where the first sum is over positive energy states and the second over those of negative energy.

Mathematical and logical propositions ( e. g. " that the square of the hypotenuse is equal to the sum of the squares of the two sides ") are examples of the first, while propositions involving some contingent observation of the world ( e. g. " the sun rises in the East ") are examples of the second.

Under the protocol signed on 7 May 1832 between Bavaria and the protecting Powers, and basically dealing with the way in which the Regency was to be managed until Otto reached his majority ( while also concluding the second Greek loan, for a sum of £ 2, 400, 000 sterling ), Greece was defined as an independent kingdom, with the Arta-Volos line as its northern frontier.

These three-wave mixing processes correspond to the nonlinear effects known as second harmonic generation, sum frequency generation, difference frequency generation and optical rectification respectively.

The sum is then a third arrow that goes directly from the start of the first to the end of the second.

We see that the second column is twice the first column, and that the fourth column equals the sum of the first and the third.

* By contrast, the event of getting a 6 the first time a die is rolled and the event that the sum of the numbers seen on the first and second trials is 8 are not independent.

The number of military personnel in Yemen is relatively high ; in sum, Yemen has the second largest military force on the Arabian Peninsula after Saudi Arabia.

There were also 16 channel energy spectra from the sum of the 2 detectors ( taken either in 1, 2, 4, 16, or 32 second integrations ).

These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous term ( proportional to the gradient of velocity ) and a pressure term.

The second measure is the " free and dutiable tariff rate " which is the tariff revenue divided by the dollar sum of both dutiable and non-dutiable imports.

For example, the following equation, from a hypothetical cipher, states the XOR sum of the first and third plaintext bits ( as in a block cipher's block ) and the first ciphertext bit is equal to the second bit of the key:

Where represents the rest mass of the first colliding body, represents the rest mass of the second colliding body, represents the initial velocity of the first collidng body, represents the initial velocity of the second colliding body, represents the velocity after collision of the first colliding body, represents the velocity after collision of the second colliding body, denotes the momentum of the first colliding body, denotes the momentum of the second colliding body and denotes the speed of light in vacuum, denotes the total energy of the system ( i. e. the sum of rest masses and kinetic energies of the colliding bodies ).

we find that the position of the first body with respect to the second is governed by the same differential equation as the position of a body with the reduced mass orbiting a body with a mass equal to the sum of the two masses, because

Also in 2008, the defendants of United States v LG Display Co., United States v. Chunghwa Picture Tubes, and United States v. Sharp Corporation heard in the Northern District of California, agreed to pay a total sum of $ 585 million to settle their prosecutions for conspiring to fix prices of liquid crystal display panels, which was the second largest amount awarded under the Sherman Act in history.

It then performs a prefix sum computation ( a second loop, over the range of possible keys ) to determine, for each key, the starting position in the output array of the items having that key.

where the first sum ranges over the vertices in the interior of, the second sum is over the boundary vertices, and is the Euler characteristic of.

Each term of the harmonic series is greater than or equal to the corresponding term of the second series, and therefore the sum of the harmonic series must be greater than the sum of the second series.

In spots such as the elbows and knees the second skin is worn off and I realized the aborigines were much darker than they appeared ; ;

Steinberg spoke with warmth and enthusiasm about Italy: `` Rome is my second home.

He must construct transitions so that a dancer who is told to lie prone one second and to leap wildly the next will have some physical preparation for the leap.

The first half of The Charles Men, ending on the climax of the battle of Poltava in 1709, is more dramatically coherent than the second.

Sibylla is pregnant with their second child when she finds the ivory tablet concealed by her husband, and the identities of mother and son are revealed.

The second reason for his popularity is his complete spontaneity with the guitar.

The idea here is one of discharge but this must stand in opposition to a second view, Plato's notion of the arousal of emotion.

There is a second feature of the influences of literature, good literature, on emotional life which may have some special value for our time.

Again omitting recent developments, E.T. Leeds' dictum of 1913 has stood unchallenged: `` So far as archaeology is concerned, there is not the least warrant for the second ( shore occupied by ) of these theories ''.

