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 element in the direct sum of two vector spaces of matrices can be represented as a direct sum of two matrices.

# 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 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 total order on X has exactly one equivalence class, X itself, because x ~ y for all x and y ;

Any commutative monoid is endowed with its algebraic preordering ≤, defined by x ≤ y if and only if there exists z such that x + z = y.

Any preordered set ( P, ≤) forms a small category, where the objects are the members of P, the morphisms are arrows pointing from x to y when x ≤ y.

A weight on a Lie algebra g over a field F is a linear map λ: g → F with λ ( y )= 0 for all x, y in g. Any weight on a Lie algebra g vanishes on the derived algebra and hence descends to a weight on the abelian Lie algebra g /.

Any transformation of the plane leaving this pattern invariant can be decomposed into a translation, ( x, y ) → ( n + x, y ), optionally followed by a reflection in either the horizontal axis, ( x, y ) → ( x ,− y ), or the vertical axis, ( x, y ) → (− x, y ), provided that this axis is chosen through or midway between two dots, or a rotation by 180 °, ( x, y ) → (− x ,− y ) ( ditto ).

Any quadratic polynomial over the complex numbers ( polynomials of the form where,, and ∈ ) can be factored into an expression with the form using the quadratic formula.

for each p ∈ U. Any differential 1-form arises this way, and by using (*) it follows that any differential 1-form α on U may be expressed in coordinates as

Any ξ ∈ gives rise to a canonical vertical vector field X < sub > ξ </ sub > by taking the derivative of the right action of the 1-parameter subgroup of H associated to ξ.

* Any Hilbert space H is a Hilbert manifold with a single global chart given by the identity function on H. Moreover, since H is a vector space, the tangent space T < sub > p </ sub > H to H at any point p ∈ H is canonically isomorphic to H itself, and so has a natural inner product, the " same " as the one on H. Thus, H can be given the structure of a Riemannian manifold with metric

Any attempt to define it ( X is good if it has property Y ) will simply shift the problem ( Why is Y-ness good in the first place?

* quantification Any morphism f: X → Y in a category with pullbacks induces a monotonous map acting by pullbacks ( A monotonous map is a functor if we consider the preorders as categories ).

Any statement that has a Y following the command letter is processed only if the match flag is set.

* Any two objects X and Y of C have a product X × Y in C.

* Any two objects Y and Z of C have an exponential Z < sup > Y </ sup > in C.

Any β perpendicular to α ( on the Y axis ) trivially fulfills both with 0, but does not define an irreducible root system.

Any vector field Y on N defines a pullback section φ < sup >*</ sup > Y of φ < sup >*</ sup > TN with ( φ < sup >*</ sup > Y )< sub > x </ sub >

* Any two maps f, g: Y → X are homotopic.

* Any map f: Y → X is null-homotopic.

Any attempt to appeal to the United Nation to stop the U. S. invasion had failed.

Any government that is willing to assist in recovery will find full co-operation on the part of the U. S. A.

Any U. S. citizen with at least one year of residence on the island is also eligible for the official citizenship certificate.

Any Chinese who left the United States had to obtain certifications for reentry, and the Act made Chinese immigrants permanent aliens by excluding them from U. S. citizenship.

In the United Kingdom, " Any weapon of whatever description designed or adapted for the discharge of any noxious liquid, gas or other thing " is a Prohibited Weapon, under S. 5 of The Firearms Act 1968.

They encouraged U. S. citizens to purchase war bonds with their rendition of Irving Berlin's song Any Bonds Today ?.

FanMail was another success for TLC, debuting at No. 1 on the Billboard 200 album sales chart and selling over 6 million copies in the U. S. The album featured the number-one hit " No Scrubs ", produced by Kevin " She ' kspere " Briggs, and the single " Unpretty ", an alternative rock-styled song about self-love written by Watkins and Dallas Austin ( another version of it sampled Dennis Edwards ' 1984 hit " Don't Look Any Further "), that also reached # 1 on the Billboard chart.

Any fears that what had now become a revolving door of band members would affect the quality of their new album were laid to rest as Helloween's new studio album, titled Keeper of the Seven Keys-The Legacy, was released on October 28, 2005 in Germany, and November 8 in the U. S. A. to commercial and critical acclaim.

To show that U ( n ) is connected, recall that any unitary matrix A can be diagonalized by another unitary matrix S. Any diagonal unitary matrix must have complex numbers of absolute value 1 on the main diagonal.

Special Inspector General for Iraq Reconstruction Stuart Bowen said that " Any doubts about how the money was handled after it left U. S. control is an Iraqi -- not U. S. government -- question ".

" The Amerasian Foundation ( AF ) and Amerasian Family Finder ( AFF ) define an Amerasian as: " Any person who was fathered by a citizen of the United States ( an American servicemen, American expatriate, or U. S. Government Employee ( Regular or Contract )) and whose mother is, or was, an Asian National Asian.

Any perfect hash functions suitable for use with a hash table require at least a number of bits that is proportional to the size of S.

Any S / 360 operating system could in fact be run under CP, but normal users ran Cambridge Monitor System ( CMS ), a simple, single-user operating system.

* Whizz for Atomms: A Guide to Survival in the 20th Century for Fellow Pupils, their Doting Maters, Pompous Paters and Any Others who are Interested, 1956 ( with Geoffrey Willans ) Published in the U. S. as Molesworth's Guide to the Atommic Age

In the southern United States at the start of the 20th century the schottische was combined with ragtime ; the most popular " ragtime schottische " of the era was " Any Rags " by Thomas S. Allen in 1902.

Any scheme S has a unique morphism to Spec ( Z ), so this attitude, part of the relative point of view, doesn't lose anything.

* Any U. S. financial institution that becomes aware that it has possession of or control over funds in which a designated FTO or its agent has an interest must retain possession of or control over the funds and report the funds to the Office of Foreign Assets Control of the U. S. Department of the Treasury.

) compete for Challenge ( best ) Dog and Challenge Bitch ; the individuals who will challenge each other for the accolade Best of Breed ( except dogs that are entered in " The import Register " or " Any Variety Not Separately Classified " classes, in these classes the dogs compete for " best import " or " best A. V. N. S. C .").

Her other television credits include appearances on Nightingales, Baywatch, The Closer, Matlock, Jake and the Fatman, The Untouchables, Any Day Now, Without a Trace, The Lyon's Den, The Division, the U. S. version of Coupling, and the science fiction television series Seven Days.

Any audit of the Union Pacific and its invoices to the U. S. Government would have revealed no evidence of fraud or profiteering.

Any affine group scheme is the spectrum of a commutative Hopf algebra ( over a base S, this is given by the relative spectrum of an O < sub > S </ sub >- algebra ).

*" Don't Look Any Further "-( 1988 )-UK # 52 ; U. S. # 64