In mathematics, the phrase up to is useful for modeling fundamental concepts within a realm of mathematical inquiry, and can be compared with the phrase " all other things being equal " in other disciplines.

In mathematics and logic, the phrase " there is one and only one " is used to indicate that exactly one object with a certain property exists.

In mathematics, the phrase sufficiently large is used in contexts such as:

In mathematics, specifically algebraic topology, the phrase semi-locally simply connected refers to a certain local connectedness condition that arises in the theory of covering spaces.

In advanced mathematics, as well as colloquially in popular culture, especially geek culture, the phrase " order of magnitude " is used to denote a change in a numeric quantity, usually a measurement, by a factor of 10 ; that is, the moving of the decimal point in a number one way or the other, possibly with the addition of significant zeros.

: This book ... is written for people who have no training in mathematics and who may not have actively thought about mathematics since high school, or even during it, but who may wish to experience an act of mathematical imagining and to consider how such an experience compares with the imaginative work involved in reading and understanding a phrase in a poem.

The station's call letters, Q. E. D., are taken from the Latin phrase, quod erat demonstrandum, commonly used in mathematics.

Use of the phrase " final solution ", even in non-Nazi contexts, e. g., " the final solution of a mathematics problem " is frowned upon in modern Germany.

The tombstone, halmos, or end of proof mark "" is used in mathematics to denote the end of a proof, in place of the traditional abbreviation " QED " for the Latin phrase " quod erat demonstrandum " ( Q. E. D .).

In mathematics, the phrase arbitrarily large, arbitrarily small, arbitrarily long is used in statements such as:

The station's call letters, Q. E. D., are taken from the Latin phrase, quod erat demonstrandum, commonly used in mathematics.

Hardy expounds by commenting about a phrase attributed to Carl Friedrich Gauss that " Mathematics is the queen of the sciences and number theory is the queen of mathematics ".

* Lyapunov stability in mathematics, from the phrase " in the sense of Lyapunov "

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.

Arithmetic or arithmetics ( from the Greek word ἀριθμός, arithmos " number ") is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations.

The physicist Richard Feynman called Euler's formula " our jewel " and " one of the most remarkable, almost astounding, formulas in all of mathematics.

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.

Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory.

Although her father refused to grant this wish of joining a convent, he agreed to let her live from that time on in an almost conventual semi-retirement, avoiding all interactions with society and devoting herself entirely to the study of mathematics.

In computer programming languages, the definitions of operator and operand are almost the same as in mathematics.

Roger Ascham thought that his pupil Robert had an uncommon talent for languages and writing, " exceed almost all other by nature ", and regretted that he had done himself harm by preferring " Euclid's pricks and lines " ( mathematics ).

For discrete time, where payments are separated by large time periods, the transform reduces to a sum, but when payments are ongoing on an almost continual basis, the mathematics of continuous functions can be used as an approximation.

It is almost certain that Hero taught at the Musaeum which included the famous Library of Alexandria, because most of his writings appear as lecture notes for courses in mathematics, mechanics, physics and pneumatics.

The College of Arts and Sciences is the largest of the eight academic units that make up the University, with almost 500 full-time faculty in 21 academic departments and seven interdepartmental programs, spanning the arts, humanities, social sciences, natural sciences, and mathematics.

The large majority of the students are admitted after two to three years of classes préparatoires, known as " mathematics superior " and " mathematics special ", which are an undergraduate cursus with almost exclusive emphasis on Math and Physics.

In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is " almost abelian ".

But this was a man so much out of the ordinary that I said ... No, you are almost thirty, you have the good fortune of being Greek, of being an architect and having studied special mathematics.

It captures, one might say, almost everything in the intuition of continuity, in a technically adequate form that can be applied in any area of mathematics.

His philosophical view of mathematics has put him in conflict with several mathematicians over the years, notably straining his relationship with Weierstrass which almost decided to leave the University in 1888.

