Models . Lattice model biophysics , a class of Ernst Ising Ising type models for the description of biomacromolecules, their transformations and binding in gene regulation and signal transduction . References references Long comment to avoid being listed on short pages disambig ... more details
In physics , the Multiplicative inverse reciprocal lattice of a lattice usually a Bravais lattice is the lattice in which the Fourier transform of the spatial wavefunction of the original lattice or direct lattice is represented. This space is also known as momentum space or less commonly k space , due ... lattice of a reciprocal lattice is the original or direct lattice . Mathematical description Consider a set of points R constituting a Bravais lattice, and a plane wave defined by math e i mathbf ... has the same periodic function periodicity as the Bravais lattice, then it satisfies the equation math ... mathbf R 1 math Mathematically, we can describe the reciprocal lattice as the set of all vector geometric vector s K that satisfy the above identity for all lattice point position vectors R . This reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice, which reveals the Pontryagin duality of their respective vector spaces. For an infinite three dimensional lattice, defined by its primitive cell primitive vector s math mathbf a 1 , mathbf a 2 , mathbf a 3 math , its reciprocal lattice can be determined by generating its three reciprocal ... s definition, comes from defining the reciprocal lattice to be math e 2 pi i mathbf K cdot mathbf R 1 math which changes the definitions of the reciprocal lattice vectors to be math mathbf b 1 frac ... manipulations, and expresses reciprocal lattice dimensions in units of spatial frequency . It is a matter of taste which definition of the lattice is used, as long as the two are not mixed. Each point hkl in the reciprocal lattice corresponds to a set of lattice planes hkl in the real space lattice. The direction of the reciprocal lattice vector corresponds to the normal mathematics normal to the real space planes. The magnitude of the reciprocal lattice vector is given in reciprocal length and is equal to the reciprocal of the interplanar spacing of the real space planes. The reciprocal lattice ... more details
File Equilateral Triangle Lattice.svg thumb right 250px A lattice in the Euclidean plane . In mathematics , especially in geometry and group theory , a lattice in R sup n sup is a discrete subgroup of R sup n sup which linear span spans the real number real vector space R sup n sup . Every lattice in R ... linear combination s with integer coefficients. A lattice may be viewed as a regular tiling of a space ... of several lattice problems , and are used in various ways in the physical sciences. For instance, in materials science and solid state physics , a lattice is a synonym for the frame work of a crystalline ... positions in a crystal . More generally, lattice model physics lattice models are studied in physics , often by the techniques of computational physics . Symmetry considerations and examples A lattice is the symmetry group of discrete translational symmetry in n directions. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. A lattice in the sense of a 3 dimension al array of regularly spaced points coinciding with e.g. ... translational symmetry, is a translate of the translation lattice a coset , which need not contain the origin, and therefore need not be a lattice in the previous sense. A simple example of a lattice in R sup n sup is the subgroup Z sup n sup . A more complicated example is the Leech lattice , which is a lattice in R sup 24 sup . The period lattice in R sup 2 sup is central to the study of elliptic ... of abelian function s. Dividing space according to a lattice A typical lattice in R sup n ... , ..., v sub n sub is a basis for R sup n sup . Different bases can generate the same lattice, but the absolute ... by d . If one thinks of a lattice as dividing the whole of R sup n sup into equal polyhedron polyhedra copies of an n dimensional parallelepiped , known as the fundamental region of the lattice ... called the covolume of the lattice. Lattice points in convex sets Minkowski s theorem relates the number ... more details
element algebraic semimodular bounded lattice is called a matroid lattice because such lattices are equivalent to simple matroid s. An atomistic semimodular bounded lattice of finite length is called a geometric lattice and corresponds to a matroid of finite rank. ref These definitions follow Stern 1999 . Some authors use the term geometric lattice for the more general matroid lattices. But most authors ... dual notion is that of a lower semimodular lattice . A finite lattice is modular lattice modular if and only if it is both upper and lower semimodular. A finite lattice, or more generally a lattice ... if it is modular lattice Modular pairs and related notions M symmetric . Some authors refer to M ... A lattice is sometimes called weakly semimodular if it satisfies the following condition due to Garrett ... b and b a b . Every semimodular lattice is weakly ... c . Every lattice satisfying Mac Lane s condition is semimodular. The converse is true for lattices ... upper continuous lattice satisfying Mac Lane s condition is M symmetric. Notes Reflist References springer title Semi modular lattice id s s084240 last Fofanova first T. S. . The article is about ... 7972 title Geometric lattice . The article is about matroid lattices. planetmath reference id 7286 title Semimodular lattice . See also Antimatroid DEFAULTSORT Semimodular Lattice Category Lattice theory ... more details
