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Uncountable set In mathematics, an uncountable set (or uncountably infinite set)[1] is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers. Characterizations There are many equivalent characterizations of uncountability. A set X is uncountable if and only if any of the following conditions hold: • There is no injective function (hence no bijection) from X to the set of natural numbers. • X is nonempty and for every ω-sequence of elements of X, there exists at least one element of X not included in it. That is, X is nonempty and there is no surjective function from the natural numbers to X. • The cardinality of X is neither finite nor equal to $\aleph _{0}$ (aleph-null, the cardinality of the natural numbers). • The set X has cardinality strictly greater than $\aleph _{0}$. The first three of these characterizations can be proven equivalent in Zermelo–Fraenkel set theory without the axiom of choice, but the equivalence of the third and fourth cannot be proved without additional choice principles. Properties • If an uncountable set X is a subset of set Y, then Y is uncountable. Examples The best known example of an uncountable set is the set R of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers and the set of all subsets of the set of natural numbers. The cardinality of R is often called the cardinality of the continuum, and denoted by ${\mathfrak {c}}$, or $2^{\aleph _{0}}$, or $\beth _{1}$ (beth-one). The Cantor set is an uncountable subset of R. The Cantor set is a fractal and has Hausdorff dimension greater than zero but less than one (R has dimension one). This is an example of the following fact: any subset of R of Hausdorff dimension strictly greater than zero must be uncountable. Another example of an uncountable set is the set of all functions from R to R. This set is even "more uncountable" than R in the sense that the cardinality of this set is $\beth _{2}$ (beth-two), which is larger than $\beth _{1}$. A more abstract example of an uncountable set is the set of all countable ordinal numbers, denoted by Ω or ω1.[1] The cardinality of Ω is denoted $\aleph _{1}$ (aleph-one). It can be shown, using the axiom of choice, that $\aleph _{1}$ is the smallest uncountable cardinal number. Thus either $\beth _{1}$, the cardinality of the reals, is equal to $\aleph _{1}$ or it is strictly larger. Georg Cantor was the first to propose the question of whether $\beth _{1}$ is equal to $\aleph _{1}$. In 1900, David Hilbert posed this question as the first of his 23 problems. The statement that $\aleph _{1}=\beth _{1}$ is now called the continuum hypothesis, and is known to be independent of the Zermelo–Fraenkel axioms for set theory (including the axiom of choice). Without the axiom of choice Main article: Dedekind-infinite set Without the axiom of choice, there might exist cardinalities incomparable to $\aleph _{0}$ (namely, the cardinalities of Dedekind-finite infinite sets). Sets of these cardinalities satisfy the first three characterizations above, but not the fourth characterization. Since these sets are not larger than the natural numbers in the sense of cardinality, some may not want to call them uncountable. If the axiom of choice holds, the following conditions on a cardinal $\kappa $ are equivalent: • $\kappa \nleq \aleph _{0};$ • $\kappa >\aleph _{0};$ and • $\kappa \geq \aleph _{1}$, where $\aleph _{1}=\|\omega _{1}\|$ and $\omega _{1}$ is the least initial ordinal greater than $\omega .$ However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriate generalization of "uncountability" when the axiom fails. It may be best to avoid using the word in this case and specify which of these one means. See also • Aleph number • Beth number • First uncountable ordinal • Injective function References 1. Weisstein, Eric W. "Uncountably Infinite". mathworld.wolfram.com. Retrieved 2020-09-05. Bibliography • Halmos, Paul, Naive Set Theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. ISBN 978-1-61427-131-4 (Paperback edition). • Jech, Thomas (2002), Set Theory, Springer Monographs in Mathematics (3rd millennium ed.), Springer, ISBN 3-540-44085-2 External links • Proof that R is uncountable Mathematical logic General • Axiom • list • Cardinality • First-order logic • Formal proof • Formal semantics • Foundations of mathematics • Information theory • Lemma • Logical consequence • Model • Theorem • Theory • Type theory Theorems (list) & Paradoxes • Gödel's completeness and incompleteness theorems • Tarski's undefinability • Banach–Tarski paradox • Cantor's theorem, paradox and diagonal argument • Compactness • Halting problem • Lindström's • Löwenheim–Skolem • Russell's paradox Logics Traditional • Classical logic • Logical truth • Tautology • Proposition • Inference • Logical equivalence • Consistency • Equiconsistency • Argument • Soundness • Validity • Syllogism • Square of opposition • Venn diagram Propositional • Boolean algebra • Boolean functions • Logical connectives • Propositional calculus • Propositional formula • Truth tables • Many-valued logic • 3 • Finite • ∞ Predicate • First-order • list • Second-order • Monadic • Higher-order • Free • Quantifiers • Predicate • Monadic predicate calculus Set theory • Set • Hereditary • Class • (Ur-)Element • Ordinal number • Extensionality • Forcing • Relation • Equivalence • Partition • Set operations: • Intersection • Union • Complement • Cartesian product • Power set • Identities Types of Sets • Countable • Uncountable • Empty • Inhabited • Singleton • Finite • Infinite • Transitive • Ultrafilter • Recursive • Fuzzy • Universal • Universe • Constructible • Grothendieck • Von Neumann Maps & Cardinality • Function/Map • Domain • Codomain • Image • In/Sur/Bi-jection • Schröder–Bernstein theorem • Isomorphism • Gödel numbering • Enumeration • Large cardinal • Inaccessible • Aleph number • Operation • Binary Set theories • Zermelo–Fraenkel • Axiom of choice • Continuum hypothesis • General • Kripke–Platek • Morse–Kelley • Naive • New Foundations • Tarski–Grothendieck • Von Neumann–Bernays–Gödel • Ackermann • Constructive Formal systems (list), Language & Syntax • Alphabet • Arity • Automata • Axiom schema • Expression • Ground • Extension • by definition • Conservative • Relation • Formation rule • Grammar • Formula • Atomic • Closed • Ground • Open • Free/bound variable • Language • Metalanguage • Logical connective • ¬ • ∨ • ∧ • → • ↔ • = • Predicate • Functional • Variable • Propositional variable • Proof • Quantifier • ∃ • ! • ∀ • rank • Sentence • Atomic • Spectrum • Signature • String • Substitution • Symbol • Function • Logical/Constant • Non-logical • Variable • Term • Theory • list Example axiomatic systems (list) • of arithmetic: • Peano • second-order • elementary function • primitive recursive • Robinson • Skolem • of the real numbers • Tarski's axiomatization • of Boolean algebras • canonical • minimal axioms • of geometry: • Euclidean: • Elements • Hilbert's • Tarski's • non-Euclidean • Principia Mathematica Proof theory • Formal proof • Natural deduction • Logical consequence • Rule of inference • Sequent calculus • Theorem • Systems • Axiomatic • Deductive • Hilbert • list • Complete theory • Independence (from ZFC) • Proof of impossibility • Ordinal analysis • Reverse mathematics • Self-verifying theories Model theory • Interpretation • Function • of models • Model • Equivalence • Finite • Saturated • Spectrum • Submodel • Non-standard model • of arithmetic • Diagram • Elementary • Categorical theory • Model complete theory • Satisfiability • Semantics of logic • Strength • Theories of truth • Semantic • Tarski's • Kripke's • T-schema • Transfer principle • Truth predicate • Truth value • Type • Ultraproduct • Validity Computability theory • Church encoding • Church–Turing thesis • Computably enumerable • Computable function • Computable set • Decision problem • Decidable • Undecidable • P • NP • P versus NP problem • Kolmogorov complexity • Lambda calculus • Primitive recursive function • Recursion • Recursive set • Turing machine • Type theory Related • Abstract logic • Category theory • Concrete/Abstract Category • Category of sets • History of logic • History of mathematical logic • timeline • Logicism • Mathematical object • Philosophy of mathematics • Supertask Mathematics portal Set theory Overview • Set (mathematics) Axioms • Adjunction • Choice • countable • dependent • global • Constructibility (V=L) • Determinacy • Extensionality • Infinity • Limitation of size • Pairing • Power set • Regularity • Union • Martin's axiom • Axiom schema • replacement • specification Operations • Cartesian product • Complement (i.e. set difference) • De Morgan's laws • Disjoint union • Identities • Intersection • Power set • Symmetric difference • Union • Concepts • Methods • Almost • Cardinality • Cardinal number (large) • Class • Constructible universe • Continuum hypothesis • Diagonal argument • Element • ordered pair • tuple • Family • Forcing • One-to-one correspondence • Ordinal number • Set-builder notation • Transfinite induction • Venn diagram Set types • Amorphous • Countable • Empty • Finite (hereditarily) • Filter • base • subbase • Ultrafilter • Fuzzy • Infinite (Dedekind-infinite) • Recursive • Singleton • Subset · Superset • Transitive • Uncountable • Universal Theories • Alternative • Axiomatic • Naive • Cantor's theorem • Zermelo • General • Principia Mathematica • New Foundations • Zermelo–Fraenkel • von Neumann–Bernays–Gödel • Morse–Kelley • Kripke–Platek • Tarski–Grothendieck • Paradoxes • Problems • Russell's paradox • Suslin's problem • Burali-Forti paradox Set theorists • Paul Bernays • Georg Cantor • Paul Cohen • Richard Dedekind • Abraham Fraenkel • Kurt Gödel • Thomas Jech • John von Neumann • Willard Quine • Bertrand Russell • Thoralf Skolem • Ernst Zermelo	Wikipedia
Landau set In voting systems, the Landau set (or uncovered set, or Fishburn set) is the set of candidates $x$ such that for every other candidate $z$, there is some candidate $y$ (possibly the same as $x$ or $z$) such that $y$ is not preferred to $x$ and $z$ is not preferred to $y$. In notation, $x$ is in the Landau set if $\forall \,z$, $\exists \,y$, $x\geq y\geq z$. The Landau set is a nonempty subset of the Smith set. It was discovered by Nicholas Miller. References • Nicholas R. Miller, "Graph-theoretical approaches to the theory of voting", American Journal of Political Science, Vol. 21 (1977), pp. 769–803. doi:10.2307/2110736. JSTOR 2110736. • Nicholas R. Miller, "A new solution set for tournaments and majority voting: further graph-theoretic approaches to majority voting", American Journal of Political Science, Vol. 24 (1980), pp. 68–96. doi:10.2307/2110925. JSTOR 2110925. • Norman J. Schofield, "Social Choice and Democracy", Springer-Verlag: Berlin, 1985. • Philip D. Straffin, "Spatial models of power and voting outcomes", in Applications of Combinatorics and Graph Theory to the Biological and Social Sciences, Springer: New York-Berlin, 1989, pp. 315–335. • Elizabeth Maggie Penn, "Alternate definitions of the uncovered set and their implications", 2004. • Nicholas R. Miller, "In search of the uncovered set", Political Analysis, 15:1 (2007), pp. 21–45. doi:10.1093/pan/mpl007. JSTOR 25791876. • William T. Bianco, Ivan Jeliazkov, and Itai Sened, "The uncovered set and the limits of legislative action", Political Analysis, Vol. 12, No. 3 (2004), pp. 256–276. doi:10.1093/pan/mph018. JSTOR 25791775.	Wikipedia
Hendecahedron A hendecahedron (or undecahedron) is a polyhedron with 11 faces. There are numerous topologically distinct forms of a hendecahedron, for example the decagonal pyramid, and enneagonal prism. Three forms are Johnson solids: augmented hexagonal prism, biaugmented triangular prism, and elongated pentagonal pyramid. Two classes, the bisymmetric and the sphenoid hendecahedra, are space-filling.[1] Name of hendecahedron The name of hendecahedron is based on its meaning. Hen- represents one. Deca represents ten, and when combined with the polyhedron suffix -hedron, the name becomes Hendecahedron. Common hendecahedron In all the convex hendecahedrons, there are a total of 440,564 convex ones with distinct differences in topology. There are significant differences in the structure of topology, which means two types of polyhedrons cannot be transformed by moving vertex positions, twisting, or scaling, such as a pentagonal pyramid and a nine diagonal column. They can't change with each other, so their topology structure is different. But the pentagonal prism and enneagonal prism can interchange by stretching out or drawing back one of the nine sides of the scale, so the triangulum prism and the triangulum pyramid have no obvious difference in topology. The common hendecahedrons are cones, cylinders, some Jason polyhedrons, and the semi-regular polyhedron. The semi-regular polyhedron here is not the Archimedean solid, but the enneagonal prism. Other hendecahedrons include enneagonal prism, Spherical octagonal pyramid, two side taper triangular prism of the duality of six, side cone Angle and bisymmetric hendecahedron, which can close shop space. Bisymmetric hendecahedron The bisymmetric hendecahedron is a space-filling polyhedron which can be assembled into layers of interpenetrating "boat-shaped" tetramers, which in turn are then stacked to fill space; it is hence a three-dimensional analogue of the Cairo pentagon. Sphenoid hendecahedron The sphenoid hendecahedron is a space-filling polyhedron which can be assembled into layers of the Floret tiling, which in turn are stacked to fill space. Hendecahedron in chemistry In the chemistry, after removing all 18 sides in borane hydrogen ions ([B11H11]), it is an Octadecahedron. If making a perpendicular to the center of gravity to the surface of a boron atom, a new polyhedron is constructed, which is 18 surface structures of the dual polyhedron, also one of hendecahedrons. Convex There are 440,564 topologically distinct convex hendecahedra, excluding mirror images, having at least 8 vertices.[2] (Two polyhedra are "topologically distinct" if they have intrinsically different arrangements of faces and vertices, such that it is impossible to distort one into the other simply by changing the lengths of edges or the angles between edges or faces.) References 1. Inchbald (1996) 2. Counting polyhedra • Thomas H. Sidebotham. The A to Z of Mathematics: A Basic Guide. John Wiley & Sons. 2003: 237. ISBN 9780471461630 • Steven Dutch: How Many Polyhedra are There? (http://www.uwgb.edu/dutchs/symmetry/POLYHOW M.HTM) • Counting polyhedra (http://www.numericana.com/data/polycount.htm) numericana.com [2016-1-10] • Inchbald, Guy. "Five Space-Filling Polyhedra." The Mathematical Gazette 80, no. 489 (November 1996): 466-475 • Space-Filling Bisymmetric Hendecahedron. [2013-04-11] • Anderson, Ian. "Constructing Tournament Designs." The Mathematical Gazette 73, no. 466 (December 1989): 284-292 • Holleman, A. F.; Wiberg, E., Inorganic Chemistry, San Diego: Academic Press: 1165, 2001, ISBN 0-12-352651-5 • Inchbald, Guy (1996). "Five space-filling polyhedra". The Mathematical Gazette. 80 (489): 466–475. doi:10.2307/3618509. ISSN 0025-5572. JSTOR 3618509. Zbl 0885.52011. External links • Weisstein, Eric W. "Undecahedron". MathWorld. Polyhedra Listed by number of faces and type 1–10 faces • Monohedron • Dihedron • Trihedron • Tetrahedron • Pentahedron • Hexahedron • Heptahedron • Octahedron • Enneahedron • Decahedron 11–20 faces • Hendecahedron • Dodecahedron • Tridecahedron • Tetradecahedron • Pentadecahedron • Hexadecahedron • Heptadecahedron • Octadecahedron • Enneadecahedron • Icosahedron >20 faces • Icositetrahedron (24) • Triacontahedron (30) • Hexecontahedron (60) • Enneacontahedron (90) • Hectotriadiohedron (132) • Apeirohedron (∞) elemental things • face • edge • vertex • uniform polyhedron (two infinite groups and 75) • regular polyhedron (9) • quasiregular polyhedron (7) • semiregular polyhedron (two infinite groups and 59) convex polyhedron • Platonic solid (5) • Archimedean solid (13) • Catalan solid (13) • Johnson solid (92) non-convex polyhedron • Kepler–Poinsot polyhedron (4) • Star polyhedron (infinite) • Uniform star polyhedron (57) prismatoid‌s • prism • antiprism • frustum • cupola • wedge • pyramid • parallelepiped	Wikipedia
