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 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 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 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 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 irreducible polynomial over k has distinct roots.

* 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.

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

# Every sequence in A has a convergent subsequence, whose limit lies in A.

* Sequentially compact: Every sequence has a convergent subsequence.

Every sequence that ran off to infinity in the real line will then converge to ∞ in this compactification.

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 page starts with the four ASCII character sequence OggS.

Lemma: Every sequence

Every synset contains a group of synonymous words or collocations ( a collocation is a sequence of words that go together to form a specific meaning, such as " car pool "); different senses of a word are in different synsets.

Every gap sequence that contains 1 yields a correct sort ; however, the properties of thus obtained versions of Shellsort may be very different.

Every variable X < sub > i </ sub > in the sequence is associated with a Bernoulli trial or experiment.

Every short exact sequence

Every sequence in principle has a generating function of each type ( except that Lambert and Dirichlet series require indices to start at 1 rather than 0 ), but the ease with which they can be handled may differ considerably.

Every time he is thrown out of the house, he is shown wearing the same shirt although he does not always wear it when he is thrown out ( the producers never shot a second sequence with Jazz being thrown out of the house, only adjusting the original scene for time purposes ; an exception is in the episode " Community Action ", where Jazz was thrown out along with a lifesize cardboard cut-out of Bill Cosby, complete with a blooper showing Jeff Townes reshooting his flying off the house several times ).

Every year, in numerical sequence starting from 2007-07-07 and continuing through 2012-12-12 hoopers dance in every city and country to raise money and donate hoops to others who can't afford them.

For example, to study the theorem “ Every bounded sequence of real numbers has a supremum ” it is necessary to use a base system which can speak of real numbers and sequences of real numbers.

Every Turán graph is a cograph ; that is, it can be formed from individual vertices by a sequence of disjoint union and complement operations.

* Metroid: Every game in the Metroid series has ended with either a timed self-destruct escape sequence, or a cut-scene showing the area explode, with the exception of Metroid 2.

:# Every element of D is the least upper bound of a countable increasing sequence of isolated elements.

* Conway's cosmological theorem: Every sequence eventually splits into a sequence of " atomic elements ", which are finite subsequences that never again interact with their neighbors.

Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.

** Well-ordering theorem: Every set can be well-ordered.

Every information exchange between living organisms — i. e. transmission of signals that involve a living sender and receiver can be considered a form of communication ; and even primitive creatures such as corals are competent to communicate.

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

* Every regular language is context-free because it can be described by a context-free grammar.

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 real number has a ( possibly infinite ) decimal representation ; i. e., it can be written as

Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable.

Every entire function can be represented as a power series that converges uniformly on compact sets.

Group actions / representations: Every group G can be considered as a category with a single object whose morphisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set, i. e. a G-set.

Every positive integer n > 1 can be represented in exactly one way as a product of prime powers:

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 holomorphic function can be separated into its real and imaginary parts, and each of these is a solution of Laplace's equation on R < sup > 2 </ sup >.

Every species can be given a unique ( and, one hopes, stable ) name, as compared with common names that are often neither unique nor consistent from place to place and language to language.

Every vector v in determines a linear map from R to taking 1 to v, which can be thought of as a Lie algebra homomorphism.

Every morpheme can be classified as either free or bound.

Every use of modus tollens can be converted to a use of modus ponens and one use of transposition to the premise which is a material implication.

Every document window is an object with which the user can work.

Every adult, healthy, sane Muslim who has the financial and physical capacity to travel to Mecca and can make arrangements for the care of his / her dependants during the trip, must perform the Hajj once in a lifetime.

Every ordered field can be embedded into the surreal numbers.

* Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological space in this way.

* Every preorder can be given a topology, the Alexandrov topology ; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set.

Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R < sup >+=</ sup >.