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Banach space or Hilbert space H {\displaystyle H} has a cyclic vector f {\displaystyle f} if the vectors f , A f , A 2 f , … {\displaystyle f,Af,A^{2}f,\dots
Cyclic_vector
Von Neumann
In mathematics, the notion of a cyclic and separating vector is important in the theory of von Neumann algebras, and, in particular, in Tomita–Takesaki
Cyclic_and_separating_vector
functional analysis, a cyclic subspace is a certain special subspace of a vector space associated with a vector in the vector space and a linear transformation
Cyclic_subspace
Result about when a matrix can be diagonalized
\lambda \mapsto \lambda } . A vector φ {\displaystyle \varphi } is called a cyclic vector for A {\displaystyle A} if the vectors φ , A φ , A 2 φ , … {\displaystyle
Spectral_theorem
Correspondence in functional analysis
called a cyclic representation. Any non-zero vector of an irreducible representation is cyclic. However, non-zero vectors in a general cyclic representation
Gelfand–Naimark–Segal construction
Gelfand–Naimark–Segal_construction
Partially unsolved problem in mathematics
for which every non-zero vector x ∈ H {\displaystyle x\in H} is a cyclic vector for T {\displaystyle T} . (Where a "cyclic vector" x {\displaystyle x} for
Invariant_subspace_problem
Operator on a Hilbert space that shifts basis vectors
}|f(z)|=0} , may or may not be cyclic. For example, f ( z ) = 1 − z {\displaystyle f(z)=1-z} is a cyclic vector. The cyclic vectors are precisely the outer functions
Unilateral_shift_operator
Type of block code
In coding theory, a cyclic code is a block code, where the circular shifts of each codeword gives another word that belongs to the code. They are error-correcting
Cyclic_code
Theorem in axiomatic quantum field theory
that the vacuum state | Ω ⟩ {\displaystyle \vert \Omega \rangle } is a cyclic vector for the field algebra A ( O ) {\displaystyle {\mathcal {A}}({\mathcal
Reeh–Schlieder_theorem
Indian academic (1946–2019)
and received a PhD in Reproducing Kernels and Operators with a Cyclic Vector (Cycle Vector Space Theory) in 1969 under doctoral advisor John L. Kelley.
Vashishtha_Narayan_Singh
Families of matrices in mathematics, physics, and quantum information
the shift matrix is just the translation operator (a cyclic permutation matrix) in that cyclic vector space, so the exponential of the momentum. They are
Generalizations of Pauli matrices
Generalizations_of_Pauli_matrices
Mathematical operation on vectors in 3D space
is that they can be deduced from any other of them by a cyclic permutation of the basis vectors. This mnemonic applies also to many formulas given in this
Cross_product
Mathematics theorem in functional analysis
non-negative z in A and f(−x* x) < 0. Consider the GNS representation πf with cyclic vector ξ. Since ‖ π f ( x ) ξ ‖ 2 = ⟨ π f ( x ) ξ ∣ π f ( x ) ξ ⟩ = ⟨ ξ ∣ π
Gelfand–Naimark_theorem
R-module which is also a finite-dimensional vector space over F, then the Jordan blocks of x acting on V are cyclic submodules. (The Jordan blocks are all
Cyclic_module
Alternative mathematical ordering
In mathematics, a cyclic order is a way to arrange a set of objects in a circle.[nb] Unlike most structures in order theory, a cyclic order is not modeled
Cyclic_order
of Type III factors. According to Tomita–Takesaki theory, every vector which is cyclic for the factor and its commutant gives rise to a 1-parameter modular
Crossed_product
Canonical form of matrices over a field
reflects a minimal decomposition of the vector space into subspaces that are cyclic for A (i.e., spanned by some vector and its repeated images under A). Since
Frobenius_normal_form
Set of a ring's prime ideals
corresponds to a reduced variety; a cyclic module (one generator) corresponds to the operator having a cyclic vector (a vector whose orbit under T spans the
Spectrum_of_a_ring
Circulation density in a vector field
In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional
Curl_(mathematics)
Concepts from linear algebra
algebra, an eigenvector (/ˈaɪɡən-/ EYE-gən-) or characteristic vector is a (nonzero) vector that has its direction unchanged (or reversed) by a given linear
