The most basic example of the binomial theorem is the formula for the square of x + y:

In its basic form, the Chinese remainder theorem will determine a number n that when divided by some given divisors leaves given remainders.

Finally, it is a basic tool for proving theorems in modern number theory, such as Lagrange's four-square theorem and the fundamental theorem of arithmetic ( unique factorization ).

Although flawed, Kempe's original purported proof of the four color theorem provided some of the basic tools later used to prove it.

Any of the several well-known axiomatisations will do ; we assume without proof all the basic well-known results about our formalism ( such as the normal form theorem or the soundness theorem ) that we need.

After the basic form of the theorem is proved, it will be easy to extend it to the case of predicate calculus with equality.

This is the most basic form of the completeness theorem.

All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.

A basic theorem about Mersenne numbers states that in order for M < sub > p </ sub > to be a Mersenne prime, the exponent p itself must be a prime number.

Gödel's second incompleteness theorem ( 1931 ) shows that no formal system extending basic arithmetic can be used to prove its own consistency.

From about the 1920s onwards, it was realised that tensors play a basic role in algebraic topology ( for example in the Künneth theorem ).

The most famous result is Gödel's incompleteness theorem ; by representing theorems about basic number theory as expressions in a formal language, and then representing this language within number theory itself, Gödel constructed examples of statements that are neither provable nor disprovable from axiomatizations of number theory.

A new approach uses a functor from filtered spaces to crossed complexes defined directly and homotopically using relative homotopy groups ; a higher homotopy van Kampen theorem proved for this functor enables basic results in algebraic topology, especially on the border between homology and homotopy, to be obtained without using singular homology or simplicial approximation.

Bertrand considers only special cases of problems from electrodynamics and heat conduction, but his article contains in distinct terms all basic ideas of modern proof of the pi – theorem and clear indication of the use of the pi – theorem for modelling physical phenomena.

From the mid 50's he moved into singularity theory, of which catastrophe theory is just one aspect, and in a series of deep ( and at the time obscure ) papers between 1960 and 1969 developed the theory of stratified sets and stratified maps, proving a basic stratified isotopy theorem describing the local conical structure of Whitney stratified sets, now known as the Thom-Mather isotopy theorem.

Thus Tychonoff's theorem joins several other basic theorems ( e. g. that every nonzero vector space has a basis ) in being equivalent to AC.

He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlet's approximation theorem.

* Book 1 contains Euclid's 10 axioms ( 5 named postulates — including the parallel postulate — and 5 named axioms ) and the basic propositions of geometry: the pons asinorum ( proposition 5 ), the Pythagorean theorem ( Proposition 47 ), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are " equal " ( have the same area ).

The bipartite graphs, line graphs of bipartite graphs, and their complements form four out of the five basic classes of perfect graphs used in the proof of the strong perfect graph theorem.

But according to a basic theorem any algebraic group is an extension of an abelian variety by a linear algebraic group.

So we may conceive the coexistence of the infinite number of universal, apparently momentary states of matter, successive one after another in consciousness, but permanent each on its own basic phase of the progressive frequencies.

I support the cosmological hypothesis which states that the development of the universe is repeated in its basic features an infinite number of times.

Under the rules, the two basic states of play during a game are ball in play and ball out of play.

For example, in England and Wales and in most states of the United States, the basic law of contracts, torts and property do not exist in statute, but only in common law ( though there may be isolated modifications enacted by statute ).

During the classical era of Buddhist philosophy in India, philosophers such as Dharmakirti argue for a dualism between states of consciousness and Buddhist atoms ( the basic building blocks that make up reality ), according to " the standard interpretation " of Dharmakirti's Buddhist metaphysics.

Buddhist dualism, in Dharmakirti ’ s sense, is different in that it is not a dualism between the mind and brain, but rather between states of consciousness ( nonphysical ) and basic building blocks ( according to the Buddhist atomism of Dharmakirti, Buddhist atoms are also nonphysical: they are unstructured points of energy ).

The foreword to the DSM-I states the US Navy had itself made some minor revisions but " the Army established a much more sweeping revision, abandoning the basic outline of the Standard and attempting to express present day concepts of mental disturbance.

In this skeptics oppose foundationalism, which states that there have to be some basic beliefs that are justified without reference to others.

For example, a basic property of addition is commutativity which states that the order of numbers being added together does not matter.

Typically, internalist foundationalists hold that basic beliefs are justified by mental events or states, such as experiences, that do not constitute beliefs ( these are called non-doxastic mental states ).

The basic design of the current flag is specified by ; outlines the addition of new stars to represent new states.

During the 17th century, the basic tenets of the Grotian or eclectic school, especially the doctrines of legal equality, territorial sovereignty, and independence of states, became the fundamental principles of the European political and legal system and were enshrined in the 1648 Peace of Westphalia.

A physical system is generally described by three basic ingredients: states ; observables ; and dynamics ( or law of time evolution ) or, more generally, a group of physical symmetries.

While the title states that it is naive, which is usually taken to mean without axioms, the book does introduce all the axioms of Zermelo – Fraenkel set theory and gives correct and rigorous definitions for basic objects.

It focuses on " basic human goods ," such as human life, knowledge, and aesthetic experience, which are self-evidently and intrinsically worthwhile, and states that these goods reveal themselves as being incommensurable with one another.

However, a 1988 article by Brian Martin in Science and Public Policy states that although their paper concluded the effects would be less severe than originally thought, with the authors describing these effects as a " nuclear autumn ", other statements by Thompson and Schneider show that they " resisted the interpretation that this means a rejection of the basic points made about nuclear winter ".

These theorists were driven by two basic questions: one, by what right or need do people form states ; and two, what the best form for a state could be.

Only the " fundamental rights " under the federal constitution apply to Puerto Rico like the Privileges and Immunities Clause ( U. S. Constitution, Article IV, Section 2, Clause 1, also known as the ' Comity Clause ') that prevents a state from treating citizens of other states in a discriminatory manner, with regard to basic civil rights.

In the 1994 issue of Foreign Affairs, National Security Advisor Anthony Lake claimed " the reality of recalcitrant and outlaw states that not only choose to remain outside the family democratic nations but also assault its basic values.

This particular adaptation of the mixed market economy is characterised by more generous welfare states ( relative to other developed countries ), which are aimed specifically at enhancing individual autonomy, ensuring the universal provision of basic human rights and stabilising the economy.

Cases of development of common law into Socialist law are unknown because of incompatibility of basic principles of these two systems ( common law presumes influential rule-making role of courts while courts in socialist states play a dependent role ).

The current notion of state sovereignty is often traced back to the Peace of Westphalia ( 1648 ), which, in relation to states, codified the basic principles:

The group's motto was God and the People, and its basic principle was the unification of the several states and kingdoms of the peninsula into a single republic as the only true foundation of Italian liberty.