In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

In mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients ( equivalently, its abelianization, which is the universal abelian quotient, is trivial ).

In mathematics, a metabelian group is a group whose commutator subgroup is abelian.

There are many distinct FFT algorithms involving a wide range of mathematics, from simple complex-number arithmetic to group theory and number theory ; this article gives an overview of the available techniques and some of their general properties, while the specific algorithms are described in subsidiary articles linked below.

In one system which is usual in physics ( r, θ, φ ) gives the radial distance, polar angle, and azimuthal angle, whereas in another system used in many mathematics books ( r, θ, φ ) gives the radial distance, azimuthal angle, and polar angle.

Internationally, the university is known for research relating to the genome of the Populus tree ( Life sciences ), contributions to the Gleason problem and function spaces on fractals ( mathematics ) and its school of industrial design which gives degree programs in English open to students from all of the world.

* What gives mathematics its hold on experience?

The applied tools of the mathematics disciplines of Celestial mechanics or its subfield Orbital mechanics ( both predict orbital paths and positions ) about a center of gravity are used to generate an ephemeris ( plural: ephemerides ; from the Greek word ephemeros = daily ) which is a table of values that gives the positions of astronomical objects in the sky at a given time or times, or a formula to calculate such given the proper time offset from the epoch.

* What gives mathematics its hold on experience?

* Multiplicative inverse, in mathematics, the number 1 / x, which multiplied by x gives the product 1, also known as a reciprocal

It gives no indication on which axiomatic system should be prefered as a foundation of mathematics.

Brouwer gives brief synopsis of his belief that the law of excluded middle cannot be " applied without reservation even in the mathematics of infinite systems " and gives two examples of failures to illustrate his assertion.

In combinatorial mathematics, Hall's marriage theorem, or simply Hall's Theorem, gives a necessary and sufficient condition for being able to select a distinct element from each of a collection of finite sets.

Some logicians, while accepting that classical mathematics is correct, still believe that the constructive approach gives a better insight into the true meaning of theorems, in much this way.

In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. It is named after George Green, and is the two-dimensional special case of the more general Stokes ' theorem.

In mathematics, a function is well-defined if it gives the same result when the form ( the way in which it is presented ) but not the value of an input is changed.

In mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain.

Meanwhile, a mathematics teacher named Elizabeth Whittaker, who was also present at the party, gives Hercule Poirot an important piece of evidence when she reveals that while the party-goers were playing Snapdragon, Elizabeth went out to hall and saw Rowena Drake coming out of the lavatory on the first floor landing.

On the other hand Brouwer gives strong counterexamples, based on properties that hold only in his constructive mathematics.

In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators.

In the mathematical fields of topology and K-theory, the Serre – Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: " projective modules over commutative rings are like vector bundles on compact spaces ".

Most importantly, through its Charitable Trust, it gives help and support to those who are less fortunate, with an emphasis on making a difference in improving education, particularly in the area of mathematics.

In mathematics, Eisenstein < nowiki >' s </ nowiki > criterion gives an easily checked sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers — that is, for it to be unfactorable into the product of lower-degree polynomials with rational coefficients.

In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.

* Euler gives up his post as director of mathematics at the Prussian Academy of Sciences and returns to Saint Petersburg.

Under his patronage, Galileo got an indication to a professorship of mathematics at the University of Pisa, in 1589.

In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent.

As a consequence of this demonstration, Descartes considers science and mathematics to be justified to the extent that their proposals are established on a similarly immediate clarity, distinctiveness, and self-evidence that presents itself to the mind.

The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.

In the same lists, MIT's strongest showings apart from in engineering are in computer science, the natural sciences, business, economics, linguistics, mathematics, and, to a lesser extent, political science and philosophy.

His work was intended as recreational mathematics, but he suggested a purely octal system of weights and measures and observed that the existing system of English units was already, to a remarkable extent, an octal system.

This means that man becomes immortal only if and to the extent that he acquires knowledge of what he can in principle know, e. g. mathematics and the natural sciences.

In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field ( or more generally any Dedekind domain ) can be described by a certain group known as an ideal class group ( or class group ).

At the end of Part One, Durham reveals the full extent of his plan to Maria: after taking his earlier self-experiments to their logical conclusion, he became convinced of something he came to call the Dust Theory, which holds that there is no difference, even in principle, between physics and mathematics, and that all mathematically possible structures exist, among them our physics and therefore our spacetime.

However, the concept of coherent states has been considerably generalized, to the extent that it has become a major topic in mathematical physics and in applied mathematics, with applications ranging from quantization to signal processing and image processing ( see Coherent states in mathematical physics ).

On science education in schools he said of all the many crises in education and science, perhaps the most serious is the disappearing species of the good teacher of physics, mathematics and to a lesser extent the other sciences and that if it is allowed to go much further, there will be no recovery for generations, comparing it to China's Cultural Revolution which he said produced a lost generation.

New Mathematics or New Math was a brief, dramatic change in the way mathematics was taught in American grade schools, and to a lesser extent in European countries, during the 1960s.

The teaching in the 10th form comprises the following subjects as obligatory subjects: Danish, mathematics, and English to an extent corresponding to a total of 14 lessons a week ( i. e. half of the minimum weekly teaching time ).

Such applications include numerical analysis, mathematics of engineering, linear programming, optimization and operations research, continuous modelling, mathematical biology and bioinformatics, information theory, game theory, probability and statistics, financial mathematics, actuarial science, cryptography and hence combinatorics and even finite geometry to some extent, graph theory as applied to network analysis, and a great deal of what is called computer science.

In terms of the philosophy and history of mathematics, the very existence of non-classical mathematics raises the question of the extent to which the foundational mathematical choices humanity has made arise from their " superiority " rather than from, say, expedience-driven concentrations of effort on particular aspects.

His early education was limited, though his talent for mathematics was discovered, and to some extent developed, by a teacher in his native town.