In particular, in the Born – Oppenheimer and harmonic approximations, i. e. when the molecular Hamiltonian corresponding to the electronic ground state can be approximated by a harmonic oscillator in the neighborhood of the equilibrium molecular geometry, the resonant frequencies are determined by the normal modes corresponding to the molecular electronic ground state potential energy surface.

The corresponding equation for pure in equilibrium with its ( pure ) vapor is:

If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.

For example, the water phase diagram has a triple point corresponding to the single temperature and pressure at which solid, liquid, and gaseous water can coexist in a stable equilibrium.

Lucas assumed that equilibrium corresponded to a unique " full employment " level ( potential output ) -- corresponding to a unique NAIRU or natural rate of unemployment.

This is done by assuming that approximate thermodynamic equilibrium can exist " locally ", and that a corresponding " local Fermi level " can be defined.

In the first-order phase transition the range of (< i > P, V, T </ i >), where the liquid phase and the gas phase are in equilibrium, it does not exhibit the empirical fact that < var > p </ var > is constant as a function of < i > V </ i > for a given temperature: although this behavior can be easily inserted into the van der Waals model ( see Maxwell's correction below ), the result is no longer a simple analytical model, and others ( such as those based on the principle of corresponding states ) achieve a better fit with roughly the same work.

The corresponding point on the potential energy curve is higher than that corresponding to the equilibrium position ( x = 0 ).

The corresponding equilibrium equation is:

This is due to a chemico-physical effect: the adsorption to the soil, which holds the contaminant back and does not allow it to progress until the quantity corresponding to the chemical adsorption equilibrium has been adsorbed.

In this situation the particles and their corresponding antiparticles do not achieve thermal equilibrium due to rapid expansion decreasing the occurrence of pair-annihilation.

The position of the equilibrium can be controlled by the acidity of the alcohol ; for example phenols typically react with alkoxides to release alcohols, giving the corresponding phenoxide.

Mutarotation is the change in the optical rotation that occurs by epimerization ( that is the change in the equilibrium between two epimers, when the corresponding stereocenters interconvert ).

At thermodynamic equilibrium, the upper state will have a lower or equivalent population and the corresponding anti-Stokes spectrum will be less intense.

For a system in equilibrium ( i. e. time-independent ) in D spatial dimensions, the corresponding statistical field theory is formally similar to a D-dimensional CFT.

As an example, might represent currency plus deposits in checking and savings accounts held by the public, real output ( which equals real expenditure in macroeconomic equilibrium ) with the corresponding price level, and the nominal ( money ) value of output.

When the level of risk in one part of the individual's life changes, there will be a corresponding rise or fall in risk elsewhere to bring the overall risk back to that individual's equilibrium.

In statics, Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and non-collinear forces, which keeps an object in static equilibrium, with the angles directly opposite to the corresponding forces.

At equilibrium three interfacial tensions corresponding to solid / gas ( γ < sub > SG </ sub >), solid / liquid ( γ < sub > SL </ sub >), and liquid / gas ( γ < sub > LG </ sub >) interfaces are counterbalanced.

He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves.

Equations for coefficients corresponding to ratios of the electric field complex-valued amplitudes of the waves ( not necessarily real-valued magnitudes ) are also called " Fresnel equations ".

; Theoretical chemical kinetics: Theoretical study of the dynamical systems associated to reactive chemicals and their corresponding differential equations.

It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding Galois group.

The hypothesis suggests that worlds corresponding to different sets of initial conditions, physical constants, or altogether different equations should be considered real.

Summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of wavefunction solutions.

It is possible to make an enumeration operator corresponding to these equations, but it will not be a recursive operator.

it is probably worth mentioning that certain non-negative ratios are readily encountered, which are appropriately labelled by similarly related indices, and which do satisfy equations corresponding to 1., 2. and 3., but which nevertheless don't satisfy the Wigner – d ' Espagnat inequality.

Hamilton's equations have another advantage over Lagrange's equations: if a system has a symmetry, such that a coordinate does not occur in the Hamiltonian, the corresponding momentum is conserved, and that coordinate can be ignored in the other equations of the set.

In the Lagrangian framework, of course the result that the corresponding momentum is conserved still follows immediately, but all the generalized velocities still occur in the Lagrangian-we still have to solve a system of equations in n coordinates.

He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves.

There may be several such diagonal lines, each corresponding to one or more branches of the curve, and the approximate equations of the branches may be found by applying this method to each line in turn.

Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts.

In the language of general relativity, de Sitter space is the maximally symmetric, vacuum solution of Einstein's field equations with a positive ( repulsive ) cosmological constant ( corresponding to a positive vacuum energy density and negative pressure ).