Eigenvalues_and_eigenvectors
M ) {\displaystyle L^{2}(M)} acted on by M {\displaystyle M} with a cyclic vector Ω {\displaystyle \Omega } . Let e N {\displaystyle e_{N}} be the projection
Subfactor
Commutative group in which all nonzero elements have the same order
non-negative integer (sometimes called the group's rank). Here, Z/pZ denotes the cyclic group of order p (or equivalently the integers mod p), and the superscript
Elementary_abelian_group
Definite integral of a scalar or vector field along a path
referred to in engineering as a cyclic integral. To establish a complete analogy with the line integral of a vector field, one must go back to the definition
Line_integral
Commutative group (mathematics)
underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler
Abelian_group
Four-dimensional number system
Quaternions can be used to represent vectors in three-dimensional space, which provides a definition of the quotient of two vectors. Quaternions were first described
Quaternion
Mathematics concept
In mathematics, cyclical monotonicity is a generalization of the notion of monotonicity to the case of vector-valued function. Let ⟨ ⋅ , ⋅ ⟩ {\displaystyle
Cyclical_monotonicity
Case in parallel computing
than the vector size. So, if the vector register is 128 bits, and the array type is 32 bits, the vector size is 128/32 = 4. All other non-cyclic dependencies
Automatic_vectorization
Representation of a tensor in Euclidean space
permutations in perpendicular directions yield the next vector in the cyclic collection of vectors: e x × e y = e z e y × e z = e x e z × e x = e y e y ×
Cartesian_tensor
matrix identity Vector space Linear combination Linear span Linear independence Scalar multiplication Basis Change of basis Hamel basis Cyclic decomposition
Outline_of_linear_algebra
Generalised alphabetical order
5 2 {\displaystyle x_{1}x_{2}^{3}x_{4}x_{5}^{2}} ) with their exponent vectors (here [1, 3, 0, 1, 2]). If n is the number of variables, every monomial
Lexicographic_order
In mathematics, vector subspace
in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply
Linear_subspace
Vector used in astronomy
corresponding cyclic coordinate in the three-dimensional Lagrangian of the system, there does not exist such a coordinate for the LRL vector. Thus, the conservation
Laplace–Runge–Lenz_vector
Partially ordered vector space, ordered as a lattice
mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice
Riesz_space
Parameterization of a rotation into a unit vector and angle
rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation
Axis–angle_representation
Open convex self-dual cones
be the restriction of L(a) to E0. T is self-adjoint and has 1 as a cyclic vector. So the commutant of T consists of polynomials in T (or a). By the spectral
Symmetric_cone
Russian mathematician (born 1940)
jfa.2008.05.011. Nikolski, Nikolai (2012). "In a shadow of the RH: Cyclic vectors of Hardy spaces on the Hilbert multidisc". Annales de l'Institut Fourier
Nikolai_Kapitonovich_Nikolski
Gallrado-Gutiérez, Eva A.; Partington, Jonathan R.; Segura, Dolores (2009), Cyclic vectors and invariant subspaces for Bergman and Dirichlet shifts (PDF), vol
Bergman_space
In linear algebra, generated subspace
linear hull or just span) of a set S {\displaystyle S} of elements of a vector space V {\displaystyle V} is the smallest linear subspace of V {\displaystyle
Linear_span
Algorithms and methods of plotting the Mandelbrot set on a computing device
subtract from n is in the interval [0, 1). For the coloring we must have a cyclic scale of colors (constructed mathematically, for instance) and containing
Plotting algorithms for the Mandelbrot set
Plotting_algorithms_for_the_Mandelbrot_set
Geometric transformation combining reflection and translation
reflection is an infinite cyclic group. Combining two equal glide reflections gives a pure translation with a translation vector that is twice that of the
Glide_reflection
American mathematician (born 1967)
completed her doctorate at Kent State in 1996; her dissertation, Cyclic Vectors and Extremal Vectors of Linear Operators, was supervised by Per Enflo. She was
