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Repeated application of an operation to a sequence
In mathematics, an iterated binary operation is an extension of a binary operation on a set S to a function on finite sequences of elements of S through
Iterated_binary_operation
Mathematical operation with two operands
a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation
Binary_operation
Mathematical operation with only one operand
Array: ,$array Unary function Binary operation Iterated binary operation Binary function Ternary operation Arity Operation (mathematics) Operator (programming)
Unary_operation
Mathematical operation that combines three elements to produce another element
odd integers from 1 through 9. Unary operation Unary function Binary operation Iterated binary operation Binary function Median algebra or Majority function
Ternary_operation
Inverse function to a tower of powers
computer science, lg* is often used to indicate the binary iterated logarithm, which iterates the binary logarithm (with base 2 {\displaystyle 2} ) instead
Iterated_logarithm
Function that takes two inputs
transformation above. Arity Unary operation Unary function Binary operation Iterated binary operation Ternary operation Clarke, Bertrand; Fokoue, Ernest;
Binary_function
Number expressed in the base-2 numeral system
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols
Binary_number
Addition of several numbers or other values
1, c, d Capital-pi notation Einstein notation Iverson bracket Iterated binary operation Kahan summation algorithm Product (mathematics) Summation by parts
Summation
Mathematical form
Infinite product Iterated binary operation – Repeated application of an operation to a sequence Multiplication – Arithmetical operation Here, "formal" means
Product_(mathematics)
Function that takes one argument
hyperbolic functions. Arity Binary function Binary operation Iterated binary operation Ternary operation Unary operation Foundations of Genetic Programming
Unary_function
Algorithm for fast exponentiation
which is equal to the number of 1s in the binary representation of n. This logarithmic number of operations is to be compared with the trivial algorithm
Exponentiation_by_squaring
Logical connective AND
a n {\displaystyle a_{1},\ldots ,a_{n}} can be denoted as an iterated binary operation using a "big wedge" ⋀ (Unicode U+22C0 ⋀ N-ARY LOGICAL AND): ⋀
Logical_conjunction
Logical connective OR
a n {\displaystyle a_{1},\ldots ,a_{n}} can be denoted as an iterated binary operation using a larger ⋁ (Unicode U+22C1 ⋁ N-ARY LOGICAL OR): ⋁ i = 1
Logical_disjunction
Variant of heap data structure
binary heap is a heap data structure that takes the form of a binary tree. Binary heaps are a common way of implementing priority queues. The binary heap
Binary_heap
Rooted binary tree data structure
complexity of operations on the binary search tree is linear with respect to the height of the tree. Binary search trees allow binary search for fast
Binary_search_tree
Family of higher-order functions
proving that iterations can be reduced to folds: y f = foldr (\_ -> f) undefined (repeat undefined) Aggregate function Iterated binary operation Catamorphism
Fold_(higher-order_function)
Result from multiplying no factors
evaluates to 4 (* 2) ; evaluates to 2 (*) ; evaluates to 1 Iterated binary operation Empty function Nešetřil, Jaroslav; Matoušek, Jiří (1998). Invitation
Empty_product
Infinite sum
compositions of analytic functions Infinite expression Infinite product Iterated binary operation List of mathematical series Prefix sum Sequence transformation
Series_(mathematics)
Set of elements in any of some sets
Set of elements common to all of some sets Iterated binary operation – Repeated application of an operation to a sequence List of set identities and relations –
Union_(set_theory)
Summation where the number of terms is zero
space V={0} to have a basis, namely the empty set. Empty product Iterated binary operation Empty function Harper, Robert (2016). Practical Foundations for
Empty_sum
Search algorithm finding the position of a target value within a sorted array
In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position
Binary_search
Mathematical concept
where it has a simple pole. Infinite products in trigonometry Iterated binary operation Infinite expression Infinite series Pentagonal number theorem
Infinite_product
Algebraic structure
In mathematics, particularly abstract algebra, a binary operation • on a set is flexible if it satisfies the flexible identity: a ∙ ( b ∙ a ) = ( a ∙ b
Flexible_algebra
Self-balancing binary search tree data structure
In computer science, a red–black tree is a self-balancing binary search tree data structure noted for fast storage and retrieval of ordered information
Red–black_tree
Software library for the C++ programming language
certain level of iterator (and therefore will work on any container that provides an interface by iterators). Searching algorithms like binary_search and lower_bound
Standard_Template_Library
Class of algorithms
before going to the next sibling. To traverse binary trees with depth-first search, the following operations are performed at each node: If the current node
Tree_traversal
Exponent of a power of two
the binary logarithm of 1 is 0, the binary logarithm of 2 is 1, the binary logarithm of 4 is 2, and the binary logarithm of 32 is 5. The binary logarithm
Binary_logarithm
Number represented as a0+1/(a1+1/...)
