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In mathematics, specifically transcendental number theory, the six exponentials theorem is a result that, given the right conditions on the exponents, guarantees the transcendence of at least one of a set of exponentials. | https://en.wikipedia.org/wiki/Six_exponentials_theorem |
In mathematics, specifically, in category theory, a 2-functor is a morphism between 2-categories. They may be defined formally using enrichment by saying that a 2-category is exactly a Cat-enriched category and a 2-functor is a Cat-functor.Explicitly, if C and D are 2-categories then a 2-functor F: C → D {\displaystyle... | https://en.wikipedia.org/wiki/2-functor |
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. The adjacency matrix of a simple undirected graph is a real symmetric m... | https://en.wikipedia.org/wiki/Graph_spectrum |
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equat... | https://en.wikipedia.org/wiki/Spectral_theory |
In mathematics, spin geometry is the area of differential geometry and topology where objects like spin manifolds and Dirac operators, and the various associated index theorems have come to play a fundamental role both in mathematics and in mathematical physics. An important generalisation is the theory of symplectic D... | https://en.wikipedia.org/wiki/Spin_geometry |
In mathematics, spin networks have been used to study skein modules and character varieties, which correspond to spaces of connections. | https://en.wikipedia.org/wiki/Spin_network |
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small va... | https://en.wikipedia.org/wiki/Stability_theory |
Various criteria have been developed to prove stability or instability of an orbit. Under favorable circumstances, the question may be reduced to a well-studied problem involving eigenvalues of matrices. A more general method involves Lyapunov functions. In practice, any one of a number of different stability criteria ... | https://en.wikipedia.org/wiki/Stability_theory |
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the Freudenthal suspension theorem, which states that given any pointed space X... | https://en.wikipedia.org/wiki/Stable_homotopy |
The first example is a standard corollary of the Hurewicz theorem, that π n ( S n ) ≅ Z {\displaystyle \pi _{n}(S^{n})\cong \mathbb {Z} } . In the second example the Hopf map, η {\displaystyle \eta } , is mapped to its suspension Σ η {\displaystyle \Sigma \eta } , which generates π 4 ( S 3 ) ≅ Z / 2 {\displaystyle \pi ... | https://en.wikipedia.org/wiki/Stable_homotopy |
According to Freudenthal's theorem, in the stable range the homotopy groups of spheres depend not on the specific dimensions of the spheres in the domain and target, but on the difference in those dimensions. With this in mind the k-th stable stem is π k s := lim n π n + k ( S n ) {\displaystyle \pi _{k}^{s}:=\lim _{n}... | https://en.wikipedia.org/wiki/Stable_homotopy |
A theorem of Goro Nishida states that all elements of positive grading in this ring are nilpotent. Thus the only prime ideals are the primes in π 0 s ≅ Z {\displaystyle \pi _{0}^{s}\cong \mathbb {Z} } . | https://en.wikipedia.org/wiki/Stable_homotopy |
So the structure of π ∗ s {\displaystyle \pi _{*}^{s}} is quite complicated. In the modern treatment of stable homotopy theory, spaces are typically replaced by spectra. | https://en.wikipedia.org/wiki/Stable_homotopy |
Following this line of thought, an entire stable homotopy category can be created. This category has many nice properties that are not present in the (unstable) homotopy category of spaces, following from the fact that the suspension functor becomes invertible. For example, the notion of cofibration sequence and fibrat... | https://en.wikipedia.org/wiki/Stable_homotopy |
In mathematics, statistics and elsewhere, sums of squares occur in a number of contexts: | https://en.wikipedia.org/wiki/Sum_of_squares |
In mathematics, statistics, and computational modelling, a grey box model combines a partial theoretical structure with data to complete the model. The theoretical structure may vary from information on the smoothness of results, to models that need only parameter values from data or existing literature. Thus, almost a... | https://en.wikipedia.org/wiki/Grey_box_model |
These have special analysis methods. In particular linear regression techniques are much more efficient than most non-linear techniques. The model can be deterministic or stochastic (i.e. containing random components) depending on its planned use. | https://en.wikipedia.org/wiki/Grey_box_model |
