In mathematics , the term latticegroup is used for two distinct notions a latticegrouplattice , 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
types as given by the crystallographic restriction theorem . Below, the wallpaper group of the lattice ... have more, but may have less symmetry than the lattice itself. If the symmetry group of a pattern .... Equivalence in the sense of generating the same lattice is represented by the modular group math T z ... in R sup n sup is a free abelian group of rank of an abelian group rank n every lattice in C sup n sup is a free abelian group of rank 2 n . In Lie groups Main Lattice discrete subgroup More generally, a lattice in a Lie group G is a discrete subgroup , such that the Quotient group quotient G ... of the modular group in SL2 R SL sub 2 sub R , which is a lattice but where the quotient isn t compact ... LatticeGroup Category Lattice points Category Discrete groups Category Lie groups Category Analytic ...See also Lattice order Lattice mathematics Unreferenced date December 2009 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 sup n sup can be generated ... coefficients. A lattice may be viewed as a regular tiling of a space by a primitive cell . Lattices ... s, number theory and group theory. They also arise in applied mathematics in connection with coding theory , in cryptography because of conjectured computational hardness of several lattice problems ... state physics , a lattice is a synonym for the frame work of a crystalline structure , a 3 dimensional ... . 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 ... more details
Infobox Defunct Company company name LatticeGroup plc company logo Image Latticelogo.PNG slogan fate Merged with National Grid plc successor National Grid plc foundation 2000 defunct 2002 location London , United Kingdom UK industry Gas Gas transmission key people Sir John Parker Chairman br Philip Nolan CEO products num employees 16,000 ref name caps http www.guardian.co.uk business 2001 oct 25 utilities.utilities Lattice accepts price caps but warns of job cuts Guardian, 25 October 2001 ref parent subsid LatticeGroup plc was a leading United Kingdom British gas gas transmission business. It was listed on the London Stock Exchange and was a constituent of the FTSE 100 Index . History The Company was established in 2000 when BG Group demerged its United Kingdom UK gas gas transmission business, formerly known as Transco , and named it LatticeGroup . ref name history http www.telegraph.co.uk finance 2760662 Grid and Lattice form utility supergroup.html Grid and Lattice form utility supergroup The Telegraph, 22 April 2002 ref In October 2002 LatticeGroup merged with National Grid plc to form National Grid Transco although in 2005 the merged entity reverted to the name National Grid plc . ref name history Operations As well as the United Kingdom UK gas transmission network, LatticeGroup owned a Telecommunication telecoms business known as 186k . ref name caps References reflist Category Companies established in 2000 Category Companies formerly listed on the London Stock Exchange Category Companies based in London Category Oil and gas companies of the United Kingdom UK company stub ... more details
wiktionarypar latticeLattice may refer to Art and design Latticework an ornamental and or structural criss crossed framework, an arrangement of crossing laths or other thin strips of material Lattice pastry Mathematics Lattice mathematics , any of the following Lattice order , a partially ordered set with unique least upper bounds and greatest lower bounds Lattice discrete subgroup , a discrete subgroup of a topological group with finite covolume Latticegroup , a repeating arrangement of points Lattice graph , a graph that can be drawn within a repeating arrangement of points Bethe lattice , a regular infinite tree structure Science A crystal structure fitting a lattice arrangement Lattice model physics , a model defined not on a continuum, but on a latticeLattice model finance , a method for evaluating stock options that divides time into discrete intervals Companies Lattice Semiconductor , a US based integrated circuit manufacturer Lattice, Incorporated, a software company and makers of Lattice C LatticeGroup , a former British gas transmission business See also Grid disambiguation Mesh disambiguation Trellis disambiguation disambig ar da Gitter de Gitter Begriffskl rung es Ret culo fr Lattice it Reticolo he pl Krata pt Ret culo ru sk Mrie ka sv Gitter ... more details
Unreferenced date December 2009 In mathematics , the n dimensional integer lattice or cubic lattice , denoted Z sup n sup , is the latticegrouplattice 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 focusses 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
