Consider for instance the open unit disc, a non-compact Riemann surface without boundary, with curvature 0 and with Euler characteristic 1: the Gauss – Bonnet formula does not work.

Consider the variance formula: e = PQ, wherein P is equal to the proportion of " 1's " or " cases " and Q is equal to ( 1-P ), the proportion of " 0's " or " noncases " in the distribution.

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

Consider the formula

Consider a probabilistic algorithm that, given a formula F ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, ..., x < sub > n </ sub >) chooses an assignment x < sub > 1 </ sub >, x < sub > 2 </ sub >, ..., x < sub > n </ sub > uniformly at random.

The Voltage Regulation formula could be visualized with the following ; " Consider power being delivered to a load such that the voltage at the load is the load's rated voltage V < sub > Rated </ sub >, if then the load disappears, the voltage at the point of the load will rise to V < sub > nl </ sub >.

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

Consider for example determining which of the following are to be considered diseases ( i. e., abnormal states requiring cure ): alcoholism, homosexuality, and chronic fatigue syndrome.

Consider the EPR thought experiment, and suppose quantum states could be cloned.

Consider a quantum ensemble of size N with occupancy numbers n < sub > 1 </ sub >, n < sub > 2 </ sub >,..., n < sub > k </ sub > corresponding to the orthonormal states, respectively, where n < sub > 1 </ sub >+...+ n < sub > k </ sub >

A fundamental part of ` Abdul-Bahá's teachings on evolution is the belief that all life came from the same origin: " the origin of all material life is one ..." He states that from this sole origin, the complete diversity of life was generated: " Consider the world of created beings, how varied and diverse they are in species, yet with one sole origin " He explains that a slow, gradual process led to the development of complex entities:

Consider, for example, the behavior of water ( the molecule H < sub > 2 </ sub > O ) which, depending on temperature and pressure, naturally exists in just one of three states: gas, liquid or ice.

Consider a single atom system with energy states.

Consider a SISO system with states ( see state space for details about MIMO systems ), if the row rank of the following observability matrix

Consider a simple exchange economy with two identical agents, one ( divisible ) good, and two potential states of the world ( which occur with some probability ).

Consider the tensor product of modules H < sub > i </ sub >( X, Z ) ⊗ A. The theorem states that there is an injective group homomorphism ι from this group to H < sub > i </ sub >( X, A ), which has cokernel Tor ( H < sub > i-1 </ sub >( X, Z ), A ).

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 this 2-qubit family of Werner states:

Consider a speech signal reduced to packets, and forced to share a link with bursty data traffic ( traffic with some large data packets ).

A few have been filmed ( in some cases rather freely adapted ): Consider Her Ways, Random Quest, Dumb Martian, Jizzle ( filmed as Maria ) and Time to Rest ( filmed as No Place Like Earth ).

Consider some set P and a binary relation ≤ on P. Then ≤ is a preorder, or quasiorder, if it is reflexive and transitive, i. e., for all a, b and c in P, we have that:

Consider the analogy between the sentence " Snow is white " and the character named Snow White, both of which can be true in some sense.

That means that some member of P ( S ), i. e., some subset of S, is not in the image of f. Consider the set:

Consider the problem of determining the index of the database entry which satisfies some search criterion.

If S is compact but not closed, then it has an accumulation point a not in S. Consider a collection consisting of an open neighborhood N ( x ) for each x ∈ S, chosen small enough to not intersect some neighborhood V < sub > x </ sub > of a.

Consider for example a sample Java fragment to represent some common farm " animals " to a level of abstraction suitable to model simple aspects of their hunger and feeding.

In some respects Look to Windward serves as a loose sequel to the first Culture novel, Consider Phlebas: the GSV Lasting Damage fought in the Idiran-Culture War, and Ziller specially composes a work to commemorate the arrival of light from a supernova triggered during the war.

Consider for example, the sharing of food in some hunter-gatherer societies, where food-sharing is a safeguard against the failure of any individual's daily foraging.

Consider the possibility of creating an ' artificial ' structure to group some otherwise unrelated, but nevertheless frequently referenced, items together.

Consider a call option and a put option with the same strike K for expiry at the same date T on some stock S, which pays no dividend.

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 the category C of algebraic structures ; these can be thought of as sets plus operations, obeying some laws.

Writing in 1960, he begins: " Consider a very long sequence of symbols ... We shall consider such a sequence of symbols to be ' simple ' and have a high a priori probability, if there exists a very brief description of this sequence-using, of course, some sort of stipulated description method.

These are set against a replay of some of the "' Consider Yourself " sequence.

Consider a topological space, that is, a space with some notion of closeness between points in the space.

Consider the general problem of inferring a distribution for a parameter θ given some datum or data x.

Consider some set of fixed cash flows.

Consider that the thermally current-limited components of a large semiconductor inverter are solid-state switches massing a few grams with a thermal time constant to their heat sinks of likely more than 100 ms, whereas the thermally current limited components of an MG are copper windings massing some hundreds of kilos which are intrinsically attached to their own large thermal mass.

Consider the case where the decision-maker is enabled to know the outcome of some additional uncertainties earlier in his / her decision situation, i. e., some u < sub > i </ sub > are moved to appear earlier in the ordering.

Consider a planar pendulum with natural frequency ω in the small angle approximation.

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

Consider this example that Adler offers: the traditional belief that people were placed deliberately on earth as God's ultimate creation is being replaced with the idea that people have evolved by natural selection.

Consider the case this way-to a cat, catching and eating mice is perfectly natural and morally sound.

Consider two such geodesics and with natural parameter,, separated by an angle.

Consider, for example, a computer chassis containing three cards with fundamental natural frequencies of aHz, bHz, and cHz, and a screen with a predicted frequency of dHz.

Consider for example the natural representation of the symmetric group S < sub > n </ sub > in n dimensions by permutation matrices, which is certainly faithful.