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 >.

The coefficients c < sub > k </ sub > must be chosen so that every valid index tuple maps to the address of a distinct element.

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 >.

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.

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.

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

* k > 1 ( supercriticality ): For every fission in the material, it is likely that there will be " k " fissions after the next mean generation time.

< 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.

All of these define the relative numbers of particles in a system as decreasing exponential functions of energy ( at the particle level ) over kT, with k representing the Boltzmann constant and T representing the temperature observed at the macroscopic level.

Then, for any given sequence of integers a < sub > 1 </ sub >, a < sub > 2 </ sub >, …, a < sub > k </ sub >, there exists an integer x solving the following system of simultaneous congruences.

Furthermore, all solutions x of this system are congruent modulo the product, N = n < sub > 1 </ sub > n < sub > 2 </ sub >… n < sub > k </ sub >.

Let x, y, z be a system of Cartesian coordinates in 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors.

where Σ represent algebraic sums and the indices k refer to the various places where heat is supplied, matter flows into the system, and boundaries are moving.

The term dV < sub > k </ sub >/ dt represents the rate of change of the system volume at position k which result in pV power done by the system.

It consists of a mass m, which experiences a single force, F, which pulls the mass in the direction of the point x = 0 and depends only on the mass's position x and a constant k. Newton's second law for the system is

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.

Kilo-( symbol: k, lowercase ) is a unit prefix in the metric system denoting multiplication of the unit by one thousand.

In certain areas of computer science, a modified base-k positional system is used, called bijective numeration, with digits 1, 2, ..., k ( k ≥ 1 ), and zero being represented by an empty string.

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

Within a planetary system, planets, dwarf planets, asteroids ( a. k. a. minor planets ), comets, and space debris orbit the barycenter in elliptical orbits.

Without altering the system, we can take multiple samples, which will have a range of values of k, but the system will always be characterized by the same λ.

A Steiner system with parameters t, k, n, written S ( t, k, n ), is an n-element set S together with a set of k-element subsets of S ( called blocks ) with the property that each t-element subset of S is contained in exactly one block.

Like its dual, the NOR operator ( a. k. a. the Peirce arrow or Quine dagger ), NAND can be used by itself, without any other logical operator, to constitute a logical formal system ( making NAND functionally complete ).