# Analytically, when the mode ( s ) of the posterior distribution can be given in closed form.

Analytically, the magnetostatic energy ( U ) of the tip-sample system can be calculated in one of two ways:

Analytically, it can be determined that each of the three segments of the LMN triangle has a length of:

Analytically, it is helpful to distinguish Russification, as a process of changing one's ethnic self-label or identity from a non-Russian ethnonym to Russian, from Russianization, the spread of the Russian language, culture, and people into non-Russian cultures and regions, distinct also from Sovietization or the imposition of institutional forms established by the Communist Party of the Soviet Union throughout the territory ruled by that party.

Analytically, the city was ranked 1st globally for a culture of innovation in 2007 and 2008, and 2nd globally after Boston in 2009 from 256 cities on an analysis of 162 indicators in the Innovation Cities Index on a 3-factor score covering culture, infrastructure and markets.

Analytically, the culture has been broken down into various stages or horizons.

This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates ( x, y, z ).

In general, if y = f ( x ), then it can be transformed into y = af ( b ( x − k )) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis.

The slope field of F ( x ) = ( x < sup > 3 </ sup >/ 3 )-( x < sup > 2 </ sup >/ 2 )- x + c, showing three of the infinitely many solutions that can be produced by varying the Constant of integration | arbitrary constant C.

Thus, all the antiderivatives of x < sup > 2 </ sup > can be obtained by changing the value of C in F ( x ) = ( x < sup > 3 </ sup >/ 3 ) + C ; where C is an arbitrary constant known as the constant of integration.

Thus, a single " rule ," like mapping every real number x to x < sup > 2 </ sup >, can lead to distinct functions and, depending on whether the images under that rule are understood to be reals or, more restrictively, non-negative reals.

Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x ∨ y := x + y + ( x · y ) and x ∧ y := x · y.

( The Taylor series indicates that − J < sub > 1 </ sub >( x ) is the derivative of J < sub > 0 </ sub >( x ), much like − sin x is the derivative of cos x ; more generally, the derivative of J < sub > n </ sub >( x ) can be expressed in terms of J < sub > n ± 1 </ sub >( x ) by the identities below.

It can be considered as a ' natural ' partner of J < sub > α </ sub >( x ).

This orthogonality relation can then be used to extract the coefficients in the Fourier – Bessel series, where a function is expanded in the basis of the functions J < sub > α </ sub >( x u < sub > α, m </ sub >) for fixed α and varying m.

The " double width " mode of the TRS-80 was also supported, so 256 x 192 pixels ( in 32x16 character mode ), or 320 x 300 pixels ( in 40x24 character mode ) were also possible.

The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportionally to 1 /√ x ( see also their asymptotic forms below ), although their roots are not generally periodic, except asymptotically for large x.

If x is a member of A, then it is also said that x belongs to A, or that x is in A.

The symbol * x * is also hereby endowed with

The term carotene ( also carotin, from the Latin carota, or carrot ) is used for several related unsaturated hydrocarbon substances having the formula C < sub > 40 </ sub > H < sub > x </ sub >, which are synthesized by plants but cannot be made by animals.

Maximum likelihood can also be used to estimate the parameters x < sub > 0 </ sub > and γ.

The CIE 1931 x, y chromaticity space, also showing the chromaticities of black body light sources of various temperatures ( Planckian locus ), and lines of constant # Correlated color temperature | correlated color temperature.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

) There are also two main conventions for the second letter of the digraph as well: voicing may be written with g and uvular affrication with x, or voicing with d and affrication with g ( a convention of Afrikaans ).

If y = f ( x ) is differentiable at a, then f must also be continuous at a.

The transform is sometimes denoted by the symbol, as in or or < ref group =" note "> As a linear transformation on a finite-dimensional vector space, the DFT expression can also be written in terms of a DFT matrix ; when scaled appropriately it becomes a unitary matrix and the X < sub > k </ sub > can thus be viewed as coefficients of x in an orthonormal basis .</ ref >

Raising both sides to the exponent of 2 ( which means, applying the function to both sides of the equation ) changes our equation into, which not only has all the previous solutions but also introduces a new set of extraneous solutions, with and x being any number.

