By setting K = ker ( f ) we immediately get the first isomorphism theorem.

As in the theory of associative rings, ideals are precisely the kernels of homomorphisms, given a Lie algebra and an ideal I in it, one constructs the factor algebra, and the first isomorphism theorem holds for Lie algebras.

Starting from finite-dimensional vector spaces ( as objects ) and the dual functor, one can define a natural isomorphism, but this requires first adding additional structure, then restricting the maps from " all linear maps " to " linear maps that respect this structure ".

Given an object X, a functor G ( taking for simplicity the first functor to be the identity ) and an isomorphism proof of unnaturality is most easily shown by giving an automorphism that does not commute with this isomorphism ( so ).

The fundamental theorem on homomorphisms ( or first isomorphism theorem ) is a theorem, again taking various forms, that applies to the quotient algebra defined by the kernel.

We first state the three isomorphism theorems in the context of groups.

By the first isomorphism theorem, the image of A under ƒ is a substructure of B isomorphic to the quotient of A by this congruence.

This is a consequence of the first isomorphism theorem, because Z ( G ) is precisely the set of those elements of G that give the identity mapping as corresponding inner automorphism ( conjugation changes nothing ).

Therefore, by the first isomorphism theorem, GL ( n, F )/ SL ( n, F ) is isomorphic to F < sup >×</ sup >.

It allows one to refine the first isomorphism theorem:

* the first isomorphism theorem states that in the above short exact sequence,

* if the sequence splits, then, and the first isomorphism theorem is just the projection onto C.

In the following diagram expressing the first isomorphism theorem, commutativity means that:

The first unitary group U ( 1 ) is topologically a circle, which is well known to have a fundamental group isomorphic to Z, and the inclusion map U ( n ) → U ( n + 1 ) is an isomorphism on π < sub > 1 </ sub >.

The standard method to construct the reciprocity homomorphism is to first construct the local reciprocity isomorphism from the multiplicative group of the completion of a global field to the Galois group of its maximal abelian extension ( this is done inside local class field theory ) and then prove that the product of all such local reciprocity maps when defined on the idele group of the global field is trivial on the image of the multiplicative group of the global field.

That is, there is a bijection between the isomorphism classes of line bundles over X and the elements of H² ( X ; Z ), which associates to a line bundle its first Chern class.

In algebraic geometry, this classification of ( isomorphism classes of ) complex line bundles by the first Chern class is a crude approximation to the classification of ( isomorphism classes of ) holomorphic line bundles by linear equivalence classes of divisors.

In other words: the kernel of π consists of all scalar multiples of the identity matrix I, and the first isomorphism theorem of group theory states that the quotient group GL ( 2, C )/(( C

Thus the coimage of f is a quotient algebra of X much as the image of f is a subalgebra of Y ; and the bijection between them becomes an isomorphism in the algebraic sense as well ( this is the most general form of the first isomorphism theorem in algebra ).

This theorem is a statement of the first isomorphism theorem of algebra to the case of vector spaces ; it generalizes to the splitting lemma.

The first Stiefel – Whitney class classifies smooth real line bundles ; in particular, the collection of ( equivalence classes of ) real line bundles are in correspondence with elements of the first cohomology with Z / 2Z coefficients ; this correspondence is in fact an isomorphism of abelian groups ( the group operations being tensor product of line bundles and the usual addition on cohomology ).

The theorem which we prove is more general than what we have described, since it works with the primary decomposition of the minimal polynomial, whether or not the primes which enter are all of first degree.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

The concept and theory of Kolmogorov Complexity is based on a crucial theorem first discovered by Ray Solomonoff, who published it in 1960, describing it in " A Preliminary Report on a General Theory of Inductive Inference " as part of his invention of algorithmic probability.

His first ( pre-IHÉS ) breakthrough in algebraic geometry was the Grothendieck – Hirzebruch – Riemann – Roch theorem, a far-reaching generalisation of the Hirzebruch – Riemann – Roch theorem proved algebraically ; in this context he also introduced K-theory.

The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

For a first order predicate calculus, with no (" proper ") axioms, Gödel's completeness theorem states that the theorems ( provable statements ) are exactly the logically valid well-formed formulas, so identifying valid formulas is recursively enumerable: given unbounded resources, any valid formula can eventually be proven.

* William McCune Argonne National Laboratory, author of Otter, the first high-performance theorem prover.

Developed original resolution and unification based first order theorem proving, co-editor of the " Handbook of Automated Reasoning ", recipient of the Herbrand Award 1996

According to (), the first historical mention of the statement of this theorem appears in ().

Following Desargues ' thinking, the sixteen-year-old Pascal produced, as a means of proof, a short treatise on what was called the " Mystic Hexagram ", Essai pour les coniques (" Essay on Conics ") and sent it — his first serious work of mathematics — to Père Mersenne in Paris ; it is known still today as Pascal's theorem.

The Cook – Levin theorem states that the Boolean satisfiability problem is NP-complete, and in fact, this was the first decision problem proved to be NP-complete.

SAT was the first known NP-complete problem, as proved by Stephen Cook in 1971 ( see Cook's theorem for the proof ).

This position is a more refined version of the theorem first discovered by David Hume.

The first proof relies on a theorem about products of limits to show that the derivative exists.

