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Mathematical binary relation
In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing
Subsequence
Computer science problem
science, the longest increasing subsequence problem aims to find a subsequence of a given sequence in which the subsequence's elements are sorted in an ascending
Longest increasing subsequence
Longest_increasing_subsequence
Algorithmic problem on pairs of sequences
A longest common subsequence (LCS) is the longest subsequence common to all sequences in a set of sequences (often just two sequences). It differs from
Longest_common_subsequence
Bounded sequence in finite-dimensional Euclidean space has a convergent subsequence
bounded sequence in R n {\displaystyle \mathbb {R} ^{n}} has a convergent subsequence. An equivalent formulation is that a subset of R n {\displaystyle \mathbb
Bolzano–Weierstrass_theorem
On when a family of real, continuous functions has a uniformly convergent subsequence
defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions
Arzelà–Ascoli_theorem
Combinatorial problem
and computer science, in the longest alternating subsequence problem, one wants to find a subsequence of a given sequence in which the elements are in
Longest alternating subsequence
Longest_alternating_subsequence
Sufficiently long sequences of numbers have long monotonic subsequences
(r-1)(s-1)+1} contains a monotonically increasing subsequence of length r or a monotonically decreasing subsequence of length s. The proof appeared in the same
Erdős–Szekeres_theorem
Sorting algorithm
the algorithm efficiently computes the length of a longest increasing subsequence in a given array. The algorithm's name derives from a simplified variant
Patience_sorting
Probabilistic data structure
made possible by maintaining a linked hierarchy of subsequences, with each successive subsequence skipping over fewer elements than the previous one (see
Skip_list
Data mining technique
repeats, finding tandem repeats, and finding unique subsequences and missing (un-spelled) subsequences. Alignment problems: that deal with comparison between
Sequential_pattern_mining
Theorem
analysis about the Cesàro convergence of a subsequence of random variables (or functions) and their subsequences to an integrable random variable (or function)
Komlós'_theorem
Generalization of the concept of subsequence to the case of nets
subnet is a generalization of the concept of subsequence to the case of nets. The analogue of "subsequence" for nets is the notion of a "subnet". The definition
Subnet_(mathematics)
Solution to algorithmic problem
known as Hunt–McIlroy algorithm, is a solution to the longest common subsequence problem. It was one of the first non-heuristic algorithms used in diff
Hunt–Szymanski_algorithm
theory, a minimal prime is a prime number for which there is no shorter subsequence of its digits in a given base that form a prime. In base 10 there are
Minimal prime (recreational mathematics)
Minimal_prime_(recreational_mathematics)
Topological space where every sequence has a convergent subsequence
if every sequence of points in X {\displaystyle X} has a convergent subsequence converging to a point in X {\displaystyle X} . Every metric space is
Sequentially_compact_space
Metric used for testing NLP models
reference summaries. ROUGE-L: Longest Common Subsequence (LCS) based statistics. Longest common subsequence problem takes into account sentence-level structure
ROUGE_(metric)
Computer science metric of string similarity
distance are obtained by restricting the set of operations. Longest common subsequence (LCS) distance is edit distance with insertion and deletion as the only
Edit_distance
Contiguous part of a sequence of symbols
substring of "It was the best of times". In contrast, "Itwastimes" is a subsequence of "It was the best of times", but not a substring. Prefixes and suffixes
Substring
Sequence of data points over time
cluster) subsequence time series clustering (single timeseries, split into chunks using sliding windows) time point clustering Subsequence time series
Time_series
Mathematical constant
that every sequence of elements of that length contains a non-empty subsequence adding up to 0. In symbols, this is D ( G ) = min { N : ∀ ( { g n } n
Davenport_constant
Subsequence of video frames
Subsequence of video frames
Group_of_pictures
Subpermutation of a longer permutation
to the number pi), then π is said to contain σ as a pattern if some subsequence of the entries of π has the same relative order as all of the entries
Permutation_pattern
Sequence that contains itself as a subsequence
mathematics, a fractal sequence is one that contains itself as a proper subsequence. An example is 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3,
Fractal_sequence
Type of continuous linear operator
usually not compact, and bounded sequences need not have convergent subsequences. Compact operators partly restore this finite-dimensional behavior by
Compact_operator
Type of mathematical space
set is compact if and only if every infinite sequence in the set has a subsequence that converges to a point of the set. Likewise, whereas every real-valued
Compact_space
Prime numbers that occupy prime-numbered positions
known as higher-order primes or prime-indexed primes (PIPs), are the subsequence of prime numbers that occupy prime-numbered positions within the sequence
Super-prime
On convergent subsequences of functions that are locally of bounded total variation
uniformly bounded sequence of monotone real functions admits a convergent subsequence. In other words, it is a sequential compactness theorem for the space
Helly's_selection_theorem
Sequences of convex sets in a bounded set have convergent subsequences
contained in a bounded set, the theorem guarantees the existence of a subsequence { K n m } {\displaystyle \{K_{n_{m}}\}} and a convex set K {\displaystyle
Blaschke_selection_theorem
Guidance and navigation computer used in Apollo spacecraft
subsequence. Simple instructions, such as TC, executed in a single subsequence of 12 pulses. More complex instructions required several subsequences.