Angela_Spalsbury
Vector space with a partial order
ordered vector space or partially ordered vector space is a real vector space equipped with a partial order that is compatible with the vector space operations
Ordered_vector_space
Order-preserving mathematical function
coefficient - measure of monotonicity in a set of data Total monotonicity Cyclical monotonicity Operator monotone function Monotone set function Absolutely
Monotonic_function
V=\mathbb {C} (X)} be the vector space over the complex numbers with a basis indexed by a finite set X {\displaystyle X} . If the cyclic group C n {\displaystyle
Cyclic_sieving
Matrix representing a Euclidean rotation
with standard coordinates v = (x, y), it should be written as a column vector, and multiplied by the matrix R: R v = [ cos θ − sin θ sin θ cos
Rotation_matrix
Square matrix constructed from a monic polynomial
F n {\displaystyle A:F^{n}\to F^{n}} makes F n {\displaystyle F^{n}} a cyclic F [ A ] {\displaystyle F[A]} -module, having a basis of the form { v , A
Companion_matrix
states that the ring of invariants of a finite group acting on a complex vector space is a polynomial ring if and only if the group is generated by pseudoreflections
Chevalley–Shephard–Todd theorem
Chevalley–Shephard–Todd_theorem
{\displaystyle (\pi ,V)} of a Banach algebra A {\displaystyle A} , a cyclic vector is a vector v ∈ V {\displaystyle v\in V} such that π ( A ) v {\displaystyle
Glossary of functional analysis
Glossary_of_functional_analysis
Four-sided polygon
to an inscribed circle. Cyclic quadrilateral: the four vertices lie on a circumscribed circle. A convex quadrilateral is cyclic if and only if opposite
Quadrilateral
Mathematical concept named for Ernst Witt
Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such
Witt_vector
topological vector space, also called an ordered TVS, is a topological vector space (TVS) X that has a partial order ≤ making it into an ordered vector space
Ordered topological vector space
Ordered_topological_vector_space
Computational problem used in cryptography
x_{n-1})} Micciancio introduced cyclic lattices in his work in generalizing the compact knapsack problem to arbitrary rings. A cyclic lattice is a lattice that
Short integer solution problem
Short_integer_solution_problem
Linear algebra matrix
n-1} . (Cyclic permutation of rows has the same effect as cyclic permutation of columns.) The last row of C {\displaystyle C} is the vector c {\displaystyle
Circulant_matrix
Type of infinitesimal in calculus
without a hole within it), then any irrotational vector field (defined as a C 1 {\displaystyle C^{1}} vector field v {\displaystyle \mathbf {v} } which curl
Exact_differential
Type of group in mathematics
group of SO(n) is cyclic of order 2, and the spin group Spin(n) is its universal cover. For n = 2 the fundamental group is infinite cyclic and the universal
Orthogonal_group
Branch of mathematics
K-theory and K-homology provide analogues of vector bundles and elliptic operators. Cyclic homology and cyclic cohomology provide noncommutative analogues
Noncommutative_geometry
Group that is also a differentiable manifold with group operations that are smooth
acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket
Lie_group
Symmetry of molecules of chemical compounds
turn divided into cyclic and dihedral groups and within a system the order of the dihedral group is twice that of the cyclic group. Cyclic groups only have
Molecular_symmetry
Fundamental theorem in condensed matter physics
same periodicity as the crystal, the wave vector k {\displaystyle \mathbf {k} } is the crystal momentum vector, e {\displaystyle e} is Euler's number, and
Bloch's_theorem
Norm on a vector space of matrices
mathematics, a norm in general is a function from a vector space to non-negative numbers. When the vector space comprises matrices, such norms are referred
Matrix_norm
Well-quasi-ordering of finite trees
theory Topics Glossary Category Key concepts Binary relation Boolean algebra Cyclic order Lattice Partially ordered set Preorder Total order Weak ordering Results
