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Suppose and V

Suppose that K is a field ( for example, the real numbers ) and V is a vector space over K. As usual, we call elements of V vectors and call elements of K scalars.

Suppose that W is a subset of V.

Suppose V and W are vector spaces over the field K. The cartesian product V × W can be given the structure of a vector space over K by defining the operations componentwise:

Suppose that exactly two of U, V, W are 0.

Suppose V is a subset of R n ( in the case of n = 3, V represents a volume in 3D space ) which is compact and has a piecewise smooth boundary S. If F is a continuously differentiable vector field defined on a neighborhood of V, then we have

Suppose we have a connected graph G = ( V, E ), The following statements are equivalent:

Suppose now that an attenuating feedback loop applies a fraction β. V out of the output to one of the subtractor inputs so that it subtracts from the circuit input voltage V in applied to the other subtractor input.

where g ( w ) = f ( w ) v. Suppose that V is finite dimensional.

Suppose V is a model of ZFC.

Suppose that ( V, ω ) and ( W, ρ ) are symplectic vector spaces.

First proof: Suppose forms a basis of ker T. We can extend this to form a basis of V:.

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 ∞( M ) ( the commutative algebra of smooth real-valued functions on M ).

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 we define V to be the set of variables on which the functions f and g operate.

Suppose V is a vector space over K, a subfield of the complex numbers ( normally C itself or R ).

Suppose the subspaces U and V are the range and null space of P respectively.

Suppose that V is a vector space with a nondegenerate bilinear form (·,·).

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.

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

) Then X i 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 i and variance σ i 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.

Suppose and only

Suppose that Af, and that the average operation requires only Af sec..

Suppose that a speaker can have the concept of water we do only if the speaker lives in a world that contains H 2 O.

Suppose that there are two agents in an economy, one that only values guns and one that only values butter.

Suppose G is an ordered abelian group, meaning an abelian group with a total ordering "<" respecting the group's addition, so that a < b if and only if a + c < b + c for all c. Let I be a well-ordered subset of G, meaning I contains no infinite descending chain.

Example: Suppose there are only three principles in our scientific theory about electrons ( those principles can be seen to be statements involving the properties ):

An example: Suppose that only Alice, Bob, and Carol have the keys to a bank safe and that, one day, the contents of the safe are missing ( without the lock being violated ).

'; he may have only had the fanciful thought, ' Suppose a rascal behaved like Hunt!

Suppose that the programmable divider, using N, is only able to operate at a maximum clock frequency of 10 MHz, but the output f is in the hundreds of MHz range ;.

Suppose that the alternatives can be partitioned into subsets, where each subset has its own cost function, and each alternative belongs to only one subset.

Suppose we want to define an extremely simple XML markup scheme for a book: a book is defined as a sequence of one or more pages ; each page contains text only.

Suppose in a single-elimination tournament the best team plays the second best team in the first round — the second best team will be eliminated right away, having only played one game.

Suppose the Hamiltonian H of interest is a self adjoint operator with only discrete spectrum.

Lewis is perhaps using Puddleglum to give a somewhat existential statement of faith when he writes, " Suppose we have only dreamed, or made up, all of those things — trees and grass and sun and moon and stars and Aslan himself.

Suppose G is a finite group, G 0 a subgroup of G. A representation U of G is induced from a representation V of G 0 if and only if there exist the following:

* Scenario 2: Suppose Gina wasn't as agile with the hammer and could only make 1 birdhouse an hour, but she took a sewing class and could print 10 t-shirts an hour.

Suppose response variable Y is binary, that is it can have only two possible outcomes which we will denote as 1 and 0.

Suppose the yield is 50 % for each reaction, the overall yield of D is only 12. 5 % from A.

Suppose there are only two groups, ( so ), and the means of each class are defined to be and the covariances are defined as.

Suppose that you are one of the Logicians and you see another colour only once.

Suppose Bob likes only chocolate, and Carol only vanilla.

Suppose 3 real-estate sites are searched, each provides a list of hyperlinked city names to click on, to see matches only in each city.

Suppose you were going to make an investment into only one of three investment vehicles: stock, mutual fund, or certificate of deposit ( CD ).

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