Consider the vectors e < sub > 1 </ sub > = ( 1, 0, 0 ), e < sub > 2 </ sub > = ( 0, 1, 0 ) and e < sub > 3 </ sub > = ( 0, 0, 1 ).

Consider the vectors ( functions ) f and g defined by f ( t ) := e < sup > it </ sup > and g ( t ) := e < sup >− it </ sup >.

The first of these three identities says that the 0 form a representation of the ordinary Lie algebra spanned by E ( Consider the 0 as vectors on which the E act.

Consider a discrete charge distribution consisting of N point charges q < sub > i </ sub > with position vectors r < sub > i </ sub >.

Consider the linear subspace of the n-dimensional Euclidean space R < sup > n </ sup > that is spanned by a collection of linearly independent vectors

Consider polynomials P ( x ), D ( x ) where degree ( D ) < degree ( P ).

< p > Consider an interesting and revealing incident that occurred during the Quasi War.

Consider the class of all regular paths from a point p to another point q.

Consider the electron ( mass m < sub > e </ sub >) and proton ( mass m < sub > p </ sub >) in the hydrogen atom.

Consider all the possible strings of p symbols, using an alphabet with a different symbols.

Consider the direct system composed of the groups Z / p < sup > n </ sup > Z and the homomorphisms Z / p < sup > n </ sup > Z → Z / p < sup > n + 1 </ sup > Z which are induced by multiplication by p. The direct limit of this system consists of all the roots of unity of order some power of p, and is called the Prüfer group Z ( p < sup >∞</ sup >).

Consider two alleles, A and a, with frequencies p and q, respectively, in the population.

Consider T to be a differentiable multilinear map of smooth sections α < sup > 1 </ sup >, α < sup > 2 </ sup >, ..., α < sup > q </ sup > of the cotangent bundle T * M and of sections X < sub > 1 </ sub >, X < sub > 2 </ sub >, ... X < sub > p </ sub > of the tangent bundle TM, written T ( α < sup > 1 </ sup >, α < sup > 2 </ sup >, ..., X < sub > 1 </ sub >, X < sub > 2 </ sub >, ...) into R. Define the Lie derivative of T along Y by the formula

A particular setting of n and p characterizes a particular probability distribution over a discrete random variable, with possible outcomes ( the support of the distribution ) ranging between 0 and n. Consider the following cases:

Consider for instance the projection p < sub > 1 </ sub >: R < sup > 2 </ sup > → R on the first component ; A =

Consider a probability density function p ( x ; a ) for a variable x, parameterized by a.

< p > Consider a regular six-sided die with four green faces and two red faces.

Consider an edge e between two input points p and q which is not an edge of a Delaunay triangulation.

Consider two bases for elements in a finite field, GF ( p < sup > m </ sup >):

Consider the line in the body defined by the two points p and q, which has the Plücker coordinates,

Let p be a point of M. Consider the space consisting of smooth maps defined in some neighborhood of p. We define an equivalence relation on as follows.

Consider any process involving an incoming particle with momentum p. For the particle to give a measurable contribution to the amplitude, it has to interact with a number of different particles with momenta via a vertex.

Consider the following reactions of acetic acid ( CH < sub > 3 </ sub > COOH ), the organic acid that gives vinegar its characteristic taste:

Consider the example ∇ < big >×</ big > < big >×</ big > F.

Consider the example of a coffee cup and a donut ( see < span class =" plainlinks "> this example </ span >).

Consider two systems ; S < sub > 1 </ sub > and S < sub > 2 </ sub > at the same temperature and capable of exchanging particles.

Consider a pseudo random number generator ( PRNG ) function P ( key ) that is uniform on the interval 2 < sup > b </ sup > − 1.

Geometric arrangement for Fresnel's calculation Consider the case of a point source located at a point P < sub > 0 </ sub >, vibrating at a frequency f. The disturbance may be described by a complex variable U < sub > 0 </ sub > known as the complex amplitude.

Consider the logarithm function: For any fixed base b, the logarithm function log < sub > b </ sub > maps from the positive real numbers R < sup >+</ sup > onto the real numbers R ; formally:

< li > Consider the group ( Z < sub > 6 </ sub >, +), the integers from 0 to 5 with addition modulo 6.

Consider a state of the system, described by the single particle states ..., n < sub > N </ sub >.

Consider the context of evaluating each one of a class of events A < sub > 1 </ sub >, A < sub > 2 </ sub >, A < sub > 3 </ sub >,..., A < sub > n </ sub > ( for example, is the occurrence of the event harmful or not ?).

Again we start with a C < sup >∞</ sup > manifold, M, and a point, x, in M. Consider the ideal, I, in C < sup >∞</ sup >( M ) consisting of all functions, ƒ, such that ƒ ( x ) = 0.

Consider, for example, the difference between strong and weak convergence of functions in the Hilbert space L < sup > 2 </ sup >( R < sup > n </ sup >).

Consider that 700 < 738, yet 700²

Consider an elastic collision in 2 dimensions of any 2 masses m < sub > 1 </ sub > and m < sub > 2 </ sub >, with respective initial velocities u < sub > 1 </ sub > and u < sub > 2 </ sub >

Consider this table: 1.

Consider what happened during World War 1,, when the Protestant churches united to push the Prohibition law through Congress.

Consider the graph of the equation y = 1 / x shown to the right.

( Consider 1 / 0, which is defined with the value of infinity, vs. 0 / 0, which is undefined.

Consider the closed intervals for all integers k ; there are countably many such intervals, each has measure 1, and their union is the entire real line.

* Chapter 7 divides humility into twelve degrees, or steps in the ladder that leads to heaven :( 1 ) Fear God ; ( 2 ) Substitute one's will to the will of God ; ( 3 ) Be obedient to one's superior ; ( 4 ) Be patient amid hardships ; ( 5 ) Confess one's sins ; ( 6 ) Accept oneself as a " worthless workman "; ( 7 ) Consider oneself " inferior to all "; ( 8 ) Follow examples set by superiors ; ( 9 ) Do not speak until spoken to ; ( 10 ) Do not laugh ; ( 11 ) Speak simply and modestly ; and ( 12 ) Be humble in bodily posture.

Consider a jar containing N lottery tickets numbered from 1 through N. If you pick a ticket randomly then you get positive integer n, with probability 1 / N if n ≤ N and with probability zero if n > N. This can be written

* Consider this example: Sarah says, " Out of every 10 apples I gather, 1 is rotten .".

Consider the function f, piecewise defined by f ( x ) = – 1 for x < 0 and f ( x ) = 1 for x ≥ 0.

Consider what happens if T < sub > 1 </ sub > fails half-way through.