In mathematics, statistics, finance, computer science, particularly in machine learning and inverse problems, regularization is a process that changes the result answer to be "simpler". It is often used to obtain results for ill-posed problems or to prevent overfitting.Although regularization procedures can be divided ... | https://en.wikipedia.org/wiki/Regularization_(machine_learning) |
The regularization term, or penalty, imposes a cost on the optimization function to make the optimal solution unique. Implicit regularization is all other forms of regularization. This includes, for example, early stopping, using a robust loss function, and discarding outliers. | https://en.wikipedia.org/wiki/Regularization_(machine_learning) |
Implicit regularization is essentially ubiquitous in modern machine learning approaches, including stochastic gradient descent for training deep neural networks, and ensemble methods (such as random forests and gradient boosted trees).In explicit regularization, independent of the problem or model, there is always a da... | https://en.wikipedia.org/wiki/Regularization_(machine_learning) |
There is a whole research branch dealing with all possible regularizations. In practice, one usually tries a specific regularization and then figures out the probability density that corresponds to that regularization to justify the choice. It can also be physically motivated by common sense or intuition. In machine le... | https://en.wikipedia.org/wiki/Regularization_(machine_learning) |
In mathematics, stochastic analysis on manifolds or stochastic differential geometry is the study of stochastic analysis over smooth manifolds. It is therefore a synthesis of stochastic analysis and differential geometry. The connection between analysis and stochastic processes stems from the fundamental relation that ... | https://en.wikipedia.org/wiki/Stochastic_analysis_on_manifolds |
Interpreting the paths of Brownian motion as characteristic curves of the operator, Brownian motion can be seen as a stochastic counterpart of a flow to a second-order partial differential operator. Stochastic analysis on manifolds investigates stochastic processes on non-linear state spaces or manifolds. Classical the... | https://en.wikipedia.org/wiki/Stochastic_analysis_on_manifolds |
In that, it is often complicated (or not possible) to formulate objects with coordinates of R d {\displaystyle \mathbb {R} ^{d}} . Thus, we require an additional structure in form of a linear connection or Riemannian metric to define martingales and Brownian motion on manifolds. Therefore, controlled by the Riemannian ... | https://en.wikipedia.org/wiki/Stochastic_analysis_on_manifolds |
However, its stochastic behaviour determines global aspects of the topology and geometry of the manifold. Brownian motion is defined to be the diffusion process generated by the Laplace-Beltrami operator 1 2 Δ M {\displaystyle {\tfrac {1}{2}}\Delta _{M}} with respect to a manifold M {\displaystyle M} and can be constru... | https://en.wikipedia.org/wiki/Stochastic_analysis_on_manifolds |
However, this obstacle can be overcome if the manifold is equipped with a connection: We can then introduce the stochastic horizontal lift of a semimartingale and the stochastic development by the so-called Eells-Elworthy-Malliavin construction.The latter is a generalisation of a horizontal lift of smooth curves to hor... | https://en.wikipedia.org/wiki/Stochastic_analysis_on_manifolds |
For example, one can apply Brownian motion to the Dirichlet problem at infinity for Cartan-Hadamard manifolds or give a probabilistic proof of the Atiyah-Singer index theorem. Stochastic differential geometry also applies in other areas of mathematics (e.g. mathematical finance). For example, we can convert classical a... | https://en.wikipedia.org/wiki/Stochastic_analysis_on_manifolds |
In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which extend to the more abstract setting of random measures. | https://en.wikipedia.org/wiki/Stochastic_geometry |
In mathematics, stratified Morse theory is an analogue to Morse theory for general stratified spaces, originally developed by Mark Goresky and Robert MacPherson. The main point of the theory is to consider functions f: M → R {\displaystyle f:M\to \mathbb {R} } and consider how the stratified space f − 1 ( − ∞ , c ] {\d... | https://en.wikipedia.org/wiki/Stratified_Morse_theory |