continued fraction formula – Mathematical identity Iterated binary operation – Repeated application of an operation to a sequence Klein polyhedron – Shape in the
Simple_continued_fraction
Operation on mathematical functions
a_{nm})).} A unary operation always commutes with itself, but this is not necessarily the case for a binary (or higher arity) operation. A binary (or higher arity)
Function_composition
Concept in order theory
meet of any non-empty finite set, by the technique described in iterated binary operations. Alternatively, if the meet defines or is defined by a partial
Join_and_meet
Limited form of tree data structure
In computer science, a binary tree is a tree data structure in which each node has at most two children, referred to as the left child and the right child
Binary_tree
Computer science data structure
a heap is the binary heap, in which the tree is a complete binary tree (see figure). The heap data structure, specifically the binary heap, was introduced
Heap_(data_structure)
Method for division with remainder
potentially cuts down the numbers of operations by up to half and lets it be executed faster. The basic algorithm for binary (radix 2) non-restoring division
Division_algorithm
Type of analog-to-digital converter
and during each iteration i produces the following approximation: ith approximation: xi = xi−1 − sgn(xi−1 − x)/2i where the binary signum function sgn
Successive-approximation_ADC
different summation rules if the series is not absolutely convergent. Iterated binary operation Infinite word Decimal expansion Power series Infinite compositions
Infinite_expression
Ordering of binary values, used for positioning and error correction
The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that
Gray_code
IEEE standard for floating-point arithmetic
of 2019[update], augmented arithmetic operations for the binary formats are also recommended. These operations, specified for addition, subtraction and
IEEE_754
Method of notation of very large integers
to represent a ↑ n b {\displaystyle a\uparrow ^{n}b} as iterated exponentiation of iterated exponentiation for any a {\displaystyle a} , n {\displaystyle
Knuth's_up-arrow_notation
Use of functions that call themselves
{ traverse(); } } This code is both recursion and iteration - the files and directories are iterated, and each directory is opened recursively. The "rtraverse"
Recursion_(computer_science)
Fractal sets in complex dynamics of mathematics
all pre-images of z under some high iterate f n {\displaystyle f^{n}} of f. Unfortunately, as the number of iterated pre-images grows exponentially, this
Julia_set
Sorting algorithm
displayed side-by-side), then using binary insertion sort may yield better performance. Binary insertion sort employs a binary search to determine the correct
Insertion_sort
Property of an iterated binary operation
telescoping refers to a property of certain algebraic expressions or iterated binary operations in which successive terms cancel each other after expansion. As
Telescoping_(mathematics)
Magma obeying the Latin square property
to itself. A binary, or 2-ary, quasigroup is an ordinary quasigroup. An example of a multiary quasigroup is an iterated group operation, y = x1 · x2 ·
Quasigroup
Generalization of addition, multiplication, exponentiation, tetration, etc.
as iterated addition and integer exponentiation is defined by iterated multiplication, the next hyperoperation, tetration, is defined by iterated exponentiation;
Hyperoperation
Class templates in the C++ programming language
self-balancing binary search trees and support bidirectional iteration. Iterators and references are not invalidated by insert and erase operations, except for
Associative_containers_(C++)
Computer approximation for real numbers
round-off error. Converting a double-precision binary floating-point number to a decimal string is a common operation, but an algorithm producing results that
Floating-point_arithmetic
Type of operation in parallel computing
balanced binary trees. All-reduce implemented with a butterfly algorithm achieves the same asymptotic runtime. The prefix-sum or scan operation is used
Collective_operation
Algorithm for computing the greatest common divisor
The binary GCD algorithm, also known as Stein's algorithm or the binary Euclidean algorithm, is an algorithm that computes the greatest common divisor
Binary_GCD_algorithm
Number of nonzero symbols in a string
instruction. The Hamming weight operation can be interpreted as a conversion from the unary numeral system to binary numbers. The population count of
Hamming_weight
Algorithms for calculating square roots
guess to use in the next round of correction. The process of updating is iterated until desired accuracy is obtained. This algorithm works equally well in
Square_root_algorithms
Topic in group theory
repeated (iterated) wreath product S 2 ≀ S 2 ≀ ⋯ ≀ S 2 {\displaystyle S_{2}\wr S_{2}\wr \cdots \wr S_{2}} is the automorphism group of a complete binary tree
Wreath_product
Open problem on 3x+1 and x/2 functions
for some odd number n, applying this operation k times yields the number 1 (that is, fk(n) = 1). Then in binary, the number n can be written as the concatenation
