Consider the positive real numbers R < sup >+</ sup >, a Lie group under the usual multiplication.

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

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 first representations of the Lie algebra

They are maps from flat 3-space into the Lie group G. Consider now glueing these two balls together at their boundary S².

* Consider the C *- algebra of complex square matrices.

Consider a finite group G, a field k and the group algebra kG.

Consider the tropical semiring ( also known as the min-plus algebra due to the definition of the semiring ).

Consider the operator algebra of a system of N identical particles.

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

Consider a periodic group G with the additional property that there exists a single integer n such that for all g in G, g < sup > n </ sup >

Consider the functions f and g from the natural numbers to the natural numbers defined as follows:

Consider an experiment to study the effect of three different levels of a factor on a response ( e. g. three levels of a fertilizer on plant growth ).

Consider a large group of items in which a very few are different in a particular way ( for e. g. Defective products or infected test subjects ).

Consider e. g. " A differs from B ".

Consider gas in a one-dimensional container ( e. g., a long thin tube ).

Consider two waveforms f and g. By calculating the convolution, we determine how much a reversed function g must be shifted along the x-axis to become identical to function f. The convolution function essentially reverses and slides function g along the axis, and calculates the integral of their ( f and the reversed and shifted g ) product for each possible amount of sliding.

Consider the cubic polynomial equation 4t < sup > 3 </ sup > − g < sub > 2 </ sub > t − g < sub > 3 </ sub > =

Consider the simplified case of a vehicle with constant mass accelerating vertically upwards with a constant thrust per unit mass a in a gravitational field of strength g. The actual acceleration of the craft is a-g and it is using delta-v at a rate of a per unit time.

Consider a date field, e. g. " September 19, 2004 ".

Consider a group G and subgroups H and K, with K contained in H. Then the left cosets of H in G are each the union of left cosets of K. Not only that, but translation ( on one side ) by any element g of G respects this decomposition.

Consider a virus which has a genetic identity modeled by a string of ones and zeros ( e. g. 11010001011101 ....).

Consider the communications process over a discrete channel.

Consider the ratio of the difference of two positions of a particle divided by the time interval, which is called the average velocity over that time interval.

Consider the following commutative diagram in any abelian category ( such as the category of abelian groups or the category of vector spaces over a given field ) or in the category of groups.

Pope Pelagius II ( 578-590 ): " Consider the fact that whoever has not been in the peace and unity of the Church cannot have the Lord … Although given over to flames and fires, they burn, or, thrown to wild beasts, they lay down their lives, there will not be ( for them ) that crown of faith but the punishment of faithlessness … Such a one can be slain, he cannot be crowned … slain outside the Church, he cannot attain the rewards of the Church " ( Denzinger, 469 ).

* Consider a uniform layer of fluid over an infinite horizontal plane.

Consider the subset C of B, consisting of those B elements, which satisfy monic polynomial equations over A:

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 a bounded linear transformation T defined everywhere over a general Banach space.

Consider a ( hyper ) elliptic curve over a finite field where is the power of a prime number.

Consider a hyperelliptic curve of genus over the field where is the power of a prime number and define as but now over the field.

* Consider the set of all finite and infinite words over the alphabet

Consider as an example the videotape format war ; Two mechanisms independent of product quality could explain how VHS achieved dominance over Betamax from a negligible early adoption lead:

Consider the following problem in deterministic optimal control over the time period:

Consider a finite group permuting those indeterminates over K. By standard Galois theory, the set of fixed points of this group action is a subfield of, typically denoted.

Consider a symmetric game with a smooth payoff function defined over actions of two or more players.

Consider an example economy with the following wages over three years:

It was published as part of a 1961 collection with some short stories called Consider Her Ways and Others ( where it forms over a third of the book ).

Consider a typical situation: A retailer faces uncertain demand for its product from customers over time.

Consider an economic system composed of many self-interested individuals each of whom is endowed with a bundle of goods, has preferences over the available bundles and wishes to achieve a maximal satisfaction by exchanging his / her own goods with others.

Consider the functor that sends to ,-coalgebras are then finite or infinite streams over the alphabet, where is the set of states and is the state-transition function.

Consider the following expressions from first order logic over a signature containing a constant symbol 0 for the number 0, a unary function symbol s for the successor function and a binary function symbol + for addition.

Consider the argument " We ought to be altruistic because evolution has selected altruism over millions of years as a reliable guide to what is good ".