In mathematics, strict differentiability is a modification of the usual notion of differentiability of functions that is particularly suited to p-adic analysis. In short, the definition is made more restrictive by allowing both points used in the difference quotient to "move". | https://en.wikipedia.org/wiki/Strict_differentiability |
In mathematics, strict positivity is a concept in measure theory. Intuitively, a strictly positive measure is one that is "nowhere zero", or that is zero "only on points". | https://en.wikipedia.org/wiki/Strictly_positive_measure |
In mathematics, structural Ramsey theory is a categorical generalisation of Ramsey theory, rooted in the idea that many important results of Ramsey theory have "similar" logical structures. The key observation is noting that these Ramsey-type theorems can be expressed as the assertion that a certain category (or class ... | https://en.wikipedia.org/wiki/Structural_Ramsey_theory |
In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact C1-small perturbations). Examples of such qualitative properties are numbers of fixed points and periodic orbits (but not t... | https://en.wikipedia.org/wiki/Structural_stability |
Variants of this notion apply to systems of ordinary differential equations, vector fields on smooth manifolds and flows generated by them, and diffeomorphisms. Structurally stable systems were introduced by Aleksandr Andronov and Lev Pontryagin in 1937 under the name "systèmes grossiers", or rough systems. They announ... | https://en.wikipedia.org/wiki/Structural_stability |
In this case, structurally stable systems are typical, they form an open dense set in the space of all systems endowed with appropriate topology. In higher dimensions, this is no longer true, indicating that typical dynamics can be very complex (cf. strange attractor). An important class of structurally stable systems ... | https://en.wikipedia.org/wiki/Structural_stability |
In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of... | https://en.wikipedia.org/wiki/Subadditive_function |
In mathematics, subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group.Let G {\displaystyle G} be a finitely generated group. Then, for each integer n {\displaystyle n} define a n ( G ) {\displaystyle a_{n}(G)} to be the number of subgroups H {\displaystyle H} ... | https://en.wikipedia.org/wiki/Subgroup_growth |
One similarly defines m n ( G ) {\displaystyle m_{n}(G)} and s n ◃ ( G ) {\displaystyle s_{n}^{\triangleleft }(G)} to denote the number of maximal and normal subgroups of index n {\displaystyle n} , respectively. Subgroup growth studies these functions, their interplay, and the characterization of group theoretical pro... | https://en.wikipedia.org/wiki/Subgroup_growth |
In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory. Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a lin... | https://en.wikipedia.org/wiki/Subharmonic_function |
In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. They also describe the set of all possible sequences executed by a finite state machine. The most widely studied shift spaces are the subshifts of finite type... | https://en.wikipedia.org/wiki/Subshifts_of_finite_type |
In mathematics, subtle cardinals and ethereal cardinals are closely related kinds of large cardinal number. A cardinal κ is called subtle if for every closed and unbounded C ⊂ κ and for every sequence A of length κ for which element number δ (for an arbitrary δ), Aδ ⊂ δ, there exist α, β, belonging to C, with α < β, su... | https://en.wikipedia.org/wiki/Ethereal_cardinal |
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826. | https://en.wikipedia.org/wiki/Summation_by_parts |
In mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an o... | https://en.wikipedia.org/wiki/Sigma_notation |
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need of parentheses, and the result is the same irrespective of the order of the ... | https://en.wikipedia.org/wiki/Sigma_notation |
Summation of a sequence of only one element results in this element itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0. Very often, the elements of a sequence are defined, through a regular pattern, as a function of their place in the sequence. | https://en.wikipedia.org/wiki/Sigma_notation |
For simple patterns, summation of long sequences may be represented with most summands replaced by ellipses. For example, summation of the first 100 natural numbers may be written as 1 + 2 + 3 + 4 + ⋯ + 99 + 100. Otherwise, summation is denoted by using Σ notation, where ∑ {\textstyle \sum } is an enlarged capital Gree... | https://en.wikipedia.org/wiki/Sigma_notation |
