1.
Lattice (group)
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In geometry and group theory, a lattice in R n is a subgroup of R n which is isomorphic to Z n, and which spans the real vector space R n. In other words, for any basis of R n, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, a lattice may be viewed as a regular tiling of a space by a primitive cell. Lattices have many significant applications in mathematics, particularly in connection to Lie algebras, number theory. More generally, lattice models are studied in physics, often by the techniques of computational physics, a lattice is the symmetry group of discrete translational symmetry in n directions. A pattern with this lattice of translational symmetry cannot have more, as a group a lattice is a finitely-generated free abelian group, and thus isomorphic to Z n. A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e. g, a simple example of a lattice in R n is the subgroup Z n. More complicated examples include the E8 lattice, which is a lattice in R8, the period lattice in R2 is central to the study of elliptic functions, developed in nineteenth century mathematics, it generalises to higher dimensions in the theory of abelian functions. Lattices called root lattices are important in the theory of simple Lie algebras, for example, a typical lattice Λ in R n thus has the form Λ = where is a basis for R n. Different bases can generate the lattice, but the absolute value of the determinant of the vectors vi is uniquely determined by Λ. If one thinks of a lattice as dividing the whole of R n into equal polyhedra and this is why d is sometimes called the covolume of the lattice. If this equals 1, the lattice is called unimodular, minkowskis theorem relates the number d and the volume of a symmetric convex set S to the number of lattice points contained in S. The number of lattice points contained in an all of whose vertices are elements of the lattice is described by the polytopes Ehrhart polynomial. Formulas for some of the coefficients of this polynomial involve d as well, Lattice basis reduction is the problem of finding a short and nearly orthogonal lattice basis. The Lenstra-Lenstra-Lovász lattice basis reduction algorithm approximates such a basis in polynomial time, it has found numerous applications. There are five 2D lattice types as given by the crystallographic restriction theorem, below, the wallpaper group of the lattice is given in IUC notation, Orbifold notation, and Coxeter notation, along with a wallpaper diagram showing the symmetry domains. Note that a pattern with this lattice of translational symmetry cannot have more, a full list of subgroups is available. For example below the hexagonal/triangular lattice is given twice, with full 6-fold, if the symmetry group of a pattern contains an n-fold rotation then the lattice has n-fold symmetry for even n and 2n-fold for odd n. For the classification of a lattice, start with one point

2.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space

3.
Translation group
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In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same amount in a given direction. In Euclidean geometry a transformation is a correspondence between two sets of points or a mapping from one plane to another. )A translation can be described as a rigid motion. A translation can also be interpreted as the addition of a constant vector to every point, a translation operator is an operator T δ such that T δ f = f. If v is a vector, then the translation Tv will work as Tv. If T is a translation, then the image of a subset A under the function T is the translate of A by T, the translate of A by Tv is often written A + v. In a Euclidean space, any translation is an isometry, the set of all translations forms the translation group T, which is isomorphic to the space itself, and a normal subgroup of Euclidean group E. The quotient group of E by T is isomorphic to the orthogonal group O, E / T ≅ O, a translation is an affine transformation with no fixed points. Matrix multiplications always have the origin as a fixed point, similarly, the product of translation matrices is given by adding the vectors, T u T v = T u + v. Because addition of vectors is commutative, multiplication of matrices is therefore also commutative. In physics, translation is movement that changes the position of an object, for example, according to Whittaker, A translation is the operation changing the positions of all points of an object according to the formula → where is the same vector for each point of the object. When considering spacetime, a change of time coordinate is considered to be a translation, for example, the Galilean group and the Poincaré group include translations with respect to time

