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...but not on the time when the observation was started. Many large families of stochastic processes have stationary increments either by definition (e.g. ...for all <math> t \geq 0 </math> and <math> h > 0 </math>, the distribution of the random variables ...

...nt of a [[Sign (mathematics)|sign]] (positive or negative) to each element of the set. ...underlying unsigned set (possibly specified explicitly as a separate part of the representation) and whose range is a two-element set representing the s ...

{{distinguish|Family of sets}} {{Short description|Collection of objects, each associated with an element from some index set}} ...

...d|uniform bound]] on [[limit cycles]] in generic finite-parameter families of [[vector fields]] on a sphere?}} ...the 1980s.<ref name="I94">Ilyashenko, Yu. (1994). "Normal forms for local families and nonlocal bifurcations". Asterisque, Vol. 222, 233-258.</ref> ...

A family <math>E_i</math> of subsets of a topological space is said to be '''closure-preserving''' if for every sub For example, a locally finite family of subsets has this property. With this terminology, the theorem states:<ref>{ ...

...book |last=Jech |first=Thomas J.|author-link=Thomas Jech |title=The Axiom of Choice |isbn=978-0-486-46624-8 |publisher=[[Dover Publications]] |year=2008 A family of sets <math>\mathcal{F}</math> is of '''finite character''' provided it has the following properties: ...

{{Short description|Matroid with complemented basis sets}} In [[matroid theory]], the '''dual''' of a matroid <math>M</math> is another matroid <math>M^\ast</math> that has th ...

...when one can extend a probability measure to a larger [[σ-algebra]]. It is of particular interest for infinite dimensional spaces. ...trich Bierlein]], who proved the statement for [[countable set|countable]] families in 1962.<ref name="db">{{cite journal |first=Dietrich |last=Bierlein |title ...

...wing in red the undirected tree induced by the subgraph reachable from one of its vertices.]] ...[Vertex (graph theory)|vertices]], or equivalently in which the [[Glossary of graph theory#subgraph|subgraph]] reachable from any vertex induces an [[Tre ...

...th>G_0, G_1, \dots</math>, with <math>h(G)</math> defined to be the length of the longest cycle in graph <math>G</math>, the shortness exponent is define | journal = [[Journal of Combinatorial Theory]] ...

...uniform spaces. He was one of the founders of modern descriptive theory of sets and spaces.<ref>Zdeněk Frolík 1933–1989, [[Mirek Husek]], [[Jan Pelant]], ' ....E. Vaughan, Countably compact and sequentially compact spaces. ''Handbook of Set-theoretic Topology'', [[Kenneth Kunen|K. Kunen]] and J. Vaughan (ed.), ...

...f subsets of <math>\{1,\dots,n\}</math> of size {{mvar|k}}, as well as the families achieving the maximum size. ...{{tmath|\cal F}} is a collection of subsets of <math>\{1,\dots,n\}</math> of size {{mvar|k}} such that if {{tmath|A,B \in \cal F}} then <math>|A\cap B| ...

{{Short description|Model of computation}} ...isjunction, and negation gates. The values in an integer circuit are sets of integers and the gates compute set union, set intersection, and set complem ...

...t of [[statistical learning theory]], where it is used to study properties of statistical learning methods. ...y Vapnik and Chervonenkis in their 1968 paper, where they also proved many of its properties.<ref name="vc">{{Cite Vapnik Chervonenkis}}</ref> ...

...-separable'''. These concepts play a pivotal role in the mathematical area of non-adaptive [[group testing]]. ...er-imposed codes'''<ref name="DeBonisVaccaro2003"/> or '''''d''-cover-free families'''.<ref name="ErdoesEtAl1985"/> ...

{{Short description|Direct sum of uniform matroids}} ...''' or '''partitional matroid''' is a [[matroid]] that is a [[direct sum]] of [[uniform matroid]]s.<ref>{{citation ...

...be '''saturated''' if <math>\mathcal{G}</math> contains a non-empty subset of <math>X</math> and if for every <math>G \in \mathcal{G},</math> the followi # <math>\mathcal{G}</math> contains every subset of <math>G</math>; ...

{{Short description|Algebra of a branch of probability theory}} ...th a [[stopping time]] in the theory of [[stochastic process]]es, a branch of [[probability theory]].<ref name="rajeeva47" /><ref name="klenke193" /> ...

...very finite family of sets (green) there is another family of equally many sets (blue outlines) such that each set in the second family is shattered by the ...r [[Norbert Sauer]] and [[Saharon Shelah]], who published it independently of each other in 1972.{{r|sauer|shelah}} The same result was also published sl ...

...The Macdonald-Koornwinder polynomials have also been studied with the aid of [[affine Hecke algebra]]s.{{sfnm|1a1=Noumi|1y=1995|2a1=Sahi|2y=1999|3a1=Mac ...he unique [[Laurent polynomial]] invariant under permutation and inversion of variables, with '''leading monomial''' ''x''^λ, and orthogonal wi ...