 Page "Algorithms for calculating variance" ¶ 75
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Here and < 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 and n 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 and </ 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 and denotes 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.

0.175 seconds.