4.
Lattice (order)
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A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of an ordered set in which every two elements have a unique supremum and a unique infimum. An example is given by the numbers, partially ordered by divisibility, for which the unique supremum is the least common multiple. Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities, since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras and these lattice-like structures all admit order-theoretic as well as algebraic descriptions. If is an ordered set, and S ⊆ L is an arbitrary subset. A set may have many upper bounds, or none at all, an upper bound u of S is said to be its least upper bound, or join, or supremum, if u ≤ x for each upper bound x of S. A set need not have a least upper bound, but it cannot have more than one, dually, l ∈ L is said to be a lower bound of S if l ≤ s for each s ∈ S. A lower bound l of S is said to be its greatest lower bound, or meet, or infimum, a set may have many lower bounds, or none at all, but can have at most one greatest lower bound. A partially ordered set is called a join-semilattice and a meet-semilattice if each two-element subset ⊆ L has a join and a meet, denoted by a ∨ b, is called a lattice if it is both a join- and a meet-semilattice. This definition makes ∨ and ∧ binary operations, both operations are monotone with respect to the order, a1 ≤ a2 and b1 ≤ b2 implies that a1 ∨ b1 ≤ a2 ∨ b2 and a1 ∧ b1 ≤ a2 ∧ b2. It follows by an argument that every non-empty finite subset of a lattice has a least upper bound. With additional assumptions, further conclusions may be possible, see Completeness for more discussion of this subject, a bounded lattice is a lattice that additionally has a greatest element 1 and a least element 0, which satisfy 0 ≤ x ≤1 for every x in L. The greatest and least element is called the maximum and minimum, or the top and bottom element. A partially ordered set is a lattice if and only if every finite set of elements has a join. Taking B to be the empty set, ⋁ = ∨ = ∨0 = ⋁ A and ⋀ = ∧ = ∧1 = ⋀ A which is consistent with the fact that A ∪ ∅ = A. A lattice element y is said to another element x, if y > x. Here, y > x means x ≤ y and x ≠ y

5.
Bragg plane
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In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, K, at right angles. The Bragg plane is defined as part of the Von Laue condition for diffraction peaks in x-ray diffraction crystallography. We know then that for constructive interference we have, | d | cos θ + | d | cos θ ′ = d ⋅ = m λ where m ∈ Z and we can set one of the scattering centres as the origin of an array. This reciprocal space plane is the Bragg plane, x-ray crystallography Reciprocal lattice Bravais lattice Powder diffraction Kikuchi line Brillouin zone

6.
Bravais lattice
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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, when the discrete points are atoms, ions, or polymer strings of solid matter, the Bravais lattice concept is used to formally define a crystalline arrangement and its frontiers. A crystal is made up of an arrangement of one or more atoms repeated at each lattice point. Consequently, the crystal looks the same when viewed from any equivalent lattice point, two Bravais lattices are often considered equivalent if they have isomorphic symmetry groups. In this sense, there are 14 possible Bravais lattices in three-dimensional space, the 14 possible symmetry groups of Bravais lattices are 14 of the 230 space groups. In two-dimensional space, there are 5 Bravais lattices, grouped into four crystal families, the unit cells are specified according to the relative lengths of the cell edges and the angle between them. The area of the cell can be calculated by evaluating the norm || a × b ||. The properties of the families are given below, In three-dimensional space. These are obtained by combining one of the six families with one of the centering types. 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-centred lattices can be described either by a C- or P-centering. This reduces the number of combinations to 14 conventional Bravais lattices, the unit cells are specified according to the relative lengths of the cell edges and the angles between them. The volume of the cell can be calculated by evaluating the triple product a ·, where a, b. The properties of the families are given below, In four dimensions. Of these,23 are primitive and 41 are centered, ten Bravais lattices split into enantiomorphic pairs. Bravais, A. Mémoire sur les systèmes formés par les points distribués régulièrement sur un plan ou dans lespace, hahn, Theo, ed. International Tables for Crystallography, Volume A, Space Group Symmetry. Catalogue of Lattices Smith, Walter Fox