See also Lattice disambiguation In mathematics , the term lattice can mean A partially ordered set poset in which any two elements have a supremum and an infimum &mdash see lattice order . The Hasse diagram s of these posets look in some simple cases like the lattices of ordinary language. These lattices can also be defined as algebra ic structures. A discrete subgroup of R sup n sup that linear span spans it as a real vector space , or its translate&mdash see latticegroup . The elements of a lattice are regularly spaced, reminiscent of the intersection points of a lath lattice. Special types include unimodular lattice s, the Leech lattice , and the Niemeier lattice . See also translational symmetry . A lattice graph is a planar graph that can be drawn with regularly spaced points in the plane. More generally, a lattice in a locally compact space locally compact topological group is a discrete subgroup of finite covolume&mdash see lattice discrete subgroup . The lattice method or form also known as the jalousie method is an algorithm used to teach basic multiplication to elementary grades. mathdab he ... 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 Latticegrouplattice 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 group s its symmetry group is known symbolically as Wallpaper groupGroup p4m p4m . Two orientations 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 there are three possibilities None. This is wallpaper group p4. In four directions. This is wallpaper group p4m. In two perpendicular directions. This is wallpaper group p4g. The points of intersection ... 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 of two subgroups is the subgroup generating set of a group generated by their union set theory union , 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 group, an idea that goes back to the work of harvs first ystein last Ore authorlink ystein Ore year 1937 year2 1938 txt . For instance, as Ore proved, a group is Locally cyclic group locally cyclic if and only if its lattice of subgroups is Distributive lattice distributive . Lattice theoretic characterizations of this type also exist for solvable group s and perfect group s harv Suzuki 1951 ... group Central subgroups form a lattice. However, neither finite subgroups nor torsion subgroups form ... 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 ... title Structure of a Group and the Structure of its Lattice of Subgroups publisher Springer Verlag location ... is isomorphic to a lattice of subgroups of a group journal Algebra and Logic volume 13 issue 6 year ... PlanetMath entry on lattice of subgroups Category Lattice theory Category Group ... more details
. 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 ... , 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 ... 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
In mathematics , the Leech lattice is a particular latticegrouplattice sub 24 sub in 24 dimensional Euclidean space E sup 24 sup . History The Leech lattice was originally discovered by Ernst Witt ... group in the process the Conway group , Co sub 1 sub . Characterization The Leech lattice sub 24 sub is the unique lattice in E sup 24 sup with the following list of properties It is unimodular lattice ... having such a high rotational symmetry group, the Leech lattice does not possess any lines of reflection symmetry. In other words, the Leech lattice is chiral . The automorphism group was first ... . Hence the full automorphism group of the Leech lattice has order 8292375 × 4096 × 244823040 ... the deep holes of the Leech lattice. There are 23 orbits of them under the automorphism group ..., as the Voronoi cell s have a low second moment . See also Conway group Sphere packing E8 lattice E .... Many of the cross sections of the Leech lattice were known before then, including the Coxeter&ndash Todd lattice and Barnes&ndash Wall lattice , in 12 and 16 dimensions, respectively. Shortly after, in 1944, O Connor and Pall discovered a related lattice in 24 dimensions, the odd Leech lattice. The actual Leech lattice was later re discovered in 1965 by John Leech mathematician John Leech . The automorphism group was first ascertained by John Horton Conway after a mere 12 hours and 20 minutes ... lattice , hexagonal tiling and E8 lattice , respectively. It has no root system and in fact is the first unimodular lattice with no roots vectors of norm less than 4 , and therefore has a centre ... 12 12 math , one can derive its absolute density. Conway showed that the Leech lattice is isometric to the set of simple roots or the Dynkin diagram of the reflection group of the 26 dimensional even Lorentzian unimodular lattice II sub 25,1 sub . By comparison, the Dynkin diagrams of II sub 9,1 sub and II sub 17,1 sub are finite. It is a Type II self dual lattice. Constructions The Leech lattice ... more details