For example, the sum of the first n natural numbers can be denoted as ∑ i = 1 n i . {\textstyle \sum _{i=1}^{n}i.} For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find closed-form expressions for the result. | https://en.wikipedia.org/wiki/Sigma_notation |
For example, ∑ i = 1 n i = n ( n + 1 ) 2 . {\displaystyle \sum _{i=1}^{n}i={\frac {n(n+1)}{2}}.} Although such formulas do not always exist, many summation formulas have been discovered—with some of the most common and elementary ones being listed in the remainder of this article. | https://en.wikipedia.org/wiki/Sigma_notation |
In mathematics, superfunction is a nonstandard name for an iterated function for complexified continuous iteration index. Roughly, for some function f and for some variable x, the superfunction could be defined by the expression S ( z ; x ) = f ( f ( … f ( x ) … ) ) ⏟ z evaluations of the function f . {\displaystyle S(... | https://en.wikipedia.org/wiki/Superfunction |
Such a definition is valid only for a positive integer index z. The variable x is often omitted. Much study and many applications of superfunctions employ various extensions of these superfunctions to complex and continuous indices; and the analysis of existence, uniqueness and their evaluation. The Ackermann functions... | https://en.wikipedia.org/wiki/Superfunction |
In mathematics, surfaces of class VII are non-algebraic complex surfaces studied by (Kodaira 1964, 1968) that have Kodaira dimension −∞ and first Betti number 1. Minimal surfaces of class VII (those with no rational curves with self-intersection −1) are called surfaces of class VII0. Every class VII surface is biration... | https://en.wikipedia.org/wiki/Global_spherical_shell |
The name "class VII" comes from (Kodaira 1964, theorem 21), which divided minimal surfaces into 7 classes numbered I0 to VII0. However Kodaira's class VII0 did not have the condition that the Kodaira dimension is −∞, but instead had the condition that the geometric genus is 0. As a result, his class VII0 also included ... | https://en.wikipedia.org/wiki/Global_spherical_shell |
In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the shift operator. Formally, a Markov partition is ... | https://en.wikipedia.org/wiki/Symbolic_dynamics |
In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that take the cone onto itself. By the Koecher–Vinberg theorem these correspond to the cone of squares in finite-dimensional ... | https://en.wikipedia.org/wiki/Jordan_frame_(Jordan_algebra) |
In mathematics, symmetric convolution is a special subset of convolution operations in which the convolution kernel is symmetric across its zero point. Many common convolution-based processes such as Gaussian blur and taking the derivative of a signal in frequency-space are symmetric and this property can be exploited ... | https://en.wikipedia.org/wiki/Symmetric_convolution |
In mathematics, symmetrization is a process that converts any function in n {\displaystyle n} variables to a symmetric function in n {\displaystyle n} variables. Similarly, antisymmetrization converts any function in n {\displaystyle n} variables into an antisymmetric function. | https://en.wikipedia.org/wiki/Symmetric_map |
In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory. There are several insights that allow for such a reformulation. The first is that most of the analytic data for describing the class of smooth manifolds can be encoded into certain ... | https://en.wikipedia.org/wiki/Synthetic_differential_geometry |
The third insight is that over a certain category, these are representable functors. Furthermore, their representatives are related to the algebras of dual numbers, so that smooth infinitesimal analysis may be used. Synthetic differential geometry can serve as a platform for formulating certain otherwise obscure or con... | https://en.wikipedia.org/wiki/Synthetic_differential_geometry |
In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also a slow... | https://en.wikipedia.org/wiki/Systolic_geometry |
In mathematics, systolic inequalities for curves on surfaces were first studied by Charles Loewner in 1949 (unpublished; see remark at end of P. M. Pu's paper in '52). Given a closed surface, its systole, denoted sys, is defined to be the least length of a loop that cannot be contracted to a point on the surface. The s... | https://en.wikipedia.org/wiki/Systoles_of_surfaces |
