Every rational number has a unique representation as an irreducible fraction.

Every polynomial in can be factorized into polynomials that are irreducible over F. This factorization is unique up to permutation of the factors and the multiplication of the factors by nonzero constants from F ( because the ring of polynomials over a field is a unique factorization domain whose units are the nonzero constant polynomials ).

Every Heyting algebra with exactly one coatom is subdirectly irreducible, whence every Heyting algebra can be made an SI by adjoining a new top.

* Every irreducible closed subset of P < sup > n </ sup >( k ) of codimension one is a hypersurface ; i. e., the zero set of some homogeneous polynomial.

* Every irreducible polynomial over k has distinct roots.

Every irreducible non-degenerate curve of degree is a rational normal curve.

Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic.

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

Every polynomial P in x corresponds to a function, ƒ ( x )

Every output of an encoder can be described by its own transfer function, which is closely related to the generator polynomial.

* Every root of a monic polynomial whose coefficients are algebraic integers is itself an algebraic integer.

Every delta operator ' has a unique sequence of " basic polynomials ", a polynomial sequence defined by three conditions:

Every real polynomial of odd degree has at least one real number as a root.

* Every Jacobi-like polynomial sequence can have its domain shifted and / or scaled so that its interval of orthogonality is, and has Q

* Every Laguerre-like polynomial sequence can have its domain shifted, scaled, and / or reflected so that its interval of orthogonality is, and has Q =

* Every Hermite-like polynomial sequence can have its domain shifted and / or scaled so that its interval of orthogonality is, and has Q

Every polynomial function is a rational function with.

* Every polynomial over k is separable.

Every field and every polynomial ring over a field ( in arbitrarily many variables ) is a reduced ring.

In mathematics, an integer-valued polynomial ( also known as a numerical polynomial ) P ( t ) is a polynomial whose value P ( n ) is an integer for every integer n. Every polynomial with integer coefficients is integer-valued, but the converse is not true.

Water for Every Farm: A practical irrigation plan for every Australian property, K. G.

Every adult citizen of this small settlement signed the small petition ; E. K Dyer and his wife, William Johnson, Joseph Otis and his wife, Hiram Walker and his wife, Joseph Pease and R. H. Valentine.

In November 2006, Korean singer K included a cover of the Vandross song " Every Year, Every Christmas " on his Christmas single, ファースト ・ クリスマス (" First Christmas "), released in Japan.

* Every real algebraic number field K of degree n contains a PV number of degree n. This number is a field generator.

Every year, the alumni of the previous year's " Special K!

This is up to isomorphism the only indecomposable module over R. Every left R-module is a direct sum of ( finitely or infinitely many ) copies of this module K < sup > n </ sup >.

* Let K < sup > a </ sup > be an algebraic closure of K containing L. Every embedding σ of L in K < sup > a </ sup > which restricts to the identity on K, satisfies σ ( L ) = L. In other words, σ is an automorphism of L over K.

Consider a Lie algebra g over a field K. Every element x of g defines the adjoint endomorphism ad ( x ) ( also written as ad < sub > x </ sub >) of g with the help of the Lie bracket, as

* Every finite p-local spectrum X has the property that K ( n )< sub >∗</ sub >( X ) = 0 if and only if n is less than a certain number N, called the type of the spectrum X.

( 3 ) Every unitary representation of G that has an ( ε, K )- invariant unit vector for any ε > 0 and any compact subset K, has a non-zero invariant vector.

Soon after, the band recorded three songs at a friend's rehearsal space ; the energetic tracks found early airplay on local radio and by KROQ's Rodney Bingenheimer in L. A. As their music further filtered across to the U. K. and Radio One, Every Move a Picture could be found opening hometown shows for the likes of the Kaiser Chiefs and Louis XIV.

Before any official release appeared overseas, Every Move a Picture played their first U. K. gigs to overwhelming response.