Apollo_Guidance_Computer
Computer science problem
data deduplication and plagiarism detection. Unlike the longest common subsequence problem, which finds insertions or deletions within the common text,
Longest_common_substring
Mathematics concept
are mathematical constants that describe the lengths of longest common subsequences of random strings. Although the existence of these constants has been
Chvátal–Sankoff_constants
Shell command for comparing file content
z From a longest common subsequence, it is only a small step to get diff-like output: if an item is absent in the subsequence but present in the first
Diff
Limit of some subsequence
mathematics, a subsequential limit of a sequence is the limit of some subsequence. Every subsequential limit is a cluster point, but not conversely. In
Subsequential_limit
Finite or infinite ordered list of elements
above and bounded from below, then the sequence is said to be bounded. A subsequence of a given sequence is a sequence formed from the given sequence by deleting
Sequence
Type of comparison sorting algorithm
into a subsequence of S {\displaystyle S} of length at most three. First, y 4 {\displaystyle y_{4}} is inserted into the three-element subsequence ( x 1
Merge-insertion_sort
finite alphabet Σ {\displaystyle \Sigma } , as partially ordered by the subsequence relation, is a well partial order. That is, if w 1 , w 2 , … ∈ Σ ∗ {\displaystyle
Higman's_lemma
Space-filling curve
{\displaystyle c} is replaced by a contiguous subsequence of the centers of these nine smaller squares. This subsequence is formed by grouping the nine smaller
Peano_curve
shortest sequence which has X and Y as subsequences. This is a problem closely related to the longest common subsequence problem. Given two sequences X = <
Shortest_common_supersequence
String metric for measuring edit distance
m+n-2L,} where L {\displaystyle L} is the length of a longest common subsequence. This formula does not apply unchanged to classical Levenshtein distance
Levenshtein_distance
Probability distribution
appears in the distribution of the length of the longest increasing subsequence of random permutations, as large-scale statistics in the Kardar-Parisi-Zhang
Tracy–Widom_distribution
Line-breaking algorithm used in the TeX typesetting package
optimum can be shown to be a special case of the convex least-weight subsequence problem, which can be solved in O ( n ) {\displaystyle O(n)} time. Methods
Knuth–Plass line-breaking algorithm
Knuth–Plass_line-breaking_algorithm
Hungarian-born American mathematician
University of Florida Thesis Exact and Asymptotic Enumeration of Permutations with Subsequence Conditions (1997) Doctoral advisor Richard P. Stanley
Miklós_Bóna
pattern is order-isomorphic to the subsequence. For instance, if π is the permutation 25314, then it has ten subsequences of length three, forming the following
Superpattern
Mathematical concept for comparing objects
starting point of an infinite increasing subsequence. The existence of such infinite increasing subsequences is sometimes taken as a definition for well-quasi-ordering
Well-quasi-ordering
{\displaystyle \mathbb {R} ^{n}} has a convergent subsequence, by the Bolzano–Weierstrass theorem. If these subsequences all have the same limit, then the original
Convergence_proof_techniques
Algorithm for solving the partition problem
is exactly 2 − 1 k {\displaystyle 2-{\frac {1}{k}}} . In the min-max subsequence problem, the input is a multiset of n numbers and an integer parameter
Largest_differencing_method
Sequence in which every finite string appears as a subsequence
Sequence in which every finite string appears as a subsequence
Disjunctive_sequence
Algorithm that generates an approximation of a random number sequence
next bit a one (or zero) with probability one-half; and any selected subsequence contains no information about the next element(s) in the sequence. K3
Pseudorandom_number_generator
Online database of integer sequences
representation of the sequence. The database is searchable by keyword, by subsequence, or by any of 16 fields. There is also an advanced search function called
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Notion of convergence in mathematics
structure). For in a topological space, when every subsequence of a sequence has itself a subsequence with the same subsequential limit, the sequence itself
Pointwise_convergence
Mathematics optimization problem
sequence of matrices and separate it into two subsequences. Find the minimum cost of multiplying out each subsequence. Add these costs together, and add in the
Matrix_chain_multiplication
term counts the maximal number of repeated blocks at the end of the subsequence from 1 to n−1 A090822 Carol numbers −1, 7, 47, 223, 959, 3967, 16127
List_of_integer_sequences
Data structure
Parsing Pattern matching Compressed pattern matching Longest common subsequence Longest common substring Sequential pattern mining Sorting String rewriting
Substring_index
Hybrid sorting algorithm based on insertion sort and merge sort
