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Concept in mathematics
In mathematics, a multiplicative sequence or m-sequence is a sequence of polynomials associated with a formal group structure. They have application in
Multiplicative_sequence
Ring homomorphism from the cobordism ring of manifolds to another ring
In mathematics, a genus of a multiplicative sequence is a ring homomorphism from the ring of smooth compact manifolds up to the equivalence of bounding
Genus of a multiplicative sequence
Genus_of_a_multiplicative_sequence
Finite or infinite ordered list of elements
In other instances, sequences are often called multiplicative, if an = na1 for all n. Moreover, a multiplicative Fibonacci sequence satisfies the recursion
Sequence
Numbers obtained by adding the two previous ones
Fibonacci sequence is a sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known
Fibonacci_sequence
Number which when multiplied by x equals 1
is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a b {\displaystyle {\tfrac {a}{b}}}
Multiplicative_inverse
Mathematical formula
0, 3, 6, 9, 2, 5, 8, 2, 8, 4, 0. (sequence A031347 in the OEIS) Multiplicative digital roots are the multiplicative equivalent of digital roots, with
Multiplicative_digital_root
Arithmetical operation
generalizations See Multiplication in group theory, above, and multiplicative group, which for example includes matrix multiplication. A very general, and
Multiplication
Topics referred to by the same term
(mathematics), a classifying property of a mathematical object Genus of a multiplicative sequence Geometric genus In graph embedding, the genus of the graph is the
Genus_(disambiguation)
Property of a number
is the smallest number of multiplicative persistence 3. In base 10, there is thought to be no number with a multiplicative persistence greater than 11;
Persistence_of_a_number
Gives the signature of a smooth compact oriented manifold in terms of Pontryagin numbers
Hirzebruch–Riemann–Roch theorem. The L-genus is the genus for the multiplicative sequence of polynomials associated to the characteristic power series x
Hirzebruch_signature_theorem
Way to write a number as a product of other numbers
pointwise. Although the study of multiplicative partitions has been ongoing since at least 1923, the name "multiplicative partition" appears to have been
Multiplicative_partition
Group of units of the ring of integers modulo n
the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity. Finally, given a, the multiplicative inverse
Multiplicative group of integers modulo n
Multiplicative_group_of_integers_modulo_n
Characteristic class in algebraic topology
\operatorname {td} _{j}} defines the Todd polynomials: they form a multiplicative sequence with Q {\displaystyle Q} as characteristic power series. If E {\displaystyle
Todd_class
Natural number
generally, in algebra, it denotes the multiplicative identity in any unital ring or field. An element with a multiplicative inverse is called a unit, generalizing
1
Telecommunication device that obscures signals
systems. A multiplicative scrambler is recursive, and a multiplicative descrambler is non-recursive. Unlike additive scramblers, multiplicative scramblers
Scrambler
Mathematical operation in linear algebra
as matrix multiplication (up to a multiplicative constant), the computational complexity of matrix multiplication appears throughout numerical linear
Matrix_multiplication
Two raised to an integer power
is the multiplicative order of 2 modulo 5k, which is φ(5k) = 4 × 5k−1 (see Multiplicative group of integers modulo n).[citation needed] (sequence A140300
Power_of_two
Algorithm to multiply two numbers
such a short sequence. In addition to the standard long multiplication, there are several other methods used to perform multiplication by hand. Such
Multiplication_algorithm
Number of "holes" of a surface
structure of biomolecules. Arithmetic genus Geometric genus Genus of a multiplicative sequence Genus of a quadratic form Group (mathematics) Spinor genus Popescu-Pampu
Genus_(mathematics)
Number used for counting
objects "larger", than the other. A sequence is a list of objects in a specific order. More precisely, a sequence is a function that assigns an object
Natural_number
Online database of integer sequences
more – More terms of the sequence are wanted. Readers can submit an extension. mult – The sequence corresponds to a multiplicative function. Term a(1) should
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Number used in combinatorial game theory
Nimber multiplication is associative and commutative, with the ordinal 1 as the multiplicative identity element. Moreover, nimber multiplication distributes
Nimber
Recursive integer sequence
The Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named
Catalan_number
Patterns of nucleic acids that occur in multiple copies throughout the genome
based on the length of the repeated sequence and/or the mode of multiplication. While some repeated DNA sequences are important for cellular functioning
Repeated_sequence_(DNA)
