This reduction has been accomplished by the general methods of linear algebra, i.e., by the primary decomposition theorem.

which in turn is a special case of the general linear model.

* linear ( general formula ) wherein the carbon atoms are joined in a snake-like structure

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.

* In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL ( V ).

A bilinear transformation, like any binary function, can be interpreted as a function from X × Y to Z, but this function in general won't be linear.

In general, a linear mapping on a normed space is continuous if and only if it is bounded on the closed unit ball.

* More generally, compact groups such as an orthogonal group are compact, while groups such as a general linear group are not.

Context-sensitive grammars are more general than context-free grammars but still orderly enough to be parsed by a linear bounded automaton.

This generated a flurry of interest in the topic, during which Maxwell's classmate Edward John Routh generalized Maxwell's results for the general class of linear systems.

Then is a derivation and is linear, i. e., and, and a Lie algebra homomorphism, i. e.,, but it is not always an algebra homomorphism, i. e. the identity does not hold in general.

Consider first-order linear ODEs of the general form:

For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about endomorphisms in any category.

The Earley parser executes in cubic time in the general case, where n is the length of the parsed string, quadratic time for unambiguous grammars, and linear time for almost all LR ( k ) grammars.

The general form of a linear equation with one variable, can be written as:

In general, a quadratic equation can be expressed in the form, where is not zero ( if it were zero, then the equation would not be quadratic but linear ).

* He constructed the general linear group over a prime field, GL ( ν, p ) and computed its order, in studying the Galois group of the general equation of degree p < sup > ν </ sup >.

However the general concept of linear filtering is broader, also used in statistics, data analysis, and mechanical engineering among other fields and technologies.

Von Baer ’ s embryo drawings display that individual development proceeds from general features of the developing embryo in early stages through differentiation into special features specific to the species, establishing that linear evolution could not occur.

Any field may be used as the scalars for a vector space, which is the standard general context for 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.

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.

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.

Then in 1875, apparently in response to the nationalizing influence of the Civil War, Congress first gave the lower federal courts general authority -- concurrently with state tribunals -- to decide cases involving federal-right questions.

MacKie's theory has fallen from favour too, mainly because from the 1970s there was a general move away from ' diffusionist ' explanations in archaeology towards those involving exclusively indigenous development.

In a number of modern states, circumcision is presumed to be legal, but under certain circumstances, more general laws, such as laws about assault or child custody, may sometimes be interpreted as applying to situations involving circumcision.

In a more general form, the first law describes the internal energy with additional terms involving the chemical potential and the number of particles of various types.

There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory ; this article gives an overview of the available techniques and some of their general properties, while the specific algorithms are described in subsidiary articles linked below.

This enables the network to offer general traffic alongside virtual “ private ” network paths for projects, such as the Large Hadron Collider, which have particular requirements involving dedicated bandwidth, security and flexibility.

Beginning as a religious conflict between Protestants and Catholics in Bohemia, it gradually developed into a general war involving much of Europe, for reasons not necessarily related to religion.

Prevalence is difficult to generalize, but research has estimated 10 – 15 % of the general population as having at least one such sexual contact, with less than 2 % involving intercourse or attempted intercourse.

Some general film festivals accept machinima, and game companies, such as Epic Games, Blizzard Entertainment and Jagex, have sponsored contests involving it.

The amount and type of carbohydrates consumed, along with some types of amino acid, can influence processes involving insulin, glucagon, and other hormones ; therefore the ratio of omega-3 versus omega-6 has wide effects on general health, and specific effects on immune function and inflammation, and mitosis ( i. e. cell division ).

In 1824, Niels Henrik Abel proved the striking result that there can be no general ( finite ) formula, involving only arithmetic operations and radicals, that expresses the roots of a polynomial of degree 5 or greater in terms of its coefficients ( see Abel-Ruffini theorem ).

More complex experiments, such as those involving stochastic processes defined in continuous-time, may demand the use of more general probability measures.

This term also includes general computer science graduates with a few years of practical on the job experience involving software engineering.

Processes involving phase transitions, such as the drying of adhesives or melting of alloys, liquid transport, chemical reactions, and solid-air-gas systems, in general cannot be observed.

Thackeray has Rosalind using their as a polite circumlocution, perhaps avoiding the directness of she ... her, and generic his in a context involving only women ; or perhaps with Rosalind meaning the statement to apply to people in general with Becky Sharp as an example.

Gradually, it developed into a more general conflict involving most of the great powers of the time.

However, in the pure Gauss – Bonnet gravity ( a modification to general relativity involving extra spatial dimensions which is sometimes studied in the context of brane cosmology ) exotic matter is not needed in order for wormholes to exist — they can exist even with no matter.

During a federal election, the governor general will curtail these public duties, so as not to appear as though they are involving themselves in political affairs.

Although Newton's theory has been superseded, most modern non-relativistic gravitational calculations are still made using Newton's theory because it is a much simpler theory to work with than general relativity, and gives sufficiently accurate results for most applications involving sufficiently small masses, speeds and energies.

In general, only lifelong interactions involving close physical and biochemical contact can properly be considered symbiotic.

His other contributions include a simplified proof of the positive energy theorem involving spinors in general relativity, his work relating supersymmetry and Morse theory, his introduction of topological quantum field theory and related work on mirror symmetry, knot theory, twistor theory and D-branes and their intersections.

Insensitivity to stimuli such as touch, heat, cold, and pain are often an adjunct to a more general physical impairment involving neural pathways and is very commonly associated with paralysis ( in which the motor neural circuits are also affected ).

The amount and type of carbohydrates consumed, along with some types of fat, can influence processes involving insulin, glucagon, and other hormones ; therefore the ratio of omega-3 versus fat has wide effects on general health, and specific effects on immune function and inflammation, and mitosis ( cell division ).

The range of scholarly interest in creativity includes a multitude of definitions and approaches involving several disciplines ; psychology, cognitive science, education, philosophy ( particularly philosophy of science ), technology, theology, sociology, linguistics, business studies, and economics, taking in the relationship between creativity and general intelligence, mental and neurological processes associated with creativity, the relationships between personality type and creative ability and between creativity and mental health, the potential for fostering creativity through education and training, especially as augmented by technology, and the application of creative resources to improve the effectiveness of learning and teaching processes.