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In mathematics, a sparse polynomial (also lacunary polynomial or fewnomial) is a polynomial that has far fewer terms than its degree and number of variables
Sparse_polynomial
In mathematics, a polynomial with two terms
a binomial is a polynomial that is the sum of two terms, each of which is a monomial. It is the simplest kind of a sparse polynomial after the monomials
Binomial_(polynomial)
the CRC of the message modulo a sparse polynomial which is a multiple of the CRC polynomial. For CRC-32, the polynomial x123 + x111 + x92 + x84 + x64 +
Computation of cyclic redundancy checks
Computation_of_cyclic_redundancy_checks
Topics referred to by the same term
measure of the extent that a pattern contains gaps Lacunary polynomial, or sparse polynomial Petrovsky lacuna, in mathematics Laguna (disambiguation) This
Lacuna
Problem of determining whether polynomials are identical
runtime. A sparse PIT has at most m {\displaystyle m} nonzero monomial terms. A sparse PIT can be deterministically solved in polynomial time of the
Polynomial_identity_testing
Polynomial with only one term
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: A monomial, also
Monomial
Class of pseudorandom number generators
efficient implementation in software without the excessive use of sparse polynomials. They generate the next number in their sequence by repeatedly taking
Xorshift
Sparse binary polynomial hashing (SBPH) is a generalization of Bayesian spam filtering that can match mutating phrases as well as single words. SBPH is
Sparse binary polynomial hashing
Sparse_binary_polynomial_hashing
Greatest common divisor of polynomials
GCD or gcd) of two polynomials is a polynomial, of the highest possible degree, which is a factor of both the two original polynomials. This concept is
Polynomial greatest common divisor
Polynomial_greatest_common_divisor
languages in P/poly are sparse, there is a polynomial-time Turing reduction from any language in P/poly to a sparse language. There is a Turing reduction (as
Sparse_language
Tool used in probabilistic polynomial identity testing
probabilistic polynomial identity testing. Identity testing is the problem of determining whether a given multivariate polynomial is the 0-polynomial, the polynomial
Schwartz–Zippel_lemma
Algebraic encoding of graph connectivity
The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays
Tutte_polynomial
Problem of sorting pairs of numbers by their sum
of the problem include transit fare minimisation, VLSI design, and sparse polynomial multiplication. As with comparison sorting and integer sorting more
X_+_Y_sorting
Theorem in computational complexity theory
there exists a sparse language, such that a polynomial-time algorithm exists to solve the SAT problem by making O(1) queries to the sparse language oracle
Mahaney's_theorem
Class of problems solvable in polynomial time
exists a sparse language that is P-complete, then L = P. P is contained in BQP; it is unknown whether this containment is strict. Polynomial-time algorithms
P_(complexity)
Finding the roots of polynomials is a long-standing problem that has been extensively studied throughout the history and substantially influenced the
Polynomial_root-finding
Function in algebraic graph theory
The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a
Chromatic_polynomial
Signal processing technique
Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and
Compressed_sensing
Data-driven algorithm
Sparse identification of nonlinear dynamics (SINDy) is a data-driven algorithm for obtaining dynamical systems from data. Given a series of snapshots of
Sparse identification of non-linear dynamics
Sparse_identification_of_non-linear_dynamics
Mathematical construct in computer algebra
of polynomial equations because FGML does not take into account the sparsity of involved matrices. This has been fixed by the introduction of sparse FGLM
Gröbner_basis
Concepts from linear algebra
the roots of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree n is the characteristic polynomial of some companion
Eigenvalues_and_eigenvectors
Graph with almost the max amount of edges
4)-sparse. Streinu and Theran show that testing (k,l)-sparsity may be performed in polynomial time when k and l are integers and 0 ≤ l < 2k. For a graph
Dense_graph