Mathematics optimization problem
chain multiplication (or the matrix chain ordering problem) is an optimization problem concerning the most efficient way to multiply a given sequence of
Matrix_chain_multiplication
Method for computing the relation of two integers with their greatest common divisor
With that provision, x is the modular multiplicative inverse of a modulo b, and y is the modular multiplicative inverse of b modulo a. Similarly, the
Extended_Euclidean_algorithm
Mathematical table
columns for multiplication by 1, the multiplicative identity, which satisfies a × 1 = a. The traditional rote learning of multiplication was based on
Multiplication_table
Tool in homological algebra
algebra to H(E; R). The multiplicative structure can be very useful for calculating differentials on the sequence. Spectral sequences can be constructed by
Spectral_sequence
Function equal to the product of its values on coprime factors
not multiplicative. However, r 2 ( n ) / 4 {\displaystyle r_{2}(n)/4} is multiplicative. In the On-Line Encyclopedia of Integer Sequences, sequences of
Multiplicative_function
Arithmetic operation
invertible elements in a multiplicative monoid, that is, an algebraic structure, with an associative multiplication and a multiplicative identity denoted 1
Exponentiation
Number that when multiplied by another number moves its last digit to its front
University Press UK, 2000. Sequence OEIS: A092697 in the On-Line Encyclopedia of Integer Sequences. Bernstein, Leon (1968), "Multiplicative twins and primitive
Parasitic_number
Complex-valued mathematical sequence
{\tilde {u}}} is the multiplicative inverse of u modulo N ZC {\displaystyle N_{\text{ZC}}} . 3. The auto correlation of a Zadoff–Chu sequence with a cyclically
Zadoff–Chu_sequence
Cycle through all length-k sequences
In combinatorial mathematics, a de Bruijn sequence of order n on a size-k alphabet A is a cyclic sequence in which every possible length-n string on A
De_Bruijn_sequence
Performing order of mathematical operations
is replaced with multiplication by the reciprocal (multiplicative inverse), then the associative and commutative laws of multiplication allow the factors
Order_of_operations
Iterative algorithm on numbers
-\beta } to produce the next number of the sequence. Repeat step 2. The sequence is called a Kaprekar sequence and the function K b ( n ) = α − β {\displaystyle
Kaprekar's_routine
Sequence of integers
In number theory, the Padovan sequence is the sequence of integers P(n) defined by the initial values: P ( 0 ) = P ( 1 ) = P ( 2 ) = 1 , {\displaystyle
Padovan_sequence
Sequence of homomorphisms such that each kernel equals the preceding image
(multiplicative notation). Consider the sequence 0 → A → B {\displaystyle 0\to A\to B} . The image of the leftmost map is 0. Therefore the sequence is
Exact_sequence
2017 research paper by Google
others. These multiplicative units are conceptually distinct from the additive attention mechanism later introduced for sequence-to-sequence models. Neural
Attention_Is_All_You_Need
Integer invariant of certain classes of topological manifolds
structure is divisible by 16. Hirzebruch signature theorem Genus of a multiplicative sequence Rokhlin's theorem Hatcher, Allen (2003). Algebraic topology (PDF)
Signature_(topology)
Number used to approximate the square root of 2
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational
Pell_number
Numbers with a certain property involving recursive summation
1^{2}+0^{2}=1} . On the other hand, 4 is not a happy number because the sequence starting with 4 2 = 16 {\displaystyle 4^{2}=16} and 1 2 + 6 2 = 37 {\displaystyle
Happy_number
Binary search variation with simplified midpoint calculation
permutation used by multiplicative binary search places the optimal number of keys in the first (root) block, regardless of block size. Multiplicative binary search
Multiplicative_binary_search
Theorem about natural numbers
proved by Reuben Goodstein in 1944, which states that every Goodstein sequence (as defined below) eventually terminates at 0. Laurence Kirby and Jeff
Goodstein's_theorem
Infinite integer series where the next number is the sum of the two preceding it
Lucas sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the
Lucas_number
In number theory, two positive integers a and b are said to be multiplicatively independent if their only common integer power is 1. That is, for integers
Multiplicative_independence
Vector space of infinite sequences
of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with
Sequence_space
Spectral sequence
In mathematics, the Adams spectral sequence is a spectral sequence introduced by J. Frank Adams (1958) which computes the stable homotopy groups of topological
Adams_spectral_sequence
Modulation technique to reduce signal interference
end. This is commonly implemented by the element-wise multiplication with the spreading sequence, followed by summation over a message symbol period. This
Direct-sequence spread spectrum
Direct-sequence_spread_spectrum
Number sequence 3,0,2,3,2,5,5,7,10,...
