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 some 2-dimensional Riemannian manifold ( not necessarily compact ), and we specify a " triangle " on M formed by three geodesics.

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.

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 that V is only a representation of the subgroup H and not necessarily the larger group G. Let be the space of V-valued differential k-forms on P. In the presence of a Cartan connection, there is a canonical isomorphism

Suppose that a mathematician is studying geometry and shapes, and she wishes to prove certain theorems about them.

Suppose John Jones, who, for 1960, filed on the basis of a calendar year, died June 20, 1961.

Take as an example a program that looks up a specific entry in a sorted list of size n. Suppose this program were implemented on Computer A, a state-of-the-art machine, using a linear search algorithm, and on Computer B, a much slower machine, using a binary search algorithm.

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 that on these sets X and Y, there are two binary operations and that happen to constitute the groups ( X ,) and ( Y ,).

: Suppose ƒ is a continuous complex-valued function defined on the real interval.

Suppose that a student performed poorly on a test and guesses that the cause was his not studying.

As they discussed Shelvocke's book, Wordsworth proffers the following developmental critique to Coleridge, which importantly contains a reference to tutelary spirits: " Suppose you represent him as having killed one of these birds on entering the south sea, and the tutelary spirits of these regions take upon them to avenge the crime.

Suppose the thimble were screwed out so that graduation 2, and three additional sub-divisions, were visible ( as shown in the image ), and that graduation 1 on the thimble coincided with the axial line on the frame.

Suppose that the thimble were screwed out so that graduation 5, and one additional 0. 5 subdivision were visible ( as shown in the image ), and that graduation 28 on the thimble coincided with the axial line on the sleeve.

Suppose a particle starts at point A, and there is a force F acting on it.

Suppose that we have already constructed Lebesgue measure on the real line: denote this measure space by ( R, B, λ ).

Suppose you want to find the shortest path between two intersections on a city map, a starting point and a destination.

Suppose someone told you they had a nice conversation with someone on the train.

* 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 Lloyds TSB is trading on the market at 410p bid, and 411p offer.

: Suppose there are three cottages on a plane ( or sphere ) and each needs to be connected to the gas, water, and electric companies.

Suppose U is a simply connected open subset of the complex plane, and a < sub > 1 </ sub >,..., a < sub > n </ sub > are finitely many points of U and f is a function which is defined and holomorphic on U

Suppose f is an analytic function defined on an open subset U of the complex plane.

Suppose that you start with $ 10 in poker chips, and you repeatedly wager $ 1 on a ( fair ) coin toss indefinitely, or until you lose all of your poker chips.

Suppose that you have a coin purse containing five quarters, five nickels and five dimes, and one-by-one, you randomly draw coins from the purse and set them on a table.