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a separable permutation is a permutation that can be obtained from the trivial permutation 1 by direct sums and skew sums. Separable permutations may
Separable_permutation
Subpermutation of a longer permutation
theoretical computer science, a (classical) permutation pattern is a sub-permutation of a longer permutation. Any permutation may be written in one-line notation
Permutation_pattern
Topics referred to by the same term
independent variables Separable permutation, a permutation that can be obtained by direct sums and skew sums of the trivial permutation Separable polynomial, a
Separability
Josephus permutation Parity of a permutation Separable permutation Stirling permutation Superpattern Transposition (mathematics) Unpredictable permutation Bijection
List_of_permutation_topics
Polynomial coprime with its derivative
In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct
Separable_polynomial
mathematics and computer science, a stack-sortable permutation (also called a tree permutation) is a permutation whose elements may be sorted by an algorithm
Stack-sortable_permutation
Process for achieving system stability
square if and only if the inactive cells have the pattern of a separable permutation. In any higher dimension, for any threshold, there is an analogous
Bootstrap_percolation
called separable permutations; they arise in the study of sortability theory, and can also be characterized as permutations avoiding the permutation patterns
Skew and direct sums of permutations
Skew_and_direct_sums_of_permutations
Mathematical connection between field theory and group theory
equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is by definition solvable by radicals
Galois_theory
Discrete mathematics decomposition
known bijections between equivalence classes of rectangulations and permutation classes. In the following table p 1 {\displaystyle p_{1}} and p 2 {\displaystyle
Rectangulations
pattern 312. Like the stack-sortable permutations, they are also a subset of the separable permutations, the permutations formed by recursive combinations
Layered_permutation
has a realizer in which one of the two permutations is the identity and the other is a separable permutation. It is known that a partial order P has
Series-parallel_partial_order
Graph formed by complementation and disjoint union
graph of a series-parallel partial order. A cograph is a permutation graph of a separable permutation. A cograph is a graph all of whose minimal chordal completions
Cograph
Quantum entanglement of more than 2 qubits
fully separable states and fully entangled states, there also exists the notion of partially separable states. The definitions of fully separable and fully
Multipartite_entanglement
Cryptographic hash function
through a 5→4 bit S-box. Mix adjacent 4-bit blocks using a maximum distance separable code over GF(24). Permute 4-bit blocks so that they will be adjacent to
JH_(hash_function)
Invariant of polynomial roots
a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the
Resolvent_(Galois_theory)
Mathematical integer sequence
cuts: The Schröder number S n {\displaystyle S_{n}} also counts the separable permutations of length n − 1. {\displaystyle n-1.} Schröder numbers are sometimes
Schröder_number
Chemical compounds with the same atoms but arranged and connected differently
can be switched with any other, either by a permutation that swaps just those two atoms, or by a permutation that swaps the two carbons and each hydrogen
Structural_isomer
the idea was first noticed, is that of finite groups (see primitive permutation group). Consider a group G and subgroups H and K, with K contained in
System_of_imprimitivity
Difficulties arising when analyzing data with many aspects ("dimensions")
lead to calculating all permutations of gene pairs for each individual or row. Given the formula for calculating the permutations of n items with a group
Curse_of_dimensionality
Criterion in quantum information theory
mechanical systems A {\displaystyle A} and B {\displaystyle B} , to be separable. It is also called the PPT criterion, for positive partial transpose.