Impossible object An impossible object (also known as an impossible figure or an undecidable figure) is a type of optical illusion that consists of a two-dimensional figure which is instantly and naturally understood as representing a projection of a three-dimensional object but cannot exist as a solid object. Impossible objects are of interest to psychologists, mathematicians and artists without falling entirely into any one discipline. Notable examples Notable impossible objects include: • Borromean rings — although conventionally drawn as three linked circles in three-dimensional space, any realization must be non-circular[1] • Impossible cube — invented by M.C. Escher for Belvedere, a lithograph in which a boy seated at the foot of the building holds an impossible cube.[2][3] • Penrose stairs – created by Oscar Reutersvärd and later independently devised and popularised by Lionel Penrose and his mathematician son Roger Penrose.[4] A variation on the Penrose triangle, it is a two-dimensional depiction of a staircase in which the stairs make four 90-degree turns as they ascend or descend yet form a continuous loop, so that a person could climb them forever and never get any higher. • Penrose triangle (Tribar) – first created by the Swedish artist Oscar Reutersvärd in 1934. Roger Penrose independently devised and popularised it in the 1950s, describing it as "impossibility in its purest form". • Impossible trident (or devil's tuning fork) – The Blivet, has three cylindrical prongs at one end which then mysteriously transform into two rectangular prongs at the other end.[5] • L'egsistential Quandary – Created by Roger Shepard in 1990, is a drawing of an elephant whose four feet are drawn at the bottom of the white space between legs, instead of on the legs themselves.[6] Examples of impossible objects • Borromean rings • Oscar Reutersvärd's optical illusion (1934) • Penrose stairs • Penrose triangle • Impossible trident • Impossible waterfall • A variant of L'egsistential Quandary Explanations Impossible objects can be unsettling because of our natural desire to interpret 2D drawings as three-dimensional objects. This is why a drawing of a Necker cube would most likely be seen as a cube, rather than "two squares connected with diagonal lines, a square surrounded by irregular planar figures, or any other planar figure". Looking at different parts of an impossible object makes one reassess the 3D nature of the object, which confuses the mind.[7] In most cases the impossibility becomes apparent after viewing the figure for a few seconds. However, the initial impression of a 3D object remains even after it has been contradicted. There are also more subtle examples of impossible objects where the impossibility does not become apparent spontaneously and it is necessary to consciously examine the geometry of the implied object to determine that it is impossible. Roger Penrose wrote about describing and defining impossible objects mathematically using the algebraic topology concept of cohomology.[8][9] History An early example of an impossible object comes from Apolinère Enameled, a 1916 advertisement painted by Marcel Duchamp. It depicts a girl painting a bed-frame with white enamelled paint, and deliberately includes conflicting perspective lines, to produce an impossible object. To emphasise the deliberate impossibility of the shape, a piece of the frame is missing. Swedish artist Oscar Reutersvärd was one of the first to deliberately design many impossible objects. He has been called "the father of impossible figures".[10] In 1934, he drew the Penrose triangle, some years before the Penroses. In Reutersvärd's version, the sides of the triangle are broken up into cubes. In 1956, British psychiatrist Lionel Penrose and his son, mathematician Roger Penrose, submitted a short article to the British Journal of Psychology titled "Impossible Objects: A Special Type of Visual Illusion". This was illustrated with the Penrose triangle and Penrose stairs. The article referred to Escher, whose work had sparked their interest in the subject, but not Reutersvärd, of whom they were unaware. The article was published in 1958.[4] From the 1930s onwards, Dutch artist M.C. Escher produced many drawings featuring paradoxes of perspective gradually working towards impossible objects.[10] In 1957, he produced his first drawing containing a true impossible object: Cube with Magic Ribbons. He produced many further drawings featuring impossible objects, sometimes with the entire drawing being an impossible object. Waterfall and Belvedere are good examples of impossible constructions. His work did much to draw the attention of the public to impossible objects. Some contemporary artists are also experimenting with impossible figures, for example, Jos de Mey, Shigeo Fukuda, Sandro del Prete, István Orosz (Utisz), Guido Moretti, Tamás F. Farkas, Mathieu Hamaekers, and Kokichi Sugihara. Constructed impossible objects Although possible to represent in two dimensions, it is not geometrically possible for such an object to exist in the physical world. However some models of impossible objects have been constructed, such that when they are viewed from a very specific point, the illusion is maintained. Rotating the object or changing the viewpoint breaks the illusion, and therefore many of these models rely on forced perspective or having parts of the model appearing to be further or closer than they actually are. The notion of an "interactive impossible object" is an impossible object that can be viewed from any angle without breaking the illusion.[11] As the viewing angle changes of this sculpture in East Perth, Australia, a Penrose triangle appears to form. See also • Four-dimensional space – Geometric space with four dimensions • Mathematics and art – Relationship between mathematics and art • Möbius strip – Non-orientable surface with one edge • Multistable perception – Perceptual phenomenon • Necker cube – Form of perceptual phenomena • Non-Euclidean geometry – Two geometries based on axioms closely related to those specifying Euclidean geometry • Paradox – Statement that apparently contradicts itself • Pareidolia – Perception of meaningful patterns or images in random or vague stimuli • Puzzle – Problem or enigma that tests the ingenuity of the solver • Strange loop – Cyclic structure that goes through several levels in a hierarchical system • Surrealism – International cultural movement active from the 1920s to the 1950s • Tesseract – Four-dimensional analogue of the cube • Tritone paradox – An auditory illusion perceived by some people to be rising in pich and by others to be falling References 1. Aigner, Martin; Ziegler, Günter M. (2018). "Chapter 15: The Borromean Rings Don't Exist". Proofs from THE BOOK (6th ed.). Springer. pp. 99–106. doi:10.1007/978-3-662-57265-8_15. ISBN 978-3-662-57265-8. 2. Bruno Ernst (Hans de Rijk) (2003). "Selection is Distortion". In Schattschneider, D.; Emmer, M. (eds.). M. C. Escher's Legacy: A Centennial Celebration. Springer. pp. 5–16. ISBN 978-3-540-28849-7. 3. Barrow, John D (1999). Impossibility: The Limits of Science and the Science of Limits. Oxford University Press. p. 14. ISBN 9780195130829. 4. Penrose, LS; Penrose, R. (1958). "Impossible objects: A special type of optical illusion". British Journal of Psychology. 49 (1): 31–33. doi:10.1111/j.2044-8295.1958.tb00634.x. PMID 13536303. 5. "Impossible Fork". Wolfram Research. Retrieved 10 February 2014. 6. Honeycutt, Brad (9 March 2012). "Impossible Elephant". anopticalillusion.com. Retrieved 11 March 2019. ..one of the most famous and classic optical illusions of all time. While most people know it simply as the "impossible elephant", the actual title of the work is "L'egs-istential Quandary". 7. "Impossible Figures in Perceptual Psychology". Fink.com. Retrieved 11 February 2014. 8. Phillips, Tony. "The Topology of Impossible Spaces". American Mathematical Society. 9. Penrose, Roger (1992). "On the Cohomology of Impossible Figures". Leonardo. The MIT Press. 25 (3, 4): 245–247. doi:10.2307/1575844. JSTOR 1575844. S2CID 125905129. 10. Seckel, Al (2004). Masters of Deception: Escher, Dalí & the Artists of Optical Illusion. Sterling Publishing Company. p. 261. ISBN 1402705778. 11. Khoh, Chih W.; Kovesi, Peter (February 1999). "Animating Impossible Objects". Archived from the original on 28 May 2015. Retrieved 10 February 2014. {{cite journal}}: Cite journal requires \|journal= (help) Further reading • Bower, Gordon H. (editor), (1990). Psychology of Learning & Motivation. Academic Press. Volume 26. p. 107. ISBN 0080863779 • Mathematical Circus, Martin Gardner 1979 ISBN 0-14-022355-X (Chapter 1 – Optical Illusions) • Optical Illusions, Bruno Ernst 2006 ISBN 3-8228-5410-7 External links Wikimedia Commons has media related to Impossible objects and Optical illusions. • Impossible World • The M.C. Escher Project • Art of Reutersvard • "Escher for Real" (3D objects) • Inconsistent Images • Echochrome, a video game that incorporates impossible objects into its gameplay Optical illusions (list) Illusions • Afterimage • Ambigram • Ambiguous image • Ames room • Autostereogram • Barberpole • Bezold • Café wall • Checker shadow • Chubb • Cornsweet • Delboeuf • Ebbinghaus • Ehrenstein • Flash lag • Fraser spiral • Gravity hill • Grid • Hering • Impossible trident • Jastrow • Lilac chaser • Mach bands • McCollough • Müller-Lyer • Necker cube • Oppel-Kundt • Orbison • Penrose stairs • Penrose triangle • Peripheral drift • Poggendorff • Ponzo • Rubin vase • Sander • Schroeder stairs • Shepard tables • Spinning dancer • Ternus • Vertical–horizontal • White's • Wundt • Zöllner Popular culture • Op art • Trompe-l'œil • Spectropia (1864 book) • Ascending and Descending (1960 drawing) • Waterfall (1961 drawing) • The dress (2015 photograph) Related • Accidental viewpoint • Auditory illusions • Tactile illusions • Temporal illusion	Wikipedia
Undefined (mathematics) In mathematics, the term undefined is often used to refer to an expression which is not assigned an interpretation or a value (such as an indeterminate form, which has the possibility of assuming different values).[1] The term can take on several different meanings depending on the context. For example: • In various branches of mathematics, certain concepts are introduced as primitive notions (e.g., the terms "point", "line" and "plane" in geometry). As these terms are not defined in terms of other concepts, they may be referred to as "undefined terms". • A function is said to be "undefined" at points outside of its domain – for example, the real-valued function $f(x)={\sqrt {x}}$ is undefined for negative $x$ (i.e., it assigns no value to negative arguments). • In algebra, some arithmetic operations may not assign a meaning to certain values of its operands (e.g., division by zero). In which case, the expressions involving such operands are termed "undefined".[2] Undefined terms In ancient times, geometers attempted to define every term. For example, Euclid defined a point as "that which has no part". In modern times, mathematicians recognize that attempting to define every word inevitably leads to circular definitions, and therefore leave some terms (such as "point") undefined (see primitive notion for more). This more abstract approach allows for fruitful generalizations. In topology, a topological space may be defined as a set of points endowed with certain properties, but in the general setting, the nature of these "points" is left entirely undefined. Likewise, in category theory, a category consists of "objects" and "arrows", which are again primitive, undefined terms. This allows such abstract mathematical theories to be applied to very diverse concrete situations. In arithmetic The expression ${\frac {n}{0}},n\neq 0$ is undefined in arithmetic, as explained in division by zero (the ${\frac {0}{0}}$ expression is used in calculus to represent an indeterminate form). Mathematicians have different opinions as to whether 00 should be defined to equal 1, or be left undefined. Further information: Zero to the power of zero Values for which functions are undefined The set of numbers for which a function is defined is called the domain of the function. If a number is not in the domain of a function, the function is said to be "undefined" for that number. Two common examples are $ f(x)={\frac {1}{x}}$, which is undefined for $x=0$, and $f(x)={\sqrt {x}}$, which is undefined (in the real number system) for negative $x$. In trigonometry In trigonometry, for all $n\in \mathbb {Z} $, the functions $\tan \theta $ and $\sec \theta $ are undefined for all $ \theta =\pi \left(n-{\frac {1}{2}}\right)$, while the functions $\cot \theta $ and $\csc \theta $ are undefined for all $\theta =\pi n$. In complex analysis In complex analysis, a point $z\in \mathbb {C} $ where a holomorphic function is undefined is called a singularity. One distinguishes between removable singularities (i.e., the function can be extended holomorphically to $z$), poles (i.e., the function can be extended meromorphically to $z$), and essential singularities (i.e., no meromorphic extension to $z$ can exist). In computer science Notation using ↓ and ↑ In computability theory, if $f$ is a partial function on $S$ and $a$ is an element of $S$, then this is written as $f(a)\downarrow $, and is read as "f(a) is defined."[3] If $a$ is not in the domain of $f$, then this is written as $f(a)\uparrow $, and is read as "$f(a)$ is undefined". The symbols of infinity In analysis, measure theory and other mathematical disciplines, the symbol $\infty $ is frequently used to denote an infinite pseudo-number, along with its negative, $-\infty $. The symbol has no well-defined meaning by itself, but an expression like $\left\{a_{n}\right\}\rightarrow \infty $ is shorthand for a divergent sequence, which at some point is eventually larger than any given real number. Performing standard arithmetic operations with the symbols $\pm \infty $ is undefined. Some extensions, though, define the following conventions of addition and multiplication: • $x+\infty =\infty $ for all $x\in \mathbb {R} \cup \{\infty \}$. • $-\infty +x=-\infty $ for all $x\in \mathbb {R} \cup \{-\infty \}$. • $x\cdot \infty =\infty $ for all $x\in \mathbb {R} ^{+}$. No sensible extension of addition and multiplication with $\infty $ exists in the following cases: • $\infty -\infty $ • $0\cdot \infty $ (although in measure theory, this is often defined as $0$) • ${\frac {\infty }{\infty }}$ • $-\infty [{1(0)}]$ For more detail, see extended real number line. References 1. Weisstein, Eric W. "Undefined". mathworld.wolfram.com. Retrieved 2019-12-15. 2. "Undefined vs Indeterminate in Mathematics". www.cut-the-knot.org. Retrieved 2019-12-15. 3. Enderton, Herbert B. (2011). Computability: An Introduction to Recursion Theory. Elseveier. pp. 3–6. ISBN 978-0-12-384958-8. Further reading • Smart, James R. (1988). Modern Geometries (Third ed.). Brooks/Cole. ISBN 0-534-08310-2.	Wikipedia
Overcategory In mathematics, specifically category theory, an overcategory (and undercategory) is a distinguished class of categories used in multiple contexts, such as with covering spaces (espace etale). They were introduced as a mechanism for keeping track of data surrounding a fixed object $X$ in some category ${\mathcal {C}}$. There is a dual notion of undercategory, which is defined similarly. Definition Let ${\mathcal {C}}$ be a category and $X$ a fixed object of ${\mathcal {C}}$[1]pg 59. The overcategory (also called a slice category) ${\mathcal {C}}/X$ is an associated category whose objects are pairs $(A,\pi )$ where $\pi :A\to X$ is a morphism in ${\mathcal {C}}$. Then, a morphism between objects $f:(A,\pi )\to (A',\pi ')$ is given by a morphism $f:A\to A'$ in the category ${\mathcal {C}}$ such that the following diagram commutes ${\begin{matrix}A&\xrightarrow {f} &A'\\\pi \downarrow {\text{ }}&{\text{ }}&{\text{ }}\downarrow \pi '\\X&=&X\end{matrix}}$ There is a dual notion called the undercategory (also called a coslice category) $X/{\mathcal {C}}$ whose objects are pairs $(B,\psi )$ where $\psi :X\to B$ is a morphism in ${\mathcal {C}}$. Then, morphisms in $X/{\mathcal {C}}$ are given by morphisms $g:B\to B'$ in ${\mathcal {C}}$ such that the following diagram commutes ${\begin{matrix}X&=&X\\\psi \downarrow {\text{ }}&{\text{ }}&{\text{ }}\downarrow \psi '\\B&\xrightarrow {g} &B'\end{matrix}}$ These two notions have generalizations in 2-category theory[2] and higher category theory[3]pg 43, with definitions either analogous or essentially the same. Properties Many categorical properties of ${\mathcal {C}}$ are inherited by the associated over and undercategories for an object $X$. For example, if ${\mathcal {C}}$ has finite products and coproducts, it is immediate the categories ${\mathcal {C}}/X$ and $X/{\mathcal {C}}$ have these properties since the product and coproduct can be constructed in ${\mathcal {C}}$, and through universal properties, there exists a unique morphism either to $X$ or from $X$. In addition, this applies to limits and colimits as well. Examples Overcategories on a site Recall that a site ${\mathcal {C}}$ is a categorical generalization of a topological space first introduced by Grothendieck. One of the canonical examples comes directly from topology, where the category ${\text{Open}}(X)$ whose objects are open subsets $U$ of some topological space $X$, and the morphisms are given by inclusion maps. Then, for a fixed open subset $U$, the overcategory ${\text{Open}}(X)/U$ is canonically equivalent to the category ${\text{Open}}(U)$ for the induced topology on $U\subseteq X$. This is because every object in ${\text{Open}}(X)/U$ is an open subset $V$ contained in $U$. Category of algebras as an undercategory The category of commutative $A$-algebras is equivalent to the undercategory $A/{\text{CRing}}$ for the category of commutative rings. This is because the structure of an $A$-algebra on a commutative ring $B$ is directly encoded by a ring morphism $A\to B$. If we consider the opposite category, it is an overcategory of affine schemes, ${\text{Aff}}/{\text{Spec}}(A)$, or just ${\text{Aff}}_{A}$. Overcategories of spaces Another common overcategory considered in the literature are overcategories of spaces, such as schemes, smooth manifolds, or topological spaces. These categories encode objects relative to a fixed object, such as the category of schemes over $S$, ${\text{Sch}}/S$. Fiber products in these categories can be considered intersections, given the objects are subobjects of the fixed object. See also • Comma category References 1. Leinster, Tom (2016-12-29). "Basic Category Theory". arXiv:1612.09375 [math.CT]. 2. "Section 4.32 (02XG): Categories over categories—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-16. 3. Lurie, Jacob (2008-07-31). "Higher Topos Theory". arXiv:math/0608040.	Wikipedia
Undercut procedure The undercut procedure is a procedure for fair item assignment between two people. It provably finds a complete envy-free item assignment whenever such assignment exists. It was presented by Brams and Kilgour and Klamler[1] and simplified and extended by Aziz.[2] Assumptions The undercut procedure requires only the following weak assumptions on the people: • Each person has a weak preference relation on subsets of items. • Each preference relation is strictly monotonic: for every set $X$ and item $y\notin X$, the person strictly prefers $X\cup y$ to $X$. It is not assumed that agents have responsive preferences. Main idea The undercut procedure can be seen as a generalization of the divide and choose protocol from a divisible resource to a resource with indivisibilities. The divide-and-choose protocol requires one person to cut the resource to two equal pieces. But, if the resource contains with indivisibilities, it may be impossible to make an exactly-equal cut. Accordingly, the undercut procedure works with almost-equal-cuts. An almost-equal-cut of a person is a partition of the set of items to two disjoint subsets (X,Y) such that: • The person weakly prefers X to Y; • If any single item is moved from X to Y, then the person strictly prefers Y to X (i.e., for all x in X, the person prefers $Y\cup x$ to $X\setminus x$). Procedure Each person reports all his almost-equal-cuts. There are two cases: • Case 1: the reports are different, e.g., there is a partition (X,Y) that is an almost-equal-cut for Alice but not for George. Then, this partition is presented to George. George can either accept or reject it: • George accepts the partition if he prefers Y to X. Then Alice receives X and George receives Y and the resulting allocation is envy-free. • George rejects the partition if he prefers X to Y. By assumption, (X,Y) is not an almost-equal-cut for George. Therefore, there exists an item x in X such that George prefers $X\setminus x$ to $Y\cup x$. George reports $X\setminus x$; we say that George undercuts X. Since (X,Y) is an almost-equal-cut for Alice, Alice prefers $Y\cup x$ to $X\setminus x$. Then George receives $X\setminus x$ and Alice receives $Y\cup x$ and the resulting allocation is envy-free. • Case 2: the reports are identical, i.e., Alice and George have exactly the same set of almost-equal-cuts. Then, the procedure asks them whether one of their almost-equal-cuts is an exactly-equal-cut. By the strict-monotonicity assumption, (X,Y) is an exactly-equal-cut, if-and-only-if both (X,Y) and (Y,X) are almost-equal-cuts. Therefore, in Case 2, Alice and George have the same set of exactly-equal-cuts. There are two sub-cases: • Easy case: there exists an exactly-equal cut (X,Y). Then one person (no matter who) receives X and the other receives Y and the division is envy-free. • Hard case: there is no exactly-equal cut. Then the procedure returns and reports that "an envy-free allocation does not exist". To prove the correctness of the procedure, it is sufficient to prove that in the Hard case, an envy-free allocation does not exist. Indeed, suppose there exists an envy-free allocation (X,Y). Since we are in the Hard case, (X,Y) is not an exactly-equal cut. So one person (e.g. George) strictly prefers Y to X, while the other person (Alice) weakly prefers X to Y. If (X,Y) is not an almost-equal-cut for Alice, then we move some items from X to Y, until we get a partition (X',Y') that is an almost-equal-cut for Alice. Alice still weakly prefers X' to Y'. By the monotonicity assumption, George still strictly prefers Y' to X'. This means that (X',Y') is not an almost-equal-cut for George. But in the Hard case, both agents have the same set of almost-equal-cuts - a contradiction. Run-time complexity In the worst case, the agents may have to evaluate all possible bundles, so the run-time might be exponential in the number of items. This is not surprising, since the undercut procedure can be used to solve the partition problem: assume both agents have identical and additive valuations and run the undercut procedure; if it finds an envy-free allocation, then this allocation represents an equal partition. Since the partition problem is NP-complete, it probably cannot be solved by a polynomial-time algorithm. Unequal entitlements The undercut procedure can also work when the agents have unequal entitlements.