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searching for Empty product 25 found (83 total)

Generating set of a group (1,729 words) [view diff] exact match in snippet view article find links to article

is the trivial group {e}{\displaystyle \{e\}}, since we consider the empty product to be the identity. When there is only a single element x{\displaystyle

Liouville function (1,720 words) [view diff] exact match in snippet view article find links to article

pk are primes and the aj are positive integers. (1 is given by the empty product.) The prime omega functions count the number of primes, with (Ω) or

Mott polynomials (648 words) [view diff] exact match in snippet view article find links to article

n{\displaystyle n} into k{\displaystyle k} positive odd integers. The empty product appearing for k=n=0{\displaystyle k=n=0} equals 1. Special values, where

Symmetric monoidal category (631 words) [view diff] exact match in snippet view article find links to article

product is the direct product of objects, and any terminal object (empty product) is the unit object. The category of bimodules over a ring R is monoidal

Root test (1,764 words) [view diff] exact match in snippet view article find links to article

{e} ^{-1+O(1/n)}{\frac {1}{n^{\rho _{n}}}},&K=1.\end{cases}}} (The empty product is set to 1.) The final result follows from the integral test for convergence

Data type (3,433 words) [view diff] exact match in snippet view article find links to article

a tuple or record. A constructor with no fields corresponds to the empty product (unit type). If all constructors have no fields then the ADT corresponds

Ring (mathematics) (13,383 words) [view diff] exact match in snippet view article

category with finite products. Let pt denote a terminal object of C (an empty product). A ring object in C is an object R equipped with morphisms R×R→aR{\displaystyle

Additive category (1,469 words) [view diff] exact match in snippet view article find links to article

groups. In a preadditive category, every finitary product (including the empty product, i.e., a final object) is necessarily a coproduct (or initial object

Arithmetic function (7,508 words) [view diff] exact match in snippet view article find links to article

pk are primes and the aj are positive integers. (1 is given by the empty product.) It is often convenient to write this as an infinite product over all

Tensor algebra (3,977 words) [view diff] exact match in snippet view article find links to article

\otimes v_{\sigma (m)}} to equal 1 for p = 0 and p = m, respectively (the empty product in TV{\displaystyle TV}). The shuffle follows directly from the first

Prior probability (6,425 words) [view diff] exact match in snippet view article find links to article

no data): Bayes' rule multiplies a prior by the likelihood, and an empty product is just the constant likelihood 1. However, without starting with a

Euler's totient function (6,082 words) [view diff] exact match in snippet view article find links to article

are prime numbers and each ki ≥ 1. (The case n = 1 corresponds to the empty product.) Repeatedly using the multiplicative property of φ and the formula

Recursion (computer science) (7,255 words) [view diff] exact match in snippet view article

unsigned int factorial(unsigned int n) { unsigned int product = 1; // empty product is 1 while (n) { product *= n; --n; } return product; } Most programming

Likelihood function (8,539 words) [view diff] exact match in snippet view article find links to article

function factors into a product of individual likelihood functions. The empty product has value 1, which corresponds to the likelihood, given no event, being

Non-associative algebra (2,964 words) [view diff] exact match in snippet view article find links to article

monomials u, v is just (u)(v). The algebra is unital if one takes the empty product as a monomial. Kurosh proved that every subalgebra of a free non-associative

Jacobi symbol (2,207 words) [view diff] exact match in snippet view article find links to article

such }}x.\end{array}}\right.} Following the normal convention for the empty product, (a/1) = 1. When the lower argument is an odd prime, the Jacobi symbol

Ratio test (5,076 words) [view diff] exact match in snippet view article find links to article

_{j=1}^{k-1}\ln _{(j)}(n)}}+O\left({\frac {1}{n^{2}}}\right),} where the empty product is assumed to be 1. Then, ρKummer=n∏k=1Kln(k)⁡(n)anan+1−(n+1)[∏k=1K

List of integer sequences (27 words) [view diff] exact match in snippet view article find links to article

5040, 40320, 362880, ... n! = 1⋅2⋅3⋅4⋅ ⋯ ⋅n for n ≥ 1, with 0! = 1 (empty product). A000142 Derangements 1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496,

Exterior algebra (12,075 words) [view diff] exact match in snippet view article find links to article

\wedge v_{\sigma (k)}} to equal 1 for p = 0 and p = k, respectively (the empty product in ⋀ ( V ) {\displaystyle {\textstyle \bigwedge }(V)} ). The shuffle

Clifford algebra (8,661 words) [view diff] exact match in snippet view article find links to article

i_{1}<i_{2}<\cdots <i_{k}\leq n{\text{ and }}0\leq k\leq n\}.} The empty product (k = 0) is defined as the multiplicative identity element. For each

New York Herald Tribune (10,433 words) [view diff] exact match in snippet view article find links to article

initially stopped the Sunday paper's circulation skid, but proved an empty product. The Tribune turned a profit in 1956, but the Times was rapidly outpacing

Table of prime factors (1,304 words) [view diff] exact match in snippet view article find links to article

n are all products of some or all prime factors of n (including the empty product 1 of no prime factors). The number of divisors can be computed by increasing

Geometric algebra (13,242 words) [view diff] exact match in snippet view article find links to article

with indices in increasing order, including 1{\displaystyle 1} as the empty product, forms a basis for the entire geometric algebra (an analogue of the

Normal order (3,903 words) [view diff] exact match in snippet view article find links to article

bosonic or fermionic)). Let O^{\displaystyle {\hat {O}}} denote a non-empty product of creation and annihilation operators. Although this may satisfy ⟨0|O^|0⟩≠0

Variety of finite semigroups (1,489 words) [view diff] exact match in snippet view article find links to article

subsemigroups and under taking quotients. The second property implies that the empty product—that is, the trivial semigroup of one element—belongs to each variety