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Page "Communication complexity" ¶ 6
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Then and problem
Then he called in his friend Walton and turned over the problem to him, with instructions to work out what was best -- provided it didn't pile unnecessary burdens on the President.
Then a single server may cause a bottleneck or constraints problem.
We wish to maximize total value subject to the constraint that total weight is less than or equal to W. Then for each w ≤ W, define m to be the maximum value that can be attained with total weight less than or equal to w. m then is the solution to the problem.
Then both concepts of particle and wave may be attributed to an observation problem of the gear.
⟨ H ⟩, be the group generated by H. Then the word problem in H < sup >*</ sup > is solvable: given two words h, k in the generators H of H < sup >*</ sup >, write them as words in X and compare them using the solution to the word problem in G. It is easy to think that this demonstrates a uniform solution the word problem for the class K ( say ) of finitely generated groups that can be embedded in G. If this were the case the non-existence of a universal solvable word problem group would follow easily from Boone-Rogers.
That puts the sequential problem in P. Then, it will be in NC if and only if it is parallelizable.
Then there is the type of activity that uses the network to waste time, as indicated in SurfControl's advice on writing AUPs, of technical staff to troubleshoot a problem for which the user is the cause, corrupting or destroying other user's data, violating the privacy of others online, using the network in such a way that it denies the service to others, continuing to use software or other system for which the user has already been warned about using, and any other misuse of the network such as introduction of viruses.
Then one faces the problem of expression sin ( θ ) and cos ( θ ) as functions of x.
Then I don't have a problem to walk away and say ' you're entitled to your opinions '.
Then, as Gotti grew increasingly impatient, Conte explained that the problem now was trying to corner DiBono again.
Then due to a licensing problem with Spatial Technologies, they dropped the Modeler program from the version 5. 5 release, and renamed the package back to Electric Image Animation System.
Then there is no problem.
Then he adds, “ By the way, a problem given by the Zens for meditation is the following: If you clap your hands, does the sound come from the left hand or from the right ?”
Then the problem might be to rank these alternatives in terms of how attractive they are to the decision maker ( s ) when all the criteria are considered simultaneously.
Then they said very little about what they were doing to solve a problem that had caused more than 100 deaths — until they got called to Washington to testify before Congress.
Then, the coefficients of the function approximations are treated as optimization variables and the problem is " transcribed " to a nonlinear optimization problem of the form:
Then sheaf cohomology enables us to consider a similar extension problem while " continuously varying " the Abelian group.
Then a solution to our minimization problem is given by
Then the satisfying instances of the resulting 2SAT formula encode solutions to the vertex cover problem, and there is a satisfying assignment with true variables if and only if there is a vertex cover with vertices.
Assume we have as input array S with elements S .. S. Then the following algorithm will solve 3SUM problem in quadratic time.

Then and computing
The algorithm for deciding this is conceptually simple: it constructs ( the description of ) a new program t taking an argument n which ( 1 ) first executes program a on input i ( both a and i being hard-coded into the definition of t ), and ( 2 ) then returns the square of n. If a ( i ) runs forever, then t will never get to step ( 2 ), regardless of n. Then clearly, t is a function for computing squares if and only if step ( 1 ) terminates.
Then the Phong reflection model provides an equation for computing the illumination of each surface point:
Then by using statistics and a lot of computing ( a 35 dual-Xeon computer-cluster powers these despeckle algorithms ) the image is improved.
Then inexact line searches provide an efficient way of computing an acceptable step length that reduces the objective function ' sufficiently ', rather than minimizing the objective function over exactly.
Then, when enough computing power was available and the first steps of numerical weather prediction were taken, the value of work done by the Chicago School 10-20 years earlier was really appreciated.
Then decided to abandon mathematics for a career in software engineering and research to the skepticism of his thesis advisor because Bill thought the future was in computing.

Then and function
Then our choice function can choose the least element of every set under our unusual ordering.
Then an arithmetic function a is
Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.
Then is a group whose identity element is The group inverse of an arbitrary group element is the function inverse
Then, using the periodic Bernoulli function P < sub > n </ sub > defined above and repeating the argument on the interval, one can obtain an expression of ƒ ( 1 ).
Then a mathematical procedure is used to find a filter transfer function which can be realized ( within some constraints ) and which approximates the desired response to within some criterion.
Then an " estimator " is a function that maps the sample space to a set of sample estimates.
Let R be a domain and f a Euclidean function on R. Then:
For instance, suppose that each input is an integer z in the range 0 to N − 1, and the output must be an integer h in the range 0 to n − 1, where N is much larger than n. Then the hash function could be h
Then the probability density function f *( x ) of the size biased population is
Then ƒ is invertible if there exists a function g with domain Y and range X, with the property:
Then for a specific value x of X, the function L ( θ | x )
Then,, and the constant function turns into the identity by substitution.
Then, in 1905, to explain the photoelectric effect ( 1839 ), i. e., that shining light on certain materials can function to eject electrons from the material, Albert Einstein postulated, based on Planck ’ s quantum hypothesis, that light itself consists of individual quantum particles, which later came to be called photons ( 1926 ).
Then the output is related to the input by the transfer function as
Then because the construction of f was uniform, this is a recursive function of two variables.
Then, its electrical field traces out a helical pattern as a function of time.
Then the function
Then the function
Then, recalling that the likelihood function is defined up to a multiplicative constant, it is just as valid to say that the likelihood function is approximately
Then, on considering the lengths of the intervals to decrease to zero, the likelihood function for an observation from the discrete component is
Then has probability density function.

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