Given the first n digits of Ω and a k ≤ n, the algorithm enumerates the domain of F until enough elements of the domain have been found so that the probability they represent is within 2 < sup >-( k + 1 )</ sup > of Ω.

It is known that there exists a unique Delaunay triangulation for P, if P is a set of points in general position ; that is, there exists no k-flat containing k + 2 points nor a k-sphere containing k + 3 points, for 1 ≤ k ≤ d − 1 ( e. g., for a set of points in < big > ℝ </ big >< sup > 3 </ sup >; no three points are on a line, no four on a plane, no four are on a circle, and no five on a sphere ).

In 1966, L. J. Lander, T. R. Parkin, and John Selfridge conjectured that if, where a < sub > i </ sub > ≠ b < sub > j </ sub > are positive integers for all 1 ≤ i ≤ n and 1 ≤ j ≤ m, then m + n ≥ k.

# f < sub > ik </ sub > = f < sub > ij </ sub > f < sub > jk </ sub > for all i ≤ j ≤ k.

* Let I consist of three elements i, j, and k with i ≤ j and i ≤ k ( not directed ).

# It is k-distributed to 32-bit accuracy for every 1 ≤ k ≤ 623 ( see definition below ).

The first theorem is for continuously differentiable ( C < sup > 1 </ sup >) embeddings and the second for analytic embeddings or embeddings that are smooth of class C < sup > k </ sup >, 3 ≤ k ≤ ∞.

This is because E ( k ) can have any sign whatsoever, and the combination of creation and annihilation operators has expectation value 1 or 0.

More generally, in a k-dimensional array, the address of an element with indices i < sub > 1 </ sub >, i < sub > 2 </ sub >, …, i < sub > k </ sub > is

: B + c < sub > 1 </ sub > · i < sub > 1 </ sub > + c < sub > 2 </ sub > · i < sub > 2 </ sub > + … + c < sub > k </ sub > · i < sub > k </ sub >.

This formula requires only k multiplications and k − 1 additions, for any array that can fit in memory.

The addressing formula is completely defined by the dimension d, the base address B, and the increments c < sub > 1 </ sub >, c < sub > 2 </ sub >, …, c < sub > k </ sub >.

In general, if y = f ( x ), then it can be transformed into y = af ( b ( x − k )) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis.

The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: where p < sub > 1 </ sub > < p < sub > 2 </ sub > < ... < p < sub > k </ sub > are primes and the a < sub > j </ sub > are positive integers.

It is the coefficient of the x < sup > k </ sup > term in the polynomial expansion of the binomial power ( 1 + x )< sup > n </ sup >.

This number can be seen as equal to the one of the first definition, independently of any of the formulas below to compute it: if in each of the n factors of the power one temporarily labels the term X with an index i ( running from 1 to n ), then each subset of k indices gives after expansion a contribution X < sup > k </ sup >, and the coefficient of that monomial in the result will be the number of such subsets.

This shows in particular that is a natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients ( counting problems for which the answer is given by a binomial coefficient expression ), for instance the number of words formed of n bits ( digits 0 or 1 ) whose sum is k is given by, while the number of ways to write where every a < sub > i </ sub > is a nonnegative integer is given by.

E has two clauses ( denoted by parentheses ), four variables ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, x < sub > 3 </ sub >, x < sub > 4 </ sub >), and k = 3 ( three literals per clause ).

However, as 1 + ( k + 1 ) x + kx < sup > 2 </ sup > ≥ 1 + ( k + 1 ) x ( since kx < sup > 2 </ sup > ≥ 0 ), it follows that ( 1 + x )< sup > k + 1 </ sup > ≥ 1 + ( k + 1 ) x, which means the statement is true for r = k + 1 as required.

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

The properties of binomial coefficients have led to extending the meaning of the symbol beyond the basic case where n and k are nonnegative integers with ; such expressions are then still called binomial coefficients.