two versions in mathematics of the abstract dual lattice concept, for a given latticegrouplattice ...In crystallography , the Multiplicative inverse reciprocal lattice of a Bravais lattice is the set of all vector geometric vector s K such that math e i mathbf K cdot mathbf R 1 math 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. For an infinite three dimensional lattice, defined by its ... lattice can be determined by generating its three reciprocal primitive vectors, through the formulas ... 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 ... mathematical 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 ... lattice plays a fundamental role in most analytic studies of periodic structures, particularly ... and diffracted X rays of a crystal is a reciprocal lattice vector. The diffraction pattern of a crystal can be used to determine the reciprocal vectors of the lattice. Using this process, one can infer ... lattice. Reciprocal lattices of various crystals Reciprocal lattices for the Cubic crystal system cubic crystal system are as follows. Simple cubic lattice The simple cubic Bravais lattice , with cubic primitive cell of side math a math , has for its reciprocal a simple cubic lattice with a cubic ... matrix math in the crystallographer s definition . The cubic lattice is therefore said to be self dual ... more details
Quantum field theory Lattice QCD is a well established non 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 latticegrouplattice of points in space and time. Analytic or perturbative solutions in low energy ... when applied to the case of QCD with gauge group SU 2 QC sub 2 sub D . Lattice QCD has already made ... 1 a , where a is the lattice spacing, which regularizes the theory. As a result lattice QCD is mathematically well defined. Most importantly, lattice QCD provides a framework for investigation ..., which are intractable by means of analytic field theories. In lattice QCD, fields representing quarks are defined at lattice sites which leads to fermion doubling , while the gluon fields are defined ... 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 Carlo method s can be extremely computationally ... as non dynamic frozen variables. While this was common in early lattice QCD calculations, dynamical ... Ensemble Formulation of Lattice Gauge Theory journal Physical Review Letters volume 49 year 1982 ... J. E. Callaway and Aneesur Rahman title Lattice gauge theory in the microcanonical ensemble journal ..., 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 released to quark ... Masses year 2008 doi 10.1126 science.1163233 pmid 19023076 arxiv 0906.3599 ref Lattice QCD has ... imposes the use of Euclidean space Euclidean time , by a Wick rotation of space time . In lattice ... hadron ic propagator s and correlation functions. Fermions on the latticeLattice QCD is a way ... more details
triangle at the upper left. The hexagonal lattice or equilateral triangular lattice is one of the five 2D Latticegrouplattice types. Three nearby points form an equilateral triangle. In images four ... 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 . A set ... lattice with horizontal rows, with triangles pointing up and down. There are three ways in which ... 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 rows is a special case of a centered rectangular i.e. ... the rectangles are 3 times as wide 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 .... The centers of the hexagons form a hexagonal 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 ... related to a hexagonal lattice the centers of the hexagons of a honeycomb form a hexagonal lattice ... 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 of the number of vertices and the number of hexagons is 2 ... lattice bonds . With respect to a hexagonal lattice we can distinguish two sets or three directions the directions of the smallest distance between lattice points let us call them the main translation ... more details
in Lie groups Bethe lattices also occur as the discrete group subgroups of certain hyperbolic Lie groups , such as the Fuchsian group s. As such, they are also lattices in the sense of a latticegrouplattice in a Lie group . See also Crystal References reflist H. A. Bethe, Statistical theory of superlattices ...Image Bethe lattice.PNG thumb 225px right A Bethe lattice with coordination number z 3 A Bethe lattice or Cayley tree though the two are not completely equivalent, see below , introduced by Hans Bethe in 1935, is a tree graph theory connected cycle free graph where each node is connected to z neighbours, where z is called the coordination number . It can be seen as a tree like structure emanating from a central node, with all the nodes arranged in shells around the central one. The central node may be called the root or origin of the lattice. The number of nodes in the k th shell is given by math , N k z z 1 k 1 text for k 0. math In some situations the definition is modified to specify that the root node has z &minus 1 neighbours. Due to its distinctive topological structure, the statistical mechanics of lattice model physics