[2] Suppose each agent $i$ is entitled to a fraction $e_{i}$ of the items. Then, the definition of an almost-equal-cut (for agent $i$) should be changed as follows: • $u_{i}(X)\geq {c_{i} \over c_{-i}}u_{i}(Y)$, and • For all x in X, the $u_{i}(X\setminus x)<{c_{i} \over c_{-i}}u_{i}(Y\cup x)$ Generation phase In the original publication,[1] the undercut procedure is preceded by the following generation phase: • While there are items on the table: • Each person reports his/her best item. • If the reports are different, then each person receives his/her best item. • If the reports are identical, then the best item is put in a contested pile. The undercut procedure described above is then executed only on the contested pile. This phase may make the division procedure more efficient: the contested pile may be smaller than the original set of items, so it may be easier to calculate and report the almost-equal-cuts. AliceGeorge w91 x84 y73 z62 However, the generation phase has several disadvantages:[2] 1. It might make the procedure miss a possible envy-free allocation. For example, suppose there are four items and their valuations are as in the adjacent table. The allocation that gives {w,z} to Alice and {x,y} to George is envy-free. Indeed, it can be found by the bare undercut procedure, since the partition ({w,z}, {x,y}) is an almost-equal-cut for Alice but not for George, and George would accept this partition. But with the generation phase, initially Alice gets w and George gets x and the other items {y,z} are put in the contested pile, and there is no envy-free allocation of the contested pile, so the procedure fails. 2. It requires the people to select their "best item" without knowing what other items they are going to receive. This relies on an assumption that the items are independent goods. Alternatively, it relies on a responsiveness assumption: if, in a bundle, an item is replaced by a better item, then the resulting bundle is better (it is closely related to weakly additive preferences). 3. It does not work when the agents have unequal claims. 4. It relies on sequential allocation, which is susceptible to strategic manipulation. See also • Decreasing Demand procedure and envy-graph procedure - two additional procedures based on the ordinal ranking of bundles. References 1. Brams, Steven J.; Kilgour, D. Marc; Klamler, Christian (2011). "The undercut procedure: An algorithm for the envy-free division of indivisible items" (PDF). Social Choice and Welfare. 39 (2–3): 615. doi:10.1007/s00355-011-0599-1. S2CID 253844146. Archived (PDF) from the original on 2017-08-12. Retrieved 2019-12-11. 2. Aziz, Haris (2015). "A note on the undercut procedure". Social Choice and Welfare. 45 (4): 723–728. arXiv:1312.6444. doi:10.1007/s00355-015-0877-4. S2CID 253842795. • Brandt, Felix; Conitzer, Vincent; Endriss, Ulle; Lang, Jérôme; Procaccia, Ariel D. (2016). Handbook of Computational Social Choice. Cambridge University Press. pp. 306–307. ISBN 9781107060432. (free online version)	Wikipedia
Damping Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation.[1] Examples include viscous drag (a liquid's viscosity can hinder an oscillatory system, causing it to slow down; see viscous damping) in mechanical systems, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur in biological systems and bikes[2] (ex. Suspension (mechanics)). Not to be confused with friction, which is a dissipative force acting on a system. Friction can cause or be a factor of damping. This article is about damping in oscillatory systems. For other uses, see Damping (disambiguation). Part of a series on Classical mechanics ${\textbf {F}}={\frac {d}{dt}}(m{\textbf {v}})$ Second law of motion • History • Timeline • Textbooks Branches • Applied • Celestial • Continuum • Dynamics • Kinematics • Kinetics • Statics • Statistical mechanics Fundamentals • Acceleration • Angular momentum • Couple • D'Alembert's principle • Energy • kinetic • potential • Force • Frame of reference • Inertial frame of reference • Impulse • Inertia / Moment of inertia • Mass • Mechanical power • Mechanical work • Moment • Momentum • Space • Speed • Time • Torque • Velocity • Virtual work Formulations • Newton's laws of motion • Analytical mechanics • Lagrangian mechanics • Hamiltonian mechanics • Routhian mechanics • Hamilton–Jacobi equation • Appell's equation of motion • Koopman–von Neumann mechanics Core topics • Damping • Displacement • Equations of motion • Euler's laws of motion • Fictitious force • Friction • Harmonic oscillator • Inertial / Non-inertial reference frame • Mechanics of planar particle motion • Motion (linear) • Newton's law of universal gravitation • Newton's laws of motion • Relative velocity • Rigid body • dynamics • Euler's equations • Simple harmonic motion • Vibration Rotation • Circular motion • Rotating reference frame • Centripetal force • Centrifugal force • reactive • Coriolis force • Pendulum • Tangential speed • Rotational frequency • Angular acceleration / displacement / frequency / velocity Scientists • Kepler • Galileo • Huygens • Newton • Horrocks • Halley • Maupertuis • Daniel Bernoulli • Johann Bernoulli • Euler • d'Alembert • Clairaut • Lagrange • Laplace • Hamilton • Poisson • Cauchy • Routh • Liouville • Appell • Gibbs • Koopman • von Neumann • Physics portal • Category The damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position of static equilibrium. A mass suspended from a spring, for example, might, if pulled and released, bounce up and down. On each bounce, the system tends to return to its equilibrium position, but overshoots it. Sometimes losses (e.g. frictional) damp the system and can cause the oscillations to gradually decay in amplitude towards zero or attenuate. The damping ratio is a measure describing how rapidly the oscillations decay from one bounce to the next. The damping ratio is a system parameter, denoted by ζ (zeta), that can vary from undamped (ζ = 0), underdamped (ζ < 1) through critically damped (ζ = 1) to overdamped (ζ > 1). The behaviour of oscillating systems is often of interest in a diverse range of disciplines that include control engineering, chemical engineering, mechanical engineering, structural engineering, and electrical engineering. The physical quantity that is oscillating varies greatly, and could be the swaying of a tall building in the wind, or the speed of an electric motor, but a normalised, or non-dimensionalised approach can be convenient in describing common aspects of behavior. Oscillation cases Depending on the amount of damping present, a system exhibits different oscillatory behaviors and speeds. • Where the spring–mass system is completely lossless, the mass would oscillate indefinitely, with each bounce of equal height to the last. This hypothetical case is called undamped. • If the system contained high losses, for example if the spring–mass experiment were conducted in a viscous fluid, the mass could slowly return to its rest position without ever overshooting. This case is called overdamped. • Commonly, the mass tends to overshoot its starting position, and then return, overshooting again. With each overshoot, some energy in the system is dissipated, and the oscillations die towards zero. This case is called underdamped. • Between the overdamped and underdamped cases, there exists a certain level of damping at which the system will just fail to overshoot and will not make a single oscillation. This case is called critical damping. The key difference between critical damping and overdamping is that, in critical damping, the system returns to equilibrium in the minimum amount of time. Damped sine wave A damped sine wave or damped sinusoid is a sinusoidal function whose amplitude approaches zero as time increases. It corresponds to the underdamped case of damped second-order systems, or underdamped second-order differential equations.[3] Damped sine waves are commonly seen in science and engineering, wherever a harmonic oscillator is losing energy faster than it is being supplied. A true sine wave starting at time = 0 begins at the origin (amplitude = 0). A cosine wave begins at its maximum value due to its phase difference from the sine wave. A given sinusoidal waveform may be of intermediate phase, having both sine and cosine components. The term "damped sine wave" describes all such damped waveforms, whatever their initial phase. The most common form of damping, which is usually assumed, is the form found in linear systems. This form is exponential damping, in which the outer envelope of the successive peaks is an exponential decay curve. That is, when you connect the maximum point of each successive curve, the result resembles an exponential decay function. The general equation for an exponentially damped sinusoid may be represented as: $y(t)=Ae^{-\lambda t}\cos(\omega t-\varphi )$ where: • $y(t)$ is the instantaneous amplitude at time t; • $A$ is the initial amplitude of the envelope; • $\lambda $ is the decay rate, in the reciprocal of the time units of the independent variable t; • $\varphi $ is the phase angle at t = 0; • $\omega $ is the angular frequency. Other important parameters include: • Frequency: $f=\omega /(2\pi )$, the number of cycles per time unit. It is expressed in inverse time units $t^{-1}$, or hertz. • Time constant: $\tau =1/\lambda $, the time for the amplitude to decrease by the factor of e. • Half-life is the time it takes for the exponential amplitude envelope to decrease by a factor of 2. It is equal to $\ln(2)/\lambda $ which is approximately $0.693/\lambda $. • Damping ratio: $\zeta $ is a non-dimensional characterization of the decay rate relative to the frequency, approximately $\zeta =\lambda /\omega $, or exactly $\zeta =\lambda /{\sqrt {\lambda ^{2}+\omega ^{2}}}<1$. • Q factor: $Q=1/(2\zeta )$ is another non-dimensional characterization of the amount of damping; high Q indicates slow damping relative to the oscillation. Damping ratio definition The damping ratio is a parameter, usually denoted by ζ (Greek letter zeta),[4] that characterizes the frequency response of a second-order ordinary differential equation. It is particularly important in the study of control theory. It is also important in the harmonic oscillator. In general, systems with higher damping ratios (one or greater) will demonstrate more of a damping effect. Underdamped systems have a value of less than one. Critically damped systems have a damping ratio of exactly 1, or at least very close to it. The damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. For a damped harmonic oscillator with mass m, damping coefficient c, and spring constant k, it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient: $\zeta ={\frac {c}{c_{c}}}={\frac {\text{actual damping}}{\text{critical damping}}},$ where the system's equation of motion is $m{\frac {d^{2}x}{dt^{2}}}+c{\frac {dx}{dt}}+kx=0$. [5] and the corresponding critical damping coefficient is $c_{c}=2{\sqrt {km}}$ or $c_{c}=2m{\sqrt {\frac {k}{m}}}=2m\omega _{n}$ where $\omega _{n}={\sqrt {\frac {k}{m}}}$ is the natural frequency of the system. The damping ratio is dimensionless, being the ratio of two coefficients of identical units. Derivation Using the natural frequency of a harmonic oscillator $ \omega _{n}={\sqrt {{k}/{m}}}$ and the definition of the damping ratio above, we can rewrite this as: ${\frac {d^{2}x}{dt^{2}}}+2\zeta \omega _{n}{\frac {dx}{dt}}+\omega _{n}^{2}x=0.$ This equation is more general than just the mass–spring system, and also applies to electrical circuits and to other domains. It can be solved with the approach $x(t)=Ce^{st},$ where C and s are both complex constants, with s satisfying $s=-\omega _{n}\left(\zeta \pm i{\sqrt {1-\zeta ^{2}}}\right).$ Two such solutions, for the two values of s satisfying the equation, can be combined to make the general real solutions, with oscillatory and decaying properties in several regimes: Undamped Is the case where $\zeta =0$ corresponds to the undamped simple harmonic oscillator, and in that case the solution looks like $\exp(i\omega _{n}t)$, as expected. This case is extremely rare in the natural world with the closest examples being cases where friction was purposefully reduced to minimal values. Underdamped If s is a pair of complex values, then each complex solution term is a decaying exponential combined with an oscillatory portion that looks like $ \exp \left(i\omega _{n}{\sqrt {1-\zeta ^{2}}}t\right)$. This case occurs for $\ 0\leq \zeta <1$, and is referred to as underdamped (e.g., bungee cable). Overdamped If s is a pair of real values, then the solution is simply a sum of two decaying exponentials with no oscillation. This case occurs for $\zeta >1$, and is referred to as overdamped. Situations where overdamping is practical tend to have tragic outcomes if overshooting occurs, usually electrical rather than mechanical. For example, landing a plane in autopilot: if the system overshoots and releases landing gear too late, the outcome would be a disaster. Critically damped The case where $\zeta =1$ is the border between the overdamped and underdamped cases, and is referred to as critically damped. This turns out to be a desirable outcome in many cases where engineering design of a damped oscillator is required (e.g., a door closing mechanism). Q factor and decay rate The Q factor, damping ratio ζ, and exponential decay rate α are related such that[6] $\zeta ={\frac {1}{2Q}}={\alpha \over \omega _{n}}.$ When a second-order system has $\zeta <1$ (that is, when the system is underdamped), it has two complex conjugate poles that each have a real part of $-\alpha $; that is, the decay rate parameter $\alpha $ represents the rate of exponential decay of the oscillations. A lower damping ratio implies a lower decay rate, and so very underdamped systems oscillate for long times.[7] For example, a high quality tuning fork, which has a very low damping ratio, has an oscillation that lasts a long time, decaying very slowly after being struck by a hammer. Logarithmic decrement For underdamped vibrations, the damping ratio is also related to the logarithmic decrement $\delta $. The damping ratio can be found for any two peaks, even if they are not adjacent.[8] For adjacent peaks:[9] $\zeta ={\frac {\delta }{\sqrt {\delta ^{2}+\left(2\pi \right)^{2}}}}$ where $\delta =\ln {\frac {x_{0}}{x_{1}}}$ where x0 and x1 are amplitudes of any two successive peaks. As shown in the right figure: $\delta =\ln {\frac {x_{1}}{x_{3}}}=\ln {\frac {x_{2}}{x_{4}}}=\ln {\frac {x_{1}-x_{2}}{x_{3}-x_{4}}}$ where $x_{1}$, $x_{3}$ are amplitudes of two successive positive peaks and $x_{2}$, $x_{4}$ are amplitudes of two successive negative peaks. Percentage overshoot In control theory, overshoot refers to an output exceeding its final, steady-state value.[10] For a step input, the percentage overshoot (PO) is the maximum value minus the step value divided by the step value. In the case of the unit step, the overshoot is just the maximum value of the step response minus one. The percentage overshoot (PO) is related to damping ratio (ζ) by: $\mathrm {PO} =100\exp \left({-{\frac {\zeta \pi }{\sqrt {1-\zeta ^{2}}}}}\right)$ Conversely, the damping ratio (ζ) that yields a given percentage overshoot is given by: $\zeta ={\frac {-\ln \left({\frac {\rm {PO}}{100}}\right)}{\sqrt {\pi ^{2}+\ln ^{2}\left({\frac {\rm {PO}}{100}}\right)}}}$ Examples and applications Viscous drag When an object is falling through the air, the only force opposing its freefall is air resistance. An object falling through water or oil would slow down at a greater rate, until eventually reaching a steady-state velocity as the drag force comes into equilibrium with the force from gravity. This is the concept of viscous drag, which for example is applied in automatic doors or anti-slam doors.[11] Damping in electrical systems / resistance Electrical systems that operate with alternating current (AC) use resistors to damp the electric current, since they are periodic. Dimmer switches or volume knobs are examples of damping in an electrical system. [11] Magnetic damping and Magnetorheological damping Kinetic energy that causes oscillations is dissipated as heat by electric eddy currents which are induced by passing through a magnet's poles, either by a coil or aluminum plate. Eddy currents are a key component of electromagnetic induction where they set up a magnetic flux directly opposing the oscillating movement, creating a resistive force. [12] In other words, the resistance caused by magnetic forces slows a system down. An example of this concept being applied is the brakes on roller coasters. [13] Magnetorheological Dampers (MR Dampers) use Magnetorheological fluid, which changes viscosity when subjected to a magnetic field. In this case, Magnetorheological damping may be considered an interdisciplinary form of damping with both viscous and magnetic damping mechanisms. [14] [15] References 1. Steidel (1971). An Introduction to Mechanical Vibrations. John Wiley & Sons. p. 37. damped, which is the term used in the study of vibration to denote a dissipation of energy 2. J. P. Meijaard; J. M. Papadopoulos; A. Ruina & A. L. Schwab (2007). "Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review". Proceedings of the Royal Society A. 463 (2084): 1955–1982. Bibcode:2007RSPSA.463.1955M. doi:10.1098/rspa.2007.1857. S2CID 18309860. lean and steer perturbations die away in a seemingly damped fashion. However, the system has no true damping and conserves energy. The energy in the lean and steer oscillations is transferred to the forward speed rather than being dissipated. 3. Douglas C. Giancoli (2000). [Physics for Scientists and Engineers with Modern Physics (3rd Edition)]. Prentice Hall. p. 387 ISBN 0-13-021517-1 4. Alciatore, David G. (2007). Introduction to Mechatronics and Measurement (3rd ed.). McGraw Hill. ISBN 978-0-07-296305-2. 5. Rahman, J.; Mushtaq, M.; Ali, A.; Anjam, Y.N; Nazir, S. (2014). "Modelling damped mass spring system in MATHLAB Simulink". Journal of Faculty of Engineering & Technology. 2. 6. William McC. Siebert. Circuits, Signals, and Systems. MIT Press. 7. Ming Rao and Haiming Qiu (1993). Process control engineering: a textbook for chemical, mechanical and electrical engineers. CRC Press. p. 96. ISBN 978-2-88124-628-9. 8. "Dynamics and Vibrations: Notes: Free Damped Vibrations". 9. "Damping Evaluation". 19 October 2015. 10. Kuo, Benjamin C & Golnaraghi M F (2003). Automatic control systems (Eighth ed.). NY: Wiley. p. §7.3 p. 236–237. ISBN 0-471-13476-7. 11. "damping \| Definition, Types, & Examples". Encyclopedia Britannica. Retrieved 2021-06-09. 12. Gupta, B. R. (2001). Principles of Electrical, Electronics and Instrumentation Engineering. S. chand Limited. p. 338. ISBN 9788121901031. 13. "Eddy Currents and Magnetic Damping \| Physics". courses.lumenlearning.com. Retrieved 2021-06-09. 14. LEE, DUG-YOUNG; WERELEY, NORMAN M. (June 2000). "QUASI-STEADY HERSCHEL-BULKLEY ANALYSIS OF ELECTRO- AND MAGNETO-RHEOLOGICAL FLOW MODE DAMPERS". Electro-Rheological Fluids and Magneto-Rheological Suspensions. WORLD SCIENTIFIC. doi:10.1142/9789812793607_0066. 15. Savaresi, Sergio M.; Poussot-Vassal, Charles; Spelta, Cristiano; Sename, Oliver; Dugard, Luc (2010-01-01), Savaresi, Sergio M.; Poussot-Vassal, Charles; Spelta, Cristiano; Sename, Oliver (eds.), "CHAPTER 2 - Semi-Active Suspension Technologies and Models", Semi-Active Suspension Control Design for Vehicles, Boston: Butterworth-Heinemann, pp. 15–39, doi:10.1016/b978-0-08-096678-6.00002-x, ISBN 978-0-08-096678-6, retrieved 2023-07-15 • Britannica, Encyclopædia. “Damping.” Encyclopædia Britannica, Encyclopædia Britannica, Inc., www.britannica.com/science/damping. • OpenStax, College. “Physics.” Lumen, courses.lumenlearning.com/physics/chapter/23-4-eddy-currents-and-magnetic-damping/.	Wikipedia
