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 >.

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

Alternative notations include C ( n, k ), < sub > n </ sub > C < sub > k </ sub >, < sup > n </ sup > C < sub > k </ sub >, C < sup > k </ sup >< sub style =" position: relative ; left :-. 5em ;"> n </ sub >, C < sup > n </ sup >< sub style =" position: relative ; left :-. 5em ;"> k </ sub >, in all of which the C stands for combinations or choices.

The k < sup > th </ sup > sub-range contains the values of r for which there is a stable cycle ( a cycle which attracts a set of initial points of unit measure ) of period This sequence of sub-ranges is called a cascade of harmonics.

Applying an annihilation operator followed by its corresponding creation operator returns the number of particles in the k < sup > th </ sup > single-particle eigenstate:

The combination of operators is known as the number operator for the k < sup > th </ sup > eigenstate.

If the k < sup > th </ sup > single-particle energy eigenstate has energy and there are bosons in this state, then the total energy of these bosons is.

Since any chemical component can participate in several reactions simultaneously, the stoichiometric coefficient of the i – th component in the k – th reaction is defined as

In number theory, Waring's problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s k < sup > th </ sup > powers of natural numbers ( for example, every number is the sum of at most 4 squares, or 9 cubes, or 19 fourth powers, etc .).

For every k, we denote by g ( k ) the minimum number s of k < sup > th </ sup > powers needed to represent all integers.

Since the r < sup > th </ sup > failure comes last, it remains to choose the k trials with successes out of the remaining k + r − 1 trials.

Given a system of n-dimensional variables ( physical variables ), in k ( physical ) dimensions, write the dimensional matrix M, whose rows are the dimensions and whose columns are the variables: the ( i, j ) th entry is the power of the ith unit in the jth variable.

represent flux of i < sup > th </ sup > component of linear momentum across the x < sup > k </ sup > surface.

More generally: the k < sup > th </ sup > difference of the sequence is written as is defined recursively as

The more restrictive definition of difference equation is an equation composed of a < sub > n </ sub > and its k < sup > th </ sup > differences.

6 >< li > For all a relatively prime to k, χ ( a ) is a φ ( k )- th complex root of unity .</ ol >

# If all the particles are identical ( same energy barrier and same magnetic moment ), their easy axes are all oriented parallel to the applied field and the temperature is low enough ( T < sub > B </ sub > < T ≲ KV /( 10 k < sub > B </ sub >)), then the magnetization of the assembly is.

The idea with the above method is you first find the general case for a kth moment where you have k different x variables-and then you can simplify this accordingly.

The two diagrams below show the electric vectors of linearly and circularly polarized light, at one moment of time, for a range of positions ; the plot of the circularly polarized electric vector forms a helix along the direction of propagation ( k ).

The power of k is because moments scale as, meaning that: they are homogeneous polynomials of degree k, thus the standardized moment is scale invariant.

Unfortunately, he is captured and almost burnt at the stake by the Grand Inquisitor Enrique ( a. k. a. Magneto ), but rescued at the last moment by Carlos Javier and his students.

At the end of Thief of Time he shares an unspecified " perfect moment " with Susan Sto Helit ( a. k. a. Susan Death ), who is also a human who inherited qualities from an anthropomorphic personification.

The concentrations of individual compounds can be derived largely using the count rates of the precursor and product ions, and the reaction rate coefficients, k. Exothermic proton transfer reactions with H < sub > 3 </ sub > O < sup >+</ sup > are assumed to proceed at the collisional rate ( see Collision Theory ), the coefficient for which, k < sub > c </ sub >, is calculable using the method described by Su and Chesnavich, providing the polarizability and dipole moment are known for the reactant molecule.

Spanish / p t k /, for example, have voice onset times ( VOTs ) of about 5, 10, and 30 milliseconds, whereas English / p t k / have VOTs of about 60, 70, and 80 ms. Korean has been measured at 20, 25, and 50 ms for and 90, 95, and 125 for.

Considerable confusion exists about the meanings of the SI ( or metric ) prefixes used with the unit byte, especially concerning prefixes such as kilo ( k or K ) and mega ( M ) as shown in the chart Prefixes for bit and byte.

* Repentance a. k. a. The Confession ( 1984 ), a Georgian film about a fictional dictator which was banned until 1987.

1929's La Belle Marinière Beautiful Lady of the Canals a / k / a The Beautiful Bargewoman still has some of the excessively-poetic dialogue, but is overall a realistic play about a love triangle involving a bargeman, his wife and his best friend and companion.

where σ determines the spread of k < sub > 1 </ sub >- values about k, and N is the amplitude of the wave.

Kittinger set unbeaten () world records for: high-altitude jump ; free-fall by falling 16. 0 miles ( 25. 7 kilometers ) before opening his parachute ; and the fastest speed attained by a human being without mechanical or chemical assistance, about 982 k. p. h ( 614 m. p. h .).

There is also speculation about the relationship the song bears to " The Bold Fusilier " ( a. k. a. " Marching through Rochester "), a song sung to the same tune and dated by some back to the 18th century but first printed in 1900.

* Mars Pathfinder, Mars Exploration Rover and the Cassini probe to Saturn use a k of 15 and a rate of 1 / 6 ; this code performs about 2 dB better than the simpler k = 7 code at a cost of 256 × in decoding complexity ( compared to Voyager mission codes ).

The so-called Landauer limit implied by the laws of physics sets a lower limit on the energy required to perform a computation of per bit erased in a computation, where T is the temperature of the computing device in kelvins, k is the Boltzmann constant, and the natural logarithm of 2 is about 0. 693.

At the opening of the book, the narrator, an everyman named John ( a. k. a. Jonah ), describes a time when he was planning to write a book about what important Americans did on the day Hiroshima was bombed.

A nickel layer, sputter deposited onto one of the emitter halves, outlines the desired channel contour and determines channel height ( a. k. a. slit height, typically 1-2 μm ) and channel width ( a. k. a. slit length, ranging from 1 mm up to about 7 cm ).

Kaufman's proposed date of about 2200 BCE would require two regular phonological ( sound ) changes that are attested in all Maya languages, " r " changing to " y " and " q " to " k ", to have happened independently after the split, in both the Huastec / Chicomuceltec branch and in the branch of all other Mayan languages.

The fire district became part of Santa Clara County Fire Department, ( a. k. a. Central Fire District ,) about 1960 and is served by San Jose Fire under a contract.

In 2011, a documentary is being filmed about a local Reeseville celebrity, Ethan Schlicher ( a. k. a. The Lorax ) and all of his prized barnyard animals.

** Joseph Balsamo ( Mémoires d ' un médecin: Joseph Balsamo, 1846 – 1848 ) ( a. k. a. Memoirs of a Physician, Cagliostro, Madame Dubarry, The Countess Dubarry, or The Elixir of Life )( Joseph Balsamo is about 1000 pages long, and is usually published in two volumes in English translations: Vol 1.

distribution about zero and white noise spectrum, K ( k ) is the ( real ) Fourier amplitude expected from the