) 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 further, because this is necessary to the alleged case for our nuclear weapon as the defence of last resort, that, as in 1940, the United States was standing aloof from the contest but that, in contrast with 1940, Britain and the Warsaw Pact respectively possessed the nuclear weaponry which they do today.

* Suppose & B is equivalent to & D. If we acquire new information A and then acquire further new information B, and update all probabilities each time, the updated probabilities will be the same as if we had first acquired new information C and then acquired further new information D. In view of the fact that multiplication of probabilities can be taken to be ordinary multiplication of real numbers, this becomes a functional equation

Suppose further that in order to find another dozen articles of interest, the researcher would have to go to an additional 10 journals.

" Suppose further ," Socrates says, " that the man was compelled to look at the fire: wouldn't he be struck blind and try to turn his gaze back toward the shadows, as toward what he can see clearly and hold to be real?

Suppose further that recipients of the new spending by the builder in turn spend their new income, this will raise demand and possibly consumption further, and so on.

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

Suppose further that we are allowed to rebalance our portfolio at times but without injecting additional cash into it.

Suppose further that the dimension of B over k is finite, i. e. that B is a central simple algebra.

Suppose further that a noisy error corrupts the three-bit state so that one bit is equal to zero but the other two are equal to one.

Suppose further that a < sub > 1 </ sub >/ a < sub > 2 </ sub > and a < sub > 0 </ sub >/ a < sub > 2 </ sub > are analytic functions.

Suppose that a further percent of the citizens also vote on every issue in the local government ballots.

Suppose further that the time to cross the device is the forward transit time.

Suppose that the equivalence problem has been through the loop enough times that no further reduction is possible.

Suppose further that for any n elements x < sub > 1 </ sub >,..., x < sub > n </ sub > of F which are linearly independent over Q, the extension field Q ( x < sub > 1 </ sub >,..., x < sub > n </ sub >, e ( x < sub > 1 </ sub >),..., e ( x < sub > n </ sub >)) has transcendence degree at least n over Q.

Suppose he goes further, and foolishly testifies on direct examination " in fact, I've never possessed heroin in my life.

Suppose the goal is to find and describe the solution ( s ), if any, of the following system of linear equations:

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

Suppose that the set of indices such that is decidable ; then, there exists a function that returns if, and otherwise.

He says, " Suppose we tell the Soviets that if they invade West Berlin we will launch an out nuclear attack.

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 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.

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 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 we wish to determine if n = 221 is prime.

Suppose the muon happens to have fallen into an orbit around a deuteron initially, which it has about a 50 % chance of doing if there are approximately equal numbers of deuterons and tritons present, forming an electrically neutral muonic deuterium atom ( d-μ )< sup > 0 </ sup > that acts somewhat like a " fat, heavy neutron " due both to its relatively small size ( again, about 207 times smaller than an electrically neutral electronic deuterium atom ( d-e )< sup > 0 </ sup >) and to the very effective " shielding " by the muon of the positive charge of the proton in the deuteron.

Suppose we want to run a regression to find out if the average annual salary of public school teachers differs among three geographical regions in Country A with 51 states: ( 1 ) North ( 21 states ) ( 2 ) South ( 17 states ) ( 3 ) West ( 13 states ).

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 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 one wants to turn the inductor off: when the current goes to zero, if the gate is not fed, the TRIAC attempts to turn off, but this causes a step in the voltage across it due to the afore-mentioned phase shift.

Suppose S is an immersed submanifold of M. If the inclusion map i: S → M is closed then S is actually an embedded submanifold of M. Conversely, if S is an embedded submanifold which is also a closed subset then the inclusion map is closed.

Suppose that is a standard multivariate normal random vector ( here denotes the n-by-n identity matrix ), and if are all n-by-n symmetric matrices with.

Suppose we wish to determine if n = 221 is prime.

Suppose their consumption set, or the enumeration of all possible consumption bundles that could be selected if there are no budget constraints, has L commodities and is limited to positive amounts of consumption of each commodity.

Suppose I believe that if I flap my arms with sufficient vigor, I can take off and fly about the room ; could I defend myself against the charge of irrationality by claiming this belief is basic?

Suppose, for example, that two threads, A and B, both attempt to grab a lock, backing off if it's already taken.

Suppose φ and χ are fields such that, if x and y are spacelike-separated points and i and j represent the spinor / tensor indices,

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

Mendelssohn answered in an open letter in December 1769: " Suppose there were living among my contemporaries a Confucius or a Solon, I could, according to the principles of my faith, love and admire the great man without falling into the ridiculous idea that I must convert a Solon or a Confucius.

Suppose a man walked into a room and saw someone rolling a pair of dice.

Suppose the man should fall asleep,

Suppose the man should fall asleep?

Suppose the man should fall asleep?

: Suppose now that there were two such magic rings, and the just put on one of them and the unjust the other ; no man can be imagined to be of such an iron nature that he would stand fast in justice.

Suppose, however, for a moment, all existing space to be bounded, and that a man runs forward to the uttermost borders, and stands upon the last verge of things, and then hurls forward a winged javelin ,— suppose you that the dart, when hurled by the vivid force, shall take its way to the point the darter aimed at, or that something will take its stand in the path of its flight, and arrest it?

Suppose, for example, that a man gives a woman a ring and tells her that it is for her next birthday and to hold on to it until then.