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 >.

The stronger of two acids will have a higher K < sub > a </ sub > than the weaker acid ; the ratio of hydrogen ions to acid will be higher for the stronger acid as the stronger acid has a greater tendency to lose its proton.

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 >

If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

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

# The length of P which by definition is K < sub > 2 </ sub >( s ).

An algebraic closure K < sup > alg </ sup > of K contains a unique separable extension K < sup > sep </ sup > of K containing all ( algebraic ) separable extensions of K within K < sup > alg </ sup >.

The previous case K < nowiki ></ nowiki > X < nowiki ></ nowiki > is a special case of this.

Scholars have often speculated that Hamlet < nowiki ></ nowiki >' s Polonius might have been inspired by William Cecil ( Lord Burghley )— Lord High Treasurer and chief counsellor to Queen Elizabeth I. E. K. Chambers suggested Polonius's advice to Laertes may have echoed Burghley's to his son Robert Cecil.

* if R = K is a field, then K < nowiki ></ nowiki > X < nowiki ></ nowiki > is a discrete valuation ring.

In algebra, the ring K < nowiki ></ nowiki > X < sub > 1 </ sub >, ..., X < sub > r </ sub >< nowiki ></ nowiki > ( where K is a field ) is often used as the " standard, most general " complete local ring over K.

: HMAC ( K, m ) = H < big >< big >(</ big ></ big >( K ⊕ opad ) ∥ H < big >(</ big >( K ⊕ ipad ) ∥ m < big >)< big >)</ big ></ big >

In the ITER tokamak, it is expected that the occurrence of a limited number of major disruptions will definitively damage the chamber with no possibility to restore the device .< ref > Wurden, G., ( 2011 ) International Workshop " MFE Roadmapping in the ITER Era ", Princeton < http :// advprojects. pppl. gov / Roadmapping / presentations / MFE_POSTERS / WURDEN_Disruption_RiskPOSTER. pdf ></ ref >< ref > Baylor, L. R .; Combs, S. K .; Foust, C. R .; Jernigan, T. C.

Above about 1500 K hydrogen begins to dissociate at low pressures, or 3000 K at high pressures, a potential area of promise for greatly increasing the I < small >< sub > sp </ sub ></ small > of solid core reactors.

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< tr >< th > Element </ th >< th > T < sub > c </ sub > ( K )</ th ></ tr >

For example, if K is a field of characteristic p and if X is transcendental over K, is a non-separable algebraic field extension.

If X and Y are Banach spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B ( X, Y ).

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.

The spaceX * of all linear maps into K ( which is called the algebraic dual space to distinguish it from X ′) also induces a weak topology which is finer than that induced by the continuous dual since X ′ ⊆ X *.

Because F ( x ) is a map from X ′ to K, it is an element of X ′′.

The 21 consonant letters in the English alphabet are B, C, D, F, G, H, J, K, L, M, N, P, Q, R, S, T, V, X, Z, and usually W and Y: The letter Y stands for the consonant in " yoke ", the vowel in " myth " and the vowel in " funny ", and " yummy " for both consonant and vowel, for examples ; W almost always represents a consonant except in rare words ( mostly loanwords from Welsh ) like " crwth " " cwm ".

A subset K of a topological space X is called compact if it is compact in the induced topology.

Many Unix, GNU, BSD and Linux programs and packages have been ported to Cygwin, including the X Window System, K Desktop Environment 3, GNOME, Apache, and TeX.

This subextension is called a separable closure of K. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of K < sup > sep </ sup >, of degree > 1.

K < sub > α </ sub >, then C < sub > α </ sub > is − 1.

* Bessel functions J < sub > ν </ sub >, Y < sub > ν </ sub >, I < sub > ν </ sub > and K < sub > ν </ sub > in Librow Function handbook.

This results in constitutive cAMP production, which in turn leads to secretion of H < sub > 2 </ sub > O, Na < sup >+</ sup >, K < sup >+</ sup >, Cl < sup >−</ sup >, and HCO < sub > 3 </ sub >< sup >−</ sup > into the lumen of the small intestine and rapid dehydration.

One reason for the popularity of the dissociation constant in biochemistry and pharmacology is that in the frequently encountered case where x = y = 1, K < sub > d </ sub > has a simple physical interpretation: when = K < sub > d </ sub >, = or equivalently /(+)= 1 / 2.

That is, K < sub > d </ sub >, which has the dimensions of concentration, equals the concentration of free A at which half of the total molecules of B are associated with A.

The conversion to kelvins ( symbol: uppercase K ) is defined by using k < sub > B </ sub >, the Boltzmann constant:

Likewise, a functor from G to the category of vector spaces, Vect < sub > K </ sub >, is a linear representation of G. In general, a functor G → C can be considered as an " action " of G on an object in the category C. If C is a group, then this action is a group homomorphism.

PuP is one actinide pnictide that is a paramagnet and has cubic symmetry at room temperature, but upon cooling undergoes a lattice distortion to tetragonal when cooled to below its T < sub > c </ sub > = 125 K. PuP has an easy axis of < 100 >, so that

If G = GL < sub >*</ sub >( K ), then the set of natural numbers is a proper subset of G < sub > 0 </ sub >, since for each natural number n, there is a corresponding identity matrix of dimension n. G ( m, n ) is empty unless m = n, in which case it is the set of all nxn invertible matrices.

In particular, the identity function id < sub > S </ sub >, defined by, is idempotent, as is the constant function K < sub > c </ sub >, where c is an element of S, defined by.

* Non-selective cation channels: These let many types of cations, mainly Na < sup >+</ sup >, K < sup >+</ sup > and Ca < sup > 2 +</ sup > through the channel.

Channels differ with respect to the ion they let pass ( for example, Na < sup >+</ sup >, K < sup >+</ sup >, Cl < sup >−</ sup >), the ways in which they may be regulated, the number of subunits of which they are composed and other aspects of structure.

K < sub > 2 </ sub > CO < sub > 3 </ sub >,

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

Some authors also require the domain of the Euclidean function be the entire ring R ; this can always be accommodated by adding 1 to the values at all nonzero elements, and defining the function to be 0 at the zero element of R, but the result is somewhat awkward in the case of K. The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set ; this weakening does not affect the most important implications of the Euclidean property.

* K, the ring of polynomials over a field K. For each nonzero polynomial P, define f ( P ) to be the degree of P.

The converse does not hold since for any field K, K is a UFD but is not a PID ( to prove this look at the ideal generated by It is not the whole ring since it contains no polynomials of degree 0, but it cannot be generated by any one single element ).

All members of the vitamin K group of vitamins share a methylated naphthoquinone ring structure ( menadione ), and vary in the aliphatic side chain attached at the 3-position ( see figure 1 ).

* The maximal ideals of the polynomial ring over an algebraically closed field K are the ideals of the form.

When specializing to the preadditive categories of abelian groups or modules over a ring, this notion of kernel coincides with the ordinary notion of a kernel of a homomorphism, if one identifies the ordinary kernel K of f: A → B with its embedding K → A.