Kruskal's_tree_theorem
Open conjecture that no real circulant Hadamard matrix has order greater than 4
} The four cyclic shifts of the generating row and their negatives give the eight order-4 circulant Hadamard matrices. The all-ones vector is an eigenvector
Ryser's conjecture on circulant Hadamard matrices
Ryser's_conjecture_on_circulant_Hadamard_matrices
Representation theory of groups
regular representation λ (over a field K) is a linear representation on the K-vector space V freely generated by the elements of G, i.e. elements of G can be
Regular_representation
Correspondence between quaternions and 3D rotations
whose vector part is p, and then performing the quaternion conjugation. The vector part of the resulting pure quaternion is the desired vector r. Clearly
Quaternions and spatial rotation
Quaternions_and_spatial_rotation
Part of spectral theory
{\displaystyle (T^{n}\xi )} is dense in H, i.e. ξ {\displaystyle \xi } is a cyclic vector for T {\displaystyle T} , then the map U {\displaystyle U} defined by
Spectral theory of ordinary differential equations
Spectral_theory_of_ordinary_differential_equations
Generalization of vector spaces from fields to rings
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative)
Module_(mathematics)
Circle that passes through the vertices of a triangle
also called the circumscribed circle, is called a cyclic polygon, or in the special case n = 4, a cyclic quadrilateral. All triangles, rectangles, isosceles
Circumcircle
Formulation of classical mechanics
point particles with masses m1, m2, ..., mN, each particle has a position vector, denoted r1, r2, ..., rN. Cartesian coordinates are often sufficient, so
Lagrangian_mechanics
On chains and antichains in partial orders
theory Topics Glossary Category Key concepts Binary relation Boolean algebra Cyclic order Lattice Partially ordered set Preorder Total order Weak ordering Results
Dilworth's_theorem
functional analysis and order theory, a topological vector lattice is a Hausdorff topological vector space (TVS) X {\displaystyle X} that has a partial
Topological_vector_lattice
Antisymmetric permutation object acting on tensors
permutation, and 0 if any index is repeated. In three dimensions only, the cyclic permutations of (1, 2, 3) are all even permutations, similarly the anticyclic
Levi-Civita_symbol
Vector bundle existing over a Grassmannian
In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k {\displaystyle
Tautological_bundle
Soviet-Israeli-American mathematician
(2): 527–553. doi:10.1090/S0002-9947-1985-0792810-2. with Leon Brown: “Cyclic vectors in A–∞, Proc. Amer. Math. Soc., 1987, v. 101, 137–138. doi:10
Boris_Korenblum
Number taken as representative of a list of numbers
data set X, thought of as a vector x = (x1,…,xn), the dispersion about a point c is the "distance" from x to the constant vector c = (c,…,c) in the p-norm
Average
topics on mathematical permutations. Alternating permutation Circular shift Cyclic permutation Derangement Even and odd permutations—see Parity of a permutation
List_of_permutation_topics
Mathematical object
ideal lattices are a special class of lattices and a generalization of cyclic lattices. Ideal lattices naturally occur in many parts of number theory
Ideal_lattice
Phenomenon in crystallization
is often caused by accidents during growth. In the tetragonal system, cyclical contact twins are the most commonly observed type of twin, such as in rutile
Crystal_twinning
Visual depiction of a partially ordered set
theory Topics Glossary Category Key concepts Binary relation Boolean algebra Cyclic order Lattice Partially ordered set Preorder Total order Weak ordering Results
Hasse_diagram
Specialized notation for multivariable calculus
respect to a vector as a column vector or a row vector. Both of these conventions are possible even when the common assumption is made that vectors should be
Matrix_calculus
Group in which the order of every element is a power of p
acting on symplectic vector spaces. Richard Brauer classified all groups whose Sylow 2-subgroups are the direct product of two cyclic groups of order 4,
P-group
Abelian group related to division algebras
