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Lattice group in Euclidean space whose points are integer n-tuples
^{n}} whose lattice points are n-tuples of integers. The two-dimensional integer lattice is also called the square lattice (or grid lattice) and the three-dimensional
Integer_lattice
Periodic set of points
in the plane whose coordinates are both integers, and its higher-dimensional analogues the integer lattices Z n {\displaystyle \mathbb {Z} ^{n}} . Closure
Lattice_(group)
High-area shapes can shift to hold many grid points
includes at least ⌈ A ⌉ {\displaystyle \lceil A\rceil } points of the integer lattice. Equivalently, every bounded set of area A {\displaystyle A} contains
Blichfeldt's_theorem
Number in {..., –2, –1, 0, 1, 2, ...}
factorization of a positive integer Complex integer Hyperinteger Integer complexity Integer lattice Integer part Integer sequence Integer-valued function Mathematical
Integer
Complex number whose mapping on a coordinate plane produces a triangular lattice
The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice in the complex
Eisenstein_integer
24-dimensional repeating pattern of points
based on the integer lattice, hexagonal tiling, and E8 lattice, respectively. It has no root system and in fact is the first unimodular lattice with no roots
Leech_lattice
2-dimensional integer lattice
the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as
Square_lattice
Complex number whose real and imaginary parts are both integers
considered within the complex plane, the Gaussian integers constitute the 2-dimensional square lattice. The conjugate of a complex number a + bi is the
Gaussian_integer
Cryptographic primitives that involve lattices
certain average-case lattice problem, known as short integer solutions (SIS), is at least as hard to solve as a worst-case lattice problem. She then showed
Lattice-based_cryptography
Fourier transform of a real-space lattice, important in solid-state physics
)n} with an integer n {\displaystyle n} ) at every direct lattice vertex. One heuristic approach to constructing the reciprocal lattice in three dimensions
Reciprocal_lattice
Sequence of end-to-end vectors across points of a lattice
In combinatorics, a lattice path L in the d-dimensional integer lattice Z d {\displaystyle \mathbb {Z} ^{d}} of length k with steps in the set S,
Lattice_path
Characterization by prime factors of sums of two squares
lengths of line segments between pairs of points in the two-dimensional integer lattice. The number of representable numbers in the range from 0 to any number
Sum_of_two_squares_theorem
Algebra of formal sums
uniquely expressed as an integer combination of finitely many basis elements. For instance, the two-dimensional integer lattice forms a free abelian group
Free_abelian_group
How many integer lattice points there are in a circle
mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius
Gauss_circle_problem
Theory of a class of elliptic curves
are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice. It has an aspect belonging to the theory of special
Complex_multiplication
Triangle with integer side lengths
either is an integer or a half-integer (has a denominator of 2). If the lattice triangle has integer sides then it is Heronian with integer area. Furthermore
Integer_triangle
Mathematical operation
mathematics, the goal of lattice basis reduction is to find a basis with short, nearly orthogonal vectors when given an integer lattice basis as input. This
Lattice_reduction
Set whose pairs have minima and maxima
"lattice" is suggested by the form of the Hasse diagram depicting it. Pic. 2: Lattice of integer divisors of 60, ordered by "divides". Pic. 3: Lattice
Lattice_(order)
set of integer numbers. For a lattice Λ, Minkowski's theorem relates the number d(Λ) (the volume of a fundamental parallelepiped of the lattice) and the
Integer points in convex polyhedra
Integer_points_in_convex_polyhedra
Type of crystal structure
{\sqrt {3}}} apart in the integer lattice; the edges of the diamond structure lie along the body diagonals of the integer grid cubes. This structure
Diamond_cubic
Curved triangle with constant width
provides the largest constant-width shape avoiding the points of an integer lattice, and is closely related to the shape of the quadrilateral maximizing
Reuleaux_triangle
Optimization problem in computer science
In computer science, lattice problems are a class of optimization problems related to mathematical objects called lattices. The conjectured intractability
Lattice_problem
including integer lattices in real vector spaces, orders in algebraic number fields, and fractional ideals in integral domains. Formally, a lattice is a kind
Lattice_(module)
Algorithm in computational number theory
\mathbf {b} _{2},\dots ,\mathbf {b} _{d}\}} with n-dimensional integer coordinates, for a lattice L (a discrete subgroup of Rn) with d ≤ n {\displaystyle d\leq
Lenstra–Lenstra–Lovász lattice basis reduction algorithm
Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm
Natural number
There are 124 different polygons of length 12 formed by edges of the integer lattice, counting two polygons as the same only when one is a translated copy
124_(number)
Geometry and crystallography point array
In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (1850), is an infinite array of discrete points generated by a set of
