As ' functions ' these can take only the value 0, but they carry infinitesimal information, in purely algebraic settings.

This does not seem to be in accordance with a continuum theory and must lead to an attempt to find a purely algebraic theory for the representation of reality.

One may say that Diophantus was studying rational points — i. e., points whose coordinates are rational — on curves and algebraic varieties ; however, unlike the Greeks of the Classical period, who did what we would now call basic algebra in geometrical terms, Diophantus did what we would now call basic algebraic geometry in purely algebraic terms.

This result is quite interesting, because the statement is purely algebraic yet the simplest proof is topological.

In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals ( or some other equivalent formulation of completeness ), which is not an algebraic concept.

One can use formal power series to prove several relations familiar from analysis in a purely algebraic setting.

The " description format " of a structure ( such as adjacency list versus adjacency matrix for graphs ) is irrelevant, because species are purely algebraic.

The above paragraph generalizes in that there is a purely algebraic process to extend a given field F with a given polynomial to a larger field where this polynomial can be reduced into linear factors.

The Whitehead problem was the first purely algebraic problem to be proved undecidable.

The development of algebraic topology during the 1940s gave additional incentive for the development of a purely algebraic treatment of the tensor product.

This purely algebraic attack conveys no geometric intuition.

Note also that although applications almost always deal with convergence of sequences, the statement is purely algebraic and will work in any field.

It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings.

Some of them are of geometric nature, while some other are purely algebraic and rely on commutative algebra.

Since f is a polynomial, this definition is purely algebraic and makes no assumption about the nature of the field F, which in particular need not be the real or complex numbers.

An equivalent, but purely algebraic description of linear differential operators is as follows: an R-linear map P is a kth-order linear differential operator, if for any k + 1 smooth functions we have

An algebraic extension is a purely inseparable extension if and only if for every, the minimal polynomial of over F is not a separable polynomial ( i. e., does not have distinct roots ).

For a field F to possess a non-trivial purely inseparable extension, it must necessarily be an infinite field of prime characteristic ( i. e. specifically, imperfect ), since any algebraic extension of a perfect field is necessarily separable.

An algebraic extension is a purely inseparable extension if and only if for every, the minimal polynomial of over F is not a separable polynomial.

If K is a field of characteristic p, and if V is an algebraic variety over K of dimension greater than zero, the function field K ( V ) is a purely inseparable extension over the subfield K ( V )< sup > p </ sup > of pth powers ( this follows from condition 2 above ).

Later still, abstraction was manifest in more purely formal terms, such as color, freedom from objective context, and a reduction of form to basic geometric designs.

In Re City Equitable Fire Insurance Co Ch 407, it was expressed in purely subjective terms, where the court held that:

In reality, the terms ' Scots / Scottish ' and ' Irish ' are purely modern geographical references to a people who share a common Celtic ancestry and consequently, a common musical heritage.

Since the dawn of Newtonian science with its vision of simple mechanical principles governing the entire universe, some philosophers have been tempted by the idea that consciousness could be explained in purely physical terms.

Under its terms the Emperor of Japan is " the symbol of the State and of the unity of the people " and exercises a purely ceremonial role without the possession of sovereignty.

There is, however, no finite set of statements that are couched in purely sensory terms and can express the satisfaction of the condition of the presence of a normal observer.

Folk music may tend to have certain characteristics but it cannot clearly be differentiated in purely musical terms.

However, the findings are not as absolute, and Kinsey himself avoided and disapproved of using terms like homosexual or heterosexual to describe individuals, asserting that sexuality is prone to change over time, and that sexual behavior can be understood both as physical contact as well as purely psychological phenomena ( desire, sexual attraction, fantasy ).

Hare, and Simon Blackburn have argued in favor of the fact / norm distinction, meanwhile, with Gibbard going so far as to argue that, even if conventional English has only mixed normative terms ( that is, terms that are neither purely descriptive nor purely normative ), we could develop a nominally English metalanguage that still allowed us to maintain the division between factual descriptions and normative evaluations.

It is also the case that, while often being characterized in purely negative terms, some mental traits or states labeled as disorders can also involve above-average creativity, non-conformity, goal-striving, meticulousness, or empathy.

Tegmark writes that " abstract mathematics is so general that any Theory Of Everything ( TOE ) that is definable in purely formal terms ( independent of vague human terminology ) is also a mathematical structure.

The mind and brain can both be described as being based in the physical world, but explaining the mind, or mental illness, purely in physical terms, may not always be appropriate or useful.

By wave-particle duality the effect can be analyzed purely in terms of waves though not as conveniently.

Subsequently these terms have become purely descriptive, and have largely lost negative connotations.

Posthumanism is sometimes used as a synonym for an ideology of technology known as " transhumanism " because it affirms the possibility and desirability of achieving a " posthuman future ", albeit in purely evolutionary terms.

