The Nyquist – Shannon sampling theorem states that a signal can be exactly reconstructed from its samples if the sampling frequency is greater than twice the highest frequency of the signal ; but requires an infinite number of samples.

Experimentally, however, the total kinetic energy is found to be much greater: in particular, assuming the gravitational mass is due to only the visible matter of the galaxy, stars far from the center of galaxies have much higher velocities than predicted by the virial theorem.

In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 is either prime itself or is the product of prime numbers, and that, although the order of the primes in the second case is arbitrary, the primes themselves are not.

The unique factorization theorem says that every positive integer greater than 1 can be written in only one way as a product of prime numbers.

In 1824, Niels Henrik Abel proved the striking result that there can be no general ( finite ) formula, involving only arithmetic operations and radicals, that expresses the roots of a polynomial of degree 5 or greater in terms of its coefficients ( see Abel-Ruffini theorem ).

The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering.

In 1961 Stephen Smale shocked mathematicians by proving the Generalized Poincaré conjecture for dimensions greater than four and extended his techniques to prove the fundamental h-cobordism theorem.

According to the Rouché – Capelli theorem, the system is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix.

The local gravitational acceleration at the poles is greater than at the equator, so, by the Von Zeipel theorem, the local luminosity is also higher at the poles.

He then cemented his reputation with a proof of the Poincaré conjecture for all dimensions greater than or equal to 5, published in 1961 ; in 1962 he generalized the ideas in a 107 page paper that established the h-cobordism theorem.

) Thus, the group of conformal transformations in spaces of dimension greater than 2 are much more restricted than the planar case, where the Riemann mapping theorem provides a large group of conformal transformations.

In mathematics, Minkowski's theorem is the statement that any convex set in R < sup > n </ sup > which is symmetric with respect to the origin and with volume greater than 2 < sup > n </ sup > d ( L ) contains a non-zero lattice point.

Minkowski's theorem states that if the volume of S is strictly greater than 2 < sup > n </ sup > d ( L ), then S must contain at least one lattice point other than the origin.

For n = 2 the theorem claims that a convex figure in the plane symmetric about the origin and with area greater than 4 encloses at least one lattice point in addition to the origin.

The Nyquist – Shannon sampling theorem states that perfect reconstruction of a signal is possible when the sampling frequency is greater than twice the maximum frequency of the signal being sampled, or equivalently, when the Nyquist frequency ( half the sample rate ) exceeds the highest frequency of the signal being sampled.

By the fundamental theorem of arithmetic, every positive integer greater than 1 has a unique prime factorization.

Niels Henrik Abel ( 1802 – 1829 ), a Norwegian, and Évariste Galois, ( 1811 – 1832 ) a Frenchman, investigated into the solutions of various polynomial equations, and proved that there is no general algebraic solution to equations of degree greater than four ( Abel – Ruffini theorem ).

A sample application of Faltings ' theorem is to a weak form of Fermat's Last Theorem: for any fixed n > 4 there are at most finitely many primitive integer solutions to a < sup > n </ sup > + b < sup > n </ sup > = c < sup > n </ sup >, since for such n the curve x < sup > n </ sup > + y < sup > n </ sup > = 1 has genus greater than 1.

: A graph with n vertices ( n ≥ 3 ) is Hamiltonian if, for each pair of non-adjacent vertices, the sum of their degrees is n or greater ( see Ore's theorem ).

The Bonnet – Myers theorem states that a complete Riemannian manifold which has Ricci curvature everywhere greater than a certain positive constant must be compact.

These properties lead to greater efficiencies in theorem proving.

Because of Cantor's theorem each set in the preceding sequence has cardinality strictly greater than the one preceding it.

The long-settled areas of states like Virginia and South Carolina developed the ante-bellum culture to its richest flowering, and there the memory is more precious, and the consciousness of loss the greater.

The fact that the Americans who upheld the sovereignty of their states did this in order to keep many of their people more securely in slavery -- the antithesis of individual liberty -- made the conflict grimmer, and the greater.

He called for a greater attraction of industry and a stop to the piracy of industry by Southern states, and a strong fight against discrimination in business and industry.

Typically, elements capable of variable coordination number and / or oxidation states tend to exhibit greater numbers of allotropic forms.

However, slavery was a social and economic institution, particularly in eleven states in the American South, such that a variety of organizations were established advocating the movement of black people from the United States to locations where they would enjoy greater freedom and equality.

Concerning the liberal use of the death penalty in the Draconic code, Plutarch states: " It is said that Drakon himself, when asked why he had fixed the punishment of death for most offences, answered that he considered these lesser crimes to deserve it, and he had no greater punishment for more important ones.

In Rerum Novarum, Leo XIII states that people are likely to work harder and with greater commitment if they themselves possess the land on which they labour, which in turn will benefit them and their families as workers will be able to provide for themselves and their household.

Because of the extensive experience of the Delaware courts, Delaware has a more well-developed body of case law than other states, which serves to give corporations and their counsel greater guidance on matters of corporate governance and transaction liability issues.

It states that for all integers n and k greater than 1, if the sum of n kth powers of positive integers is itself a kth power, then n is greater than or equal to k.

Lynn states that " This rules out improvements in education, greater test sophistication, etc.

Tornadoes are infrequent at the coast but do occur, however the frequency at which they occur in inland portions of Gulf Coast states is much greater.

Unlike the Khoisan, the Bantu states were not conquered by the Afrikaners, because of population density and greater unity.

Constituted in 1966, it is one of two newly created states carved out of the greater Punjab province as a means of creating a stronger national identity.

However, the central government in India has greater power in relation to its states, and its central government is patterned after the British parliamentary system.

During the unification of Germany, the term Großdeutschland ( or greater Germany ) referred to a possible German nation consisting of the states that later comprised the Second German Empire and Austria ; the term lesser Germany, or small Germany, or Kleindeutschland, referred to a possible German state without Austria.

Zygmunt Gloger in his work Historical geography of land of ancient Poland ( Geografia historyczna ziem dawnej Polski ) states that according to a Polish custom, whenever a new village was formed next to an older one, the name of the new entity was presented with an adjective little ( or lesser ), while the old village was described as greater.

All European monarchies are constitutional ones, with the exception of the Vatican City, but sovereigns in the smaller states exercise greater political influence than in the larger.

As well as giving greater priority to Latin America and the Caribbean states.

In most states, the Coalition agreement is not in force when the parties are in opposition, allowing the two parties greater freedom of action.

Nansen and others prefer to give the cotangent 209 / 600, which is the inverse of the tangent, but the angle is greater than 45 ° and it is the tangent that Strabo states.

The Socialist's 1962 congress pushed for greater democratisation and powers for Parliament though it was only in 1969 that this possibility was examined by the member states.

Author Bill Bryson states that Perth is the most remote city on earth, which he justifies by noting that the population of metropolitan Perth is greater than the combined populations of the rest of Western Australia, the Northern Territory and South Australia, west of Adelaide.