compact topological group with the Haar measure &mu . A discretesubgroup &Gamma is called a lattice ...In Lie theory and related areas of mathematics , a lattice in a locally compact topological group is a discretesubgroup with the property that the quotient space has finite invariant measure . In the special case of subgroups of R sup n sup , this amounts to the usual geometric notion of a lattice group lattice , and both the algebraic structure of lattices and the geometry of the totality of all ... subgroups, and a subgroup &Gamma of G is discrete if &Gamma sub x sub is finite for some and hence, for any vertex x . The subgroup &Gamma is an X lattice if the suitably defined volume of math X Gamma ... group and the volume &mu G &Gamma is finite. The lattice is uniform or cocompact if the quotient space ... of a nonuniform lattice is given by the group SL 2, Z , which is a lattice in the special linear group ... conditions, and by restricting the entries to the integer s Z , one obtains a lattice G Z . Conversely, Grigory Margulis proved that under certain assumptions on G , any lattice in it essentially arises ... as the S arithmetic lattices . The first example is given by the diagonally embedded subgroup math ... . math This is a lattice in the product of algebraic groups over different local fields, both real ... Q or a more general global field over the completions of Q at the places from S . To form the discretesubgroup, instead of matrices with integer entries, one considers matrices with entries in the localization ... assumptions, this construction indeed produces a lattice. The class of S arithmetic lattices is much wider than the class of arithmetic lattices, but they share many common features. Adelic case A lattice ... of K , and is a lattice there. Unlike arithmetic lattices, G K is not finitely generated. Rigidity ... as rigidity . The Mostow rigidity theorem showed that the algebraic structure of a lattice in simple ... a generalization dealing with homomorphisms from a lattice in an algebraic group G into another algebraic ... more details
wiktionarypar 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 Latticediscretesubgroup , a discretesubgroup 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 In 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 Lattice Group , 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 ka ja Lattice pl Krata pt Ret culo ru sk Mrie ka sv Gitter ... more details
In mathematics , the term lattice group is used for two distinct notions a lattice group lattice , a discretesubgroup 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
. Lattices in complex space A lattice in C sup n sup is a discretesubgroup of C sup n sup which ... discretesubgroup More generally, a lattice in a Lie group G is a discretesubgroup , such that the Quotient ... volume 290 DEFAULTSORT Lattice Group Category Lattice points Category Discrete groups Category Lie ...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 discretesubgroup 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 ... 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
of a given latticediscretesubgrouplattice L in an abelian group abelian locally compact space locally ... vector space, and its closed subgroup L dual to L turns out to be a lattice in V . Therefore L ... continuous characters that are equal to one at each point of L . In discrete mathematics, a lattice ...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 function of the original lattice or direct lattice is represented. This space is also known as momentum space or less commonly k space , due to the relationship ... 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 ... the same periodic function periodicity as the Bravais lattice, then it satisfies the equation math ... 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 mathbf ... 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 ... more details
Nilpotent normal subgroup s form a lattice, which is part of the content of Fitting s theorem . In general, for any Fitting class F , both the subnormal subgroup subnormal F subgroups and the normal ...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 ... if and only if its lattice of subgroups is Distributive lattice distributive . Lattice theoretic ... and the Trivial group trivial subgroup . Five of the eight group elements generate subgroups of order ... elements. The lattice formed by these ten subgroups is shown in the illustration. Characteristic lattices ... 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 ... BF01181188 ref harv Cite book last Schmidt first Roland title Subgroup Lattices of Groups year 1994 ... 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 ... of the strong force . This formulation of QCD in discrete rather than continuous spacetime naturally 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 lattice ... more details
In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum ... science . Being a special instance of lattice order lattices , they are studied both in order theory ... A partially ordered set L , is a complete lattice if every subset A of L has both a greatest lower ... of binary meets and joins, complete lattices do thus form a special class of bounded lattice ... a lattice in fact, only the top element may be missing . This discussion is also found in the article on semilattice s. Complete sublattices A sublattice M of a complete lattice L is called ... is Compact space compact as a topological space if it is complete as a lattice. The non negative integer s, ordered by divisibility . The least element of this lattice is the number 1, since it divides .... If 0 is removed from this structure it remains a lattice but ceases to be complete. The subgroups ..., the supremum of a set of subgroups is the subgroup generated by the set theoretic union of the subgroups ... is the partial order minimum subgroup