It lacks the NH < sub > 2 </ sub > group because of the cyclization of the side-chain and is known as an imino acid ; it falls under the category of special structured amino acids .</ ref > where R is an organic substituent known as a " side-chain "); often the term " amino acid " is used to refer specifically to these.

½ ( pKa < sub > 1 </ sub > + pKa < sub > R </ sub >), where pKa < sub > R </ sub > is the side-chain pKa.

Cysteine also has potentially negative side-chain with pKa < sub > R </ sub >

½ ( pKa < sub > R </ sub > + pKa < sub > 2 </ sub >).

where < big ></ big > is the number of degrees of freedom divided by two, R is the universal gas constant and n is the number of moles in the system ( a constant ).

An amide is a compound with the functional group R < sub > n </ sub > E ( O )< sub > x </ sub > NR '< sub > 2 </ sub > ( R and R ' refer to H or organic groups ).

The term amide refers both to classes of compounds and to the functional group ( R < sub > n </ sub > E ( O )< sub > x </ sub > NR '< sub > 2 </ sub >) within those compounds.

Amide can also refer to the conjugate base of ammonia ( the anion H < sub > 2 </ sub > N < sup >–</ sup >) or of an organic amine ( an anion R < sub > 2 </ sub > N < sup >–</ sup >).

Closely related and even more numerous are amides derived from primary amines ( R ' NH < sub > 2 </ sub >) with the formula RC ( O ) NHR '.

* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

* Every polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra.

where k < sub > B </ sub > is Boltzmann's constant, T is the temperature, R is the resistance, and is the bandwidth of the frequency.

auxiliary regression is TR < sup > 2 </ sup >, where T is the sample size and R < sup > 2 </ sup > is the coefficient of determination.

Most models for reconstruction assume that the tip is a spherical object, and utilise empirical corrections to stereographic projection to convert detector positions back to a 2D surface embedded in R < sup > 3 </ sup >.

By sweeping this surface through R < sup > 3 </ sup > as a function of the ion sequence input data, such as via ion-ordering, a volume is generated onto which positions the 2D detector positions can be computed and placed three-dimensional space.

So, for example, while R < sup > n </ sup > is a Banach space with respect to any norm defined on it, it is only a Hilbert space with respect to the Euclidean norm.

* Take the Banach space R < sup > n </ sup > ( or C < sup > n </ sup >) with norm || x ||

The one-point compactification of R is homeomorphic to the circle ; the one-point compactification of R < sup > 2 </ sup > is homeomorphic to the sphere.

For any subset A of Euclidean space R < sup > n </ sup >, the following are equivalent:

However, the converse may fail in non-Euclidean R < sup > n </ sup >.

A binary relation is the special case of an n-ary relation R ⊆ A < sub > 1 </ sub > × … × A < sub > n </ sub >, that is, a set of n-tuples where the jth component of each n-tuple is taken from the jth domain A < sub > j </ sub > of the relation.

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

for all m in M, n in N and t in R.

If a is a point in R < sup > n </ sup >, then the higher dimensional chain rule says that:

) The exterior derivative of a k-form in R < sup > 3 </ sup > is defined as the ( k + 1 )- form from above ( and in R < sup > n </ sup > if, e. g., < math >

The space R of real numbers and the space C of complex numbers ( with the metric given by the absolute value ) are complete, and so is Euclidean space R < sup > n </ sup >, with the usual distance metric.

This is a generalization of the Heine – Borel theorem, which states that any closed and bounded subspace S of R < sup > n </ sup > is compact and therefore complete.

Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.

Likewise, the problem of computing a quantity on a manifold which is invariant under differentiable mappings is inherently global, since any local invariant will be trivial in the sense that it is already exhibited in the topology of R < sup > n </ sup >.

Differential topology also deals with questions like these, which specifically pertain to the properties of differentiable mappings on R < sup > n </ sup > ( for example the tangent bundle, jet bundles, the Whitney extension theorem, and so forth ).

We say that N is nilpotent if there is some positive integer R such that Af.

for, using the fact that N and N' commute Af and so when R is sufficiently large every term in this expression for Af will be 0.

It is very easy to establish by induction on R that if F is in Af then Af ; ;

It is difficult to measure the direction and magnitude of R directly.

This is not a constant value like Af, but varies with the thickness of the coating and the direction and magnitude of the resultants R and Af of Fig. 6.

If the rake angle **yc of the knife is high enough and the friction angle **yt between the front of the knife and the back of the chip is low enough to give a positive value for Af, the resultant vector R will lie above the plane of the substrate.

But even more important than this is the fact that the direct search by simultaneously varying all operating conditions has produced only one optimal policy, namely, that for the given feed state and R stages.

But Af is vastly larger than R.

These will be numbered in the direction opposite to the flow of the process stream, so that stage R is the T stage from the end.

In the third column is given the optimal policy for stage R, and in the fourth, the resulting state of the stream when this policy is used.

In this case the stage R operating with conditions Af transforms the state of the stream from Af to Af, but only the probability distribution of Af is known.

that is, it is known empirically that names beginning with R are more common than those beginning with Z ; ;

The study of altruism was the initial impetus behind George R. Price's development of the Price equation, which is a mathematical equation used to study genetic evolution.

Kempthorne and his students make an assumption of unit treatment additivity, which is discussed in the books of Kempthorne and David R. Cox.

An alkyl group, generally abbreviated with the symbol R, is a functional group or side-chain that, like an alkane, consists solely of single-bonded carbon and hydrogen atoms, for example a methyl or ethyl group.

Where territory is occupied in the course of hostilities by an enemy's force, even if the annexation of the occupied country is proclaimed by the enemy, there can be no change of allegiance during the progress of hostilities on the part of a citizen of the occupied country ( R v Vermaak ( 1900 ) 21 NLR 204 ( South Africa )).

A natural-born subject owes allegiance wherever they may be, so that where territory is occupied in the course of hostilities by an enemy's force, even if the annexation of the occupied country is proclaimed by the enemy, there can be no change of allegiance during the progress of hostilities on the part of a citizen of the occupied country ( R v Vermaak ( 1900 ) 21 NLR 204 ( South Africa )).

USACC is co-chaired by Tim Cejka, President of Exxon Mobil Corporation and Reza Vaziri, President of R. V.