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 replace r, the number of the stage from the end of the process, by t, a continuous variable which measures the `` distance '' of the point considered from the end of the process.

< li > gcd ( a < sup > r / 2 </ sup > ± 1, N ) is a nontrivial factor of N. We are done .</ li >

We are observing this sequence until a predefined number r of failures has occurred.

) We can be more specific in light of the definition above: Plugging in F = 1 dyne, q < sub > 1 </ sub >= q < sub > 2 </ sub >= 1 statC, and r

We can prove the cancellation law easily using Euclid's lemma, which generally states that if an integer b divides a product rs ( where r and s are integers ), and b is relatively prime to r, then b must divide s. Indeed, the equation

We will redo the record, open up the doors for it to get on the r & b charts and make the black stations to play the record ...

We wish to show ( for small enough values of n ) that it is possible to color the edges of the graph in two colors ( say red and blue ) so that there is no complete subgraph on r vertices which is monochromatic ( every edge colored the same color ).

We calculate the expected number of monochromatic subgraphs on r vertices as follows:

We prove that R ( r, s ) exists by finding an explicit bound for it.

We can obtain a formula for r by substituting estimates of the covariances and variances based on a sample into the formula above.

We define the representation Ψ: R → B ( L < sup > 2 </ sup >( R )) by Ψ ( r )

We can embed the rational numbers into the reals by identifying the rational number r with the equivalence class of the sequence ( r, r, r, …).

( We put the " r " in brackets to avoid confusion with an exponent.

We want to choose β and r < nowiki >'</ nowiki > so that the invariants described above hold.

He was referenced by Richie Rich in Tupac Shakur's song " Rather Be Ya Nigga ", in the line, " Smoke blunts, but leave them stunts up to Super Dave ," in the Ice Cube song " Wicked ", with the line " We'll have to break his ass up like Super Dave ," in the A Tribe Called Quest song " The Chase, Part II " with the lyric, " We flippin ' on niggas like we Super Dave ," in the Cool Calm Pete song " Lost " with the line " You ( r ) fake stunts that's strictly for Super Dave ".

We call the resulting graph the Turán graph T ( n, r ).

We can define an even finer equivalence relation of oriented C < sup > r </ sup > curves by requiring φ to be φ ‘( t ) > 0.

We also have the exterior derivative which maps Ω < sup > r </ sup >( M )< sup > C </ sup > to Ω < sup > r + 1 </ sup >( M )< sup > C </ sup >.