# Category: Mathematical principles

Laplace principle (large deviations theory)
In mathematics, Laplace's principle is a basic theorem in large deviations theory which is similar to Varadhan's lemma. It gives an asymptotic expression for the Lebesgue integral of exp(−θφ(x)) over
Phragmén–Lindelöf principle
In complex analysis, the Phragmén–Lindelöf principle (or method), first formulated by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf (1870–1946) in 1908, is a technique which employs an a
Vopěnka's principle
In mathematics, Vopěnka's principle is a large cardinal axiom. The intuition behind the axiom is that the set-theoretical universe is so large that in every proper class, some members are similar to o
Dirichlet's principle
In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation.
Hausdorff maximal principle
In mathematics, the Hausdorff maximal principle is an alternate and earlier formulation of Zorn's lemma proved by Felix Hausdorff in 1914 (Moore 1982:168). It states that in any partially ordered set,
Incompressibility method
In mathematics, the incompressibility method is a proof method like the probabilistic method, the counting method or the pigeonhole principle. To prove that an object in a certain class (on average) s
Uniform boundedness principle
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theor
Homotopy principle
In mathematics, the homotopy principle (or h-principle) is a very general way to solve partial differential equations (PDEs), and more generally (PDRs). The h-principle is good for underdetermined PDE
Bloch's principle
Bloch's Principle is a philosophical principle in mathematicsstated by André Bloch. Bloch states the principle in Latin as: Nihil est in infinito quod non prius fuerit in finito, and explains this as
Tilted large deviation principle
In mathematics — specifically, in large deviations theory — the tilted large deviation principle is a result that allows one to generate a new large deviation principle from an old one by "tilting", i
Schwarz reflection principle
In mathematics, the Schwarz reflection principle is a way to extend the domain of definition of a complex analytic function, i.e., it is a form of analytic continuation. It states that if an analytic
Harnack's principle
In the mathematical field of partial differential equations, Harnack's principle or Harnack's theorem is a corollary of Harnack's inequality which deals with the convergence of sequences of harmonic f
Markov's principle
Markov's principle, named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below. The principle is logically valid classicall
Contraction principle (large deviations theory)
In mathematics — specifically, in large deviations theory — the contraction principle is a theorem that states how a large deviation principle on one space "pushes forward" (via the pushforward of a p
Diamond principle
In mathematics, and particularly in axiomatic set theory, the diamond principle ◊ is a combinatorial principle introduced by Ronald Jensen in that holds in the constructible universe (L) and that impl
Hume's principle
Hume's principle or HP says that the number of Fs is equal to the number of Gs if and only if there is a one-to-one correspondence (a bijection) between the Fs and the Gs. HP can be stated formally in
In combinatorics, the addition principle or rule of sum is a basic counting principle. Stated simply, it is the intuitive idea that if we have A number of ways of doing something and B number of ways
Racetrack principle
In calculus, the racetrack principle describes the movement and growth of two functions in terms of their derivatives. This principle is derived from the fact that if a horse named Frank Fleetfeet alw
Clubsuit
In mathematics, and particularly in axiomatic set theory, ♣S (clubsuit) is a family of combinatorial principles that are a weaker version of the corresponding ◊S; it was introduced in 1975 by Adam Ost
Hopf lemma
In mathematics, the Hopf lemma, named after Eberhard Hopf, states that if a continuous real-valued function in a domain in Euclidean space with sufficiently smooth boundary is harmonic in the interior
Principle of maximum caliber
The principle of maximum caliber (MaxCal) or maximum path entropy principle, suggested by E. T. Jaynes, can be considered as a generalization of the principle of maximum entropy. It postulates that th
Maximum principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of el
Littlewood's three principles of real analysis
Littlewood's three principles of real analysis are heuristics of J. E. Littlewood to help teach the essentials of measure theory in mathematical analysis.
Courant minimax principle
In mathematics, the Courant minimax principle gives the eigenvalues of a real symmetric matrix. It is named after Richard Courant.
Square–cube law
The square–cube law (or cube–square law) is a mathematical principle, applied in a variety of scientific fields, which describes the relationship between the volume and the surface area as a shape's s
Duhamel's principle
In mathematics, and more specifically in partial differential equations, Duhamel's principle is a general method for obtaining solutions to inhomogeneous linear evolution equations like the heat equat
Principle of maximum entropy
The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy, in the context of precise
Combinatorial principles
In proving results in combinatorics several useful combinatorial rules or combinatorial principles are commonly recognized and used. The rule of sum, rule of product, and inclusion–exclusion principle
Inclusion–exclusion principle
In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finit
Well-ordering principle
In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element. In other words, the set of positive integers is well-ordered by its "natural"
Hopf maximum principle
The Hopf maximum principle is a maximum principle in the theory of second order elliptic partial differential equations and has been described as the "classic and bedrock result" of that theory. Gener
Maupertuis's principle
In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis) states that the path followed by a physical system is the one of least length (with a suitable interpretation of pa
Pigeonhole principle
In mathematics, the pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. For example, if one has three gloves
Cavalieri's principle
In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are i
Hasse principle
In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to pie
Reflection principle
In set theory, a branch of mathematics, a reflection principle says that it is possible to find sets that resemble the class of all sets. There are several different forms of the reflection principle
Rule of product
In combinatorics, the rule of product or multiplication principle is a basic counting principle (a.k.a. the fundamental principle of counting). Stated simply, it is the intuitive idea that if there ar
Splitting principle
In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles. In the theory of vector bundles, one often wishes to simplify computat
Method of distinguished element
In the mathematical field of enumerative combinatorics, identities are sometimes established by arguments that rely on singling out one "distinguished element" of a set.
Maximum modulus principle
In mathematics, the maximum modulus principle in complex analysis states that if f is a holomorphic function, then the modulus |f | cannot exhibit a strict local maximum that is properly within the do
Transfer principle
In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle
Order-extension principle
No description available.