Collatz_conjecture
Search tree data structure
search operation takes O ( m ) {\displaystyle O(m)} time, where m {\displaystyle m} is the size of the string parameter key. In a balanced binary search
Trie
Exponentation in modular arithmetic
where i is the number of times the loop has been iterated. (This makes i the next working bit of the binary exponent exponent, where the least-significant
Modular_exponentiation
Mapping function that preserves data point locality
interleaving the binary representations of its coordinate values. However, when querying a multidimensional search range in these data, using binary search is
Z-order_curve
multiplication, built-in constants and functions, numerical calculus operations, iterated operators, user defined constants, user defined functions, user defined
MXparser
Data structure holding key/value pairs
sometimes also possible to solve the problem using directly addressed arrays, binary search trees, or other more specialized structures. Many programming languages
Associative_array
Computer data structure
In computer science, a min-max heap is a complete binary tree data structure which combines the usefulness of both a min-heap and a max-heap, that is,
Min-max_heap
Sorting algorithm using the heap data structure
comparison operations. This is because each sift-down step in a ternary heap requires three comparisons and one swap, whereas in a binary heap, two comparisons
Heapsort
Abstract data type for storing distinct values
after applying Ai+1 := F(Ai, e) for each element e of S, for some binary operation F. F must be associative and commutative for this to be well-defined
Set_(abstract_data_type)
Fractal composed of triangles
3. This method is also called the chaos game, and is an example of an iterated function system. You can start from any point outside or inside the triangle
Sierpiński_triangle
Estimate of time taken for running an algorithm
Algorithms taking logarithmic time are commonly found in operations on binary trees or when using binary search. An O ( log n ) {\displaystyle O(\log n)}
Time_complexity
Sequence of n-tuples of integers
128)\rightarrow (0,0,0,0)} This 5-tuple sequence enters a period 15 binary 'loop' after 7 iterations. 15799 → 42208 → 20284 → 22642 → 04220 → 42020 → 22224 → 00022
Ducci_sequence
Process of changing beliefs to take into account a new piece of information
Specific iterated revision operators have been proposed by Spohn, Boutilier, Williams, Lehmann, and others. Williams also provided a general iterated revision
Belief_revision
Linear error correcting code
contrast, belief propagation on the binary erasure channel is particularly simple where it consists of iterative constraint satisfaction. For example
Low-density_parity-check_code
Non-standard positional numeral system
skew binary representations of the numbers from 0 to 15 are shown in following table: The advantage of skew binary is that each increment operation can
Skew_binary_number_system
Data structure for priority queues
combining features of the binary heap and binomial heap. It can be stored in an array as an implicit binary tree like a binary heap, and has the efficiency
Weak_heap
Algorithm for computing trigonometric, hyperbolic, logarithmic and exponential functions
colleague of Volder at Convair, developed conversion algorithms between binary and binary-coded decimal (BCD). In 1958, Convair finally started to build a demonstration
CORDIC
Sequence merge algorithm in computer science
merges are also referred to as binary merges. The k-way merge is also an external sorting algorithm. A 2-way merge, or a binary merge, has been studied extensively
K-way_merge_algorithm
Quadratic homogeneous polynomial in two variables
In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables q ( x , y ) = a x 2 + b x y + c y 2 , {\displaystyle q(x
Binary_quadratic_form
Arithmetic operation
extension. The sum a + b {\displaystyle a+b} can be interpreted as a binary operation that combines a {\displaystyle a} and b {\displaystyle b} algebraically
Addition
Data structure in tree form sorted for fast lookup
application stores the entire key–value pair at that particular location. A Binary Search Tree is a node-based data structure where each node contains a key
Search_tree
Type of cipher
effectively improving security by combining simple operations such as substitutions and permutations. Iterated product ciphers carry out encryption in multiple
Block_cipher
Algebraic structure with an associative operation and an identity element
In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the natural numbers with addition
Monoid
Root-finding algorithm
{\displaystyle x} to an integer as a way to compute an approximation of the binary logarithm log 2 ( x ) {\textstyle \log _{2}(x)} Use this approximation
Fast_inverse_square_root
Common Lisp function that produces formatted text
for instance: hex, binary, octal, roman numerals, and English), apply certain format specifiers only under certain conditions, iterate over data structures
Format_(Common_Lisp)
Mathematical map
of the operation of folding the unit interval in two, then stretching the resulting interval [0, 1/2] to get again the interval [0, 1]. Iterating the procedure
