In the nineteenth century, using the same methods, Cantor showed that this restriction is not necessary.

This is the essentially the reason for the restriction to exponential type of less than 2π: the function sin ( 2πnz ) grows faster than e < sup > 2π | z |</ sup > along the imaginary axis!

Every completely multiplicative function is a homomorphism of monoids and is completely determined by its restriction to the prime numbers.

An " extensional stance " and restriction to a second-order predicate logic means that a propositional function extended to all individuals such as " All ' x ' are blue " now has to list all of the ' x ' that satisfy ( are true in ) the proposition, listing them in a possibly infinite conjunction: e. g. x < sub > 1 </ sub > V x < sub > 2 </ sub > V.

* Type I enzymes () cleave at sites remote from recognition site ; require both ATP and S-adenosyl-L-methionine to function ; multifunctional protein with both restriction and methylase () activities.

* Type II enzymes () cleave within or at short specific distances from recognition site ; most require magnesium ; single function ( restriction ) enzymes independent of methylase.

* The complex conjugate function z → z * is not complex analytic, although its restriction to the real line is the identity function and therefore real analytic, and it is real analytic as a function from R² to R².

then F is called an analytic continuation of f. In other words, the restriction of F to U is the function f we started with.

* The restriction of the logarithmic function to N.

For every type of restricted partition there is a corresponding function for the number of partitions satisfying the given restriction.

In general, if is a holomorphic function whose restriction to the real numbers is real-valued, and is defined, then

Although Luttinger liquids are physically the same as Fermi liquids, the restriction to one dimension gives rise to several qualitative differences such as the absence of a quasiparticle peak in the momentum dependent spectral function and the presence of spin density waves.

An arbitrary topological space X can be considered a locally ringed space by taking O < sub > X </ sub > to be the sheaf of real-valued ( or complex-valued ) continuous functions on open subsets of X ( there may exist continuous functions over open subsets of X which are not the restriction of any continuous function over X ).

But the restriction to a subset of all available functions has another great benefit: it is possible to obtain domains that contain their own function spaces, i. e. one gets functions that can be applied to themselves.

Orifice plate is choke plate, working as restriction function.

Furthermore, in order to ensure that the cost function is bounded, the additional restriction is imposed that the pair is controllable.

Note that the above step requires that a > 0 and cos ( θ ) > 0 ; we can choose the a to be the positive square root of a < sup > 2 </ sup >; and we impose the restriction on θ to be − π / 2 < θ < π / 2 by using the arcsin function.

a maximal function ), Stein complementary series representations, Nikishin – Pisier – Stein factorization in operator theory, the Tomas – Stein restriction theorem in Fourier analysis, the Kunze – Stein phenomenon in convolution on semisimple groups, the Cotlar – Stein lemma concerning the sum of almost orthogonal operators, and the Fefferman – Stein theory of the Hardy space and the space of functions of bounded mean oscillation.

The only restriction that function places on its variables is that they implement the "+" and the "*" methods.

The expression e < sup > s </ sup > shall always denote the exponential function, which is the restriction of a principal branch of z < sup > s </ sup > to z

It generates sampling values from an arbitrary probability distribution function by using an instrumental distribution, under the only restriction that < math > f ( x )< M g ( x )</ math > where is an appropriate bound on.

This restriction also inhibits more dynamic behavior of the function.

Initially, most IRC servers formed a single IRC network, to which new servers could join without restriction, but this was soon abused by people who set up servers to sabotage other users, channels, or servers.

Earlier in the year left-wing councils had protested at Government restriction of their budgets by refusing to set budgets, resulting in a budget crisis in Militant-dominated Liverpool City Council.

His plans are to set a restriction to five foreign players and having six players from the said team's own nationality.

He states: " And yet, even the elementary form that Russell < sup > 9 </ sup > gave to the set-theoretic antinomies could have persuaded them König, Jourdain, F. Bernstein that the solution of these difficulties is not to be sought in the surrender of well-ordering but only in a suitable restriction of the notion of set ".

Charles's dissolution of the chamber of deputies, his July Ordinances, which set up rigid control of the press, and his restriction of suffrage resulted in the July Revolution of 1830.

Stronger than the determinant restriction is the fact that an orthogonal matrix can always be diagonalized over the complex numbers to exhibit a full set of eigenvalues, all of which must have ( complex ) modulus 1.

The product over the set of all prime numbers of the restriction of these functors to the category of torsion groups, is a faithful functor from the category of torsion groups to the product over all prime numbers of the categories of p-torsion groups.

* For every open set U of X, the restriction morphism res < sub > U, U </ sub >: F ( U ) → F ( U ) is the identity morphism on F ( U ).

* a family of ring homomorphisms φ < sub > V </ sub >: O < sub > Y </ sub >( V ) → O < sub > X </ sub >( f < sup >-1 </ sup >( V )) for every open set V of Y which commute with the restriction maps.

If F ( U ) is a module over the ring O < sub > X </ sub >( U ) for every open set U in X, and the restriction maps are compatible with the module structure, then we call F an O < sub > X </ sub >- module.

In a moment of weakness James had shown Gondomar the contract under which Raleigh had sailed, the restriction upon attacking Spanish settlements, in order to mollify his objections to an enterprise on which James had set his heart.

The film had been completely closed to the media, with no reporters at all permitted on set during production, a restriction greater than Beatty's previous productions.

Abulia ( poverty of motivation )-is a restriction in will or motivation, often characterized by an inability to set goals or make decisions.

The restriction of the divisibility relation to the set of all monic monomials ( in the given ring ) is a partial order, and thus makes this set to a poset.

The restriction of the latter was set to Germany's " approved peacetime needs ", which were defined to be set on the average European standard.

* Calibrated orifice, a restriction placed into a system of pipes to set the flow rate

The adding-up restriction stems from the assumption that consumption always takes place at the upper boundary of the household's opportunity set, which is only fulfilled if the household cannot completely satisfy all its wants within the boundaries of the opportunity set

A restriction for a set of coordinates to serve as generalized coordinates is that they should uniquely define any possible configuration of the system relative to the reference configuration.

Initially, most IRC servers formed a single IRC network, to which new servers could join without restriction, but this was soon abused by people who set up servers to sabotage other users, channels, or servers.

", was removed from the original CD release ( due to the 74: 33 redbook time restriction on the CD format at the time ) rather than issue it as a 2CD set, but was included on all cassette releases.

Exec Shield allows these controls to be set for both binary executables and for libraries ; if an executable loads a library requiring a given restriction relaxed, the executable will inherit that marking and have that restriction relaxed.