Statistical analysis technique
Sparse principal component analysis (SPCA or sparse PCA) is a technique used in statistical analysis and, in particular, in the analysis of multivariate
Sparse_PCA
Moving average and polynomial regression method for smoothing data
regression or local polynomial regression, also known as moving regression, is a generalization of the moving average and polynomial regression. Its most
Local_regression
Methodic assignment of colors to elements of a graph
Birkhoff introduced the chromatic polynomial to study the coloring problem, which was generalised to the Tutte polynomial by W. T. Tutte, both of which are
Graph_coloring
Polynomial that has three terms
Monomial Binomial Multinomial Simple expression Compound expression Sparse polynomial Quadratic expressions are not always trinomials, the expressions'
Trinomial
Problem in combinatorial optimization
pseudo-polynomial time algorithm using dynamic programming. There is a fully polynomial-time approximation scheme, which uses the pseudo-polynomial time
Knapsack_problem
Discrete Fourier transform algorithm
computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors. As a result, it manages to reduce the complexity
Fast_Fourier_transform
Multiparty cryptographic process
verifiable secret sharing protocol to share the results of two random polynomial functions. Every party then verifies all the shares they received. If
Distributed_key_generation
Computational problem in graph theory
pseudo-polynomial and weakly polynomial is that a pseudo-polynomial bound may be polynomial in U {\displaystyle U} , but for a weakly polynomial bound
Maximum_flow_problem
Numerical method for solving physical or engineering problems
defined with polynomial and even non-polynomial shapes (e.g., ellipse or circle). Examples of methods that use higher degree piecewise polynomial basis functions
Finite_element_method
Decision problem in computer science
This solution does not count as polynomial time in complexity theory because B − A {\displaystyle B-A} is not polynomial in the size of the problem, which
Subset_sum_problem
Unsolved problem in structural complexity theory
Turing reductions, the existence of a sparse NP-complete language would imply an unexpected collapse of the polynomial hierarchy. As evidence towards the
Berman–Hartmanis_conjecture
Factorization algorithm
number field sieve is super-polynomial but sub-exponential in the size of the input. Suppose f is a k-degree polynomial over Q {\textstyle \mathbb {Q}
General_number_field_sieve
Set of problems solved by small circuits
languages in P/poly are sparse languages, there is a polynomial-time Turing reduction from any language in P/poly to a sparse language. Adleman's theorem
P/poly
Algorithmic complexity class
machine in exponential time, i.e., in O(2p(n)) time, where p(n) is a polynomial function of n. EXPTIME is one intuitive class in an exponential hierarchy
EXPTIME
Complexity class
problem with only polynomial overhead. If P is different from co-NP, then all of the co-NP-complete problems are not solvable in polynomial time. If there
Co-NP-complete
Mapping a graph onto itself without changing edge-vertex connectivity
list of generators, is polynomial-time equivalent to the graph isomorphism problem, and therefore solvable in quasi-polynomial time, that is with running
Graph_automorphism
Game in algorithmic game theory
be done in polynomial time, and for a graph with a bounded treewidth, this is also true for finding an optimal correlated equilibrium. Sparse games are
Succinct_game
Unrelated vertices in graphs
approximated to within any approximation ratio c < 1 in polynomial time; similar polynomial-time approximation schemes exist in any family of graphs
Independent set (graph theory)
Independent_set_(graph_theory)
Mathematical result
random. If you keep rolling the dice, you will eventually obtain one in polynomial random time. The proof below is based on the course notes of Afonso Bandeira
Johnson–Lindenstrauss_lemma
Relation between algebraic varieties and polynomial ideals
conditions for the existence of solutions to systems of multivariate polynomial equations over an algebraically closed field (such as the complex numbers
Hilbert's_Nullstellensatz
Number divisible only by 1 and itself