Bravais_lattice
Family of tetrahedra on an integer lattice
vertices of a Reeve tetrahedron are integer lattice points (points whose coordinates are all integers). No other lattice points lie on the surface or in the
Reeve_tetrahedra
Natural number
chains of length six using horizontal and vertical segments of the integer lattice. 147 (disambiguation) Sloane, N. J. A. (ed.). "Sequence A005902 (Centered
147_(number)
Integer that divides another integer
} of non-negative integers into a partially ordered set that is a complete distributive lattice. The largest element of this lattice is 0 and the smallest
Divisor
Lattice formed by all integer partitions
In mathematics, Young's lattice is a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers On quantitative
Young's_lattice
Graph whose embedding in a Euclidean space forms a regular tiling
wazir form a square lattice graph. Lattice path Pick's theorem Integer triangles in a 2D lattice Regular graph Weisstein, Eric W. "Lattice graph". MathWorld
Lattice_graph
Czech mathematician (1897–1970)
developing Jarník's algorithm, he found tight bounds on the number of integer lattice points on convex curves, studied the relationship between the Hausdorff
Vojtěch_Jarník
Natural number
ring of integers yields the Leech lattice. Conway and Sloane provided constructions of the Leech lattice from all other 23 Niemeier lattices. Twenty-three
23_(number)
Mathematical optimization problem restricted to integers
An integer programming, also known as integer optimization, problem is a mathematical optimization or feasibility program in which some or all of the variables
Integer_programming
Lattice in 8-dimensional space with special properties
2*(8*7)/(2*1) = 56 All integer (can only be 0, ±1): Two ±1, six zeroes: 4*(8*7)/(2*1)=112 These form a root system of type E8. The lattice Γ8 is equal to the
E8_lattice
Number used for counting
2, 3, and so on, possibly excluding 0. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set
Natural_number
Mathematical object
discrete mathematics, ideal lattices are a special class of lattices and a generalization of cyclic lattices. Ideal lattices naturally occur in many parts
Ideal_lattice
Every symmetric convex set in R^n with volume > 2^n contains a non-zero integer point
theory called the geometry of numbers. It can be extended from the integers to any lattice L {\displaystyle L} and to any symmetric convex set with volume
Minkowski's_theorem
Application of geometry in number theory
a ring of algebraic integers is viewed as a lattice in R n , {\displaystyle \mathbb {R} ^{n},} and the study of these lattices provides fundamental information
Geometry_of_numbers
Geometric arrangements of points, foundational to Lie theory
coordinates are half-integers (a mixture of integers and half-integers is not allowed). This lattice is isomorphic to the lattice of Hurwitz quaternions
Root_system
Construction analogous to that of a dual vector space
matrix B {\textstyle B} . The dual lattice is the set of linear functionals on L {\textstyle L} which take integer values on each point of L {\textstyle
Dual_lattice
Reconstruction of binary images from a small number of their projections
problem of reconstruction of binary images (or finite subsets of the integer lattice) from a small number of their projections. In general, tomography deals
Discrete_tomography
Decomposition of an integer as a sum of positive integers
partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only
Integer_partition
Number-theoretical function
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A004018 (Theta series of square lattice, r_2(n))". The On-Line
Sum_of_squares_function
One of the five 2D Bravais lattices
below. Square lattice (see dots in a diagonal square centered) Hexagonal tiling Close-packing Centered hexagonal number Eisenstein integer Voronoi diagram
Hexagonal_lattice
Relation of an integral polytope's volume to how many integer points it encloses
each dimension, then L(P, t) is the number of integer lattice points in tP. More formally, consider a lattice L {\displaystyle {\mathcal {L}}} in Euclidean
Ehrhart_polynomial
Topics referred to by the same term
Cubic lattice may refer to: Cubic crystal system Cubic honeycomb vertex arrangement Integer lattice Z3 This disambiguation page lists articles associated
Cubic_lattice
Set of points touching all convex bodies of unit volume
this would equal the growth rate of well-spaced point sets like the integer lattice (which is not a Danzer set). An equivalent formulation involves the
Danzer_set
On centroids of sets of lattice points
conjecture states that every set of integer lattice points in the plane has a large subset whose centroid is also a lattice point. It was proved independently
Kemnitz's_conjecture
Periodic spatial graph
of the Laves graph uses one out of every eight of the points in the integer lattice as its points, and connects all pairs of these points that are nearest
Laves_graph
Positive real number which when multiplied by itself gives 7
two points of a cubic integer lattice (or equivalently, the length of the space diagonal of a rectangular cuboid with integer side lengths). 15 {\displaystyle
Square_root_of_7
Notation system for crystal lattice planes