Trefoil knot In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory. Trefoil Common nameOverhand knot Arf invariant1 Braid length3 Braid no.2 Bridge no.2 Crosscap no.1 Crossing no.3 Genus1 Hyperbolic volume0 Stick no.6 Tunnel no.1 Unknotting no.1 Conway notation[3] A–B notation31 Dowker notation4, 6, 2 Last /Next01 / 41 Other alternating, torus, fibered, pretzel, prime, knot slice, reversible, tricolorable, twist The trefoil knot is named after the three-leaf clover (or trefoil) plant. Descriptions The trefoil knot can be defined as the curve obtained from the following parametric equations: ${\begin{aligned}x&=\sin t+2\sin 2t\\y&=\cos t-2\cos 2t\\z&=-\sin 3t\end{aligned}}$ The (2,3)-torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus $(r-2)^{2}+z^{2}=1$: ${\begin{aligned}x&=(2+\cos 3t)\cos 2t\\y&=(2+\cos 3t)\sin 2t\\z&=\sin 3t\end{aligned}}$ Any continuous deformation of the curve above is also considered a trefoil knot. Specifically, any curve isotopic to a trefoil knot is also considered to be a trefoil. In addition, the mirror image of a trefoil knot is also considered to be a trefoil. In topology and knot theory, the trefoil is usually defined using a knot diagram instead of an explicit parametric equation. In algebraic geometry, the trefoil can also be obtained as the intersection in C2 of the unit 3-sphere S3 with the complex plane curve of zeroes of the complex polynomial z2 + w3 (a cuspidal cubic). A left-handed trefoil and a right-handed trefoil. If one end of a tape or belt is turned over three times and then pasted to the other, the edge forms a trefoil knot.[1] Symmetry The trefoil knot is chiral, in the sense that a trefoil knot can be distinguished from its own mirror image. The two resulting variants are known as the left-handed trefoil and the right-handed trefoil. It is not possible to deform a left-handed trefoil continuously into a right-handed trefoil, or vice versa. (That is, the two trefoils are not ambient isotopic.) Though chiral, the trefoil knot is also invertible, meaning that there is no distinction between a counterclockwise-oriented and a clockwise-oriented trefoil. That is, the chirality of a trefoil depends only on the over and under crossings, not the orientation of the curve. Nontriviality The trefoil knot is nontrivial, meaning that it is not possible to "untie" a trefoil knot in three dimensions without cutting it. Mathematically, this means that a trefoil knot is not isotopic to the unknot. In particular, there is no sequence of Reidemeister moves that will untie a trefoil. Proving this requires the construction of a knot invariant that distinguishes the trefoil from the unknot. The simplest such invariant is tricolorability: the trefoil is tricolorable, but the unknot is not. In addition, virtually every major knot polynomial distinguishes the trefoil from an unknot, as do most other strong knot invariants. Classification In knot theory, the trefoil is the first nontrivial knot, and is the only knot with crossing number three. It is a prime knot, and is listed as 31 in the Alexander-Briggs notation. The Dowker notation for the trefoil is 4 6 2, and the Conway notation is [3]. The trefoil can be described as the (2,3)-torus knot. It is also the knot obtained by closing the braid σ13. The trefoil is an alternating knot. However, it is not a slice knot, meaning it does not bound a smooth 2-dimensional disk in the 4-dimensional ball; one way to prove this is to note that its signature is not zero. Another proof is that its Alexander polynomial does not satisfy the Fox-Milnor condition. The trefoil is a fibered knot, meaning that its complement in $S^{3}$ is a fiber bundle over the circle $S^{1}$. The trefoil K may be viewed as the set of pairs $(z,w)$ of complex numbers such that $\|z\|^{2}+\|w\|^{2}=1$ and $z^{2}+w^{3}=0$. Then this fiber bundle has the Milnor map $\phi (z,w)=(z^{2}+w^{3})/\|z^{2}+w^{3}\|$ as the fibre bundle projection of the knot complement $S^{3}\setminus \mathbf {K} $ to the circle $S^{1}$. The fibre is a once-punctured torus. Since the knot complement is also a Seifert fibred with boundary, it has a horizontal incompressible surface—this is also the fiber of the Milnor map. (This assumes the knot has been thickened to become a solid torus Nε(K), and that the interior of this solid torus has been removed to create a compact knot complement $S^{3}\setminus \operatorname {int} (\mathrm {N} _{\varepsilon }(\mathbf {K} )$.) Invariants The Alexander polynomial of the trefoil knot is $\Delta (t)=t-1+t^{-1},$ and the Conway polynomial is[2] $\nabla (z)=z^{2}+1.$ The Jones polynomial is $V(q)=q^{-1}+q^{-3}-q^{-4},$ and the Kauffman polynomial of the trefoil is $L(a,z)=za^{5}+z^{2}a^{4}-a^{4}+za^{3}+z^{2}a^{2}-2a^{2}.$ The HOMFLY polynomial of the trefoil is $L(\alpha ,z)=-\alpha ^{4}+\alpha ^{2}z^{2}+2\alpha ^{2}.$ The knot group of the trefoil is given by the presentation $\langle x,y\mid x^{2}=y^{3}\rangle $ or equivalently[3] $\langle x,y\mid xyx=yxy\rangle .$ This group is isomorphic to the braid group with three strands. In religion and culture As the simplest nontrivial knot, the trefoil is a common motif in iconography and the visual arts. For example, the common form of the triquetra symbol is a trefoil, as are some versions of the Germanic Valknut. • An ancient Norse Mjöllnir pendant with trefoils • A simple triquetra symbol • A tightly-knotted triquetra • The Germanic Valknut • A metallic Valknut in the shape of a trefoil • A Celtic cross with trefoil knots • A Carolingian cross • Trefoil knot used in aTV's logo • Mathematical surface in which the boundary is the trefoil knot in different angles. In modern art, the woodcut Knots by M. C. Escher depicts three trefoil knots whose solid forms are twisted in different ways.[4] See also Wikimedia Commons has media related to Trefoil knots. • Pretzel link • Figure-eight knot (mathematics) • Triquetra symbol • Cinquefoil knot • Gordian Knot References 1. Shaw, George Russell (MCMXXXIII). Knots: Useful & Ornamental, p.11. ISBN 978-0-517-46000-9. 2. "3_1", The Knot Atlas. 3. Weisstein, Eric W. "Trefoil Knot". MathWorld. Accessed: May 5, 2013. 4. The Official M.C. Escher Website — Gallery — "Knots" External links • Wolframalpha: (2,3)-torus knot • Trefoil knot 3d model Knot theory (knots and links) Hyperbolic • Figure-eight (41) • Three-twist (52) • Stevedore (61) • 62 • 63 • Endless (74) • Carrick mat (818) • Perko pair (10161) • (−2,3,7) pretzel (12n242) • Whitehead (52 1 ) • Borromean rings (63 2 ) • L10a140 • Conway knot (11n34) Satellite • Composite knots • Granny • Square • Knot sum Torus • Unknot (01) • Trefoil (31) • Cinquefoil (51) • Septafoil (71) • Unlink (02 1 ) • Hopf (22 1 ) • Solomon's (42 1 ) Invariants • Alternating • Arf invariant • Bridge no. • 2-bridge • Brunnian • Chirality • Invertible • Crosscap no. • Crossing no. • Finite type invariant • Hyperbolic volume • Khovanov homology • Genus • Knot group • Link group • Linking no. • Polynomial • Alexander • Bracket • HOMFLY • Jones • Kauffman • Pretzel • Prime • list • Stick no. • Tricolorability • Unknotting no. and problem Notation and operations • Alexander–Briggs notation • Conway notation • Dowker–Thistlethwaite notation • Flype • Mutation • Reidemeister move • Skein relation • Tabulation Other • Alexander's theorem • Berge • Braid theory • Conway sphere • Complement • Double torus • Fibered • Knot • List of knots and links • Ribbon • Slice • Sum • Tait conjectures • Twist • Wild • Writhe • Surgery theory • Category • Commons	Wikipedia
Multigraph In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called parallel edges[1]), that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge. There are 2 distinct notions of multiple edges: • Edges without own identity: The identity of an edge is defined solely by the two nodes it connects. In this case, the term "multiple edges" means that the same edge can occur several times between these two nodes. • Edges with own identity: Edges are primitive entities just like nodes. When multiple edges connect two nodes, these are different edges. A multigraph is different from a hypergraph, which is a graph in which an edge can connect any number of nodes, not just two. For some authors, the terms pseudograph and multigraph are synonymous. For others, a pseudograph is a multigraph that is permitted to have loops. Undirected multigraph (edges without own identity) A multigraph G is an ordered pair G := (V, E) with • V a set of vertices or nodes, • E a multiset of unordered pairs of vertices, called edges or lines. Undirected multigraph (edges with own identity) A multigraph G is an ordered triple G := (V, E, r) with • V a set of vertices or nodes, • E a set of edges or lines, • r : E → {{x,y} : x, y ∈ V}, assigning to each edge an unordered pair of endpoint nodes. Some authors allow multigraphs to have loops, that is, an edge that connects a vertex to itself,[2] while others call these pseudographs, reserving the term multigraph for the case with no loops.[3] Directed multigraph (edges without own identity) A multidigraph is a directed graph which is permitted to have multiple arcs, i.e., arcs with the same source and target nodes. A multidigraph G is an ordered pair G := (V, A) with • V a set of vertices or nodes, • A a multiset of ordered pairs of vertices called directed edges, arcs or arrows. A mixed multigraph G := (V, E, A) may be defined in the same way as a mixed graph. Directed multigraph (edges with own identity) A multidigraph or quiver G is an ordered 4-tuple G := (V, A, s, t) with • V a set of vertices or nodes, • A a set of edges or lines, • $s:A\rightarrow V$, assigning to each edge its source node, • $t:A\rightarrow V$, assigning to each edge its target node. This notion might be used to model the possible flight connections offered by an airline. In this case the multigraph would be a directed graph with pairs of directed parallel edges connecting cities to show that it is possible to fly both to and from these locations. In category theory a small category can be defined as a multidigraph (with edges having their own identity) equipped with an associative composition law and a distinguished self-loop at each vertex serving as the left and right identity for composition. For this reason, in category theory the term graph is standardly taken to mean "multidigraph", and the underlying multidigraph of a category is called its underlying digraph. Labeling Multigraphs and multidigraphs also support the notion of graph labeling, in a similar way. However there is no unity in terminology in this case. The definitions of labeled multigraphs and labeled multidigraphs are similar, and we define only the latter ones here. Definition 1: A labeled multidigraph is a labeled graph with labeled arcs. Formally: A labeled multidigraph G is a multigraph with labeled vertices and arcs. Formally it is an 8-tuple $G=(\Sigma _{V},\Sigma _{A},V,A,s,t,\ell _{V},\ell _{A})$ where • $V$ is a set of vertices and $A$ is a set of arcs. • $\Sigma _{V}$ and $\Sigma _{A}$ are finite alphabets of the available vertex and arc labels, • $s\colon A\rightarrow \ V$ and $t\colon A\rightarrow \ V$ are two maps indicating the source and target vertex of an arc, • $\ell _{V}\colon V\rightarrow \Sigma _{V}$ and $\ell _{A}\colon A\rightarrow \Sigma _{A}$ are two maps describing the labeling of the vertices and arcs. Definition 2: A labeled multidigraph is a labeled graph with multiple labeled arcs, i.e. arcs with the same end vertices and the same arc label (note that this notion of a labeled graph is different from the notion given by the article graph labeling). See also • Multidimensional network • Glossary of graph theory terms • Graph theory Notes 1. For example, see Balakrishnan 1997, p. 1 or Chartrand and Zhang 2012, p. 26. 2. For example, see Bollobás 2002, p. 7 or Diestel 2010, p. 28. 3. For example, see Wilson 2002, p. 6 or Chartrand and Zhang 2012, pp. 26-27. References • Balakrishnan, V. K. (1997). Graph Theory. McGraw-Hill. ISBN 0-07-005489-4. • Bollobás, Béla (2002). Modern Graph Theory. Graduate Texts in Mathematics. Vol. 184. Springer. ISBN 0-387-98488-7. • Chartrand, Gary; Zhang, Ping (2012). A First Course in Graph Theory. Dover. ISBN 978-0-486-48368-9. • Diestel, Reinhard (2010). Graph Theory. Graduate Texts in Mathematics. Vol. 173 (4th ed.). Springer. ISBN 978-3-642-14278-9. • Gross, Jonathan L.; Yellen, Jay (1998). Graph Theory and Its Applications. CRC Press. ISBN 0-8493-3982-0. • Gross, Jonathan L.; Yellen, Jay, eds. (2003). Handbook of Graph Theory. CRC. ISBN 1-58488-090-2. • Harary, Frank (1995). Graph Theory. Addison Wesley. ISBN 0-201-41033-8. • Janson, Svante; Knuth, Donald E.; Luczak, Tomasz; Pittel, Boris (1993). "The birth of the giant component". Random Structures and Algorithms. 4 (3): 231–358. arXiv:math/9310236. Bibcode:1993math.....10236J. doi:10.1002/rsa.3240040303. ISSN 1042-9832. MR 1220220. S2CID 206454812. • Wilson, Robert A. (2002). Graphs, Colourings and the Four-Colour Theorem. Oxford Science Publ. ISBN 0-19-851062-4. • Zwillinger, Daniel (2002). CRC Standard Mathematical Tables and Formulae (31st ed.). Chapman & Hall/CRC. ISBN 1-58488-291-3. External links • This article incorporates public domain material from Paul E. Black. "Multigraph". Dictionary of Algorithms and Data Structures. NIST.	Wikipedia
Forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signature, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the underlying set of a structure. Because many structures in mathematics consist of a set with an additional added structure, a forgetful functor that maps to the underlying set is the most common case. Overview As an example, there are several forgetful functors from the category of commutative rings. A (unital) ring, described in the language of universal algebra, is an ordered tuple $(R,+,\times ,a,0,1)$ satisfying certain axioms, where $+$ and $\times $ are binary functions on the set $R$, $a$ is a unary operation corresponding to additive inverse, and 0 and 1 are nullary operations giving the identities of the two binary operations. Deleting the 1 gives a forgetful functor to the category of rings without unit; it simply "forgets" the unit. Deleting $\times $ and 1 yields a functor to the category of abelian groups, which assigns to each ring $R$ the underlying additive abelian group of $R$. To each morphism of rings is assigned the same function considered merely as a morphism of addition between the underlying groups. Deleting all the operations gives the functor to the underlying set $R$. It is beneficial to distinguish between forgetful functors that "forget structure" versus those that "forget properties". For example, in the above example of commutative rings, in addition to those functors that delete some of the operations, there are functors that forget some of the axioms. There is a functor from the category CRing to Ring that forgets the axiom of commutativity, but keeps all the operations. Occasionally the object may include extra sets not defined strictly in terms of the underlying set (in this case, which part to consider the underlying set is a matter of taste, though this is rarely ambiguous in practice). For these objects, there are forgetful functors that forget the extra sets that are more general. Most common objects studied in mathematics are constructed as underlying sets along with extra sets of structure on those sets (operations on the underlying set, privileged subsets of the underlying set, etc.) which may satisfy some axioms. For these objects, a commonly considered forgetful functor is as follows. Let ${\mathcal {C}}$ be any category based on sets, e.g. groups—sets of elements—or topological spaces—sets of 'points'. As usual, write $\operatorname {Ob} ({\mathcal {C}})$ for the objects of ${\mathcal {C}}$ and write $\operatorname {Fl} ({\mathcal {C}})$ for the morphisms of the same. Consider the rule: For all $A$ in $\operatorname {Ob} ({\mathcal {C}}),A\mapsto \|A\|=$ the underlying set of $A,$ For all $u$ in $\operatorname {Fl} ({\mathcal {C}}),u\mapsto \|u\|=$ the morphism, $u$, as a map of sets. The functor $\|\cdot \|$ is then the forgetful functor from ${\mathcal {C}}$ to Set, the category of sets. Forgetful functors are almost always faithful. Concrete categories have forgetful functors to the category of sets—indeed they may be defined as those categories that admit a faithful functor to that category. Forgetful functors that only forget axioms are always fully faithful, since every morphism that respects the structure between objects that satisfy the axioms automatically also respects the axioms. Forgetful functors that forget structures need not be full; some morphisms don't respect the structure. These functors are still faithful however because distinct morphisms that do respect the structure are still distinct when the structure is forgotten. Functors that forget the extra sets need not be faithful, since distinct morphisms respecting the structure of those extra sets may be indistinguishable on the underlying set. In the language of formal logic, a functor of the first kind removes axioms, a functor of the second kind removes predicates, and a functor of the third kind remove types. An example of the first kind is the forgetful functor Ab → Grp. One of the second kind is the forgetful functor Ab → Set. A functor of the third kind is the functor Mod → Ab, where Mod is the fibred category of all modules over arbitrary rings. To see this, just choose a ring homomorphism between the underlying rings that does not change the ring action. Under the forgetful functor, this morphism yields the identity. Note that an object in Mod is a tuple, which includes a ring and an abelian group, so which to forget is a matter of taste. Left adjoints of forgetful functors Forgetful functors tend to have left adjoints, which are 'free' constructions. For example: • free module: the forgetful functor from $\mathbf {Mod} (R)$ (the category of $R$-modules) to $\mathbf {Set} $ has left adjoint $\operatorname {Free} _{R}$, with $X\mapsto \operatorname {Free} _{R}(X)$, the free $R$-module with basis $X$. • free group • free lattice • tensor algebra • free category, adjoint to the forgetful functor from categories to quivers • universal enveloping algebra For a more extensive list, see (Mac Lane 1997). As this is a fundamental example of adjoints, we spell it out: adjointness means that given a set X and an object (say, an R-module) M, maps of sets $X\to \|M\|$ correspond to maps of modules $\operatorname {Free} _{R}(X)\to M$: every map of sets yields a map of modules, and every map of modules comes from a map of sets. In the case of vector spaces, this is summarized as: "A map between vector spaces is determined by where it sends a basis, and a basis can be mapped to anything." Symbolically: $\operatorname {Hom} _{\mathbf {Mod} _{R}}(\operatorname {Free} _{R}(X),M)=\operatorname {Hom} _{\mathbf {Set} }(X,\operatorname {Forget} (M)).$ The unit of the free–forgetful adjunction is the "inclusion of a basis": $X\to \operatorname {Free} _{R}(X)$. Fld, the category of fields, furnishes an example of a forgetful functor with no adjoint. There is no field satisfying a free universal property for a given set. See also • Adjoint functors • Functors • Projection (set theory) References • Mac Lane, Saunders. Categories for the Working Mathematician, Graduate Texts in Mathematics 5, Springer-Verlag, Berlin, Heidelberg, New York, 1997. ISBN 0-387-98403-8 • Forgetful functor at the nLab Functor types • Additive • Adjoint • Conservative • Derived • Diagonal • Enriched • Essentially surjective • Exact • Forgetful • Full and faithful • Logical • Monoidal • Representable • Smooth	Wikipedia