lattice models on this graph are often exactly solvable. The solutions are related to the often used Bethe approximation for these systems. Relation to Cayley graphs see Cayley graph The Bethe lattice where each node is joined to 2 n others is essentially the Cayley graph of a free group on n generators. A presentation of a group G by n generators corresponds to a surjective map from the free group on n generators to the group G, and at the level of Cayley graphs to a map from the Cayley tree to the Cayley graph. This can also be interpreted in algebraic topology as the universal cover of the Cayley graph, which is not in general simply connected . The distinction between a Bethe lattice and a Cayley tree is that the former is the thermodynamic ... Press year 1982 isbn 0 12 083182 1 External links Category Lattice models Category Trees graph theory ... more details
In mathematics , a unimodular lattice is a latticegrouplattice of determinant 1 or &minus 1. The E8 mathematics E sub 8 sub lattice and the Leech lattice are two famous examples. Definitions A lattice is a free abelian group of finite rank of an abelian group rank with an integral symmetric bilinear form , . A lattice is even if a , a is always even. The dimension of a lattice is the same as its rank as a Z module mathematics module . A lattice is positive definite if a , a is always positive for non zero a . The discriminant of a lattice is the determinant of the matrix with entries a sub i sub , a sub j sub , where the elements a sub i sub form a basis for the lattice. A lattice is unimodular ... bilinear form. The lattice is positive definite , Lorentzian , and so on if its vector space is. The signature of a lattice is the signature quadratic form signature of the form on the vector space. Examples The three most important examples of unimodular lattices are The lattice Z , in one dimension. The E8 lattice E sub 8 sub lattice , an even 8 dimensional lattice, The Leech lattice , the 24 dimensional even unimodular lattice with no roots. Classification For indefinite lattices, the classification ... &minus ... &minus a sub m n sub b sub m n sub . In R sup m,n sup there is one odd unimodular lattice ... is even. The lattice II sub 8,0 sub is the same as the E sub 8 sub lattice. Positive definite unimodular ... by the Leech lattice, and this accounts for their unusually good behavior in these dimensions ... up to 25 can be naturally identified with a configuration of vectors in the Leech lattice. The wild ... controlled by the Leech lattice. Even positive definite unimodular lattice exist only in dimensions divisible by 8. There is one in dimension 8 the E sub 8 sub lattice , two in dimension 16 E sub 8 sub sup 2 sup and II sub 16,0 sub , and 24 in dimension 24, called the Niemeier lattice s examples the Leech lattice , II sub 24,0 sub , II sub 16,0 sub II sub 8,0 sub , II sub 8,0 sub sup 3 sup ... 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
Image Tile 3636.svg thumb left 2d Kagome lattice Image p12fig1.jpg thumb right Japanese basket showing the kagome pattern A kagome lattice is an arrangement of lines arrangement of lath s composed of interlaced triangles such that each point where two laths cross has four neighboring points. Although called a lattice, it is more closely related to the trihexagonal tiling than to a latticegroup mathematical lattice . Its name derives from two separate Japanese language Japanese words, meaning the pattern of holes me , literally eyes in a basket kago . There has been some debate about the proper way to write this term. Lately, it is understood that it is not strictly a Japanese word in itself and it is not the name of a person. Hence, it is proper for kagome to be written in roman font, not italic foreign words , with a lower case k, and without an unnecessary pronunciation acute accent on the last e. Some Mineral minerals , namely Jarosite jarosites and herbertsmithite , contain layers with kagome lattice arrangement of Atom atoms in their crystal structure . These minerals display novel physical properties connected with Geometrically frustrated magnet geometrically frustrated magnetism . The term is much in use nowadays in the scientific literature, especially by theorists studying the magnetic properties of a theoretical kagome lattice in two or three dimensions. The term was first coined by Japanese physicist K ji Fushimi, who was working with Ichir Sh ji. The first paper ref cite web title I. Sy zi, Prog. Theor. Phys. 6, 306 1951 . url http ptp.ipap.jp link?PTP 6 306 ref on the subject appeared in 1951. There is an article ref cite web title Physics Today article on the word kagome url http scitation.aip.org journals doc PHTOAD ft vol 56 iss 2 12 1.shtml ref in Physics Today about all this. See also Percolation threshold Star of David Footnotes reflist Category Japanese crafts Category Crystallography Category Condensed matter physics condensedmatter stub ja ... more details
