Whether you experienced the passion of desire I have, of course, no way of knowing, nor indeed have I wished with even the most fleeting fragment of a wish to know, for the fact that one constitutes by one's mere existence so to speak the proof of some sort of passion makes any speculation upon this part of one's parents' experience more immodest, more scandalizing, more deeply unwelcome than an obscenity from a stranger.

A credulousness, a distaste for documentation, an uncritical reliance on contemporary accounts, and a proneness to assume a theory as true before adequate proof was provided were all evidences of his failure to comprehend the use of the scientific method or to evaluate the responsibilities of the historian to his reading public.

The first is the strictly scientific, which demands concrete proof and therefore may err on the conservative side by waiting for evidence in the flesh.

In the notation of the proof of Theorem 12, let us take a look at the special case in which the minimal polynomial for T is a product of first-degree polynomials, i.e., the case in which each Af is of the form Af.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

Attorneys for the eight other defendants said only that there was no proof of their clients' guilt.

( A formal proof for all finite sets would use the principle of mathematical induction to prove " for every natural number k, every family of k nonempty sets has a choice function.

Similarly, all the statements listed below which require choice or some weaker version thereof for their proof are unprovable in ZF, but since each is provable in ZF plus the axiom of choice, there are models of ZF in which each statement is true.

There are several results in category theory which invoke the axiom of choice for their proof.

A grant application to build a proof of concept prototype was submitted in March 1939 to the Agronomy department which was also interested in speeding up computation for economic and research analysis.

* Drinking songs: According to the grammarian Athenaeus, Alcaeus made every occasion an excuse for drinking and he has provided posterity several quotes in proof of it.

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.

Mordell's theorem had an ad hoc proof ; Weil began the separation of the infinite descent argument into two types of structural approach, by means of height functions for sizing rational points, and by means of Galois cohomology, which was not to be clearly named as that for two more decades.

Among his major accomplishments were the 1940 proof, of the Riemann hypothesis for zeta-functions of curves over finite fields, and his subsequent laying of proper foundations for algebraic geometry to support that result ( from 1942 to 1946, most intensively ).

This key result opened the way for a proof of the Weil conjectures, ultimately completed by his student Pierre Deligne.

The " heuristic " approach of the Logic Theory Machine tried to emulate human mathematicians, and could not guarantee that a proof could be found for every valid theorem even in principle.

For the frequent case of propositional logic, the problem is decidable but Co-NP-complete, and hence only exponential-time algorithms are believed to exist for general proof tasks.

* Alan Bundy, University of Edinburgh, meta-level reasoning for guiding inductive proof, proof planning and recipient of 2007 IJCAI Award for Research Excellence, Herbrand Award, and 2003 Donald E. Walker Distinguished Service Award.

* Metamath-a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems ; while the Metamath language is not accompanied with an automated theorem prover, it can be regarded as important because the formal language behind it allows development of such a software ; as of March, 2012, there is no " widely " known such software, so it is not a subject of " automated theorem proving " ( it can become such a subject ), but it is a proof assistant.

Smale's original proof was indirect: he identified ( regular homotopy ) classes of immersions of spheres with a homotopy group of the Stiefel manifold.

In principle the proof can be unwound to produce an explicit regular homotopy, but this is not easy to do.

The homotopy principle generalizes such results as Smale's proof of sphere eversion.

This statement is a special case of a homotopical excision theorem, involving induced modules for n > 2 ( crossed modules if n = 2 ), which itself is deduced from a higher homotopy van Kampen theorem for relative homotopy groups, whose proof requires development of techniques of a cubical higher homotopy groupoid of a filtered space.

The proof of this theorem uses a higher homotopy van Kampen type theorem for triadic homotopy groups, which requires a notion of the fundamental cat < sup > n </ sup >- group of an n-cube of spaces.

The proof involves the concept of homotopy double groupoid of a pointed pair of spaces.

A consequence is that the Generalized Poincaré conjecture is true in PL for dimensions greater than four – the proof is to take a homotopy sphere, remove two balls, apply the h-cobordism theorem to conclude that this is a cylinder, and then attach cones to recover a sphere.

The formal proof of the theorem utilizes the condition of invariance to derive an expression for a current associated with a conserved physical quantity.

The most celebrated result was his proof of the topological invariance of dimension.

This has led to the abandonment of BAN-family logics in favor of proof methods based on standard invariance reasoning.

