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Page "Examples of differential equations" ¶ 38
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Since and μ
Since only the positions in space – time are being warped, not the fields, the Ψ are all zero and the X < sub > μ </ sub >< sup > ν </ sup > equal the Kronecker delta δ < sub > μ </ sub >< sup > ν </ sup >, where we have used μ instead of r for the index.
Since μ < sub > X </ sub >
Since the analysis must typically go beyond two-loop perturbation theory, the definition of the running coupling α < sub > TC </ sub >( μ ), it ’ s fixed point value α < sub > IR </ sub >, and the strength α < sub > χ SB </ sub > necessary for chiral symmetry breaking depend on the particular renormalization scheme adopted.
Since G is compact in this topology, there is a unique Haar measure μ on G. For every prime v of K not in S there is an associated Frobenius conjugacy class F < sub > v </ sub >.
Since the estimate does not depend on Σ, we can just substitute it for μ in the likelihood function, getting

Since and is
Since the Supreme Court's decision of that year this is more doubtful ; ;
( Since the time-span of the nation-state coincides roughly with the separate existence of the United States as an independent entity, it is perhaps natural for Americans to think of the nation as representative of the highest form of order, something permanent and unchanging.
Since it is not far from Viareggio, he will visit Puccini's house, as he never fails to do, to pay his respects to the memory of the composer of La Boheme, which he considers one of Puccini's masterpieces.
Since attack serves to stimulate interest in broadcasts, I added to my opening statement a sentence in which I claimed that German youth seemed to lack the enthusiasm which is a necessary ingredient of anger, and might be classified as uninterested and bored rather than angry.
Since the air is a continuum, the network of communication remains intact regardless of the positions or motions of the points ( the people ) in the net.
Since the hazards of poor communication are so great, p can be justified as a habitable site only on the basis of unusual productivity such as is made available by a waterfall for milling purposes, a mine, or a sugar maple camp.
Since the difficulty of drawing the net is great, we will merely discuss it.
Since a civilizational crisis involves also a crisis in private interests and in the ruling class, reaction is normally found among those who feel themselves to be among the ruling class.
Since civilizational change is the most difficult to perceive and analyze, it seldom is given adequate attention.
Since the slogans have little application to reality and are sanctimonious to boot, the applause is faint even in areas of the world where we should expect to find the greatest affection for free government.
Since more is known about Quiney than about any other acquaintance of Shakespeare in Stratford, his career may be followed to its sudden end in 1602.
Since the great flood of these dystopias has appeared only in the last twelve years, it seems fairly reasonable to assume that the chief impetus was the 1949 publication of Nineteen Eighty-Four, an assumption which is supported by the frequent echoes of such details as Room 101, along with education by conditioning from Brave New World, a book to which science-fiction writers may well have returned with new interest after reading the more powerful Orwell dystopia.
Since Laos is of no more purely military value to Moscow itself than it is to Washington, this approach might be expected to head off Mr. Khrushchev for the moment.
Since little is known about autism, and almost nothing has been written for the layman, we'd like to share one experienced mother's comments.
Since the work is done by hand, the only limitation, it is said, `` is that of human conception ''.
Since the apparatus is new, it requires experimentation and changes in technique.
Since the obvious is not always true, the Republican National Committee wisely analyzed its defeat of last autumn and finds that it occurred, as suspected, in the larger cities.
Since he has just shown who is top dog, he may not be ready to receive this highest honor in the gift of the Soviet people.
Since appeals to morality, to humanity, and to sanity have had such small effect, perhaps our last recourse is the deterrent example.
Since the goal of our international planners is a World Government, this Atlantic Community would mark a giant step in that direction for, once American economic autonomy is absorbed, a larger grouping is a question of time.

Since and function
Since Af are linearly independent functions and the exponential function has no zeros, these R functions Af, form a basis for the space of solutions.
Since the removal force is a function of coating thickness, a differential transformer pickup has been incorporated into the instrument to accurately measure film thickness.
Since the 12th century, courts have had parallel and co-equal authority to make law -- " legislating from the bench " is a traditional and essential function of courts, which was carried over into the U. S. system as an essential component of the " judicial power " specified by Article III of the U. S. constitution.
A set S is called countable if there exists an injective function f from S to the natural numbers Since there is an obvious bijection between and it makes no difference whether one considers 0 to be a natural number of not.
: Since the result is a function of, this function can be defined as.
Since tangents and inverse tangents are both continuous, our identification function is a homeomorphism between the real line and the unit circle without ∞.
Since such life utilised strong nuclear forces instead of electromagnetic interactions, it was posited that life might function millions of times faster than typical on Earth.
Since these devices are used to run countless different types of programs, these CPU designs are not specifically targeted at one type of application or one function.
Since the separation of variables in this case involves dividing by y, we must check if the constant function y = 0 is a solution of the original equation.
Since all structural levels of the protein determines its function, the protein can no longer perform its function once it has been denatured.
Since the latter is valid over the entire cycle, this gave Clausius the hint that at each stage of the cycle, work and heat would not be equal, but rather their difference would be a state function that would vanish upon completion of the cycle.
Since 2001, 40 non-natural amino acids have been added into protein by creating a unique codon ( recoding ) and a corresponding transfer-RNA: aminoacyl – tRNA-synthetase pair to encode it with diverse physicochemical and biological properties in order to be used as a tool to exploring protein structure and function or to create novel or enhanced proteins.
Since its inception it has broadened to find applications in many other areas, including statistical inference, natural language processing, cryptography, neurobiology, the evolution and function of molecular codes, model selection in ecology, thermal physics, quantum computing, plagiarism detection and other forms of data analysis.
Since the identity element of a monoid is unique, one can alternately define the identity function on M to be this identity element.
Since terms of linear equations cannot contain products of distinct or equal variables, nor any power ( other than 1 ) or other function of a variable, equations involving terms such as xy, x < sup > 2 </ sup >, y < sup > 1 / 3 </ sup >, and sin ( x ) are nonlinear.
Since the names of functions are largely a convenience, the lambda calculus has no means of naming a function.
Since any fluorescence image is made up of a large number of such small fluorescent light sources, the image is said to be " convolved by the point spread function ".
Since the energy of the photoelectrons emitted is exactly the energy of the incident photon minus the material's work function or binding energy, the work function of a sample can be determined by bombarding it with a monochromatic X-ray source or UV source, and measuring the kinetic energy distribution of the electrons emitted.
Since each sample is a random variable, the sample mean is a function of random variables and hence a random variable itself, whose distribution can be computed and properties determined.
Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation ( that assigns to each function of bounded variation the corresponding Lebesgue-Stieltjes measure, and the integral with respect to the Lebesgue-Stieltjes measure agrees with the Riemann-Stieltjes integral for continuous functions ), the above stated theorem generalizes the original statement of F. Riesz.
Since this variable is not passed as a parameter but can be altered, the results of subsequent calls to the function can differ even if the parameters are identical.

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