We first define a function b{t} as follows: given the set of squares such that each has three corners on C and vertex at t, b{t} is the corresponding set of positive parametric differences between T and the backward corner points.

We define these values as Af, and define g{t} in the same way for each T.

Articles of faith are sets of beliefs usually found in creeds, sometimes numbered, and often beginning with " We believe ...", which attempt to more or less define the fundamental theology of a given religion, and especially in the Christian Church.

We then define f ( x, y ) to be this z.

We can then define the differential map d: C < sup >∞</ sup >( M ) → T < sub > x </ sub >< sup >*</ sup > M at a point x as the map which sends f to df < sub > x </ sub >.

We can now define the value

We should not define ' wisdom ' as the absence of folly, or a healthy thing as whatever is not sick.

We cannot define a point except as ' something with no parts ', nor blindness except as ' the absence of sight in a creature that is normally sighted '.

We define the periodic Bernoulli functions P < sub > n </ sub > by

We then define a contravariant functor F from C to D as a mapping that

We need a " mark " to define where we are and which direction we are heading to see if we ever get back to exactly the same pixel.

Verner wrote, " We can conclude that although the ancient Egyptians could not precisely define the value of π, in practice they used it ".

We define the kernel of h to be the set of elements in G which are mapped to the identity in H

We define the degree of to be the number of universal quantifier blocks, separated by existential quantifier blocks as shown above, in the prefix of.

We define the inverse limit of the inverse system (( A < sub > i </ sub >)< sub > i ∈ I </ sub >, ( f < sub > ij </ sub >)< sub > i ≤ j ∈ I </ sub >) as a particular subgroup of the direct product of the A < sub > i </ sub >' s:

We wish to maximize total value subject to the constraint that total weight is less than or equal to W. Then for each w ≤ W, define m to be the maximum value that can be attained with total weight less than or equal to w. m then is the solution to the problem.

We could also define a Lie algebra structure on T < sub > e </ sub > using right invariant vector fields instead of left invariant vector fields.

* We then define truth-functional operators, beginning with negation.

We can define addition, subtraction, and multiplication on by the following rules:

We can define methods to manipulate these new complex types.

We assume that A is an m-by-n matrix over either the real numbers or the complex numbers, and we define the linear map f by f ( x ) = Ax as above.

We can define a multiplication on the set S using the Steiner triple system by setting aa

We may define a possibly different topology on X using the continuous ( or topological ) dual space X < sup >*</ sup >.

We can also define it as a second component of business which includes all activities, functions and institutions involved in transferring goods from producers to consumer.

We note that two such curves C and Af, cannot coincide at more than a finite number of points ; ;

* The nominative case indicates the subject of a finite verb: We went to the store.

We say V is finite-dimensional if the dimension of V is finite.

We can only concede that there are things that lie beyond the finite ken of the human mind.

We may also define it differently when working with finite sets.

We can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite ; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former ; and finally the attributes " equal ," greater ," and " less ," are not applicable to infinite, but only to finite, quantities.

We are not interested in properties of the positive integers that have no descriptive meaning for finite man.

We cannot eliminate the condition that one point sets cannot be open as a finite space given the discrete topology shows.

We call a field E a splitting field for A if A ⊗ E is isomorphic to a matrix ring over E. Every finite dimensional CSA has a splitting field: indeed, in the case when A is a division algebra, then a maximal subfield of A is a splitting field.

We know, therefore, that this language cannot be accepted correctly by any finite state machine, and is thus not a regular language.

We set if there is no path of finite length from to.

We can appreciate the sense of fulfillment we find in serving a larger whole and form our characters progressively upon the ways in which those experiences of fulfillment point us ever outwards, beyond the finite self, but we are not so constitute as to experience the greater Whole to which our experiences belong.

We will see that for compact operators, the proof of the main theorem uses essentially the same idea from the finite dimensional argument.

We therefore have a finite extension

We will assume for the moment that all state spaces of the systems considered, classical or quantum, are finite dimensional.

We assume X is finite so C ( X ) can be identified with the n-dimensional Euclidean space with entry-wise multiplication.

We are not related to finite Universe enclave in a sky and earth.

We are not infinite but are not that sort of finite as the universe is.

We say that A is of finite type if all of its principal minors are positive, that A is of affine type if its proper principal minors are positive and A has determinant 0, and that A is of indefinite type otherwise.

We say that a learner can identify in the limit a class of languages if given any presentation of any language in the class the learner will produce a finite number of wrong representations, and therefore converge on the correct representation in a finite number of steps, without however necessarily being able to announce its correctness since a counterexample to that representation could appear as an element arbitrarily long after.

We prove the finite version, using Radon's theorem as in the proof by.

We assume the charges to be clustered around the origin, so that for all i: r < sub > i </ sub > < r < sub > max </ sub >, where r < sub > max </ sub > has some finite value.