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Formally and let 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 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, let p ( x, y ) be a complex polynomial in the complex variables x and y. 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

Formally and S Formally, a binary operation on a set S is called associative if it satisfies the associative law: Formally, a set S is called finite if there exists a bijection Informally, G has the above presentation if it is the " freest group " generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R. Formally, cl ( S ) denotes the smallest subset Y of M that contains S such that for each reaction ( A, B ) Formally, it is a coalition of U. S. and European CEOs, largely from transnational corporations, launched at the urging of former United States Secretary of Commerce Ron Brown in Seville ( Spain ) in late 1995 at a conference attended by CEOs from more than 100 U. S. and EU companies and by top government representatives led by the Commissioners for Trade and Industry and by the U. S. Secretary of Commerce. * Citibank Banamex USA – Formally California Commerce Bank, Banamex's U. S. banking division. Formally designated as the Manhattan Engineer District, it refers specifically to the period of the project from 1941 – 1946 under the control of the U. S. Army Corps of Engineers, under the administration of General Leslie R. Groves. Formally, a Hopf algebra is a ( associative and coassociative ) bialgebra H over a field K together with a K-linear map S: H → H ( called the antipode ) such that the following diagram commutes: Formally, an unlabelled state transition system is a tuple ( S, →) where S is a set ( of states ) and → ⊆ S × S is a binary relation over S ( of transitions ). Formally, given a partially ordered set ( P, ≤), then an element g of a subset S of P is the greatest element of S if Formally, a partially ordered set ( P, ≤) is bounded complete if the following holds for any subset S of P: Formally, the theorem states that, for every Sperner family S over an n-set, Formally, given two partially ordered sets ( S, ≤) and ( T, ≤), a function f: ST 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 and T Formally, the ith row, jth column element of A < sup > T </ sup > is the jth row, ith column element of A: Formally, if T is a complete sufficient statistic for θ and E ( g ( T )) = τ ( θ ) then g ( T ) is the minimum-variance unbiased estimator ( MVUE ) of τ ( θ ). Formally, the bounded Borel functional calculus of a self adjoint operator T on Hilbert space H is a mapping defined on the space of bounded complex-valued Borel functions f on the real line, Formally, we define a bad field as a structure of the form ( K, T ), where K is an algebraically closed field and T is an infinite, proper, distinguished subgroup of K, such that ( K, T ) is of finite Morley rank in its full language.

Formally and be 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, 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 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, 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.

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