Tent_map
Study of resources used by an algorithm
that grow extremely slowly: (binary) iterated logarithm (log*) is less than 5 for all practical data (265536 bits); (binary) log-log (log log n) is less
Analysis_of_algorithms
Property of operations
relation – a generalization of idempotence to binary relations Idempotent (ring theory) Involution (mathematics) Iterated function List of matrices Nilpotent Pure
Idempotence
C++ Standard Library header providing algorithm implementations
various functions that perform algorithmic operations on containers and other sequences, represented by Iterators. The C++ standard provides some standard
Algorithm_(C++)
Computer algorithm
that it used binary punch cards to denote machine language. In note C, this point is furthered by the fact that simultaneous and iterated actions can be
Note_G
Generalization of associativity properties
originate in algebraic topology; they were introduced to characterize iterated loop spaces by J. Michael Boardman and Rainer M. Vogt in 1968 and by J
Operad
Detects a given configuration (or pattern) in a binary image
morphology, hit-or-miss transform is an operation that detects a given configuration (or pattern) in a binary image, using the morphological erosion operator
Hit-or-miss_transform
Array data structure that compactly stores bits
functions and an extensive number of logical operations is supported. Arithmetic logic unit Binary code Binary numeral system Bitboard Chess and similar
Bit_array
Fraction with denominator a power of two
In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example,
Dyadic_rational
Mathematical term in group theory
and automata groups, and it has important connections with the theory of iterated monodromy groups. The growth of a finitely generated group measures the
Grigorchuk_group
Random process of binary (boolean) random variables
implementations which are resistant to timing attacks. Von Neumann–Peres (iterated) main operation pseudocode: if (Bit1 ≠ Bit2) { output(1, Sequence1) output(Bit1)
Bernoulli_process
Concept in modular arithmetic
and altering the binary operation appropriately. As with the analogous operation on the real numbers, a fundamental use of this operation is in solving,
Modular multiplicative inverse
Modular_multiplicative_inverse
Optimization technique
that, for general problems, the GPS approach can be applied iteratively to sequences of binary problems, using their now ubiquitous alpha-expansion algorithm
Graph cuts in computer vision and artificial intelligence
Graph_cuts_in_computer_vision_and_artificial_intelligence
Process in machine learning and statistics
2014.02.004. hdl:10533/196878. Zhang, Y.; Wang, S.; Phillips, P. (2014). "Binary PSO with Mutation Operator for Feature Selection using Decision Tree applied
Feature_selection
Data structure for storing strings
being edited, so that operations such as insertion, deletion, and random access can be done efficiently. A rope is a type of binary tree where each leaf
Rope_(data_structure)
Binary operation in graph theory
{\displaystyle G,H} , denoted by G ∘ H {\displaystyle G\circ H} , is a binary operation which takes a large graph ( G {\displaystyle G} ) and a small graph
Zig-zag_product
Assigning or adjusting addresses at runtime
tables once again become mandatory because of the possibility that the binary may be dynamically moved before being executed, even though they are still
Relocation_(computing)
Doubling map on the unit interval
arises because, if the value of an iterate is written in binary notation, the next iterate is obtained by shifting the binary point one bit to the right, and
Dyadic_transformation
Sequentially looking in an array
rarely practical because other search algorithms and schemes, such as the binary search algorithm and hash tables, allow significantly faster searching for
Linear_search
Arithmetic in a field with a finite number of elements
of GF(pn) as binary numbers, with the coefficient of each term in a polynomial represented by one bit in the corresponding element's binary expression.
Finite_field_arithmetic
Microprocessor instruction set
microprocessor introduced in 1976. The instruction set was designed to be upward binary compatible with the Intel 8080. Intel 8080 instructions are one to three
Z80_instruction_set
Family of related bitwise operations on machine words
complementary operation that finds the index or position of the most significant set bit is log base 2, so called because it computes the binary logarithm
Find_first_set
Floating-point number formats
supports an extended 40-bit variant of the floating-point format Microsoft Binary Format (MBF) since 1977. The IEEE 754 floating-point standard recommends
Extended_precision
Data structure
to binary number addition—two adjacent trees of the same height can be linked on constant time, with at most a logarithmic number of carry operations. This
Conc-tree_list
ITERATED BINARY-OPERATION
ITERATED BINARY-OPERATION
Girl/Female
Tamil
Aamuktha | ஆமà¯à®•à¯à®¤à®¾Â
Liberated
Aamuktha | ஆமà¯à®•à¯à®¤à®¾Â
Male
Scandinavian
Scandinavian form of Old Norse Einarr, EINAR means "lone warrior."
