Any Banach algebra ( whether it has an identity element or not ) can be embedded isometrically into a unital Banach algebra so as to form a closed ideal of.

Any change to the information processing environment introduces an element of risk.

# Any pure element ( even if it forms diatomic molecules like chlorine, Cl < sub > 2 </ sub >) has an oxidation state ( OS ) of zero.

Any translating element was omitted, since it does not affect the FWHM.

Any document element could be given new " methods " ( capabilities ), and could respond to changes in the content or structure of the document itself.

Any monoid ( any algebraic structure with a single associative binary operation and an identity element ) forms a small category with a single object x.

* Any principal ideal domain, such as the integers, is Noetherian since every ideal is generated by a single element.

Any formal series with has a composition inverse provided is an invertible element of R. The coefficients are found recursively from the above formula for the coefficients of a composition, equating them with those of the composition identity X ( that is 1 at degree 1 and 0 at every degree greater than 1 ).

* Frozen: Any waterfall which has some element of ice.

# Any group G may be regarded as an " abstract " category with one object,, and one morphism for each element of the group.

Any element from the holiday special that is not referenced in other works is considered S-Canon, which means that it is canon, and that it " happened ," but its canonicity is not set in stone.

Any set of sequences can be represented as paths in a tree, by forming a tree node for every prefix of a sequence and making the parent of one of these nodes represent the sequence with one fewer element ; the tree formed in this way for a set of strings is called a trie.

:: Theorem ( Lefschetz theorem on ( 1, 1 )- classes ) Any element of H < sup > 2 </ sup >( X, Z ) ∩ H < sup > 1, 1 </ sup >( X ) is the cohomology class of a divisor on X.

Any antichain can intersect any chain in at most one element, so, if we can partition the elements of an order into k chains then the width of the order must be at most k. Dilworth's theorem states that this bound can always be reached: there always exists an antichain, and a partition of the elements into chains, such that the number of chains equals the number of elements in the antichain, which must therefore also equal the width.

Any point x gives rise to a continuous function p < sub > x </ sub > from the one element topological space 1 ( all subsets of which are open ) to the space X by defining p < sub > x </ sub >( 1 ) = x. Conversely, any function from 1 to X clearly determines one point: the element that it " points " to.

Any element can be expressed as a word in the T-alphabet

Any commutative semigroup is a medial magma, and a medial magma has an identity element if and only if it is a commutative monoid.

* Any nonempty set S of submodules of the module has a maximal element ( with respect to set inclusion.

Any difference is amplified and used to control the regulation element in such a way as to reduce the voltage error.

Any element y ∈ Y := Ran ( S − I ) has a distance larger than or equal to 1 from u ( otherwise y < sub > i </ sub > = x < sub > i + 1 </ sub >-x < sub > i </ sub > would be positive and bounded away from zero, whence x < sub > i </ sub > could not be bounded ).

Any C *- algebra generated by a unitary element is the homomorphic image of this universal C *- algebra.

* Any directed set A may be made into a Cauchy space by declaring a filter F to be Cauchy if, given any element n of A, there is an element U of F such that U is either a singleton or a subset of the tail

Any direct relation however with the Byzantine formula is very unlikely.

Any increase in power at the supranational level, he argued, results from a direct decision by governments.

Any mutual recursion can be converted to direct recursion by inlining the code of one procedure into the other.

Any real mode operating system ( such as Xenix ) could also be made to run using OS / 2's virtual machine capabilities, subject to certain direct hardware access limitations.

Raimi may direct By Any Means Necessary, the next film based on the " Jack Ryan " CIA character created by Tom Clancy.

Any subcontractor will have a direct contractual relationship with the main contractor.

Any commando or small group of commandos or a similar unit, agents, and saboteurs not in uniform, who fell into the hands of the German military forces by some means other than direct combat ( through the police in occupied territories, for instance ) were to be handed over immediately to the Sicherheitsdienst ( SD, Security Service ).

Any state able to direct its military forces outside the limited bounds of its territory might be said to have some level of power projection capability, but the term itself is used most frequently in reference to militaries with a worldwide reach ( or at least significantly broader than a state's immediate area ).

Any increase in power at supranational level, he argues, results from a direct decision by governments.

Any medium that can be used to deliver a communication to a customer can be employed in direct marketing, including:

Any state able to direct its military forces outside the limited bounds of its territory might be said to have some level of power projection capability, but the term itself is used most frequently in reference to militaries with a worldwide reach ( or at least significantly broader than a state's immediate area ).

Any decomposition of the generalized Cartan matrix leads to the direct sum decomposition of the corresponding Kac – Moody algebra:

* Any group which is a direct product of two groups both of which have generic polynomials.

Any messages received by such an MSA are therefore traceable to an author who has a direct relationship with the MSA, and who can be held accountable for his actions.

Any dispute that is not settled by direct negotiation or by mediation shall be finally settled under the Rules of Arbitration of the International Chamber of Commerce by one or more arbitrators appointed in accordance with the said Rules .”

Any collapse of the U. S. Housing Bubble has a direct impact not only on home valuations, but the nation's mortgage markets, home builders, real estate, home supply retail outlets, Wall Street hedge funds held by large institutional investors, and foreign banks, increasing the risk of a nationwide recession.

Any student who wants to direct a play is automatically accepted.

Any mention of airways might include a very small number of ' direct ' segments to allow for situations when there are no convenient airway junctions.

Any direct debit instruction that has not been used to collect funds for over 13 months is automatically cancelled by the customer's bank ( this is known as a " dormancy period ").