Image Lattice reduction.svg thumb right 300px Lattice reduction in two dimensions the black vectors are the given basis for the lattice represented by blue dots , the red vectors are the reduced basis In mathematics, the goal of lattice basis reduction is given an integer latticegrouplattice basis as input, to find a basis linear algebra basis with short, nearly orthogonal vectors. This is realized using different algorithms, whose running time is usually at least exponential in the dimension of the lattice. Nearly Orthogonal One measure of nearly orthogonal is the orthogonality defect . This compares the product of lengths of the basis vectors with the volume of the parallelepiped they define. For perfectly orthogonal basis vectors, these quantities would be the same. Any particular basis of math n math vectors may be represented by a Matrix mathematics matrix math B math , whose columns are the basis vectors math b i, i 1, ldots, n math . In the fully dimensional case where the number ... det B T B math . For a given lattice math Lambda math , this volume is the same to within a sign for any basis, and hence is referred to as the determinant of the lattice math det Lambda math or lattice ... if and only if the basis is orthogonal. If the lattice reduction problem is defined as finding ... by adding or subtracting an integer multiple of the smaller vector. Applications Lattice reduction ... problem, algorithms such as the Lenstra Lenstra Lov sz lattice basis reduction algorithm LLL algorithm .... Lenstra Lenstra Lov sz lattice basis reduction algorithm LLL is widely used in the cryptanalysis ... 538 548 ref . Algorithms The following algorithms reduce lattice bases. They can be compared in terms ... lattice. If there are public implementations of these algorithms this should also be noted here. Year Algorithm Name Implementation 1982 Lenstra Lenstra Lov sz lattice basis reduction algorithm LLL ... Theory of cryptography Category Computational number theory Category Lattice points Category Linear ... more details
Translational symmetry Latticegroup Crystal system Classification of lattices classification of lattices ...In geometry and crystallography , a Bravais lattice , studied by harvs txt first Auguste last Bravais ... i sub are known as the primitive vectors which lie in different planes and span the lattice. For any choice of position vector R , the lattice looks exactly the same. A crystal is made up of a periodic arrangement of one or more atoms the basis repeated at each lattice point. Consequently, the crystal looks the same when viewed from any of the lattice points. Two Bravais lattices are often considered ... dimensional space there is just one type of Bravais lattice. In two dimensions, there are five Bravais ... The 14 Bravais lattices in 3 dimensions are arrived at by combining one of the seven lattice system s or axial systems with one of the lattice centerings. Each Bravais lattice refers to a distinct lattice type. The lattice centerings are Primitive centering P lattice points on the cell corners only Body centered I one additional lattice point at the center of the cell Face centered F one additional lattice point at center of each of the faces of the cell Base centered A, B or C one additional lattice point at the center of each of one pair of the cell faces. Not all combinations of the crystal systems and lattice centerings are needed to describe the possible lattices. There are in total 7 ... to each other. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different ... in the table below. align left border 1 style margin left 1em The 7 lattice systems colspan 4 ..., body centered rowspan 2 align center rhombohedral lattice system rhombohedral br align center P Image Rhombohedral.svg 80px Rhombohedral rowspan 2 align center Hexagonal lattice system hexagonal align ... centered The volume of the unit cell can be calculated by evaluating a b c where a , b , and c are the lattice ... margin left 1em Lattice system colspan 4 align center Volume Triclinic math abc sqrt 1 cos 2 alpha ... more details
No footnotes date May 2009 See also LatticegroupLattice mathematics File Lattice of partitions of an order 4 set.svg thumb 360px The name lattice is suggested by the form of the Hasse diagram depicting it. Shown here is the lattice of partition set theory partition s of a four element set 1,2,3,4 , ordered by the relation is a refinement of . In mathematics , a lattice is a partially ordered ... mathematics module , and the lattice of normal subgroup s of a group mathematics group . Semimodularity ... identity mathematics identities . Since the two definitions are equivalent, lattice theory draws ... Heyting algebra Heyting and Boolean algebra structure Boolean algebra s. These lattice like structures ... set poset L , is a lattice if it satisfies the following two axioms. Existence of binary joins ... of a