three integers h, k, and l, the Miller indices. They are written (hkl), and denote the family of (parallel) lattice planes (of the given Bravais lattice) orthogonal
Miller_index
Special type of lattice
A lattice-ordered vector space is a distributive lattice. Young's lattice given by the inclusion ordering of Young diagrams representing integer partitions
Distributive_lattice
Centered figurate number representing an octahedron
is a figurate number that counts the points of a three-dimensional integer lattice that lie inside an octahedron centered at the origin. The same numbers
Centered_octahedral_number
Topics referred to by the same term
reducing the number of random variables under consideration Lattice reduction, given an integer lattice basis as input, to find a basis with short, nearly orthogonal
Reduction
Set of distances defined from a set of points
numbers in the distance set of the two-dimensional integer lattice: they are the square roots of integers whose prime factorization does not contain an odd
Distance_set
Image comprising exactly two colors, typically black and white
Binary images can be interpreted as subsets of the two-dimensional integer lattice Z 2 {\displaystyle \mathbb {Z} ^{2}} ; the field of morphological
Binary_image
Number of paths between grid corners, allowing diagonal steps
{\displaystyle m} and n {\displaystyle n} , the points in an m-dimensional integer lattice or cross polytope which are at most n steps from the origin, and, in
Delannoy_number
Rational number equal to an integer plus 1/2
from the integers to the half-integers: f : x → x + 0.5 {\displaystyle f:x\to x+0.5} , where x {\displaystyle x} is an integer. The densest lattice packing
Half-integer
Tessellation of Euclidean space
are unit squares or unit cubes, and the vertices are points on the integer lattice. A rectilinear grid is a tessellation by rectangles or rectangular
Regular_grid
Doignon's theorem in geometry is an analogue of Helly's theorem for the integer lattice. It states that, if a family of convex sets in d {\displaystyle d}
Doignon's_theorem
Generalization of algebraic integers
Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd integers; a mixture of integers and half-integers
Hurwitz_quaternion
Integral lattice of determinant 1 or –1
The lattice is integral if (·,·) takes integer values. The dimension of a lattice is the same as its rank (as a Z-module). The norm of a lattice element
Unimodular_lattice
Vector quantization algorithm minimizing the sum of squared deviations
O ( d n 4 M 2 ) {\displaystyle O(dn^{4}M^{2})} for n points in an integer lattice { 1 , … , M } d {\displaystyle \{1,\dots ,M\}^{d}} . Lloyd's algorithm
K-means_clustering
Shape with four equal sides and angles
{\displaystyle c} . The Gaussian integers, complex numbers with integer real and imaginary parts, form a square lattice in the complex plane. The construction
Square
Crystallographic concept
parallel lattice planes that, taken together, intersect all lattice points. Every family of lattice planes can be described by a set of integer Miller indices
Lattice_plane
Classification of crystalline materials by their three-dimensional structural geometry
integers and a1, a2, and a3 are three non-coplanar vectors, called primitive vectors. These lattices are classified by the space group of the lattice
Crystal_system
Exponential function of an exponential function
The maximal volume of a polytope in a d-dimensional integer lattice with k ≥ 1 interior lattice points is at most k ⋅ ( 8 d ) d ⋅ 15 d ⋅ 2 2 d + 1 ,
Double_exponential_function
Combinitorics of Polyhedra
convex polytopes does not form a convex subset of the four-dimensional integer lattice, and much remains unknown about the possible values of these vectors
Polyhedral_combinatorics
Mathematics of varieties with integer coordinates
therefore are the primary consideration; but integral solutions (i.e., integer lattice points) can be treated in the same way as an affine variety may be
Diophantine_geometry
Simple Lie group; the automorphism group of the octonions
(−1, 2), however the integer lattice spanned by those is not the one pictured above (from obvious reason: the hexagonal lattice on the plane cannot be
G2_(mathematics)
Type of plane curve
points of the integer lattice. If the curve has length L {\displaystyle L} , then according to a theorem of Vojtěch Jarník, the number of lattice points that
Convex_curve
2007 mathematics textbook
interplay between the volume of convex polytopes and the number of integer lattice points they contain. It was written by Matthias Beck and Sinai Robins
Computing the Continuous Discretely
Computing_the_Continuous_Discretely
Property of being an even or odd number
the face-centered cubic lattice and its higher-dimensional generalizations (the Dn lattices) consist of all of the integer points whose coordinates have
Parity_(mathematics)
Geometric object with flat sides
differs, in terms of integer lattice points, from a t {\displaystyle t} -dilate of P {\displaystyle {\mathcal {P}}} only by lattice points gained on the
Polytope
Process forming a path from many random steps
example is the random walk on the d-dimensional integer lattice (sometimes called the hypercubic lattice) Z d {\displaystyle \mathbb {Z} ^{d}} . If, in
Random_walk
Computational problem used in cryptography