And the second requirement for convincing people without their knowledge is artistic talent to prepare the words and pictures which persuade by using the principles which the scientists have discovered.

Even to be `` from hope and fear set free '' is at least better than to have lost the first without having got rid of the second.

Obviously, if this club is going to move from second to first in the American League, it will have to show improvement someplace.

Hank Foiles, backed up by Frank House who will be within calling distance in the minors, make up better second line catching than the Birds had all last year, but Gus is still that big man you need when you start talking pennant.

The 15th Street deposit is not to be confused with the nearby famous Mayflower Hotel cypress swamp on 17th Street reported in The Washington Post, August 2, 1955, which was probably formed during the second interglacial period and is therefore much younger.

The second mistake is Tibet.

Even apart from the fact that now at the age of 31 my personal life is being totally disrupted for the second time for no very compelling reason -- I cannot help looking around at the black leather jacket brigades standing idly on the street corners and in the taverns of every American city and asking myself if our society has gone mad.

He is interested in public opinion for two reasons: first, because it is important in itself, and, second, because he knows that the American public cares about a decent respect for the opinions of mankind.

The second capability is represented by our deployed ground, naval, and air forces in essential forward areas, together with ready reserves capable of effecting early emergency reinforcement.

A second fundamental principle is that involved particularly in the present proceeding -- the difference between nighttime and daytime propagation conditions with respect to the standard broadcast frequencies.

A second and also good practice is to shear off the tops, leaving an inch high stub with just a leaf or two on each branch.

I like to shear half my plants at a time, leaving one half of them to blossom while the second half is getting started on its new round of blooming.

If the plants are cared for and protected over the winter, the second year is more prolific than the first.

Then the second Cauchy – Riemann equation ( 1b ) asserts that is irrotational:

In fact, Smale's list contains some of the original Hilbert problems, including the Riemann hypothesis and the second half of Hilbert's sixteenth problem, both of which are still unsolved.

These are unproven ; in 1967, Hooley published a conditional proof for the second conjecture, assuming certain cases of the Generalized Riemann hypothesis.

For Riemann surfaces, Rado's theorem implies that the surface is automatically second countable.

The second definition just compensates the sign and it works together with the second definition of the Riemann tensor ( see e. g. Barrett O ' Neill's Semi-riemannian geometry ).

An improper Riemann integral of the second kind.

The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten:

The Schwarzian derivative and associated second order ordinary differential equation can be used to determine the Riemann mapping between the upper half-plane or unit circle and any bounded polygon in the complex plane, the edges of which are circular arcs or straight lines.

This was proved by Riemann, using a delicate technique of double formal integration ; and showing that the resulting sum has some generalized kind of second derivative using Toeplitz operators.

Equivalently, is conjugate to in if and only if and satisfy the Cauchy – Riemann equations in As an immediate consequence of the latter equivalent definition, if is any harmonic function on the function is conjugate to, for then the Cauchy – Riemann equations are just and the symmetry of the mixed second order derivatives, Therefore an harmonic function admits a conjugated harmonic function if and only if the holomorphic function has a primitive in in which case a conjugate of is, of course, So any harmonic function always admits a conjugate function whenever its domain is simply connected, and in any case it admits a conjugate locally at any point of its domain.

In recent years, pseudomathematicians have devoted their energies to disproving Gödel's second incompleteness theorem ( efforts that fall in the first category mentioned above ) and to proving Fermat's Last Theorem or the Riemann hypothesis using elementary mathematical techniques ( second category ).

Namely, if φ is a Riemann map between D and a second domain E then

If you view the Riemann tensor as a second rank tensor acting on bivectors, the vanishing of invariants is analogous to the fact that a nonzero null vector has vanishing squared length.

As one might expect from the Ricci decomposition of the Riemann tensor into the Weyl tensor plus a sum of fourth-rank tensors constructed from the second rank Ricci tensor and from the Ricci scalar, these two sets of invariants are related: