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 are trying to study a linear operator T on the finite-dimensional space V, by decomposing T into a direct sum of operators which are in some sense elementary.

We know that is opaque and thus follows that is opaque, so in the above equation, each operator can be written as a convex combination:

We now give an operator theoretic proof for the Cauchy – Schwarz inequality which passes to the C *- algebra setting.

We also note that a two-body Dirac equation composed of a Dirac operator for each of the two point particles interacting via the Coulomb interaction can be exactly separated in the ( relativistic ) center of momentum frame and the resulting ground state eigenvalue has been obtained very accurately using the Finite element methods of J. Shertzer.

We can also define a number operator N which has the following property:

We are provided with ( quantum black box ) access to a subroutine in the form of a unitary operator, U < sub > ω </ sub >, which acts as:

He further wrote: " We didn't learn for years — until the reactor vessel was physically opened — that by the time the plant operator called the NRC at about 8 am, roughly ½ of the uranium fuel had already melted.

We now utilize quantum mechanical ladder operators, which are defined for a general angular momentum operator as

We consider generalizations of this operator below.

We see that there is a bijection between symmetric extensions of an operator and isometric extensions of its Cayley transform.

An operator grimly replied, " We put two rounds in their forehead ; the dead don't need handcuffs.

We also define the total number operator for the field which is a sum of number operators of each mode:

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 differentiated according to the product rule and noted that is the time derivative of the operator we started with.

Take X to be a complex manifold with a complex vector bundle V. We let the vector bundles E and F be the sums of the bundles of differential forms with coefficients in V of type ( 0, i ) with i even or odd, and we let the differential operator D be the sum

There were worries that if Arriva Trains Wales were granted this service, it would push Wrexham – London operator Wrexham & Shropshire out of business, with Wrexham & Shropshire Managing Director, Andy Hamilton, saying " We believe that this proposal – if approved – would push the date of profitability of WSMR by at least a year.

We drop the t < sub > 0 </ sub > index in the time evolution operator with the convention that and write it as U ( t ).

And when, the next day, Woodward — this is probably Sunday or maybe Monday, because the burglary was Saturday morning early — called the number and asked to speak to Mr. Hunt, and the operator said, ‘ Well, he's not here now ; he's over at ,’ such-and-such a place, gave him another number, and Woodward called him up, and Hunt answered the phone, and Woodward said, ‘ We want to know why your name was in the address book of the Watergate burglars .’ And there is this long, deathly hush, and Hunt said, ‘ Oh my God !’ and hung up.

We call hyperreflexive if there is a constant K such that for every operator T in B ( H ),

We say that the operator mutation is cis-dominant, it is dominant to wild type but affects only the copy of the operon which is immediately adjacent to it.

We can measure the spin of this electron's state by applying an operator to the state, and we will always obtain ( or spin 1 / 2 ) since the spin-up and spin-down states are both eigenstates of this operator with the same eigenvalue.

We can also define the functional calculus for not necessarily bounded Borel functions h ; the result is an operator which in general fails to be bounded.