Overshoot (signal) In signal processing, control theory, electronics, and mathematics, overshoot is the occurrence of a signal or function exceeding its target. Undershoot is the same phenomenon in the opposite direction. It arises especially in the step response of bandlimited systems such as low-pass filters. It is often followed by ringing, and at times conflated with the latter. Definition Maximum overshoot is defined in Katsuhiko Ogata's Discrete-time control systems as "the maximum peak value of the response curve measured from the desired response of the system."[1] Control theory In control theory, overshoot refers to an output exceeding its final, steady-state value.[2] For a step input, the percentage overshoot (PO) is the maximum value minus the step value divided by the step value. In the case of the unit step, the overshoot is just the maximum value of the step response minus one. Also see the definition of overshoot in an electronics context. For second-order systems, the percentage overshoot is a function of the damping ratio ζ and is given by [3] $\mathrm {PO} =100\exp \left({\frac {-\zeta \pi }{\sqrt {1-\zeta ^{2}}}}\right)$ The damping ratio can also be found by $\zeta ={\frac {-\ln \left({\frac {\rm {PO}}{100}}\right)}{\sqrt {\pi ^{2}+\ln ^{2}\left({\frac {\rm {PO}}{100}}\right)}}}$ Electronics In electronics, overshoot refers to the transitory values of any parameter that exceeds its final (steady state) value during its transition from one value to another. An important application of the term is to the output signal of an amplifier.[4] Usage: Overshoot occurs when the transitory values exceed final value. When they are lower than the final value, the phenomenon is called "undershoot". A circuit is designed to minimize rise time while containing distortion of the signal within acceptable limits. 1. Overshoot represents a distortion of the signal. 2. In circuit design, the goals of minimizing overshoot and of decreasing circuit rise time can conflict. 3. The magnitude of overshoot depends on time through a phenomenon called "damping." See illustration under step response. 4. Overshoot often is associated with settling time, how long it takes for the output to reach steady state; see step response. Also see the definition of overshoot in a control theory context. Mathematics Main article: Gibbs phenomenon In the approximation of functions, overshoot is one term describing quality of approximation. When a function such as a square wave is represented by a summation of terms, for example, a Fourier series or an expansion in orthogonal polynomials, the approximation of the function by a truncated number of terms in the series can exhibit overshoot, undershoot and ringing. The more terms retained in the series, the less pronounced the departure of the approximation from the function it represents. However, though the period of the oscillations decreases, their amplitude does not;[5] this is known as the Gibbs phenomenon. For the Fourier transform, this can be modeled by approximating a step function by the integral up to a certain frequency, which yields the sine integral. This can be interpreted as convolution with the sinc function; in signal processing terms, this is a low-pass filter. Signal processing In signal processing, overshoot is when the output of a filter has a higher maximum value than the input, specifically for the step response, and frequently yields the related phenomenon of ringing artifacts. This occurs for instance in using the sinc filter as an ideal (brick-wall) low-pass filter. The step response can be interpreted as the convolution with the impulse response, which is a sinc function. The overshoot and undershoot can be understood in this way: kernels are generally normalized to have integral 1, so they send constant functions to constant functions – otherwise they have gain. The value of a convolution at a point is a linear combination of the input signal, with coefficients (weights) the values of the kernel. If a kernel is non-negative, such as for a Gaussian kernel, then the value of the filtered signal will be a convex combination of the input values (the coefficients (the kernel) integrate to 1, and are non-negative), and will thus fall between the minimum and maximum of the input signal – it will not undershoot or overshoot. If, on the other hand, the kernel assumes negative values, such as the sinc function, then the value of the filtered signal will instead be an affine combination of the input values, and may fall outside of the minimum and maximum of the input signal, resulting in undershoot and overshoot. Overshoot is often undesirable, particularly if it causes clipping, but is sometimes desirable in image sharpening, due to increasing acutance (perceived sharpness). Related concepts A closely related phenomenon is ringing, when, following overshoot, a signal then falls below its steady-state value, and then may bounce back above, taking some time to settle close to its steady-state value; this latter time is called the settle time. In ecology, overshoot is the analogous concept, where a population exceeds the carrying capacity of a system. See also • Step response • Ringing (signal) • Settling time • Overmodulation • Integral windup References and notes 1. Ogata, Katsuhiko (1987). Discrete-time control systems. Prentice-Hall. p. 344. ISBN 0-13-216102-8. 2. Kuo, Benjamin C & Golnaraghi M F (2003). Automatic control systems (Eighth ed.). NY: Wiley. p. §7.3 pp. 236–237. ISBN 0-471-13476-7. 3. Modern Control Engineering (3rd Edition), Katsuhiko Ogata, page 153. 4. Phillip E Allen & Holberg D R (2002). CMOS analog circuit design (Second ed.). NY: Oxford University Press. Appendix C2, p. 771. ISBN 0-19-511644-5. 5. Gerald B Folland (1992). Fourier analysis and its application. Pacific Grove, Calif.: Wadsworth: Brooks/Cole. pp. 60–61. ISBN 0-534-17094-3. External links • Percentage overshoot calculator	Wikipedia
Underwood Dudley Underwood Dudley (born January 6, 1937) is an American mathematician and writer. His popular works include several books describing crank mathematics by pseudomathematicians who incorrectly believe they have squared the circle or done other impossible things. Career Dudley was born in New York City. He received bachelor's and master's degrees from the Carnegie Institute of Technology and a PhD from the University of Michigan. His academic career consisted of two years at Ohio State University followed by 37 at DePauw University, from which he retired in 2004. He edited the College Mathematics Journal and the Pi Mu Epsilon Journal, and was a Pólya Lecturer for the Mathematical Association of America (MAA) for two years. He is the discoverer of the Dudley triangle. Publications Dudley's popular books include Mathematical Cranks (MAA 1992, ISBN 0-88385-507-0), The Trisectors (MAA 1996, ISBN 0-88385-514-3), and Numerology: Or, What Pythagoras Wrought (MAA 1997, ISBN 0-88385-524-0). Dudley won the Trevor Evans Award for expository writing from the MAA in 1996. Dudley has also written and edited straightforward mathematical works such as Readings for Calculus (MAA 1993, ISBN 0-88385-087-7) and Elementary Number Theory (W.H. Freeman 1978, ISBN 0-7167-0076-X). In 2009, he authored "A Guide to Elementary Number Theory" (MAA, 2009, ISBN 978-0-88385-347-4), published under Mathematical Association of America's Dolciani Mathematical Expositions. Lawsuit In 1995, Dudley was one of several people sued by William Dilworth for defamation because Mathematical Cranks included an analysis of Dilworth's "A correction in set theory",[1] an attempted refutation of Cantor's diagonal method. The suit was dismissed in 1996 due to failure to state a claim. The dismissal was upheld on appeal in a decision written by jurist Richard Posner. From the decision: "A crank is a person inexplicably obsessed by an obviously unsound idea—a person with a bee in his bonnet. To call a person a crank is to say that because of some quirk of temperament he is wasting his time pursuing a line of thought that is plainly without merit or promise ... To call a person a crank is basically just a colorful and insulting way of expressing disagreement with his master idea, and it therefore belongs to the language of controversy rather than to the language of defamation."[2] See also • Pseudomathematics References 1. Dilworth, William (1974), "A correction in Set Theory" (PDF), Transactions of the Wisconsin Academy of Sciences, Arts and Letters, 62: 205–216, retrieved June 16, 2016 2. Caselaw: United States Court of Appeals, Seventh Circuit, ruling on Dillworth vs. Dudley, 1996 External links Wikiquote has quotations related to Underwood Dudley. • Underwood Dudley at the Mathematics Genealogy Project • DePauw University News story on Underwood Dudley and his "crank file"" (with photo) • Review of Hans Walser's The Golden Section by Underwood Dudley Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • Belgium • United States • Japan • Czech Republic • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef	Wikipedia
Unduloid In geometry, an unduloid, or onduloid, is a surface with constant nonzero mean curvature obtained as a surface of revolution of an elliptic catenary: that is, by rolling an ellipse along a fixed line, tracing the focus, and revolving the resulting curve around the line. In 1841 Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid.[1] Formula Let $\operatorname {sn} (u,k)$ represent the normal Jacobi sine function and $\operatorname {dn} (u,k)$ be the normal Jacobi elliptic function and let $\operatorname {F} (z,k)$ represent the normal elliptic integral of the first kind and $\operatorname {E} (z,k)$ represent the normal elliptic integral of the second kind. Let a be the length of the ellipse's major axis, and e be the eccentricity of the ellipse. Let k be a fixed value between 0 and 1 called the modulus. Given these variables, $\operatorname {x} (u)=-a(1-e)(\operatorname {F} (\operatorname {sn} (u,k),k)+\operatorname {F} (1,k))-a(1+e)(\operatorname {E} (\operatorname {sn} (u,k),k)+\operatorname {E} (1,k))\,$ $\operatorname {y} (u)=a(1+e)\operatorname {dn} (u,k)\,$ The formula for the surface of revolution that is the unduloid is then $\operatorname {X} (u,v)=\langle \operatorname {x} (u),\operatorname {y} (u)\cos(v),\operatorname {y} (u)\sin(v)\rangle \,$ Properties One interesting property of the unduloid is that the mean curvature is constant. In fact, the mean curvature across the entire surface is always the reciprocal of twice the major axis length: 1/(2a). Also, geodesics on an unduloid obey the Clairaut relation, and their behavior is therefore predictable. Occurrence in material science Unduloids are not a common pattern in nature. However, there are a few circumstances in which they form. First documented in 1970, passing a strong electric current through a thin (0.16—1.0mm), horizontally mounted, hard-drawn (non-tempered) silver wire will result in unduloids forming along its length.[2] This phenomenon was later discovered to also occur in molybdenum wire.[3] Unduloids have also been formed with ferrofluids.[4] By passing a current axially through a cylinder coated with a viscous magnetic fluid film, the magnetic dipoles of the fluid interact with the magnetic field of the current, creating a droplet pattern along the cylinder’s length. References 1. Delaunay, Ch. (1841). "Sur la surface de révolution dont la courbure moyenne est constante". Journal de Mathématiques Pures et Appliquées. 6: 309–314. 2. Lipski, T.; Furdal, A. (1970), "New observations on the formation of unduloids on wires", Proceedings of the Institution of Electrical Engineers, 117 (12): 2311-2314, doi:10.1049/piee.1970.0425 3. “Periodic Videos, Exploding wires” on YouTube 4. Weidner, D.E. (2017), "Drop formation in a magnetic fluid coating a horizontal cylinder carrying an axial electric current", Physics of Fluids, 29 (5): 052103, doi:10.1063/1.4982618	Wikipedia
Unfoldable cardinal In mathematics, an unfoldable cardinal is a certain kind of large cardinal number. Formally, a cardinal number κ is λ-unfoldable if and only if for every transitive model M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model with the critical point of j being κ and j(κ) ≥ λ. A cardinal is unfoldable if and only if it is an λ-unfoldable for all ordinals λ. A cardinal number κ is strongly λ-unfoldable if and only if for every transitive model M of cardinality κ of ZFC-minus-power set such that κ is in M and M contains all its sequences of length less than κ, there is a non-trivial elementary embedding j of M into a transitive model "N" with the critical point of j being κ, j(κ) ≥ λ, and V(λ) is a subset of N. Without loss of generality, we can demand also that N contains all its sequences of length λ. Likewise, a cardinal is strongly unfoldable if and only if it is strongly λ-unfoldable for all λ. These properties are essentially weaker versions of strong and supercompact cardinals, consistent with V = L. Many theorems related to these cardinals have generalizations to their unfoldable or strongly unfoldable counterparts. For example, the existence of a strongly unfoldable implies the consistency of a slightly weaker version of the proper forcing axiom. Relations between large cardinal properties Assuming V = L, the least unfoldable cardinal is greater than the least totally indescribable cardinal.[1]p.14 Assuming a Ramsey cardinal exists, it is less than the least Ramsey cardinal.[1]p.3 A Ramsey cardinal is unfoldable, and will be strongly unfoldable in L. It may fail to be strongly unfoldable in V, however. In L, any unfoldable cardinal is strongly unfoldable; thus unfoldables and strongly unfoldables have the same consistency strength. A cardinal k is κ-strongly unfoldable, and κ-unfoldable, if and only if it is weakly compact. A κ+ω-unfoldable cardinal is totally indescribable and preceded by a stationary set of totally indescribable cardinals. References • Hamkins, Joel David (2001). "Unfoldable cardinals and the GCH". The Journal of Symbolic Logic. 66 (3): 1186–1198. arXiv:math/9909029. doi:10.2307/2695100. JSTOR 2695100. S2CID 6269487. • Johnstone, Thomas A. (2008). "Strongly unfoldable cardinals made indestructible". Journal of Symbolic Logic. 73 (4): 1215–1248. doi:10.2178/jsl/1230396915. S2CID 30534686. • Joel David Hamkins; Džamonja, Mirna (2004). "Diamond (On the regulars) can fail at any strongly unfoldable cardinal". arXiv:math/0409304. Bibcode:2004math......9304H. {{cite journal}}: Cite journal requires \|journal= (help) Citations 1. Villaveces, Andres (1996). "Chains of End Elementary Extensions of Models of Set Theory". arXiv:math/9611209.	Wikipedia
