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.

The equation was eventually solved by Euler in the early 18th century, who also solved a number of other Diophantine equations.

The Euler – Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives ƒ < sup >( k )</ sup > at the end points of the interval m and n. Explicitly, for any natural number p, we have

For closed ( orientable or non-orientable ) surfaces with positive genus, the maximum number p of colors needed depends on the surface's Euler characteristic χ according to the formula

Euler worked in almost all areas of mathematics: geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics.

The Swiss mathematician Leonhard Euler pioneered the modern approach to congruence in about 1750, when he explicitly introduced the idea of congruence modulo a number N.

In number theory, the Euler numbers are a sequence E < sub > n </ sub > of integers defined by the following Taylor series expansion:

The Euler number E < sub > 2n </ sub > can be expressed as a sum over the even partitions of 2n,

Alternatively, it is possible to show that any bridgeless bipartite planar graph with n vertices and m edges has by combining the Euler formula ( where f is the number of faces of a planar embedding ) with the observation that the number of faces is at most half the number of edges ( because each face has at least four edges and each edge belongs to exactly two faces ).

This means that as the number of contours increases, Euler diagrams are typically less visually complex than the equivalent Venn diagram, particularly if the number of non-empty intersections is small.

The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere.

In number theory, Euler's theorem ( also known as the Fermat – Euler theorem or Euler's totient theorem ) states that if n and a are coprime positive integers, then

If the Euler criterion formula is used modulo a composite number, the result may or may not be the value of the Jacobi symbol.

Orientation-free metrics of a group of connected or surrounded pixels include the Euler number, the perimeter, the area, the compactness, the area of holes, the minimum radius, the maximum radius.

In number theory, an odd composite integer n is called an Euler – Jacobi pseudoprime to base a, if a and n are coprime, and

Note that for orientable compact surfaces without boundary, the Euler characteristic equals, where is the genus of the surface: Any orientable compact surface without boundary is topologically equivalent to a sphere with some handles attached, and counts the number of handles.

More generally, if the polyhedron has Euler characteristic ( where g is the genus, meaning " number of holes "), then the sum of the defect is

The difference between the nth harmonic number and the natural logarithm of n converges to the Euler – Mascheroni constant.

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic ( or Euler – Poincaré characteristic ) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.

where B < sub > k </ sub > is a Bernoulli number and R < sub > m, n </ sub > is the remainder term in the Euler – Maclaurin formula.

Bridge 8: Euler walks are possible if exactly zero or two nodes have an odd number of edges.

Amongst the fruits of his industry may be mentioned a laborious investigation of the disturbances of Jupiter by Saturn, the results of which were employed and confirmed by Euler in his prize essay of 1748 ; a series of lunar observations extending over fifty years ; some interesting researches in terrestrial magnetism and atmospheric electricity, in the latter of which he detected a regular diurnal period ; and the determination of the places of a great number of stars, including at least twelve separate observations of Uranus, between 1750 and its discovery as a planet.

But even if that other plant employs the same number of workers and makes the same product, there are other facts to consider.

Alpha decay is by far the most common form of cluster decay where the parent atom ejects a defined daughter collection of nucleons, leaving another defined product behind ( in nuclear fission, a number of different pairs of daughters of approximately equal size are formed ).

It is often convenient to write this as an infinite product over all the primes, where all but a finite number have a zero exponent.

A number of formal and industry standards exist for bicycle components to help make spare parts exchangeable and to maintain a minimum product safety.

The inner product of two vectors is a complex number.

# for each line, the number of product groups is equal to.

Now setting all of the X < sub > s </ sub > equal to the unlabeled variable X, so that the product becomes, the term for each k-combination from S becomes X < sup > k </ sup >, so that the coefficient of that power in the result equals the number of such k-combinations.

While at university, Gauss independently rediscovered several important theorems ; his breakthrough occurred in 1796 when he showed that any regular polygon with a number of sides which is a Fermat prime ( and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a power of 2 ) can be constructed by compass and straightedge.

The type of end product resulting from a condensation polymerization is dependent on the number of functional end groups of the monomer which can react.

Furthermore, if b < sub > 1 </ sub > and b < sub > 2 </ sub > are both coprime with a, then so is their product b < sub > 1 </ sub > b < sub > 2 </ sub > ( modulo a it is a product of invertible elements, and therefore invertible ); this also follows from the first point by Euclid's lemma, which states that if a prime number p divides a product bc, then p divides at least one of the factors b, c.

The total amount of scattering in a sparse medium is determined by the product of the scattering cross-section and the number of particles present.

This space is homeomorphic to the product of a countable number of copies of the discrete space S.

For example, a sales transaction can be broken up into facts such as the number of products ordered and the price paid for the products, and into dimensions such as order date, customer name, product number, order ship-to and bill-to locations, and salesperson responsible for receiving the order.

Here n is the number of electrons / mole product, F is the Faraday constant ( coulombs / mole ), and ΔE is cell potential.

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.

In algebraic number theory 2 is called irreducible ( only divisible by itself or a unit ) but not prime ( if it divides a product it must divide one of the factors ).

The gamma function must alternate sign between the poles because the product in the forward recurrence contains an odd number of negative factors if the number of poles between and is odd, and an even number if the number of poles is even.

The number 54 can be expressed as a product of two other integers in several different ways: