If we denote by the number of occurrences of an event < font style =" vertical-align: 25 %;"></ font > in trials, then if we say that

Let H be a Hilbert space, and let H * denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H, then the function φ < sub > x </ sub >, defined by

If V is a real vector space, then we replace V by its complexification V ⊗< sub > R </ sub > C and let g denote the induced bilinear form on V ⊗< sub > R </ sub > C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g |< sub > W </ sub > = 0.

If ƒ and g are elements of a C *- algebra, f * and g * denote their respective adjoints.

If we denote by, the set of all problems that can be solved by Turing machines using at most space for some function of the input size, then we can define PSPACE formally as

If we denote this unique identity element by e then for each x, there exists an element i such that x * i = e = i * x.

If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is " not too large ", in the sense that if R is Noetherian, the same must be true for R. Formally,

If we denote by exp ( X ) the formal power series

If we denote the-fold product of with itself by, then morphisms from to are m-by-n matrices with entries from the ring.

If neither convergence nor divergence is required in the prescription, " plano " is used to denote a refractive power of zero.

If we denote this error, we may specify an empirical model for our observations,

Gilbert Murray summarizes the derivation from pelas gē, (" neighboring land "):" If Pelasgoi is connected with πέλας, ' near ', the word would mean ' neighbor ' and would denote the nearest strange people to the invading Greeks ..."

If a is an ideal of O < sub > K </ sub >, other than the zero ideal we denote its norm by Na.

If the symbol ℰ is used to denote electron energy level measured relative to the bottom of the conduction band, then in general we have ℰ =

If we denote the length of the altitude by h, we then have the relation

As above, let f and g denote measurable real-or complex-valued functions defined on S. If || fg ||< sub > 1 </ sup > is finite, then the products of f with g and its complex conjugate function, respectively, are μ-integrable, the estimates

If S is the complement of a prime ideal I the localization is denoted by R < sub > I </ sub >, and R < sub > f </ sub > is used to denote the localization by the powers of an element f. The two latter cases are fundamental in algebraic theory and scheme theory.

If is the twisting sheaf of Serre on, we let denote the pullback of to ; that is, for the canonical map

If we denote acceleration ( the second derivative of x ( t ) with respect to time ) as, the equations of motion are:

If is a real number, let denote where i the least integer.

If we denote by S the FOR / DOM exchange rate ( i. e. 1 unit of foreign currency is worth S units of domestic currency ) we can observe that paying out 1 unit of the domestic currency if the spot at maturity is above or below the strike is exactly like a cash-or nothing call and put respectively.

If is a unital ring, we denote by the group of invertible-by-matrices with elements in.

If we let N ( x ) denote the number of Harshad numbers ≤ x, then for any given ε > 0,

If " econophysics " is taken to denote the principle of applying statistical mechanics to economic analysis, as opposed to a particular literature or network, priority of innovation is probably due to Farjoun and Machover ( 1983 ).

If α is an integer, the limit has to be calculated.

If n ≥ 1 and is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation ; it is written as ( Z / nZ )< sup >×</ sup > or Z < sub > n </ sub >< sup >*</ sup >.

If λ < sub > 1 </ sub >, ..., λ < sub > ν </ sub > are the eigenvalues of J they will be resonant if one eigenvalue is an integer linear combination of two or more of the others.

:“ If an integer n is greater than 2, then has no solutions in non-zero integers a, b, and c. I have a truly marvelous proof of this proposition which this margin is too narrow to contain .”

If implemented using remainders of Euclidean division rather than subtractions, Euclid's algorithm computes the GCD of large numbers efficiently: it never requires more division steps than five times the number of digits ( base 10 ) of the smaller integer.

If the radix point is omitted then it is implicitly assumed to lie at the right ( least significant ) end of the string ( that is, the number is an integer ).

* If m is a non-negative integer, then gcd ( m · a, m · b ) = m · gcd ( a, b ).

* If m is any integer, then gcd ( a + m · b, b ) = gcd ( a, b ).

If the datum to be hashed is small enough, one can use the datum itself ( reinterpreted as an integer in binary notation ) as the hashed value.

If a spatial extra dimension is of radius R, the invariant mass of such standing waves would be M < sub > n </ sub > = nh / Rc with n an integer, h being Planck's constant and c the speed of light.

In arithmetic and number theory, the least common multiple ( also called the lowest common multiple or smallest common multiple ) of two integers a and b, usually denoted by LCM ( a, b ), is the smallest positive integer that is divisible by both a and b. If either a or b is 0, LCM ( a, b ) is defined to be zero.

If a and b are any integers and n is a positive integer, then the congruence

# If a is any integer, then 0 + a

If lexicographic ordering is used to sort variable-length integer representations, then the representations of the numbers from 1 to 10 would be output as 1, 10, 2, 3, 4, 5, 6, 7, 8, 9, as if the shorter keys were left-justified and padded on the right with blank characters to make the shorter keys as long as the longest key for the purpose of determining sorted order.

If all of the are drawn from the same distribution and the integer weights indicate frequency of occurrence in the sample, then the unbiased estimator of the weighted population variance is given by

If one used Planck's energy quanta, and demanded that electromagnetic radiation at a given frequency could only transfer energy to matter in integer multiples of an energy quantum hν, then the photoelectric effect could be explained very simply.

If B is a circle, then these eigenfunctions have an angular component that is a trigonometric function of the polar angle θ, multiplied by a Bessel function ( of integer order ) of the radial component.

If scatterers are arranged symmetrically with a separation d, these spherical waves will be in sync ( add constructively ) only in directions where their path-length difference 2d sin θ equals an integer multiple of the wavelength λ.

If integer factorization is difficult ( as is suspected ) then B. B. S.

If one pursues this heuristic, one might expect the total number of ways to write a large even integer n as the sum of two odd primes to be roughly

Consider a jar containing N lottery tickets numbered from 1 through N. If you pick a ticket randomly then you get positive integer n, with probability 1 / N if n ≤ N and with probability zero if n > N. This can be written

Fermat's little theorem states that if p is prime and a is coprime to p, then a < sup > p − 1 </ sup > − 1 is divisible by p. If a composite integer x is coprime to an integer a > 1 and x divides a < sup > x − 1 </ sup > − 1, then x is called a Fermat pseudoprime to base a.

Sylows ' test: Let n be a positive integer that is not prime, and let p be a prime divisor of n. If 1 is the only divisor of n that is equal to 1 modulo p, then there does not exist a simple group of order n.

The Gaussian integers form a principal ideal domain with units 1, − 1, i, and − i. If x is a Gaussian integer, the four numbers x, ix, − x, and − ix are called the associates of x.