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 polynomial in K which has a root in L factors into linear factors in L.

* Every quadratic Bézier curve is also a cubic Bézier curve, and more generally, every degree n Bézier curve is also a degree m curve for any m > n. In detail, a degree n curve with control points P < sub > 0 </ sub >, …, P < sub > n </ sub > is equivalent ( including the parametrization ) to the degree n + 1 curve with control points P '< sub > 0 </ sub >, …, P '< sub > n + 1 </ sub >, where.

Every space filling curve hits some points multiple times, and does not have a continuous inverse.

Every point on the Lorenz curve represents a statement like " the bottom 20 % of all households have 10 % of the total income.

Every immersed curve in the plane admits two possible orientations.

Every study has found that instead of people's MBTI scores clustering around two opposite poles, such as intuition vs. sensation, with few people scoring in the middle, people's scores actually cluster around the middle of each scale in a bell curve.

Every Edwards curve with a point of order 3 can be written in the ways shown above.

Every algebraic curve C of genus g ≥ 1 is associated with an abelian variety J of dimension g, by means of an analytic map of C into J.

Every closed curve c on X is homologous to for some simple closed curves c < sub > i </ sub >, that is,

Every minimal projective ruled surface other than the projective plane is the projective bundle of a 2-dimensional vector bundle over some curve.

Every algebraic plane curve has a degree, which can be defined, in case of an algebraically closed field, as number of intersections of the curve with a generic line.

Every aspect of life was regulated to some degree by the party, and the will of its founding-president, Mobutu Sese Seko.

Every hyperbola is congruent to the origin-centered East-West opening hyperbola sharing its same eccentricity ε ( its shape, or degree of " spread "), and is also congruent to the origin-centered North-South opening hyperbola with identical eccentricity ε — that is, it can be rotated so that it opens in the desired direction and can be translated ( rigidly moved in the plane ) so that it is centered at the origin.

Every game in the main Zelda series has consisted of three principal areas: an overworld in which movement is multidirectional, allowing the player some degree of freedom of action ; areas of interaction with other characters ( merely caves or hidden rooms in the first game, but expanding to entire towns and cities in subsequent games ) in which the player gains special items or advice ; and dungeons, areas of labyrinthine layout, usually underground, comprising a wide range of difficult enemies, bosses, and items.

Every student of the engineering trainee program receives the degree of “ ingénieur de l ' École centrale de Nantes ”, more commonly called “ ingénieur centralien ”.

# Every document can contain a royalty mechanism at any desired degree of granularity to ensure payment on any portion accessed, including virtual copies (" transclusions ") of all or part of the document.

Every year more than 5, 000 research publications go to print and an average of 260 PhD students are awarded their PhD degree.

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

Every vertex of this graph has an even degree, therefore this is an Eulerian graph.

Every cylinder bore has a nikasil coating giving it a high degree of wear resistance.

Every commissioned officer in the United States armed forces is expected to have a post-graduate degree and Joint Professional Military Education prior to promotion to lieutenant colonel or commander.

Every degree of freedom in the energy is quadratic and, thus, should contribute k < sub > B </ sub > T to the total average energy, and k < sub > B </ sub > to the heat capacity.

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

* Every character value is a sum of n m < sup > th </ sup > roots of unity, where n is the degree ( that is, the dimension of the associated vector space ) of the representation with character χ and m is the order of g. In particular, when F is the field of complex numbers, every such character value is an algebraic integer.

Every riparian owner enjoys this right to the same extent and degree, and each such owner maintains a qualified right to use the water for domestic purposes, such as drinking and bathing.

Every big band drummer after Lewis has been influenced by him to some degree.

Every student of the Scuola Normale must enroll in a degree programme at the University of Pisa, and their degree programme must correspond to the educational areas of the Scuola Normale in the Faculty of Arts and the Faculty of Sciences.

Every stage in the origin of clays, sands and gravels can be seen in process around us, but where these have been converted into coherent shales, sandstone and conglomerates, and still more where they have experienced some degree of metamorphism, there are many obscure points about their history upon which experiment may yet throw light.

“ Every man's own satisfaction ” Tucker holds to be the ultimate end of action ; and satisfaction or pleasure is one and the same in kind, however much it may vary in degree.

Every legislator from Brasstown Bald to Folkston is going to have his every vote subjected to the closest scrutiny as a test of his political allegiances, not his convictions.

Every detail in his interpretation has been beautifully thought out, and of these I would especially cite the delicious laendler touch the pianist brings to the fifth variation ( an obvious indication that he is playing with Viennese musicians ), and the gossamer shading throughout.

Every taxpayer is well aware of the vast size of our annual defense budget and most of our readers also realize that a large portion of these expenditures go for military electronics.

Every single problem touched on thus far is related to good marketing planning.

Every few days, in the early morning, as the work progressed, twenty men would appear to push it ahead and to shift the plank foundation that distributed its weight widely on the Rotunda pavement, supported as it is by ancient brick vaulting.

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 man in every one of these houses is a Night Rider.

Every library borrower, or at least those whose taste goes beyond the five-cent fiction rentals, knows what it is to hear the librarian say apologetically, `` I'm sorry, but we don't have that book.

Every community, if it is alive has a spirit, and that spirit is the center of its unity and identity.

The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.

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

Every natural-born citizen of a foreign state who is also an American citizen and every natural-born American citizen who is a citizen of a foreign land owes a double allegiance, one to the United States, and one to his homeland ( in the event of an immigrant becoming a citizen of the US ), or to his adopted land ( in the event of an emigrant natural born citizen of the US becoming a citizen of another nation ).

Every line of written text is a mere reflection of references from any of a multitude of traditions, or, as Barthes puts it, " the text is a tissue of quotations drawn from the innumerable centres of culture "; it is never original.

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

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

Every year, on the last Sunday in April, there is an ice fishing competition in the frozen estuarine waters of the Anadyr River's mouth.

Every lattice element of the structure is in its proper place, whether it is a single atom or a molecular grouping.