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For multi-loop integrals that will depend on several variables we can make a change of variables to polar coordinates and then replace the integral over the angles by a sum so we have only a divergent integral, that will depend on the modulus and then we can apply the zeta regularization algorithm, the main idea for multi-loop integrals is to replace the factor after a change to hyperspherical coordinates so the UV overlapping divergences are encoded in variable r. In order to regularize these integrals one needs a regulator, for the case of multi-loop integrals, these regulator can be taken as so the multi-loop integral will converge for big enough's ' using the Zeta regularization we can analytic continue the variable's ' to the physical limit where s = 0 and then regularize any UV integral.
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For and integrals
For 25 years, he worked on theories about elementary particles, played a role in the discovery of the quark, and researched the analytic and high-energy properties of Feynman integrals and the foundations of S-Matrix theory.
For integration limits of, the integrals for rectifying radius and mean radius evaluate to the same result, which, for Earth, amounts to 6, 367. 4491 km (≈ 3, 956. 545 mi ; ≈ 3, 438. 147 nmi ).
For indefinite integrals, the sum rule states
For a list of definite integrals see List of definite integrals.
For example, in the nineteenth century there was much interest in hyperelliptic integrals that may be expressed in terms of elliptic integrals.
For every closed curve c on X ( i. e., smooth function ), we can associate a differential form with the pleasant property that integrals along c can be calculated by integrals over X:
For this reason path integrals were used in the study of Brownian motion and diffusion a while before they were introduced in quantum mechanics.
For a particle in a smooth potential, the path integral is approximated by zig-zag paths, which in one dimension is a product of ordinary integrals.
For example, " Introduction to the theory of series ," " Elements of the theory of determinants ," " Course on definite integrals " ( 1903 ), and " Elements of algebraic analysis " ( 1912 ).
For integrals of scalar fields, the answer to this question is simple, the value of the surface integral will be the same no matter what parametrization one uses.
For integrals of vector fields things are more complicated, because the surface normal is involved.
For non-periodic functions, however, methods with unequally spaced points such as Gaussian quadrature and Clenshaw – Curtis quadrature are generally far more accurate ; Clenshaw – Curtis quadrature can be viewed as a change of variables to express arbitrary integrals in terms of periodic integrals, at which point the trapezoidal rule can be applied accurately.
For purposes of numeric computations, being in closed form is not in general necessary, as many limits and integrals can be efficiently computed.
For instance, in 1971, Charles Fefferman was able to use the Besicovitch set construction to show that in dimensions greater than 1, truncated Fourier integrals taken over balls centered at the origin with radii tending to infinity need not converge in L < sup > p </ sup > norm when p ≠ 2 ( this is in contrast to the one-dimensional case where such truncated integrals do converge ).
For a thermostatted steady state, time integrals of the dissipation function are related to the dissipative flux, J, by the equation
For example, the first example's integral is expressible using incomplete elliptic integrals of the first kind, the second and third use the logarithmic integral, the fourth the exponential integral, and the fifth the error function.
For and will
For three straight years, Tom Horn patrolled the southern Wyoming pastures, and how many men he killed after Lewis and Powell ( if he killed Lewis and Powell ) will never be known.
For pride's sake, I will not say that the coy and leering vade mecum of those verses insinuated itself into my soul.
William Wimsatt and Cleanth Brooks, it seems to me, have a penetrating insight into the way in which this control is effected: `` For if we say poetry is to talk of beauty and love ( and yet not aim at exciting erotic emotion or even an emotion of Platonic esteem ) and if it is to talk of anger and murder ( and yet not aim at arousing anger and indignation ) -- then it may be that the poetic way of dealing with these emotions will not be any kind of intensification, compounding, or magnification, or any direct assault upon the affections at all.
For them a box will be lodgment enough.
For that reason any democratic reform and effort to bring genuine representative government to the Dominican Republic will need the greatest sympathy and help.
`` For if ye forgive men their trespasses, your heavenly Father will also forgive you: but if ye forgive not men their trespasses, neither will your Father forgive -- your trespasses ''.
For the policy officer will know that action can almost never be secret and that in general the effectiveness of policy will be conditioned by the readiness of the country to sustain it.
For example, it probably will be necessary for the Corps to have authority to pay medical expenses of volunteers.
For me, these will belong more completely to their surroundings if they are conceived in this early stage, though I freely admit that I do not hesitate to add or eliminate figures on the full sheet when it serves my final purpose.
For the most part, this discussion will be confined to results obtained since the introduction of the reference standard.
For any such square the middle corner of these will be called the vertex of the square and the corner not on the curve will be called the diagonal point of the square.