mathematics, the Perrin numbers are a doubly infinite constant-recursive integer sequence with characteristic equation x3 = x + 1. The Perrin numbers, named after
Perrin_number
Algebraic ring without a multiplicative identity
same properties as a ring, but without assuming the existence of a multiplicative identity. The term rng is meant to suggest that it is a ring without
Rng_(algebra)
Algorithm for modelling sequential data
others. These multiplicative units are conceptually distinct from the additive attention mechanism later introduced for sequence-to-sequence models. Neural
Transformer_(deep_learning)
Multiplicative partitions of factorials are expressions of values of the factorial function as products of powers of prime numbers. They have been studied
Multiplicative partitions of factorials
Multiplicative_partitions_of_factorials
Open problem on 3x+1 and x/2 functions
after receiving his doctorate. The sequence of numbers involved is sometimes referred to as the hailstone sequence, hailstone numbers or hailstone numerals
Collatz_conjecture
Mathematical sequence
integer sequence devised by and named after Stanisław Ulam, who introduced it in 1964. The standard Ulam sequence (the (1, 2)-Ulam sequence) starts with
Ulam_number
Generalised alphabetical order
monoid are the finite sequences (words) of elements of A (including the empty sequence, of length 0), and the operation (multiplication) is the concatenation
Lexicographic_order
Integral using products instead of sums
the multiplicative Lorenz system", Chaos, Solitons & Fractals Volume 25, Issue 1, July 2005, pages 79–90. Fernando Córdova-Lepe. "The multiplicative derivative
Product_integral
Algorithmic technique
SDPs), and game theory. "Multiplicative weights" implies the iterative rule used in algorithms derived from the multiplicative weight update method. It
Multiplicative weight update method
Multiplicative_weight_update_method
Type of number introduced by Mike Keith
True sequence = [] y = x while y > 0: sequence.append(y % b) y = y // b digit_count = len(sequence) sequence.reverse() while sequence[len(sequence) - 1]
Keith_number
Algorithm for integer multiplication
The Karatsuba algorithm is a fast multiplication algorithm for integers. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. It is a
Karatsuba_algorithm
Computation modulo a fixed integer
a modular multiplicative inverse of a modulo m. If a ≡ b (mod m) and a−1 exists, then a−1 ≡ b−1 (mod m) (compatibility with multiplicative inverse, and
Modular_arithmetic
Algorithm that generates an approximation of a random number sequence
generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generated sequence is not truly
Pseudorandom_number_generator
Mathematical sequences
In mathematics, the Fibonacci numbers form a sequence defined recursively by: F n = { 0 n = 0 1 n = 1 F n − 1 + F n − 2 n > 1 {\displaystyle
Generalizations of Fibonacci numbers
Generalizations_of_Fibonacci_numbers
Mathematical sequence of numbers
A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by
Geometric_progression
Branch of elementary mathematics
{\displaystyle 48\div 8=48\times {\tfrac {1}{8}}} . The multiplicative identity element is 1 and the multiplicative inverse of a number is the reciprocal of that
Arithmetic
Mapping arbitrary data to fixed-size values
(modulo) by a constant can be inverted to become a multiplication by the word-size multiplicative-inverse of that constant. This can be done by the programmer
Hash_function
Irreducible polynomial whose roots are nth roots of unity
with the multiplicative order modulo a prime number. More precisely, given a prime number p and an integer b coprime with p, the multiplicative order of
Cyclotomic_polynomial
Figurate number
The triangular numbers or triangle numbers are the sequence of positive integers that can be represented as a lattice of points arranged in an equilateral
Triangular_number
Ten raised to an integer power
ten are: 1, 10, 100, 1,000, 10,000, 100,000, 1,000,000, 10,000,000... (sequence A011557 in the OEIS) In decimal notation the nth power of ten is written
Power_of_10
Integers occurring in the coefficients of the Taylor series of 1/cosh t
In mathematics, the Euler numbers are a sequence En of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion 1 cosh t = 2 e
Euler_numbers
Number that is the result of operation on its own digits
2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2592, 2737, 2916, ... (sequence A036057 in the OEIS). Friedman numbers are named after Erich Friedman,
Friedman_number
Multiplicative function in number theory
The Möbius function μ ( n ) {\displaystyle \mu (n)} is a multiplicative function in number theory introduced by the German mathematician August Ferdinand
Möbius_function
Algebraic structure with addition and multiplication
defined to have a multiplicative identity, while a structure with the same axiomatic definition but without the requirement for a multiplicative identity is
Ring_(mathematics)
Count of the possible partitions of a set
numbers, then B n {\displaystyle B_{n}} gives the number of different multiplicative partitions of N {\displaystyle N} . These are factorizations of N {\displaystyle
Bell_number
Type of Poulet number
and a super-Poulet number. The super-Poulet numbers below 10,000 are (sequence A050217 in the OEIS): It is relatively easy to get super-Poulet numbers
Super-Poulet_number
Positive integer whose divisors have a harmonic mean that is an integer
