The runes for a, n, s and t are the old Danish unsimplified forms which should have been out of use for a long time the 14th century ... I suggest that posited 14th centurycreator must at some time or other in his life have been familiar with an inscription ( or inscriptions ) composed at a time when these unsimplified forms were still in use " and that he " was not a professional runic scribe before he left his homeland ".

Traditional Chinese characters are also known as Hanja in Korean ( in the 20th century almost completely replaced with Hangul ), and many Kanji ( used in Japanese ) are unsimplified.

However, the use of runes in the medieval period is far from systematic or coherent: in defence of the possibility of the genuineness of the stone, S. N. Hagen wrote: " The Kensington alphabet is a synthesis of older unsimplified runes, later dotted runes, and a number of Latin letters ...

Solutions of Einstein's equations that violate this inequality exist, but they do not possess an event horizon.

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 ).

Methods that do not include any empirical or semi-empirical parameters in their equations – being derived directly from theoretical principles, with no inclusion of experimental data – are called ab initio methods.

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

Although several versions of many-worlds have been proposed since Hugh Everett's original work, they all contain one key idea: the equations of physics that model the time evolution of systems without embedded observers are sufficient for modelling systems which do contain observers ; in particular there is no observation-triggered wave function collapse which the Copenhagen interpretation proposes.

For fluids which are sufficiently dense to be a continuum, do not contain ionized species, and have velocities small in relation to the speed of light, the momentum equations for Newtonian fluids are the Navier-Stokes equations, which is a non-linear set of differential equations that describes the flow of a fluid whose stress depends linearly on velocity gradients and pressure.

Since Einstein's equations are non-linear, arbitrarily strong gravitational waves do not obey linear superposition, making their description difficult.

Linear equations do not include exponents.

Since energy is dependent on reference frame ( upon the observer ) it is convenient to formulate the equations of physics in a way such that mass values are invariant ( do not change ) between observers, and so the equations are independent of the observer.

Finally, any phenomenon involving individual photons, such as the photoelectric effect, Planck's law, the Duane – Hunt law, single-photon light detectors, etc., would be difficult or impossible to explain if Maxwell's equations were exactly true, as Maxwell's equations do not involve photons.

In the special theory of relativity, Einstein keeps the postulate that the equations of motion do not depend on the reference frame, but assumes that the speed of light is invariant.

It also accounted for anomalous observations, including the properties of black body radiation, that other physicists, most notably Max Planck, had sought to explain using semiclassical models, in which light is still described by Maxwell's equations, but the material objects that emit and absorb light, do so in amounts of energy that are quantized ( i. e., they change energy only by certain particular discrete amounts and cannot change energy in any arbitrary way ).

Work on Hilbert's 10th problem led in the late twentieth century to the construction of specific Diophantine equations for which it is undecidable whether they have a solution, or even if they do, whether they have a finite or infinite number of solutions.

It is straightforward to find algebraic equations the solutions of which have the topology of a Möbius strip, but in general these equations do not describe the same geometric shape that one gets from the twisted paper model described above.

Because relativity treats space and time as a whole, a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically, as they do in the Maxwell equations that govern the behavior of light — the equations must be differentially of the same order in space and time.

For example, equations used to develop models of the origin do not in themselves explain how the conditions of the universe that the equations model came to be in the first place.

Ancient Greek mathematicians calculated not with quantities and equations as we do today, but instead they used proportionalities to express the relationship between quantities.

We have chosen to give it at the end of the section since it deals with differential equations and thus is not purely linear algebra.

-- 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.

If the differential equations are nonlinear and have a known solution, it may be possible to linearize the nonlinear differential equations at that solution.

If the resulting linear differential equations have constant coefficients one can take their Laplace transform to obtain a transfer function.

Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds.

Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all 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.

Diophantine geometry, which is the application of techniques from algebraic geometry in this field, has continued to grow as a result ; since treating arbitrary equations is a dead end, attention turns to equations that also have a geometric meaning.

Looking at the original charges and currents that are the cause of the wave brings into play terms involving charges and currents (" sources ") in Maxwell ’ s equations that produce a local type of electromagnetic field near sources that does not have the behavior of EMR.

The is related to Hilbert's tenth problem, which asks for an algorithm to decide whether Diophantine equations have a solution.

The simplest equations to solve are linear equations that have only one variable.

All quadratic equations will have two solutions in the complex number system, but need not have any in the real number system.

Therefore, a number of more accurate equations of state have been developed for gases and liquids.

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.

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.

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

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.

Otherwise the more general compressible flow equations must be used.