In mathematics, t-norms are a special kind of binary operations on the real unit interval . Various constructions of t-norms, either by explicit definition or by transformation from previously known functions, provide a plenitude of examples and classes of t-norms. This is important, e.g., for finding counter-examples ... | https://en.wikipedia.org/wiki/Construction_of_t-norms |
In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering. The calculation of mathematical tables was an important area of study, which led to the development of the first mec... | https://en.wikipedia.org/wiki/Trigonometric_tables |
Often, these libraries use pre-calculated tables internally, and compute the required value by using an appropriate interpolation method. Interpolation of simple look-up tables of trigonometric functions is still used in computer graphics, where only modest accuracy may be required and speed is often paramount. Another... | https://en.wikipedia.org/wiki/Trigonometric_tables |
In this case, calling generic library routines every time is unacceptably slow. One option is to call the library routines once, to build up a table of those trigonometric values that will be needed, but this requires significant memory to store the table. The other possibility, since a regular sequence of values is re... | https://en.wikipedia.org/wiki/Trigonometric_tables |
In mathematics, taking the nth root is an operation involving two numbers, the radicand and the index or degree. Taking the nth root is written as x n {\displaystyle {\sqrt{x}}} , where x is the radicand and n is the index (also sometimes called the degree). This is pronounced as "the nth root of x". The definition the... | https://en.wikipedia.org/wiki/Seventh_root |
{\displaystyle r^{n}=x.} A root of degree 2 is called a square root (usually written without the n as just x {\displaystyle {\sqrt {x}}} ) and a root of degree 3, a cube root (written x 3 {\displaystyle {\sqrt{x}}} ). Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. ... | https://en.wikipedia.org/wiki/Seventh_root |
For example, 3 is a square root of 9, since 32 = 9, and −3 is also a square root of 9, since (−3)2 = 9. Any non-zero number considered as a complex number has n different complex nth roots, including the real ones (at most two). The nth root of 0 is zero for all positive integers n, since 0n = 0. | https://en.wikipedia.org/wiki/Seventh_root |
In particular, if n is even and x is a positive real number, one of its nth roots is real and positive, one is negative, and the others (when n > 2) are non-real complex numbers; if n is even and x is a negative real number, none of the nth roots is real. If n is odd and x is real, one nth root is real and has the same... | https://en.wikipedia.org/wiki/Seventh_root |
Roots of real numbers are usually written using the radical symbol or radix {\displaystyle {\sqrt {{~^{~}}^{~}\!\!}}} , with x {\displaystyle {\sqrt {x}}} denoting the positive square root of x if x is positive; for higher roots, x n {\displaystyle {\sqrt{x}}} denotes the real nth root if n is odd, and the positive nth... | https://en.wikipedia.org/wiki/Seventh_root |
When complex nth roots are considered, it is often useful to choose one of the roots, called principal root, as a principal value. The common choice is to choose the principal nth root of x as the nth root with the greatest real part, and when there are two (for x real and negative), the one with a positive imaginary p... | https://en.wikipedia.org/wiki/Seventh_root |
A difficulty with this choice is that, for a negative real number and an odd index, the principal nth root is not the real one. For example, − 8 {\displaystyle -8} has three cube roots, − 2 {\displaystyle -2} , 1 + i 3 {\displaystyle 1+i{\sqrt {3}}} and 1 − i 3 . {\displaystyle 1-i{\sqrt {3}}.} | https://en.wikipedia.org/wiki/Seventh_root |
The real cube root is − 2 {\displaystyle -2} and the principal cube root is 1 + i 3 . {\displaystyle 1+i{\sqrt {3}}.} An unresolved root, especially one using the radical symbol, is sometimes referred to as a surd or a radical. | https://en.wikipedia.org/wiki/Seventh_root |
Any expression containing a radical, whether it is a square root, a cube root, or a higher root, is called a radical expression, and if it contains no transcendental functions or transcendental numbers it is called an algebraic expression. The positive root of a number is the inverse operation of Exponentiation with po... | https://en.wikipedia.org/wiki/Seventh_root |
