Therefore, given any positive integer n, it produces a string with Kolmogorov complexity at least as great as n. The program itself has a fixed length U. The input to the program GenerateComplexString is an integer n. Here, the size of n is measured by the number of bits required to represent n, which is log < sub > 2 </ sub >( n ).

Here K denotes the field of real numbers or complex numbers, I is a closed and bounded interval b and p, q are real numbers with 1 < p, q < ∞ so that

Here k is first-order rate constant having dimension 1 / time, ( t ) is concentration at a time t and < sub > 0 </ sub > is the initial concentration.

Here I < sub > 2 </ sub > is reduced to I < sup >–</ sup > and S < sub > 2 </ sub > O < sub > 3 </ sub >< sup > 2 –</ sup > ( thiosulfate anion ) is oxidized to S < sub > 4 </ sub > O < sub > 6 </ sub >< sup > 2 –</ sup >.

The simplest, and original, implementation of the protocol uses the multiplicative group of integers modulo p, where p is prime and g is primitive root mod p. Here is an example of the protocol, with non-secret values in < span style =" color: blue "> blue </ span >, and secret values in < span style =" color: red "> boldface red </ span >:

Here is the center of the ellipse, and is the angle between the < math > X </ Math >- axis and the major axis of the ellipse.

Here C < sub > p </ sub > is the heat capacity at constant pressure and α is the coefficient of ( cubic ) thermal expansion

Here ( Z / 2Z ) is the polynomial ring of Z / 2Z and ( Z / 2Z )/( T < sup > 2 </ sup >+ T + 1 ) are the equivalence classes of these polynomials modulo T < sup > 2 </ sup >+ T + 1.

Here the kets and columns are identified with the vectors of V and the bras and rows with the dual vectors or linear functionals of the dual space V < sup >*</ sup >, with conjugacy associated with duality.

Here, R < sub > ij </ sub > is the Ricci tensor.

# If A is a cartesian product of intervals I < sub > 1 </ sub > × I < sub > 2 </ sub > × ... × I < sub > n </ sub >, then A is Lebesgue measurable and Here, | I | denotes the length of the interval I.

Here n is the number of electrons / mole product, F is the Faraday constant ( coulombs / mole ), and ΔE is cell potential.

As explained above, Gaussian elimination writes a given m × n matrix A uniquely as a product of an invertible m × m matrix S and a row-echelon matrix T. Here, S is the product of the matrices corresponding to the row operations performed.

Here n is the refractive index of the medium in which the measurement is made ; and f is the measured frequency of the source.

Here n is the number of digits of precision at which the natural logarithm is to be evaluated and M ( n ) is the computational complexity of multiplying two n-digit numbers.

Here, the coefficients χ < sup >( n )</ sup > are the n-th order susceptibilities of the medium and the presence of such a term is generally referred to as an n-th order nonlinearity.

Here Z is atomic number and n is a number which varies between 4 and 5.

The first question was answered in the negative when in 1963, Eggan gave examples of regular languages of star height n for every n. Here, the star height h ( L ) of a regular language L is defined as the minimum star height among all regular expressions representing L. The first few languages found by are described in the following, by means of giving a regular expression for each language:

Here d is the spacing between diffracting planes, is the incident angle, n is any integer, and λ is the wavelength of the beam.

Here are the concrete instances of these equations for n = 2 and n = 3:

Here Δk is the field-dependent difference between LCP and RCP absorption, α is the electric permeability, n is the index of refraction, H is the applied magnetic field, k is the Boltzmann constant and T is the temperature.

Here the last product means that a first electron, r < sub > 1 </ sub >, is in an atomic hydrogen-orbital centered at the second nucleus, whereas the second electron runs around the first nucleus.

Here the term yoga denotes a kind of " meta-theory " that can be used heuristically ; Michel Raynaud writes the other terms " Ariadne's thread " and " philosophy " as effective equivalents.

Here highbit ( S ) denotes the most significant bit of S ; the '< tt >*</ tt >' operator denotes unsigned integer multiplication with lost overflow ; '< tt >^</ tt >' is the bitwise exclusive or operation applied to words ; and P is a suitable fixed word.

Here is the particle's velocity and × denotes the cross product.

Let ρ, θ, and φ be spherical coordinates for the source point P. Here θ denotes the angle with the vertical axis, which is contrary to the usual American mathematical notation, but agrees with standard European and physical practice.

Here denotes the maximum-likelihood estimate, and is the sample mean of the observations.

Here, denotes the natural logarithm.

Here C < sub > b </ sub >( X ) denotes the C *- algebra of all continuous bounded functions on X with sup-norm.

Here denotes vertical composition of natural transformations, and denotes horizontal composition.

Here the t denotes the matrix transpose.

( Here V0, V1, and V2 denote verbs and Ny denotes a noun.

Here y *( w ) denotes the value of the linear functional y * ( which is an element of the dual space of W ) when evaluated at the element w ∈ W. This scalar in turn is multiplied by x to give as the final result an element of the space V.

Here, || ||< sub > 2 </ sup > is the matrix 2-norm, c < sub > n </ sub > is a small constant depending on n, and ε denotes the unit round-off.

Here 0 denotes the trivial abelian group with a single element, the map from Z to Z is multiplication by 2, and the map from Z to the factor group Z / 2Z is given by reducing integers modulo 2.

Here, denotes the radical of J and I ( U ) is the ideal of all polynomials which vanish on the set U.

Here, " prequel " denotes status as a " franchise-renewing original " that depicts events earlier in the ( internally inconsistent ) narrative cycle than those of a previous installment.

Here N < sub > m </ sub > denotes the number of turns in loop m, Φ < sub > m </ sub > the magnetic flux through this loop, and L < sub > m, n </ sub > are some constants.

( Here K ( a ) denotes the smallest subfield of L containing K and a )

Here denotes the standard complex inner product on and.

Here it also denotes a hand gesture, now linked to three other hand mudrās — the action ( karma ), pledge ( samaya ), and dharma mudrās — but also involves " mantra recitations and visualizations that symbolize and help to effect one ’ s complete identification with a deity ’ s divine form or awakening mind ( bodhicitta ).

Here, denotes the resolution limit in arcseconds and is in millimeters.