and the AKS primality test, which always produces the correct answer in polynomial time but is too slow to be practical. Particularly fast methods are available
Prime_number
including linear (standard and generalized) and nonlinear (quadratic, polynomial and general), as well as the SVD. Recent versions also include support
SLEPc
Algorithms for computing Gröbner bases
Jean-Charles Faugère, computes the Gröbner basis of an ideal of a multivariate polynomial ring. The algorithm uses the same mathematical principles as the Buchberger
Faugère's F4 and F5 algorithms
Faugère's_F4_and_F5_algorithms
Matrix equal to its conjugate-transpose
Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular
Hermitian_matrix
Partition of a graph's nodes into 2 disjoint subsets
minimum cut that separates the source and the sink are equal. There are polynomial-time methods to solve the min-cut problem, notably the Edmonds–Karp algorithm
Cut_(graph_theory)
Family of graphs whose shallow minors are sparse graphs
minors are sparse graphs. Many natural families of sparse graphs have bounded expansion. A closely related but stronger property, polynomial expansion
Bounded_expansion
On collapse of the polynomial hierarchy if NP is in non-uniform polynomial time class
Schuler, If NP has Polynomial-Size Circuits, then MA = AM Kannan, R. (1982). "Circuit-size lower bounds and non-reducibility to sparse sets". Information
Karp–Lipton_theorem
Mathematical optimization problem restricted to integers
a fixed constant, then the feasibility problem can be solved in time polynomial in m and log V. This is trivial for the case n=1. The case n=2 was solved
Integer_programming
delta-squared process — most useful for linearly converging sequences Minimum polynomial extrapolation — for vector sequences Richardson extrapolation Shanks transformation
List of numerical analysis topics
List_of_numerical_analysis_topics
Function defined by multiple sub-functions
function composed of power-law sub-functions Spline, a function composed of polynomial sub-functions, often constrained to be smooth at the joints between pieces
Piecewise_function
German mathematician
Zbl 0944.65131 Gatermann, Karin; Huber, Birkett (2002), "A family of sparse polynomial systems arising in chemical reaction systems", Journal of Symbolic
Karin_Gatermann
Function in discrete mathematics
converting between sample values and the coefficients of a trigonometric polynomial that interpolates those values. It is therefore a basic tool for numerical
Discrete_Fourier_transform
Computer system for solving algebra problems
fundamental integer and polynomial operations, such as the Schönhage–Strassen algorithm for fast multiplication of integers and polynomials. Integer factorization
Magma (computer algebra system)
Magma_(computer_algebra_system)
Computer algebra system
finite field elements, multivariable polynomials, rational functions, or polynomials modulo other polynomials. The main areas of application are multivariate
Fermat (computer algebra system)
Fermat_(computer_algebra_system)
Task of computing complete subgraphs
the maximum as can be found in polynomial time. Although much of this work has focused on independent sets in sparse graphs, a case that does not make
Clique_problem
Branch of mathematics
graphs, and especially the chromatic polynomial, the Tutte polynomial and knot invariants. The chromatic polynomial of a graph, for example, counts the
Algebraic_graph_theory
the discrete logarithm problem Polynomial long division: an algorithm for dividing a polynomial by another polynomial of the same or lower degree Risch
List_of_algorithms
Unsolved problem in the mathematics of graph coloring
conjecture is an unsolved problem on the distance between pairs of colorings of sparse graphs. It states that, for two different colorings of a graph of degeneracy
Cereceda's_conjecture
Discrete analog of a derivative
the polynomial is 36x. Subtracting out the third term: Without any pairwise differences, it is found that the 4th and final term of the polynomial is the
Finite_difference
Approximation method
hierarchical matrices (H-matrices) are used as data-sparse approximations of non-sparse matrices. While a sparse matrix of dimension n {\displaystyle n} can be
Hierarchical_matrix
Concept in computational complexity theory
as verifiers. A language L is in NEXPTIME if and only if there exist polynomials p and q, and a deterministic Turing machine M, such that For all x and
NEXPTIME
Method of cryptanalysis
for instance, that the linear polynomials in the key bits that are obtained during the attack will be unusually sparse. He has not yet supplied evidence
Cube_attack
Partition of a graph's nodes into cliques
number in perfect graphs in polynomial time. Another class of graphs in which the minimum clique cover can be found in polynomial time are the triangle-free
Clique_cover
Topics referred to by the same term
Routh–Hurwitz matrix, a square matrix constructed with coefficients of a real polynomial Parity-check matrix is often called H-matrix. This disambiguation page
H-matrix
Public key cryptosystem
The key point of this attack is to recover the private key as sparse univariate polynomials over the extension field F q n {\displaystyle \mathbb {F} _{q^{n}}}
Hidden_Field_Equations
Sparse graph with strong connectivity
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander
Expander_graph
Indian mathematician (born 1956)
an ISI highly cited researcher. He invented one of the first provably polynomial time algorithms for linear programming, which is generally referred to
Narendra_Karmarkar
Subfield of mathematical optimization
over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard
Convex_optimization
UP: Unambiguous Polynomial-Time. Hemaspaandra, Lane A.; Rothe, Jörg (June 1997). "Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete
UP_(complexity)
Methods of calculating definite integrals
interpolating functions are polynomials. In practice, since polynomials of very high degree tend to oscillate wildly, only polynomials of low degree are used
Numerical_integration
Method for solving certain optimization problems
restricted isometry property, which is generally a sufficient condition for sparse solutions. To find the parameters β = (β1, …,βk)T which minimize the Lp
Iteratively reweighted least squares
Iteratively_reweighted_least_squares
Mathematical concept
positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k2 − k + n produces a prime number. When k is equal to n, the value cannot
Lucky_numbers_of_Euler
Property in graph theory
The biclique-free graph families form one of the most general types of sparse graph family. They arise in incidence problems in discrete geometry, and
Biclique-free_graph
class, a non-uniform analogue of the class NP of problems solvable in polynomial time by a non-deterministic Turing machine. It is the non-deterministic
NP/poly
Array of numbers
the eigenvalues of a square matrix are the roots of its characteristic polynomial, det ( λ I − A ) {\displaystyle \det(\lambda I-A)} . Matrix theory is
Matrix_(mathematics)
Cycles in a graph that generate all cycles
positive weights, the minimum weight cycle basis may be constructed in polynomial time. In planar graphs, the set of bounded cycles of an embedding of the
Cycle_basis
Computational complexity of quantum algorithms
as the set of problems solvable by a (deterministic) Turing machine in polynomial time. Similarly, quantum complexity classes may be defined using quantum
Quantum_complexity_theory
Property of artificial neural networks
hidden layer. It states that if the layer's activation function is non-polynomial (which is true for common choices like the sigmoid function or ReLU),
Universal approximation theorem
Universal_approximation_theorem
Algorithm to be run on quantum computers
integer factorization problem in polynomial time, whereas the best known classical algorithms take super-polynomial time. It is unknown whether these
Quantum_algorithm
Method to solve optimization problems
polynomial-time algorithm? Does LP admit a strongly polynomial-time algorithm to find a strictly complementary solution? Does LP admit a polynomial-time
Linear_programming
Type of pseudoprime
The quadratic polynomial x 2 − 3 x − 5 {\displaystyle x^{2}-3x-5} , i.e. ( P , Q ) = ( 3 , − 5 ) {\displaystyle (P,Q)=(3,-5)} , has sparser pseudoprimes
Frobenius_pseudoprime
Routines for performing common linear algebra operations
chronological order of definition and publication, as well as the degree of the polynomial in the complexities of algorithms; Level 1 BLAS operations typically take
Basic Linear Algebra Subprograms
Basic_Linear_Algebra_Subprograms
Flaw in mathematical modelling