containing all roots of unity. The Brauer group BrR of the real numbers is the cyclic group of order two. There are just two non-isomorphic real division algebras
Brauer_group
Polyhedron with 20 faces
coordinates of the 12 vertices can be defined by the vectors defined by all the possible cyclic permutations and sign-flips of coordinates of the form
Icosahedron
order to generate a cyclic Hadamard core, the vector (of coefficients of) g ( x ) {\displaystyle g(x)} when operated upon with the cyclic shift operation
Coding theory approaches to nucleic acid design
Coding_theory_approaches_to_nucleic_acid_design
Special type of lattice
is a Boolean algebra if and only if n is square-free. A lattice-ordered vector space is a distributive lattice. Young's lattice given by the inclusion
Distributive_lattice
Method of encoding digital data on multiple carrier frequencies
which suffers from poor power efficiency Loss of efficiency caused by cyclic prefix/guard interval In OFDM, the subcarrier frequencies are chosen so
Orthogonal frequency-division multiplexing
Orthogonal_frequency-division_multiplexing
Matrices important in quantum mechanics and the study of spin
0 {\displaystyle \sigma _{0}} ), the Pauli matrices form a basis of the vector space of 2 × 2 {\displaystyle 2\times 2} Hermitian matrices over the real
Pauli_matrices
Geometry concept
groups, except for C2 and D1, which share abstract group Z2. All of the cyclic groups are abelian or commutative, but only two of the dihedral groups are:
Point groups in two dimensions
Point_groups_in_two_dimensions
Special subset of a partially ordered set
lattice of vector subspaces of a given vector space, ordered by inclusion. Explicitly, a linear filter on a vector space X is a family B of vector subspaces
Filter_(mathematics)
Picking a generator in each cyclic direct summand of B creates a p-basis of B, which is analogous to a basis of a vector space or a free abelian group
Basic_subgroup
Algebraic object with an ordered structure
of redirect targets Ordered ring Ordered topological vector space Ordered vector space – Vector space with a partial order Partially ordered ring – Ring
Ordered_field
Computing joint values of a kinematic chain from a known end position
usually support joint constraints. The most popular heuristic algorithms are cyclic coordinate descent (CCD) and forward and backward reaching inverse kinematics
Inverse_kinematics
Simple Lie group; the automorphism group of the octonions
equivalently, as the subgroup of SO(7) that preserves any chosen particular vector in its 8-dimensional real spinor representation (a spin representation)
G2_(mathematics)
Deformation mechanism in crystallines
operating slip depending on its (mis)orientation. Formation of slip bands under cyclic conditions is addressed as persistent slip bands (PSBs) where formation
Slip_bands_in_metals
Set with associative invertible operation
so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1 {\displaystyle 1} . Any cyclic group with n {\displaystyle
Group_(mathematics)
Concept in mathematics
reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that
Complex_reflection_group
Overview of mechanics based on the least action principle
considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system; it can also be called vectorial mechanics
Analytical_mechanics
Family of linear transformations
taking cyclic permutations of x, y, z components (i.e. change x to y, y to z, and z to x, repeat). These commutation relations, and the vector space of
Lorentz_transformation
Size of subsets in order theory
theory Topics Glossary Category Key concepts Binary relation Boolean algebra Cyclic order Lattice Partially ordered set Preorder Total order Weak ordering Results
Cofinality
Concept in mathematics
is cyclic; this implies that its Sylow subgroups are cyclic or generalized quaternion groups. Any group such that all Sylow subgroups are cyclic is called
Frobenius_group
Transformations induced by a mathematical group
polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify
Group_action
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