Net (polyhedron) In geometry, a net of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane which can be folded (along edges) to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard.[1] An early instance of polyhedral nets appears in the works of Albrecht Dürer, whose 1525 book A Course in the Art of Measurement with Compass and Ruler (Unterweysung der Messung mit dem Zyrkel und Rychtscheyd ) included nets for the Platonic solids and several of the Archimedean solids.[2] These constructions were first called nets in 1543 by Augustin Hirschvogel.[3] Existence and uniqueness Many different nets can exist for a given polyhedron, depending on the choices of which edges are joined and which are separated. The edges that are cut from a convex polyhedron to form a net must form a spanning tree of the polyhedron, but cutting some spanning trees may cause the polyhedron to self-overlap when unfolded, rather than forming a net.[4] Conversely, a given net may fold into more than one different convex polyhedron, depending on the angles at which its edges are folded and the choice of which edges to glue together.[5] If a net is given together with a pattern for gluing its edges together, such that each vertex of the resulting shape has positive angular defect and such that the sum of these defects is exactly 4π, then there necessarily exists exactly one polyhedron that can be folded from it; this is Alexandrov's uniqueness theorem. However, the polyhedron formed in this way may have different faces than the ones specified as part of the net: some of the net polygons may have folds across them, and some of the edges between net polygons may remain unfolded. Additionally, the same net may have multiple valid gluing patterns, leading to different folded polyhedra.[6] Unsolved problem in mathematics: Does every convex polyhedron have a simple edge unfolding? (more unsolved problems in mathematics) In 1975, G. C. Shephard asked whether every convex polyhedron has at least one net, or simple edge-unfolding.[7] This question, which is also known as Dürer's conjecture, or Dürer's unfolding problem, remains unanswered.[8][9][10] There exist non-convex polyhedra that do not have nets, and it is possible to subdivide the faces of every convex polyhedron (for instance along a cut locus) so that the set of subdivided faces has a net.[4] In 2014 Mohammad Ghomi showed that every convex polyhedron admits a net after an affine transformation.[11] Furthermore, in 2019 Barvinok and Ghomi showed that a generalization of Dürer's conjecture fails for pseudo edges,[12] i.e., a network of geodesics which connect vertices of the polyhedron and form a graph with convex faces. A related open question asks whether every net of a convex polyhedron has a blooming, a continuous non-self-intersecting motion from its flat to its folded state that keeps each face flat throughout the motion.[13] Shortest path The shortest path over the surface between two points on the surface of a polyhedron corresponds to a straight line on a suitable net for the subset of faces touched by the path. The net has to be such that the straight line is fully within it, and one may have to consider several nets to see which gives the shortest path. For example, in the case of a cube, if the points are on adjacent faces one candidate for the shortest path is the path crossing the common edge; the shortest path of this kind is found using a net where the two faces are also adjacent. Other candidates for the shortest path are through the surface of a third face adjacent to both (of which there are two), and corresponding nets can be used to find the shortest path in each category.[14] The spider and the fly problem is a recreational mathematics puzzle which involves finding the shortest path between two points on a cuboid. Higher-dimensional polytope nets A net of a 4-polytope, a four-dimensional polytope, is composed of polyhedral cells that are connected by their faces and all occupy the same three-dimensional space, just as the polygon faces of a net of a polyhedron are connected by their edges and all occupy the same plane. The net of the tesseract, the four-dimensional hypercube, is used prominently in a painting by Salvador Dalí, Crucifixion (Corpus Hypercubus) (1954).[15] The same tesseract net is central to the plot of the short story "—And He Built a Crooked House—" by Robert A. Heinlein.[16] The number of combinatorially distinct nets of $n$-dimensional hypercubes can be found by representing these nets as a tree on $2n$ nodes describing the pattern by which pairs of faces of the hypercube are glued together to form a net, together with a perfect matching on the complement graph of the tree describing the pairs of faces that are opposite each other on the folded hypercube. Using this representation, the number of different unfoldings for hypercubes of dimensions 2, 3, 4, ... have been counted as 1, 11, 261, 9694, 502110, 33064966, 2642657228, ...[17] See also • Paper model • Cardboard modeling • UV mapping References 1. Wenninger, Magnus J. (1971), Polyhedron Models, Cambridge University Press 2. Dürer, Albrecht (1525), Unterweysung der Messung mit dem Zyrkel und Rychtscheyd, Nürnberg: München, Süddeutsche Monatsheft, pp. 139–152. English translation with commentary in Strauss, Walter L. (1977), The Painter's Manual, New York{{citation}}: CS1 maint: location missing publisher (link) 3. Friedman, Michael (2018), A History of Folding in Mathematics: Mathematizing the Margins, Science Networks. Historical Studies, vol. 59, Birkhäuser, p. 8, doi:10.1007/978-3-319-72487-4, ISBN 978-3-319-72486-7 4. Demaine, Erik D.; O'Rourke, Joseph (2007), "Chapter 22. Edge Unfolding of Polyhedra", Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, pp. 306–338 5. Malkevitch, Joseph, "Nets: A Tool for Representing Polyhedra in Two Dimensions", Feature Columns, American Mathematical Society, retrieved 2014-05-14 6. Demaine, Erik D.; Demaine, Martin L.; Lubiw, Anna; O'Rourke, Joseph (2002), "Enumerating foldings and unfoldings between polygons and polytopes", Graphs and Combinatorics, 18 (1): 93–104, arXiv:cs.CG/0107024, doi:10.1007/s003730200005, MR 1892436, S2CID 1489 7. Shephard, G. C. (1975), "Convex polytopes with convex nets", Mathematical Proceedings of the Cambridge Philosophical Society, 78 (3): 389–403, Bibcode:1975MPCPS..78..389S, doi:10.1017/s0305004100051860, MR 0390915, S2CID 122287769 8. Weisstein, Eric W., "Shephard's Conjecture", MathWorld 9. Moskovich, D. (June 4, 2012), "Dürer's conjecture", Open Problem Garden 10. Ghomi, Mohammad (2018-01-01), "Dürer's Unfolding Problem for Convex Polyhedra", Notices of the American Mathematical Society, 65 (1): 25–27, doi:10.1090/noti1609 11. Ghomi, Mohammad (2014), "Affine unfoldings of convex polyhedra", Geom. Topol., 18 (5): 3055–3090, arXiv:1305.3231, Bibcode:2013arXiv1305.3231G, doi:10.2140/gt.2014.18.3055, S2CID 16827957 12. Barvinok, Nicholas; Ghomi, Mohammad (2019-04-03), "Pseudo-Edge Unfoldings of Convex Polyhedra", Discrete & Computational Geometry, 64 (3): 671–689, arXiv:1709.04944, doi:10.1007/s00454-019-00082-1, ISSN 0179-5376, S2CID 37547025 13. Miller, Ezra; Pak, Igor (2008), "Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings", Discrete & Computational Geometry, 39 (1–3): 339–388, doi:10.1007/s00454-008-9052-3, MR 2383765 14. O’Rourke, Joseph (2011), How to Fold It: The Mathematics of Linkages, Origami and Polyhedra, Cambridge University Press, pp. 115–116, ISBN 9781139498548 15. Kemp, Martin (1 January 1998), "Dali's dimensions", Nature, 391 (6662): 27, Bibcode:1998Natur.391...27K, doi:10.1038/34063, S2CID 5317132 16. Henderson, Linda Dalrymple (November 2014), "Science Fiction, Art, and the Fourth Dimension", in Emmer, Michele (ed.), Imagine Math 3: Between Culture and Mathematics, Springer International Publishing, pp. 69–84, doi:10.1007/978-3-319-01231-5_7 17. Sloane, N. J. A. (ed.), "Sequence A091159 (Number of distinct nets for the n-hypercube)", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation External links • Weisstein, Eric W., "Net", MathWorld • Weisstein, Eric W., "Unfolding", MathWorld • Regular 4d Polytope Foldouts • Editable Printable Polyhedral Nets with an Interactive 3D View • Paper Models of Polyhedra • Unfolder for Blender • Unfolding package for Mathematica Mathematics of paper folding Flat folding • Big-little-big lemma • Crease pattern • Huzita–Hatori axioms • Kawasaki's theorem • Maekawa's theorem • Map folding • Napkin folding problem • Pureland origami • Yoshizawa–Randlett system Strip folding • Dragon curve • Flexagon • Möbius strip • Regular paperfolding sequence 3d structures • Miura fold • Modular origami • Paper bag problem • Rigid origami • Schwarz lantern • Sonobe • Yoshimura buckling Polyhedra • Alexandrov's uniqueness theorem • Blooming • Flexible polyhedron (Bricard octahedron, Steffen's polyhedron) • Net • Source unfolding • Star unfolding Miscellaneous • Fold-and-cut theorem • Lill's method Publications • Geometric Exercises in Paper Folding • Geometric Folding Algorithms • Geometric Origami • A History of Folding in Mathematics • Origami Polyhedra Design • Origamics People • Roger C. Alperin • Margherita Piazzola Beloch • Robert Connelly • Erik Demaine • Martin Demaine • Rona Gurkewitz • David A. Huffman • Tom Hull • Kôdi Husimi • Humiaki Huzita • Toshikazu Kawasaki • Robert J. Lang • Anna Lubiw • Jun Maekawa • Kōryō Miura • Joseph O'Rourke • Tomohiro Tachi • Eve Torrence	Wikipedia
Ungula In solid geometry, an ungula is a region of a solid of revolution, cut off by a plane oblique to its base.[1] A common instance is the spherical wedge. The term ungula refers to the hoof of a horse, an anatomical feature that defines a class of mammals called ungulates. The volume of an ungula of a cylinder was calculated by Grégoire de Saint Vincent.[2] Two cylinders with equal radii and perpendicular axes intersect in four double ungulae.[3] The bicylinder formed by the intersection had been measured by Archimedes in The Method of Mechanical Theorems, but the manuscript was lost until 1906. A historian of calculus described the role of the ungula in integral calculus: Grégoire himself was primarily concerned to illustrate by reference to the ungula that volumetric integration could be reduced, through the ductus in planum, to a consideration of geometric relations between the lies of plane figures. The ungula, however, proved a valuable source of inspiration for those who followed him, and who saw in it a means of representing and transforming integrals in many ingenious ways.[4]: 146 Cylindrical ungula A cylindrical ungula of base radius r and height h has volume $V={2 \over 3}r^{2}h$,.[5] Its total surface area is $A={1 \over 2}\pi r^{2}+{1 \over 2}\pi r{\sqrt {r^{2}+h^{2}}}+2rh$, the surface area of its curved sidewall is $A_{s}=2rh$, and the surface area of its top (slanted roof) is $A_{t}={1 \over 2}\pi r{\sqrt {r^{2}+h^{2}}}$. Proof Consider a cylinder $x^{2}+y^{2}=r^{2}$ bounded below by plane $z=0$ and above by plane $z=ky$ where k is the slope of the slanted roof: $k={h \over r}$. Cutting up the volume into slices parallel to the y-axis, then a differential slice, shaped like a triangular prism, has volume $A(x)\,dx$ where $A(x)={1 \over 2}{\sqrt {r^{2}-x^{2}}}\cdot k{\sqrt {r^{2}-x^{2}}}={1 \over 2}k(r^{2}-x^{2})$ is the area of a right triangle whose vertices are, $(x,0,0)$, $(x,{\sqrt {r^{2}-x^{2}}},0)$, and $(x,{\sqrt {r^{2}-x^{2}}},k{\sqrt {r^{2}-x^{2}}})$, and whose base and height are thereby ${\sqrt {r^{2}-x^{2}}}$ and $k{\sqrt {r^{2}-x^{2}}}$, respectively. Then the volume of the whole cylindrical ungula is $V=\int _{-r}^{r}A(x)\,dx=\int _{-r}^{r}{1 \over 2}k(r^{2}-x^{2})\,dx$ $\qquad ={1 \over 2}k{\Big (}[r^{2}x]_{-r}^{r}-{\Big [}{1 \over 3}x^{3}{\Big ]}_{-r}^{r}{\Big )}={1 \over 2}k(2r^{3}-{2 \over 3}r^{3})={2 \over 3}kr^{3}$ which equals $V={2 \over 3}r^{2}h$ after substituting $rk=h$. A differential surface area of the curved side wall is $dA_{s}=kr(\sin \theta )\cdot r\,d\theta =kr^{2}(\sin \theta )\,d\theta $, which area belongs to a nearly flat rectangle bounded by vertices $(r\cos \theta ,r\sin \theta ,0)$, $(r\cos \theta ,r\sin \theta ,kr\sin \theta )$, $(r\cos(\theta +d\theta ),r\sin(\theta +d\theta ),0)$, and $(r\cos(\theta +d\theta ),r\sin(\theta +d\theta ),kr\sin(\theta +d\theta ))$, and whose width and height are thereby $r\,d\theta $ and (close enough to) $kr\sin \theta $, respectively. Then the surface area of the wall is $A_{s}=\int _{0}^{\pi }dA_{s}=\int _{0}^{\pi }kr^{2}(\sin \theta )\,d\theta =kr^{2}\int _{0}^{\pi }\sin \theta \,d\theta $ where the integral yields $-[\cos \theta ]_{0}^{\pi }=-[-1-1]=2$, so that the area of the wall is $A_{s}=2kr^{2}$, and substituting $rk=h$ yields $A_{s}=2rh$. The base of the cylindrical ungula has the surface area of half a circle of radius r: ${1 \over 2}\pi r^{2}$, and the slanted top of the said ungula is a half-ellipse with semi-minor axis of length r and semi-major axis of length $r{\sqrt {1+k^{2}}}$, so that its area is $A_{t}={1 \over 2}\pi r\cdot r{\sqrt {1+k^{2}}}={1 \over 2}\pi r{\sqrt {r^{2}+(kr)^{2}}}$ and substituting $kr=h$ yields $A_{t}={1 \over 2}\pi r{\sqrt {r^{2}+h^{2}}}$. ∎ Note how the surface area of the side wall is related to the volume: such surface area being $2kr^{2}$, multiplying it by $dr$ gives the volume of a differential half-shell, whose integral is ${2 \over 3}kr^{3}$, the volume. When the slope k equals 1 then such ungula is precisely one eighth of a bicylinder, whose volume is ${16 \over 3}r^{3}$. One eighth of this is ${2 \over 3}r^{3}$. Conical ungula A conical ungula of height h, base radius r, and upper flat surface slope k (if the semicircular base is at the bottom, on the plane z = 0) has volume $V={r^{3}kHI \over 6}$ where $H={1 \over {1 \over h}-{1 \over rk}}$ is the height of the cone from which the ungula has been cut out, and $I=\int _{0}^{\pi }{2H+kr\sin \theta \over (H+kr\sin \theta )^{2}}\sin \theta \,d\theta $. The surface area of the curved sidewall is $A_{s}={kr^{2}{\sqrt {r^{2}+H^{2}}} \over 2}I$. As a consistency check, consider what happens when the height of the cone goes to infinity, so that the cone becomes a cylinder in the limit: $\lim _{H\rightarrow \infty }{\Big (}I-{4 \over H}{\Big )}=\lim _{H\rightarrow \infty }{\Big (}{2H \over H^{2}}\int _{0}^{\pi }\sin \theta \,d\theta -{4 \over H}{\Big )}=0$ so that $\lim _{H\rightarrow \infty }V={r^{3}kH \over 6}\cdot {4 \over H}={2 \over 3}kr^{3}$, $\lim _{H\rightarrow \infty }A_{s}={kr^{2}H \over 2}\cdot {4 \over H}=2kr^{2}$, and $\lim _{H\rightarrow \infty }A_{t}={1 \over 2}\pi r^{2}{{\sqrt {1+k^{2}}} \over 1+0}={1 \over 2}\pi r^{2}{\sqrt {1+k^{2}}}={1 \over 2}\pi r{\sqrt {r^{2}+(rk)^{2}}}$, which results agree with the cylindrical case. Proof Let a cone be described by $1-{\rho \over r}={z \over H}$ where r and H are constants and z and ρ are variables, with $\rho ={\sqrt {x^{2}+y^{2}}},\qquad 0\leq \rho \leq r$ and $x=\rho \cos \theta ,\qquad y=\rho \sin \theta $. Let the cone be cut by a plane $z=ky=k\rho \sin \theta $. Substituting this z into the cone's equation, and solving for ρ yields $\rho _{0}={1 \over {1 \over r}+{k\sin \theta \over H}}$ which for a given value of θ is the radial coordinate of the point common to both the plane and the cone that is farthest from the cone's axis along an angle θ from the x-axis. The cylindrical height coordinate of this point is $z_{0}=H{\Big (}1-{\rho _{0} \over r}{\Big )}$. So along the direction of angle θ, a cross-section of the conical ungula looks like the triangle $(0,0,0)-(\rho _{0}\cos \theta ,\rho _{0}\sin \theta ,z_{0})-(r\cos \theta ,r\sin \theta ,0)$. Rotating this triangle by an angle $d\theta $ about the z-axis yields another triangle with $\theta +d\theta $, $\rho _{1}$, $z_{1}$ substituted for $\theta $, $\rho _{0}$, and $z_{0}$ respectively, where $\rho _{1}$ and $z_{1}$ are functions of $\theta +d\theta $ instead of $\theta $. Since $d\theta $ is infinitesimal then $\rho _{1}$ and $z_{1}$ also vary infinitesimally from $\rho _{0}$ and $z_{0}$, so for purposes of considering the volume of the differential trapezoidal pyramid, they may be considered equal. The differential trapezoidal pyramid has a trapezoidal base with a length at the base (of the cone) of $rd\theta $, a length at the top of ${\Big (}{H-z_{0} \over H}{\Big )}rd\theta $, and altitude ${z_{0} \over H}{\sqrt {r^{2}+H^{2}}}$, so the trapezoid has area $A_{T}={r\,d\theta +{\Big (}{H-z_{0} \over H}{\Big )}r\,d\theta \over 2}{z_{0} \over H}{\sqrt {r^{2}+H^{2}}}=r\,d\theta {(2H-z_{0})z_{0} \over 2H^{2}}{\sqrt {r^{2}+H^{2}}}$. An altitude from the trapezoidal base to the point $(0,0,0)$ has length differentially close to ${rH \over {\sqrt {r^{2}+H^{2}}}}$. (This is an altitude of one of the side triangles of the trapezoidal pyramid.) The volume of the pyramid is one-third its base area times its altitudinal length, so the volume of the conical ungula is the integral of that: $V=\int _{0}^{\pi }{1 \over 3}{rH \over {\sqrt {r^{2}+H^{2}}}}{(2H-z_{0})z_{0} \over 2H^{2}}{\sqrt {r^{2}+H^{2}}}r\,d\theta =\int _{0}^{\pi }{1 \over 3}r^{2}{(2H-z_{0})z_{0} \over 2H}d\theta ={r^{2}k \over 6H}\int _{0}^{\pi }(2H-ky_{0})y_{0}\,d\theta $ where $y_{0}=\rho _{0}\sin \theta ={\sin \theta \over {1 \over r}+{k\sin \theta \over H}}={1 \over {1 \over r\sin \theta }+{k \over H}}$ Substituting the right hand side into the integral and doing some algebraic manipulation yields the formula for volume to be proven. For the sidewall: $A_{s}=\int _{0}^{\pi }A_{T}=\int _{0}^{\pi }{(2H-z_{0})z_{0} \over 2H^{2}}r{\sqrt {r^{2}+H^{2}}}\,d\theta ={kr{\sqrt {r^{2}+H^{2}}} \over 2H^{2}}\int _{0}^{\pi }(2H-z_{0})y_{0}\,d\theta $ and the integral on the rightmost-hand-side simplifies to $H^{2}rI$. ∎ As a consistency check, consider what happens when k goes to infinity; then the conical ungula should become a semi-cone. $\lim _{k\rightarrow \infty }{\Big (}I-{\pi \over kr}{\Big )}=0$ $\lim _{k\rightarrow \infty }V={r^{3}kH \over 6}\cdot {\pi \over kr}={1 \over 2}{\Big (}{1 \over 3}\pi r^{2}H{\Big )}$ which is half of the volume of a cone. $\lim _{k\rightarrow \infty }A_{s}={kr^{2}{\sqrt {r^{2}+H^{2}}} \over 2}\cdot {\pi \over kr}={1 \over 2}\pi r{\sqrt {r^{2}+H^{2}}}$ which is half of the surface area of the curved wall of a cone. Surface area of top part When $k=H/r$, the "top part" (i.e., the flat face that is not semicircular like the base) has a parabolic shape and its surface area is $A_{t}={2 \over 3}r{\sqrt {r^{2}+H^{2}}}$. When $k<H/r$ then the top part has an elliptic shape (i.e., it is less than one-half of an ellipse) and its surface area is $A_{t}={1 \over 2}\pi x_{max}(y_{1}-y_{m}){\sqrt {1+k^{2}}}\Lambda $ where $x_{max}={\sqrt {{k^{2}r^{4}H^{2}-k^{4}r^{6} \over (k^{2}r^{2}-H^{2})^{2}}+r^{2}}}$, $y_{1}={1 \over {1 \over r}+{k \over H}}$, $y_{m}={kr^{2}H \over k^{2}r^{2}-H^{2}}$, $\Lambda ={\pi \over 4}-{1 \over 2}\arcsin(1-\lambda )-{1 \over 4}\sin(2\arcsin(1-\lambda ))$, and $\lambda ={y_{1} \over y_{1}-y_{m}}$. When $k>H/r$ then the top part is a section of a hyperbola and its surface area is $A_{t}={\sqrt {1+k^{2}}}(2Cr-aJ)$ where $C={y_{1}+y_{2} \over 2}=y_{m}$, $y_{1}$ is as above, $y_{2}={1 \over {k \over H}-{1 \over r}}$, $a={r \over {\sqrt {C^{2}-\Delta ^{2}}}}$, $\Delta ={y_{2}-y_{1} \over 2}$, $J={r \over a}B+{\Delta ^{2} \over 2}\log {\Biggr \|}{{r \over a}+B \over {-r \over a}+B}{\Biggr \|}$, where the logarithm is natural, and $B={\sqrt {\Delta ^{2}+{r^{2} \over a^{2}}}}$. See also • Spherical wedge • Steinmetz solid References 1. Ungula at Webster Dictionary.org 2. Gregory of St. Vincent (1647) Opus Geometricum quadraturae circuli et sectionum coni 3. Blaise Pascal Lettre de Dettonville a Carcavi describes the onglet and double onglet, link from HathiTrust 4. Margaret E. Baron (1969) The Origins of the Infinitesimal Calculus, Pergamon Press, republished 2014 by Elsevier, Google Books preview 5. Solids - Volumes and Surfaces at The Engineering Toolbox External links • William Vogdes (1861) An Elementary Treatise on Measuration and Practical Geometry via Google Books	Wikipedia