lattice has a join supremum and a meet infimum . With additional assumptions, further conclusions ... theoretic approach to lattices. A bounded lattice has a greatest element greatest or maximum and least ... lattice can be converted into a bounded lattice by adding a greatest and least element, and every non empty finite lattice is bounded, by taking the join resp., meet of all elements, denoted by math ... A a 1, ldots,a n math . A poset is a bounded lattice if and only if every finite set of elements including ... math , on L is a lattice if the following axiomatic identities hold for all elements a, b, c of L ..., the only axioms above in which both meet and join appear, distinguish a lattice from a random pair ... semilattice is the Duality order theory dual of the other. A bounded lattice is an algebraic structure ... is a lattice, 0 the lattice s bottom is the identity element for the join operation math lor math , and 1 the lattice s top is the identity element for the meet operation math land math . See semilattice for further details. Lattices have some connections to the family of magma algebra group like algebraic structures . Because meet and join both commute and associate, a lattice can be viewed as consisting ... more details
of Young s lattice, but also rotational symmetry the rotation group of order n ...File Young s lattice.svg thumb 300px A Hasse diagram of Young s lattice In mathematics , Young s lattice is a partially ordered set and a lattice order lattice that is formed by all partition number theory ... substitutional analysis developed representation theory of the symmetric group . In Young s theory .... Young s lattice prominently figures in algebraic combinatorics , forming the simplest example of a differential ... base s for affine Lie algebra s. Definition Young s lattice is a partially ordered set Y formed ... The traditional application of Young s lattice is to the description of the irreducible representations ... n 1 sub with partition q if and only if q covers p in Young s lattice. Iterating this procedure ... of n have rank n . This means that given two partitions that are comparable in the lattice, their ranks ... of each intermediate rank. The poset Y is a lattice. The meet and the join of two partitions is given by the intersection and the union of the corresponding Young diagrams. Because it is a lattice ... lattice . If a partition p covering relation covers k elements of Young s lattice for some k ... algebra M bius function of Young s lattice takes values 0, 1. It is given by the formula ..., Young s lattice is depicted in a Hasse diagram with all elements of the same rank shown at the same height above the bottom. File Young5.svg 400px thumb The portion of Young s lattice lying below 1 ... 2002 has shown that a different way of depicting some subsets of Young s lattice shows some unexpected ... only one element immediately below them in Young s lattice. Suter showed that the set of all elements ... have dihedral symmetry the n 1 th dihedral groupgroup action acts faithfully on this set ... of Young s lattice has both bilateral and rotational symmetry, and hence dihedral symmetry. For example ... Ferrers diagrams are 1 1 1 1 2 2 2 3 3 4 The subset of Young s lattice lying below these partitions ... more details
Image tonnetz.jpg thumb right On the neo Riemmanian Tonnetz, pitches are connected by lines if they are separated by minor third , major third , or perfect fifth . Image Equilateral Triangle Lattice.svg thumb right A lattice in the Euclidean plane . In musical tuning , a lattice is a way of modeling the tuning relationships in a just intonation system. It is an array of points in a periodic multidimensional pattern. Each point on the lattice group lattice corresponds to a ratio i.e., a pitch music pitch , or an interval music interval with respect to some other point on the lattice . The lattice can be two , three , or n dimensional, with each dimension corresponding to a different prime number partial music partial ref name Gilmore Gilmore, Bob 2006 . Introduction , p.xviii, Maximum Clarity and Other Writings on Music , edited by Bob Gilmore. Urbana University of Illinois Press. ISBN 0252030982. ref or pitch class chroma . For example Hugo Riemann s Tonnetz 1739 , and Ben Johnston composer Ben Johnston s tuning systems. Adriaan Fokker s Fokker periodicity blocks are used to mathematically relate musical intervals in just intonation to those in equal temperament equal tuning . The limit music limit is the highest prime number partial used in a tuning. Thus Pythagorean tuning, which uses only the perfect fifth 3 2 and octave 2 1 and their multiples exponentiation powers of 2 and 3 , is represented through a two dimensional lattice, while standard 5 limit just intonation, which adds the use of the just major third 5 4 , may be represented through a three dimensional lattice though a twelve note chromatic scale may be represented as a two dimensional 3,5 projection plane within the three dimensional 2,3,5 space needed to map the scale. Octave equivalents would appear on an axis at right angles to the other two, but this arrangement is not really necessary graphically. ref name Gilmore . In other words the circle of fifths on one dimension and a series of major thirds o ... more details