1997). All of the terms in this formula are multiplicative, so that the harmonic mean H(n) is also multiplicative. It follows that, for any positive integer
Harmonic_divisor_number
Number equal to the sum of its proper divisors
function s(n) = σ(n) − n, and the aliquot sequence associated with a perfect number is a constant sequence. All perfect numbers are also S {\displaystyle
Perfect_number
Abundant number whose proper divisors are all deficient numbers
abundant numbers are: 20, 70, 88, 104, 272, 304, 368, 464, 550, 572 ... (sequence A071395 in the OEIS) The smallest odd primitive abundant number is 945
Primitive_abundant_number
Repeated sum of a number's digits
_{b}(a)\cdot \operatorname {dr} _{b}(c)).} This is a consequence of multiplicative compatibility modulo b − 1 {\displaystyle b-1} . Compatibility with
Digital_root
Number of orderings allowing ties
2^{n-1}} ordered multiplicative partitions. Numbers that are neither squarefree nor prime powers have a number of ordered multiplicative partitions that
Ordered_Bell_number
Pair of integers related by their divisors
10856), (12285, 14595), (17296, 18416), (63020, 76084), and (66928, 66992) (sequence A259180 in the OEIS). It is unknown if there are infinitely many pairs
Amicable_numbers
Function that can be written as a sum over prime factors
Totally additive is also used in this sense by analogy with totally multiplicative functions. Every completely additive function is additive, but not vice
Additive_function
Integer having a non-trivial divisor
15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36. (sequence A002808 in the OEIS) Every composite number can be written as the product
Composite_number
Square of numbers with equal row, column and diagonal totals
some other operation. For example, a multiplicative magic square has a constant product of numbers. A multiplicative magic square can be derived from an
Magic_square
Number that remains the same when its digits are reversed
131, 151, ... (sequence A002385 in the OEIS). The palindromic square numbers are 0, 1, 4, 9, 121, 484, 676, 10201, 12321, ... (sequence A002779 in the
Palindromic_number
Algorithm for multiplying large numbers
\end{array}}} This sequence requires five addition/subtraction operations, one less than the straightforward evaluation. Moreover the multiplication by 4 {\displaystyle
Toom–Cook_multiplication
Discrete Fourier transform algorithm
computes the discrete Fourier transform (DFT), or its inverse (IDFT), of a sequence. A Fourier transform converts a signal from its original domain (often
Fast_Fourier_transform
Space of bounded sequences
^{\infty }} is a sequence space whose elements are the bounded sequences. The vector space operations, addition and scalar multiplication, are applied coordinate
L-infinity
Algorithm for generating pseudo-randomized numbers
that specify the generator. If c = 0, the generator is often called a multiplicative congruential generator (MCG), or Lehmer RNG. If c ≠ 0, the method is
Linear_congruential_generator
Mathematical operator acting on sequences
In mathematics, a sequence transformation is an operator acting on a given space of sequences (a sequence space). Sequence transformations include linear
Sequence_transformation
Identity obeyed by many special functions related to the gamma function
obeying the multiplication theorem from any totally multiplicative function. Let f ( n ) {\displaystyle f(n)} be totally multiplicative; that is, f (
Multiplication_theorem
Product of numbers from 1 to n
numbers from 1 to n {\displaystyle n} in sequence is inefficient, because it involves n {\displaystyle n} multiplications, a constant fraction of which take
Factorial
Semiring defined over probabilities
{\displaystyle \max(0,x)=x} for any x ∈ [ 0 , 1 ] {\displaystyle x\in [0,1]} . Multiplicative operation ( ⊗ {\displaystyle \otimes } ): defined as the standard product
Viterbi_semiring
Axiomatic definition of a class of L-functions
exponentiation of Dirichlet series, one can deduce that an is a multiplicative sequence and that F p ( s ) = ∑ n = 0 ∞ a p n p n s for Re ( s ) > 1. {\displaystyle
Selberg_class
fast multiplication algorithm for large integers Toom–Cook multiplication: (Toom3) a multiplication algorithm for large integers Multiplicative inverse
List_of_algorithms
Type of short exact sequence in mathematics
sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way. A short exact sequence of
Split_exact_sequence
Number of integers coprime to and less than n
1 ) = 1 {\displaystyle \gcd(1,1)=1} . Euler's totient function is a multiplicative function, meaning that if two numbers m {\displaystyle m} and n {\displaystyle
Euler's_totient_function
Spectral sequence in algebraic topology
Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important
Serre_spectral_sequence
Special semigroup of positive rational numbers
of the multiplicative semigroup of all positive rational numbers. The elements of a generating set of this semigroup are related to the sequence of numbers
3x_+_1_semigroup
Topics referred to by the same term
In mathematics, characteristic power series may refer to: Multiplicative sequence Iwasawa algebra This disambiguation page lists mathematics articles
Characteristic_power_series
Electronic circuit used to multiply binary numbers
summed together using binary adders. This process is similar to long multiplication, except that it uses a base-2 (binary) numeral system. Between 1947
Binary_multiplier
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