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Polynomial function with logarithm terms
In mathematics, a polylogarithmic function in n is a polynomial in the logarithm of n, a k ( log n ) k + a k − 1 ( log n ) k − 1 + ⋯ + a 1 ( log
Polylogarithmic_function
Special mathematical function
Lerch transcendent. Polylogarithms should not be confused with polylogarithmic functions, nor with the offset logarithmic integral Li(z), which has the
Polylogarithm
Mathematical approximation of a function
of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the
Taylor_series
Notation describing limiting behavior in computational number theory
c]=e^{(c+o(1))\ln \ln n}=(\ln n)^{c+o(1)}\,} is a polylogarithmic function (a polynomial function of ln n); When α {\displaystyle \alpha } is 1 then
L-notation
Transcendental single-variable function
particularly in relation to the evaluation of many classes of logarithmic and polylogarithmic integrals, both definite and indefinite. They also have numerous applications
Clausen_function
Describes approximate behavior of a function
science is O ~ {\displaystyle {\tilde {O}}} (read soft-O), which hides polylogarithmic factors. There are two definitions in use: some authors use f ( n )
Big_O_notation
Estimate of time taken for running an algorithm
algorithm gets closer to the target word. An algorithm is said to run in polylogarithmic time if its time T ( n ) {\displaystyle T(n)} is O ( ( log n ) k
Time_complexity
Subexponential bound in computational complexity
is bounded by an exponential function of a polylogarithmic function. This generalizes the polynomials and the functions of polynomial growth, for which
Quasi-polynomial_growth
Complexity class of decision problems
machine by an algorithm whose space complexity is bounded by a polylogarithmic function in the size of the input. In other words, polyL = DSPACE((log n)O(1))
PolyL
Algorithmic problem on point-line incidence
factor, can reduce the given problem to subproblems whose size is a polylogarithmic function of n {\displaystyle n} , in time O ( n 4 / 3 ) {\displaystyle O(n^{4/3})}
Hopcroft's_problem
algorithm Pollard's rho algorithm for logarithms Polylogarithm Polylogarithmic function Prime number theorem Richter magnitude scale Grégoire de Saint-Vincent
Index_of_logarithm_articles
Stauffer, Allan (2019). "Definite Integral of Arctangent and Polylogarithmic Functions Expressed as a Series". Mathematics. 7 (1099): 1099. doi:10.3390/math7111099
List_of_definite_integrals
of RATMs has been advanced through the exploration of deterministic polylogarithmic time and space and two-sorted logic, a concept explored in depth by
Random-access_Turing_machine
Sum of the inverses of the positive cubes
1007/s11139-013-9528-5, S2CID 120943474. Broadhurst, D.J. (1998), "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ (
Apéry's_constant
Number, approximately 3.14
Plouffe, Simon (April 1997). "On the Rapid Computation of Various Polylogarithmic Constants" (PDF). Mathematics of Computation. 66 (218): 903–913. Bibcode:1997MaCom
Pi
Class in computational complexity theory
(for "Nick's Class") is the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors
NC_(complexity)
Conditions for switching order of integration in calculus
The integral of the product of the reciprocal function and the natural logarithm is a polylogarithmic integral. Fubini's theorem allows this to be evaluated
Fubini's_theorem
Difference between two successive prime numbers
2 ) , {\displaystyle g_{n}=O\!\left((\log p_{n})^{2}\right)\!,} a polylogarithmic growth rate slower than any exponent θ > 0. Cramér's model, under which
Prime_gap
Data structures used to track continuously moving geometric bodies
measures of performance. We say a quantity is small if it is a polylogarithmic function of n {\displaystyle n} , or is O ( n ϵ ) {\displaystyle O(n^{\epsilon
Kinetic_data_structure
Computer science metric for string similarity
S2CID 207046453. Andoni, Alexandr; Krauthgamer, Robert; Onak, Krzysztof (2010). Polylogarithmic approximation for edit distance and the asymmetric query complexity
Levenshtein_distance
Formula for computing the nth base-16 digit of π
Peter B.; Plouffe, Simon (1997). "On the Rapid Computation of Various Polylogarithmic Constants". Mathematics of Computation. 66 (218): 903–913. doi:10
Bailey–Borwein–Plouffe formula
Bailey–Borwein–Plouffe_formula
Class in computational complexity theory
If we use NC reductions, that is, reductions that can operate in polylogarithmic time on a parallel computer with a polynomial number of processors
P-complete
Type of Turing reduction
{\displaystyle AC_{0}} or N C 0 {\displaystyle NC_{0}} circuits, or polylogarithmic projections where each subsequent reduction notion is weaker than the
