Here, < sub > n </ sub > denotes the sample mean of the first n samples ( x < sub > 1 </ sub >, ..., x < sub > n </ sub >), s < sup > 2 </ sup >< sub > n </ sub > their sample variance, and σ < sup > 2 </ sup >< sub > n </ sub > their population variance.

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.

For example, the notion of gauge invariance forms the basis of the well-known Mattis spin glasses, which are systems with the usual spin degrees of freedom for i = 1 ,..., N, with the special fixed " random " couplings Here the ε < sub > i </ sub > and ε < sub > k </ sub > quantities can independently and " randomly " take the values ± 1, which corresponds to a most-simple gauge transformation This means that thermodynamic expectation values of measurable quantities, e. g. of the energy are invariant.

Here the sum is extended over all ( k, j ) with and with

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 V is the molar volume of a substance, T < sub > C </ sub > is the critical temperature and k is a constant valid for almost all substances.

Here, Joseph Lynch Martin ( a. k. a. " Greenbrier Joe ") established a trading post called Pensacola.

Here k may be negative.

Here we see that the period of 3 < sup > k </ sup > modulo 7 is 6.

Here we show that the size of the ( sub ) tree rooted at any node x of degree d in the heap must have size at least F < sub > d + 2 </ sub >, where F < sub > k </ sub > is the kth Fibonacci number.

Given a homogeneous prime ideal P of, let X be a subset of P < sup > n </ sup >( k ) consisting of all roots of polynomials in P .< ref > The definition makes sense since if and only if for any nonzero λ in k .</ ref > Here we show X admits a structure of variety by showing locally it is an affine variety.

Here, S ( P, Ei ) represents the similarity between the prototype and the ith exemplar, k is the subscript for the dimensions ( k

Here the maps i: A ∩ B ↪ A, j: A ∩ B ↪ B, k: A ↪ X, and l: B ↪ X are inclusion maps and denotes the direct sum of abelian groups.

Here ω < sup > α </ sup >< sub > β </ sub > defines a k × k matrix of one-forms on U. In fact, given any such matrix the above expression defines a connection on E restricted to U. This is because ω < sup > α </ sup >< sub > β </ sub > determines a one-form ω with values in End ( E ) and this expression defines ∇ to be the connection d + ω, where d is the trivial connection on E over U defined by differentiating the components of a section using the local frame.

Here Z < sub > l </ sub > denotes the ℓ-adic integers, but the definition is by means of the system of ' constant ' sheaves with the finite coefficients Z / l < sup > k </ sup > Z.

" Here A is a polynomial ring over a field, that is, A = k ..., x < sub > n </ sub >.

Here the rank of Q < sub > i </ sub > should be interpreted as meaning the rank of the matrix B < sup >( i )</ sup >, with elements B < sub > j, k </ sub >< sup >( i )</ sup >, in the representation of Q < sub > i </ sub > as a quadratic form:

Here X < sub > i </ sub > are any vector fields on P. Dϕ is a tensorial ( k + 1 )- form on P.

Here, if we define the sequence p < sup > k </ sup > by

Here to be a global norm means to be an element k of K such that there is an element l of L with ; in other words k is a relative norm of some element of the extension field L. To be a local norm means that for some prime p of K and some prime P of L lying over K, then k is a norm from L < sub > P </ sub >; here the " prime " p can be an archimedean valuation, and the theorem is a statement about completions in all valuations, archimedean and non-archimedean.

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, is the thickness of the oceanic mantle lithosphere, is the thermal diffusivity ( approximately 10 < sup >− 6 </ sup > m < sup > 2 </ sup >/ s ), and is time.

Here Z < sub > GUE </ sub >( n ) = 2 < sup > n / 2 </ sup > < sup > n < sup > 2 </ sup >/ 2 </ sup > is a normalization constant, chosen so that the integral of the density is equal to one.

Here, H is the coatings removal force measured parallel to the surface of the substrate and T is the film thickness.

Here, purines form hydrogen bonds to pyrimidines, with A bonding only to T, and C bonding only to G. This arrangement of two nucleotides binding together across the double helix is called a base pair.

Here ΔG is change in Gibbs free energy, T is absolute temperature, R is the gas constant and Q is reaction quotient.

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 < sub > x </ sub > is the number of x atoms per volume, λ the electron escape depth, θ the analyzer angle, T the transmission of the analyzer, I ( t ) the electron excitation flux at depth t, dΩ the solid angle, and δt is the thickness of the layer being probed.

Here T, ΔS and ΔH are respectively the temperature at the melting point, change of entropy of melting and the change of enthalpy of melting.

* 2007: Here and Now Tour ( with Belinda Carlisle, The Human League, Paul Young, The Weather Girls, T ' Pau, etc.

Here orthogonality is important not only for reducing A < sup > T </ sup > A = ( R < sup > T </ sup > Q < sup > T </ sup >) QR to R < sup > T </ sup > R, but also for allowing solution without magnifying numerical problems.

Here Q is the molar activation energy, R is the ideal gas constant, T is absolute temperature, and K < sub > 0 </ sub > is a material dependent factor.

Here the trail reaches a height of 1811 meters at İncegeriş T .. Then on to Belos, Finike, Kumluca, Mavikenic, Karaöz, Olympos, Çıralı.

Here T is the stress – energy tensor, G is the Einstein tensor, and c is the speed of light,

In 2008, T ' Pau were part of the Here and Now 80s nostalgia tour.

Here is a summary of the main differences between Intel syntax and AT & T syntax:

Here, the class that consists of T's children would be closed at the time of the instrument as it is impossible for T to have any children after T dies.