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

* 1799: Doctoral dissertation on the Fundamental theorem of algebra, with the title: Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse (" New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors ( i. e., polynomials ) of the first or second degree ")

The first case was done by the Gorenstein – Walter theorem which showed that the only simple groups are isomorphic to L < sub > 2 </ sub >( q ) for q odd or A < sub > 7 </ sub >, the second and third cases were done by the Alperin – Brauer – Gorenstein theorem which implies that the only simple groups are isomorphic to L < sub > 3 </ sub >( q ) or U < sub > 3 </ sub >( q ) for q odd or M < sub > 11 </ sub >, and the last case was done by Lyons who showed that U < sub > 3 </ sub >( 4 ) is the only simple possibility.

The day before Hilbert pronounced these phrases at the 1930 annual meeting of the Society of German Scientists and Physicians, Kurt Gödel — in a roundtable discussion during the Conference on Epistemology held jointly with the Society meetings — tentatively announced the first expression of his incompleteness theorem.

Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous finiteness theorem.

In 1961 the first important legislative victory of the Kennedy Administration came when the principle of national responsibility for local economic distress won out over a `` state's-responsibility '' proposal -- provision was made for payment for unemployment relief by nation-wide taxation rather than by a levy only on those states afflicted with manpower surplus.

In his recognition of his impersonal self the dancer moves, and this self, in the `` first revealed stroke of its existence '', states the theme from which all else must follow.

Although we are still far from a complete understanding of these problems, as a first approximation, it is suggested that alterations in the hypothalamic balance with consequent changes in the hypothalamic-cortical discharges account for major changes in behavior seen in various moods and states of emotions in man and beast under physiological circumstances, in experimental and clinical neurosis, and as the result of psychopharmacological agents.

The first meeting was held in Faneuil Hall, a great big place where we were able to meet members from all the other states.

Much of its shock value derives from the fact that the first portion of the essay describes the plight of starving beggars in Ireland, so that the reader is unprepared for the surprise of Swift's solution when he states, " A young healthy child well nursed, is, at a year old, a most delicious nourishing and wholesome food, whether stewed, roasted, baked, or boiled ; and I make no doubt that it will equally serve in a fricassee, or a ragoust.

The Articles of Confederation, formally the Articles of Confederation and Perpetual Union, was an agreement among the 13 founding states that established the United States of America as a confederation of sovereign states and served as its first constitution.

where the first sum is over positive energy states and the second over those of negative energy.

He was the first German noble to support Luther's ideas and in 1544 founded the University of Königsberg ( the so called Albertina ) as a rival to the Roman Catholic Cracow Academy ; it was only the second Lutheran university in the German states, after Marburg.

In early March Congress, led in part by Radical Republicans, passed the first in a series of four Reconstruction Acts, initially providing for the recognition of provisional governments to be established thereunder by the Southern states, on the condition that each state ratify the Fourteenth Amendment and assure suffrage for freedmen.

While Stradivari's first known violin states that he was a pupil of Amati, the validity of his statement is questioned.

124-126 .</ ref > While other philosophers also recognized such transitions in states of matter, Anaximenes was the first to associate the quality pairs hot / dry and cold / wet with the density of a single material and add a quantitative dimension to the Milesian monistic system .< ref > Kirk, G. S., J. E.

In the Fort Worth Star-Telegram daily newspaper ( morning edition ) 19 September 1970, J. Howard " Doc " DeCelles states that he was actually the victim of the first skyjacking in December 1929.

Armida was translated into German and widely performed, especially in the northern German states, where it helped to establish Salieri's reputation as an important and innovative modern composer It would also be the first opera to receive a serious preparation in a piano and vocal reduction by Carl Friedrich Cramer in 1783.

The ANFC ran the Championship of Australia, the first national club competition, which commenced in 1888 and saw clubs from different states compete on an even playing field.

The Calendar of Saints states that her first husband was poisoned by the holder of real power, his successor, Berengar of Ivrea, who attempted to cement his political power by forcing her to marry his son, Adalbert ; when she refused and fled, she was tracked down and imprisoned for four months at Como.

A coin by a contender for the imperial throne, Pacatianus, explicitly states " Year one thousand and first ", which is an indication that the citizens of the Empire had a sense of the beginning of a new era, a Saeculum Novum.

http :// uk. news. yahoo. com /​ oklahoma-says-running-death ​- penalty-drug-001238468. ht ​ ml reads: " Oklahoma was the first state in the country to use pentobarbital in 2010 after a shortage of another AESTHETIC, sodium thiopental, caused penal officials in death penalty states to look for an alternative .... Oklahoma could resort to another AESTHETIC never used before in executions, Massie said, or it could try to tap existing supplies of pentobarbital.

For example, in the U. S. states of California, Pennsylvania, Virginia, and Washington, barratry is a misdemeanor ; in Texas, a misdemeanor on the first conviction, but a felony on subsequent convictions.

Soviet-style centralised planning in five-year blocks had more immediate benefits there than in the other European states where it was first applied in the early 1950s.

Unlike the communist parties in most other East European states, the BCP ( changing its name to Bulgarian Socialist Party ) retained majority power after the transition in Bulgaria by winning the first free national elections in June 1990.

The energy released from transferring the electrons from high-energy states in NADH and quinol is conserved first as proton gradient and converted to ATP via ATP synthase.

The cell theory, first developed in 1839 by Matthias Jakob Schleiden and Theodor Schwann, states that all organisms are composed of one or more cells, that all cells come from preexisting cells, that vital functions of an organism occur within cells, and that all cells contain the hereditary information necessary for regulating cell functions and for transmitting information to the next generation of cells.