If it were wholly random and unrelated, it would be 2.0, assuming the five classes were equal in n, which approximately they are.

If denotes the quantum state of a particle ( n ) with momentum p, spin J whose component in the z-direction is σ, then one has

If each node additionally records the size of its subtree ( including itself and its descendants ), then the nodes can be retrieved by index in O ( log n ) time as well.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

If ( m, n ) is regular and M and N have i and j prime factors respectively, then ( m, n ) is said to be of type ( i, j ).

* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.

More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient

If a is a point in R < sup > n </ sup >, then the higher dimensional chain rule says that:

If k, m, and n are 1, so that and, then the Jacobian matrices of f and g are.

If the affected bits are independently chosen at random, the probability of a two-bit error being undetected is 1 / n.

If S is an arbitrary set, then the set S < sup > N </ sup > of all sequences in S becomes a complete metric space if we define the distance between the sequences ( x < sub > n </ sub >) and ( y < sub > n </ sub >) to be, where N is the smallest index for which x < sub > N </ sub > is distinct from y < sub > N </ sub >, or 0 if there is no such index.

If F ≥ F < sub > Critical </ sub > ( Numerator DF, Denominator DF, α )

If the ranges of the morphisms of the inverse system of abelian groups ( A < sub > i </ sub >, f < sub > ij </ sub >) are stationary, that is, for every k there exists j ≥ k such that for all i ≥ j: one says that the system satisfies the Mittag-Leffler condition.

If an equation can be put into the form f ( x ) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x < sub > 1 </ sub > in the basin of attraction of x, and let x < sub > n + 1 </ sub > = f ( x < sub > n </ sub >) for n ≥ 1, and the sequence

If ( x < sub > α </ sub >) is a net from a directed set A into X, and if Y is a subset of X, then we say that ( x < sub > α </ sub >) is eventually in Y ( or residually in Y ) if there exists an α in A so that for every β in A with β ≥ α, the point x < sub > β </ sub > lies in Y.

If Φ is a unital positive map, then for every normal element a in its domain, we have Φ ( a * a ) ≥ Φ ( a *) Φ ( a ) and Φ ( a * a ) ≥ Φ ( a ) Φ ( a *).

If M is an Hermitian positive-semidefinite matrix, one sometimes writes M ≥ 0 and if M is positive-definite one writes M > 0 .< ref > This may be confusing, as sometimes nonnegative matrices are also denoted in this way.

If M ≥ N > 0 then N < sup >− 1 </ sup > ≥ M < sup >− 1 </ sup > > 0.

< li > If M = ( m < sub > ij </ sub >) ≥ 0 then the diagonal entries m < sub > ii </ sub > are real and non-negative.

* If f: R → R is a Lebesgue integrable function and f ( x ) ≥ 0 almost everywhere, then

If there exists a K ≥ 1 with

If we restrict attention to real-valued W then the relation is defined only for x ≥ − 1 / e, and is double-valued on (− 1 / e, 0 ); the additional constraint W ≥ − 1 defines a single-valued function W < sub > 0 </ sub >( x ).

If n ≥ 2, then the group GL ( n, F ) is not abelian.

If is a monotone sequence of real numbers ( i. e., if a < sub > n </ sub > ≤ a < sub > n + 1 </ sub > or a < sub > n </ sub > ≥ a < sub > n + 1 </ sub > for every n ≥ 1 ), then this sequence has a finite limit if and only if the sequence is bounded.

If X is a set, a diffeology on X is a set of maps, called plots, from open subsets of R < sup > n </ sup > ( n ≥ 0 ) to X such that the following hold:

If the chosen anti-aliasing filter ( a low-pass filter in this case ) has a transition band of 2000 Hz, then the cut-off frequency should be ≤ 20050 Hz to yield a signal with negligible power at frequencies ≥ 22050 Hz and complete pass of frequencies ≤ 20 kHz ( within the human hearing range ).

