Suppose that one particle is in the state n < sub > 1 </ sub >, and another is in the state n < sub > 2 </ sub >.

Suppose we have a system of N bosons ( fermions ) in the symmetric ( antisymmetric ) state

Suppose Alice has a qubit in some arbitrary quantum state.

Suppose we have a material in its normal state, containing a constant internal magnetic field.

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

Suppose we are given a Hidden Markov Model ( HMM ) with state space, initial probabilities of being in state and transition probabilities of transitioning from state to state.

Suppose we have five state functions, and.

Suppose that the state space is two dimensional and any of the five quantities are exact differentials.

That is, since log | f ( z )| is harmonic, it is thus the steady state of a heat flow on the region D. Suppose a strict maximum was attained on the interior of D, the heat at this maximum would be dispersing to the points around it, which would contradict the assumption that this represents the steady state of a system.

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 a two-level atom whose ground state is shown by g and excited state by e. Efficient laser cooling occurs when the frequency of the laser beam is tuned to the red sideband i. e.

Suppose also that we know the corresponding state | ψ >.

Suppose some system is presently in a quantum state.

Suppose if this one state were to be attacked, who's to say whether the other states would respond?

Suppose we have N particles with quantum numbers n < sub > 1 </ sub >, n < sub > 2 </ sub >, ..., n < sub > N </ sub >.

Suppose a is thrown that contains six sides and that the numeric result of the throw corresponds to a quantum mechanics observable.

Suppose instead of quantum mechanical systems, the two systems A and B are classical.

Suppose each set of quantum numbers specify f states where f is

Suppose each set of quantum numbers specify states where is the number of internal degrees of freedom of the particle that can be altered by collision.

Suppose we encode the quantum numbers into the single index or.

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

Suppose we were to increase the concentration of CO in the system.

Suppose this fuzzy system has the following rule base:

Suppose we have a preparation procedure for a system in a physics

Suppose that we want to solve a well-conditioned system of linear equations.

Suppose this system of N qubits undergoes a dephasing process, a rotation around the eigenstates of, for example.

Suppose that takes the form of the incommensurate quantity, the base of the natural system of logarithms.

Suppose the system has one external variable x.

Suppose that the gain of the control system is reduced drastically and it is restarted.

Suppose a system is subdivided into N sub-systems with negligible interaction energy.

Suppose that is input to system and gives output.

Suppose that the system is a discrete-time, linear, time-invariant ( LTI ) system described by the impulse response.

Definition 2: Suppose is the impulse response of the system.

Suppose S is in a microstate indexed by m. From the above definitions, the total energy of the system S * is given by

Suppose we represent a system by an operator.

Suppose that we use a pub / sub system to log problems in a factory: any application that senses an error publishes an appropriate message, and the messages are displayed on a console by the logger daemon, which subscribes to the " errors " topic.

Suppose that the data is generated by a first-order system and let denote the solution of the filter's Riccati equation.

Suppose a pendulum is swinging with a particular period T. For such a system, it is advantageous to perform calculations relating to the swinging relative to T. In some sense, this is normalizing the measurement with respect to the period.

Suppose for simplicity that a certain system is characterized by two variables-a dependent variable x and an independent variable t, where x is a function of t. Both x and t represent quantities with units.

Suppose a system is defined by the Lagrangian L with generalized coordinates q.

Suppose they both had ventured into realms which their colleagues thought infidel: is this the way gentlemen settle frank differences of opinion??

Suppose that u and v are real-differentiable at a point in an open subset of, which can be considered as functions from to.

Suppose that is a complex-valued function which is differentiable as a function.

Suppose n < sub > 1 </ sub >, n < sub > 2 </ sub >, …, n < sub > k </ sub > are positive integers which are pairwise coprime.

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 a number of scientists are assessing the probability of a certain outcome ( which we shall call ' success ') in experimental trials.

* Suppose that there is a compression algorithm that transforms every file into a distinct file which is no longer than the original file, and that at least one file will be compressed into something that is shorter than itself.

* Suppose G and H are topologically finitely-generated profinite groups which are isomorphic as discrete groups by an isomorphism ι.

Suppose that the signal to be sent ( called the modulating or message signal ) is and the carrier onto which the signal is to be modulated is

Suppose, for concreteness, that we have an algorithm for examining a program p and determining infallibly whether p is an implementation of the squaring function, which takes an integer d and returns d < sup > 2 </ sup >.

: 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 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 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.

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 I is an ideal which is not R ( respectively, A is a right ideal which is not R ).

Suppose it is also known that 75 % of women have long hair, which we denote as

Suppose P is a definable binary relation ( which may be a proper class ) such that for every set x there is a unique set y such that P ( x, y ) holds.

Suppose f were injective, which means the pieces of S cut out by the squares stack up in a non-overlapping way.

Suppose that a particle of charge experiences an acceleration which is collinear with its velocity ( this is the relevant case for linear accelerators ).

Suppose you recognize one student — call her Anna — from a prior course in which Anna either excelled or did poorly.

Suppose that C is a twice continuously differentiable immersed plane curve, which here means that there exists parametric representation of C by a pair of functions such that the first and second derivatives of x and y both exist and are continuous, and

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 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