Peres–Horodecki_criterion
Quantum state
|j\rangle \langle i|_{B}} is the permutation or flip operator that exchanges the two subsystems A and B. Werner states are separable for p ≥ 1⁄2 and entangled
Werner_state
*-algebra of bounded operators on a Hilbert space
Neumann algebra that acts on a separable Hilbert space is called separable. Note that such algebras are rarely separable in the norm topology. The von
Von_Neumann_algebra
Algorithm for modelling sequential data
less important tokens to be diminished. Because self-attention alone is permutation-invariant, transformers inject positional information, typically through
Transformer_(deep_learning)
Branch of mathematics that studies algebraic structures
Algebraically closed field Algebraic element Algebraic closure Separable extension Separable polynomial Normal extension Galois extension Abelian extension
List of abstract algebra topics
List_of_abstract_algebra_topics
Graph with tight clique-coloring relation
sequence and its permutation. The complement of a permutation graph is another permutation graph, for the reverse of the given permutation. Therefore, as
Perfect_graph
Specific algebraic group
smooth for large enough r {\displaystyle r} . In general one has to use separable closures instead of algebraic closures. If F {\displaystyle F} is a field
Algebraic_torus
Unique von Neumann algebra
In mathematics, there are up to isomorphism exactly two separably acting hyperfinite type II factors; one infinite and one finite. Murray and von Neumann
Hyperfinite_type_II_factor
Type of mathematical group
that the group of permutations on an infinite set with finite support is not residually finite as the subgroup with the permutations of signature 1 is
Residually_finite_group
Computational tool
arise in practice. As an example, a separable Hilbert space can only have a countable Schauder basis, but a non-separable Hilbert space may have an uncountable
Schauder_basis
Set of methods for supervised statistical learning
space, it often happens that the sets to discriminate are not linearly separable in that space. For this reason, it was proposed that the original finite-dimensional
Support_vector_machine
Correspondence between subfields and subgroups
equivalently, Galois extension, since any subextension of a separable extension is separable) if and only if H {\displaystyle H} is a normal subgroup of
Fundamental theorem of Galois theory
Fundamental_theorem_of_Galois_theory
Special low-energy state in quantum mechanics
the vicinity of SU(2)-singlet states with collective measurements. For separable states of N {\displaystyle N} spin- j {\displaystyle j} particles ( Δ
Singlet_state
Quantum state
...,N} is the number of 1's, and the summation is over all distinct permutations. A W-state is given as | W ⟩ = 1 N ( | 1000...000 ⟩ + | 0100...000 ⟩
Dicke_state
Process of producing small rectangular items of fixed dimensions
right, such that x1 ≤ ... ≤ xm. There is a permutation p on the indices such that, with this permutation, the rectangles would be ordered from bottom
Guillotine_cutting
Cryptographic scheme
preimage. Note that since we do not know how to construct a one-way permutation from any one-way function, this section reduces the strength of the cryptographic
Commitment_scheme
Collection of statistical models
closely approximate the permutation test's p-values: The approximation is particularly close when the design is balanced. Such permutation tests characterize
Analysis_of_variance
Class of error-correcting code
separable or MDS. Such codes, when they exist, are in some sense best possible. If C1 and C2 are two codes of length n and if there is a permutation p
Linear_code
Mathematics timeline
as Kneser earlier had, that manifolds are second countable. The term "separable manifold", to distinguish second countable manifolds, survived into the
Timeline_of_manifolds
Notion of convergence of random variables
sample which is used to estimate Q P ( h ) {\displaystyle Q_{P}(h)} . Permutation: Since r {\displaystyle r} and s {\displaystyle s} are picked identically
Uniform convergence in probability
Uniform_convergence_in_probability
Term in quantum mechanics
{F}}(\rho ,\sigma ).} It is possible to carry out the optimization over separable states, rather than over general quantum states and define F S , s e p
Fidelity_of_quantum_states
Weak form of quantum entanglement
metrology than separable states. Families of bound entangled states known analytically even for high dimension that outperform separable states for metrology
Bound_entanglement
they can be taken η j = ± 1 {\displaystyle \eta _{j}=\pm 1} . By cyclic permutation, there are four classes of solutions. Writing η = η 1 η 2 η 3 {\displaystyle
Three-wave_equation
Intrinsic quantum property of particles