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

Thus, if ( M, d ) and ( N, d ) are two metric spaces, and, then there is a constant k such that

As a consequence of the first point, if a and b are coprime, then so are any powers a < sup > k </ sup > and b < sup > l </ sup >.

If φ is C < sup > k </ sup >, then the inhomogeneous equation is explicitly solvable in any bounded domain D, provided φ is continuous on the closure of D. Indeed, by the Cauchy integral formula,

If X < sub > k </ sub > and Y < sub > k </ sub > are the DFTs of x < sub > n </ sub > and y < sub > n </ sub > respectively then the Plancherel theorem states:

If the expression that defines the DFT is evaluated for all integers k instead of just for, then the resulting infinite sequence is a periodic extension of the DFT, periodic with period N.

... = p < sub > k </ sub >), then the weighted average turns into the simple average.

It states that for all integers n and k greater than 1, if the sum of n kth powers of positive integers is itself a kth power, then n is greater than or equal to k.

( under the conditions given above ) then n ≥ k − 1.

For if k, m, and n are integers, and k is a common factor of two integers A and B, then A = nk and B = mk implies A − B = ( n − m ) k, therefore k is also a common factor of the difference.

If every formula in R of degree k is either refutable or satisfiable, then so is every formula in R of degree k + 1.

Let φ be a formula of degree k + 1 ; then we can write it as

In mathematics, a contraction mapping, or contraction, on a metric space ( M, d ) is a function f from M to itself, with the property that there is some nonnegative real number < math > k < 1 </ math > such that for all x and y in M,

Every contraction mapping is Lipschitz continuous and hence uniformly continuous ( for a Lipschitz continuous function, the constant k is no longer necessarily less than 1 ).

The exterior derivative is defined to be the unique R-linear mapping from k-forms to ( k + 1 )- forms satisfying the following properties:

This re-indexing of n is called the Ruritanian mapping ( also Good's mapping ), while this re-indexing of k is called the CRT mapping.

( One could instead use the Ruritanian mapping for the output k and the CRT mapping for the input n, or various intermediate choices.

Now, given any arbitrary mapping T from a product of k copies of and l copies of into C < sup >∞</ sup >( M ), it turns out that it arises from a tensor field on M if and only if it is a multilinear over C < sup >∞</ sup >( M ).

Let E and F be two vector bundles over a differentiable manifold M. An R-linear mapping of sections is said to be a kth-order linear differential operator if it factors through the jet bundle J < sup > k </ sup >( E ).

Suppose that n is the dimensionality of the oriented inner product space and k is an integer such that 0 ≤ k ≤ n, then the Hodge star operator establishes a one-to-one mapping from the space of k-vectors and the space of ( n − k )- vectors.

The Hodge star operator on a vector space V with a nondegenerate symmetric bilinear form ( herein aka inner product ) is a linear operator on the exterior algebra of V, mapping k-vectors to ( n − k )- vectors where, for.

where ENC is the encryption function, DEC the decryption function DEC is defined as ENC < sup >- 1 </ sup > ( inverse mapping ) and k < sub > 1 </ sub > and k < sub > 2 </ sub > are two keys.

The Hodge conjecture, may be neatly reformulated using motives: it holds iff the Hodge realization mapping any pure motive with rational coefficients ( over a subfield k of ℂ ) to its Hodge structure is a full functor H: M ( k )< sub > ℚ </ sub > → HS < sub > ℚ </ sub > ( rational Hodge structures ).

More precisely, there is a cross product operation by which an i-cycle on X and a j-cycle on Y can be combined to create an ( i + j )- cycle on X × Y ; so that there is an explicit linear mapping defined from the direct sum to H < sub > k </ sub >( X × Y ).

The k-th jet is the Taylor series of the mapping truncated at degree k and deleting the constant term.

The usual exterior derivative defines a mapping of sections d: E < sup > k </ sup >→ E < sup > k + 1 </ sup >.