of G , while the partial order maximum subgroup is the group G ... of topologies. The lattice of all transitive relation s on a set. The lattice of all sub multisets of a multiset . The lattice of all equivalence relation s on a set the equivalence relation is considered to be smaller or finer than if x y always implies x y . Any finite lattice is trivially a complete lattice. Morphisms of complete lattices The traditional morphisms between complete lattices are the complete homomorphisms or complete lattice homomorphisms . These are characterized as functions ... lattice over a generating set S is a complete lattice L together with a function i S L , such that any function f from S to the underlying set of some complete lattice M can be factored uniquely ... very easily the complete lattice generated by some set S is just the powerset 2 sup S sup , i.e. the set ... to the one for the case of lattice order lattices , but the collection of all possible word problem ... more details
element discrete poset is not a lattice. Although the set 1,2,3,6 partially ordered by divisibility ... of submodules of a module mathematics module , and the lattice of normal subgroup s of a group mathematics ...No footnotes date May 2009 See also Lattice group 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 ... is a refinement of . In mathematics , a lattice is a partially ordered set in which any two ... the two definitions are equivalent, lattice theory draws on both order theory and universal algebra ... structure Boolean algebra s. These lattice like structures all admit order theoretic as well as algebraic descriptions. Lattices as posets A Partially ordered set poset L , is a lattice if it satisfies ... induction argument that every non empty finite subset of a lattice has a join supremum and a meet ... lattice has a greatest element greatest or maximum and least element least or minimum element, denoted 1 and 0 by convention also called top , and bottom . Any 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 ... lattice if and only if every finite set of elements including the empty set has a join and a meet ... structures General lattice An algebraic structure L , math lor, land math , consisting of a set L and two binary Operation mathematics operations math lor math , and math land math , on L is a lattice ... in which both meet and join appear, distinguish a lattice from a random pair of semilattices and assure ... order theory dual of the other. Bounded lattice A bounded lattice is an algebraic structure of the form L , math lor, land math , 1, 0 such that L , math lor, land math 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 . Identity mathematics ... more details
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 discretesubgroup 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 ... or symmetry group of a lattice in R sup n sup is defined as the subgroup of the orthogonal group O ...DISPLAYTITLE E sub 8 sub lattice In mathematics , the E sub 8 sub lattice is a special lattice group lattice 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 ... 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 ... 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 8 sup with the following properties It is unimodular lattice unimodular , meaning that it can be generated ... of the lattice is 1 . Equivalently, sub 8 sub is self dual , meaning it is equal to its dual lattice . It is even , meaning that the norm ref name norm of any lattice vector is even. Even ... . In dimension 24 there are 24 such lattices, called Niemeier lattice s. The most important of these is the Leech ... more details
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 ... 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 lattice group lattice in a Lie group . See also Crystal References reflist H. A. Bethe, Statistical theory of superlattices ... Press year 1982 isbn 0 12 083182 1 External links Category Lattice models Category Trees graph theory ... 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 mathematics , the lattice theorem , sometimes referred to as the fourth isomorphism theorem or the correspondence theorem , states that if math N math is a normal subgroup of a Group mathematics group math G math , then there exists a bijection from the set of all subgroups math A math of math G math such that math A math contains math N math , onto the set of all subgroups of the quotient group math G N math . The structure of the subgroups of math G N math is exactly the same as the structure of the subgroups of math G math containing math N, math with math N math collapsed to the identity element . This establishes a Galois connection monotone Galois connection between the lattice of subgroups of math G math and the lattice of subgroups of math G N math , where the associated closure operator on subgroups of math G math is math bar H HN. math Specifically, If G is a group, N is a normal subgroup of G , math mathcal G math is the set of all subgroups A of G such that math N subseteq A subseteq G math , and math mathcal N math is the set of all subgroups of G N , then there is a bijective map math phi mathcal G to mathcal N math such that math phi A A N math for all math A in mathcal G . math One further has that if A and B are in math mathcal G math , and A A N and B B N , then math A subseteq B math if and only if math A subseteq B math if math A subseteq B math then math B A B A math , where B A is the index group theory index of A in B the number of coset s bA of A in B math langle A,B rangle N langle A ,B rangle, math where math langle A,B rangle math is the subgroup of math G math Generating set of a group generated by math A cup B math math A cap B N A cap B math , and math A math is a normal subgroup of math G math if and only if math A math is a normal subgroup of math G N math . This list is far from exhaustive. In fact, most properties of subgroups are preserved ... to include rings, etc. See also Modular lattice References W.R. Scott Group Theory , Prentice ... more details