Girl/Female
Indian
Liberated
Boy/Male
Hindu
Liberated
Female
English
English pet form of German Belinda, possibly BINDY means "bright serpent" or "bright linden tree."
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Sanskrit, Telugu
Liberated
Boy/Male
Tamil
Muktananda | à®®à¯à®•à¯à®¤à®¾à®¨à®‚தா
Liberated
Muktananda | à®®à¯à®•à¯à®¤à®¾à®¨à®‚தா
Girl/Female
Indian
Modesty
Female
Hebrew
(×‘Ö¼Ö´×™× Ö¸×”) Hebrew name BINA means "intelligence, wisdom."Â
Surname or Lastname
English (chiefly South Yorkshire)
English (chiefly South Yorkshire) : topographic name for someone who lived on land enclosed by a bend in a river, from Old English binnan ēa ‘within the river’, or a habitational name from places in Kent called Binney and Binny, which have this origin.Scottish : habitational name from Binney or Binniehill near Falkirk, named in Gaelic as Beinnach, from beinn ‘hill’ + the locative suffix -ach.
Boy/Male
Tamil
Nirvanin | நீரà¯à®µà®¨à¯€à®¨
Liberated
Nirvanin | நீரà¯à®µà®¨à¯€à®¨
Male
English
English unisex form of Latin Hilarius and Hilaria, HILARY means "joyful; happy."Â Originally, this was strictly a masculine name.
Girl/Female
Hindu, Indian
Liberated
Female
Hebrew
Variant spelling of Hebrew Bina, BINAH means "intelligence, wisdom."Â
Female
Turkish
Turkish name PINAR means "spring."
Boy/Male
Indian, Punjabi, Sikh
Blessing
Boy/Male
American, Australian, French, German, Greek, Latin, Polish, Swedish
Cheerful; Happy; Joyful; Similar to Hilary
Male
Hindi/Indian
(विनय) Hindi name VINAY means "leading asunder."
Male
Hindi/Indian
Variant spelling of Hindi Vijay, BIJAY means "victory."
Girl/Female
English
Originally a diminutive used for names ending in -bina, like Albina, Columbina, and Robina, now...
ITERATED BINARY-OPERATION
ITERATED BINARY-OPERATION
Boy/Male
Hindu
Drifting about, Revolution
Girl/Female
Arabic, Iranian, Muslim, Parsi
A Flower; Poppy
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Mythological, Sanskrit, Tamil, Telugu, Thai
The God of Ramayana
Girl/Female
Australian, Indian, Tamil
Sleep
Boy/Male
Gaelic Scottish Irish
Slender.
Male
Arthurian
, (Sir), brother of Palamedes.
Girl/Female
Muslim
Partner, Chaste woman
Boy/Male
Tamil
Dipanshu | தீபாஂஷà¯Â
The Sun
Boy/Male
Muslim
Newborn child.
Girl/Female
Indian
One of the names of Goddess Durga
ITERATED BINARY-OPERATION
ITERATED BINARY-OPERATION
ITERATED BINARY-OPERATION
ITERATED BINARY-OPERATION
ITERATED BINARY-OPERATION
a.
lasting for one day; as, a diary fever.
n.
A binary compound of phosphorus.
a.
Of or pertaining to the Canary Islands; as, canary wine; canary birds.
n.
A canary bird.
imp. & p. p.
of Iterate
a.
Relating or belonging to bile; conveying bile; as, biliary acids; biliary ducts.
a.
Of or pertaining to the urine; as, the urinary bladder; urinary excretions.
n.
A register of daily events or transactions; a daily record; a journal; a blank book dated for the record of daily memoranda; as, a diary of the weather; a physician's diary.
v. t.
To utter or do a second time or many times; to repeat; as, to iterate advice.
n.
A binary compound of zinc.
n.
A binary compound of iodine, or one which may be regarded as binary; as, potassium iodide.
n.
A pale yellow color, like that of a canary bird.
n.
See Finery.
n.
A binary compound of silicon, or one regarded as binary.
n.
Wine made in the Canary Islands; sack.
a.
Containing ten; tenfold; proceeding by tens; as, the denary, or decimal, scale.
n.
A binary compound of selenium, or a compound regarded as binary; as, ethyl selenide.
a.
Standardized; determined or analyzed by titration; as, titrated solutions.
a.
Of a pale yellowish color; as, Canary stone.
v. i.
To perform the canary dance; to move nimbly; to caper.