special case of a model that consists of a polynomial function, these parameters represent the degree of a polynomial. The essence of overfitting is to unknowingly
Overfitting
Function used in signal processing
discrete-time windows. A kth-order B-spline basis function is a piece-wise polynomial function of degree k − 1 that is obtained by k-fold self-convolution of
Window_function
Mapping arbitrary data to fixed-size values
division by a polynomial modulo 2 instead of an integer to map n bits to m bits. In this approach, M = 2m, and we postulate an mth-degree polynomial Z(x) = xm
Hash_function
Subset of artificial intelligence
polynomial time. There are two kinds of time complexity results: Positive results show that a certain class of functions can be learned in polynomial
Machine_learning
Vector quantization algorithm minimizing the sum of squared deviations
corroborated by the fact that the smoothed running time of k-means is polynomial. The "assignment" step is referred to as the "expectation step", while
K-means_clustering
Square matrix used to represent a graph or network
and A2 are similar and therefore have the same minimal polynomial, characteristic polynomial, eigenvalues, determinant and trace. These can therefore
Adjacency_matrix
Copy of a directed graph with redundant edges removed
NP-hard to construct, while transitive reductions can be constructed in polynomial time. Transitive reduction can be defined for an abstract binary relation
Transitive_reduction
Numerical integration method
roots of a Chebyshev polynomial and these values are used to construct a polynomial approximation for the function. This polynomial is then integrated exactly
Clenshaw–Curtis_quadrature
Class in computational complexity theory
affect the exact set of problems. Generically, reductions stricter than polynomial-time reductions are used, since all languages in P (except the empty language
P-complete
Property of graphs that depends only on abstract structure
of integers, such as the degree sequence of a graph. A polynomial, such as the Tutte polynomial of a graph. Easily computable graph invariants are instrumental
Graph_property
Software for approximating the roots of a polynomial with arbitrarily high precision
use of multiprecision. "Mpsolve takes advantage of sparsity, and has special hooks for polynomials that can be evaluated efficiently by straight-line
MPSolve
Number of stacked spheres in a pyramid
{\displaystyle n} positive square numbers, or as the values of a cubic polynomial. They can be used to solve several other counting problems, including
Square_pyramidal_number
Optimization problem
exchange for a sparser x, basis pursuit denoising is preferred. Basis pursuit problems can be converted to linear programming problems in polynomial time and
Basis_pursuit
Integers occurring in the coefficients of the Taylor series of 1/cosh t
function. The Euler numbers are related to a special value of the Euler polynomials, namely E n = 2 n E n ( 1 2 ) . {\displaystyle E_{n}=2^{n}E_{n}({\tfrac
Euler_numbers
Economical computational problem
polymatrix games, approximating a Nash equilibrium with polynomial precision is PPAD-hard, even for sparse win-lose games. Rank-1 bimatrix games have payoff
Nash_equilibrium_computation
Hilbert matrix A structured grid of rational values formed by the sum of polynomial denominators, modulated symmetrically and positively as approximation
List_of_named_matrices
SPARSE POLYNOMIAL
SPARSE POLYNOMIAL
Surname or Lastname
English
English : variant of Spear.
Female
English
English variant form of French Cerise, SHARISE means "cherry."Â
Surname or Lastname
Portuguese
Portuguese : occupational name from soeiro ‘swineherd’, Latin suerius.English : patronymic from a nickname for someone with reddish hair, from Anglo-Norman French sor ‘chestnut (color)’.
Surname or Lastname
English
English : nickname for a frugal person, from Middle English spare ‘sparing’, ‘frugal’.
Surname or Lastname
English
English : from the Norman personal name Serlo, Germanic Sarilo, Serilo. This was probably originally a byname cognate with Old Norse Sorli, and akin to Old English searu ‘armor’, meaning perhaps ‘defender’, ‘protector’.
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Telugu
Feel; Healthy; Touch
Surname or Lastname
English
English : patronymic from Spear.