Cubic lattice may refer to Cubic crystal system Cubic honeycomb Integer lattice Short pages monitor This long comment was added to the page to prevent it being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Longcomment. Please do not remove the monitor template without removing the comment as well. disambig ... more details
mergefrom Ionized impurity scattering date September 2011 mergeto Phonon scattering date September 2011 Lattice scattering is the scattering of ions by interaction with atoms in a lattice. ref cite book author Bube, Richard H. title Electrons in Solids an introductory survey pages 176 177 publisher Academic Press 1992 isbn 0121385531 ref This effect can be qualitatively understood as phonons colliding with charge carriers. In the current Classical and quantum conductivity quantum mechanical picture of conductivity the ease with which electrons traverse a crystal lattice is dependent on the near perfectly regular spacing of ions in that lattice. Only when a lattice contains perfectly regular spacing can the ion lattice interaction scattering lead to almost transparent behavior of the lattice. ref cite book author Kip, Arthur F. title Fundamentals of Electricity and Magnetism pages 211 213 publisher McGraw Hill isbn 070347808 ref In the quantum understanding, an electron is viewed as a wave traveling through a medium. When the wavelength of the electrons is larger than the crystal spacing, the electrons will propagate freely throughout the metal without collision. References reflist See also Ionized impurity scattering External links cite book author Kundstrom, Mark title Fundamentals of carrier transport publisher Cambridge University Press 2000 isbn 0521631343 DEFAULTSORT Lattice Scattering Category Quantum mechanics ... more details
Quantum field theory Lattice QCD is a well established non Perturbation theory quantum mechanics perturbative approach to solving the quantum chromodynamics QCD theory of quark s and gluon s. It is a lattice gauge theory formulated on a grid or lattice group lattice of points in space and time. Analytic ... introduces a momentum cut off at the order 1 a , where a is the lattice spacing, which regularizes the theory. As a result lattice QCD is mathematically well defined. Most importantly, lattice .... In lattice QCD, fields representing quarks are defined at lattice sites which leads to fermion ... approaches continuum QCD as the spacing between lattice sites is reduced to zero. Because the computational cost of numerical simulations can increase dramatically as the lattice spacing decreases, results are often extrapolation extrapolated to a 0 by repeated calculations at different lattice spacings a that are large enough to be tractable. Numerical lattice QCD calculations using Monte ... lattice QCD calculations, dynamical fermions are now standard. ref cite journal author A. Bazavov ... Callaway David J. E. Callaway and Aneesur Rahman title Microcanonical Ensemble Formulation of Lattice .... Callaway and Aneesur Rahman title Lattice gauge theory in the microcanonical ensemble journal Physical .....28.1506C ref At present, lattice QCD is primarily applicable at low densities where the numerical sign problem does not interfere with calculations. Lattice QCD predicts that confined quarks will become ... from the sign problem when applied to the case of QCD with gauge group SU 2 QC sub 2 sub D . Lattice ... bibcode 2008Sci...322.1224D ref Lattice QCD has also been used as a benchmark for high performance ... of space time . In lattice Monte Carlo simulations the aim is to calculate correlation function ... gauge configurations to calculate hadron ic propagator s and correlation functions. Fermions on the latticeLattice QCD is a way to solve the theory exactly from first principles, without any assumptions ... more details
File Fano plane Hasse diagram.svg thumb Hasse diagram of a complemented lattice br A point and a line ... of order theory , a complemented lattice is a lattice order bounded lattice in which every element ... b 0. A relatively complemented lattice is a lattice such that every interval mathematics ... lattice is an involution mathematics involution which is order reversing and maps each element to a complement. An orthocomplemented lattice satisfying a weak form of the modular lattice modular law is called an orthomodular lattice . In distributive lattices, complements are unique. Every complemented distributive lattice has a unique orthocomplementation and is in fact a Boolean algebra structure Boolean algebra . Definition and basic properties A complemented lattice is a bounded lattice with least element 0 and greatest element 1 , in which every element a has a complement , i.e. ... an element may have more than one complement. However, in a bounded distributive lattice every element ... 9.3 p. 25. ref A lattice in which every element has exactly one complement is called a uniquely complemented lattice . A lattice with the property that every interval is complemented is called a relatively complemented lattice . In other words, a relatively complemented lattice is characterized ... of a relative to the interval. A distributive lattice is complemented if and only if it is bounded and relatively ... An orthocomplementation on a bounded lattice is a function that maps each element a to an orthocomplement ... a sup sup . An orthocomplemented lattice or ortholattice is a bounded lattice which is equipped with an orthocomplementation ..., distributive. ref http unapologetic.wordpress.com 2009 05 07 orthogonal complements and the lattice of subspaces The Unapologetic Mathematician Orthogonal Complements and the Lattice of Subspaces . ref gallery Caption Some complemented lattices Image Smallest nonmodular lattice 1.svg In the pentagon lattice N sub 5 sub , the node on the right hand side has two complements. Image Diamond ... more details