Roots can also be defined as special cases of exponentiation, where the exponent is a fraction: x n = x 1 / n . {\displaystyle {\sqrt{x}}=x^{1/n}.} Roots are used for determining the radius of convergence of a power series with the root test. The nth roots of 1 are called roots of unity and play a fundamental role in v... | https://en.wikipedia.org/wiki/Seventh_root |
In mathematics, tautness is a rigidity property of foliations. A taut foliation is a codimension 1 foliation of a closed manifold with the property that every leaf meets a transverse circle. : 155 By transverse circle, is meant a closed loop that is always transverse to the tangent field of the foliation. If the foliat... | https://en.wikipedia.org/wiki/Rummler–Sullivan_theorem |
Equivalently, by a result of Dennis Sullivan, a codimension 1 foliation is taut if there exists a Riemannian metric that makes each leaf a minimal surface. Furthermore, for compact manifolds the existence, for every leaf L {\displaystyle L} , of a transverse circle meeting L {\displaystyle L} , implies the existence of... | https://en.wikipedia.org/wiki/Rummler–Sullivan_theorem |
In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, it was used by Albert Einstein to develop his general theory of re... | https://en.wikipedia.org/wiki/Tensor_calculus |
Working with a main proponent of the exterior calculus Elie Cartan, the influential geometer Shiing-Shen Chern summarizes the role of tensor calculus:In our subject of differential geometry, where you talk about manifolds, one difficulty is that the geometry is described by coordinates, but the coordinates do not have ... | https://en.wikipedia.org/wiki/Tensor_calculus |
In mathematics you have a function, you write down the function, you calculate, or you add, or you multiply, or you can differentiate. You have something very concrete. In geometry the geometric situation is described by numbers, but you can change your numbers arbitrarily. So to handle this, you need the Ricci calculu... | https://en.wikipedia.org/wiki/Tensor_calculus |
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though ↑↑ {\displaystyle \uparrow \uparrow } and the left-exponent xb are common. Under the definition as repeated exponentiation, n a {\displaystyle {^{n}a}} means a a ⋅ ⋅... | https://en.wikipedia.org/wiki/Iterated_exponentiation |
It would be read as "the nth tetration of a". It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration. | https://en.wikipedia.org/wiki/Iterated_exponentiation |
Tetration is also defined recursively as a ↑↑ n := { 1 if n = 0 , a a ↑↑ ( n − 1 ) if n > 0 , {\displaystyle {a\uparrow \uparrow n}:={\begin{cases}1&{\text{if }}n=0,\\a^{a\uparrow \uparrow (n-1)}&{\text{if }}n>0,\end{cases}}} allowing for attempts to extend tetration to non-natural numbers such as real and complex numb... | https://en.wikipedia.org/wiki/Iterated_exponentiation |
In mathematics, the "happy ending problem" (so named by Paul Erdős because it led to the marriage of George Szekeres and Esther Klein) is the following statement: This was one of the original results that led to the development of Ramsey theory. The happy ending theorem can be proven by a simple case analysis: if four ... | https://en.wikipedia.org/wiki/Erdős–Szekeres_conjecture |
See Peterson (2000) for an illustrated explanation of this proof, and Morris & Soltan (2000) for a more detailed survey of the problem. The Erdős–Szekeres conjecture states precisely a more general relationship between the number of points in a general-position point set and its largest subset forming a convex polygon,... | https://en.wikipedia.org/wiki/Erdős–Szekeres_conjecture |
In mathematics, the "strong law of small numbers" is the humorous law that proclaims, in the words of Richard K. Guy (1988): There aren't enough small numbers to meet the many demands made of them. In other words, any given small number appears in far more contexts than may seem reasonable, leading to many apparently s... | https://en.wikipedia.org/wiki/Strong_law_of_small_numbers |
In mathematics, the 'extension' of a mathematical concept C {\displaystyle C} is the set that is specified by C {\displaystyle C} . (That set might be empty, currently.) For example, the extension of a function is a set of ordered pairs that pair up the arguments and values of the function; in other words, the function... | https://en.wikipedia.org/wiki/Extension_(semantics) |
