In mathematics, the Lyusternik – Schnirelmann category ( or, Lusternik – Schnirelmann category, LS-category, or simply, category ) of a topological space is the homotopical invariant defined to be the smallest integer number such that there is an open covering of with the property that each inclusion map is nullhomotopic.

His works, published in Proceedings of the London Mathematical Society, were " Density integration " ( 53, 1951 ); " On the measure of sum sets ( I ) The theorems of Brunn, Minkowski, and Lusternik, ( with A. M. McBeath )" ( 3, 1953 ); " Linear functions with domain a real countably infinite dimensional space " ( 5, 1955 ); " Linear and bilinear functions with domain contained in a real countably infinite dimensional space " ( 6, 1956 ); " The use of convergence factors in Ward integration " ( 10, 1960 ); " The equivalence of generalized forms of the Ward, variational, Denjoy-Stieltjes, and Perron-Stieltjes integrals " ( 10, 1960 ); " N-variation and N-variational integrals of set functions " ( 11, 1961 ); " Definitions of Riemann type of the variational integrals " ( 11, 1961 ); " Difference-sets and the Banach – Steinhaus theorem " ( 13, 1963 ); " Generalized integrals of vector-valued functions ( 19 1969 )

* 1905 – Lev Schnirelmann, Russian mathematician ( d. 1938 )

** Lyusternik – Schnirelmann category, sometimes called LS-category or simply category

He was granted a doctorate in mathematics in 1969 under the supervision of Nicolaas Kuiper for a thesis entitled The minimal number of critical points of a function on a compact manifold and the Lyusternik – Schnirelmann category.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

** The Nielsen – Schreier theorem, that every subgroup of a free group is free.

** The Hahn – Banach theorem in functional analysis, allowing the extension of linear functionals

** The Banach – Alaoglu theorem about compactness of sets of functionals.

** The numbers and are not algebraic numbers ( see the Lindemann – Weierstrass theorem ); hence they are transcendental.

This is a result of Galois theory ( see Quintic equations and the Abel – Ruffini theorem ).

Transmission, Gregory Chaitin also presents this theorem in J. ACM – Chaitin's paper was submitted October 1966 and revised in December 1968, and cites both Solomonoff's and Kolmogorov's papers.

This began in his doctoral work leading to the Mordell – Weil theorem ( 1928, and shortly applied in Siegel's theorem on integral points ).

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 ).

His ' matrix divisor ' ( vector bundle avant la lettre ) Riemann – Roch theorem from 1938 was a very early anticipation of later ideas such as moduli spaces of bundles.

* Borel – Weil theorem

* De Rham – Weil theorem

* Mordell – Weil theorem

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.

In 1956, he applied the same thinking to the Riemann – Roch theorem, which had already recently been generalized to any dimension by Hirzebruch.

Statements such as the Banach – Tarski paradox can be rephrased as conditional statements, for example, " If AC holds, the decomposition in the Banach – Tarski paradox exists.

If amber is heated under the right conditions, oil of amber is produced, and in past times this was combined carefully with nitric acid to create " artificial musk " – a resin with a peculiar musky odor.

Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that

If this etymology is combined with the tradition reported by Geoffrey of Monmouth stating that Ambrosius Aurelianus ordered the building of Stonehenge – which is located within the parish of Amesbury ( and where Ambrosius was supposedly buried ) – and with the presence of an Iron Age hill fort also in that parish, then it may be tempting to connect Ambrosius with Amesbury.

If anything, her poems suggest the opposite – they speak of quietly experienced but intensely felt emotions, hidden from others, without any indication of being requited.

If Acorn had thought to include this small modification in the original Electron design it is likely the machine would have had a much greater impact as it would have nearly doubled the amount of motion possible in games and saved modes 0 – 3 ( including the only 16 colour mode ) from being nearly useless due to contended memory timings.

If during the 14 – 15th centuries armour seldom weighed more than 15 kg, then by the late 16th century it weighed 25 kg.

If this article / noun pair is used as the object of a verb, it ( usually ) changes to the accusative case, which entails an article shift in German – Ich sehe den Wagen.

* If the third baseman fields a ball and throws it to first, it is recorded as a 5 – 3 out.

If the score reaches 20-all, then the game continues until one side gains a two point lead ( such as 24 – 22 ), up to a maximum of 30 points ( 30 – 29 is a winning score ).

But this destruction, coupled with a protracted siege of Rain ( 9 – 16 July ), caused Prince Eugene to lament "... since the Donauwörth action I cannot admire their performances ", and later to conclude " If he has to go home without having achieved his objective, he will certainly be ruined.

If a " clinch " – a defensive move in which a boxer wraps his or her opponents arms and holds on to create a pause – is broken by the referee, each fighter must take a full step back before punching again ( alternatively, the referee may direct the fighters to " punch out " of the clinch ).

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula for details.

If a cabal can be implicated in a sequence of events, there is always the hope, however tenuous, of being able to break the cabal's power – or joining it and exercising some of that power oneself.

