There are several ways to make this intuition mathematically rigorous.

Gilbert Newton Lewis ForMemRS ( October 23, 1875 – March 23, 1946 ) was an American physical chemist known for the discovery of the covalent bond ( see his Lewis dot structures and his 1916 paper " The Atom and the Molecule "), his purification of heavy water, his reformulation of chemical thermodynamics in a mathematically rigorous manner accessible to ordinary chemists, his theory of Lewis acids and bases, and his photochemical experiments.

This is a mathematically rigorous reformulation of the calculus in terms of infinitesimals.

Some are more complicated or more mathematically rigorous than others ; some have been shown to be incorrect.

In the mathematically rigorous formulation of quantum mechanics developed by Paul Dirac David Hilbert, and John von Neumann, the possible states of a quantum mechanical system are represented by unit vectors ( called " state vectors ").

Although Marshall took economics to a more mathematically rigorous level, he did not want mathematics to overshadow economics and thus make economics irrelevant to the layman.

These depend on applying mathematically rigorous serial numbers and serial number arithmetic, again not identifying a single instance of the content being protected.

This article treats the notion of ensembles in a mathematically rigorous fashion, although relevant physical aspects will be mentioned.

For example, one can specify the density operators describing microcanonical, canonical, and grand canonical ensembles of quantum mechanical systems, in a mathematically rigorous fashion.

In topological field theory, he introduced the moduli space of stable maps, which may be considered a mathematically rigorous formulation of the Feynman integral for topological string theory.

The term " mathematical physics " is sometimes used to denote research aimed at studying and solving problems inspired by physics within a mathematically rigorous framework.

The effort to put physical theories on a mathematically rigorous footing has inspired many mathematical developments.

In computer science, operational semantics is a way to give meaning to computer programs in a mathematically rigorous way.

The result may be seen as a series of scaled, shifted delta functions ( although this is not mathematically rigorous ):

In recent years, Borcherds has been attempting to construct quantum field theory in a mathematically rigorous manner.

In physics the Wightman axioms are an attempt at a mathematically rigorous formulation of quantum field theory.

In particular, the Standard model of particle physics has no mathematically rigorous foundations.

The formula is however not mathematically rigorous, since is usually not even defined.

However, the first mathematically rigorous proof of this fact was obtained only in the 19th century.

This provided another way of putting calculus on a mathematically rigorous foundation that was in a similar spirit to the way calculus was done before the ( ε, δ )- definition of limit had been fully developed.

Unlike, for example, Karl Popper's informal inductive inference theory, however, Solomonoff's is mathematically rigorous.

While it seems very straightforward to say that has two holes, it is surprisingly hard to formulate this in a mathematically rigorous way ; this is a central purpose of homology theory.

Second, there is the question regarding the role and status of mathematically rigorous results and relations.

This form is more mathematically rigorous, but leads to a hyperbolic groundwater flow equation, which is more difficult to solve and is only useful at very small times, typically out of the realm of practical use.

Uses of the information obtained from the analysis vary from highlighting possible coding errors ( e. g., the lint tool ) to formal methods that mathematically prove properties about a given program ( e. g., its behavior matches that of its specification ).

Ecologists collect data on trophic levels and food webs to statistically model and mathematically calculate parameters, such as those used in other kinds of network analysis ( e. g., graph theory ), to study emergent patterns and properties shared among ecosystems.

The mathematically formal treatment is the subject of functional analysis.

Zipf's law attempts to state this analysis mathematically.

Moreover, Levin's team, on 12 April 2012, reported a statistical speculation, based on old data — reinterpreted mathematically through complexity analysis — of the Labeled Release experiments, that may suggest evidence of " extant microbial life on Mars.

Kriging is mathematically closely related to regression analysis.

Systematic political science, as developed by Dallas F. Bell Jr., basically is the use of game theory methods to mathematically unify the anthropocentric academic disciplines of theology, epistemology, psychology, sociology and eschatology for computerized analysis and predictions after verification and validation methods are employed, such as red team procedures.

However, the remainder is still mathematically relevant, being used, for example, in oscillator design and stability analysis.

Implementations of algorithms such as AES and triple DES that are believed to be mathematically strong may be trivially breakable using power analysis attacks.

The equation arises not by statistical analysis of all the individual positions and momenta of each particle in the fluid ; rather by considering the probability that a number of particles all occupy a very small region of space ( mathematically written d < sup > 3 </ sup > r, where d means " differential ", a very small change ) centered at the tip of the position vector r, and have very nearly equal small changes in momenta from a momentum vector p, at an instant of time.

The usefulness of the von Mises distribution is twofold: it is the most mathematically tractable of all circular distributions, allowing simpler statistical analysis, and it is a close approximation to the wrapped normal distribution, which, analogously the linear normal distribution, is important because it is the limiting case for the sum of a large number of small angular deviations.

The composition of spectral music is concerned with timbral structures, especially when decisions about timbre are mathematically informed by Fourier analysis using the computer-efficient fast Fourier transform ( FFT ).

The Renaissance tried to extract and codify the system of proportions in the orders as used by the ancients, believing that with analysis a mathematically absolute ideal of beauty would emerge.

* Anwar Shaikh has modelled the formation and change of production prices mathematically using iterative methods.

* Eudoxus of Cnidus develops the method of exhaustion for mathematically determining the area under a curve.

The D ' Hondt method ( mathematically but not operationally equivalent to Jefferson's method, and Bader – Ofer method ) is a highest averages method for allocating seats in party-list proportional representation.

Note that this method of computing vacuum energy is mathematically equivalent to having a quantum harmonic oscillator at each point and, therefore, suffers the same renormalization problems.

While it is not known if the two problems are mathematically equivalent, factoring is currently the only method of directly breaking RSA.

It is mathematically possible with discontinuous functions for the method to fail to converge to a zero limit or sign change, but this is not a problem in practice since it would require an infinite sequence of coincidences for both endpoints to get stuck converging to discontinuities where the sign does not change ( for example at x =± 1 in f ( x )= 1 /( x-1 ) ² + 1 /( x + 1 ) ² ).

The refractive index gradient of GRIN lenses can be mathematically modelled according to the method of production used.

Using this method either actual historical rainfalls or hypothetical " design storms " can be modeled mathematically to confirm characteristics of historical floods, or to predict a stream's reaction to a predited storm.

Virtually every artistic technique and method used by Renaissance artists 1, 900 year later, had been demonstrated by Ancient Greek artists, with the notable exceptions of oil colors and mathematically accurate perspective.

Hornblower finds that the most mathematically sound method for the duel is to take an " even chance " by having the duelists select from two pistols, only one of which has been loaded, and taking fire only a few feet apart.

An important reason for this is that computing power has become far more accessible, making simulation the preferred analytical method for problems that are not easily solved mathematically.

These polynomials mathematically generalize to a Maclaurin series about an arbitrary external parameter ; which gives the solution method more flexibly than direct Taylor series expansion.

The chain method and the EAC method give mathematically equivalent answers.