This case is called Potential flow and allows the differential equations used to be a simplified version of the governing equations of fluid dynamics, thus making available to the aerodynamicist a range of quick and easy solutions.

This is in fact a special case of a more general result on the regularity of solutions of hypoelliptic partial differential equations.

In this case, the solutions are and.

The reduced cubic equation of state yields in that case 3 solutions.

In the case of plane geometry ( valid for small areas on the Earth's surface ) the solutions to both problems reduce to simple trigonometry.

Many solutions were proposed for how to determine longitude at the end of an exploratory sea voyage and hence the longitude of the place that was visited ( in case one would want to revisit it, place it on a map, or more urgently, avoid known marine hazards ).

When this is the case, and x < sub > 0 </ sub > is one solution of ( 1 ), then the set of all solutions is given by

In this case, " event horizons disappear " means when the solutions are complex for, or < math >

A more challenging task, typically requiring a computer search, is to count the total number of solutions in each case.

Electrons behave as beams of energy, and in the presence of a potential U ( z ), assuming 1-dimensional case, the energy levels ψ < sub > n </ sub >( z ) of the electrons are given by solutions to Schrödinger ’ s equation,

It is nearly always the case that measurements are made at nominally 1 atmosphere ( 101. 325 kPa the variations caused by changing weather patterns ) but as specific gravity usually refers to highly incompressible aqueous solutions or other incompressible substances ( such as petroleum products ) variations in density caused by pressure are usually neglected at least where apparent specific gravity is being measured.

In that case the remainder of the Laplacian can be evaluated to its characteristic equation by considering solutions of the form

In the general case, if a complete basis set of solutions is chosen, any real laser beam can be described as a superposition of solutions from this set.

Instead of describing a process which can only evolve in one way ( as in the case, for example, of solutions of an ordinary differential equation ), in a stochastic or random process there is some indeterminacy: even if the initial condition ( or starting point ) is known, there are several ( often infinitely many ) directions in which the process may evolve.

In this case, the solutions are the pairs of the form ( 5, 2kπ ) and (− 5 ,( 2k + 1 ) π ), where k ranges over all integers.

Although the issue of existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the Picard – Lindelöf theorem, that is far from the case for partial differential equations.

where HA is a generic acid that dissociates by splitting into A < sup >−</ sup >, known as the conjugate base of the acid, and the hydrogen ion or proton, H < sup >+</ sup >, which, in the case of aqueous solutions, exists as a solvated hydronium ion.

Greenpeace's Cool IT Leaderboard of February 2012 " evaluates global IT companies on their leadership in the fight to stop climate change " and ranks Fujitsu 3rd out of 21 leading manufacturers, on the strength of " well-developed case study data of its solutions with transparent methodology " and " out in the Leaderboard for scoring high in the Future Savings Goal criterion.

Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave.

The null space of A is precisely the set of solutions to these equations ( in this case, a line through the origin in R < sup > 3 </ sup >).

This case describes the standing wave solutions of the time-dependent equation, which are the states with definite energy ( instead of a probability distribution of different energies ).

For the case of electrolyte solutions, see the article: Conductivity ( electrolytic ).

The solutions given by the two lists may be different, in which case the solver must decide at the outset which list they are going to follow, or the solutions may be identical, in which case the straight clues offer additional help for a solver having difficulty with the cryptic clues.

But in our case -- and neither my wife nor I have extreme views on integration, nor are we given to emotional outbursts -- the situation has ruined one or two valued friendships and come close to wrecking several more.

( There are two receivers in case one should be dropped and damaged.

Some years ago Julian Huxley proposed to an audience made up of members of the British Association for the Advancement of Science that `` man's supernormal or extra-sensory faculties are ( now ) in the same case as were his mathematical faculties during the ice age ''.

But, here again, comparative benchmarks are lacking, and we do not know, in any case, what measure of profoundity and intensity to expect from healthy, young, secure and relatively inexperienced persons ; ;

The novels and stories like Pohl's Drunkard's Walk ( 1960 ), with the focus on adventure and with the dystopian elements only a dim background -- in this case an uneasy, overpopulated world in which the mass of people do uninteresting routine jobs while a carefully selected, university-trained elite runs everything -- are in all likelihood as numerous as dystopias.

That fact is very clearly illustrated in the case of the many present-day intellectuals who were Communists or near-Communists in their youth and are now so extremely conservative ( or reactionary, as many would say ) that they can define no important political conviction that does not seem so far from even a centrist position as to make the distinction between Mr. Nixon and Mr. Khrushchev for them hardly worth noting.

The very rules and regulations in every city are the primary case of slum conditions.

Our litterbug ordinances are not enforced and I have yet to read of a conviction in a littering case.

The replies from each individual town are not given in detail because the questions asked the personal opinion of the several assessors and are not necessarily the established policy of the town in each case.

All payments authorized under Section 7 of this Title shall be disbursed from the proper fund, as the case may be, and all amounts covered into the Treasury to the credit of the aforesaid funds are hereby permanently appropriated for the making of the payments authorized by Section 7 of this Title.

As is the case with the allotment provisions for support of vocational rehabilitation services, the matching requirements are also based on a statutory formula.

The District Courts, in the framing of equitable decrees, are clothed ' with large discretion to model their judgements to fit the exigencies of the particular case.

The same restrictions apply after local sunset in the case of class 3, stations operating on regional channels, which after that time are required to operate under nighttime restrictions in order to protect each other.

This conformational entropy is, in this case, equal to the usual entropy, for there are no other changes or other energies involved.

As is the case with the surface-active agent, the requirements for builders to be used in detergent compositions for washing textiles are also high.

The reader will find it helpful to think of the special case when the primes are of degree 1, and even more particularly, to think of the proof of Theorem 10, a special case of this theorem.

In each case, lines of the bundles are transformed by involutions within the pencils they determine with the multiple secant.

In the first case the fixed elements within each pencil are the multiple secant and the line joining the vertex, P, to the intersection of **zg and the plane of the pencil which does not lie on the multiple secant.

The multiple secants, of course, are exceptional and in each case are transformed into cones of order Af.

Thus, in the example cited above Fromm rests his whole case on the premise that the workers are being deprived unconsciously, unknowingly, of fulfillment, and then supports this with survey data reporting conscious, experienced frustrations.

Presumably the interfacial bond strength and gross cohesive properties are identical in each case.

Merchant has found that the same basic relationships which describe the geometry and force systems in the case of the cutting mechanism can also be applied to the discontinuous chip formation provided the proper values of instantaneous shear angle and instantaneous chip thickness or cross-sectional area are used.