These formulas give the explicit numbers of rational curves of all degrees in a large class of Calabi – Yau manifolds, in terms of the Picard – Fuchs equations of the corresponding mirror manifolds.

In other respects, the properties of monic polynomials and of their corresponding monic polynomial equations depend crucially on the coefficient ring A.

For example, given the four points the pencil of conics through them can be parameterized as yielding the following pencil ; in all cases the center is at the origin :< ref group =" note "> A simpler parametrization is given by which are the affine combinations of the equations and corresponding the parallel vertical lines and horizontal lines, and results in the degenerate conics falling at the standard points of

The equations represent conservation of mass ( continuity ), momentum, and energy, corresponding to the Navier – Stokes equations with zero viscosity and heat conduction terms.

This introduces an additional factor of 8π into Newton's law of universal gravitation but simplifies Einstein's equations, the Einstein-Hilbert action, the Friedmann equations and the Newtonian Poisson equation by removing the corresponding factor.

On the contrary, even in the heart of `` the Bible belt '' itself, as can be attested by any one who is called to work there, the industrial and technological revolutions have long been under way, together with the corresponding changes in man's picture of himself and his world.

In spaces that are compact in this latter sense, it is often possible to patch together information that holds locally — that is, in a neighborhood of each point — into corresponding statements that hold throughout the space, and many theorems are of this character.

When a switch is pressed, it connects the corresponding x and y lines together.

Start with a square, and then glue together corresponding coloured edges, in the following diagram, so that the arrows match.

The non-abelian character of the SU ( 3 ) corresponds thereby to the non-trivial " chemical links “, which glue different loop segments together, and " asymptotic freedom " means in the polymer analogy simply the fact that in the short-wave limit, i. e. for ( where R < sub > c </ sub > is a characteristic correlation-length for the glued loops, corresponding to the above-mentioned " bag radius ", while λ < sub > w </ sub > is the wavelength of an excitation ) any non-trivial correlation vanishes totally, as if the system had crystallized.

Such a neighborhood, together with the corresponding homeomorphism, is known as a ( coordinate ) chart.

The spinor / quaternion representation of rotations in 3D is becoming increasingly prevalent in computer geometry and other applications, because of the notable brevity of the corresponding spin matrix, and the simplicity with which they can be multiplied together to calculate the combined effect of successive rotations about different axes.

Other provinces corresponding to modern Ecuador, the eastern and southern parts of today's Venezuela, and Panama came together in a political unit under the jurisdiction of Bogotá, confirming that city as one of the principal administrative centers of the Spanish possessions in the New World, along with Lima and Mexico City.

Rings, for example, can be viewed as abelian groups ( corresponding to addition ) together with a second operation ( corresponding to multiplication ).

As the dynamic linker loads modules and joins them together, it writes actual addresses into the IAT slots, so that they point to the memory locations of the corresponding library functions.

This " bank " or kisad, together with the corresponding western bank of the Tigris ( according to Fritz Hommel the modern Shatt el-Uai ), gave its name to the land of Chesed, whence the Kasdim / Kasdin of the Old Testament.

He compared them to a living being, with physics the meat and blood, logic the bones and tendons holding the organism together, and finally ethics – the most important part – corresponding to the soul.

Ixchel figures in a Verapaz myth related by Las Casas, according to which she, together with her spouse, Itzamna, had thirteen sons, two of whom ( probably corresponding to the Howler Monkey Gods ) created heaven and earth and all that belongs to it.

In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together.

Since the product of diagonal matrices amounts to simply multiplying corresponding diagonal elements together, the power k of a diagonal matrix A will have entries raised to the power.

For instance, XLR connectors have a notch to ensure proper orientation, while Mini-DIN plugs have a plastic projection, which fits into a corresponding hole in the socket and prevent different connectors from being pushed together ( they also have a notched metal skirt to provide secondary keying ).

* Moroccans also give one kiss on each cheek ( to corresponding genders ) together with the handshake.

Normally, these are paired structures, but in birds, crocodilians, and some cartilaginous fishes, one or the other side fails to develop ( together with the corresponding ovary ), and only one functional oviduct is found.

One computes holonomies of the connection along every link of the graph, determines representation matrices corresponding to every link, multiplies all matrices and intertwiners together, and contracts indices in a prescribed way.

Sensations which are often associated together become each associated with the ideas corresponding to the others ; and the ideas corresponding to the associated sensations become associated together, sometimes so intimately that they form what appears to be a new simple idea, not without careful analysis resolvable into its component parts.