bosons the prefactor (−1)2s will reduce to +1, for fermions to −1. This permutation postulate for N-particle state functions has most important consequences
Spin_(physics)
Measure of distinguishability between two quantum states
the family of separable states. A physical interpretation of the quantity is the optimal distinguishability of the state ρ from separable states. Clearly
Quantum_relative_entropy
Representation of a group or algebra that is a direct sum of simple representations
a semisimple operator) if and only if the minimal polynomial of T is separable; i.e., a product of distinct irreducible polynomials. Given a finite-dimensional
Semisimple_representation
Correspondence between quantum channels and quantum states
are (generalised) permutations since they are Clifford gates, which preserve the Pauli group. A generalised permutation is a permutation that also allows
Choi–Jamiołkowski_isomorphism
Mathematical function
closely related to algebraic curves and their function fields; in the separable case, they may also be studied via finite or ramified covers of the projective
Algebraic_function
Mathematical structures that allow quantum mechanics to be explained
state of the composite system is called a separable state. The density matrix of a bipartite system in a separable state can be expressed as ρ = ∑ k p k ρ
Mathematical formulation of quantum mechanics
Mathematical_formulation_of_quantum_mechanics
Figure of speech or rhetorical figure
which he calls "separable statements", are an integral part of the novelistic text and participate fully in its stylistics: "These separable statements are
Epiphrase
Construction in algebra
interesting projection usually denoted by ΠR(a) or εs(a) with image a separable subalgebra denoted by HR or Hs); a ( 1 ) S ( a ( 2 ) ) = ϵ ( 1 ( 1 ) a
Hopf_algebra
devoted to the problem of isomorphism and the algorithmic theory of permutation groups. In particular, a number of algorithms (which became already classical)
Sergei_Evdokimov
German mathematician (1882–1935)
Richard Brauer and received her degree in June 1935, with a thesis on separable normal extensions. After her doctorate, Stauffer worked as a teacher for
Emmy_Noether
Distance function defined between probability distributions
{\displaystyle \gamma } . The metric space (P∞(M), W∞) is complete if (M, d) is separable and complete. Here, P∞ is the space of all probability measures with bounded
Wasserstein_metric
In mathematics, a quantitative measure of the shape of a set of points
(For technical reasons, it is also convenient to assume that M is a separable space with respect to the metric d.) Let 1 ≤ p ≤ ∞. The pth central moment
Moment_(mathematics)
Describes approximate behavior of a function
{\mathcal {O}}(N\log N)} Fast Direct Solver for Partial Hierarchically Semi-Separable Matrices, J. Scientific Computing 57 (2013), no. 3, 477–501. Saket Saurabh
Big_O_notation
{\displaystyle n} -vertex oriented tree? Szymanski's conjecture: every permutation on the n {\displaystyle n} -dimensional doubly-directed hypercube graph
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Finite or infinite ordered list of elements
Ordinal-indexed sequence Recursion (computer science) Set (mathematics) Tuple Permutation If the inequalities are replaced by strict inequalities then this is
Sequence
Book by Marvin Minsky and Seymour Papert
perceptron network was "Network Synthesis". Other names included linearly separable logic, linear-input logic, threshold logic, majority logic, and voting
Perceptrons_(book)
Statistical ensemble of particles in thermodynamic equilibrium
phase space contains Ni! copies of the same actual state (formed by the permutation of the particles' different exact states). The chosen width of the orbital
Grand_canonical_ensemble
"Highly entangled" quantum state of 3 or more qubits
contrast, for the 3-qubit GHZ state, discarding a qubit results in a separable mixed state, and a Z measurement on a qubit will always leave behind a
Greenberger–Horne–Zeilinger state
Greenberger–Horne–Zeilinger_state
Method of data analysis
similar problems as principal component analysis, but finds additively separable components rather than successive approximations. Given a matrix E {\displaystyle
Principal_component_analysis
West Germanic language
fahren (to drive), erfahren (to experience). Many German verbs have a separable prefix, often with an adverbial function. In finite verb forms, it is
German_language
Distance between probability distributions
type is the plane with the taxicab metric. All Euclidean spaces and even separable Hilbert spaces have strong negative type. In the literature on kernel
Energy_distance
Algebraic structure with addition and multiplication
Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting
Ring_(mathematics)
(\mathbb {Z} /q\mathbb {Z} )^{\times }} and calculates the signature of a permutation. The shortest known proof also uses a simplified version of double counting
Proofs of quadratic reciprocity
Proofs_of_quadratic_reciprocity
Metric on a complex projective space endowed with Hermitian form
_{ij}=2(n+1)g_{ij}.} The common notions of separability apply for the Fubini–Study metric. More precisely, the metric is separable on the natural product of projective
Fubini–Study_metric
American mathematician (1922–2007)
"On the automorphisms of the group of Baire equivalences of a complete separable metric space", Bulletin of the Institute of Mathematics Academia Sinica
Hing_Tong
1854 = number of permutations of 7 elements with no fixed points, Mertens function zero 1855 = rencontres number: number of permutations of [7] with exactly
1000_(number)
algebra of H, and one shows as an exercise that Q is isomorphic to the permutation module on the right cosets. The 2013 paper referenced below proves that
Depth of noncommutative subrings
Depth_of_noncommutative_subrings
Term in mathematics
τ ( I ∙ ) = I ∙ {\displaystyle \tau (I_{\bullet })=I_{\bullet }} The permutation τ | I ∙ {\displaystyle \tau |_{I_{\bullet }}} coincides with − w ∙ {\displaystyle
Satake_diagram
preparative, preparatory, prepare, repair, reparable, reparation, reparative, separable, separate, separation, separator, sever, severable, several, severance
List of Latin verbs with English derivatives
List_of_Latin_verbs_with_English_derivatives
Mathematical term in group theory
automorphisms Aut(T2) can thus be thought of as the group of all length-preserving permutations σ of Σ* that also respect the initial segment relation: whenever a string x
Grigorchuk_group
Enterprise architecture framework
Gerrit Blaauw described how one could think about computer design as separable domains: architecture, implementation and realization. However, the concepts
DYA_framework
follows that A ( α ) {\displaystyle A(\alpha )} is the permutation matrix corresponding to the permutation of G which sends each σ i {\displaystyle \sigma _{i}}
Normal_basis
Crystalline chemical element or compound formed by geologic processes
metals, and congealed juices), compound (intimately mixed) and composite (separable). An early classification of minerals was given by Carl Linnaeus in his
Mineral
Class of statistics in estimation theory
U-statistics for exchangeable random elements taking values in a vector space (separable Banach space). Borovskikh, Yu. V. (1996). U-statistics in Banach spaces
U-statistic
Austrian composer and conductor (1883–1945)
Tell me, can one at all denote thinking and feeling as things entirely separable? I cannot imagine a sublime intellect without the ardor of emotion. — Webern
Anton_Webern
Prime-factor FFT algorithm Rader's FFT algorithm Bit-reversal permutation — particular permutation of vectors with 2m entries used in many FFTs. Butterfly diagram
List of numerical analysis topics
List_of_numerical_analysis_topics
Infinite sum
unconditionally summable, then the series remains convergent after any permutation σ : N → N {\displaystyle \sigma :\mathbb {N} \to \mathbb {N} } of the
Series_(mathematics)
Signal processing technique
Li, Jian (January 2011). "New Method of Sparse Parameter Estimation in Separable Models and Its Use for Spectral Analysis of Irregularly Sampled Data"
Spectral_density_estimation
Grammar of the Spanish language
the Spanish sentence without affecting its structure – although each permutation would, to a native speaker, give a subtly different shading of emphasis
Spanish_grammar
Grouping a set of objects by similarity
clusters in a data set Parallel coordinates Structured data analysis Linear separability Driver and Kroeber (1932). "Quantitative Expression of Cultural Relationships"
Cluster_analysis
General concept and operation in mathematics
The same duality pattern holds for a smooth projective variety over a separably closed field, using l-adic cohomology with Qℓ-coefficients instead. This
Duality_(mathematics)
Fourier-related mathematical transform
multidimensional algorithms. Three such algorithms follow. In pursuit of separability for efficiency, we consider the following transform (Bracewell, 1983)
Discrete_Hartley_transform
Method used in statistics, pattern recognition, and other fields
Talagrand's concentration inequality for product probability spaces). Data separability by classical linear discriminants simplifies the problem of error correction
Linear_discriminant_analysis
Function from sets to numbers
finite and translation-invariant Borel measure on an infinite-dimensional separable normed space is the trivial measure. However, it is possible to define
Set_function
Algorithms for matrix decomposition
exact NMF that works for the case where one of the factors W satisfies a separability condition. In Learning the parts of objects by non-negative matrix factorization
Non-negative matrix factorization
Non-negative_matrix_factorization
Theme in Tolkien's fiction
plant forms, as he had a "special fascination ... in the variations and permutations of flowers that are the evident kin of those I know". Among his artworks
Plants_in_Middle-earth
one must distinguish between an action by the cyclic group C3 or the permutation group S3. Alternatively, an index can be represented using the name of
List of irreducible Tits indices
List_of_irreducible_Tits_indices
Function for integral Fourier-like transform
unevenly spaced data Morlet wavelet Multiresolution analysis Noiselet Non-separable wavelet Scale space Scaled correlation Shearlet Short-time Fourier transform
Wavelet
American mathematician and educator (1921–2008)
property of a quantum system is defined by a Hermitian operator on a separable Hilbert space, that the only observable values of the property are the
Andrew_M._Gleason
set permutation A rearrangement of the elements of a set or sequence, where the structure of the set changes but the elements do not. permutation model
Glossary_of_set_theory
and machine learning as an example of a function that is not linearly separable. The table below represents a simple dataset where the relationship between
Multifactor dimensionality reduction
Multifactor_dimensionality_reduction
Type of fair division
required that partners' valuations over subsets of the cake be additively separable. Consider now the consensus 1/k-division case: any k>1 and equal weights
Consensus_splitting
SEPARABLE PERMUTATION
SEPARABLE PERMUTATION
Girl/Female
Arabic, Muslim, Sindhi
Inseparable Friend
Boy/Male
Sikh
Triumph for gods name, Triumph of the inseparable creator
Girl/Female
Muslim/Islamic
Inseparable friend
Boy/Male
Indian, Marathi
Separate
Girl/Female
Indian, Punjabi, Sikh
Love of the Inseparable Creator
Boy/Male
Muslim/Islamic
Inseparable friend
Boy/Male
Sikh
Triumph for gods name, Triumph of the inseparable creator
Girl/Female
Arabic, Muslim
Inseparable Friend
Biblical
a parable; governing
Girl/Female
Arabic, Muslim
Example; Allegory; Parable
Girl/Female
Indian
Inseparable
Girl/Female
Arabic
Separate
Girl/Female
Muslim
Example, Allegory, Parable
Girl/Female
Biblical
A parable, governing.
Boy/Male
Muslim
Considerate, Inseparable friend
Surname or Lastname
English
English : variant spelling of Rimer 1.German : variant of Riemer.German : habitational name for someone from Riem (now a suburb of Munich; formerly a separate town).
Girl/Female
Muslim
Inseparable friend
Surname or Lastname
English
English : occupational name for a maker of arms and armor, from Anglo-Norman French armer ‘arms-maker’ (Old French armier). Originally this was a separate name from Armour, but in due course the two became inextricably confused.
Girl/Female
Indian, Punjabi, Sikh
Triumph of the Inseparable Creator
Boy/Male
Arabic, Australian, Muslim
Considerate; Inseparable Friend
SEPARABLE PERMUTATION
SEPARABLE PERMUTATION
Boy/Male
Tamil
A Prince
Boy/Male
Biblical
Father of the wall; father of uprightness.
Girl/Female
Indian
Future, Futuristic
Female
English
Pet form of English Nancy, NANNA means "favor; grace." Compare with another form of Nanna.
Girl/Female
Persian
Name of a princess.
Boy/Male
Muslim
Good luck
Boy/Male
Hindu
An efficient horse rider
Girl/Female
Latin
Beloved.
Boy/Male
Tamil
Future
Boy/Male
Hindu
Loved by Vishnu, The curl on vishnus breast
SEPARABLE PERMUTATION
SEPARABLE PERMUTATION
SEPARABLE PERMUTATION
SEPARABLE PERMUTATION
SEPARABLE PERMUTATION
a.
Able to speak.
a.
Capable of being spoken; fit to be spoken.
a.
Invariably attached to some word, stem, or root; as, the inseparable particle un-.
adv.
In a reparable manner.
a.
Capable of being prepared.
n.
A kind of small nail used by shoemakers.
a.
Capable of being, or proper to be , repaid; due; as, a loan repayable in ten days; services repayable in kind.
a.
Inseparable.
n.
See Sperable.
a.
Capable of being separated, disjoined, disunited, or divided; as, the separable parts of plants; qualities not separable from the substance in which they exist.
a.
Separable.
a.
Capable of being severed.
a.
Reparable.
adv.
In an inseparable manner or condition; so as not to be separable.
a.
Not separable; incapable of being separated or disjoined.
a.
That may be secured.
p. a.
Disunited from the body; disembodied; as, a separate spirit; the separate state of souls.
a.
Capable of being overcome or conquered; surmountable.
v. t.
To represent by parable.
a.
Capable of being repaired, restored to a sound or good state, or made good; restorable; as, a reparable injury.