where we use the symbol k to denote the quantum numbers p and σ of the previous section and the sign of the energy, E ( k ), and a < sub > k </ sub > denotes the corresponding annihilation operators.

Analysis of the properties of a < sub > k </ sub > and b < sub > k </ sub > shows that one is the annihilation operator for particles and the other for antiparticles.

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

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.

The region of supercriticality between k

The region of supercriticality for k > 1 /( 1-β ) is known as prompt supercriticality ( or prompt criticality ), which is the region in which nuclear weapons operate.

In names for familiar relatives, where both genders are taken into account, either the words for each gender are put together (" son ": seme ; " daughter ": alaba ; " children "( meaning son ( s ) and daughter ( s )): seme-alaba ( k )) or there is a noun that includes both: " father ": aita ; " mother ": ama ; " father " ( both genders ): guraso.

* Id < sub > k </ sub >( n ): the power functions, defined by Id < sub > k </ sub >( n ) =

* gcd ( n, k ): the greatest common divisor of n and k, as a function of n, where k is a fixed integer.

* < sub > k </ sub >( n ): the divisor function, which is the sum of the k-th powers of all the positive divisors of n ( where k may be any complex number ).

* k < 1 ( subcriticality ): The system cannot sustain a chain reaction, and any beginning of a chain reaction dies out over time.

* k = 1 ( criticality ): Every fission causes an average of one more fission, leading to a fission ( and power ) level that is constant.

< li > A finite table ( occasionally called an action table or transition function ) of instructions ( usually quintuples: q < sub > i </ sub > a < sub > j </ sub >→ q < sub > i1 </ sub > a < sub > j1 </ sub > d < sub > k </ sub >, but sometimes 4-tuples ) that, given the state ( q < sub > i </ sub >) the machine is currently in and the symbol ( a < sub > j </ sub >) it is reading on the tape ( symbol currently under the head ) tells the machine to do the following in sequence ( for the 5-tuple models ):

That is the activation energy is defined to be (- R ) times the slope of a plot of ln :( k ): vs. :( 1 / T ):

* Ukrainian — Karenne, Diana ( worked mainly in Italy, Germany, and France ): Pierrot a. k. a. Story of a Pierrot ( 1917 ; still from Pierrot ).

* American — Craton, John: Pierrot and Pierrette a. k. a. Le Mime solitaire ( 2009 ; ballet ); Muller, Jennifer ( head of three-member Works Dance Company, New York ): Pierrot ( 1986 ; music and scenario by Thea Musgrave below under # Western classical | Western classical: Instrumental ); Russillo, Joseph ( works mainly in France ): Pierrot ( 1975 ; ballet ).

* Dutch — Boer, Eduard de ( a. k. a. Alexander Comitas ): Pierrot: Scherzo for String Orchestra ( 1992 ).

Solutions of the Laplace equation in two dimensions are intimately connected with analytic functions of a complex variable ( a. k. a. holomorphic functions ): the real and imaginary parts of any analytic function are conjugate harmonic functions: they both satisfy the Laplace equation, and their gradients are orthogonal.

Now, considering S ( k + 1 ):

Dalmatian is the only Romance language that palatalised / k / and / g / before / i /, but not before / e / ( others palatalise in both situations, except Sardinian, which did not palatalise ): > Vegliot: (), > Vegliot: ().

For homogeneous turbulence ( i. e., statistically invariant under translations of the reference frame ) this is usually done by means of the energy spectrum function, where k is the modulus of the wavevector corresponding to some harmonics in a Fourier representation of the flow velocity field u ( x ):

* Site-specific theatre ( a. k. a. environmental theatre ): The stage and audience either blend together, or are in numerous or oddly shaped sections.

* Assibilation of Indo-European ( IE ) palatals ( Satem change ): * ekwo-( the original k was palatal ) to Luwian á-zú-wa / i -, Lycian esbe, " horse.

For example, a comma-shaped letter represented g, d, y, k, or j.

For very large distances, cosmological redshift complicates the relation between absolute and apparent magnitude, and an additional k correction might be required.

For natural numbers ( taken to include 0 ) n and k, the binomial coefficient can be defined as the coefficient of the monomial X < sup > k </ sup > in the expansion of.

For example, a poker hand can be described as a 5-combination ( k = 5 ) of cards from a 52 card deck ( n = 52 ).

For example, the three consonants क ्, न ्, and य ्, ( k, n, y ), when written consecutively without virāma form कनय, as shown above.

* Prediction: For every state in S ( k ) of the form ( X → α • Y β, j ) ( where j is the origin position as above ), add ( Y → • γ, k ) to S ( k ) for every production in the grammar with Y on the left-hand side ( Y → γ ).

* Completion: For every state in S ( k ) of the form ( X → γ •, j ), find states in S ( j ) of the form ( Y → α • X β, i ) and add ( Y → α X • β, i ) to S ( k ).

For example, some references, and Wolfram's Mathematica software, define the complete elliptic integral of the first kind in terms of the parameter m, instead of the elliptic modulus k.

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.

For k = 3, this is approximately equal to 1. 3684.

For k = 4, it is approximately 1. 1106.

For such data, a function that extracts the numeric part k of the file name and returns k mod n would be nearly optimal.

For every fission that is induced in the system, an average total of 1 /( 1 − k ) fissions occur.

For a given mass of fissile material the value of k can be increased by increasing the density.

For example, in English, the " k " sounds in the words kit and skill are not identical ( as described below ), but they are distributional variants of a single phoneme,.