Formally, bending modulus is defined as the energy required to deform a membrane from its intrinsic curvature to some other curvature.

Formally organized vocational programs supported by federal funds allow high school students to gain experience in a field of work which is likely to lead to a full-time job on graduation.

Formally, a binary operation on a set S is called associative if it satisfies the associative law:

Formally, their designation is the letter Ž and the number.

Formally, a topological space X is called compact if each of its open covers has a finite subcover.

Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata ( PDA ).

Formally, the derivative of the function f at a is the limit

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, the base is known as Naval Support Facility Diego Garcia ( the US activity ) or Permanent Joint Operating Base ( PJOB ) Diego Garcia ( the UK's term ).

Formally, there is a clear distinction: " DFT " refers to a mathematical transformation or function, regardless of how it is computed, whereas " FFT " refers to a specific family of algorithms for computing DFTs.

Formally, oxidation state is the hypothetical charge that an atom would have if all bonds to atoms of different elements were 100 % ionic.

Formally, a bifunctor is a functor whose domain is a product category.

Formally, a set S is called finite if there exists a bijection

Formally, the system is said to have memory.

Formally, an inner product space is a vector space V over the field together with an inner product, i. e., with a map

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, when working over the reals, as here, this is accomplished by considering the limit as ε → 0 ; but the " infinitesimal " language generalizes directly to Lie groups over general rings.

Formally, a profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space.

Formally, this sharing of dynamics is referred to as universality, and systems with precisely the same critical exponents are said to belong to the same universality class.

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, Φ = kx − ωt is the phase.

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 speaking, a collation method typically defines a total order on a set of possible identifiers, called sort keys, which consequently produces a total preorder on the set of items of information ( items with the same identifier are not placed in any defined order ).

Formally, we start with a category C with finite products ( i. e. C has a terminal object 1 and any two objects of C have a product ).

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, however, a face is any of the lower dimensional boundaries of the polytope, more specifically called an n-face.

Formally, dependence refers to any situation in which random variables do not satisfy a mathematical condition of probabilistic independence.

Formally, the word is applied to persons who are publicly accepted in a recognised capacity, such as professional employment, graduation from a course of study, etc., to give critical commentaries in one or any of a number of specific fields of public or private achievement or endeavour.

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

Formally, an absolute coequalizer of a pair in a category C is a coequalizer as defined above but with the added property that given any functor F ( Q ) together with F ( q ) is the coequalizer of F ( f ) and F ( g ) in the category D. Split coequalizers are examples of absolute coequalizers.

Formally titled ' An Act to protect all Persons in the United States in their Civil Rights, and furnish the Means of their vindication, the Act declared that people born in the United States and not subject to any foreign power are entitled to be citizens, without regard to race, color, or previous condition of slavery or involuntary servitude.

Formally, the complexity class MA is the set of decision problems that can be decided in polynomial time by an Arthur – Merlin protocol where Merlin's only move precedes any computation by Arthur.

Formally, a deterministic algorithm computes a mathematical function ; a function has a unique value for any given input, and the algorithm is a process that produces this particular value as output.

Formally, a partially ordered set ( P, ≤) is bounded complete if the following holds for any subset S of P:

Formally, a system is said to be observable if, for any possible sequence of state and control vectors, the current state can be determined in finite time using only the outputs ( this definition is slanted towards the state space representation ).

Formally, the definition only requires some invertibility, so we can substitute for Q any matrix M whose eigenvalues do not include − 1.

Formally, if f is a harmonic function, then f cannot exhibit a true local maximum within the domain of definition of f. In other words, either f is a constant function, or, for any point inside the domain of f, there exist other points arbitrarily close to at which f takes larger values.

Formally, P is a symmetric polynomial, if for any permutation σ of the subscripts 1, 2, ..., n one has P ( X < sub > σ ( 1 )</ sub >, X < sub > σ ( 2 )</ sub >, …, X < sub > σ ( n )</ sub >) = P ( X < sub > 1 </ sub >, X < sub > 2 </ sub >, …, X < sub > n </ sub >).

Formally, a redistribution of all agents ’ initial endowments of goods among all agents in the system is a core allocation if no group of agents can redistribute their own initial endowments among themselves so as to improve the satisfaction of someone in the group without impairing that of any other in the group.

Formally, if and only if, for any given, there exists an such that

Formally, a natural number n is called superabundant precisely when, for any m < n,

Formally however they define it as any variable that does not directly affect the fundamentals of the economy.

Formally, a market is complete with respect to a trading strategy,, if there exists a self-financing trading strategy, such that at any time, the returns of the two strategies, and are equal.

Formally, this meant that the whole peloton finished out of time limits, but the referees understandably used a rule saying that they could give clemency to any group of more than 20 % of the stage's starting riders, officially citing the weather conditions as their reason to do so.

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