Short integer solution (SIS) and ring-SIS problems are two average-case problems that are used in lattice-based cryptography constructions. Lattice-based
Short integer solution problem
Short_integer_solution_problem
Tiling of the plane with 60° rhombi
to embed the rhombille tiling into a subset of a three-dimensional integer lattice, consisting of the points (x,y,z) with |x + y + z| ≤ 1, in such a way
Rhombille_tiling
Mathematical formula expressing equality
curve, algebraic surface, or more general object, and ask about the integer lattice points on it. The word Diophantine refers to the Hellenistic mathematician
Equation
Type of metric geometry
3D balls of radii 1 (red) and 2 (blue) are regular octahedrons: the number of integer lattice points enclosed form the centered octahedral numbers
Taxicab_geometry
Number, approximately 3.14
(optimal) upper bound on the volume of a convex body containing only one integer lattice point. The Riemann zeta function ζ(s) is used in many areas of mathematics
Pi
Three linked but pairwise separated rings
of ropelength, the shortest representation using only edges of the integer lattice, the minimum length for the Borromean rings is exactly 36 {\displaystyle
Borromean_rings
Matrix form in linear algebra
normal form is an analogue of reduced echelon form for matrices over the integers Z {\displaystyle \mathbb {Z} } . Just as reduced echelon form can be used
Hermite_normal_form
Theorem about admissible crystal symmetries
B' must both be lattice points. Due to periodicity of the crystal, the new vector r' which connects them must be equal to an integer multiple of r: r
Crystallographic restriction theorem
Crystallographic_restriction_theorem
Power series derived from a discrete probability distribution
variable taking values (x1, ..., xd) in the d-dimensional non-negative integer lattice {0,1, ...}d, then the probability generating function of X is defined
Probability generating function
Probability_generating_function
Turing machine on a two-dimensional grid
Hutton have also investigated one-dimensional relative turmites on the integer lattice, which Brady termed flippers. (One-dimensional absolute turmites are
Turmite
apply types of double counting. One by Gotthold Eisenstein counts integer lattice points. Another applies Zolotarev's lemma to ( Z / p q Z ) × {\displaystyle
Proofs of quadratic reciprocity
Proofs_of_quadratic_reciprocity
Mathematics book by Piper Harron
The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields: An Artist's Rendering is a mathematics book
The_Equidistribution_of_Lattice_Shapes_of_Rings_of_Integers_of_Cubic,_Quartic,_and_Quintic_Number_Fields
Cryptography secured against quantum computers
algorithms rely on the difficulty of one of three mathematical problems: the integer factorization problem, the discrete logarithm problem, or the elliptic-curve
Post-quantum_cryptography
Lattice in universal algebra
false-preserving functions. Post's lattice consists of 9 named clones, two countably infinite families of clones indexed by the positive integers, and all finite intersections
Post's_lattice
Natural number, composite number
(n-iamonds, polyiamonds) in the 2-dimensional hexagonal lattice.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-05-15. Grünbaum
14_(number)
Convex polytope whose vertices all have integer Cartesian coordinates
whose vertices all have integer Cartesian coordinates. That is, it is a polytope that equals the convex hull of its integer points. Integral polytopes
Integral_polytope
Integer side lengths of a right triangle
over all positive and negative integers. Any Pythagorean triangle with triple (a, b, c) can be drawn within a 2D lattice with vertices at coordinates (0
Pythagorean_triple
Type of subatomic particle
subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin 3/2, etc.) and obey the Pauli exclusion principle
Fermion
Mathematical term
a lattice word (or lattice permutation) is a string composed of positive integers, in which every prefix contains at least as many positive integers i
Lattice_word
Mathematical model of the time dependence of a point in space
M, Φ), with T a lattice such as the integers or a higher-dimensional integer grid, M is a set of functions from an integer lattice (again, with one or
Dynamical_system
Doughnut-shaped surface of revolution
any coordinate. That is, the n-torus is Rn modulo the action of the integer lattice Zn (with the action being taken as vector addition). Equivalently,
Torus
Mathematical expression
fraction in canonical form for the irrational real number α, and the way integer lattice points in two dimensions lie to either side of the line y = αx. Generalizing
Continued_fraction
Number of stacked spheres in a pyramid
with lattice points as its vertices. Specifically, the Ehrhart polynomial L(P,t) of an integer polyhedron P is a polynomial that counts the integer points
Square_pyramidal_number
Graph that can be embedded in the plane
sets of quadratic size, formed by taking a rectangular subset of the integer lattice. Every simple outerplanar graph admits an embedding in the plane such
Planar_graph
INTEGER LATTICE
INTEGER LATTICE
Surname or Lastname
English
English : variant of Fretter, an occupational name for a maker of ornaments (especially for the hair) consisting of jewels set in a lattice network, from an agent derivative of Middle English frette, Old French frete ‘interlaced work’.
