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Suppose and instead

Suppose now that instead of one particle in this box we have N particles in the box and that these particles are fermions with spin 1 / 2.

The covariant derivative can also be constructed from the Cartan connection η on P. In fact, constructing it in this way is slightly more general in that V need not be a fully fledged representation of G. Suppose instead that that V is a (, H )- module: a representation of the group H with a compatible representation of the Lie algebra.

Suppose there is instead the following two values:

Suppose instead of quantum mechanical systems, the two systems A and B are classical.

Suppose that, instead of letting the gas undergo a free expansion in which the volume is doubled, a free expansion in allowed in which the volume expands by a very small amount δV.

Suppose that, instead of being fixed, the positions of the two ends of the ideal chain are now controlled by an operator.

Suppose and exact

Suppose that one is summing n values x < sub > i </ sub >, for i = 1 ,..., n. The exact sum is:

Suppose that the state space is two dimensional and any of the five quantities are exact differentials.

Suppose we are given a covariant left exact functor F: A → B between two abelian categories A and B.

Suppose P is an exact category ; associated to P a new category QP is defined, objects of which are those of P and morphisms from M ′ to M ″ are isomorphism classes of diagrams

Suppose an adversary knows the exact content of all or part of one of our messages.

Suppose and observation

Suppose some theory T implies an observation O ( observation meaning here the result of the observation, rather than the process of observation per se ):

Suppose each observation is y < sub > xi </ sub > where x indicates the category that observation is in and i is the label of the particular observation.

Suppose and x

Suppose that R ( x, y ) is a relation in the xy plane.

Suppose random variable X can take value x < sub > 1 </ sub > with probability p < sub > 1 </ sub >, value x < sub > 2 </ sub > with probability p < sub > 2 </ sub >, and so on, up to value x < sub > k </ sub > with probability p < sub > k </ sub >.

Suppose an array A with elements indexed 1 to n is to be searched for a value x.

Suppose M is a C k manifold ( k ≥ 1 ) and x is a point in M. Pick a chart φ: U → R n where U is an open subset of M containing x.

Suppose many points are close to the x axis and distributed along it.

Suppose we wish to find the maximum likelihood estimate of β for a single observed value x.

Suppose that the distribution consists of a number of discrete probability masses p < sub > k </ sub >( θ ) and a density f ( x | θ ), where the sum of all the ps added to the integral of f is always one.

Suppose that x and y are real numbers and that y is a function of x, that is, for every value of x, there is a corresponding value of y.

Suppose P is a definable binary relation ( which may be a proper class ) such that for every set x there is a unique set y such that P ( x, y ) holds.

Suppose that L is a lattice of determinant d ( L ) in the n-dimensional real vector space R n and S is a convex subset of R n that is symmetric with respect to the origin, meaning that if x is in S then − x is also in S.

Suppose that C is a twice continuously differentiable immersed plane curve, which here means that there exists parametric representation of C by a pair of functions such that the first and second derivatives of x and y both exist and are continuous, and

Suppose u ( x ) and v ( x ) are two continuously differentiable functions.

Suppose that for every real number x.

Suppose we are given boundary conditions, i. e., a specification of the value of φ at the boundary if M is compact, or some limit on φ as x approaches ∞.

Suppose and is

Suppose Af is defined in the sub-interval Af.

Suppose they both had ventured into realms which their colleagues thought infidel: is this the way gentlemen settle frank differences of opinion??

Suppose, says Dr. Lyttleton, the proton has a slightly greater charge than the electron ( so slight it is presently immeasurable ).

Suppose it is something right on the planet, native to it.

Suppose there is a program

Suppose there is a chain at 1A, 2A, 3A, and 4A, along with another chain at 6A and 7A.

If two players tie for minority, they will share the minority shareholder bonus. Suppose Festival is the chain being acquired.

Alex is the majority shareholder, and Betty is the minority shareholder. Suppose now that Worldwide is the chain being acquired.

) Then X < sub > i </ sub > is the value ( or realization ) produced by a given run of the process at time i. Suppose that the process is further known to have defined values for mean μ < sub > i </ sub > and variance σ < sub > i </ sub > 2 for all times i. Then the definition of the autocorrelation between times s and t is

Suppose that a car is driving up a tall mountain.

Suppose that the car is ascending at 2. 5 km / h.

Suppose the vector field describes the velocity field of a fluid flow ( such as a large tank of liquid or gas ) and a small ball is located within the fluid or gas ( the centre of the ball being fixed at a certain point ).

Suppose that F is a partial function that takes one argument, a finite binary string, and possibly returns a single binary string as output.

Suppose, says Searle, that this computer performs its task so convincingly that it comfortably passes the Turing test: it convinces a human Chinese speaker that the program is itself a live Chinese speaker.

; Dennett's reply from natural selection: Suppose that, by some mutation, a human being is born that does not have Searle's " causal properties " but nevertheless acts exactly like a human being.

Suppose that is a complex-valued function which is differentiable as a function.

Suppose a mass is attached to a spring which exerts an attractive force on the mass proportional to the extension / compression of the spring.

Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.

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