The extension of a set is the set itself. That a set can capture the notion of the extension of anything is the idea behind the axiom of extensionality in axiomatic set theory. This kind of extension is used so constantly in contemporary mathematics based on set theory that it can be called an implicit assumption. A ty... | https://en.wikipedia.org/wiki/Extension_(semantics) |
In mathematics, the (exponential) shift theorem is a theorem about polynomial differential operators (D-operators) and exponential functions. It permits one to eliminate, in certain cases, the exponential from under the D-operators. | https://en.wikipedia.org/wiki/Shift_theorem |
In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield. | https://en.wikipedia.org/wiki/Relative_norm |
In mathematics, the (linear) Peetre theorem, named after Jaak Peetre, is a result of functional analysis that gives a characterisation of differential operators in terms of their effect on generalized function spaces, and without mentioning differentiation in explicit terms. The Peetre theorem is an example of a finite... | https://en.wikipedia.org/wiki/Peetre_theorem |
In mathematics, the (right) Ziegler spectrum of a ring R is a topological space whose points are (isomorphism classes of) indecomposable pure-injective right R-modules. Its closed subsets correspond to theories of modules closed under arbitrary products and direct summands. Ziegler spectra are named after Martin Ziegle... | https://en.wikipedia.org/wiki/Ziegler_spectrum |
In mathematics, the (signed and unsigned) Lah numbers are coefficients expressing rising factorials in terms of falling factorials and vice versa. They were discovered by Ivo Lah in 1954. Explicitly, the unsigned Lah numbers L ( n , k ) {\displaystyle L(n,k)} are given by the formula involving the binomial coefficient ... | https://en.wikipedia.org/wiki/Lah_number |
Lah numbers are related to Stirling numbers.For n ≥ 1 {\textstyle n\geq 1} , the Lah number L ( n , 1 ) {\textstyle L(n,1)} is equal to the factorial n ! {\textstyle n!} | https://en.wikipedia.org/wiki/Lah_number |
in the interpretation above, the only partition of { 1 , 2 , 3 } {\textstyle \{1,2,3\}} into 1 set can have its set ordered in 6 ways: L ( 3 , 2 ) {\textstyle L(3,2)} is equal to 6, because there are six partitions of { 1 , 2 , 3 } {\textstyle \{1,2,3\}} into two ordered parts: L ( n , n ) {\textstyle L(n,n)} is always... | https://en.wikipedia.org/wiki/Lah_number |
In mathematics, the 15 theorem or Conway–Schneeberger Fifteen Theorem, proved by John H. Conway and W. A. Schneeberger in 1993, states that if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers. The proof was complicated, and was ne... | https://en.wikipedia.org/wiki/15_and_290_theorems |
In mathematics, the 2π theorem of Gromov and Thurston states a sufficient condition for Dehn filling on a cusped hyperbolic 3-manifold to result in a negatively curved 3-manifold. Let M be a cusped hyperbolic 3-manifold. Disjoint horoball neighborhoods of each cusp can be selected. The boundaries of these neighborhoods... | https://en.wikipedia.org/wiki/2π_theorem |
A slope, i.e. unoriented isotopy class of simple closed curves on these boundaries, thus has a well-defined length by taking the minimal Euclidean length over all curves in the isotopy class. The 2π theorem states: a Dehn filling of M with each filling slope greater than 2π results in a 3-manifold with a complete metri... | https://en.wikipedia.org/wiki/2π_theorem |
The basic idea of the proof is to explicitly construct a negatively curved metric inside each horoball neighborhood that matches the metric near the horospherical boundary. This construction, using cylindrical coordinates, works when the filling slope is greater than 2π. See Bleiler & Hodgson (1996) for complete detail... | https://en.wikipedia.org/wiki/2π_theorem |
According to the geometrization conjecture, these negatively curved 3-manifolds must actually admit a complete hyperbolic metric. A horoball packing argument due to Thurston shows that there are at most 48 slopes to avoid on each cusp to get a hyperbolic 3-manifold. For one-cusped hyperbolic 3-manifolds, an improvement... | https://en.wikipedia.org/wiki/2π_theorem |
This result was later improved independently by Ian Agol (2000) and Marc Lackenby (2000) with the 6 theorem. The "6 theorem" states that Dehn filling along slopes of length greater than 6 results in a hyperbolike 3-manifold, i.e. an irreducible, atoroidal, non-Seifert-fibered 3-manifold with infinite word hyperbolic fu... | https://en.wikipedia.org/wiki/2π_theorem |
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