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.

The ( unique ) representable functor F: → is the Cayley representation of G. In fact, this functor is isomorphic to and so sends to the set which is by definition the " set " G and the morphism g of ( i. e. the element g of G ) to the permutation F < sub > g </ sub > of the set G. We deduce from the Yoneda embedding that the group G is isomorphic to the group

" We are not witnessing an end of writing which, to follow McLuhan's ideological representation, would restore a transparency or immediacy of social relations ; but indeed a more and more powerful historical unfolding of a general writing of which the system of speech, consciousness, meaning, presence, truth, etc., would only be an effect, to be analyzed as such.

We begin with the representation of a binary image, where the image may be thought of as a subset of.

We see from these two equations that in order to have non-zero, the electric and magnetic dipole moment operators ( and ) must transform as the same irreducible representation.

We consider the group as a G-set, which can be shown to have permutation representation, say.

We should distinguish at the outset between the L-series, an infinite series representation ( for example the Dirichlet series for the Riemann zeta-function ), and the L-function, the function in the complex plane that is its analytic continuation.

We can, however, construct a representation of the covering group of the Poincare group, called the inhomogeneous SL ( 2, C ); this has elements ( a, A ) where as before, a is a four-vector, but now A is a complex 2 × 2 matrix with unit determinant.

We denote the unitary operators we get by U ( a, A ), and these give us a continuous, unitary and true representation in that the collection of U ( a, A ) obey the group law of the inhomogeneous SL ( 2, C ).

Thus we define the representations of G on an Hilbert space H to be those group homomorphisms, ρ, which arise from continuous actions of G on H. We say that a representation ρ is unitary if ρ ( g ) is a unitary operator for all g ∈ G ; i. e., for all v, w ∈ H. ( I. e.

We have two cases, one where irreps are described by an integral multiple of 1 / 2, called the helicity and the other called the " continuous spin " representation.

We assume there is a pure state called the vacuum such that the Hilbert space associated with it is a unitary representation of the Poincaré group compatible with the Poincaré covariance of the net such that if we look at the Poincaré algebra, the spectrum with respect to energy-momentum ( corresponding to spacetime translations ) lies on and in the positive light cone.

We may ask whether a given affine representation has a fixed point in the given affine space A.

We can write g by means of its cycle representation, which gives a ' cycle type ' c ( g ), again a partition of n.

" We the Children " is a statistical representation of the minimal progress made in the decade.

Bruce Schneier and Niels Ferguson write, " We have one criticism of AES: we don't quite trust the security … What concerns us the most about AES is its simple algebraic structure … No other block cipher we know of has such a simple algebraic representation.

We say that ρ is a real representation of G if the matrices are real.

We call the resulting representation the quotient representation of ρ by σ.

We know that any irreducible representation can be turned into a unitary representation.

We can mechanize the above transformation by defining a function, called the standard representation of V with respect to B, that takes every vector to its coordinate representation:.

We have, then, a natural instinct for representation and for tune and rhythm — and starting with these instincts men very gradually developed them until they produced poetry out of their improvisations.

Humans had long distinguished ourselves from the rest of the animal kingdom as “ Man the Toolmaker .” In response to Goodall ’ s revolutionary findings, Louis Leaky wrote, “ We must now redefine man, redefine tool, or accept chimpanzees as human !” Over the course of her study, Goodall found evidence of mental traits in chimpanzees such as reasoned thought, abstraction, generalization, symbolic representation, and even the concept of self, all previously thought to be uniquely human abilities.