2002 for use in the Python programming language. The algorithm finds subsequences of the data that are already ordered (runs) and uses them to sort the
Timsort
Russian mathematician (1933–2024)
infinite symmetric groups and applications to the longest increasing subsequences. Vershik studied at Leningrad State University (later renamed to Saint
Anatoly_Vershik
theorem is a result from probabilistic combinatorics. It deals with the subsequences of a randomly uniformly drawn permutation from the set { 1 , 2 , … ,
Baik–Deift–Johansson_theorem
Eigenvalue algorithm
direction of an eigenvector associated with the dominant eigenvalue, then a subsequence ( b k ) {\displaystyle \left(b_{k}\right)} converges to an eigenvector
Power_iteration
number of increasing subsequences of the sequence. This is a measure of presortedness, and in particular measures how many subsequences must be merged to
Run_of_a_sequence
Functional programming construct
Parsing Pattern matching Compressed pattern matching Longest common subsequence Longest common substring Sequential pattern mining Sorting String rewriting
Pattern_matching
Parallel sorting algorithm
of the (green) subsequence with the element of the other (orange) subsequence at the respective index produces two bitonic subsequences. These two bitonic
Bitonic_sorter
Mathematical formula for the number of Young tableaux
and algorithm analysis; for example, the problem of longest increasing subsequences. A related formula gives the number of semi-standard Young tableaux,
Hook_length_formula
Resampling method
a subsequence, and there are M such subsequences (phases) multiplexed together. The dot product is the sum of the dot products of each subsequence with
Downsampling (signal processing)
Downsampling_(signal_processing)
Continuous real function on a closed interval has a maximum and a minimum
that there exists a subsequence that converges to a point in the domain. Use continuity to show that the image of the subsequence converges to the supremum
Extreme_value_theorem
Bijective correspondence in mathematics
increasing subsequence of σ1, ..., σn is equal to the length of the first row of P (and of Q). The length of the longest decreasing subsequence of σ1, .
Robinson–Schensted correspondence
Robinson–Schensted_correspondence
Relates three different kinds of weak compactness in a Banach space
following statements are equivalent: each sequence of elements of A has a subsequence that is weakly convergent in X each sequence of elements of A has a weak
Eberlein–Šmulian_theorem
Type of mathematical sequence
sequence with the property that for all values of N {\displaystyle N} , its subsequence x 1 , … , x N {\displaystyle x_{1},\ldots ,x_{N}} has a low discrepancy
Low-discrepancy_sequence
Sequence of points that get progressively closer to each other
x_{N}} ). In any metric space, a Cauchy sequence which has a convergent subsequence with limit s is itself convergent (with the same limit), since, given
Cauchy_sequence
Property of a partially ordered set
xn of real numbers in a closed interval [a, b] must have a convergent subsequence. This theorem can be proved by considering the set S = {s ∈ [a, b]
Least-upper-bound_property
Animated television series
Flammarion. It was produced by Paris-based Planet Nemo Animation and Subsequence Entertainment, in association with SRC Radio-Canada, TVOntario, Knowledge
Bali_(TV_series)
Compact embedding theorem concerning Sobolev spaces
theorem implies that any uniformly bounded sequence in W1,p(Ω; R) has a subsequence that converges in Lq(Ω; R). Stated in this form, in the past the result
Rellich–Kondrachov_theorem
Theorem in measure theory
{\displaystyle m} -dimensional Euclidean space), then there exist a subsequence ( μ n k ) {\displaystyle (\mu _{n_{k}})} and a probability measure μ
Prokhorov's_theorem
Series of clashes in Sagaing Region, Myanmar in (2021-present)
The Kalay clashes are a series of clashes between the Tatmadaw and armed civilians in the town of Kalay and surrounding villages in Kale Township during
Kalay_clashes
Normed vector space that is complete
in a Banach space. Either {xn}n∈ℕ has a weakly Cauchy subsequence, or it admits a subsequence equivalent to the standard unit vector basis of ℓ1. A complement
Banach_space
Characterizes the height of any finite partially ordered set
colorings in graphs, and to the Erdős–Szekeres theorem on monotonic subsequences. The height of a partially ordered set is defined to be the maximum cardinality
Mirsky's_theorem
Property of certain normed spaces
spaces stating that every bounded sequence of points in the space has a subsequence that is convergent in the mean (also known as Cesàro summation or limesable)
Banach-Saks_property
Contiguous sequence of errors occurring in a communications channel
the first and last symbols are in error and there exists no contiguous subsequence of m correctly received symbols within the error burst. The integer parameter
Burst_error
Concept in mathematics
