In fact, the space of all Lipschitz functions on a compact metric space is dense in the Banach space of continuous functions, an elementary consequence of the Stone – Weierstrass theorem.

In 1882, the task was proven to be impossible, as a consequence of the Lindemann – Weierstrass theorem which proves that pi () is a transcendental, rather than an algebraic irrational number ; that is, it is not the root of any polynomial with rational coefficients.

Ω is in fact transcendental as the direct consequence of Lindemann – Weierstrass theorem.

This is a consequence of the Born-Oppenheimer approximation.

As a consequence of its definition, the Sobel operator can be implemented by simple means in both hardware and software: only eight image points around a point are needed to compute the corresponding result and only integer arithmetic is needed to compute the gradient vector approximation.

This is a foundational result in diophantine approximation, showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational α, the inequality

As a consequence of the Hahn – Banach theorem, this map is injective, and isometric.

* The Hilbert cube is compact, again a consequence of Tychonoff's theorem.

In this situation, the chain rule represents the fact that the derivative of is the composite of the derivative of f and the derivative of g. This theorem is an immediate consequence of the higher dimensional chain rule given above, and it has exactly the same formula.

As a consequence of the Nyquist – Shannon sampling theorem, any spatial waveform that can be displayed must consist of at least two pixels, which is proportional to image resolution.

According to Noether's theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time.

As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere ( complex plane and the point at infinity ) is constant.

Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic.

It is deduced from the model existence theorem as follows: if there is no formal proof of a formula then adding its negation to the axioms gives a consisten theory, which has thus a model, so that the formula is not a semantic consequence of the initial theory.

An important consequence of the completeness theorem is that it is possible to recursively enumerate the semantic consequences of any effective first-order theory, by enumerating all the possible formal deductions from the axioms of the theory, and use this to produce an enumeration of their conclusions.

The compactness theorem says that if a formula φ is a logical consequence of a ( possibly infinite ) set of formulas Γ then it is a logical consequence of a finite subset of Γ.

This is an immediate consequence of the completeness theorem, because only a finite number of axioms from Γ can be mentioned in a formal deduction of φ, and the soundness of the deduction system then implies φ is a logical consequence of this finite set.

Conversely, for many deductive systems, it is possible to prove the completeness theorem as an effective consequence of the compactness theorem.

As a consequence of the polynomial remainder theorem, the entries in the third row are the coefficients of the second-degree polynomial, the quotient of on division by.

An important consequence of the KAM theorem is that for a large set of initial conditions the motion remains perpetually quasiperiodic.

A multiplicative function is completely determined by its values at the powers of prime numbers, a consequence of the fundamental theorem of arithmetic.

That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position ; this is a special case of Noether's theorem.

is a consequence of Stone's theorem on one-parameter unitary groups.

The embedding property is a consequence of a result known as Desargues ' theorem.

Quines are possible in any programming language that has the ability to output any computable string, as a direct consequence of Kleene's recursion theorem.

As a consequence of the noisy-channel coding theorem, noise limits the information content of an analog information carrier to be finite.

One consequence of Toda's theorem is that a polynomial-time machine with a # P oracle ( P < sup ># P </ sup >) can solve all problems in PH, the entire polynomial hierarchy.

As a consequence of the tensions thus produced in his thoughts and feelings, he wrote on the one hand sketches of idealized hunting trips and on the other an anecdote of the village of Hardscrabble, Arkansas, where no one had ever seen a piano ; ;

Since the 1946 disaster there have been 15 tsunami in the Pacific, but only one was of any consequence.

As an obvious consequence, obstacles to genuine interfaith communication have grown more formidable in one important area: relations between Christians and non-Christians in these lands.

We should not allow the image of an immanent end brought about indirectly by our own action in the continuing human struggle for a just endurable order of existence to blind us to the fact that in some measure accelerating the end of our lease may be one consequence among others of many other of mankind's thrusts toward we know not what future.

As an unintended consequence, the soundtrack to one of the films — Beethoven's Ninth Symphony — renders Alex unable to listen to his beloved classical music.

As a consequence, Johnson assumed an attitude of white supremacy typical of one in his position in his town, and he was unable to shed this perspective during his life.

However, one consequence of this shift in emphasis was that during the last years of his life, Dürer produced comparatively little as an artist.

This star was seen to possess an apparent motion similar to that which would be a consequence of the nutation of the Earth's axis ; but since its declination varied only one half as much as in the case of γ Draconis, it was obvious that nutation did not supply the requisite solution.

As a consequence, substituted alkenes may exist as one of two isomers, called cis or trans isomers.

Deane and J. R. Thomson write this valid conclusion, “ The Book of Obadiah is occupied with one subject – the punishment of Edom for its cruel and unbrotherly love conduct towards Judah ...” One can link this idea of punishment to one of the major prophets “ Ezekiel ” who “... interprets the exile to Babylon and the destruction of Jerusalem as deserved punishments for the sins of those who themselves committed them .” Verses 3-7 in Obadiah explain to the reader the reason for the punishment theme, “ Confidence in one ’ s power, intelligence, allies, or the topographical features of one ’ s territory is often mentioned as an attribute of those who foolishly confront the Lord and are consequently punished .” Although destruction is vital to understanding Obadiah, it is of note to understand the destruction being a consequence of action.

Thus, from a consequentialist standpoint, a morally right act ( or omission ) is one that will produce a good outcome, or consequence.

Preaching the Crusade had one important consequence.

According to Douglass, smallpox inoculation was " a medical experiment of consequence ," one not to be undertaken lightly.

The administration never made use of many Republicans of consequence whose services in one form or another would have been available for the asking.

Thus, from a consequentialist standpoint, a morally right action is one that produces a good outcome, or consequence.

In deontology, an act may be considered right even if the act produces a bad consequence, if it follows the rule that “ one should do unto others as they would have done unto them ”, and even if the person who does the act lacks virtue and had a bad intention in doing the act.

As a consequence, he has limited sight in one eye and lacks normal stereoscopic vision.

" Flash Crowd " is a 1973 English language novella by science fiction author Larry Niven, one of a series about the social consequence of inventing an instant, practically free transfer booth that could take one anywhere on Earth in milliseconds.

Under Hippocrates ’ bodily humors theory, differences in human moods come as a consequence of imbalances in one of the four bodily fluids: blood, yellow bile, black bile, and phlegm.

but observe that one cannot just tell by scrutinizing a property what it entails ; one might be surprised at a consequence.