In physics, the most common orbital descriptions are based on the solutions to the hydrogen atom, where orbitals are given by the product between a radial function and a pure spherical harmonic.

Emotions have also been described as biologically given and a result of evolution because of they provided good solutions to ancient and recurring problems that faced our ancestors.

The first part ( Forward Elimination ) reduces a given system to either triangular or echelon form, or results in a degenerate equation, indicating the system has no unique solution but may have multiple solutions ( rank < order ).

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

A precise set of evaluation criteria, which includes mainly evaluation data and evaluation metrics, enables several teams to compare their solutions to a given NLP problem.

η is the pump efficiency, and may be given by the manufacturer's information, such as in the form of a pump curve, and is typically derived from either fluid dynamics simulation ( i. e. solutions to the Navier-stokes for the particular pump geometry ), or by testing.

where n is a given nonsquare integer and integer solutions are sought for x and y. Trivially, x = 1 and y = 0 always solve this equation.

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,

The wave equation is unchanged under rotations of the spatial coordinates, because the Laplacian operator is invariant under rotation, and therefore one may expect to find solutions that depend only on the radial distance from a given point.

The final equation and one of its solutions is given for each of the 288 problems.

When comparing the predicted expansion rate of the universe as given by the tracker solutions with cosmological data, a main feature of tracker solutions is that one needs four parameters to properly describe the behavior of their equation of state, whereas it has been shown that at most a two-parameter model can optimally be constrained by mid-term future data ( horizon 2015-2020 ).

* For specific optimization problems and problem instances, other optimization algorithms may find better solutions than genetic algorithms ( given the same amount of computation time ).

As such, consideration should be given to using frits in such solutions and they should be rapidly and thoroughly rinsed when cleaning the glass with bases like KOH.

Bernoulli also wrote a large number of papers on various mechanical questions, especially on problems connected with vibrating strings, and the solutions given by Brook Taylor and by Jean le Rond d ' Alembert.

Somewhat surprisingly, given their wide range of practical uses, mathematicians have not yet proven that in three dimensions solutions always exist ( existence ), or that if they do exist, then they do not contain any singularity ( smoothness ).

Additionally, it is not fundamental for modern algebra ; its name was given at a time when the study of algebra was mainly concerned with the solutions of polynomial equations with real or complex coefficients.

It is also a name sometimes given to solutions of carbon dioxide in water ( carbonated water ), because such solutions contain small amounts of H < sub > 2 </ sub > CO < sub > 3 </ sub >.

Standard home solutions such as salted rice water, salted yogurt drinks, vegetable and chicken soups with salt can be given.

Two constructions, called the category of Eilenberg – Moore algebras and the Kleisli category are two extremal solutions to the problem of constructing an adjunction that gives rise to a given monad.

It is beneficial to confront the students with unexpected solutions to problems posed in Lat Sao, as an exercise and to demonstrate that each Lat Sao drill is just one of very many possible solutions to a given problem.

Then the solutions to the homogeneous system given in parametric form with x < sub > 3 </ sub >, x < sub > 5 </ sub >, and x < sub > 6 </ sub > as free variables are

Since these satisfy the Schrödinger equation-the solutions to Schrödinger's equation for a given situation will not only be the plane waves used to obtain it, but any wavefunctions which satisfy the Schrödinger's equation prescribed by the system, in addition to the relevant boundary conditions.

Among current problems that require solutions, participated in by citizens as well as government, are:

The solutions were not arrived at by any theoreticians of the Karl Marx stripe but by men of government -- lawyers, most of them -- and men of business.

This has an interesting analogy with the assumption stated by Philippoff that `` the deformational mechanics of elastic solids can be applied to flowing solutions ''.

These cells were thawed at 37-degrees-C for 30 min and were deglycerolized by alternately centrifuging and mixing with descending concentrations of glycerol solutions ( 20, 18, 10, 8, 4 and 2% ).

more effective solutions include explanations, the example of another child, or conditioning by associating the feared object, place, or person with something pleasant.

One of the most desirable solutions is achieved by the use of a non-linear amplifier for Af.

Heydrich, in opening the Conference, followed the reasoning and even the phraseology of the order issued earlier by Goering which authorized the Final Solution as `` a complement to '' previous `` solutions '' for eliminating the Jews from German living space through violence, economic strangulation, forced emigration, and evacuation.

Changemaker Central has signature programs, including Innovation Challenge and 10, 000 Solutions, that support students in their journey to become changemakers by creating communities of support around new solutions / ideas and increasing access to early stage seed funding.

The slope field of F ( x ) = ( x < sup > 3 </ sup >/ 3 )-( x < sup > 2 </ sup >/ 2 )- x + c, showing three of the infinitely many solutions that can be produced by varying the Constant of integration | arbitrary constant C.

This view was held in particular by Vladimir Belinsky, Isaak Khalatnikov, and Evgeny Lifshitz, who tried to prove that no singularities appear in generic solutions.

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y ( x ) of Bessel's differential equation:

The Bessel functions of the second kind, denoted by Y < sub > α </ sub >( x ), occasionally denoted instead by N < sub > α </ sub >( x ), are solutions of the Bessel differential equation.

Another important formulation of the two linearly independent solutions to Bessel's equation are the Hankel functions H < sub > α </ sub >< sup >( 1 )</ sup >( x ) and H < sub > α </ sub >< sup >( 2 )</ sup >( x ), defined by:

Because Bessel's equation becomes Hermitian ( self-adjoint ) if it is divided by x, the solutions must satisfy an orthogonality relationship for appropriate boundary conditions.

Cross section view of the structures that can be formed by phospholipids in aqueous solutions

DWNT synthesis on the gram-scale was first proposed in 2003 by the CCVD technique, from the selective reduction of oxide solutions in methane and hydrogen.

Because the size of the dispersed phase may be difficult to measure, and because colloids have the appearance of solutions, colloids are sometimes identified and characterized by their physico-chemical and transport properties.

The cosmological solutions of general relativity were found by Alexander Friedmann in the early 1920s.

In particular, the 1997 observation by Skulan and DePaolo that calcium minerals are isotopically lighter than the solutions from which the minerals precipitate is the basis of analogous applications in medicine and in paleooceanography.

The total energy is determined by approximate solutions of the time-dependent Schrödinger equation, usually with no relativistic terms included, and by making use of the Born – Oppenheimer approximation, which allows for the separation of electronic and nuclear motions, thereby simplifying the Schrödinger equation.