7.
Diamond cubic
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The diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. Diamond cubic is in the Fd3m space group, which follows the face-centered cubic Bravais lattice, the diamond lattice can be viewed as a pair of intersecting face-centered cubic lattices, with each separated by 1/4 of the width of the unit cell in each dimension. Many compound semiconductors such as gallium arsenide, β-silicon carbide, and indium antimonide adopt the analogous zincblende structure, zincblendes space group is F43m, but many of its structural properties are quite similar to the diamond structure. The atomic packing factor of the cubic structure is π√3/16 ≈0.34. Zincblende structures have higher packing factors than 0.34 depending on the sizes of their two component atoms. The first-, second-, third-, fourth- and fifth-nearest-neighbor distances in units of the lattice constant are √3/4, √2/2, √11/4,1 and √19/4. Mathematically, the points of the cubic structure can be given coordinates as a subset of a three-dimensional integer lattice by using a cubic unit cell four units across. With these coordinates, the points of the structure have coordinates satisfying the equations x = y = z, and x + y + z =0 or 1. There are eight points that satisfy these conditions, All of the points in the structure may be obtained by adding multiples of four to the x, y. Adjacent points in this structure are at distance √3 apart in the integer lattice and this structure may be scaled to a cubical unit cell that is some number a of units across by multiplying all coordinates by a/4. Alternatively, each point of the cubic structure may be given by four-dimensional integer coordinates whose sum is either zero or one. Two points are adjacent in the structure if and only if their four-dimensional coordinates differ by one in a single coordinate. The total difference in values between any two points gives the number of edges in the shortest path between them in the diamond structure. These four-dimensional coordinates may be transformed into three-dimensional coordinates by the formula →, because the diamond structure forms a distance-preserving subset of the four-dimensional integer lattice, it is a partial cube. Yet another coordinatization of the diamond cubic involves the removal of some of the edges from a grid graph. The diamond cubic is sometimes called the lattice but it is not, mathematically. However, it is still a highly symmetric structure, any incident pair of a vertex, moreover the diamond crystal as a network in space has a strong isotropic property. Another crystal with this property is the Laves graph, the compressive strength and hardness of diamond and various other materials, such as boron nitride, is attributed to the diamond cubic structure

8.
Divisor summatory function
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In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the behaviour of the Riemann zeta function. The various studies of the behaviour of the function are sometimes called divisor problems. The divisor summatory function is defined as D = ∑ n ≤ x d = ∑ j, k j k ≤ x 1 where d = σ0 = ∑ j, k j k = n 1 is the divisor function. The divisor function counts the number of ways that the integer n can be written as a product of two integers. More generally, one defines D k = ∑ n ≤ x d k = ∑ m n ≤ x d k −1 where dk counts the number of ways that n can be written as a product of k numbers. This quantity can be visualized as the count of the number of lattice points fenced off by a surface in k dimensions. Thus, for k=2, D = D2 counts the number of points on a lattice bounded on the left by the vertical-axis, on the bottom by the horizontal-axis. Roughly, this shape may be envisioned as a hyperbolic simplex, finding a closed form for this summed expression seems to be beyond the techniques available, but it is possible to give approximations. The leading behaviour of the series is not difficult to obtain, peter Gustav Lejeune Dirichlet demonstrated that D = x log x + x + Δ where γ is the Euler–Mascheroni constant, and the non-leading term is Δ = O. Here, O denotes Big-O notation. The Dirichlet divisor problem, precisely stated, is to find the smallest value of θ for which Δ = O holds true, as of today, this problem remains unsolved. Many of the same work for this problem and for Gausss circle problem. Section F1 of Unsolved Problems in Number Theory surveys what is known, in 1904, G. Voronoi proved that the error term can be improved to O. In 1916, G. H. Hardy showed that inf θ ≥1 /4. In particular, he demonstrated that for some constant K, there exist values of x for which Δ > K x 1 /4, in 1922, J. van der Corput improved Dirichlets bound to inf θ ≤33 /100. In 1928, J. van der Corput proved that inf θ ≤27 /82, in 1950, Chih Tsung-tao and independently in 1953 H. E. Richert proved that inf θ ≤15 /46. In 1969, Grigori Kolesnik demonstrated that inf θ ≤12 /37, in 1973, Grigori Kolesnik demonstrated that inf θ ≤346 /1067. In 1982, Grigori Kolesnik demonstrated that inf θ ≤35 /108, in 1988, H. Iwaniec and C. J. Mozzochi proved that inf θ ≤7 /22