In mathematics, a lattice word or lattice permutation is a sequence of integer s such that in every initial part of the sequence any number i occurs at least as often as the number i 1. A reverse lattice word , or Yamanouchi word , is a sequence whose reversal is a lattice word. References Citation last1 Fulton first1 William author1 link William Fulton mathematician title Young tableaux publisher Cambridge University Press series London Mathematical Society Student Texts isbn 978 0 521 56144 0 978 0 521 56724 4 id MathSciNet id 1464693 year 1997 volume 35 Category Algebraic combinatorics Category Combinatorics on words fr Mot de Yamanouchi ... more details
Image Tile 3,6.svg thumb Triangular tiling . The vertices form a hexagonal lattice with horizontal rows ... lattice or equilateral triangular lattice is one of the five 2D Lattice group lattice types ... to point into two oblique directions. Two orientations of an image of the lattice are by far the most common. They can conveniently be referred to as hexagonal lattice with horizontal rows like in the figure below , with triangles pointing up and down, and hexagonal lattice with vertical rows , with triangles pointing left and right. They differ by an angle of 30 . The hexagonal lattice with horizontal ... as high. Its symmetry category is wallpaper group p6m. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. Image Tile 6,3.svg ... lattice with vertical rows for each color the centers of the hexagons of that color form a hexagonal lattice with horizontal rows which is in linear scale 3 times as large. The vertices of all hexagons together with their centers form a hexagonal lattice with horizontal rows which is in linear scale 3 times as small as the first mentioned lattice. For an image of a Honeycomb geometry honeycomb ... of 90 , or equivalently 30 . A honeycomb structure is in two ways related to a hexagonal lattice the centers of the hexagons of a honeycomb form a hexagonal lattice, with the rows oriented the same the vertices of a honeycomb, together with their centers, form a hexagonal lattice, rotated by 30 or equivalently 90 , and scaled by a factor math frac 1 3 sqrt 3 math , relative to the other lattice The ratio ... is 3, the reciprocal of the square of the scale factor. The term honeycomb lattice could mean a corresponding hexagonal lattice, or a structure which is not a lattice in the Group mathematics group sense, but e.g. one in the sense of a lattice model mathematics lattice model . A set of points ... of them, the sides of the hexagons may be shown depending on application they may be called lattice ... more details
Image Dih4 subgroups.svg thumb 360px The lattice of subgroups of the dihedral group Dihedral group of order 8 Dih sub 4 sub , represented as groups of rotations and reflections of a plane figure. The lattice is shown as a Hasse diagram . In mathematics , the lattice of subgroups of a Group mathematics group math G math is the Lattice order lattice whose elements are the subgroup s of math G math , with the partial order Relation mathematics relation being set inclusion . In this lattice, the join ... , and the meet of two subgroups is their intersection set theory intersection . Lattice theoretic information about the lattice of subgroups can sometimes be used to infer information about the original ... if and only if its lattice of subgroups is Distributive lattice distributive . Lattice theoretic ... elements. The lattice formed by these ten subgroups is shown in the illustration. Characteristic lattices ... Nilpotent normal subgroup s form a lattice, which is part of the content of Fitting s theorem . In general ... group Central subgroups form a lattice. However, neither finite subgroups nor torsion subgroups form a lattice for instance, the free product math mathbf Z 2 mathbf Z mathbf Z 2 mathbf Z math is generated ... lemma , an isomorphism between certain quotients in the lattice of subgroups Complemented group , a group with a complemented lattice of subgroups Lattice theorem , a Galois connection between the lattice of subgroups of a group and of its quotient Example v Symmetric group S4 Lattice of subgroups Lattice of subgroups of the symmetric group S4 References cite journal title The significance ... by Ralph Freese in Bull. AMS 33 4 487 492. cite journal title On the lattice of subgroups of finite ... Structure of a Group and the Structure of its Lattice of Subgroups publisher Springer Verlag location Berlin year 1956 cite journal author Yakovlev, B. V. title Conditions under which a lattice is isomorphic to a lattice of subgroups of a group journal Algebra and Logic volume 13 issue 6 year 1974 doi ... more details
