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...expanders”, 2002) in order to conveniently define the [[zig-zag product]] and prove its properties. ...the rotation map returns the <math>i</math>'th neighbor of <math>v</math> and the edge label that would lead back to <math>v</math>. ...

{{Short description|Graph with directed and undirected edges}} ...a set of (undirected) [[edge (graph theory)|edges]] {{mvar|E}}, and a set of [[Directed graph|directed]] edges (or arcs) {{mvar|A}}.<ref name="BeckBlado ...

...pretation in the spirit of [[color refinement algorithm|color refinement]] and a connection to logic. There are several versions of the test (e.g. k-WL and k-FWL) referred to in the literature by various names, which easily leads t ...

...e pairs of entities are associated by directed relationships, and entities and relationships can have properties. ...elationships. Each arc has an identifier, a source node and a target node, and may have properties. ...

...'. The graphon is shown as a magenta heatmap (lower right). A random graph of size <math>n</math> is generated by independently assigning to each vertex <math>U_k \sim \mathrm{U}(0,1)</math> (values along vertical axis) and ...

{{short description|Generalizations in graph theory}} ...-JGT2>3.0.CO;2-V|issn=1097-0118}}</ref> [[Roy Meshulam]],<ref name=":7" /> and others. ...

| conjectured by = van den Berg and [[Harry Kesten|Kesten]] | generalizations = ...

...groups]].''' Bass–Serre theory can be regarded as one-dimensional version of the [[orbifold|orbifold theory]]. ...antially to the formalization and development of basic tools of the theory and currently the term "Bass–Serre theory" is widely used to describe the subje ...

...of two [[hyperbolic metric space]]s extending a discrete isometric actions of the group on those spaces. ...iginated from a seminal 1980s preprint of [[James W. Cannon|James Cannon]] and [[William Thurston]] "Group-invariant Peano curves" (eventually published i ...

...t description|Deterministic finite automaton accepting set of all suffixes of particular string}} ...hat paths from the initial vertex to final vertices represent the suffixes of the string. ...

{{for|Roth's theorem on Diophantine approximation of algebraic numbers|Roth's theorem}} {{short description|On the existence of arithmetic progressions in subsets of the natural numbers}} ...

...man]] extended this method to arbitrary fixed-length keys. Generalizations of tabulation hashing have also been developed that can handle variable-length ...slower variants of tabulation hashing extend the method to higher degrees of independence. ...

| birth_date = {{birth date and age|1943|01|30}} | work_institutions = [[University of Wisconsin-Madison]] <br>[[Brigham Young University]] ...

...e such example is <math>\ell_1</math> regularization (also known as Lasso) of the form ...rangle)^2+ \lambda \|w\|_1, \quad \text{ where } x_i\in \mathbb{R}^d\text{ and } y_i\in\mathbb{R}.</math> ...

{{short description|Branch of mathematics}} ...rg/details/geometrylanguage0000taba |title=Geometry: the language of space and form |date=2014 |publisher=Infobase Publishing |isbn=978-0-8160-4953-0 |pag ...

{{Short description|Branch of applied mathematics}} ...Republished with revisions from 1986, "Theoretic Models: Mathematical Form and Economic Content", ''Econometrica'', 54(6), pp. [https://www.jstor.org/pss/ ...