* 25 Ta Life, 100 Demons, A Life Once Lost, A Thousand Falling Skies, The Acacia Strain, Adelphi, Agnostic Front, The A. K. A. s, Alexisonfire, All Else Failed, All That Remains, Andrew W. K., Anterrabae, As I Lay Dying, The Ataris, The Autumn Offering, Avail, Bad Luck 13 Riot Extravaganza, Bane, The Banner, Bear Vs. Shark, Beloved, Between The Buried And Me, BigWig, Blacklisted, Bleeding Through, Blood for Blood, Breath of Silence, Brokeneck, The Bronx, Burnt by the Sun, Caliban, The Calico System, Champion, Comeback Kid, Converge, Day of Contempt, The Dead Season, Dead to Fall, Death by Stereo, XdeathstarX, Death Threat, The Dillinger Escape Plan, Draw Blood, DRI, Drive Without, Drowingman, Dry Kill Logic, E-Town Concrete, Embrace Today, Endwell, Engineer, Ensign, Evergreen Terrace, Every Time I Die, Fear Before the March of Flames, Fear Factory, Flat Earth Society, Folly, For the Love of, Fordirelifesake, Forever is Forgotten, From a Second Story Window, Found Dead Hanging, Full Blown Chaos, Glasseater, Halifax, Hawthorne Heights, The Hope Conspiracy, HORSE the Band, If Hope Dies, I Killed The Prom Queen, In Flames, It Dies Today, The Judas Cradle, The Juliana Theory, The Killing, Killswitch Engage, Last Perfection, Let It Die, Life of Agony, Love is Red, Malice Afterthought, Martyr AD, Mastodon, Merauder, Mest, The Minor Times, Misery Signals, Misfits, Misery Index, Most Precious Blood, Nag Hamadi, No Hollywood Ending, No Redeeming Social Value, No Warning, NORA, Norma Jean, Nothing Left to Mourn, The Number Twelve Looks Like You, Odd Project, Only Crime, Park, Pig Destroyer, Planes Mistaken for Stars, Prayer for Cleansing, Premonitions of War, The Promise, Psyopus, The Red Chord, Reflux, Remembering Never, Roses are Red, Scars of Tomorrow, Shadows Fall, Shai Hulud, Shattered Realm, Sick of it All, Spitalfield, Stabbed by Words, Stars Turn Cold, Stretch Arm Strong, Strike Anywhere, Suffocate Faster, Suffocation, Sworn Enemy, Sycamore Dreams, Terror, Time in Malta, To The Grave, Today Is The Day, Through the Discipline, Throwdown, Underoath, Undying, Unearth, Until The End, Walk the Line, Walls of Jericho, Wings of Scarlet, With Dead Hands Rising, With Honor, World Inferno Friendship Society, Your Enemies Friends, Zao, Zombie Apocalypse

In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes the matrix algebra by extending scalars (= tensoring with a field extension ), i. e. for a suitable field extension K of F, is isomorphic to the 2 × 2 matrix algebra over K.

Every new scandal which would provide more `` copy '' for Marshall's pen would thus mean more publicity for Welch.

Every dream, and this is true of a mental image of any type even though it may be readily interpreted into its equivalent of wakeful thought, is a psychic phenomenon for which no explanation is available.

Every family of Riviera Presbyterian Church has been asked to read the Bible and pray together daily during National Christian Family Week and to undertake one project in which all members of the family participate.

** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.

* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).

** Every infinite game in which is a Borel subset of Baire space is determined.

Every telephone company, whether large or small, determines its own ANAC for each individual central office, which tends to perpetuate the current situation of a mess of overlapping and / or spotty areas of coverage.

: Every non-empty set A contains an element B which is disjoint from A.

Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.

Every year since 1972 the BVI has hosted the Spring Regatta, which is a seven-day collection of sailing races throughout the islands.

* Every real Banach algebra which is a division algebra is isomorphic to the reals, the complexes, or the quaternions.

* Every unital real Banach algebra with no zero divisors, and in which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions.

Every character is automatically continuous from A to C, since the kernel of a character is a maximal ideal, which is closed.

Every computer contains an internal clock that regulates the rate at which instructions are executed and synchronizes all the various computer components.

* Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by " reversing all the arrows ".

Every context-sensitive grammar which does not generate the empty string can be transformed into an equivalent one in Kuroda normal form.

According to Every, one example may be " the myth of St. George " and other stories about saints battling dragons, which were " modelled no doubt in many cases on older representations of the creator and preserver of the world in combat with chaos ".

Every argues that " the disparagement of myth in our own civilization " stems partly from objections to perceived idolatry, objections which intensified in the Reformation, both among Protestants and among Catholics reacting against the classical mythology revived during the Renaissance.

Every grammar in Chomsky normal form is context-free, and conversely, every context-free grammar can be transformed into an equivalent one which is in Chomsky normal form.

Every four years, during which an extra 24 hours have accumulated, one extra day is added to keep the count coordinated with the sun's apparent position.

Every node has a location, which is a number between 0 and 1.

Every sequence can, thus, be read in three reading frames, each of which will produce a different amino acid sequence ( in the given example, Gly-Lys-Pro, Gly-Asn, or Glu-Thr, respectively ).

Every time the boy had an injury which caused him internal or external bleeding, the Tsarina called on Rasputin, and the Tsarevich subsequently got better.

Every week before June 19, the strip focuses on Garfield's birthday, which he dreads because of his fear of getting older.

::: Every Bill which shall have passed the House of Representatives and the Senate, shall, before it become a Law, be presented to the President of the United States ; If he approves he shall sign it, but if not he shall return it, with his Objections to that House in which it shall have originated ...