Mathematical operators and symbols in Unicode The Unicode Standard encodes almost all standard characters used in mathematics.[1] Unicode Technical Report #25 provides comprehensive information about the character repertoire, their properties, and guidelines for implementation.[1] Mathematical operators and symbols are in multiple Unicode blocks. Some of these blocks are dedicated to, or primarily contain, mathematical characters while others are a mix of mathematical and non-mathematical characters. This article covers all Unicode characters with a derived property of "Math".[2][3] Dedicated blocks Mathematical Operators block The Mathematical Operators block (U+2200–U+22FF) contains characters for mathematical, logical, and set notation. Mathematical Operators[1] Official Unicode Consortium code chart (PDF) 0123456789ABCDEF U+220x ∀ ∁ ∂ ∃ ∄ ∅ ∆ ∇ ∈ ∉ ∊ ∋ ∌ ∍ ∎ ∏ U+221x ∐ ∑ − ∓ ∔ ∕ ∖ ∗ ∘ ∙ √ ∛ ∜ ∝ ∞ ∟ U+222x ∠ ∡ ∢ ∣ ∤ ∥ ∦ ∧ ∨ ∩ ∪ ∫ ∬ ∭ ∮ ∯ U+223x ∰ ∱ ∲ ∳ ∴ ∵ ∶ ∷ ∸ ∹ ∺ ∻ ∼ ∽ ∾ ∿ U+224x ≀ ≁ ≂ ≃ ≄ ≅ ≆ ≇ ≈ ≉ ≊ ≋ ≌ ≍ ≎ ≏ U+225x ≐ ≑ ≒ ≓ ≔ ≕ ≖ ≗ ≘ ≙ ≚ ≛ ≜ ≝ ≞ ≟ U+226x ≠ ≡ ≢ ≣ ≤ ≥ ≦ ≧ ≨ ≩ ≪ ≫ ≬ ≭ ≮ ≯ U+227x ≰ ≱ ≲ ≳ ≴ ≵ ≶ ≷ ≸ ≹ ≺ ≻ ≼ ≽ ≾ ≿ U+228x ⊀ ⊁ ⊂ ⊃ ⊄ ⊅ ⊆ ⊇ ⊈ ⊉ ⊊ ⊋ ⊌ ⊍ ⊎ ⊏ U+229x ⊐ ⊑ ⊒ ⊓ ⊔ ⊕ ⊖ ⊗ ⊘ ⊙ ⊚ ⊛ ⊜ ⊝ ⊞ ⊟ U+22Ax ⊠ ⊡ ⊢ ⊣ ⊤ ⊥ ⊦ ⊧ ⊨ ⊩ ⊪ ⊫ ⊬ ⊭ ⊮ ⊯ U+22Bx ⊰ ⊱ ⊲ ⊳ ⊴ ⊵ ⊶ ⊷ ⊸ ⊹ ⊺ ⊻ ⊼ ⊽ ⊾ ⊿ U+22Cx ⋀ ⋁ ⋂ ⋃ ⋄ ⋅ ⋆ ⋇ ⋈ ⋉ ⋊ ⋋ ⋌ ⋍ ⋎ ⋏ U+22Dx ⋐ ⋑ ⋒ ⋓ ⋔ ⋕ ⋖ ⋗ ⋘ ⋙ ⋚ ⋛ ⋜ ⋝ ⋞ ⋟ U+22Ex ⋠ ⋡ ⋢ ⋣ ⋤ ⋥ ⋦ ⋧ ⋨ ⋩ ⋪ ⋫ ⋬ ⋭ ⋮ ⋯ U+22Fx ⋰ ⋱ ⋲ ⋳ ⋴ ⋵ ⋶ ⋷ ⋸ ⋹ ⋺ ⋻ ⋼ ⋽ ⋾ ⋿ Notes 1.^ As of Unicode version 15.0 Supplemental Mathematical Operators block The Supplemental Mathematical Operators block (U+2A00–U+2AFF) contains various mathematical symbols, including N-ary operators, summations and integrals, intersections and unions, logical and relational operators, and subset/superset relations. Supplemental Mathematical Operators[1] Official Unicode Consortium code chart (PDF) 0123456789ABCDEF U+2A0x ⨀ ⨁ ⨂ ⨃ ⨄ ⨅ ⨆ ⨇ ⨈ ⨉ ⨊ ⨋ ⨌ ⨍ ⨎ ⨏ U+2A1x ⨐ ⨑ ⨒ ⨓ ⨔ ⨕ ⨖ ⨗ ⨘ ⨙ ⨚ ⨛ ⨜ ⨝ ⨞ ⨟ U+2A2x ⨠ ⨡ ⨢ ⨣ ⨤ ⨥ ⨦ ⨧ ⨨ ⨩ ⨪ ⨫ ⨬ ⨭ ⨮ ⨯ U+2A3x ⨰ ⨱ ⨲ ⨳ ⨴ ⨵ ⨶ ⨷ ⨸ ⨹ ⨺ ⨻ ⨼ ⨽ ⨾ ⨿ U+2A4x ⩀ ⩁ ⩂ ⩃ ⩄ ⩅ ⩆ ⩇ ⩈ ⩉ ⩊ ⩋ ⩌ ⩍ ⩎ ⩏ U+2A5x ⩐ ⩑ ⩒ ⩓ ⩔ ⩕ ⩖ ⩗ ⩘ ⩙ ⩚ ⩛ ⩜ ⩝ ⩞ ⩟ U+2A6x ⩠ ⩡ ⩢ ⩣ ⩤ ⩥ ⩦ ⩧ ⩨ ⩩ ⩪ ⩫ ⩬ ⩭ ⩮ ⩯ U+2A7x ⩰ ⩱ ⩲ ⩳ ⩴ ⩵ ⩶ ⩷ ⩸ ⩹ ⩺ ⩻ ⩼ ⩽ ⩾ ⩿ U+2A8x ⪀ ⪁ ⪂ ⪃ ⪄ ⪅ ⪆ ⪇ ⪈ ⪉ ⪊ ⪋ ⪌ ⪍ ⪎ ⪏ U+2A9x ⪐ ⪑ ⪒ ⪓ ⪔ ⪕ ⪖ ⪗ ⪘ ⪙ ⪚ ⪛ ⪜ ⪝ ⪞ ⪟ U+2AAx ⪠ ⪡ ⪢ ⪣ ⪤ ⪥ ⪦ ⪧ ⪨ ⪩ ⪪ ⪫ ⪬ ⪭ ⪮ ⪯ U+2ABx ⪰ ⪱ ⪲ ⪳ ⪴ ⪵ ⪶ ⪷ ⪸ ⪹ ⪺ ⪻ ⪼ ⪽ ⪾ ⪿ U+2ACx ⫀ ⫁ ⫂ ⫃ ⫄ ⫅ ⫆ ⫇ ⫈ ⫉ ⫊ ⫋ ⫌ ⫍ ⫎ ⫏ U+2ADx ⫐ ⫑ ⫒ ⫓ ⫔ ⫕ ⫖ ⫗ ⫘ ⫙ ⫚ ⫛ ⫝̸ ⫝ ⫞ ⫟ U+2AEx ⫠ ⫡ ⫢ ⫣ ⫤ ⫥ ⫦ ⫧ ⫨ ⫩ ⫪ ⫫ ⫬ ⫭ ⫮ ⫯ U+2AFx ⫰ ⫱ ⫲ ⫳ ⫴ ⫵ ⫶ ⫷ ⫸ ⫹ ⫺ ⫻ ⫼ ⫽ ⫾ ⫿ Notes 1.^ As of Unicode version 15.0 Mathematical Alphanumeric Symbols block Main article: Mathematical Alphanumeric Symbols (Unicode block) The Mathematical Alphanumeric Symbols block (U+1D400–U+1D7FF) contains Latin and Greek letters and decimal digits that enable mathematicians to denote different notions with different letter styles. The reserved code points (the "holes") in the alphabetic ranges up to U+1D551 duplicate characters in the Letterlike Symbols block.[4] Mathematical Alphanumeric Symbols[1][2] Official Unicode Consortium code chart (PDF) 0123456789ABCDEF U+1D40x 𝐀 𝐁 𝐂 𝐃 𝐄 𝐅 𝐆 𝐇 𝐈 𝐉 𝐊 𝐋 𝐌 𝐍 𝐎 𝐏 U+1D41x 𝐐 𝐑 𝐒 𝐓 𝐔 𝐕 𝐖 𝐗 𝐘 𝐙 𝐚 𝐛 𝐜 𝐝 𝐞 𝐟 U+1D42x 𝐠 𝐡 𝐢 𝐣 𝐤 𝐥 𝐦 𝐧 𝐨 𝐩 𝐪 𝐫 𝐬 𝐭 𝐮 𝐯 U+1D43x 𝐰 𝐱 𝐲 𝐳 𝐴 𝐵 𝐶 𝐷 𝐸 𝐹 𝐺 𝐻 𝐼 𝐽 𝐾 𝐿 U+1D44x 𝑀 𝑁 𝑂 𝑃 𝑄 𝑅 𝑆 𝑇 𝑈 𝑉 𝑊 𝑋 𝑌 𝑍 𝑎 𝑏 U+1D45x 𝑐 𝑑 𝑒 𝑓 𝑔 𝑖 𝑗 𝑘 𝑙 𝑚 𝑛 𝑜 𝑝 𝑞 𝑟 U+1D46x 𝑠 𝑡 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 𝑨 𝑩 𝑪 𝑫 𝑬 𝑭 𝑮 𝑯 U+1D47x 𝑰 𝑱 𝑲 𝑳 𝑴 𝑵 𝑶 𝑷 𝑸 𝑹 𝑺 𝑻 𝑼 𝑽 𝑾 𝑿 U+1D48x 𝒀 𝒁 𝒂 𝒃 𝒄 𝒅 𝒆 𝒇 𝒈 𝒉 𝒊 𝒋 𝒌 𝒍 𝒎 𝒏 U+1D49x 𝒐 𝒑 𝒒 𝒓 𝒔 𝒕 𝒖 𝒗 𝒘 𝒙 𝒚 𝒛 𝒜 𝒞 𝒟 U+1D4Ax 𝒢 𝒥 𝒦 𝒩 𝒪 𝒫 𝒬 𝒮 𝒯 U+1D4Bx 𝒰 𝒱 𝒲 𝒳 𝒴 𝒵 𝒶 𝒷 𝒸 𝒹 𝒻 𝒽 𝒾 𝒿 U+1D4Cx 𝓀 𝓁 𝓂 𝓃 𝓅 𝓆 𝓇 𝓈 𝓉 𝓊 𝓋 𝓌 𝓍 𝓎 𝓏 U+1D4Dx 𝓐 𝓑 𝓒 𝓓 𝓔 𝓕 𝓖 𝓗 𝓘 𝓙 𝓚 𝓛 𝓜 𝓝 𝓞 𝓟 U+1D4Ex 𝓠 𝓡 𝓢 𝓣 𝓤 𝓥 𝓦 𝓧 𝓨 𝓩 𝓪 𝓫 𝓬 𝓭 𝓮 𝓯 U+1D4Fx 𝓰 𝓱 𝓲 𝓳 𝓴 𝓵 𝓶 𝓷 𝓸 𝓹 𝓺 𝓻 𝓼 𝓽 𝓾 𝓿 U+1D50x 𝔀 𝔁 𝔂 𝔃 𝔄 𝔅 𝔇 𝔈 𝔉 𝔊 𝔍 𝔎 𝔏 U+1D51x 𝔐 𝔑 𝔒 𝔓 𝔔 𝔖 𝔗 𝔘 𝔙 𝔚 𝔛 𝔜 𝔞 𝔟 U+1D52x 𝔠 𝔡 𝔢 𝔣 𝔤 𝔥 𝔦 𝔧 𝔨 𝔩 𝔪 𝔫 𝔬 𝔭 𝔮 𝔯 U+1D53x 𝔰 𝔱 𝔲 𝔳 𝔴 𝔵 𝔶 𝔷 𝔸 𝔹 𝔻 𝔼 𝔽 𝔾 U+1D54x 𝕀 𝕁 𝕂 𝕃 𝕄 𝕆 𝕊 𝕋 𝕌 𝕍 𝕎 𝕏 U+1D55x 𝕐 𝕒 𝕓 𝕔 𝕕 𝕖 𝕗 𝕘 𝕙 𝕚 𝕛 𝕜 𝕝 𝕞 𝕟 U+1D56x 𝕠 𝕡 𝕢 𝕣 𝕤 𝕥 𝕦 𝕧 𝕨 𝕩 𝕪 𝕫 𝕬 𝕭 𝕮 𝕯 U+1D57x 𝕰 𝕱 𝕲 𝕳 𝕴 𝕵 𝕶 𝕷 𝕸 𝕹 𝕺 𝕻 𝕼 𝕽 𝕾 𝕿 U+1D58x 𝖀 𝖁 𝖂 𝖃 𝖄 𝖅 𝖆 𝖇 𝖈 𝖉 𝖊 𝖋 𝖌 𝖍 𝖎 𝖏 U+1D59x 𝖐 𝖑 𝖒 𝖓 𝖔 𝖕 𝖖 𝖗 𝖘 𝖙 𝖚 𝖛 𝖜 𝖝 𝖞 𝖟 U+1D5Ax 𝖠 𝖡 𝖢 𝖣 𝖤 𝖥 𝖦 𝖧 𝖨 𝖩 𝖪 𝖫 𝖬 𝖭 𝖮 𝖯 U+1D5Bx 𝖰 𝖱 𝖲 𝖳 𝖴 𝖵 𝖶 𝖷 𝖸 𝖹 𝖺 𝖻 𝖼 𝖽 𝖾 𝖿 U+1D5Cx 𝗀 𝗁 𝗂 𝗃 𝗄 𝗅 𝗆 𝗇 𝗈 𝗉 𝗊 𝗋 𝗌 𝗍 𝗎 𝗏 U+1D5Dx 𝗐 𝗑 𝗒 𝗓 𝗔 𝗕 𝗖 𝗗 𝗘 𝗙 𝗚 𝗛 𝗜 𝗝 𝗞 𝗟 U+1D5Ex 𝗠 𝗡 𝗢 𝗣 𝗤 𝗥 𝗦 𝗧 𝗨 𝗩 𝗪 𝗫 𝗬 𝗭 𝗮 𝗯 U+1D5Fx 𝗰 𝗱 𝗲 𝗳 𝗴 𝗵 𝗶 𝗷 𝗸 𝗹 𝗺 𝗻 𝗼 𝗽 𝗾 𝗿 U+1D60x 𝘀 𝘁 𝘂 𝘃 𝘄 𝘅 𝘆 𝘇 𝘈 𝘉 𝘊 𝘋 𝘌 𝘍 𝘎 𝘏 U+1D61x 𝘐 𝘑 𝘒 𝘓 𝘔 𝘕 𝘖 𝘗 𝘘 𝘙 𝘚 𝘛 𝘜 𝘝 𝘞 𝘟 U+1D62x 𝘠 𝘡 𝘢 𝘣 𝘤 𝘥 𝘦 𝘧 𝘨 𝘩 𝘪 𝘫 𝘬 𝘭 𝘮 𝘯 U+1D63x 𝘰 𝘱 𝘲 𝘳 𝘴 𝘵 𝘶 𝘷 𝘸 𝘹 𝘺 𝘻 𝘼 𝘽 𝘾 𝘿 U+1D64x 𝙀 𝙁 𝙂 𝙃 𝙄 𝙅 𝙆 𝙇 𝙈 𝙉 𝙊 𝙋 𝙌 𝙍 𝙎 𝙏 U+1D65x 𝙐 𝙑 𝙒 𝙓 𝙔 𝙕 𝙖 𝙗 𝙘 𝙙 𝙚 𝙛 𝙜 𝙝 𝙞 𝙟 U+1D66x 𝙠 𝙡 𝙢 𝙣 𝙤 𝙥 𝙦 𝙧 𝙨 𝙩 𝙪 𝙫 𝙬 𝙭 𝙮 𝙯 U+1D67x 𝙰 𝙱 𝙲 𝙳 𝙴 𝙵 𝙶 𝙷 𝙸 𝙹 𝙺 𝙻 𝙼 𝙽 𝙾 𝙿 U+1D68x 𝚀 𝚁 𝚂 𝚃 𝚄 𝚅 𝚆 𝚇 𝚈 𝚉 𝚊 𝚋 𝚌 𝚍 𝚎 𝚏 U+1D69x 𝚐 𝚑 𝚒 𝚓 𝚔 𝚕 𝚖 𝚗 𝚘 𝚙 𝚚 𝚛 𝚜 𝚝 𝚞 𝚟 U+1D6Ax 𝚠 𝚡 𝚢 𝚣 𝚤 𝚥 𝚨 𝚩 𝚪 𝚫 𝚬 𝚭 𝚮 𝚯 U+1D6Bx 𝚰 𝚱 𝚲 𝚳 𝚴 𝚵 𝚶 𝚷 𝚸 𝚹 𝚺 𝚻 𝚼 𝚽 𝚾 𝚿 U+1D6Cx 𝛀 𝛁 𝛂 𝛃 𝛄 𝛅 𝛆 𝛇 𝛈 𝛉 𝛊 𝛋 𝛌 𝛍 𝛎 𝛏 U+1D6Dx 𝛐 𝛑 𝛒 𝛓 𝛔 𝛕 𝛖 𝛗 𝛘 𝛙 𝛚 𝛛 𝛜 𝛝 𝛞 𝛟 U+1D6Ex 𝛠 𝛡 𝛢 𝛣 𝛤 𝛥 𝛦 𝛧 𝛨 𝛩 𝛪 𝛫 𝛬 𝛭 𝛮 𝛯 U+1D6Fx 𝛰 𝛱 𝛲 𝛳 𝛴 𝛵 𝛶 𝛷 𝛸 𝛹 𝛺 𝛻 𝛼 𝛽 𝛾 𝛿 U+1D70x 𝜀 𝜁 𝜂 𝜃 𝜄 𝜅 𝜆 𝜇 𝜈 𝜉 𝜊 𝜋 𝜌 𝜍 𝜎 𝜏 U+1D71x 𝜐 𝜑 𝜒 𝜓 𝜔 𝜕 𝜖 𝜗 𝜘 𝜙 𝜚 𝜛 𝜜 𝜝 𝜞 𝜟 U+1D72x 𝜠 𝜡 𝜢 𝜣 𝜤 𝜥 𝜦 𝜧 𝜨 𝜩 𝜪 𝜫 𝜬 𝜭 𝜮 𝜯 U+1D73x 𝜰 𝜱 𝜲 𝜳 𝜴 𝜵 𝜶 𝜷 𝜸 𝜹 𝜺 𝜻 𝜼 𝜽 𝜾 𝜿 U+1D74x 𝝀 𝝁 𝝂 𝝃 𝝄 𝝅 𝝆 𝝇 𝝈 𝝉 𝝊 𝝋 𝝌 𝝍 𝝎 𝝏 U+1D75x 𝝐 𝝑 𝝒 𝝓 𝝔 𝝕 𝝖 𝝗 𝝘 𝝙 𝝚 𝝛 𝝜 𝝝 𝝞 𝝟 U+1D76x 𝝠 𝝡 𝝢 𝝣 𝝤 𝝥 𝝦 𝝧 𝝨 𝝩 𝝪 𝝫 𝝬 𝝭 𝝮 𝝯 U+1D77x 𝝰 𝝱 𝝲 𝝳 𝝴 𝝵 𝝶 𝝷 𝝸 𝝹 𝝺 𝝻 𝝼 𝝽 𝝾 𝝿 U+1D78x 𝞀 𝞁 𝞂 𝞃 𝞄 𝞅 𝞆 𝞇 𝞈 𝞉 𝞊 𝞋 𝞌 𝞍 𝞎 𝞏 U+1D79x 𝞐 𝞑 𝞒 𝞓 𝞔 𝞕 𝞖 𝞗 𝞘 𝞙 𝞚 𝞛 𝞜 𝞝 𝞞 𝞟 U+1D7Ax 𝞠 𝞡 𝞢 𝞣 𝞤 𝞥 𝞦 𝞧 𝞨 𝞩 𝞪 𝞫 𝞬 𝞭 𝞮 𝞯 U+1D7Bx 𝞰 𝞱 𝞲 𝞳 𝞴 𝞵 𝞶 𝞷 𝞸 𝞹 𝞺 𝞻 𝞼 𝞽 𝞾 𝞿 U+1D7Cx 𝟀 𝟁 𝟂 𝟃 𝟄 𝟅 𝟆 𝟇 𝟈 𝟉 𝟊 𝟋 𝟎 𝟏 U+1D7Dx 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖 𝟗 𝟘 𝟙 𝟚 𝟛 𝟜 𝟝 𝟞 𝟟 U+1D7Ex 𝟠 𝟡 𝟢 𝟣 𝟤 𝟥 𝟦 𝟧 𝟨 𝟩 𝟪 𝟫 𝟬 𝟭 𝟮 𝟯 U+1D7Fx 𝟰 𝟱 𝟲 𝟳 𝟴 𝟵 𝟶 𝟷 𝟸 𝟹 𝟺 𝟻 𝟼 𝟽 𝟾 𝟿 Notes 1.^ As of Unicode version 15.0 2.^ Grey areas indicate non-assigned code points Letterlike Symbols block The Letterlike Symbols block (U+2100–U+214F) includes variables. Most alphabetic math symbols are in the Mathematical Alphanumeric Symbols block shown above. The math subset of this block is U+2102, U+2107, U+210A–U+2113, U+2115, U+2118–U+211D, U+2124, U+2128–U+2129, U+212C–U+212D, U+212F–U+2131, U+2133–U+2138, U+213C–U+2149, and U+214B.[5] Letterlike Symbols[1] Official Unicode Consortium code chart (PDF) 0123456789ABCDEF U+210x ℀ ℁ ℂ ℃ ℄ ℅ ℆ ℇ ℈ ℉ ℊ ℋ ℌ ℍ ℎ ℏ U+211x ℐ ℑ ℒ ℓ ℔ ℕ № ℗ ℘ ℙ ℚ ℛ ℜ ℝ ℞ ℟ U+212x ℠ ℡ ™ ℣ ℤ ℥ Ω ℧ ℨ ℩ K Å ℬ ℭ ℮ ℯ U+213x ℰ ℱ Ⅎ ℳ ℴ ℵ ℶ ℷ ℸ ℹ ℺ ℻ ℼ ℽ ℾ ℿ U+214x ⅀ ⅁ ⅂ ⅃ ⅄ ⅅ ⅆ ⅇ ⅈ ⅉ ⅊ ⅋ ⅌ ⅍ ⅎ ⅏ Notes 1.^ As of Unicode version 15.0 Miscellaneous Mathematical Symbols-A block The Miscellaneous Mathematical Symbols-A block (U+27C0–U+27EF) contains characters for mathematical, logical, and database notation. Miscellaneous Mathematical Symbols-A[1] Official Unicode Consortium code chart (PDF) 0123456789ABCDEF U+27Cx ⟀ ⟁ ⟂ ⟃ ⟄ ⟅ ⟆ ⟇ ⟈ ⟉ ⟊ ⟋ ⟌ ⟍ ⟎ ⟏ U+27Dx ⟐ ⟑ ⟒ ⟓ ⟔ ⟕ ⟖ ⟗ ⟘ ⟙ ⟚ ⟛ ⟜ ⟝ ⟞ ⟟ U+27Ex ⟠ ⟡ ⟢ ⟣ ⟤ ⟥ ⟦ ⟧ ⟨ ⟩ ⟪ ⟫ ⟬ ⟭ ⟮ ⟯ Notes 1.^ As of Unicode version 15.0 Miscellaneous Mathematical Symbols-B block Main article: Miscellaneous Mathematical Symbols-B (Unicode block) The Miscellaneous Mathematical Symbols-B block (U+2980–U+29FF) contains miscellaneous mathematical symbols, including brackets, angles, and circle symbols. Miscellaneous Mathematical Symbols-B[1] Official Unicode Consortium code chart (PDF) 0123456789ABCDEF U+298x ⦀ ⦁ ⦂ ⦃ ⦄ ⦅ ⦆ ⦇ ⦈ ⦉ ⦊ ⦋ ⦌ ⦍ ⦎ ⦏ U+299x ⦐ ⦑ ⦒ ⦓ ⦔ ⦕ ⦖ ⦗ ⦘ ⦙ ⦚ ⦛ ⦜ ⦝ ⦞ ⦟ U+29Ax ⦠ ⦡ ⦢ ⦣ ⦤ ⦥ ⦦ ⦧ ⦨ ⦩ ⦪ ⦫ ⦬ ⦭ ⦮ ⦯ U+29Bx ⦰ ⦱ ⦲ ⦳ ⦴ ⦵ ⦶ ⦷ ⦸ ⦹ ⦺ ⦻ ⦼ ⦽ ⦾ ⦿ U+29Cx ⧀ ⧁ ⧂ ⧃ ⧄ ⧅ ⧆ ⧇ ⧈ ⧉ ⧊ ⧋ ⧌ ⧍ ⧎ ⧏ U+29Dx ⧐ ⧑ ⧒ ⧓ ⧔ ⧕ ⧖ ⧗ ⧘ ⧙ ⧚ ⧛ ⧜ ⧝ ⧞ ⧟ U+29Ex ⧠ ⧡ ⧢ ⧣ ⧤ ⧥ ⧦ ⧧ ⧨ ⧩ ⧪ ⧫ ⧬ ⧭ ⧮ ⧯ U+29Fx ⧰ ⧱ ⧲ ⧳ ⧴ ⧵ ⧶ ⧷ ⧸ ⧹ ⧺ ⧻ ⧼ ⧽ ⧾ ⧿ Notes 1.^ As of Unicode version 15.0 Miscellaneous Technical block The Miscellaneous Technical block (U+2300–U+23FF) includes braces and operators. The math subset of this block is U+2308–U+230B, U+2320–U+2321, U+237C, U+239B–U+23B5, 23B7, U+23D0, and U+23DC–U+23E2. Miscellaneous Technical[1][2] Official Unicode Consortium code chart (PDF) 0123456789ABCDEF U+230x ⌀ ⌁ ⌂ ⌃ ⌄ ⌅ ⌆ ⌇ ⌈ ⌉ ⌊ ⌋ ⌌ ⌍ ⌎ ⌏ U+231x ⌐ ⌑ ⌒ ⌓ ⌔ ⌕ ⌖ ⌗ ⌘ ⌙ ⌚ ⌛ ⌜ ⌝ ⌞ ⌟ U+232x ⌠ ⌡ ⌢ ⌣ ⌤ ⌥ ⌦ ⌧ ⌨ 〈 〉 ⌫ ⌬ ⌭ ⌮ ⌯ U+233x ⌰ ⌱ ⌲ ⌳ ⌴ ⌵ ⌶ ⌷ ⌸ ⌹ ⌺ ⌻ ⌼ ⌽ ⌾ ⌿ U+234x ⍀ ⍁ ⍂ ⍃ ⍄ ⍅ ⍆ ⍇ ⍈ ⍉ ⍊ ⍋ ⍌ ⍍ ⍎ ⍏ U+235x ⍐ ⍑ ⍒ ⍓ ⍔ ⍕ ⍖ ⍗ ⍘ ⍙ ⍚ ⍛ ⍜ ⍝ ⍞ ⍟ U+236x ⍠ ⍡ ⍢ ⍣ ⍤ ⍥ ⍦ ⍧ ⍨ ⍩ ⍪ ⍫ ⍬ ⍭ ⍮ ⍯ U+237x ⍰ ⍱ ⍲ ⍳ ⍴ ⍵ ⍶ ⍷ ⍸ ⍹ ⍺ ⍻ ⍼ ⍽ ⍾ ⍿ U+238x ⎀ ⎁ ⎂ ⎃ ⎄ ⎅ ⎆ ⎇ ⎈ ⎉ ⎊ ⎋ ⎌ ⎍ ⎎ ⎏ U+239x ⎐ ⎑ ⎒ ⎓ ⎔ ⎕ ⎖ ⎗ ⎘ ⎙ ⎚ ⎛ ⎜ ⎝ ⎞ ⎟ U+23Ax ⎠ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⎧ ⎨ ⎩ ⎪ ⎫ ⎬ ⎭ ⎮ ⎯ U+23Bx ⎰ ⎱ ⎲ ⎳ ⎴ ⎵ ⎶ ⎷ ⎸ ⎹ ⎺ ⎻ ⎼ ⎽ ⎾ ⎿ U+23Cx ⏀ ⏁ ⏂ ⏃ ⏄ ⏅ ⏆ ⏇ ⏈ ⏉ ⏊ ⏋ ⏌ ⏍ ⏎ ⏏ U+23Dx ⏐ ⏑ ⏒ ⏓ ⏔ ⏕ ⏖ ⏗ ⏘ ⏙ ⏚ ⏛ ⏜ ⏝ ⏞ ⏟ U+23Ex ⏠ ⏡ ⏢ ⏣ ⏤ ⏥ ⏦ ⏧ ⏨ ⏩ ⏪ ⏫ ⏬ ⏭ ⏮ ⏯ U+23Fx ⏰ ⏱ ⏲ ⏳ ⏴ ⏵ ⏶ ⏷ ⏸ ⏹ ⏺ ⏻ ⏼ ⏽ ⏾ ⏿ Notes 1.^ As of Unicode version 15.0 2.^ Unicode code points U+2329 and U+232A are deprecated as of Unicode version 5.2 Geometric Shapes block The Geometric Shapes block (U+25A0–U+25FF) contains geometric shape symbols. The math subset of this block is U+25A0–25A1, U+25AE–25B7, U+25BC–25C1, U+25C6–25C7, U+25CA–25CB, U+25CF–25D3, U+25E2, U+25E4, U+25E7–25EC, and U+25F8–25FF. Geometric Shapes[1] Official Unicode Consortium code chart (PDF) 0123456789ABCDEF U+25Ax ■ □ ▢ ▣ ▤ ▥ ▦ ▧ ▨ ▩ ▪ ▫ ▬ ▭ ▮ ▯ U+25Bx ▰ ▱ ▲ △ ▴ ▵ ▶ ▷ ▸ ▹ ► ▻ ▼ ▽ ▾ ▿ U+25Cx ◀ ◁ ◂ ◃ ◄ ◅ ◆ ◇ ◈ ◉ ◊ ○ ◌ ◍ ◎ ● U+25Dx ◐ ◑ ◒ ◓ ◔ ◕ ◖ ◗ ◘ ◙ ◚ ◛ ◜ ◝ ◞ ◟ U+25Ex ◠ ◡ ◢ ◣ ◤ ◥ ◦ ◧ ◨ ◩ ◪ ◫ ◬ ◭ ◮ ◯ U+25Fx ◰ ◱ ◲ ◳ ◴ ◵ ◶ ◷ ◸ ◹ ◺ ◻ ◼ ◽ ◾ ◿ Notes 1.^ As of Unicode version 15.0 Arrows block The Arrows block (U+2190–U+21FF) contains line, curve, and semicircle arrows and arrow-like operators. The math subset of this block is U+2190–U+21A7, U+21A9–U+21AE, U+21B0–U+21B1, U+21B6–U+21B7, U+21BC–U+21DB, U+21DD, U+21E4–U+21E5, U+21F4–U+21FF.[6] Arrows[1] Official Unicode Consortium code chart (PDF) 0123456789ABCDEF U+219x ← ↑ → ↓ ↔ ↕ ↖ ↗ ↘ ↙ ↚ ↛ ↜ ↝ ↞ ↟ U+21Ax ↠ ↡ ↢ ↣ ↤ ↥ ↦ ↧ ↨ ↩ ↪ ↫ ↬ ↭ ↮ ↯ U+21Bx ↰ ↱ ↲ ↳ ↴ ↵ ↶ ↷ ↸ ↹ ↺ ↻ ↼ ↽ ↾ ↿ U+21Cx ⇀ ⇁ ⇂ ⇃ ⇄ ⇅ ⇆ ⇇ ⇈ ⇉ ⇊ ⇋ ⇌ ⇍ ⇎ ⇏ U+21Dx ⇐ ⇑ ⇒ ⇓ ⇔ ⇕ ⇖ ⇗ ⇘ ⇙ ⇚ ⇛ ⇜ ⇝ ⇞ ⇟ U+21Ex ⇠ ⇡ ⇢ ⇣ ⇤ ⇥ ⇦ ⇧ ⇨ ⇩ ⇪ ⇫ ⇬ ⇭ ⇮ ⇯ U+21Fx ⇰ ⇱ ⇲ ⇳ ⇴ ⇵ ⇶ ⇷ ⇸ ⇹ ⇺ ⇻ ⇼ ⇽ ⇾ ⇿ Notes 1.^ As of Unicode version 15.0 Supplemental Arrows-A block The Supplemental Arrows-A block (U+27F0–U+27FF) contains arrows and arrow-like operators. Supplemental Arrows-A[1] Official Unicode Consortium code chart (PDF) 0123456789ABCDEF U+27Fx ⟰ ⟱ ⟲ ⟳ ⟴ ⟵ ⟶ ⟷ ⟸ ⟹ ⟺ ⟻ ⟼ ⟽ ⟾ ⟿ Notes 1.^ As of Unicode version 15.0 Supplemental Arrows-B block The Supplemental Arrows-B block (U+2900–U+297F) contains arrows and arrow-like operators (arrow tails, crossing arrows, curved arrows, and harpoons). Supplemental Arrows-B[1] Official Unicode Consortium code chart (PDF) 0123456789ABCDEF U+290x ⤀ ⤁ ⤂ ⤃ ⤄ ⤅ ⤆ ⤇ ⤈ ⤉ ⤊ ⤋ ⤌ ⤍ ⤎ ⤏ U+291x ⤐ ⤑ ⤒ ⤓ ⤔ ⤕ ⤖ ⤗ ⤘ ⤙ ⤚ ⤛ ⤜ ⤝ ⤞ ⤟ U+292x ⤠ ⤡ ⤢ ⤣ ⤤ ⤥ ⤦ ⤧ ⤨ ⤩ ⤪ ⤫ ⤬ ⤭ ⤮ ⤯ U+293x ⤰ ⤱ ⤲ ⤳ ⤴ ⤵ ⤶ ⤷ ⤸ ⤹ ⤺ ⤻ ⤼ ⤽ ⤾ ⤿ U+294x ⥀ ⥁ ⥂ ⥃ ⥄ ⥅ ⥆ ⥇ ⥈ ⥉ ⥊ ⥋ ⥌ ⥍ ⥎ ⥏ U+295x ⥐ ⥑ ⥒ ⥓ ⥔ ⥕ ⥖ ⥗ ⥘ ⥙ ⥚ ⥛ ⥜ ⥝ ⥞ ⥟ U+296x ⥠ ⥡ ⥢ ⥣ ⥤ ⥥ ⥦ ⥧ ⥨ ⥩ ⥪ ⥫ ⥬ ⥭ ⥮ ⥯ U+297x ⥰ ⥱ ⥲ ⥳ ⥴ ⥵ ⥶ ⥷ ⥸ ⥹ ⥺ ⥻ ⥼ ⥽ ⥾ ⥿ Notes 1.^ As of Unicode version 15.0 Miscellaneous Symbols and Arrows block The Miscellaneous Symbols and Arrows block (U+2B00–U+2BFF Arrows) contains arrows and geometric shapes with various fills. The math subset of this block is U+2B30–U+2B44, U+2B47–U+2B4C.[7] Miscellaneous Symbols and Arrows[1][2] Official Unicode Consortium code chart (PDF) 0123456789ABCDEF U+2B0x ⬀ ⬁ ⬂ ⬃ ⬄ ⬅ ⬆ ⬇ ⬈ ⬉ ⬊ ⬋ ⬌ ⬍ ⬎ ⬏ U+2B1x ⬐ ⬑ ⬒ ⬓ ⬔ ⬕ ⬖ ⬗ ⬘ ⬙ ⬚ ⬛ ⬜ ⬝ ⬞ ⬟ U+2B2x ⬠ ⬡ ⬢ ⬣ ⬤ ⬥ ⬦ ⬧ ⬨ ⬩ ⬪ ⬫ ⬬ ⬭ ⬮ ⬯ U+2B3x ⬰ ⬱ ⬲ ⬳ ⬴ ⬵ ⬶ ⬷ ⬸ ⬹ ⬺ ⬻ ⬼ ⬽ ⬾ ⬿ U+2B4x ⭀ ⭁ ⭂ ⭃ ⭄ ⭅ ⭆ ⭇ ⭈ ⭉ ⭊ ⭋ ⭌ ⭍ ⭎ ⭏ U+2B5x ⭐ ⭑ ⭒ ⭓ ⭔ ⭕ ⭖ ⭗ ⭘ ⭙ ⭚ ⭛ ⭜ ⭝ ⭞ ⭟ U+2B6x ⭠ ⭡ ⭢ ⭣ ⭤ ⭥ ⭦ ⭧ ⭨ ⭩ ⭪ ⭫ ⭬ ⭭ ⭮ ⭯ U+2B7x ⭰ ⭱ ⭲ ⭳ ⭶ ⭷ ⭸ ⭹ ⭺ ⭻ ⭼ ⭽ ⭾ ⭿ U+2B8x ⮀ ⮁ ⮂ ⮃ ⮄ ⮅ ⮆ ⮇ ⮈ ⮉ ⮊ ⮋ ⮌ ⮍ ⮎ ⮏ U+2B9x ⮐ ⮑ ⮒ ⮓ ⮔ ⮕ ⮗ ⮘ ⮙ ⮚ ⮛ ⮜ ⮝ ⮞ ⮟ U+2BAx ⮠ ⮡ ⮢ ⮣ ⮤ ⮥ ⮦ ⮧ ⮨ ⮩ ⮪ ⮫ ⮬ ⮭ ⮮ ⮯ U+2BBx ⮰ ⮱ ⮲ ⮳ ⮴ ⮵ ⮶ ⮷ ⮸ ⮹ ⮺ ⮻ ⮼ ⮽ ⮾ ⮿ U+2BCx ⯀ ⯁ ⯂ ⯃ ⯄ ⯅ ⯆ ⯇ ⯈ ⯉ ⯊ ⯋ ⯌ ⯍ ⯎ ⯏ U+2BDx ⯐ ⯑ ⯒ ⯓ ⯔ ⯕ ⯖ ⯗ ⯘ ⯙ ⯚ ⯛ ⯜ ⯝ ⯞ ⯟ U+2BEx ⯠ ⯡ ⯢ ⯣ ⯤ ⯥ ⯦ ⯧ ⯨ ⯩ ⯪ ⯫ ⯬ ⯭ ⯮ ⯯ U+2BFx ⯰ ⯱ ⯲ ⯳ ⯴ ⯵ ⯶ ⯷ ⯸ ⯹ ⯺ ⯻ ⯼ ⯽ ⯾ ⯿ Notes 1.^ As of Unicode version 15.0 2.^ Grey areas indicate non-assigned code points Combining Diacritical Marks for Symbols block The Combining Diacritical Marks for Symbols block contains arrows, dots, enclosures, and overlays for modifying symbol characters. The math subset of this block is U+20D0–U+20DC, U+20E1, U+20E5–U+20E6, and U+20EB–U+20EF. Combining Diacritical Marks for Symbols[1][2] Official Unicode Consortium code chart (PDF) 0123456789ABCDEF U+20Dx ◌⃐ ◌⃑ ◌⃒ ◌⃓ ◌⃔ ◌⃕ ◌⃖ ◌⃗ ◌⃘ ◌⃙ ◌⃚ ◌⃛ ◌⃜ ◌⃝ ◌⃞ ◌⃟ U+20Ex ◌⃠ ◌⃡ ◌⃢ ◌⃣ ◌⃤ ◌⃥ ◌⃦ ◌⃧ ◌⃨ ◌⃩ ◌⃪ ◌⃫ ◌⃬ ◌⃭ ◌⃮ ◌⃯ U+20Fx ◌⃰ Notes 1.^ As of Unicode version 15.0 2.^ Grey areas indicate non-assigned code points Arabic Mathematical Alphabetic Symbols block The Arabic Mathematical Alphabetic Symbols block (U+1EE00–U+1EEFF) contains characters used in Arabic mathematical expressions. Arabic Mathematical Alphabetic Symbols[1][2] Official Unicode Consortium code chart (PDF) 0123456789ABCDEF U+1EE0x 𞸀 𞸁 𞸂 𞸃 𞸅 𞸆 𞸇 𞸈 𞸉 𞸊 𞸋 𞸌 𞸍 𞸎 𞸏 U+1EE1x 𞸐 𞸑 𞸒 𞸓 𞸔 𞸕 𞸖 𞸗 𞸘 𞸙 𞸚 𞸛 𞸜 𞸝 𞸞 𞸟 U+1EE2x 𞸡 𞸢 𞸤 𞸧 𞸩 𞸪 𞸫 𞸬 𞸭 𞸮 𞸯 U+1EE3x 𞸰 𞸱 𞸲 𞸴 𞸵 𞸶 𞸷 𞸹 𞸻 U+1EE4x 𞹂 𞹇 𞹉 𞹋 𞹍 𞹎 𞹏 U+1EE5x 𞹑 𞹒 𞹔 𞹗 𞹙 𞹛 𞹝 𞹟 U+1EE6x 𞹡 𞹢 𞹤 𞹧 𞹨 𞹩 𞹪 𞹬 𞹭 𞹮 𞹯 U+1EE7x 𞹰 𞹱 𞹲 𞹴 𞹵 𞹶 𞹷 𞹹 𞹺 𞹻 𞹼 𞹾 U+1EE8x 𞺀 𞺁 𞺂 𞺃 𞺄 𞺅 𞺆 𞺇 𞺈 𞺉 𞺋 𞺌 𞺍 𞺎 𞺏 U+1EE9x 𞺐 𞺑 𞺒 𞺓 𞺔 𞺕 𞺖 𞺗 𞺘 𞺙 𞺚 𞺛 U+1EEAx 𞺡 𞺢 𞺣 𞺥 𞺦 𞺧 𞺨 𞺩 𞺫 𞺬 𞺭 𞺮 𞺯 U+1EEBx 𞺰 𞺱 𞺲 𞺳 𞺴 𞺵 𞺶 𞺷 𞺸 𞺹 𞺺 𞺻 U+1EECx U+1EEDx U+1EEEx U+1EEFx 𞻰 𞻱 Notes 1.^ As of Unicode version 15.0 2.^ Grey areas indicate non-assigned code points Characters in other blocks Mathematical characters also appear in other blocks. Below is a list of these characters as of Unicode version 15.0: • Basic Latin block U+002B+PLUS SIGN U+002D-HYPHEN-MINUS[8] U+003C<LESS-THAN SIGN U+003D=EQUALS SIGN U+003E>GREATER-THAN SIGN U+005E^CIRCUMFLEX ACCENT U+007C\|VERTICAL LINE U+007E~TILDE • Latin-1 Supplement block U+00AC¬NOT SIGN U+00B0°DEGREE SIGN[9] U+00B1±PLUS-MINUS SIGN U+00D7×MULTIPLICATION SIGN U+00F7÷DIVISION SIGN • Greek and Coptic block U+03D0ϐGREEK BETA SYMBOL U+03D1ϑGREEK THETA SYMBOL U+03D2ϒGREEK UPSILON WITH HOOK SYMBOL U+03D5ϕGREEK PHI SYMBOL U+03F0ϰGREEK KAPPA SYMBOL U+03F1ϱGREEK RHO SYMBOL U+03F4ϴGREEK CAPITAL THETA SYMBOL U+03F5ϵGREEK LUNATE EPSILON SYMBOL U+03F6϶GREEK REVERSED LUNATE EPSILON SYMBOL • Arabic block U+0606؆ARABIC-INDIC CUBE ROOT U+0607؇ARABIC-INDIC FOURTH ROOT U+0608؈ARABIC RAY • General Punctuation block U+2016‖DOUBLE VERTICAL LINE U+2032′PRIME U+2033″DOUBLE PRIME U+2034‴TRIPLE PRIME U+2040⁀CHARACTER TIE U+2044⁄FRACTION SLASH U+2052⁒COMMERCIAL MINUS SIGN U+2061noteFUNCTION APPLICATION U+2062noteINVISIBLE TIMES U+2063noteINVISIBLE SEPARATOR U+2064noteINVISIBLE PLUS Note: non-marking character • Superscripts and Subscripts block U+207A⁺SUPERSCRIPT PLUS SIGN U+207B⁻SUPERSCRIPT MINUS U+207C⁼SUPERSCRIPT EQUALS SIGN U+207D⁽SUPERSCRIPT LEFT PARENTHESIS U+207E⁾SUPERSCRIPT RIGHT PARENTHESIS U+208A₊SUBSCRIPT PLUS SIGN U+208B₋SUBSCRIPT MINUS U+208C₌SUBSCRIPT EQUALS SIGN U+208D₍SUBSCRIPT LEFT PARENTHESIS U+208E₎SUBSCRIPT RIGHT PARENTHESIS • Miscellaneous Symbols block U+2605★BLACK STAR U+2606☆WHITE STAR U+2640♀FEMALE SIGN U+2642♂MALE SIGN U+2660♠BLACK SPADE SUIT U+2661♡WHITE HEART SUIT U+2662♢WHITE DIAMOND SUIT U+2663♣BLACK CLUB SUIT U+266D♭MUSIC FLAT SIGN U+266E♮MUSIC NATURAL SIGN U+266F♯MUSIC SHARP SIGN • Alphabetic Presentation Forms block U+FB29﬩HEBREW LETTER ALTERNATIVE PLUS SIGN • Small Form Variants block U+FE61﹡SMALL ASTERISK U+FE62﹢SMALL PLUS SIGN U+FE63﹣SMALL HYPHEN-MINUS U+FE64﹤SMALL LESS-THAN SIGN U+FE65﹥SMALL GREATER-THAN SIGN U+FE66﹦SMALL EQUALS SIGN U+FE68﹨SMALL REVERSE SOLIDUS • Halfwidth and Fullwidth Forms block U+FF0B＋FULLWIDTH PLUS SIGN U+FF1C＜FULLWIDTH LESS-THAN SIGN U+FF1D＝FULLWIDTH EQUALS SIGN U+FF1E＞FULLWIDTH GREATER-THAN SIGN U+FF3C＼FULLWIDTH REVERSE SOLIDUS U+FF3E＾FULLWIDTH CIRCUMFLEX ACCENT U+FF5C｜FULLWIDTH VERTICAL LINE U+FF5E～FULLWIDTH TILDE U+FFE2￢FULLWIDTH NOT SIGN U+FFE9￩HALFWIDTH LEFTWARDS ARROW U+FFEA￪HALFWIDTH UPWARDS ARROW U+FFEB￫HALFWIDTH RIGHTWARDS ARROW U+FFEC￬HALFWIDTH DOWNWARDS ARROW See also • Glossary of mathematical symbols • List of logic symbols • Greek letters used in mathematics, science, and engineering • List of letters used in mathematics and science • List of mathematical uses of Latin letters • Unicode subscripts and superscripts • Unicode symbols • CJK Compatibility Unicode symbols includes symbols for SI units • Units for order of magnitude shows position of SI units References 1. "Unicode Technical Report #25: Unicode Support for Mathematics" (PDF). The Unicode Consortium. 2 April 2012. Retrieved 14 August 2014. 2. "Unicode Character Database: Derived Core Properties". The Unicode Consortium. 19 February 2014. Retrieved 14 August 2014. 3. "Unicode Technical Annex #44: Unicode Character Database" (PDF). The Unicode Consortium. 25 September 2013. Retrieved 14 August 2014. 4. In order, these are ℎ / ℬ ℰ ℱ ℋ ℐ ℒ ℳ ℛ / ℯ ℊ ℴ / ℭ ℌ ℑ ℜ ℨ / ℂ ℍ ℕ ℙ ℚ ℝ ℤ. 5. See https://www.unicode.org/Public/UCD/latest/ucd/DerivedCoreProperties.txt 6. More symbols are supported by TeX math packages, see e.g. Will Robertson, Symbols defined by unicode-math. 7. The quadruple arrows U+2B45 and U+2B46 are supported by TeX math packages, per Will Robertson, Symbols defined by unicode-math. 8. As per Unicode 15.0.0, the ASCII hyphen-minus is not a mathematical symbol. To express the minus sign in math, U+2212 − MINUS SIGN is used instead. 9. As per Unicode 15.0.0, the degree sign is not a mathematical symbol, since it is a measurement unit symbol rather than a math symbol. Consistently, TeX packages support many non-math symbols. But this article is designed to cover only Unicode characters with a derived property of "Math". External links • Mathematical Markup Language (MathML) W3C Recommendation. 3.0 (2nd ed.). W3C. 10 April 2014. • Images of glyphs in section 6.3.3 of the Mathematical Markup Language (MathML) W3C Recommendation. 2.0 (2nd ed.). W3C. 21 February 2001. 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of mathematical symbols • List of mathematical constants • Physical constants • Table of mathematical symbols by introduction date • List of typographical symbols and punctuation marks	Wikipedia
Unicoherent space In mathematics, a unicoherent space is a topological space $X$ that is connected and in which the following property holds: For any closed, connected $A,B\subset X$ with $X=A\cup B$, the intersection $A\cap B$ is connected. For example, any closed interval on the real line is unicoherent, but a circle is not. If a unicoherent space is more strongly hereditarily unicoherent (meaning that every subcontinuum is unicoherent) and arcwise connected, then it is called a dendroid. If in addition it is locally connected then it is called a dendrite. The Phragmen–Brouwer theorem states that, for locally connected spaces, unicoherence is equivalent to a separation property of the closed sets of the space. References • Charatonik, Janusz J. (2003). "Unicoherence and Multicoherence". Encyclopedia of General Topology. pp. 331–333. doi:10.1016/B978-044450355-8/50088-X. ISBN 9780444503558. External links • Insall, Matt. "Unicoherent Space". MathWorld.	Wikipedia
