) 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 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 that Y is the sum of n identically distributed independent random variables all with the same distribution as X.

: Suppose X is a compact Hausdorff space and A is a subalgebra of C ( X, R ) which contains a non-zero constant function.

Suppose that X is a topological space.

Suppose that U: D → C is a functor from a category D to a category C, and let X be an object of C. Consider the following dual ( opposite ) notions:

Suppose ( A < sub > 1 </ sub >, φ < sub > 1 </ sub >) is an initial morphism from X < sub > 1 </ sub > to U and ( A < sub > 2 </ sub >, φ < sub > 2 </ sub >) is an initial morphism from X < sub > 2 </ sub > to U. By the initial property, given any morphism h: X < sub > 1 </ sub > → X < sub > 2 </ sub > there exists a unique morphism g: A < sub > 1 </ sub > → A < sub > 2 </ sub > such that the following diagram commutes:

An uncountable subset of the real numbers with the standard ordering ≤ cannot be a well-order: Suppose X is a subset of R well-ordered by ≤.

Suppose that X is a regular space.

Suppose that X and Y are a pair of commuting vector fields.

Suppose we are given a topological space X.

Suppose that X and Y are two plane projective curves defined over a field F that do not have a common component ( this condition is true if both X and Y are defined by different irreducible polynomials, in particular, it holds for a pair of " generic " curves ).

Suppose that we vary the complex structure of X over a simply connected base.

Suppose that X is a non-singular n-dimensional projective algebraic variety over the field F < sub > q </ sub > with q elements.

Suppose that F is a collection of continuous linear operators from X to Y.

Suppose C is a category, and f: X → Y is a morphism in C. The morphism f is called a constant morphism ( or sometimes left zero morphism ) if for any object W in C and any g, h: W → X, fg

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.

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 that T is a bounded operator on the normed vector space X.

Suppose we now consider a slightly more complicated vector field:

Suppose that u and v satisfy the Cauchy – Riemann equations in an open subset of R < sup > 2 </ sup >, and consider the vector field

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

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 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 V is a subset of R < sup > n </ sup > ( 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 everything in the universe undergoes an improper rotation described by the rotation matrix R, so that a position vector x is transformed to x ′

Suppose the system starts in state 2, represented by the vector.

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

Suppose that a tangent vector to the sphere S is given at the north pole, and we are to define a manner of consistently moving this vector to other points of the sphere: a means for parallel transport.

Suppose that P ( t ) is a curve in S. Naïvely, one may consider a vector field parallel if the coordinate components of the vector field are constant along the curve.

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 the random column vectors X, Y live in R < sup > n </ sup > and R < sup > m </ sup > respectively, and the vector ( X, Y ) in R < sup > n + m </ sup > has a multivariate normal distribution whose variance is the symmetric positive-definite matrix

Suppose we parallel transport the vector first along the equator until P and then ( keeping it parallel to itself ) drag it along a meridian to the pole N and ( keeping the direction there ) subsequently transport it along another meridian back to Q.

Let M be a differentiable manifold and p a point of M. Suppose that f is a function defined in a neighborhood of p, and differentiable at p. If v is a tangent vector to M at p, then the directional derivative of f along v, denoted variously as ( see covariant derivative ), ( see Lie derivative ), or ( see Tangent space # Definition via derivations ), can be defined as follows.

Suppose further that we can generate a sample of replications of the random vector.

Suppose that we have a sample of realizations of the random vector.

Suppose that a sample is taken from a distribution depending on a parameter vector of length, with prior distribution.

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