Nevertheless, the theory that the determining influence of the hypothalamic balance has a profound influence on the clinical behavior of neuropsychiatric patients has not yet been tested on an adequate number of patients.

Connes has applied his work in areas of mathematics and theoretical physics, including number theory, differential geometry and particle physics.

The exact number and placement of Endosymbiotic theory | endosymbiotic events is currently unknown, so this diagram can be taken only as a general guide It represents the most parsimonious way of explaining the three types of endosymbiotic origins of plastids.

They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous second incompleteness theorem.

In number theory, if P ( n ) is a property of positive integers, and if p ( N ) denotes the number of positive integers n less than N for which P ( n ) holds, and if

This is an example of renormalization in quantum field theory — the field theory being necessary because the number of particles changes from one to two and back again.

Wallace was one of the leading evolutionary thinkers of the 19th century and made a number of other contributions to the development of evolutionary theory besides being co-discoverer of natural selection.

Supporting literature includes: the work of social impact theory, which discusses persuasion in part through the number of persons engaging in influence ; as well as studies made on the relative influence of communicator credibility in different kinds of persuasion ; and examinations of the trustworthiness of the speaker.

It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory.

He is especially known for his foundational work in number theory and algebraic geometry.

He made substantial contributions in many areas, the most important being his discovery of profound connections between algebraic geometry and number theory.

Atle Selberg ( 14 June 1917 – 6 August 2007 ) was a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory.

Sir Andrew John Wiles, KBE, FRS ( born 11 April 1953 ) is a British mathematician and a Royal Society Research Professor at Oxford University, specializing in number theory.

His construction of new cohomology theories has left deep consequences for algebraic number theory, algebraic topology, and representation theory.

Alexander Grothendieck's work during the ` Golden Age ' period at IHÉS established several unifying themes in algebraic geometry, number theory, topology, category theory and complex analysis.

In that setting one can use birational geometry, techniques from number theory, Galois theory and commutative algebra, and close analogues of the methods of algebraic topology, all in an integrated way.

* abc conjecture, a concept in number theory

On a more abstract level, model theoretic arguments hold that a given set of symbols in a theory can be mapped onto any number of sets of real-world objects — each set being a " model " of the theory — providing the interrelationships between the objects are the same.

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

* The Mordell conjecture that a curve of genus greater than 1 over a number field has only finitely many rational points ;

In 1930-1960 research on the geometry of numbers was conducted by many number theorists ( including Louis Mordell, Harold Davenport and Carl Ludwig Siegel ).

In mathematics, the Chowla – Mordell theorem is a result in number theory determining cases where a Gauss sum is the square root of a prime number, multiplied by a root of unity.

He was awarded the Fields Medal in 1986 for proving the Mordell conjecture, which states that any non-singular projective curve of genus g > 1 defined over a number field K contains only finitely many K-rational points.

He made a number of conjectures in diophantine geometry: Mordell – Lang conjecture, Bombieri – Lang conjecture, Lang's integral point conjecture, Lang – Trotter conjecture, Lang conjecture on Gamma values, Lang conjecture on analytically hyperbolic varieties.

Louis Joel Mordell ( 28 January 1888 – 12 March 1972 ) was a British mathematician, known for pioneering research in number theory.

In mathematics, any number of cases supporting a conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample would immediately bring down the conjecture.

For example, the Riemann hypothesis is a conjecture from number theory that ( amongst other things ) makes predictions about the distribution of prime numbers.

Because many outstanding problems in number theory, such as Goldbach's conjecture are equivalent to solving the halting problem for special programs ( which would basically search for counter-examples and halt if one is found ), knowing enough bits of Chaitin's constant would also imply knowing the answer to these problems.

Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert – Pólya conjecture, for reasons that are anecdotal.

The most celebrated single question in the field, the conjecture known as Fermat's Last Theorem, was solved by Andrew Wiles but using tools from algebraic geometry developed during the last century rather than within number theory where the conjecture was originally formulated.

If the four-color conjecture were false, there would be at least one map with the smallest possible number of regions that requires five colors.

In 1973 the number theorist Hugh Montgomery was visiting the Institute for Advanced Study and had just made his pair correlation conjecture concerning the distribution of the zeros of the Riemann zeta function.

For instance, Goldbach's conjecture is the assertion that every even number ( greater than 2 ) is the sum of two prime numbers.

In the second edition of his book on number theory ( 1808 ) he then made a more precise conjecture, with A = 1 and B = − 1. 08366.

For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample ( i. e., a natural number n for which the Mertens function M ( n ) equals or exceeds the square root of n ) is known: all numbers less than 10 < sup > 14 </ sup > have the Mertens property, and the smallest number which does not have this property is only known to be less than the exponential of 1. 59 × 10 < sup > 40 </ sup >, which is approximately 10 to the power 4. 3 × 10 < sup > 39 </ sup >.

The Arnold conjecture on the number of fixed points of Hamiltonian symplectomorphisms and Lagrangian intersections were also a major motivation in the development of Floer homology.

The conjecture is that no matter what number you start with, you will always eventually reach 1.

The Collatz conjecture is: This process will eventually reach the number 1, regardless of which positive integer is chosen initially.

If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence which does not contain 1.

In 1849 de Polignac made the more general conjecture that for every natural number k, there are infinitely many prime pairs p and p ′ such that p ′ − p

A stronger form of the twin prime conjecture, the Hardy – Littlewood conjecture, postulates a distribution law for twin primes akin to the prime number theorem.

This conjecture can be justified ( but not proven ) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem.

Polignac's conjecture from 1849 states that for every positive even natural number k, there are infinitely many consecutive prime pairs p and p ′ such that p ′ − p = k ( i. e. there are infinitely many prime gaps of size k ).

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics.

Statistical considerations which focus on the probabilistic distribution of prime numbers present informal evidence in favour of the conjecture ( in both the weak and strong forms ) for sufficiently large integers: the greater the integer, the more ways there are available for that number to be represented as the sum of two or three other numbers, and the more " likely " it becomes that at least one of these representations consists entirely of primes.