# If the injectivity radius of a compact n-dimensional Riemannian manifold is ≥ π then the average scalar curvature is at most n ( n-1 ).

If X is a topological space, we say that an α-indexed sequence of elements of X converges to a limit x when it converges as a net, in other words, when given any neighborhood U of x there is an ordinal β < α such that x < sub > ι </ sub > is in U for all ι ≥ β.

If X is a connected cell complex with homotopy groups π < sub > n </ sub >( X ) = 0 for all n ≥ 2, then the universal covering space T of X is contractible, as follows from applying the Whitehead theorem to T. In this case X is a classifying space or K ( G, 1 ) for G = π < sub > 1 </ sub >( X ).

If a function f is concave, and f ( 0 ) ≥ 0, then f is subadditive.

If we are to believe the list of titles printed in Malraux's latest book, La Metamorphose Des Dieux, Vol. 1 ( ( 1957 ), he is still engaged in writing a large novel under his original title.

If desired, sprinkle with 1 teaspoon drained crushed pineapple.

If the Af bond is linear then there are three reasonable positions for the hydrogen atoms: ( 1 ) The hydrogen atoms are centered and hence all lie on a sheet midway between the oxygen sheets ; ;

If, in a certain part of the range, it starts life 1 foot longer than do any of the other ( relatively large ) giants, and reaches maturity at, let us guess, 18 inches longer than the others, a quadrupling of the maturity length would result in a maximum of ( nearly ) 40 feet.

If, at any time during the assignment pass, the compiler finds that there are no more index words available for assignment, the warning message `` No More Index Words Available '' will be placed in the object program listing, the table will be altered to show that index words 1 through 96 are available, and the assignment will continue as before.

If the compiler finds that there are no more electronic switches available for assignment, the warning message `` No More Electronic Switches Available '' will be placed in the object program listing, the table will be altered to show that electronic switches 1 through 30 are available, and assignment will continue as before.

If the resulting four kilobyte table size is too large for a given target platform, the table lookup operation can be performed with a single 256-entry 32-bit ( i. e. 1 kilobyte ) table by the use of circular rotates.

Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that

If more than one player selects a tile in that row, then the player whose tile is in the leftmost column ( closest to 1 ) goes first.

If the color were fully green, its RGBA would be ( 0, 1, 0, 0. 5 ).

* If the balance factor of R is + 1, two different rotations are needed.

* If the balance factor of L is + 1, a single right rotation ( with P as the root ) is needed ( Left-Left case ).

If there is refraction at a collective spherical surface, or through a thin positive lens, O ' 2 will lie in front of O ' 1 so long as the angle u2 is greater than u1 ( under correction ); and conversely with a dispersive surface or lenses ( over correction ).

If the angle u1 is very small, O ' 1 is the Gaussian image ; and O ' 1 O ' 2 is termed the longitudinal aberration, and O ' 1R the lateral aberration of the pencils with aperture u2.

If the pencil with the angle u2 is that of the maximum aberration of all the pencils transmitted, then in a plane perpendicular to the axis at O ' 1 there is a circular disk of confusion of radius O ' 1R, and in a parallel plane at O ' 2 another one of radius O ' 2R2 ; between these two is situated the disk of least confusion.

If the object point O is infinitely distant, u1 and u2 are to be replaced by h1 and h2, the perpendicular heights of incidence ; the sine condition then becomes sin u ' 1 / h1 = sin u ' 2 / h2.

If the function R is well-defined, its value must lie in the range, with 1 indicating perfect correlation and − 1 indicating perfect anti-correlation.

If there is no number to its left, simply look at the column headed " 1 " in the previous row.

If a detector was placed at a distance of 1 m, the ion flight times would be X and Y ns.

If the boat floats, the mass of the boat ( plus contents ) as a whole divided by the volume below the waterline is equal to the density of water ( 1 kg / l ).