With the development of efficient computer technology in the 1940s, the solutions of elaborate wave equations for complex atomic systems began to be a realizable objective.

With this additional condition — the covariant divergence of the energy – momentum tensor, and hence of whatever is on the other side of the equation, is zero — the simplest set of equations are what are called Einstein's ( field ) equations:

With the advent of general relativity, the classical equations of motion became modified.

With the inclusion of a variable for the density of these magnetic charges, say ρ < sub > m </ sub >, there will also be a " magnetic current density " variable in the equations, j < sub > m </ sub >.

With the addition of the three compatibility equations the number of independent equations is reduced to three, matching the number of unknown displacement components.

With the point as the midpoint of the arc, we may write the following two equations:

With Fourier transformed initial conditions and forcing, this coupled system of ordinary differential equations may be integrated in time ( using, e. g., a Runge Kutta technique ) to find a solution.

With C < sub > 1 </ sub >, use the parametric equations: x

With C < sub > 3 </ sub >, use the parametric equations: x

With this mentality, it also isn't too surprising that general non-autonomous equations of second order can't be solved explicitly, since these can also be chaotic ( an example of this is a periodically forced pendulum ).

With this definition of the dissipation function the precise statement of the FT simply replaces entropy production with the dissipation function in each of the FT equations above.

With his field equations Einstein ensured that general relativity is consistent with this conservation condition.

With the addition theorems above and for a given k with 0 < k < 1 they therefore are solutions to the following nonlinear ordinary differential equations:

With these criteria in mind and based on then prevailing psychophysical theory Roberts proposed the following equations:

With his brother Heinz, he made a significant contribution to understanding electromagnetic properties of superconductors ( see London equations ).

With some change of symbols ( and units ): r → J, R → − E and the material constant E < sup >− 2 </ sup > → 4π ε < sub > r </ sub > ε < sub > 0 </ sub > these equations take the familiar form:

With these definitions of coordinate rotation and reflection, the following four equations are true:

With adjustments for velocity changes, the initial estimate of the hypocenter is made, then a series of linear equations is set up, one for each station.

With no charges or currents, these field equations take the form of wave equations

With his students, he has developed a widely-used mathematical software package ( SPARSPAK ) for solving sparse systems of equations and sparse least squares problems.

With these conservation laws, Miura showed a connection ( called the Miura transformation ) between solutions of the KdV and MKdV equations.

With the above results we can make the following remarks about the graph of F.

With respect to this type of item, one might properly apply the language of the Committee Report, quoted above, which cautions against using section 381 as a basis for treating other tax attributes not mentioned therein.

With naked gas jets below and leaky windows above, enough to ruin wall paintings in any medium, they have survived, in a building long unheated in winter, hot and damp under the iron dome in summer.

With cash receipts from marketings expected to be slightly above 1960, farmers' gross income is estimated at $39.5 billion, $1.5 billion above 1960's record high.

With a $ 1. 92-billion-a-year tourism industry, Costa Rica stands as the most visited nation in the Central American region, with 1. 9 million foreign visitors in 2007, thus reaching a rate of foreign tourists per capita of 0. 46, one of the highest in the Caribbean Basin, and above other popular destinations such as Mexico ( 0. 21 ), Dominican Republic ( 0. 38 ), and Brazil ( 0. 03 ).

With a vastly inferior goal difference and with the two teams directly above them ( Southampton and Nottingham Forest ) still having to play each other, it was effectively an impossible task for Charlton to avoid relegation.

With rates above 75-120 frames / s no improvement in realism or smoothness is perceivable due to the way the eye and brain process images.

With the above product, the set of all homotopy classes of loops with base point x < sub > 0 </ sub > forms the fundamental group of X at the point x < sub > 0 </ sub > and is denoted

With the above parameterization the cross section is a 2: 1 Lissajous curve.

With an elevation of above sea level, the lake reaches a depth of.

With a price increase, price elasticity tends to increase, and in the optimum case above it will be greater than one for most customers.

With the opponent's right arm across his own torso, the athlete uses his left hand to keep the pressure on the opponent's right arm by grabbing and pressing down on it just above the wrist.

With the above definition in each step always a single symbol ( top of the stack ) is popped, replacing it with as many symbols as necessary.

: " With proper use of the analytical results, the rotor efficiency using laminar flow can be very high, even above 95 %.

With rule 1 above, this rule ensures that users can take on only roles for which they are authorized.

With this additional kinetic energy the exciton may lie above the band-gap.

With solid plugs, a tube or area exists above and below the bevel and turning the plug controls access.

With some chemicals, above all with caustic alkalis such as sodium hydroxide, there may be necrosis of the conjunctiva with a deceptively white eye due to vascular closure, followed by sloughing of the dead epithelium.

With the requirement of consent, the fourth type of interpretation above would usually not fall under the definition of " victimless crime ".

With two mallets in one hand, by putting one mallet beneath the key and another above it, and by moving the wrist vertically, the percussionist can make a rolling sound with just one hand as opposed to using both.

With the social and economic development of the 19th and 20th centuries, and above all, the massive migrations to the Americas and Western Europe, that jus soli was established in a greater and greater number of countries.

With the highest point of four metres above sea level, the threat of severe cyclones is extremely disturbing, and severe water shortages will further threaten the livelihoods of people in many islands.

With the development of low self-discharge NiMHs ( see section above ), many occasional-use and very low-power applications are now candidates for NiMH cells.