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 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 statements contained therein that attribute fault or causation to an individual will generally NOT be admissible under this exception.

Any case is a member of very many classes, in which the frequency of the attribute of interest ( such as crashing ) differs, and reference class problem discusses which is the most appropriate to use.

Any place in that uses an instance of the object and either sets or gets the attribute value ( x

Any claimant whose claim is denied, or is approved for less than the full amount of such claim, shall be entitled, under such regulations as the Commission may prescribe, to a hearing before the Commission, or its duly authorized representatives, with respect to such claim.

Any rational number with a denominator whose only prime factors are 2 and / or 5 may be precisely expressed as a decimal fraction and has a finite decimal expansion.

Any configuration of charges or currents has a ' dipole moment ', which describes the dipole whose field is the best approximation, at large distances, to that of the given configuration.

Any algebra whose elements are idempotent is also power-associative.

Any symmetry group whose elements have a common fixed point, which is true for all finite symmetry groups and also for the symmetry groups of bounded figures, can be represented as a subgroup of orthogonal group O ( n ) by choosing the origin to be a fixed point.

Any collection of objects and morphisms defines a ( possibly large ) directed graph G. If we let J be the free category generated by G, there is a universal diagram F: J → C whose image contains G. The limit ( or colimit ) of this diagram is the same as the limit ( or colimit ) of the original collection of objects and morphisms.

Any class can be viewed as a category whose only morphisms are the identity morphisms.

: Any magistrate, or in the absence of any magistrate any commissioned officer in Her Majesty's Naval, Military or Air Force Service or any police officer above the rank of inspector, in whose view a riot is being committed, or who apprehends that a riot is about to be committed by persons being assembled within his view, may make or cause to be made a proclamation in the Queen's name, in such form as he thinks fit, commanding the rioters or persons so assembled to disperse peaceably.

Any sort of radio transmitter whose presence is suspected is detectable with suitable equipment.

Any source whose appearance or behavior indicates that he is willing to talk should be noted by the screeners.

Any conjectures are largely based on the date of her father's death ( 1006 ), the approximate date of her betrothal to Cnut ( 1004 × 1016, see below ) and the time by which she had borne him sons, whose ages are themselves difficult to establish.

Any timelike curve admits a comoving observer whose " time axis " corresponds to that curve, and, since no observer is privileged, we can always find a local coordinate system in which lightcones are inclined at 45 degrees to the time axis.

Any oscillator whose oscillations are driven by a system that almost always is dissipating energy may be called a relaxation oscillator.

Any misuse of the arm by the employee of the company in whose custody the arm is placed is the responsibility of the company / agency.

" 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 physical object whose radius R becomes less than or equal to the Schwarzschild radius will undergo gravitational collapse and become a black hole.

** " Common variation ": Any player whose score hits exactly " 45 " is set back to Zero — The dreaded Forty-Five Rule.

Any man whose name appeared on the list was ipso facto stripped of his citizenship and excluded from all protection under law ; reward money was given to any informer who gave information leading to the death of a proscribed man, and any person who killed a proscribed man was entitled to keep part of his estate ( the remainder went to the state ).

-- Any attempt to access a member whose existence depends

Any node whose lower bound is greater than can be discarded.

Any human with courage and presence of mind can outwit a troll, and those whose faith is strong can even challenge them to mortal combat.

Any Iraqi whose citizenship was withdrawn for political, religious, racial, or sectarian reasons has the right to reclaim his Iraqi citizenship, and each Iraqi is guaranteed the right to carry more than one citizenship.

An introductory title screen – repeated as the Foreword to the 1946 novelization by Eric Warman – contains an explicit statement: " This is the story of two worlds, the one we know and another which exists only in the mind of a young airman whose life and imagination have been violently shaped by war ", adding " Any resemblance to any other world known or unknown is purely coincidental ".

Any skew semistandard tableau of shape / with positive integer entries gives rise to a sequence of partitions ( or Young diagrams ), by starting with, and taking for the partition places further in the sequence the one whose diagram is obtained from that of by adding all the boxes that contain a value ≤ in ; this partition eventually becomes equal to.

Any pair of successive shapes in such a sequence is a skew shape whose diagram contains at most one box in each column ; such shapes are called horizontal strips.