If M is a Turing Machine which, on input w, outputs string x, then the concatenated string < M > w is a description of x.

Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that

Thus, if P is a program in L < sub > 2 </ sub > which is a minimal description of s, then InterpretLanguage ( P ) returns the string s. The length of this description of s is the sum of

Concordant with the above understanding, any assertion about the motion of a body boils down to a description over time in which the body under consideration is at t < sub > 1 </ sub > found in the vicinity of one group of " landmark " bodies and at some t < sub > 2 </ sub > is found in the vicinity of some other " landmark " body or bodies.

In 1927, the first mathematically complete quantum description of a simple chemical bond, i. e. that produced by one electron in the hydrogen molecular ion, H < sub > 2 </ sub >< sup >+</ sup >, was derived by the Danish physicist Oyvind Burrau.

) The notation " h / 2 " refers to the M < sub > 2 </ sub > C type structure described above, which is only an approximate description of the actual structures.

The heterotic SO ( 32 ) and the heterotic E < sub > 8 </ sub >× E < sub > 8 </ sub > theories are also related by T-duality ; the heterotic SO ( 32 ) description of a circle of radius R is exactly the same as the heterotic E < sub > 8 </ sub >× E < sub > 8 </ sub > description of a circle of radius 1 / R.

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The above description is of activation by slow neutrons, slow neutrons are fully moderated within the reactor and have KE < 0. 5 eV.

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The International Time Bureau ( BIH ) began a time scale, T < sub > m </ sub > or AM, in July 1955, using both local caesium clocks and comparisons to distant clocks using the phase of VLF radio signals.

Algeria has always been a source of inspiration for different painters who tried to immortalize the prodigious diversity of the sites it offers and the profusion of the facets that passes its population, which offers for Orientalists between the 19 < sup > th </ sup > century and the 20 < sup > th </ sup > century, a striking inspiration for a very rich artistic creation like Eugène Delacroix with his famous painting women of Algiers in their apartment or Etienne Dinet or other painters of world fame like Pablo Picasso with his painting women of Algiers, or painters issued from the Algiers school.

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All 128 ASCII characters, including non-printable characters ( represented by their abbreviations ). The 95 ASCII graphic characters are numbered from 20 < sub > hexadecimal | hex </ sub > to 7E < sub > hexadecimal | hex </ sub > ( decimal 32 to 126 ).

The " space " character had to come before graphics to make sorting easier, so it became position 20 < sub > hex </ sub >; for the same reason, many special signs commonly used as separators were placed before digits.

To keep options available for lower case letters and other graphics, the special and numeric codes were arranged before the letters, and the letter " A " was placed in position 41 < sub > hex </ sub > to match the draft of the corresponding British standard.

The @ symbol was not used in continental Europe and the committee expected it would be replaced by an accented À in the French variation, so the @ was placed in position 40 < sub > hex </ sub > next to the letter A.

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.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

When there are only two means to compare, the t-test and the ANOVA F-test are equivalent ; the relation between ANOVA and t is given by F = t < sup > 2 </ sup >.

The number of carbon atoms is used to define the size of the alkane ( e. g., C < sub > 2 </ sub >- alkane ).

The simplest possible alkane ( the parent molecule ) is methane, CH < sub > 4 </ sub >.

The reason why pHs of acids are less than 7 is that the concentration of hydronium ions is greater than 10 < sup >− 7 </ sup > moles per liter.

An Arrhenius acid is a substance that increases the concentration of the hydronium ion, H < sub > 3 </ sub > O < sup >+</ sup >, when dissolved in water.

BF < sub > 3 </ sub > is a Lewis acid because it accepts the electron pair from fluoride.

The species that gains the electron pair is the Lewis acid ; for example, the oxygen atom in H < sub > 3 </ sub > O < sup >+</ sup > gains a pair of electrons when one of the H — O bonds is broken and the electrons shared in the bond become localized on oxygen.

The Brønsted-Lowry definition is the most widely used definition ; unless otherwise specified acid-base reactions are assumed to involve the transfer of a proton ( H < sup >+</ sup >) from an acid to a base.

Reactions of acids are often generalized in the form HA H < sup >+</ sup > + A < sup >−</ sup >, where HA represents the acid and A < sup >−</ sup > is the conjugate base.

The acid dissociation constant K < sub > a </ sub > is generally used in the context of acid-base reactions.

The numerical value of K < sub > a </ sub > is equal to the concentration of the products divided by the concentration of the reactants, where the reactant is the acid ( HA ) and the products are the conjugate base and H < sup >+</ sup >.

Because the range of possible values for K < sub > a </ sub > spans many orders of magnitude, a more manageable constant, pK < sub > a </ sub > is more frequently used, where pK < sub > a </ sub >