Theorem: Given A1 and A2, A3 ’ and A4 ’ imply A3 and A4.

Theorem: Given A1 and A2, A3 ” and A4 ” imply A3 and A4.

* Theorem ( by Mackey ): Given a dual pair, the bounded sets under any dual topology are identical.

In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.

: Theorem: Armstrong's axioms are sound and complete ; given a header and a set of FDs that only contain subsets of, if and only if holds in all relation universes over in which all FDs in hold.

On 13 August, 2012, this project was officially announced to be The Zero Theorem, set to start shooting in Bucharest on October 22, produced by Dean Zanuck ( son to the late Richard D. Zanuck who was to originally produce in 2009 ), worldwide sales handled by Voltage Pictures, Toronto and starring Academy Award winner Christoph Waltz in the lead, replacing Billy Bob Thornton who had been attached to the project in 2009.

In social choice theory, Arrow ’ s impossibility theorem, the General Possibility Theorem, or Arrow ’ s paradox, states that, when voters have three or more distinct alternatives ( options ), no rank order voting system can convert the ranked preferences of individuals into a community-wide ( complete and transitive ) ranking while also meeting a specific set of criteria.

This book reprints much of Boolos's work on the rehabilitation of Frege, as well as a number of his papers on set theory, second-order logic and nonfirstorderizability, plural quantification, proof theory, and three short insightful papers on Gödel's Incompleteness Theorem.

The Second Fundamental Theorem implies that the set of deficient values of a function meromorphic in the plane is at most countable and the following relation holds:

In his dissertation Sur les fonctions de variable réelles (" On the Functions of Real Variables "), Baire studied a combination of set theory and analysis topics to arrive at the Baire Category Theorem and define the nowhere dense set.

A summary statement is given by Veblen & Young as Theorem 10: " The set of points on a line, with removed, forms a field with respect to the operations previously defined ".

: Theorem: If S is any set then S cannot contain elements of all cardinalities.

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.

: Theorem ( A. Korselt 1899 ): A positive composite integer is a Carmichael number if and only if is square-free, and for all prime divisors of, it is true that ( where means that divides ).

Image: Thales ' Theorem Simple. svg | Thales ' theorem: if AC is a diameter, then the angle at B is a right angle.

To see the equivalence, note first that if Theorem 1 holds, and φ is not satisfiable in any structure, then ¬ φ is valid in all structures and therefore provable, thus φ is refutable and Theorem 2 holds.

In the General Possibility Theorem, Kenneth Arrow argues that if a legislative consensus can be reached through a simple majority, then minimum conditions must be satisfied, and these conditions must provide a superior ranking to any subset of alternative votes ( Arrow 1963 ).

#* Note: This fact provides a proof of the infinitude of primes distinct from Euclid's Theorem: if there were finitely many primes, with p being the largest, we reach an immediate contradiction since all primes dividing 2 < sup > p </ sup > − 1 must be larger than p .</ li >

Urysohn's Theorem can be restated as: A topological space is separable and metrizable if and only if it is regular, Hausdorff and second-countable.

: Theorem: A superkey holds in a relation universe over if and only if and holds in.

: Theorem: An FD is trivial under a header if and only if.

* Birkhoff's HSP Theorem, which states that a class of algebras is a variety if and only if it is closed under homomorphic images, subalgebras, and arbitrary direct products.

The Coase Theorem states that assigning property rights will lead to an optimal solution, regardless of who receives them, if transaction costs are trivial and the number of parties negotiating is limited.

Theorem: The angle may be trisected if and only if is reducible over the field extension Q.

He began research under the supervision of Ernest William Barnes, who suggested that he attempt to prove the Riemann hypothesis: Littlewood showed that if the Riemann hypothesis is true then the Prime Number Theorem follows and obtained the error term.

" The Implicit Function Theorem states that if is defined on an open disk containing, where,, and and are continuous on the disk, then the equation defines as a function of near the point and the derivative of this function is given by ..."

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

It follows now that we need only prove Theorem 2 for formulas φ in normal form.

By the Chinese Remainder Theorem, this ring factors into the direct product of rings of integers mod p. Now each of these factors is a field, so it is clear that the only idempotents will be 0 and 1.

There is no formal distinction between a lemma and a theorem, only one of intention – see Theorem # Terminology.

Her brilliant theorem is known only because of the footnote in Legendre's treatise on number theory, where he used it to prove Fermat's Last Theorem for p = 5 ( see Correspondence with Legendre ).

If n is not a prime power, then every Sylow subgroup is proper, and, by Sylow's Third Theorem, we know that the number of Sylow p-subgroups of a group of order n is equal to 1 modulo p and divides n. Since 1 is the only such number, the Sylow p-subgroup is unique, and therefore it is normal.

For Theorem I the data are generated in full range, while in Theorem II data is only generated when it surpasses a certain threshold, called Peak Over Threshold models ( POT ).

However, the link between the Riemann hypothesis and the Prime Number Theorem had been known before in Continental Europe, and Littlewood also wrote later in his book A mathematician ’ s miscellany that his actually only rediscovered result did not shed a bright light on the isolated state of British mathematics at the time.

* Case g > 1: according to the Mordell conjecture, now Faltings ' Theorem, C has only a finite number of rational points.

Good on what is now called the prime-factor FFT algorithm ( PFA ); although Good's algorithm was initially mistakenly thought to be equivalent to the Cooley – Tukey algorithm, it was quickly realized that PFA is a quite different algorithm ( only working for sizes that have relatively prime factors and relying on the Chinese Remainder Theorem, unlike the support for any composite size in Cooley – Tukey ).

# The First Fundamental Theorem of Asset Pricing: A discrete market on a discrete probability space ( Ω,, ) is arbitrage-free if and only if there exists at least one risk neutral probability measure that is equivalent to the original probability measure, P.

# The Second Fundamental Theorem of Asset Pricing: An arbitrage-free market ( S, B ) consisting of a collection of stocks S and a risk-free bond B is complete if and only if there exists a unique risk-neutral measure that is equivalent to P and has numeraire B.

In 1945 Prigogine ( see also Prigogine ( 1947 )) proposed a “ Theorem of Minimum Entropy Production ” which applies only to the linear regime near a stationary thermodynamically non-equilibrium state.

However, 14 kbit / s is only 40 % of the theoretical maximum bit rate predicted by Shannon's Theorem for POTS lines ( approximately 35 kbit / s ).

* Bertrand's Theorem, a theorem in classical mechanics stating only two types of central force potentials produce stable, closed orbits