Alphabets can also be infinite ; e. g. first-order logic is often expressed using an alphabet which, besides symbols such as ∧, ¬, ∀ and parentheses, contains infinitely many elements x < sub > 0 </ sub >, x < sub > 1 </ sub >, x < sub > 2 </ sub >, … that play the role of variables.

* This 12 " was also released with a 12 " x 24 " poster ( 12 ZTAQ 22 )

The argument hinges on the nature of statements such as " Anything that is pleasant is also good " and the possibility of asking questions such as " Is it good that x is pleasant?

We also say that x necessarily exists if for every essence P the following is true: in every possible world, there is an element y with P ( y ).

The arithmetic mean of a variable is often denoted by a bar, for example ( read " x bar ") would be the mean of some sample space.

In German, words starting with sch-( constituting the German phoneme ) would be intercalated between words with initial sca-and sci-( all incidentally loanwords ) instead of this graphic cluster appearing after the letter s, as though it were a single letter — a lexicographical policy which would be de rigueur in a dictionary of Albanian, i. e. dh -, ë -, gj -, ll -, rr -, th -, xh-and zh-( all representing phonemes and considered separate single letters ) would follow the letters d, e, g, l, n, r, t, x and z respectively.

The usual order relation ≤ on the real numbers is antisymmetric: if for two real numbers x and y both inequalities x ≤ y and y ≤ x hold then x and y must be equal.

These points form a line, and y = x is said to be the equation for this line.

The function ƒ ( x ) may or may not be defined at a, and its precise value at the point x = a does not affect the asymptote.

The oblique asymptote, for the function f ( x ), will be given by the equation y = mx + n.

Some mathematicians, especially in set theory, do not consider the sets and to be part of the relation, and therefore define a binary relation as being a subset of x, that is, just the graph.

We then define f ( x, y ) to be this z.

The natural logarithm of a number x is the power to which e would have to be raised to equal x.

In general, when a binomial like x + y is raised to a positive integer power we have:

At x = 1, the y-value equals the base because any number raised to the power of 1 is the number itself.

In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals x

: A candidate x should not be harmed < nowiki > change from being a winner to a loser </ nowiki > if x is raised on some ballots without changing the orders of the other candidates.

If the user selects " Dec " mode, and types " 16 " and then presses the " x ^ y " button and then types " 16 " again, and then presses "=" the result will be ( decimal ) 18446744073709551616, which is correct, as that is the decimal number for " 16 raised to the 16th power ".

The mosque is built on a platform measuring about 75 m x 75 m ( 245 ft x 245 ft ) that is raised by 3 metres ( 9 ft ) above the level of the marketplace.

This addition raised total seating capacity to 45, 674 and included the 80 x 30 ft. ( 24 x 9 m ) ProStar Digital VideoPlus Display screen.

The grand stand which will seat nearly 1200 people, is 151 x, and in the rear is raised.

Standard equipment included a Champagne White exterior with red and blue stripes, a roof rack, a sports steering wheel, and Goodyear G70 x 15 raised white letter tires mounted on wider steel wheels.

On October 3, 2006, Trump raised a 20 ' x 30 ' American flag on an 80 ' flagpole at Mar-a-Lago.

The Hunnewell Pine, Pinus x hunnewellii, is a hybrid between Eastern White Pine and Japanese White Pine first raised at the Walter Hunnewell Arboretum in 1952.

A simple 4 ' x 8 ' ( 32 square feet ) raised bed garden based on the principles of bio-intensive planting and square foot gardening uses fewer nutrients and less water, and could keep a family, or community, supplied with an abundance of healthy, nutritious organic greens, while promoting a more sustainable way of living.

The names of well-known functions, such as sin x ( the trigonometric function sine ) and exp x ( the constant e raised to the power of x ) are written in lowercase upright letters ( and often, as shown here, without parentheses around the argument ).