For the vertex at Af, these values will be denoted by Af.
For winning Larson will receive a $100 U.S. Savings Bond from the Junior Achievement national organization.
For most of them, it will be their first experience in membership training, since this is a recent development in many churches.
For the army of compulsive eaters -- from the nibblers and the gobblers to the downright gluttons -- reducing is a war with the will that is rarely won.
" Swift extends the metaphor to get in a few jibes at England ’ s mistreatment of Ireland, noting that " For this kind of commodity will not bear exportation, and flesh being of too tender a consistence, to admit a long continuance in salt, although perhaps I could name a country, which would be glad to eat up our whole nation without it.
For example, if an author is paid a modest advance of $ 2000. 00, and their royalty rate is 10 % of a book priced at $ 20. 00-that is, $ 2. 00 per book-the book will need to sell 1000 copies before any further payment will be made.
For Tarrou, plague is the destructive impulse within every person, the will and the capacity to do harm, and it is everyone's duty to be on guard against this tendency within themselves, lest they infect someone else with it.
For an example of its use, analysis of the concentration of elements is important in managing a nuclear reactor, so nuclear scientists will analyze neutron activation to develop discrete measurements within vast samples.
For reception, a local oscillator will typically restore the suppressed carrier so the signal can be demodulated with a product detector.
For the living know that they will die ; But the dead know nothing, And they have no more reward, For the memory of them is forgotten.
For and depend
For irradiation of food, the results obtained depend upon the dose rather than the specific type of radiation, and X-ray, gamma, and high-energy electron radiation are suitable.
For example, the rules of the Hebrew calendar depend on the seven-day week cycle ( a very simple calendar ), so the week is one of the cycles of the Hebrew calendar.
For these reasons, rasterization is usually the approach of choice when interactive rendering is required ; however, the pixel-by-pixel approach can often produce higher-quality images and is more versatile because it does not depend on as many assumptions about the image as rasterization.
For a given material, it can have a positive or negative sign or exceptionally it can be zero, and this can depend on the temperature, as it does for water about 4 C. The concept of latent heat with respect to volume was perhaps first recognized by Joseph Black in 1762.
For those gases, the relative radiative forcing will depend upon abundance and hence upon the future scenario adopted.
For example, the result of a significance test depends on the probability of a result as extreme or more extreme than the observation, and that probability may depend on the design of the experiment.
For instance, the " distance " between two points on such an object, may depend on the order in which you consider the pair of points.
For example, systems that depend heavily on immutable objects such as many functional programming languages can suffer an efficiency penalty due to frequent copies.
For several reasons, modern scholars depend more heavily on the Arian accounts than the Catholic accounts.
For monopropellant rockets that depend on a chemical reaction, the power for the propulsive reaction and resultant thrust is provided by the chemical itself.
For our purposes, this means that yes / no questions that depend on countably many coordinates have a probabilistic answer.
For hydrogen-like atoms ( ions ) only, the Rydberg levels depend only on the principal quantum number.
For a wide variety of materials and conditions, V and I are directly proportional to each other, and therefore R and G are constant ( although they can depend on other factors like temperature or strain ).
For example, entrepreneurs on the internet have also created entirely new models that depend entirely on existing or emergent technology.
For example, Drew S. Days has written: " we have developed criteria for evaluating the constitutionality of racial classifications that do not depend upon findings of psychic harm or social science evidence.
For illustration, consider a Lagrangian that does not depend on time, i. e., that is invariant ( symmetric ) under changes t → t + δt, without any change in the coordinates q.
For illustration, consider a physical system of fields that behaves the same under translations in time and space, as considered above ; in other words, the fields do not depend on the absolute position in space and time.
For instance, the stiffness of a composite panel will often depend upon the orientation of the applied forces and / or moments.
For this reason established Churches and civil authorities fear that the hierarchy on which they depend is undermined by Bogomilism.
* For a given semi-major axis the orbital period does not depend on the eccentricity ( See also: Kepler's third law ).
* For large spots the sizes depend on the differential rotation of the star, but there are some indications that large spots which give rise to light variations can survive for many years even in stars with differential rotation.
For example, if the Lagrangian L does not depend on q < sub > i </ sub > itself, then the generalized momentum:
For example, density is an intensive property of a substance because it does not depend on the amount of that substance ; mass and volume, which are measures of the amount of the substance, are extensive properties.
For classical propositional logic this does not yield a problem, since the conclusions that one can draw from a collection of premises does not depend on these data.
For example, the oysters in the Apalachicola Bay, Florida, depend on the brackish water mixture of river and ocean water, and the alternating freshwater and saltwater flows that the river and the tides provide.
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