employing an ultrafilter to bypass the need for repeated consideration of subsequences to ensure convergence. Ultralimits generalize Gromov–Hausdorff convergence
Ultralimit
Mathematical term in complex analysis
called a normal family if every sequence of functions in F contains a subsequence which converges uniformly on compact subsets of X to a continuous function
Normal_family
Graph representing a permutation
permutation graphs by using a longest decreasing subsequence algorithm. likewise, an increasing subsequence in a permutation corresponds to an independent
Permutation_graph
Family of graphs based on the Fibonacci sequence
contiguous subsequences. Within these two subsequences, the path can be constructed recursively by the same rule, linking the two subsequences at the ends
Fibonacci_cube
(containing subsequence 'a-b-c'), leading to production of Dicer substrate B·C targeting mRNA silencing target Y (containing independent subsequence 'w-x-y-z')
Small_conditional_RNA
Binary tree derived from a sequence of numbers
sequence, and recursively construct its left and right subtrees from the subsequences before and after this number. It is uniquely defined as a min-heap whose
Cartesian_tree
Measure on group representations
probabilistic problems, especially in the study of longest increasing subsequence of a random permutation. As a result of its importance in that area,
Plancherel_measure
Property of some mathematical functions
\limsup _{n}{\frac {a_{n}}{n}}\leq s^{*}} . If not, then there exists a subsequence ( a n k ) k {\displaystyle (a_{n_{k}})_{k}} , and an ϵ > 0 {\displaystyle
Subadditivity
Monoid of all words in the alphabet of positive integers modulo Knuth equivalence
given by Craige Schensted (1961) in his study of the longest increasing subsequence of a permutation. It was named the "monoïde plaxique" by Lascoux & Schützenberger
Plactic_monoid
On convergent subsequences of regulated functions
that is uniformly bounded in the total variation norm, there exists a subsequence ( f n ( k ) ) ⊆ ( f n ) ⊂ B V ( [ 0 , T ] ; X ) {\displaystyle
Fraňková–Helly selection theorem
Fraňková–Helly_selection_theorem
String distance measure
and the transposition of two adjacent characters; the longest common subsequence (LCS) distance allows only insertion and deletion, not substitution;
Jaro–Winkler_distance
Integer sequence
splits ("decays") into a sequence of "atomic elements", which are finite subsequences that never again interact with their neighbors. There are 92 elements
Look-and-say_sequence
American bible scholar (1934–2022)
in 1985, "Baptism in the Holy Spirit: The Issue of Separability and Subsequence", he argued that there is little biblical evidence for the classical
Gordon_Fee
Self-balancing binary search tree data structure
must be divided into subsequences S according to the insertion position of each element. For example sn,𝑙𝑒𝑓𝑡 is the subsequence of I that contains the
Red–black_tree
Fair item allocation procedure
sub-sequences: the first subsequence starts at agent 1 and ends at agent i − 1 {\displaystyle i-1} ; the latter subsequences start at i {\displaystyle
Round-robin_item_allocation
Mathematical version of an order change
An ascending run of a permutation is a nonempty increasing contiguous subsequence that cannot be extended at either end; it corresponds to a maximal sequence
Permutation
Speed of convergence of a mathematical sequence
for this sequence because one subsequence of error quotients starting from odd steps converges to 1 and another subsequence of quotients starting from even
Rate_of_convergence
Intersection graph for intervals on the real number line
non-overlapping times. Other applications include assembling contiguous subsequences in DNA mapping, and temporal reasoning. An interval graph is an undirected
Interval_graph
Fundamental theorem in probability theory and statistics
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample
Central_limit_theorem
2011 film
December 29, 2013. Tony Vilgotsky - Incest with sister led to unpleassant subsequences "AWARD WINNERS 2011". TADFF. Archived from the original on November 1
Father's_Day_(2011_film)
On distances between points on a circle
1 {\displaystyle n+1} distinct consecutive subsequences of length n {\displaystyle n} . Each subsequence occurs infinitely often with a certain frequency
Three-gap_theorem
Well-quasi-ordering of finite trees
of letters x i , … , x 2 i {\displaystyle x_{i},\ldots ,x_{2i}} is a subsequence of any later block x j , … , x 2 j {\displaystyle x_{j},\ldots ,x_{2j}}
Kruskal's_tree_theorem
Topics referred to by the same term
provides location information Longest increasing subsequence, algorithm to find the longest increasing subsequence in an array of numbers Laser Isotope Separation
LIS
applications to percolations and longest increasing subsequence. To study the longest increasing subsequence of a random permutation π, we generate it in an
Kingman's subadditive ergodic theorem
Kingman's_subadditive_ergodic_theorem
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