Many-one_reduction
Topics referred to by the same term
group. NC (complexity), the set of decision problems decidable in polylogarithmic time on a parallel computer with a polynomial number of processors
NC
1999 paper titled Computationally Private Information Retrieval with Polylogarithmic Communication, where it was used in a private information retrieval
Phi-hiding_assumption
Number, approximately 0.916
mathworld.wolfram.com. Retrieved 2024-10-02. Broadhurst, D. J. (1998). "Polylogarithmic ladders, hypergeometric series and the ten millionth digits of ζ(3)
Catalan's_constant
Type of error-correcting code
Using LDCs with polynomial codeword length and polylogarithmic query complexity, one can take a function L : { 0 , 1 } n → { 0 , 1 } {\displaystyle L:\{0
Locally_decodable_code
Model of computation
are restricted not only to having polynomial-size but also to having polylogarithmic depth. The class AC is defined similarly to NC, however gates are allowed
Boolean_circuit
Mathematical constant
Goyanes Guillera (2007). "Construction of binomial sums for π and polylogarithmic constants inspired by BBP formulas" (PDF). Applied Math. E-Notes. 7:
Natural_logarithm_of_2
Leonid A.; Szegedy, Mario (1991-01-01). "Checking computations in polylogarithmic time". Proceedings of the twenty-third annual ACM symposium on Theory
Verifiable_computing
Algorithm checking for prime numbers
twelfth power of the number of digits in n times a factor that is polylogarithmic in the number of digits. However, this upper bound was rather loose;
AKS_primality_test
Model of computational complexity
R_{0}(f)=O(R_{2}(f)^{2}\log R_{2}(f))} . This relationship is optimal up to polylogarithmic factors. As for quantum decision tree complexities, D ( f ) = O ( Q
Decision_tree_model
Algorithmic problem of finding non-crossing drawings
logarithmic update-time lower bound by Pătrașcu and Demaine, and a polylogarithmic update-time algorithm by Holm and Rotenberg, improving on sub-linear
Planarity_testing
Optimization problem in computer science
high-dimensional Euclidean space using polynomial preprocessing and polylogarithmic search time. The simplest solution to the NNS problem is to compute
Nearest_neighbor_search
Algorithm for shuffling a finite sequence
Fisher-Yates shuffle to get a parallel algorithm with linear work and polylogarithmic depth. The asymptotic time and space complexity of the Fisher–Yates
Fisher–Yates_shuffle
Set of problems in computational complexity theory
are restricted not only to having polynomial-size but also to having polylogarithmic depth. The class AC is defined similarly to NC, however gates are allowed
Complexity_class
Computational problem of interest in computer science
Nikhil; Buchbinder, Niv; Madry, Aleksander; Naor, Joseph (2015). "A polylogarithmic-competitive algorithm for the k-server problem" (PDF). Journal of the
K-server_problem
Algorithm used by Google Search to rank web pages
each node processes and sends a number of bits per round that are polylogarithmic in n, the network size. The Google Toolbar long had a PageRank feature
PageRank
Information retrieval using cryptography
Markus (1999). "Computationally Private Information Retrieval with Polylogarithmic Communication". Advances in Cryptology – EUROCRYPT '99. Prague, Czech
Private_information_retrieval
polynomial time by a Merlin–Arthur protocol NC Solvable efficiently (in polylogarithmic time) on parallel computers NE Solvable by a non-deterministic machine
List_of_complexity_classes
of polylogarithmic fan-in at the inputs, connected to a single gate computing some symmetric (not depending on the order of the inputs) function. These
ACC0
System with multiple networked computers
in parallel (see speedup). If a decision problem can be solved in polylogarithmic time by using a polynomial number of processors, then the problem is
Distributed_computing
Complexity class
NP-Complete problems such as SAT are known to be complete even under polylogarithmic time projections. It is known, however, that AC0 reductions define
NP-completeness
Type of algorithm
ISSN 0895-4801. Halperin, Eran; Krauthgamer, Robert (June 9, 2003). "Polylogarithmic inapproximability". Proceedings of the thirty-fifth annual ACM symposium
Parameterized approximation algorithm
Parameterized_approximation_algorithm
Theorem in computational complexity theory
Lance; Levin, Leonid; Szegedy, Mario (1991), "Checking computations in polylogarithmic time", STOC '91: Proceedings of the twenty-third annual ACM symposium
PCP_theorem
Problem of finding obscured edges in a wire-frame 3D model
hidden-line problem is in the complexity class NC, i.e., it can be solved in polylogarithmic time by using a polynomial number of processors. Hidden-surface algorithms
Hidden-line_removal
