These axioms are sufficient for many proofs in elementary mathematical analysis, and are consistent with some principles, such as the Lebesgue measurability of all sets of reals, that are disprovable from the full axiom of choice.

J. Desaulx suggested in 1877 that the phenomenon was caused by the thermal motion of water molecules, and in 1905 Albert Einstein produced the first mathematical analysis of the motion.

Grothendieck's early mathematical work was in functional analysis.

The use of aerodynamics through mathematical analysis, empirical approximations, wind tunnel experimentation, and computer simulations form the scientific basis for heavier-than-air flight and a number of other technologies.

B *- algebras were mathematical structures studied in functional analysis.

A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis.

In addition to mathematical analysis of cryptographic algorithms, cryptanalysis also includes the study of side-channel attacks that do not target weaknesses in the cryptographic algorithms themselves, but instead exploit weaknesses in their implementation.

Compactness in this more general situation plays an extremely important role in mathematical analysis, because many classical and important theorems of 19th century analysis, such as the extreme value theorem, are easily generalized to this situation.

It is also a tool used in branches of mathematics including combinatorics, abstract algebra, and mathematical analysis.

A system can be mechanical, electrical, fluid, chemical, financial and even biological, and the mathematical modeling, analysis and controller design uses control theory in one or many of the time, frequency and complex-s domains, depending on the nature of the design problem.

In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain state space representation, a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations.

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated.

He contributed to the development of the rigorous analysis of the computational complexity of algorithms and systematized formal mathematical techniques for it.

The mathematical study of Diophantine problems Diophantus initiated is now called " Diophantine analysis ".

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function.

Bombieri's research in number theory, algebraic geometry, and mathematical analysis have earned him many international prizes --- a Fields Medal in 1974 and the Balzan Prize in 1980.

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure ( e. g. inner product, norm, topology, etc.

Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects ( for example, numbers and functions ) from all the traditional areas of mathematics ( such as algebra, analysis and topology ) in a single theory, and provides a standard set of axioms to prove or disprove them.

He worked on a great variety of mathematical topics, including series, number theory, mathematical analysis, geometry, algebra, combinatorics, and probability.

Galileo, however, felt that the descriptive content of the technical disciplines warranted philosophical interest, particularly because mathematical analysis of astronomical observations — notably the radical analysis offered by astronomer Nicolaus Copernicus concerning the relative motions of the Sun, Earth, Moon, and planets — indicated that philosophers ' statements about the nature of the universe could be shown to be in error.

In mathematics, specifically general topology and metric topology, a compact space is a mathematical space in which any infinite collection of points sampled from the space must — as a set — be arbitrarily close to some point of the space.

Thus the mathematical formulation of the Alcubierre metric does not contradict the conventional claim that the laws of relativity do not allow a slower-than-light object to accelerate to faster-than-light speeds.

However, theories of analysis can be applied to any space of mathematical objects that has a definition of nearness ( a topological space ) or, more specifically, distance ( a metric space ).

The motivation for studying mathematical analysis in the wider context of topological or metric spaces is threefold:

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold ( such as a surface in space ) which takes as input a pair of tangent vectors v and w and produces a real number ( scalar ) g ( v, w ) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.

In theoretical and mathematical physics, twistor theory maps the geometric objects of conventional 3 + 1 space-time ( Minkowski space ) into geometric objects in a 4 dimensional space with metric signature ( 2, 2 ).

In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space ; that is, a space homeomorphic to a complete metric space that has a countable dense subset.

In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance ( such functions are always continuous ).

In the mathematical study of metric spaces, one can consider the arclength of paths in the space.

A caveat: many terms in Riemannian and metric geometry, such as convex function, convex set and others, do not have exactly the same meaning as in general mathematical usage.

In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian manifold whose Ricci tensor is proportional to the metric.

They are unusual in that they were probably the most " scientific " design of screw, with a basis in metric size ( the 1 mm pitch and 6 mm diameter of 0BA ) and with a mathematical relationship between the ' number ' ( e. g. 2BA with K = 2 ) and the corresponding pitch, major diameter, and then spanner size.

In mathematical physics, a metric describes the arrangement of relative distances within a surface or volume, usually measured by signals passing through the region – essentially describing the intrinsic geometry of the region.

The mathematical object introduced, the Minkowski metric, changes form from one coordinate system to another, but it isn't part of the dynamics, it doesn't obey equations of motion.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

* Ambient space, a mathematical concept

The " space " in cyberspace has more in common with the abstract, mathematical meanings of the term ( see space ) than physical space.

His work was a key aspect of Hermann Weyl and John von Neumann's work on the mathematical equivalence of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation and his namesake Hilbert space plays an important part in quantum theory.

This observation motivates the theoretical concept of an abstract data type, a data structure that is defined indirectly by the operations that may be performed on it, and the mathematical properties of those operations ( including their space and time cost ).

This is essentially no different than any other data processing, except DSP mathematical techniques ( such as the FFT ) are used, and the sampled data is usually assumed to be uniformly sampled in time or space.

The probability " wave " can be said to " pass through space " because the probability values that one can compute from its mathematical representation are dependent on time.

This technique employs the photoelectric effect to measure the reciprocal space — a mathematical representation of periodic structures that is used to infer the original structure.

This new class of preferred motions, too, defines a geometry of space and time — in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential.

In his graphic art, he portrayed mathematical relationships among shapes, figures and space.

Conservation of momentum is a mathematical consequence of the homogeneity ( shift symmetry ) of space ( position in space is the canonical conjugate quantity to momentum ).

Newton's three laws of motion, along with his law of universal gravitation, explain Kepler's laws of planetary motion, which were the first to accurately provide a mathematical model or understanding orbiting bodies in outer space.

The more precise mathematical definition is that there is never translational symmetry in more than n – 1 linearly independent directions, where n is the dimension of the space filled ; i. e. the three-dimensional tiling displayed in a quasicrystal may have translational symmetry in two dimensions.

The items may be stored individually as records in a database ; or may be elements of a search space defined by a mathematical formula or procedure, such as the roots of an equation with integer variables ; or a combination of the two, such as the Hamiltonian circuits of a graph.

In physics, spacetime ( or space – time, space time or space – time continuum ) is any mathematical model that combines space and time into a single continuum.