lattice site of point symmetry group C sub 3 sub . The lattice energy defining reaction then reads ...Image Sodium chloride 3D ionic.png thumb right Sodium chloride crystal lattice The lattice energy of an Ionic ... gas eous ion s and as such is invariably exothermic . The concept of lattice energy was initially ... occupy high symmetry crystal lattice sites. In the case of NaCl, the lattice energy is the energy ... ref name Johnson . Some older textbooks define lattice energy as the energy required to convert the ionic compound into gaseous ions which is an endothermic process, and following this definition the lattice energy of NaCl would be 787 kJ mol. The precise value of the lattice energy may not be determined ..., the value of the lattice energy may either be derived theoretically from electrostatics or from a thermodynamic ... and Alfred Land Land proposed that the lattice energy could be derived from the electric potential of the ionic lattice and a repulsive potential energy term. ref name Johnson David Arthur Johnson ... Chemistry 2d Edn. New York Wiley Interscience. ref The Born Land equation gives a reasonable fit to the lattice energy. ref name Johnson class wikitable border 1 Compound Calculated Lattice Energy Experimental Lattice Energy NaCl 756 kJ mol 787 kJ mol LiF 1007 kJ mol 1046 kJ mol CaCl sub 2 sub 2170 kJ mol 2255 kJ mol From the Born Land equation it can be seen that the lattice energy of a compound is dependent on a number of factors as the charges on the ions increase the lattice energy increases becomes more negative , when ions are closer together the lattice energy increases becomes more negative ... constant, has a bond radius of 275 picometers and a lattice energy of 3054 kJ mol, while sodium chloride NaCl has a bond radius of 283 picometers and a lattice energy of 786 kJ mol. Kapustinskii equation The Kapustinskii equation can be used as a simpler way of deriving lattice energies where high ... lattice sites with crystallographic point groups C sub 1 sub , C sub 1 sub sub h sub , C sub ... more details
DISPLAYTITLE E sub 8 sub lattice In mathematics , the E sub 8 sub lattice is a special latticegrouplattice in R sup 8 sup . It can be characterized as the unique positive definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the root lattice of the E8 mathematics E ... or symmetry group of a lattice in R sup n sup is defined as the subgroup of the orthogonal group O n that preserves the lattice. The symmetry group of the E sub 8 sub lattice is the Weyl group Weyl Coxeter ... s in the hyperplanes orthogonal to the 240 roots of the lattice. Its order group theory order is given ... 4 lattice points. The 17520 norm 8 lattice points fall into two classes two orbit group theory orbit s under the action of the E sub 8 sub automorphism group 240 are twice the norm 2 lattice points while ... length squared the square of the ordinary norm mathematics norm . ref of the E sub 8 sub lattice divided ... form can be used to construct a positive definite, even, unimodular lattice of rank 8. The existence .... In 1877 they constructed the corresponding E sub 8 sub lattice explicitly as part of a study of sphere ... doi 10.1007 BF01442667 ref The E sub 8 sub lattice is also called the Gosset lattice after Thorold Gosset who was one of the first to study the geometry of the lattice itself around 1900. ref name ... of n dimensions journal Messenger of Mathematics volume 29 pages 43 48 year 1900 ref Lattice points The E sub 8 sub lattice is a discrete subgroup of R sup 8 sup of full rank i.e. it spans all of R sup ... . math It is not hard to check that the sum of two lattice points is another lattice point, so that sub 8 sub is indeed a subgroup. An alternative description of the E sub 8 sub lattice which is sometimes ... one to the other by changing the signs of any odd number of coordinates. The lattice sub 8 sub is sometimes called the even coordinate system for E sub 8 sub while the lattice sub 8 sub is called .... Properties The E sub 8 sub lattice sub 8 sub can be characterized as the unique lattice in R sup ... more details
Image CherryLatticePastry.jpg 300px right A cherry lattice pastry The criss crossing pattern of strips in this pastry is reminiscent of the lath s in a garden trellis , as well as a Hasse diagram of a lattice order lattice in mathematics . The idea of latticed pastry is used as a lid to many different tarts or pies. See also pastry lattice Category pastry dessert stub ... more details
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