Formally, let S and T be finite sets and let F =

If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is " not too large ", in the sense that if R is Noetherian, the same must be true for R. Formally,

Formally, let P be an expression in which the variable x is free.

Formally, let

Formally, let be a surjective homomorphism.

Formally, let be a stochastic process and let represent the cumulative distribution function of the joint distribution of at times.

Formally, let f: < sup > n </ sup > → be the cost function which must be minimized.

Formally, let P and Q be abelian categories, and let

Formally, let A be a real matrix of which we want to compute the eigenvalues, and let A < sub > 0 </ sub >:= A.

Formally, let G be a Coxeter group with reduced root system R and k < sub > v </ sub > a multiplicity function on R ( so k < sub > u </ sub > = k < sub > v </ sub > whenever the reflections σ < sub > u </ sub > and σ < sub > v </ sub > corresponding to the roots u and v are conjugate in G ).

Formally, for received words, let denote the Hamming distance between and, that is, the number of positions in which and differ.

Formally, let p and q be two nonzero polynomials, respectively of degree m and n. Thus:

Formally, let P be a poset ( partially ordered set ), and let F be a filter on P ; that is, F is a subset of P such that:

Formally, let denote the pairwise score for against.

Formally, let X be any scheme and S be a sheaf of graded-algebras ( the definition of which is similar to the definition of-modules on a locally ringed space ): that is, a sheaf with a direct sum decomposition

More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,

Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero and non-unit x of R can be written as a product ( including an empty product ) of irreducible elements p < sub > i </ sub > of R and a unit u:

Formally, the outcomes Y < sub > i </ sub > are described as being Bernoulli-distributed data, where each outcome is determined by an unobserved probability p < sub > i </ sub > that is specific to the outcome at hand, but related to the explanatory variables.

Formally, the problem can be stated as follows: given a desired property, expressed as a temporal logic formula p, and a structure M with initial state s, decide if.

Formally, an EGARCH ( p, q ):

Formally, he says, the piece consists of two parts of nearly equal length, the end of the first section being bars 24 – 28 ( p. 77 ).

Formally, a cumulativity predicate CUM can be defined as follows, where capital X is a variable over sets, U is the universe of discourse, p is a mereological part structure on U, and is the mereological sum operation.

Formally, a detailed balanced process is the one on which the probability to be in the state x and transit to x ' must be the same as to be in the state x ' and transit to x.

Formally, a function ƒ is real analytic on an open set D in the real line if for any x < sub > 0 </ sub > in D one can write

( Formally speaking, this then satisfies the premises of an axiom of well-founded induction, which asserts that these two conditions are sufficient for the proposition to hold for all x.

Formally, if we write F < sub > Δ </ sub >( x ) to mean the f-polynomial of Δ, then the h-polynomial of Δ is

Formally, this is achieved by mapping the input x to a two dimensional vector u ( x )

Formally, given two partially ordered sets ( S, ≤) and ( T, ≤), a function f: S → T is an order-embedding if f is both order-preserving and order-reflecting, i. e. for all x and y in S, one has

Formally, f < sub > X, Y </ sub >( x, y ) is the probability density function of ( X, Y ) with respect to the product measure on the respective supports of X and Y.

Formally, an algebraic function in n variables over the field K is an element of the algebraic closure of the field of rational functions K ( x < sub > 1 </ sub >,..., x < sub > n </ sub >).

Formally, this is described in algebraic notation like this: ( 19 + 1 ) + ( 15 − 1 ) = x, but even a young student might use this technique without calling it algebra.

Formally, in Euclidean space, the wave front set of ƒ is defined as the complement of the set of all pairs ( x < sub > 0 </ sub >, v ) such that there exists a test function with φ ( x < sub > 0 </ sub >) ≠ 0 and an open cone Γ containing v such that the estimate

Formally, the equation above defines the macroscopic neutron cross-section ( for reaction x ) as the proportionality constant between a neutron flux incident on a ( thin ) piece of material and the number of reactions that occur ( per unit volume ) in that material.

* Spanish Exploration: Hezeta ( Heceta ) and Bodega y Quadra Expedition of 1775 to Formally Claim the Pacific Northwest for Spain

However, shortly after this positive result, Kurt Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems ( 1931 ), showing that in any sufficiently strong axiomatic system there are true statements which cannot be proved in the system.

Formally the algorithm can be described as:

Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies

Formally, a frame is defined to be a lattice L in which finite meets distribute over arbitrary joins, i. e. every ( even infinite ) subset

Formally a random variable is considered to be a function on the possible outcomes.

Formally the " head of state " can also personally be the " head of government " ( ex officio or by ad hoc cumulation such as an absolute monarch nominating himself ) but otherwise has formal precedence over the Head of Government and other ministers, whether he is their actual political superior ( absolute monarch, executive president ) or rather theoretical or ceremonial in character.

Formally, in the finite-dimensional case, if the linear map is represented as a multiplication by a matrix A and the translation as the addition of a vector, an affine map acting on a vector can be represented as

Formally, the theorem can be stated as follows:

Formally, the convex hull may be defined as the intersection of all convex sets containing X or as the set of all convex combinations of points in X.

Formally, a decision problem is P-complete ( complete for the complexity class P ) if it is in P and that every problem in P can be reduced to it by using an appropriate reduction.

Formally, powers with positive integer exponents may be defined by the initial condition

Formally, the problem of supervised pattern recognition can be stated as follows: Given an unknown function ( the ground truth ) that maps input instances to output labels, along with training data assumed to represent accurate examples of the mapping, produce a function that approximates as closely as possible the correct mapping.

Formally the self-inductance of a wire loop would be given by the above equation with i

Formally, a government would lay out its reasons for going to war, as well as its intentions in prosecuting it and the steps that might be taken to avert it.

Formally, if is any non-zero polynomial, it must be writable as.

Formally, a Menger sponge can be defined as follows:

Formally, the case where only a subset of parameters is defined is still a composite hypothesis ; nonetheless, the term point hypothesis is often applied in such cases, particularly where the hypothesis test can be structured in such a way that the distribution of the test statistic ( the distribution under the null hypothesis ) does not depend on the parameters whose values have not been specified under the point null hypothesis.