Orphan date December 2010 No footnotes date September 2011 Infobox disease Name Lattice degeneration Image Caption DiseasesDB ICD10 ICD9 ICDO OMIM MedlinePlus eMedicineSubj oph eMedicineTopic 397 MeshID Lattice degeneration is a disease of the human eye eye where peripheral retina becomes atrophic in a lattice pattern and may develop tears breaks holes, which may further progress to retinal detachment . It is an important cause of retinal detachment in young myopic individuals. The cause is not known but pathology reveals vascular insufficiency resulting in ischemia and fibrosis . It is present in about 8 of the general population and occurs in about 40 of eyes with retinal detachment. Similar lesions are seen in patients with Marfan syndrome , Stickler syndrome and Ehlers Danlos syndrome , all of which are associated with an increased risk of retinal detachment . Signs Typical lattice consists of sharply demarcated spindle shaped areas of retinal thinning usually located between the equator of the retina and the posterior border of the vitreous base. This is more frequently located in the temporal half of the retina and is seen more superiorly than inferiorly. Atypical lattice is characterised ... around a hole or tear associated with lattice degeneration in an eye at risk of developing ... in patients with asymptomatic lattice degeneration. So prophylactic treatment for asymptomatic lattice ... risks of retinal detachment in symptomatic lattice degeneration. However there are documented ... with laser. Prognosis No complications are encountered in most patients with lattice degeneration ... off retinal detachment in patients with asymptomatic lattice degeneration. Partial or complete ... or cure for lattice degeneration. It is not known if research is being carried out to find a cure for lattice degeneration. References reflist External links http eyewiki.aao.org Lattice Degeneration http cms.revoptom.com handbook sect5e.htm DEFAULTSORT Lattice Degeneration Category Diseases of the eye ... more details
Unreferenced date September 2009 Infobox Software name Lattice C logo File lattice logo.png screenshot caption collapsible author developer Steve Krueger et al. released 1982 latest release version latest release date latest preview version latest preview date frequently updated programming language C, Assembler operating system DOS, OS 2, MVS, VMS, UNIX, AmigaDOS, Sinclair QDOS platform size language status genre Compiler license Closed source website http www.sas.com products sasc Lattice C according to its author, Lattice, Incorporated was the first C programming language C compiler for MS DOS on the International Business Machines IBM personal computer PC , in 1982 . It was ported to many other platforms, such as Mainframe computer mainframes MVS , minicomputer s Virtual Memory System VMS , workstation s Unix UNIX , OS 2 , the Commodore International Commodore Amiga , Atari ST and the Sinclair Research Sinclair Sinclair QL QL . The compiler was subsequently repackaged by Microsoft under a distribution agreement as Microsoft C. Lattice was purchased by SAS Institute in 1987 . After this, support for other platforms dwindled until compiler development ceased for all platforms except IBM mainframes. The product is still available in versions that run on other platforms, but these are Cross compilation cross compilers that only produce mainframe code. Some of the early 1982 commercial software for the IBM PC was ported from CP M where it was written for the BDS C subset of the C language to MS DOS using Lattice C including Perfect Writer , PerfectCalc , PerfectSpeller and PerfectFiler . This suite was bundled with the Seequa Chameleon and Columbia Data Products . LMK, make software make tool LSE, screen editor TMN, text management utilities External links http web.archive.org ... 24 Category C compilers Category Amiga development software Category Atari ST software de Lattice C it Lattice C ru Lattice C ... more details
Image Smallest nonmodular lattice 1.svg thumb right Hasse diagram of N sub 5 sub , the smallest non modular lattice. In the branch of mathematics called order theory , a modular lattice is a lattice order lattice that satisfies the following self dual condition Modular law x b implies x ... and called join and meet respectively are the operations of the lattice ... lattice. Every distributive lattice distributive lattice is modular. In a not necessarily modular lattice, there may still be elements b for which the modular law holds in connection with arbitrary ... generalizations of modularity related to this notion and to semimodular lattice semimodularity ... law that connects the two lattice operations similarly to the way in which the associative law ... x b is clearly necessary, since it follows from x a b x a b . Image Smallest nonmodular lattice ... x a b in every lattice. Therefore the modular law can also be stated as Modular law variant x b implies ... of modular lattices are again modular. The smallest non modular lattice is the pentagon lattice ... to x or to b . For this lattice x a b x 0 x b 1 b x a b holds, contradicting the modular law. Every non modular lattice contains a copy of N sub 5 sub as a sublattice. Modular lattices are sometimes ... theorem For any two elements a , b of a modular lattice, one can consider the intervals a b ... In a modular lattice, the maps and indicated by the arrows are mutually inverse isomorphisms. Image Not a modular pair.svg Failure of the diamond isomorphism theorem in a non modular lattice .... In a modular lattice, however, equality holds. Since the dual of a modular lattice is again ... lattices. A lattice is modular if and only if the diamond isomorphism theorem holds for every pair ... theorem in algebra, and it is a generalization of the lattice theorem . Modular pairs and related notions Image Centred hexagon lattice D2.svg thumb right The centred hexagon lattice S sub 7 sub ... more details
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