In the classical account strictly and purely in terms of cyclic processes, the spatial interior of the ' working body ' of a cyclic process is not considered ; the ' working body ' thus does not have a defined internal thermodynamic state of its own because no assumption is made that it should be in thermodynamic equilibrium ; only its inputs and outputs of energy as heat and work are considered.

Carver Mead's Collective Electrodynamics: Quantum Foundations of Electromagnetism ( 2000 ) analyzes the behavior of electrons and photons purely in terms of electron wave functions, and attributes the apparent particle-like behavior to quantization effects and eigenstates.

Unlike many other major civilizations where written law was held in honor and often attributed to divine origin, law in early China was viewed in purely secular terms and its initial appearance was greeted with hostility by Confucian thinkers as indicative of a serious moral decline, a violation of human morality, and even a disturbance of the total cosmic order.

However, his controversial beliefs that religion and morality could be explained in terms purely of social interaction earned him many critics.

The reason is that the axioms of the identity element and inversion are not stated purely in terms of equational laws but also have clauses involving the phrase " there exists ... such that ...".

Since Laos is of no more purely military value to Moscow itself than it is to Washington, this approach might be expected to head off Mr. Khrushchev for the moment.

The pressure for our entry to the Common Market is mounting and we will proceed towards this amalgamated trade union by way of a purely `` economic thoroughfare '', or garden path, with the political ramifications kept neatly in the background.

But for purely definition purposes -- used in conjunction with your regular Squatting, Leg Curling, Leg Extensor programs -- a heavy weight is not needed.

For example, the level of improvement noted in a recent experiment with a short course of immediate treatment for parent-child relationship problems compared favorably with the results reported by typical child guidance clinics where the hours spent in purely diagnostic study may equal or exceed the number of hours devoted to actual treatment interviews in the experimental project.

When I pressed for a purely religious definition, I encountered the familiar blend of liberal piety, interfaith good will, and a small residue of ethnic loyalty.

This is not to say that the only explanation of the present infatuation with Norman Vincent Peale's `` cult of reassurance '' or the other types of a purely cultural Christianity is the ever-present need for a demythologized gospel.

The tremendous emphasis on the 5 in the Lo Shu square -- for purely mathematical reasons -- and the fact that this number so neatly symbolized the heart and center of the universe, could well explain why the Old Chinese seem to have so revered the number 5, and why they put so much stress on the concept of Centrality.

They would become tagged as men not interested in being purely real estate `` professionals '' but agitators for some kind of `` cause '' or `` reform '', and this was no longer to be a `` pro ''.

Many ancient civilizations alloyed metals for purely aesthetic purposes.

Arminius states that " Justification, when used for the act of a Judge, is either purely the imputation of righteousness through mercy … or that man is justified before God … according to the rigor of justice without any forgiveness.

" Stephen Ashby clarifies: " Arminius allowed for only two possible ways in which the sinner might be justified: ( 1 ) by our absolute and perfect adherence to the law, or ( 2 ) purely by God's imputation of Christ's righteousness.

More purely Hellenic myth would have Amathus settled instead by one of the sons of Heracles, thus accounting for the fact that he was worshiped there.

EMS and disaster relief helicopters will be required to fly in unpleasant conditions, this may require more aircraft sensors, some of which were until recently considered purely for military aircraft.

On occasion SPAAGs have been used as very effective direct fire weapons against infantry, for example by American forces during late World War II, in Korea against mass infantry assault, and extensively during the Vietnam War, where for example the U. S. M42 Duster SPAAG ( based on a light tank ) was employed purely for this purpose.

The curtain-wall and towers of the Mycenaean citadel, its gate with heraldic lions, and the great " Treasury of Atreus " had borne silent witness for ages before Heinrich Schliemann's time ; but they were supposed only to speak to the Homeric, or, at farthest, a rude Heroic beginning of purely Hellenic civilization.

The stage is set for the concept of God as the cause of all existence, who, as the Almighty, holds everything actual without reason or explanation as an act purely of will.

Vocalist Gaahl said that during its early years, " black metal was never meant to reach an audience, it was purely for our own satisfaction ".

Its subfields can be divided into practical techniques for its implementation and application in computer systems and purely theoretical areas.

Possibly the most common type of collector is the hobbyist, who amasses a collection purely for fun with no real expectation of profit.

Its message is purely a spiritual one, an allegory of the spiritual path, which is equally relevant in all ages and for all people.

However, purely negative play is rarely a winning strategy: successful players ( in all versions other than golf croquet ) will use all four balls to set up a break for themselves, rather than simply making the game as difficult as possible for their opponents.

Cavalry or mounted gendarmerie units continue to be maintained for purely or primarily ceremonial purposes by the United States, British, French, Italian, Danish, Swedish, Dutch, Chilean, Portuguese, Moroccan, Nepalese, Nigerian, Venezuelan, Brazilian, Peruvian, Paraguayan, Polish, Argentine, Senegalese, Jordanian, Pakistani, Indian, Spanish and Bulgarian armed forces.