-- The theory of elasticity of Gaussian networks has been developed on a more general basis and the equations of state relating variables of pressure, volume, temperature, stress and strain have been precisely formulated.

The eleventh century Persian mathematician Omar Khayyám saw a strong relationship between geometry and algebra, and was moving in the right direction when he helped to close the gap between numerical and geometric algebra with his geometric solution of the general cubic equations, but the decisive step came later with Descartes.

In general, linear equations involving x and y specify lines, quadratic equations specify conic sections, and more complicated equations describe more complicated figures.

More general equations of fluid flow-the Euler equations-were published by Leonhard Euler in 1757.

Dark energy in its simplest formulation takes the form of the cosmological constant term in Einstein's field equations of general relativity, but its composition and mechanism are unknown and, more generally, the details of its equation of state and relationship with the Standard Model of particle physics continue to be investigated both observationally and theoretically.

Ten years later, Alexander Friedmann, a Russian cosmologist and mathematician, derived the Friedmann equations from Albert Einstein's equations of general relativity, showing that the Universe might be expanding in contrast to the static Universe model advocated by Einstein at that time.

However, the law of mass action is valid only for concerted one-step reactions that proceed through a single transition state and is not valid in general because rate equations do not, in general, follow the stoichiometry of the reaction as Guldberg and Waage had proposed ( see, for example, nucleophilic aliphatic substitution by S < sub > N </ sub > 1 or reaction of hydrogen and bromine to form hydrogen bromide ).

The equations of motion governing the universe as a whole are derived from general relativity with a small, positive cosmological constant.

As with much algorithmic music, and algorithmic art in general, more depends on the way in which the parameters are mapped to aspects of these equations than on the equations themselves.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

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

The same type of construction works in the general case of congruence equations.

While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations ( beyond the theory of quadratic forms ) was an achievement of the twentieth century.

The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as equivalently described as recursively enumerable.

Such equations do not have a general theory ; particular cases such as Catalan's conjecture have been tackled.

Electromagnetic waves as a general phenomenon were predicted by the classical laws of electricity and magnetism, known as Maxwell's equations.

By 1951, 87 subroutines in the following categories were available for general use: floating point arithmetic ; arithmetic operations on complex numbers ; checking ; division ; exponentiation ; routines relating to functions ; differential equations ; special functions ; power series ; logarithms ; miscellaneous ; print and layout ; quadrature ; read ( input ); nth root ; trigonometric functions ; counting operations ( simulating repeat until loops, while loops and for loops ); vectors ; and matrices.

The unsimplified equations do not have a general closed-form solution, so they are primarily of use in Computational Fluid Dynamics.

Otherwise the more general compressible flow equations must be used.

But, in general, we may argue that the student can direct the primary emphasis of his attention toward one or the other.

The publication of Father Connolly's The Man Has Wings has made more of the group available in print so that a general picture of what it contained can now be had without difficulty.

Meanwhile he has been thinking about the facts surrounding the problem, facts which he knows can never be complete, and the general background, much of which has already been lost to history.

For the policy officer will know that action can almost never be secret and that in general the effectiveness of policy will be conditioned by the readiness of the country to sustain it.

In Missouri ( which we are including in our general Midwest region ) you can glance into Mark Twain's birthplace at Hannibal, see the landmarks of his life and writings and visualize where Huck Finn hatched his boyish mischief.

What can we do with the general T??

This can only be for one of two reasons: either the two are quite different and will require totally different theory ( and hence techniques ), or our existing theories are insufficiently general.

While this influence is a complex matter, depending upon personality factors in the individual as well as upon his social-class experience, there probably are some general statements about social-class background and educational policy that can be made with a fair degree of truth.

We can see the general characteristics of the earlier decade if we look at two poems of very different qualities: `` Revulsion '' ( 1866 ) and `` Neutral Tones '' ( 1867 ).

Here again laboratory approaches are being evolved, for it is recognized how `` elastic '' these readings can be, how they can apply to many people, and are often stated in general terms all too easily applied to any individual's own case.

Motion-picture exhibitions took place in stores in a general atmosphere like that of the penny arcade which can still be found in such urban areas as Times Square.

The tendency to reciprocate can even generalize so people become more helpful toward others in general after being helped.

Direct reciprocity and cooperation in a group can be increased by changing the focus and incentives from intra-group competition to larger scale competitions such as between groups or against the general population.

Altruism that ultimately serves selfish gains is thus differentiated from selfless altruism, but the general conclusion has been that empathy-induced altruism can be genuinely selfless.

The adoption of a standard recognizable type for a long time, is probably because nature gives preference in survival of a type which has long be adopted by the climatic conditions, and also due to the general Greek belief that nature expresses itself in ideal forms that can be imagined and represented.

The exact number and placement of Endosymbiotic theory | endosymbiotic events is currently unknown, so this diagram can be taken only as a general guide It represents the most parsimonious way of explaining the three types of endosymbiotic origins of plastids.

general case, " The analysis of variance can also be applied to

Saturated hydrocarbons can also combine any of the linear, cyclic ( e. g., polycyclic ) and branching structures, and they are still alkanes ( no general formula ) as long as they are acyclic ( i. e., having no loops ). They also have single covalent bonds between their carbons.

While the Ural – Altaic hypothesis can still be found in encyclopedias, atlases, and similar general references, it has not had any adherents in the linguistics community for decades.

This is not the most general situation of a Cartesian product of a family of sets, where a same set can occur more than once as a factor ; however, one can focus on elements of such a product that select the same element every time a given set appears as factor, and such elements correspond to an element of the Cartesian product of all distinct sets in the family.

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

For example, in some groups, the group operation is commutative, and this can be asserted with the introduction of an additional axiom, but without this axiom we can do quite well developing ( the more general ) group theory, and we can even take its negation as an axiom for the study of non-commutative groups.