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

Suppose two curves γ < sub > 1 </ sub >: (- 1, 1 ) → M and γ < sub > 2 </ sub >: (- 1, 1 ) → M with γ < sub > 1 </ sub >( 0 )

Suppose M is a C < sup >∞</ sup > manifold.

Suppose block M is a dominator with several incoming edges, some of them being back edges ( so M is a loop header ).

Suppose M is an m × n matrix whose entries come from the field K, which is either the field of real numbers or the field of complex numbers.

Suppose we have an n-dimensional oriented Riemannian manifold, M and a target manifold T. Let be the configuration space of smooth functions from M to T. ( More generally, we can have smooth sections of a fiber bundle over M .)

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 Ω is given in the standard form and let M be a 2n × 2n block matrix given by

Suppose that x < sup > i </ sup > are local coordinates on the base manifold M. In terms of these base coordinates, there are fibre coordinates p < sub > i </ sub >: a one-form at a particular point of T * M has the form p < sub > i </ sub > dx < sup > i </ sup > ( Einstein summation convention implied ).

Suppose we zero-pad to a length M ≥ 2N – 1.

Suppose we are given an element e < sub > 0 </ sub > ∈ E < sub > P </ sub > at P = γ ( 0 ) ∈ M, rather than a section.

Suppose f is bounded: i. e. there exists a constant M such that | f ( z )| ≤ M for all z.

Suppose that f is entire and | f ( z )| is less than or equal to M | z |, for M a positive real number.

Suppose M is a compact smooth manifold, and a V is a smooth vector bundle over M. The space of smooth sections of V is then a module over C < sup >∞</ sup >( M ) ( the commutative algebra of smooth real-valued functions on M ).

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.

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

) 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 >< sup > 2 </ sup > 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.

Suppose the formula for some given function is known, but too complex to evaluate efficiently.

It is frequently stated in the following equivalent form: Suppose that is continuous and that u is a real number satisfying or Then for some c ∈ b, f ( c ) = u.

Suppose Alice has a qubit in some arbitrary quantum state.

# Suppose that P is some piece of knowledge.

:: “ Suppose that a sheriff were faced with the choice either of framing a Negro for a rape that had aroused hostility to the Negroes ( a particular Negro generally being believed to be guilty but whom the sheriff knows not to be guilty )— and thus preventing serious anti-Negro riots which would probably lead to some loss of life and increased hatred of each other by whites and Negroes — or of hunting for the guilty person and thereby allowing the anti-Negro riots to occur, while doing the best he can to combat them.

* Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all have Lipschitz constant bounded by some K. If ƒ < sub > n </ sub > converges to a mapping ƒ uniformly, then ƒ is also Lipschitz, with Lipschitz constant bounded by the same K. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the Banach space of continuous functions.

Suppose some given data points each belong to one of two classes, and the goal is to decide which class a new data point will be in.

Suppose that one has a table listing the population of some country in 1970, 1980, 1990 and 2000, and that one wanted to estimate the population in 1994.

Suppose homo economicus thinks about exerting some extra effort to defend the nation.

Suppose some particle has a mass m which is 3. 4 times the mass of electron.

Suppose, however, that we have some matrix Q that is not a pure rotation — due to round-off errors, for example — and we wish to find the quaternion q that most accurately represents Q.

Suppose two people who once loved each other come to be on bad terms ; they must make some condition of reconciliation before the love they previously enjoyed can be revived.

Suppose that the government finances some extra spending through deficits ; i. e. it chooses to tax later.

Suppose we can use some number, to index the quality of used cars, where is uniformly distributed over the interval.

Suppose that hunting requires also some arrows, with input coefficients equal to, meaning that to catch for instance one beaver you need to use arrows, besides hours of labour.

Suppose that ζ is an th root of unity for some odd prime.

Suppose that ζ is an lth root of unity for some odd regular prime l. Since l is regular, we can extend the symbol

Suppose for some unknown constants and unobserved random variables, where and, where < math > k < p </ math >, we have

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 that we have statements, denoted by some formal sequence of symbols, about some objects ( for example, numbers, shapes, patterns ).