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Formally and we
Formally we mean that is an ideal if it satisfies the following conditions:
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 it is precisely in allowing quantification over class variables α, β, etc., that we assume a range of values for these variables to refer to.
Formally, we are given a set of hypotheses and a set of manifestations ; they are related by the domain knowledge, represented by a function that takes as an argument a set of hypotheses and gives as a result the corresponding set of manifestations.
Formally, we have for the approximation to the full solution A, a series in the small parameter ( here called ), like the following:
Formally, we start with a metric space M and a subset 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 means that we want a function to be monotonic.
Formally, for a countable set of events A < sub > 1 </ sub >, A < sub > 2 </ sub >, A < sub > 3 </ sub >, ..., we have
Formally, we define
Formally, we begin by considering some family of distributions for a random variable X, that is indexed by some θ.
Formally, let A be a real matrix of which we want to compute the eigenvalues, and let A < sub > 0 </ sub >:= A.
Formally we have:
Formally, we have
Formally, an antihomomorphism between X and Y is a homomorphism, where equals Y as a set, but has multiplication reversed: denoting the multiplication on Y as and the multiplication on as, we have.
Formally, the definition only requires some invertibility, so we can substitute for Q any matrix M whose eigenvalues do not include − 1.
Formally, given a finite set X, a collection C of subsets of X, all of size n, has Property B if we can partition X into two disjoint subsets Y and Z such that every set in C meets both Y and Z.
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, we want:.
Formally we can write the factor as,
Formally, if we denote the set of stable functions by S ( D ) and the stability radius by r ( f, D ), then:

Formally and define
Formally, the issue is that interfertile " able to interbreed " is not a transitive relation – if A can breed with B, and B can breed with C, it does not follow that A can breed with C – and thus does not define an equivalence relation.
Formally, define the set of lines in the plane P as L ( P ); then a rigid motion of the plane takes lines to lines – the group of rigid motions acts on the set of lines – and one may ask which lines are unchanged by an action.
Formally however they define it as any variable that does not directly affect the fundamentals of the economy.

Formally and inductively
Formally, the sets of free and bound names of a process in π – calculus are defined inductively as follows.
Formally, the set of LTL formulas over AP is inductively defined as follows:

Formally and using
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 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 ).

we and define
How do we define it ''??
But first, we must define two terms so that their meaning will be clearly understood: form -- any unique sequence of alphabetic characters that can appear in a language preceded and followed by a space ; ;
For a single stage we may define Af where the maximization is by choice of Af.
In summary, while we stress that constructive engagement between anthropology and the military is possible, CEAUSSIC suggests that the AAA emphasize the incompatibility of HTS with disciplinary ethics and practice for job seekers and that it further recognize the problem of allowing HTS to define the meaning of “ anthropology ” within DoD.
To define angles in an abstract real inner product space, we replace the Euclidean dot product ( · ) by the inner product, i. e.
To define the Kolmogorov complexity, we must first specify a description language for strings.
How best to define the term “ art ” is a subject of constant contention ; many books and journal articles have been published arguing over even the basics of what we mean by the term “ art ”.
Time efficiency estimates depend on what we define to be a step.
If we define r < sub > i </ sub > as the displacement of particle i from the center of mass, and v < sub > i </ sub > as the velocity of particle i with respect to the center of mass, then we have
If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).
Some philosophers deny that the concept of " being " has any meaning at all, since we only define an object's existence by its relation to other objects, and actions it undertakes.
If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.
The real line R with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it.
For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies
Starting from any ket in this Hilbert space, we can define a complex scalar function of r, known as a wavefunction:
Christopher Hitchens was offended by the notion of Clinton as the first black president noting " we can still define blackness by the following symptoms: alcoholic mothers, under-the-bridge habits ... the tendency to sexual predation and shameless perjury about the same ".
" In a general way, we can define computing to mean any goal-oriented activity requiring, benefiting from, or creating computers.
In fact, what we have done is define a category of categories and functors – the objects are categories, and the morphisms ( between categories ) are functors.
These categories surely have some objects that are " special " in a certain way, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered to be atomic, i. e., we do not know whether an object A is a set, a topology, or any other abstract concept – hence, the challenge is to define special objects without referring to the internal structure of those objects.
But how can we define the empty set without referring to elements, or the product topology without referring to open sets?
If we define tangent covectors in terms of equivalence classes of smooth maps vanishing at a point then the definition of the pullback is even more straightforward.
First we must also define where our view point is, that is, from what vantage point will the scene be drawn.
If S is an arbitrary set, then the set S < sup > N </ sup > of all sequences in S becomes a complete metric space if we define the distance between the sequences ( x < sub > n </ sub >) and ( y < sub > n </ sub >) to be, where N is the smallest index for which x < sub > N </ sub > is distinct from y < sub > N </ sub >, or 0 if there is no such index.

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