In mathematics, an almost perfect number ( sometimes also called slightly defective number ) is a natural number n such that the sum of all divisors of n ( the divisor function σ ( n )) is equal to 2n-1, the sum of all proper divisors of n, s ( n ) = σ ( n )-n, then being equal to n-1.

The body of research in reverse mathematics has established that weak subsystems of second-order arithmetic suffice to formalize almost all undergraduate-level mathematics.

One of the most important aspects of functions of bounded variation is that they form an algebra of discontinuous functions whose first derivative exists almost everywhere: due to this fact, they can and frequently are used to define generalized solutions of nonlinear problems involving functionals, ordinary and partial differential equations in mathematics, physics and engineering.

As mathematics is a part of primary education in almost all countries, almost all educated people have some exposure to pure mathematics.

The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory.

In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X

The Platonist seemed to outweigh the Aristotelian in Alan, but he felt strongly that the divine is all intelligibility and argued this notion through much Aristotelian logic combined with Pythagorean mathematics.

Two aspects of this attitude deserve to be mentioned: 1 ) he did not only study science from books, as other academics did in his day, but actually observed and experimented with nature ( the rumours starting by those who did not understand this are probably at the source of Albert's supposed connections with alchemy and witchcraft ), 2 ) he took from Aristotle the view that scientific method had to be appropriate to the objects of the scientific discipline at hand ( in discussions with Roger Bacon, who, like many 20th century academics, thought that all science should be based on mathematics ).

Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution ; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.

The axiom of regularity is arguably the least useful ingredient of Zermelo – Fraenkel set theory, since virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity ( see chapter 3 of ).

On November 29, 1921, the trustees declared it to be the express policy of the Institute to pursue scientific research of the greatest importance and at the same time " to continue to conduct thorough courses in engineering and pure science, basing the work of these courses on exceptionally strong instruction in the fundamental sciences of mathematics, physics, and chemistry ; broadening and enriching the curriculum by a liberal amount of instruction in such subjects as English, history, and economics ; and vitalizing all the work of the Institute by the infusion in generous measure of the spirit of research.

Mathematical historian Eric Temple Bell estimated that, had Gauss published all of his discoveries in a timely manner, he would have advanced mathematics by fifty years.

In mathematics, a contraction mapping, or contraction, on a metric space ( M, d ) is a function f from M to itself, with the property that there is some nonnegative real number < math > k < 1 </ math > such that for all x and y in M,

In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group.

These can include shi ( 史, historical works ), zi ( 子, philosophical works belonging to schools of thought other than the Confucian, but also works of agriculture, medicine, mathematics, astronomy, divination, art criticism, and all sorts of miscellaneous writings ) and ji ( 集, literary works ) as well as jing.

Note that unlike Hypatia he did not study ' mathematics, philosophy and astronomy ', thus he and his followers came into conflict with the ancient University of Alexandria which pursued all forms of knowledge including science and human anatomy, politics and history according to the model inaugurated by Alexander the Great, the founder of Alexandria.

In mathematics, any vector space, V, has a corresponding dual vector space ( or just dual space for short ) consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors.

# all of mathematics follows from a correctly chosen finite system of axioms ; and

Using a branch of mathematics known as tuple calculus, he demonstrated that such a system could support all the operations of normal databases ( inserting, updating etc.

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.

Certain forms treat all knowledge as empirical, while some regard disciplines such as mathematics and logic as exceptions.

In mathematics, given a set and an equivalence relation on, the equivalence class of an element in is the subset of all elements in which are equivalent to.

It is, properly speaking, the most universal mathematics of all.

Unsurprisingly, Galois ' collected works amount to only some 60 pages, but within them are many important ideas that have had far-reaching consequences for nearly all branches of mathematics.

Mill's empiricism went a significant step beyond Hume in still another respect: in maintaining that induction is necessary for all meaningful knowledge including mathematics.

In logic and the foundations of mathematics, formal languages are used to represent the syntax of axiomatic systems, and mathematical formalism is the philosophy that all of mathematics can be reduced to the syntactic manipulation of formal languages in this way.