Theoretically, all algorithms in PAM are work-efficient and have polylogarithmic depth. PAM uses underlying persistent tree structure such that multi-versioning
PAM_library
Hypothesis in computational complexity theory
implication is that "fast" parallel computers (i.e. those that run in polylogarithmic time) recognize exactly the languages in polyL. It was proven in 1978
Parallel_computation_thesis
Algorithmic optimization method
time is polylogarithmic, leading to a total time for the parametric search that is slower than the decision algorithm by only a polylogarithmic factor
Parametric_search
Set of points touching all convex bodies of unit volume
possible to construct a Danzer set of growth rate that is within a polylogarithmic factor of O ( r d ) {\displaystyle O(r^{d})} . For instance, overlaying
Danzer_set
Hypothesis in computational complexity theory
Markus (1999). "Computationally Private Information Retrieval with Polylogarithmic Communication". In Stern, Jacques (ed.). Advances in Cryptology — EUROCRYPT
Computational hardness assumption
Computational_hardness_assumption
Long dense subsets of the integers contain arbitrarily large arithmetic progressions
Ben; Tao, Terence (2017). "New bounds for Szemerédi's theorem, III: A polylogarithmic bound for r4(N)". Mathematika. 63 (3): 944–1040. arXiv:1705.01703.
Szemerédi's_theorem
Mathematical method in graph theory
M. R.; King, V. (1995). "Randomized dynamic graph algorithms with polylogarithmic time per operation". Proceedings of the twenty-seventh annual ACM symposium
Euler_tour_technique
Data structure that maintains info about the connected components of a graph
M.; King, V.; Mountjoy, B. (2013). Dynamic graph connectivity in polylogarithmic worst case time. Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium
Dynamic_connectivity
\Omega (N)} worst-case access overheads. Some ORAM constructions with polylogarithmic worst-case computational overheads are. The constructions of were in
Oblivious_RAM
Filling in missing entries of a matrix
Tao. They achieve bounds that differ from the optimal bounds only by polylogarithmic factors by strengthening the assumptions. Instead of the incoherence
Matrix_completion
Open-source library for pattern matching in text
2020. Andoni, Alexandr; Krauthgamer, Robert; Onak, Krzysztof (2010). Polylogarithmic approximation for edit distance and the asymmetric query complexity
TRE_(computing)
O(m\log n)} . One possible parallelisation of this algorithm yields a polylogarithmic time complexity, i.e. T ( m , n , p ) ⋅ p ∈ O ( m log n ) {\displaystyle
Parallel algorithms for minimum spanning trees
Parallel_algorithms_for_minimum_spanning_trees
Fast summation method in mathematics
P. B. Borwein and S. Plouffe, On the rapid computation of various polylogarithmic constants. Math. Comp., Vol. 66 (1997). R. P. Brent and E. M. McMillan
FEE_method
On unit fractions adding to 4/n
over the prime numbers up to n {\displaystyle n} ) is upper bounded polylogarithmically in n {\displaystyle n} . For some other Diophantine problems, the
Erdős–Straus_conjecture
Varying methods used to calculate pi
Plouffe, Simon (April 1997). "On the Rapid Computation of Various Polylogarithmic Constants" (PDF). Mathematics of Computation. 66 (218): 903–913. Bibcode:1997MaCom
Approximations_of_pi
Computer science award
Zuckerman, David (June 2016). "Explicit two-source extractors and resilient functions". Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing
Gödel_Prize
Type of error-correcting code
{\displaystyle c\in (0,1)} . The next nearly linear goal is linear up to a polylogarithmic factor; n = poly ( log k ) ∗ k {\displaystyle n={\text{poly}}(\log
Locally_testable_code
Fewest cliques covering a graph's edges
better than the trivial O ( n 2 ) {\displaystyle O(n^{2})} by only a polylogarithmic factor. Researchers in this area have also investigated the computational
Intersection number (graph theory)
Intersection_number_(graph_theory)
Technique in parallel algorithms
a variety of problems, with the goal of designing highly parallel (polylogarithmic depth), work-efficient (linear in the sequential running time) algorithms
Tree_contraction
Mathematical proof about the permanent of matrices
(informally, whether every polynomially-solvable problem can be solved by a polylogarithmic-time parallel algorithm) and Ketan Mulmuley has suggested an approach
♯P-completeness of 01-permanent
♯P-completeness_of_01-permanent
algorithm, which performs a linear number of violation tests and a polylogarithmic number of basis evaluations. The basis evaluations for g may be performed
LP-type_problem
POLYLOGARITHMIC FUNCTION
POLYLOGARITHMIC FUNCTION
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Male
Egyptian
, the son of the functionary Heknofre.
