Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.

Dissatisfaction with older philosophical approaches had begun earlier and had produced other changes in society, such as the Protestant Reformation, but the revolution in science began when natural philosophers began to mount a sustained attack on the Scholastic philosophical program and supposed that mathematical descriptive schemes adopted from such fields as mechanics and astronomy could actually yield universally valid characterizations of motion and other concepts.

There is some question as to whether advanced mathematical skills or talent are in fact necessary for fields such as history, English, philosophy, or art.

However, a second definition and usage has historically been in practice in many fields of computer science and information technology, which defines the prefix kilo when used with byte or bit units of data as 1024 ( 2 < sup > 10 </ sup >); this is due to the mathematical coincidence that Thus, in these fields 1 kilobyte is equal to 1 kibibyte, a new unit standardized as part of the binary prefixes to resolve the ambiguity.

Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of the imposing problems presented in related scientific fields.

The border lines between these fields, and the lines between mathematical logic and other fields of mathematics, are not always sharp.

In mathematics, model theory is the study of ( classes of ) mathematical structures ( e. g. groups, fields, graphs, universes of set theory ) using tools from mathematical logic.

It is the underlying mathematical framework of many fields of physics and chemistry, including condensed matter physics, solid-state physics, atomic physics, molecular physics, computational physics, computational chemistry, quantum chemistry, particle physics, nuclear chemistry, and nuclear physics.

A major break-through came with the introduction of what are now known as Ashtekar variables, which represent geometric gravity using mathematical analogues of electric and magnetic fields.

The development of technology may draw upon many fields of knowledge, including scientific, engineering, mathematical, linguistic, and historical knowledge, to achieve some practical result.

Theories in various fields of study are expressed in natural language, but are always constructed in such a way that their general form is identical to a theory as it is expressed in the formal language of mathematical logic.

Since the 1960s Kripke has been a central figure in a number of fields related to mathematical logic, philosophy of language, philosophy of mathematics, metaphysics, epistemology, and set theory.

The concept appears in most fields of mathematics, including geometry and mathematical analysis, much like the concept of groups does.

Within pure mathematics it is the starting point of more advanced asymptotic analysis, and it is commonly used in more applied fields of numerics as well as in mathematical physics.

The most famous straightedge-and-compass problems have been proven impossible in several cases by Pierre Wantzel, using the mathematical theory of fields.

Python expressions can also be typed directly into number entry fields, allowing mathematical expressions to be used to specify values.

A large number of other curves have been studied in multiple mathematical fields.

The medal was originally to honor scientists in the fields of the " physical, biological, mathematical, or engineering sciences ".

He was known for his contributions to topology -- notably the Mayer-Vietoris sequence -- and other fields of mathematics, his interest in mathematical history and for being a keen alpinist.

Pearson's work was all-embracing in the wide application and development of mathematical statistics, and encompassed the fields of biology, epidemiology, anthropometry, medicine and social history.

He has been a consultant on chaos theory and its applications in numerous fields, such as medical physiology, ecology, mathematical economics, psychotherapy, etc.

The theory of atomic spectra ( and, later, quantum mechanics ) developed almost concurrently with the mathematical fields of linear algebra, the spectral theory of operators, and more broadly, functional analysis.

The mathematical structure of quantum mechanics is based in large part on linear algebra:

The transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant solution of the nonlinear differential equations describing the system.

Differential geometry is a mathematical discipline that uses the techniques of differential calculus and integral calculus, as well as linear algebra and multilinear algebra, to study problems in geometry.

From a mathematical viewpoint, continuous-time IIR LTI filters may be described in terms of linear differential equations, and their impulse responses considered as Green's functions of the equation.

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces ; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.

Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear ones.

Linear prediction is a mathematical operation where future values of a discrete-time signal are estimated as a linear function of previous samples.

In system analysis ( a subfield of mathematics ), linear prediction can be viewed as a part of mathematical modelling or optimization.

It was developed in parallel with a new approach to the mathematical spectral theory based on linear operators rather than the quadratic forms that were David Hilbert's approach a generation earlier.

In the computer science subfields of computer-aided design and computer graphics, the term B-spline frequently refers to a spline curve parametrized by spline functions that are expressed as linear combinations of B-splines ( in the mathematical sense above ).

" The mathematical behaviour then becomes linear, allowing optical components and systems to be described by simple matrices.

Operational amplifiers had their origins in analog computers where they were used to do mathematical operations in many linear, non-linear and frequency-dependent circuits.

Slide rules come in a diverse range of styles and generally appear in a linear or circular form with a standardized set of markings ( scales ) essential to performing mathematical computations.

One major mathematical application of the construction of spinors is to make possible the explicit construction of linear representations of the Lie algebras of the special orthogonal groups, and consequently spinor representations of the groups themselves.

For a brief period in fact, complete normed linear spaces where referred to as " Banach-Wiener " spaces in mathematical literature, based on terminology introduced by Wiener himself.

A transfer function ( also known as the system function or network function ) is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system with zero initial conditions and zero-point equilibrium.

In mathematical terms, Hamming codes are a class of binary linear codes.

The mathematical model of the multipath can be presented using the method of the impulse response used for studying linear systems.

Han-era mathematical achievements include solving problems with right-angle triangles, square roots, cube roots, and matrix methods, finding more accurate approximations for pi, providing mathematical proof of the Pythagorean theorem, use of the decimal fraction, Gaussian elimination to solve linear equations, and continued fractions to find the roots of equations.

Linear programming ( LP, or linear optimization ) is a mathematical method for determining a way to achieve the best outcome ( such as maximum profit or lowest cost ) in a given mathematical model for some list of requirements represented as linear relationships.