Girl/Female
Hindu, Indian
Touch
Male
English
Short form of English unisex Paisley, PAISE means "church."Â
Surname or Lastname
English
English : metonymic occupational name for someone who made bags or purses or for an official in charge of expenditure, from Middle English purse (via Old English from Latin bursa).Scottish : variant of Purser.
Surname or Lastname
Irish (Kerry)
Irish (Kerry) : Anglicized form of Gaelic Mac Saoghair, which in turn may be a patronymic from a Gaelicized form of the Old English personal name Saeger (see 2 below).English : patronymic from a Middle English personal name Saher or Seir (see Sayer 1).Americanized form of French Cyr.Richard Sears came to Plymouth, MA, from England about 1630.
Surname or Lastname
English
English : variant of Sparks.
Boy/Male
Anglo Saxon Welsh
Spares.
Surname or Lastname
English
English : variant of Speake.
Surname or Lastname
English
English : patronymic from Spire 1.
Surname or Lastname
English
English : variant spelling of Pass.French : possibly a nickname from passe ‘sparrow’.
Surname or Lastname
English (Suffolk)
English (Suffolk) : unexplained.
Boy/Male
Afghan, Arabic, Iranian, Muslim, Parsi
Pious; Pure; Chaste; Holy
Boy/Male
American, British, English
Gallant
Surname or Lastname
English
English : patronymic from Spark 1.
SPARSE POLYNOMIAL
SPARSE POLYNOMIAL
Boy/Male
Muslim/Islamic
First Former
Girl/Female
Indian, Marathi, Sindhi, Tamil, Traditional
Precious Girl
Boy/Male
Tamil
Pramsu | பà¯à®°à®®à¯à®¸à¯‚
A scholar
Girl/Female
Muslim
Star in the Sky
Boy/Male
Tamil
Talaketu | தலாகேதà¯
Bhishma pitamaha
Girl/Female
Hindu, Indian
Grand; Stately
Girl/Female
Indian, Malayalam, Sanskrit
Fair
Boy/Male
British, English
Shouting Man's Meadow
Girl/Female
Muslim
Honorable
Boy/Male
Latin Welsh
From the cultivated land.
SPARSE POLYNOMIAL
SPARSE POLYNOMIAL
SPARSE POLYNOMIAL
SPARSE POLYNOMIAL
SPARSE POLYNOMIAL
n.
The right of bowling again at a full set of pins, after having knocked all the pins down in less than three bowls. If all the pins are knocked down in one bowl it is a double spare; in two bowls, a single spare.
imp. & p. p.
of Spare
n.
One who spares.
adv.
Sparsely; scatteredly; here and there.
v. t.
To sift through a sarse.
n.
Brilliancy; luster; as, the sparkle of a diamond.
superl.
Not refined; rough; rude; unpolished; gross; indelicate; as, coarse manners; coarse language.
v. t.
To sprinkle; to moisten by sprinkling; as, to sparge paper.
imp. & p. p.
of Parse
n.
One who parses.
superl.
Large in bulk, or composed of large parts or particles; of inferior quality or appearance; not fine in material or close in texture; gross; thick; rough; -- opposed to fine; as, coarse sand; coarse thread; coarse cloth; coarse bread.
n.
To emit sparks; to throw off ignited or incandescent particles; to shine as if throwing off sparks; to emit flashes of light; to scintillate; to twinkle; as, the blazing wood sparkles; the stars sparkle.
v. t.
Held in reserve, to be used in an emergency; as, a spare anchor; a spare bed or room.
v. t.
To emit in the form or likeness of sparks.
v. t.
Scanty; not abundant or plentiful; as, a spare diet.
superl.
Thinly scattered; set or planted here and there; not being dense or close together; as, a sparse population.
adv.
In a scattered or sparse manner.
v. t.
To inclose in a hearse; to entomb.
n.
A fine sieve; a searce.
v. t.
Being over and above what is necessary, or what must be used or reserved; not wanted, or not used; superfluous; as, I have no spare time.