real, dimension (:), allocatable :: points

:: See computer insecurity for an example of the failure of this approach in the real world.

:: When, therefore, this court adjudges, as it does now adjudge, that Congress cannot impose a duty or tax upon personal property, or upon income arising either from rents of real estate or from personal property, including invested personal property, bonds, stocks, and investments of all kinds, except by apportioning the sum to be so raised among the States according to population, it practically decides that, without an amendment of the Constitution — two-thirds of both Houses of Congress and three-fourths of the States concurring — such property and incomes can never be made to contribute to the support of the national government.

:: For any open neighborhood N of A, there is a positive constant T such that f ( t, b ) ∈ N for all real t > T.

:: " My hand feels touched as well as it touches ; real means this, and nothing more "

:: If any person liable to pay any tax neglects or refuses to pay the same after demand, the amount ( including any interest, additional amount, addition to tax, or assessable penalty, together with any costs that may accrue in addition thereto ) shall be a lien in favor of the United States upon all property and rights to property, whether real or personal, belong to such person.

:: Due in part to the sensational death of Reg Calvert and lurid tales depicting real swashbuckling piracy, the British government were finally able to bring in legislation to make offshore broadcasting a part of British criminal law in the United Kingdom.

:: Hot afternoons are real ; afternoons are ; places, things, thoughts, feelings are ; poetry is ;

real :: A ( m, n )!

real :: AP (- kl: ku, n )!

:: One of Dorai's best students, he is now an up-and-coming martial arts movie star whose insistence on " making the fight real " rapidly becomes detrimental to production, due mainly to damaged props and public property as well as injuries to the stunt doubles.

:: Then we'll all live together with a love that is real.

:: so anything real includes it all.

:: Wife: Umm ul-Banin — which means mother of many sons her real name Fatima bint Hizam al-Qilabiyya.

:: The balebast cook promised me a real gedila

As an example, the field of real numbers is not algebraically closed, because the polynomial equation x < sup > 2 </ sup > + 1 = 0 has no solution in real numbers, even though all its coefficients ( 1 and 0 ) are real.

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

This subset I is referred to as the domain of f. Possible choices include I = R, the whole set of real numbers, an open interval

Then I < sub > x </ sub > and I < sub > x </ sub >< sup > 2 </ sup > are real vector spaces and the cotangent space is defined as the quotient space T < sub > x </ sub >< sup >*</ sup > M = I < sub > x </ sub > / I < sub > x </ sub >< sup > 2 </ sup >.

Typically u and v are taken to be the real and imaginary parts respectively of a complex-valued function of a single complex variable z = x + iy, f ( x + iy ) = u ( x, y ) + iv ( x, y ).

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.

It is frequently stated in the following equivalent form: Suppose that is continuous and that u is a real number satisfying or Then for some c ∈ b, f ( c ) = u.

Alboni's voice, an exceptionally fine contralto with a seamless compass of two and one-half octaves, extending as high as the soprano range < ref name = C243 >" She was a real contralto from f < sub > 3 </ sub > to g < sub > 5 </ sub >, then, up to c < sub > 6 </ sub >, she was an excellent soprano.

The natural logarithm can be defined for any positive real number a as the area under the curve y = 1 / x from 1 to a.

* the field of real rational functions, where p ( x ) and q ( x ), are polynomials with real coefficients, can be made into an ordered field where the polynomial p ( x ) = x is greater than any constant polynomial, by defining that whenever, for.

The white spot at s = 1 is the pole of the zeta function ; the black spots on the negative real axis and on the critical line Re ( s ) = 1 / 2 are its zeros.

Furthermore, the non-trivial zeros are symmetric about the real axis and the line σ = 1 / 2 and, according to the Riemann Hypothesis, they all lie in the line σ = 1 / 2.

We assume that A is an m-by-n matrix over either the real numbers or the complex numbers, and we define the linear map f by f ( x ) = Ax as above.

Thus the ( real < ref > The complex spinors are obtained as the representations of the tensor product H ⊗< sub > R </ sub > C = Mat < sub > 2 </ sub >( C ).

If V is a real vector space, then we replace V by its complexification V ⊗< sub > R </ sub > C and let g denote the induced bilinear form on V ⊗< sub > R </ sub > C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g |< sub > W </ sub > = 0.

For instance, division of real numbers is a partial function, because one can't divide by zero: a / 0 is not defined for any real a.

where the integration limits indicate integration along a contour that can be chosen as follows: from −∞ to 0 along the negative real axis, from 0 to ± iπ along the imaginary axis, and from ± iπ to +∞± iπ along a contour parallel to the real axis.

for every integer r ≥ 0 and every real number x ≥ − 1.

for every integer r ≥ 2 and every real number x ≥ − 1 with x ≠ 0.

For any real numbers x, r > 0, one has

The economy remained sluggish until 2003, when it began to show clear signs of recovery, achieving 4. 0 % real GDP growth.

After robust growth rates in the 1980s ( average annual growth was 6. 1 %), economic performance in the 1990s was mixed: real GDP growth was 9. 7 % in 1992, 1. 7 % in 1993, 6. 0 % in 1994, 6. 0 % in 1995, 1. 9 % in 1996 and 2. 3 % in 1997.

The scores range from 0, which signifies the beer is in such poor condition it cannot be drunk or that no real ale is available ; through 2, which signifies an average beer that is drunk without calling attention to itself in either a positive or negative manner ; up to 5, which signifies a perfect beer.

This is an infinite sum which has one summand for every p in the domain of F. The requirement that the domain be prefix-free, together with Kraft's inequality, ensures that this sum converges to a real number between 0 and 1.

* The cofinality of the real numbers with their usual ordering is ℵ < sub > 0 </ sub >, since N is cofinal in R. The usual ordering of R is not order isomorphic to c, the cardinality of the real numbers, which has cofinality strictly greater than ℵ < sub > 0 </ sub >.

If this limit exists, then it may be computed by taking the limit as h → 0 along the real axis or imaginary axis ; in either case it should give the same result.