Image Tamari lattice.svg thumb 250px A Tamari lattice In mathematics, a Tamari lattice , introduced by harvs txt authorlink Dov Tamari mathematician first Dov last Tamari year 1962 , is a partially ordered set in which the elements consist of different ways of grouping a sequence of objects into pairs using parentheses for instance, for a sequence of four objects abcd , the five possible groupings are ab c d , ab cd , a bc d , a bc d , and a b cd . Each grouping describes a different order in which the objects may be combined by a binary operation in the Tamari lattice, one grouping is ordered before another if the second grouping may be obtained from the first by only rightward applications of the associativity associative law xy z x yz . For instance, applying this law with x a , y bc , and z d gives the expansion a bc d a bc d , so in the ordering of the Tamari lattice a bc d a bc d . In this partial order, any two groupings g sub 1 sub and g sub 2 sub have a greatest common predecessor, the meet g sub 1 sub g sub 2 sub , and a least common successor, the join g sub 1 sub g sub 2 sub . Thus, the Tamari lattice has the structure of a Lattice order lattice . The Hasse diagram of this lattice ... . The number of elements in a Tamari lattice for a sequence of n 1 objects is the n th Catalan number . The Tamari lattice can also be described in several other equivalent ways It is the poset ... 43 title Chain lengths in the Tamari lattice volume 8 year 2004 . citation doi 10.1016 S0021 9800 67 ... 1967 . citation last Geyer first Winfried mr 1298967 issue 1 3 journal Discrete Mathematics journal Discrete Mathematics pages 99 122 title On Tamari lattices volume 133 year 1994 . citation last1 Huang ... of Combinatorial Theory, Series A pages 7 13 title Problems of associativity A simple proof for the lattice ... and their enumeration volume 10 year 1962 . Category Lattice theory ... more details
the lattice. This discrete set of vectors must be closed under vector addition and subtraction. 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 ...In geometry and crystallography , a Bravais lattice , studied by harvs txt first Auguste last Bravais ... of discrete points generated by a set of discrete translation geometry translation operations described ... space there is just one type of Bravais lattice. In two dimensions, there are five Bravais ... 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 ... equivalent to each other. For example, the monoclinic I lattice can be described by a monoclinic C lattice by different choice of crystal axes. Similarly, all A or B centered lattices can be described ... Bravais lattices, shown in the table below. align left border 1 style margin left 1em The 7 lattice ... 80px Tetragonal, 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 ... a , b , and c are the lattice vector s. The volumes of the Bravais lattices are given below align left border 1 style margin left 1em Lattice system colspan 4 align center Volume Triclinic math abc ... more details
Mergeto Lattice based cryptography date July 2009 In computer science , lattice problems are a class of optimization problems on Lattice group lattices . The conjectured intractability of such problems is central to construction of secure Lattice based cryptography lattice based cryptosystems . For applications ... vector in the lattice L math lambda L mathbf min v N v in mathbf L , v neq 0 math . Shortest vector ... N often Norm mathematics Euclidean norm L sup 2 sup are given for a lattice L and one must find the shortest ... math SVP gamma math , one must find a non zero lattice vector of length at most math gamma lambda ... of computing short vectors in a lattice. Tech. rep., University of Amsterdam, Department ... lattice basis reduction algorithm produces a relatively short vector in polynomial time, but does not solve ... function of math n math , the number of vectors. Given a basis for the lattice, the algorithm ... caption Lattice problems by example widths 200px heights 200px Image Svp09.png The SVP by example ... metric M often Euclidean distance L sup 2 sup are given for a lattice L , as well as a vector v ... . In the math gamma math approximation version math CVP gamma math , one must find a lattice vector ... vector problem. It is easy ref Daniele Micciancio and Shafi Goldwasser , Complexity of lattice problems ... 0 is itself a lattice vector and the algorithm could potentially output 0. The reduction from ... math problem is the basis for lattice math B b 1,b 2, ldots,b n math . Consider the basis math B ... in the given lattice. Known results Goldreich et al. ref O. Goldreich et al., Approximating shortest lattice vectors is not harder than approximating closet lattice vectors, Inf. Process. Lett. 71 .... and Pohst, M., Improved Methods for Calculating Vectors of Short Length in a Lattice, Including a Complexity ... is similar to the GapSVP problem. For math GapCVP beta math , the input consists of a lattice basis and a vector math v math and the algorithm must answer whether there is a lattice vector such that the distance ... more details
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
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
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
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 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 symmetry combinations centered square number Gaussian integer DEFAULTSORT Square Lattice Category Euclidean ... 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
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
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 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 ... more details
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