Boy/Male
German, Norse, Swedish
Guarded by Ing; Ing's Beauty
Boy/Male
Muslim
To wait
Female
Swedish
Swedish contracted form of Scandinavian Ingegerd, INGER means "Ing's enclosure."
Girl/Female
American, Australian, Danish, Finnish, German, Scandinavian, Swedish, Teutonic
Guarded by Ing; Ing is Beautiful; Daughter of Hero; Enclosure
Girl/Female
Scandinavian Teutonic Danish Swedish
Ing's abundance. Feminine of Ing who was Norse mythological god of the earth's fertility.
Boy/Male
Norse
Son's army.
Female
Scandinavian
Scandinavian form of Old Norse Ingigerðr, INGEGERD means "Ing's enclosure."
Girl/Female
Danish, Finnish, German, Swedish
Guarded by Ing; Ing's Beauty; Ing's Place
Boy/Male
Arabic, Muslim
To Wait
INTEGER LATTICE
INTEGER LATTICE
Boy/Male
Hindu, Indian, Telugu
Famous; Scholar; Lord Ganesh; One with Fame
Girl/Female
Bengali, Indian, Tamil, Telugu
Favour
Girl/Female
Norse
Holy.
Girl/Female
Tamil
A creeper
Boy/Male
Tamil
Help, Lord Shiva
Boy/Male
Tamil
Viswavardan | விஸà¯à®µà®µà®¾à®°à¯à®¤à®¨Â
Boy/Male
Indian, Sanskrit
Follower; Companion
Boy/Male
Hindu
Thought, Devotion, Another name of the Sun, Lord Shiva
Boy/Male
American, Bengali, British, Christian, Danish, English, French, German, Gujarati, Hebrew, Hindu, Indian, Irish, Jamaican, Kannada, Malayalam, Marathi, Netherlands, Polish, Swedish, Swiss, Tamil, Telugu
Gift from God; God has Given; Given; Husband; Controller; God; Giver; Gift; Given by God
Boy/Male
Hindu, Indian, Marathi
The Best King
INTEGER LATTICE
INTEGER LATTICE
INTEGER LATTICE
INTEGER LATTICE
INTEGER LATTICE
v. t.
To bury; to inter; to entomb; as, obscurely sepulchered.
n.
One who inters.
n.
A complete entity; a whole number, in contradistinction to a fraction or a mixed number.
v. t.
To deposit, as a dead body, in the earth; to bury; to inter.
n.
That number placed below the line in vulgar fractions which shows into how many parts the integer or unit is divided.
v. t.
To inhume; to bury; to inter.
v. t.
To inter again.
imp. & p. p.
of Inter
a.
Essential to completeness; constituent, as a part; pertaining to, or serving to form, an integer; integrant.
p. pr. & vb. n.
of Inter
v. t.
To deposit or inter in a chapel; to enshrine.
n.
One who makes an index.
n.
One who gathers the vintage.
n.
One who makes an entrance or beginning.
v. t.
To deposit and cover in the earth; to bury; to inhume; as, to inter a dead body.
v. t.
To inter.
v. t.
To place in a tomb; to bury; to inter; to entomb.
v. t.
To inter with funeral rites; to bury.
n.
One who intends.