The grammarian Athenaeus quoted some verses about perfumed ointments to prove just how unwarlike Alcaeus could be and he quoted his description of the armour adorning the walls of his house as proof that he could be unusually warlike for a lyric poet.

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.

She defined faith as " the acceptance of allegations without evidence or proof, either apart from or against the evidence of one's senses and reason ... Mysticism is the claim to some non-sensory, non-rational, non-definable, non-identifiable means of knowledge, such as ' instinct ,' ' intuition ,' ' revelation ,' or any form of ' just knowing.

* John Morgan and Gang Tian posted a paper on the arXiv in July 2006 which gave a detailed proof of just the Poincaré Conjecture ( which is somewhat easier than the full geometrization conjecture ) and expanded this to a book.

The proof works just as well if we have an algorithm for deciding any other nontrivial property of programs, and will be given in general below.

Muhammad Subuh saw the present age as one that demands personal evidence and proof of religious or spiritual realities, as people no longer just believe in words.

A rigorous proof was published by Argand in 1806 ; it was here that, for the first time, the fundamental theorem of algebra was stated for polynomials with complex coefficients, rather than just real coefficients.

Gödel's original statement and proof of the incompleteness theorem requires the assumption that the theory is not just consistent but ω-consistent.

Actually, the machinery from analytic function theory enters only in a formal way in this proof, in that what is really needed is just some property of the formal residue, and a more direct formal proof is available.

The theorem is thus generally credited to Niels Henrik Abel, who published a proof that required just six pages in 1824.

The fact that Duse was just coming out of a relationship with rebellious young lesbian feminist Lina Poletti fueled speculation as to the nature of Duncan and Duse's relationship, but there has never been definite proof that the two were involved romantically.

The ASI report said there is sufficient proof of existence of a massive and monumental structure having a minimum dimension of 50x30 metres in north-south and east-west directions respectively just below the disputed structure.

This proof uses just the kind of objects needed to formulate the Cayley – Hamilton theorem: matrices with polynomials as entries.

This proof, of course, is just another, more systematic ( even if outmoded ) way of proving the theorem in terms of limits, and is therefore equivalent to the first proof above – and even reduces to it, if you make the right substitutions in the right places.

He was greatly mourned ; Senator Daniel Patrick Moynihan stated " Henry Jackson is proof of the old belief in the Judaic tradition that at any moment in history goodness in the world is preserved by the deeds of 36 just men who do not know that this is the role the Lord has given them.

There is a distinction between a simple mistake and a mathematical fallacy in a proof: a mistake in a proof leads to an invalid proof just in the same way, but in the best-known examples of mathematical fallacies, there is some concealment in the presentation of the proof.

From the point of view of the Muslim rulers, jizya was a material proof of the non-Muslims ' acceptance of subjection to the state and its laws, " just as for the inhabitants it was a concrete continuation of the taxes paid to earlier regimes.

The proof is somewhat constructive — it shows that the singularity can be found by following light-rays from a surface just inside the horizon.

In that circumstance, the defendant manufacturer or distributor may move to dismiss the case on the basis of spoliation ( instead of just having to rely on the plaintiff's usual burden of proof, the argument being that any testimony of plaintiff's witnesses would not overcome the spoliation inference born of the lost evidentiary value of the missing product itself ).

In the 2010 romantic-comedy movie How Do You Know finale, Paul Rudd's character tells the story of Play-Doh to Reese Witherspoon's character, saying: " I have kept this for a long time as proof that we're all just one small adjustment away from making our lives work.

He had introduced the adele ring in the late 1930s, following Claude Chevalley's lead with the ideles, and given a proof of the Riemann – Roch theorem with them ( a version appeared in his Basic Number Theory in 1967 ).

He had been well educated, and was versed in grammar and rhetoric, and had already, while still a young man, and before reaching the episcopate, given proof to those who dwelt with him of his wisdom and acumen " ( Soz., II, xvii ).

This is not a rejection of existence by Gilson, a leading modern metaphysician in the classical tradition: " philosophers are wholly justified in taking existence for granted ... and in never mentioning it again ...." In Gilson's view, the participial being is a given, a primitive of experience, not subject to proof or investigation, as it is the grounds of proof.

The first proof was given by (), where the formulation of the problem was attributed to Ulam.

An " elementary " proof can be given using the fact that geometric mean of positive numbers is less than arithmetic mean

The higher-dimensional chain rule can be proved using a technique similar to the second proof given above.

The proof of this fact relies on an algorithm which, given the first n digits of Ω, solves Turing's halting problem for programs of length up to n. Since the halting problem is undecidable, Ω can not be computed.

The date given is usually the publication date of the complete proof of a result, which is sometimes several years later than the proof or first announcement of the result, so some of the items appear in the " wrong " order.

A proof from Euclid's elements that, given a line segment, an equilateral triangle exists that includes the segment as one of its sides.

One proof was given by Alfred Kempe in 1879, which was widely acclaimed ; another was given by Peter Guthrie Tait in 1880.

A simpler proof considering only 633 configurations was given twenty years later by Robertson, Seymour, Sanders and Thomas.

In spite of the early proof that Agricola had given of the tolerance of his own religious attitude, he was not suffered to end his days in peace.

The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 ( and a rewritten version of the dissertation, published as an article in 1930 ) is not easy to read today ; it uses concepts and formalism that are outdated and terminology that is often obscure.

The version given below attempts to represent all the steps in the proof and all the important ideas faithfully, while restating the proof in the modern language of mathematical logic.

Hilbert produced an innovative proof by contradiction using mathematical induction ; his method does not give an algorithm to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist.

( Indirect proof via the inverse power series technique was given by Newton and Leibniz between 1670 and 1680.

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true for all natural numbers ( positive integers ).

I think that in all essential respects you have given the best possible proof & this does not happen frequently.

The remaining, more difficult part, is to prove that for there is no equivalent regular expression of star height less than n ; a proof is given in.

In Chapter Four of Utilitarianism Mill considers what proof can be given for the Principle of Utility.

The proof follows from DTFT # Periodic_data, which indicates that can be written as:

The reports sum up the issue by saying that " the proof that Amenhotep III and Queen Tiye are the parents of KV55, combined with this anthropological and archaeological evidence, indicates that the mummy in KV55 is almost certainly Akhenaten ".

" SPG " here is short for " shortest proof game " and the " 9. 0 " indicates how many moves must be played to reach the position ; 9. 0 means the position is reached after black's ninth move, 7. 5 would mean the position is reached after seven and a half moves ( that is, after white's eighth move ) and so on.

A PR70 coin is the highest grade possible for a proof coin and indicates a perfect example, with PR69 and lower grades reflecting some deficiency in the strike, centering, details, or other aspect of the coin.

In a note (" Notiz ") that appeared in the following issue of the Journal, Schönemann points this out to Eisenstein, and indicates that the latter's method is not essentially different from the one he used in the second proof.

This mark indicates that the device is designed and manufactured so as to not compromise the safety of patients, but no independent testing is required as proof.

A man who wears jasmine on his left ear indicates that he is single and in addition, offering white jasmine is seen as a proof of love while on the contrary, offering odorless winter jasmine is a sign of insolence.