Male
Egyptian
, a great functionary.
Male
Egyptian
, Functionary of the Interior.
Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, an Egyptian functionary.
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Biblical
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Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Male
Celtic
, great justiciary, or functionary.
Male
Egyptian
, a high Egyptian functionary.
POLYLOGARITHMIC FUNCTION
POLYLOGARITHMIC FUNCTION
Girl/Female
Scottish
Scottish surname and place name.
Girl/Female
Hindu
Wife of Lord Shiva, Goddess Parvati
Boy/Male
German
Wise Man; Old Man
Boy/Male
Hindu
Surname or Lastname
English
English : variant of Pearce.
Boy/Male
Indian
Sun
Girl/Female
Hindu, Indian, Kannada, Tamil
Goddess Parvati; Sun; The First; Lord of the Sun
Girl/Female
Arabic, Muslim
Sunnah; Practice
Male
German
Low German form of Old High German Gebhard, GEBBERT means "gift of strength."
Surname or Lastname
English
English : patronymic from the Norman personal name Hamo (see Hammond).
POLYLOGARITHMIC FUNCTION
POLYLOGARITHMIC FUNCTION
POLYLOGARITHMIC FUNCTION
POLYLOGARITHMIC FUNCTION
POLYLOGARITHMIC FUNCTION
n.
The doctrine that all the functions of a living organism are due to an unknown vital principle distinct from all chemical and physical forces.
n.
One deputed or authorized to perform the functions of another; a substitute in office; a deputy.
a.
Having relation to growth or nutrition; partaking of simple growth and enlargement of the systems of nutrition, apart from the sensorial or distinctively animal functions; vegetal.
pl.
of Functionary
n.
Fig.: Any cavity, or hollow place, in which any function may be conceived of as operating.
n.
One charged with the performance of a function or office; as, a public functionary; secular functionaries.
a.
Destitute of function, or of an appropriate organ. Darwin.
v. i.
Alt. of Functionate
a.
Of or pertaining to the vessels of animal and vegetable bodies; as, the vascular functions.
a.
Belonging or relating to life, either animal or vegetable; as, vital energies; vital functions; vital actions.
prep.
Acting as a substitute; -- said of abnormal action which replaces a suppressed normal function; as, vicarious hemorrhage replacing menstruation.
a.
Pertaining to the function of an organ or part, or to the functions in general.
adv.
In a functional manner; as regards normal or appropriate activity.
v. i.
To execute or perform a function; to transact one's regular or appointed business.
n.
The appropriate action of any special organ or part of an animal or vegetable organism; as, the function of the heart or the limbs; the function of leaves, sap, roots, etc.; life is the sum of the functions of the various organs and parts of the body.
a.
Of, pertaining to, or designating, certain secret tribunals which flourished in Germany from the end of the 12th century to the middle of the 16th, usurping many of the functions of the government which were too weak to maintain law and order, and inspiring dread in all who came within their jurisdiction.
n.
A quantity so connected with another quantity, that if any alteration be made in the latter there will be a consequent alteration in the former. Each quantity is said to be a function of the other. Thus, the circumference of a circle is a function of the diameter. If x be a symbol to which different numerical values can be assigned, such expressions as x2, 3x, Log. x, and Sin. x, are all functions of x.
n.
A certain function relating to a system of forces and their points of application, -- first used by Clausius in the investigation of problems in molecular physics.
v. t.
To assign to some function or office.
a.
Pertaining to, or connected with, a function or duty; official.