The proof of the invariance under ( ii ) is as follows:

A singular independence of spirit, a breadth of mind which refused to be contracted by party formulas, a sanity which was proof against the contagion of national delirium, were equally characteristic of uncle and nephew.

In Image and Logic, Galison explored the fundamental rift rising in the physical sciences: whether singular, visual accounts of scientific phenomenon would be accepted as the dominant language of proof, or whether statistically significant, frequently repeated results would dominate the field.

In the campaign of 217 BC we find him commanding a body of ten thousand select troops, and just before the battle of Raphia he gave a singular proof of daring by penetrating with only two companions into the heart of the Egyptian camp, in order to assassinate Ptolemy himself.

The great width of the lane, from the road to Ulverley, and its singular narrowness from thence to Hogg's moat, is another proof of its antiquity.

In 1990, Dupont and Sah provided a simpler proof of Sydler's result by reinterpreting it as a theorem about the homology of certain classical groups.

It was later noticed that this lemma provides a direct proof of the Brouwer fixed-point theorem without explicit use of homology.

His book L ' analysis situs et la géométrie algébrique from 1924, though opaque foundationally given the current technical state of homology theory, was in the long term very influential ( one could say that it was one of the sources for the eventual proof of the Weil conjectures, through SGA7 ).

Ultimately, this approximation technique is a standard ingredient in the proof that simplicial homology groups are topological invariants.

First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite dimensional Morse homology.

Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification of quasithin groups.

( For groups of low 2-rank the proof of this breaks down, because theorems such as the signalizer functor theorem only work for groups with elementary abelian subgroups of rank at least 3.

Emery, an ethnologist for Honolulu's Bernice P. Bishop Museum, indicated that settlers on Manra Island were apparently of two distinct groups, one Polynesian and the other Micronesian, hence the same might have been true on Howland Island, though no proof of this has been forthcoming.

From May to July 2006, several groups presented papers that filled in the details of Perelman's proof of the Poincaré conjecture, as follows:

The proof uses the classification of finite simple groups.

The classification of finite simple groups is regarded by some to be the longest proof of a theorem ; it comprises tens of thousands of pages in 500 journal articles by some 100 authors.

This proof does not prove the existence of a uniform algorithm for solving the word problem for this class of groups.

There are two threads in the history of finite simple groups – the discovery and construction of specific simple groups and families, which took place from the work of Galois in the 1820s to the construction of the Monster in 1981 ; and proof that this list was complete, which began in the 19th century, most significantly took place 1955 through 1983 ( when victory was initially declared ), but was only generally agreed to be finished in 2004., work on improving the proofs and understanding continues ; see for 19th century history of simple groups.

Soon after the construction of the Monster in 1981, a proof, totaling more than 10, 000 pages, was supplied that group theorists had successfully listed all finite simple groups, with victory declared in 1983 by Daniel Gorenstein.

Dunwoody found a graph-theoretic proof of Stallings ' theorem about ends of groups in 1982, by constructing certain tree-like automorphism invariant graph decompositions.

Dan Olinger, a professor at the fundamentalist Bob Jones University in Greenville said, “ We want to be good citizens and participants, but we ’ re not really interested in using the iron fist of the law to compel people to everything Christians should do .” Bob Marcaurelle, interim pastor at Mountain Springs Baptist Church in Piedmont, said the Middle Ages were proof enough that Christian ruling groups are almost always corrupted by power.

Evolutionary developmental biology ( evo-devo ) springs from clear proof that development is closely controlled by special genetic systems, and the hope that comparison of these systems will tell us much about the evolutionary history of different groups.

By our supposed existing proof, since these are groups of four, all horses in them must be the same color.

Horse theft was at this stage not only a proof of courage, but often a desperate contribution to survival, for many ethnic groups competed for hunting in the grasslands.

He is also credited for categorizing logic into two separate groups, the first being " idea " and the second being " proof ".

Like the open mapping theorem in functional analysis, the proof in the setting of topological groups uses the Baire category theorem.

Some of the Resistance groups he spoke of had never existed, and there was no proof of any of his claimed exploits.

Many Kashmiri militant groups also maintain their headquarters in Pakistan-administered Kashmir, which is cited as further proof by the Indian government.

To see this for any particular order is usually not difficult ( for example, there is, up to isomorphism, one non-solvable group and 12 solvable groups of order 60 ) but the proof of this for all orders uses the classification of finite simple groups.