If the corporate President is not the COO ( such as Richard Parsons of Time Warner from 1995 – 2001 ), then many division heads report directly to the CEO themselves, with the President taking on special assignments from the CEO.

If their existence past the K – T boundary can be confirmed, these hadrosaurids would be considered a Dead Clade Walking.

: " If anyone kills person – unless it be ( a punishment ) for murder or for spreading mischief in the land — it would be as if he killed all people.

* If the Venerable was not a martyr – all non-martyrs are " confessors " as they " confessed " or bore witness to their faith by how they lived their lives – it must be proven that a miracle has taken place by his or her intercession: that is, that God has shown a sign that the person is enjoying the Beatific Vision by God performing a miracle in response to the Blessed's prayers.

The idea that immediately suggested itself was that the star's declination varied because of short-term changes in the orientation of the Earth's axis relative to the celestial sphere – a phenomenon known as nutation.

His first " continuous tension – discontinuous compression " geodesic dome ( full sphere in this case ) was constructed at the University of Oregon Architecture School in 1959 with the help of students.

* Private sphere –

* Public sphere –

In the aftermath of the 1917 – 18 crisis and civil war, Finland passed from Russian rule to the German Empire's sphere of power.

Early in Classical Greece, that the earth is a sphere (" round "), was generally known by all, and around 240 BCE, Eratosthenes ( 276 BCE – 194 BCE ) accurately estimated its circumference.

Some writers have proposed building megastructures that have some similarities to a hollow Earth – see Dyson sphere, Globus Cassus.

* Academic discipline – body of knowledge given to-or received by-a disciple ( student ); a branch or sphere of knowledge, or field of study, that an individual has chosen to specialise in.

* Academic discipline – body of knowledge given to-or received by-a disciple ( student ); a branch or sphere of knowledge, or field of study, that an individual has chosen to specialise in.

Recognition by the United Kingdom in the 1904 Entente Cordiale of France's " sphere of influence " in Morocco provoked a German reaction ; the " crisis " of 1905 – 1906 was resolved at the Algeciras Conference ( 1906 ), which formalized France's " special position " and entrusted policing of Morocco jointly to France and Spain.

Eleven days after the Soviet invasion of the polish Kresy, the secret protocol of the Molotov – Ribbentrop Pact was modified by the German – Soviet Treaty of Friendship, Cooperation and Demarcation ,) allotting Germany a larger part of Poland and transferring Lithuania's territory ( with the exception of left bank of river Scheschupe, the " Lithuanian Strip ") from the envisioned German sphere to the Soviets.

Convex polytopes are equivalent to tilings of the sphere, while others may be tilings of other elliptic, flat or toroidal surfaces – see elliptic tiling and toroidal polyhedron.

This is merely the math behind the line – sphere intersection and the subsequent determination of the colour of the pixel being calculated.

In the cultural sphere, Scott's Waverley novels played a significant part in the movement ( begun with James Macpherson's Ossian cycle ) in rehabilitating the public perception of the culture of the Scottish Highlands and its culture, which had been formally suppressed as barbaric – and viewed in the southern mind as a breeding ground of hill bandits, religious fanaticsim, and Jacobite rebellions.

* Academic discipline – body of knowledge given to-or received by-a disciple ( student ); a branch or sphere of knowledge, or field of study, that an individual has chosen to specialise in.

* 1094 – The Chinese mechanical engineer and astronomer Su Song incorporates an escapement mechanism and the world's first known chain drive to operate the armillary sphere, the astronomical clock, and the striking clock jacks of his clock tower in Kaifeng.

After the widespread Göktürk revolt of Shabolüe Khan ( d. 658 ) was put down at Issyk Kul in 657 by Su Dingfang ( 591 – 667 ), Emperor Gaozong established several protectorates governed by a Protectorate General or Grand Protectorate General, which extended the Chinese sphere of influence as far as Herat in Western Afghanistan.

Using a water clock, waterwheel and a series of gears, the Court Astronomer Zhang Heng ( 78 – 139 CE ) was able to mechanically rotate his metal-ringed armillary sphere.

If Halley was once a long-period comet, it is likely to have originated in the Oort Cloud, a sphere of cometary bodies that has an inner edge of 20, 000 – 50, 000 AU.

During the first three years after the end of the war ( 1945 – 1948 ), however, literature maintained a certain degree of freedom, although the strengthening of the extreme left gradually pushed out of the public sphere first the Catholic authors ( Deml, Durych, Čep, Zahradníček ), then the moderate Communists.

The Earth is not a perfect sphere, it is an oblate spheroid or ellipsoid ( i. e. – slightly compressed at the poles ), which means that the shortest distance between two points ( a geodesic ) is not quite a great circle.

| 7 | Earlier the French had incorporated Tunisia into their sphere ( 1881 – 1956 ), preceded by many Italian settlers, merchants and farmers.

Between 1590 – 1712 the Dutch also possessed one of the strongest navies in the world, allowing for their varied conquests including breaking the Portuguese sphere of influence on the Indian Ocean and in the Orient.