Unified strength theory The unified strength theory (UST).[1][2][3][4] proposed by Yu Mao-Hong is a series of yield criteria (see yield surface) and failure criteria (see Material failure theory). It is a generalized classical strength theory which can be used to describe the yielding or failure of material begins when the combination of principal stresses reaches a critical value.[5][6][7] Mathematical Formulation Mathematically, the formulation of UST is expressed in principal stress state as $F={\sigma _{1}}-{\frac {\alpha }{1+b}}(b{\sigma _{2}}+{\sigma _{3}})={\sigma _{t}},\,{\text{ when }}{\sigma _{2}}\leqslant {\frac {{\sigma _{1}}+\alpha {\sigma _{3}}}{1+\alpha }}$ (1a) $F'={\frac {1}{1+b}}({\sigma _{1}}+b{\sigma _{3}})-\alpha {\sigma _{3}}={\sigma _{t}},\,{\text{ when }}{\sigma _{2}}\geqslant {\frac {{\sigma _{1}}+\alpha {\sigma _{3}}}{1+\alpha }}$ (1b) where ${\sigma _{1}},{\sigma _{2}},{\sigma _{3}}$ are three principal stresses, ${\sigma _{t}}$is the uniaxial tensile strength and $\alpha $ is tension-compression strength ratio ($\alpha ={\sigma _{t}}/{\sigma _{c}}$). The unified yield criterion (UYC) is the simplification of UST when $\alpha =1$, i.e. $f={\sigma _{1}}-{\frac {1}{1+b}}(b{\sigma _{2}}+{\sigma _{3}})={\sigma _{s}},{\text{ when }}{\sigma _{2}}\leqslant {\frac {1}{2}}({\sigma _{1}}+{\sigma _{3}})$ (2a) $f'={\frac {1}{1+b}}({\sigma _{1}}+b{\sigma _{2}})-{\sigma _{3}}={\sigma _{s}},{\text{ when }}{\sigma _{2}}\geqslant {\frac {1}{2}}({\sigma _{1}}+{\sigma _{3}})$ (2b) Limit surfaces of Unified Strength Theory The limit surfaces of the unified strength theory in principal stress space are usually a semi-infinite dodecahedron cone with unequal sides. The shape and size of the limiting dodecahedron cone depends on the parameter b and $\alpha $. The limit surfaces of UST and UYC are shown as follows. Derivation of Unified Strength Theory Due to the relation (${\tau _{13}}={\tau _{12}}+{\tau _{23}}$), the principal stress state (${\sigma _{1}},{\sigma _{2}},{\sigma _{3}}$) may be converted to the twin-shear stress state (${\tau _{13}},{\tau _{12}};{\sigma _{13}},{\sigma _{12}}$) or (${\tau _{13}},{\tau _{23}};{\sigma _{13}},{\sigma _{23}}$). Twin-shear element models proposed by Mao-Hong Yu are used for representing the twin-shear stress state.[1] Considering all the stress components of the twin-shear models and their different effects yields the unified strength theory as $F={\tau _{13}}+b{\tau _{12}}+\beta ({\sigma _{13}}+b{\sigma _{12}})=C,{\text{ when }}{\tau _{12}}+\beta {\sigma _{12}}\geqslant {\tau _{23}}+\beta {\sigma _{23}}$ (3a) $F'={\tau _{13}}+b{\tau _{23}}+\beta ({\sigma _{13}}+b{\sigma _{23}})=C,{\text{ when }}{\tau _{12}}+\beta {\sigma _{12}}\leqslant {\tau _{23}}+\beta {\sigma _{23}}$ (3b) The relations among the stresses components and principal stresses read ${\tau _{13}}={\frac {1}{2}}\left({{\sigma _{1}}-{\sigma _{3}}}\right)$, ${\sigma _{13}}={\frac {1}{2}}\left({{\sigma _{1}}+{\sigma _{3}}}\right)$ (4a) ${\tau _{12}}={\frac {1}{2}}\left({{\sigma _{1}}-{\sigma _{2}}}\right)$, ${\sigma _{12}}={\frac {1}{2}}\left({{\sigma _{1}}+{\sigma _{2}}}\right)$ (4b) ${\tau _{23}}={\frac {1}{2}}\left({{\sigma _{2}}-{\sigma _{3}}}\right)$, ${\sigma _{23}}={\frac {1}{2}}\left({{\sigma _{2}}+{\sigma _{3}}}\right)$ (4c) The $\beta $ and C should be obtained by uniaxial failure state ${\sigma _{1}}={\sigma _{t}},{\sigma _{2}}={\sigma _{3}}=0$ (5a) ${\sigma _{1}}={\sigma _{2}}=0,{\sigma _{3}}=-{\sigma _{\text{c}}}$ (5b) By substituting Eqs.(4a), (4b) and (5a) into the Eq.(3a), and substituting Eqs.(4a), (4c) and (5b) into Eq.(3b), the $\beta $ and C are introduced as $\beta ={\frac {{\sigma _{\text{c}}}-{\sigma _{\text{t}}}}{{\sigma _{\text{c}}}+{\sigma _{\text{t}}}}}={\frac {1-\alpha }{1+\alpha }}$, $C={\frac {1+b{\sigma _{\text{c}}}{\sigma _{\text{t}}}}{{\sigma _{\text{c}}}+{\sigma _{\text{t}}}}}={\frac {1+b}{1+\alpha }}{\sigma _{t}}$ (6) History of Unified Strength Theory The development of the unified strength theory can be divided into three stages as follows. 1. Twin-shear yield criterion (UST with $\alpha =1$ and $b=1$)[8][9] $f={\sigma _{1}}-{\frac {1}{2}}({\sigma _{2}}+{\sigma _{3}})={\sigma _{t}},{\text{ when }}{\sigma _{2}}\leqslant {\frac {{\sigma _{1}}+{\sigma _{3}}}{2}}$ (7a) $f={\frac {1}{2}}({\sigma _{1}}+{\sigma _{2}})-{\sigma _{3}}={\sigma _{t}},{\text{ when }}{\sigma _{2}}\geqslant {\frac {{\sigma _{1}}+{\sigma _{3}}}{2}}$ (7b) 2. Twin-shear strength theory (UST with $b=1$)[10]. $F={\sigma _{1}}-{\frac {\alpha }{2}}({\sigma _{2}}+{\sigma _{3}})={\sigma _{t}},{\text{ when }}{\sigma _{2}}\leqslant {\frac {{\sigma _{1}}+\alpha {\sigma _{3}}}{1+\alpha }}$ (8a) $F={\frac {1}{2}}({\sigma _{1}}+{\sigma _{2}}){\text{ - }}\alpha {\sigma _{3}}={\sigma _{t}},{\text{ when }}{\sigma _{2}}\geqslant {\frac {{\sigma _{1}}+\alpha {\sigma _{3}}}{1+\alpha }}$ (8b) 3. Unified strength theory[1]. Applications of the Unified Strength theory Unified strength theory has been used in Generalized Plasticity,[11] Structural Plasticity,[12] Computational Plasticity[13] and many other fields[14][15] References 1. Yu M. H., He L. N. (1991) A new model and theory on yield and failure of materials under the complex stress state. Mechanical Behaviour of Materials-6 (ICM-6). Jono M and Inoue T eds. Pergamon Press, Oxford, (3), pp. 841–846. https://doi.org/10.1016/B978-0-08-037890-9.50389-6 2. Yu M. H. (2004) Unified Strength Theory and Its Applications. Springer: Berlin. ISBN 978-3-642-18943-2 3. Zhao, G.-H.; Ed., (2006) Handbook of Engineering Mechanics, Rock Mechanics, Engineering Structures and Materials (in Chinese), China's Water Conservancy Resources and Hydropower Press, Beijing, pp. 20-21 4. Yu M. H. (2018) Unified Strength Theory and Its Applications (second edition). Springer and Xi'an Jiaotong University Press, Springer and Xi'an. ISBN 978-981-10-6247-6 5. Teodorescu, P.P. (Bucureşti). (2006). Review: Unified Strength Theory and its applications, Zentralblatt MATH Database 1931 – 2009, European Mathematical Society,Zbl 1059.74002, FIZ Karlsruhe & Springer-Verlag 6. Altenbach, H., Bolchoun, A., Kolupaev, V.A. (2013). Phenomenological Yield and Failure Criteria, in Altenbach, H., Öchsner, A., eds., Plasticity of Pressure-Sensitive Materials, Serie ASM, Springer, Heidelberg, pp. 49-152. 7. Kolupaev, V. A., Altenbach, H. (2010). Considerations on the Unified Strength Theory due to Mao-Hong Yu (in German: Einige Überlegungen zur Unified Strength Theory von Mao-Hong Yu), Forschung im Ingenieurwesen, 74(3), pp. 135-166. 8. Yu M. H. (1961) Plastic potential and flow rules associated singular yield criterion. Res. Report of Xi'an Jiaotong University. Xi'an, China (in Chinese) 9. Yu MH (1983) Twin shear stress yield criterion. International Journal of Mechanical Sciences, 25(1), pp. 71-74. https://doi.org/10.1016/0020-7403(83)90088-7 10. Yu M. H., He L. N., Song L. Y. (1985) Twin shear stress theory and its generalization. Scientia Sinica (Sciences in China), English edn. Series A, 28(11), pp. 1174–1183. 11. Yu M. H. et al., (2006) Generalized Plasticity. Springer: Berlin. ISBN 978-3-540-30433-3 12. Yu M. H., Ma G. W., Li J. C. (2009) Structural Plasticity: Limit, Shakedown and Dynamic Plastic Analyses of Structures. ZJU Press and Springer: Hangzhou and Berlin. ISBN 978-3-540-88152-0 13. Yu M. H., Li J. C. (2012) Computational Plasticity, Springer and ZJU Press: Berlin and Hangzhou. ISBN 978-3-642-24590-9 14. Fan, S. C., Qiang, H. F. (2001). Normal high-velocity impaction concrete slabs-a simulation using the meshless SPH procedures. Computational Mechanics-New Frontiers for New Millennium, Valliappan S. and Khalili N. eds. Elsevier Science Ltd, pp. 1457-1462 15. Guowei, M., Iwasaki, S., Miyamoto, Y. and Deto, H., 1998. Plastic limit analyses of circular plates with respect to unified yield criterion. International journal of mechanical sciences, 40(10), pp.963-976. https://doi.org/10.1016/S0020-7403(97)00140-9	Wikipedia
Markov information source In mathematics, a Markov information source, or simply, a Markov source, is an information source whose underlying dynamics are given by a stationary finite Markov chain. Formal definition An information source is a sequence of random variables ranging over a finite alphabet $\Gamma $, having a stationary distribution. A Markov information source is then a (stationary) Markov chain $M$, together with a function $f:S\to \Gamma $ that maps states $S$ in the Markov chain to letters in the alphabet $\Gamma $. A unifilar Markov source is a Markov source for which the values $f(s_{k})$ are distinct whenever each of the states $s_{k}$ are reachable, in one step, from a common prior state. Unifilar sources are notable in that many of their properties are far more easily analyzed, as compared to the general case. Applications Markov sources are commonly used in communication theory, as a model of a transmitter. Markov sources also occur in natural language processing, where they are used to represent hidden meaning in a text. Given the output of a Markov source, whose underlying Markov chain is unknown, the task of solving for the underlying chain is undertaken by the techniques of hidden Markov models, such as the Viterbi algorithm. See also • Entropy rate References • Robert B. Ash, Information Theory, (1965) Dover Publications. ISBN 0-486-66521-6	Wikipedia
Discrete uniform distribution In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of n values has equal probability 1/n. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen". discrete uniform Probability mass function n = 5 where n = b − a + 1 Cumulative distribution function Notation ${\mathcal {U}}\{a,b\}$ or $\mathrm {unif} \{a,b\}$ Parameters $a,b$ integers with $b\geq a$ $n=b-a+1$ Support $k\in \{a,a+1,\dots ,b-1,b\}$ PMF ${\frac {1}{n}}$ CDF ${\frac {\lfloor k\rfloor -a+1}{n}}$ Mean ${\frac {a+b}{2}}$ Median ${\frac {a+b}{2}}$ Mode N/A Variance ${\frac {n^{2}-1}{12}}$ Skewness $0$ Ex. kurtosis $-{\frac {6(n^{2}+1)}{5(n^{2}-1)}}$ Entropy $\ln(n)$ MGF ${\frac {e^{at}-e^{(b+1)t}}{n(1-e^{t})}}$ CF ${\frac {e^{iat}-e^{i(b+1)t}}{n(1-e^{it})}}$ PGF ${\frac {z^{a}-z^{b+1}}{n(1-z)}}$ A simple example of the discrete uniform distribution is throwing a fair die. The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of a given score is 1/6. If two dice are thrown and their values added, the resulting distribution is no longer uniform because not all sums have equal probability. Although it is convenient to describe discrete uniform distributions over integers, such as this, one can also consider discrete uniform distributions over any finite set. For instance, a random permutation is a permutation generated uniformly from the permutations of a given length, and a uniform spanning tree is a spanning tree generated uniformly from the spanning trees of a given graph. The discrete uniform distribution itself is inherently non-parametric. It is convenient, however, to represent its values generally by all integers in an interval [a,b], so that a and b become the main parameters of the distribution (often one simply considers the interval [1,n] with the single parameter n). With these conventions, the cumulative distribution function (CDF) of the discrete uniform distribution can be expressed, for any k ∈ [a,b], as $F(k;a,b)={\frac {\lfloor k\rfloor -a+1}{b-a+1}}$ Estimation of maximum Main article: German tank problem This example is described by saying that a sample of k observations is obtained from a uniform distribution on the integers $1,2,\dotsc ,N$, with the problem being to estimate the unknown maximum N. This problem is commonly known as the German tank problem, following the application of maximum estimation to estimates of German tank production during World War II. The uniformly minimum variance unbiased (UMVU) estimator for the maximum is given by ${\hat {N}}={\frac {k+1}{k}}m-1=m+{\frac {m}{k}}-1$ where m is the sample maximum and k is the sample size, sampling without replacement.[1] This can be seen as a very simple case of maximum spacing estimation. This has a variance of[1] ${\frac {1}{k}}{\frac {(N-k)(N+1)}{(k+2)}}\approx {\frac {N^{2}}{k^{2}}}{\text{ for small samples }}k\ll N$ so a standard deviation of approximately ${\tfrac {N}{k}}$, the (population) average size of a gap between samples; compare ${\tfrac {m}{k}}$ above. The sample maximum is the maximum likelihood estimator for the population maximum, but, as discussed above, it is biased. If samples are not numbered but are recognizable or markable, one can instead estimate population size via the capture-recapture method. Random permutation See rencontres numbers for an account of the probability distribution of the number of fixed points of a uniformly distributed random permutation. Properties The family of uniform distributions over ranges of integers (with one or both bounds unknown) has a finite-dimensional sufficient statistic, namely the triple of the sample maximum, sample minimum, and sample size, but is not an exponential family of distributions, because the support varies with the parameters. For families whose support does not depend on the parameters, the Pitman–Koopman–Darmois theorem states that only exponential families have a sufficient statistic whose dimension is bounded as sample size increases. The uniform distribution is thus a simple example showing the limit of this theorem. See also • Dirac delta distribution • Continuous uniform distribution References 1. Johnson, Roger (1994), "Estimating the Size of a Population", Teaching Statistics, 16 (2 (Summer)): 50–52, CiteSeerX 10.1.1.385.5463, doi:10.1111/j.1467-9639.1994.tb00688.x Probability distributions (list) Discrete univariate with finite support • Benford • Bernoulli • beta-binomial • binomial • categorical • hypergeometric • negative • Poisson binomial • Rademacher • soliton • discrete uniform • Zipf • Zipf–Mandelbrot with infinite support • beta negative binomial • Borel • Conway–Maxwell–Poisson • discrete phase-type • Delaporte • extended negative binomial • Flory–Schulz • Gauss–Kuzmin • geometric • logarithmic • mixed Poisson • negative binomial • Panjer • parabolic fractal • Poisson • Skellam • Yule–Simon • zeta Continuous univariate supported on a bounded interval • arcsine • ARGUS • Balding–Nichols • Bates • beta • beta rectangular • continuous Bernoulli • Irwin–Hall • Kumaraswamy • logit-normal • noncentral beta • PERT • raised cosine • reciprocal • triangular • U-quadratic • uniform • Wigner semicircle supported on a semi-infinite interval • Benini • Benktander 1st kind • Benktander 2nd kind • beta prime • Burr • chi • chi-squared • noncentral • inverse • scaled • Dagum • Davis • Erlang • hyper • exponential • hyperexponential • hypoexponential • logarithmic • F • noncentral • folded normal • Fréchet • gamma • generalized • inverse • gamma/Gompertz • Gompertz • shifted • half-logistic • half-normal • Hotelling's T-squared • inverse Gaussian • generalized • Kolmogorov • Lévy • log-Cauchy • log-Laplace • log-logistic • log-normal • log-t • Lomax • matrix-exponential • Maxwell–Boltzmann • Maxwell–Jüttner • Mittag-Leffler • Nakagami • Pareto • phase-type • Poly-Weibull • Rayleigh • relativistic Breit–Wigner • Rice • truncated normal • type-2 Gumbel • Weibull • discrete • Wilks's lambda supported on the whole real line • Cauchy • exponential power • Fisher's z • Kaniadakis κ-Gaussian • Gaussian q • generalized normal • generalized hyperbolic • geometric stable • Gumbel • Holtsmark • hyperbolic secant • Johnson's SU • Landau • Laplace • asymmetric • logistic • noncentral t • normal (Gaussian) • normal-inverse Gaussian • skew normal • slash • stable • Student's t • Tracy–Widom • variance-gamma • Voigt with support whose type varies • generalized chi-squared • generalized extreme value • generalized Pareto • Marchenko–Pastur • Kaniadakis κ-exponential • Kaniadakis κ-Gamma • Kaniadakis κ-Weibull • Kaniadakis κ-Logistic • Kaniadakis κ-Erlang • q-exponential • q-Gaussian • q-Weibull • shifted log-logistic • Tukey lambda Mixed univariate continuous- discrete • Rectified Gaussian Multivariate (joint) • Discrete: • Ewens • multinomial • Dirichlet • negative • Continuous: • Dirichlet • generalized • multivariate Laplace • multivariate normal • multivariate stable • multivariate t • normal-gamma • inverse • Matrix-valued: • LKJ • matrix normal • matrix t • matrix gamma • inverse • Wishart • normal • inverse • normal-inverse • complex Directional Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham Degenerate and singular Degenerate Dirac delta function Singular Cantor Families • Circular • compound Poisson • elliptical • exponential • natural exponential • location–scale • maximum entropy • mixture • Pearson • Tweedie • wrapped • Category • Commons	Wikipedia
Uniform 10-polytope In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge. Graphs of three regular and related uniform polytopes. 10-simplex Truncated 10-simplex Rectified 10-simplex Cantellated 10-simplex Runcinated 10-simplex Stericated 10-simplex Pentellated 10-simplex Hexicated 10-simplex Heptellated 10-simplex Octellated 10-simplex Ennecated 10-simplex 10-orthoplex Truncated 10-orthoplex Rectified 10-orthoplex 10-cube Truncated 10-cube Rectified 10-cube 10-demicube Truncated 10-demicube A uniform 10-polytope is one which is vertex-transitive, and constructed from uniform facets. Regular 10-polytopes Regular 10-polytopes can be represented by the Schläfli symbol {p,q,r,s,t,u,v,w,x}, with x {p,q,r,s,t,u,v,w} 9-polytope facets around each peak. There are exactly three such convex regular 10-polytopes: 1. {3,3,3,3,3,3,3,3,3} - 10-simplex 2. {4,3,3,3,3,3,3,3,3} - 10-cube 3. {3,3,3,3,3,3,3,3,4} - 10-orthoplex There are no nonconvex regular 10-polytopes. Euler characteristic The topology of any given 10-polytope is defined by its Betti numbers and torsion coefficients.[1] The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 10-polytopes, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.[1] Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.[1] Uniform 10-polytopes by fundamental Coxeter groups Uniform 10-polytopes with reflective symmetry can be generated by these three Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams: # Coxeter group Coxeter-Dynkin diagram 1A10[39] 2B10[4,38] 3D10[37,1,1] Selected regular and uniform 10-polytopes from each family include: 1. Simplex family: A10 [39] - • 527 uniform 10-polytopes as permutations of rings in the group diagram, including one regular: 1. {39} - 10-simplex - 2. Hypercube/orthoplex family: B10 [4,38] - • 1023 uniform 10-polytopes as permutations of rings in the group diagram, including two regular ones: 1. {4,38} - 10-cube or dekeract - 2. {38,4} - 10-orthoplex or decacross - 3. h{4,38} - 10-demicube . 3. Demihypercube D10 family: [37,1,1] - • 767 uniform 10-polytopes as permutations of rings in the group diagram, including: 1. 17,1 - 10-demicube or demidekeract - 2. 71,1 - 10-orthoplex - The A10 family The A10 family has symmetry of order 39,916,800 (11 factorial). There are 512+16-1=527 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. 31 are shown below: all one and two ringed forms, and the final omnitruncated form. Bowers-style acronym names are given in parentheses for cross-referencing. # Graph Coxeter-Dynkin diagram Schläfli symbol Name Element counts 9-faces8-faces7-faces6-faces5-faces4-facesCellsFacesEdgesVertices 1 t0{3,3,3,3,3,3,3,3,3} 10-simplex (ux) 11551653304624623301655511 2 t1{3,3,3,3,3,3,3,3,3} Rectified 10-simplex (ru) 49555 3 t2{3,3,3,3,3,3,3,3,3} Birectified 10-simplex (bru) 1980165 4 t3{3,3,3,3,3,3,3,3,3} Trirectified 10-simplex (tru) 4620330 5 t4{3,3,3,3,3,3,3,3,3} Quadrirectified 10-simplex (teru) 6930462 6 t0,1{3,3,3,3,3,3,3,3,3} Truncated 10-simplex (tu) 550110 7 t0,2{3,3,3,3,3,3,3,3,3} Cantellated 10-simplex 4455495 8 t1,2{3,3,3,3,3,3,3,3,3} Bitruncated 10-simplex 2475495 9 t0,3{3,3,3,3,3,3,3,3,3} Runcinated 10-simplex 158401320 10 t1,3{3,3,3,3,3,3,3,3,3} Bicantellated 10-simplex 178201980 11 t2,3{3,3,3,3,3,3,3,3,3} Tritruncated 10-simplex 66001320 12 t0,4{3,3,3,3,3,3,3,3,3} Stericated 10-simplex 323402310 13 t1,4{3,3,3,3,3,3,3,3,3} Biruncinated 10-simplex 554404620 14 t2,4{3,3,3,3,3,3,3,3,3} Tricantellated 10-simplex 415804620 15 t3,4{3,3,3,3,3,3,3,3,3} Quadritruncated 10-simplex 115502310 16 t0,5{3,3,3,3,3,3,3,3,3} Pentellated 10-simplex 415802772 17 t1,5{3,3,3,3,3,3,3,3,3} Bistericated 10-simplex 970206930 18 t2,5{3,3,3,3,3,3,3,3,3} Triruncinated 10-simplex 1108809240 19 t3,5{3,3,3,3,3,3,3,3,3} Quadricantellated 10-simplex 623706930 20 t4,5{3,3,3,3,3,3,3,3,3} Quintitruncated 10-simplex 138602772 21 t0,6{3,3,3,3,3,3,3,3,3} Hexicated 10-simplex 346502310 22 t1,6{3,3,3,3,3,3,3,3,3} Bipentellated 10-simplex 1039506930 23 t2,6{3,3,3,3,3,3,3,3,3} Tristericated 10-simplex 16170011550 24 t3,6{3,3,3,3,3,3,3,3,3} Quadriruncinated 10-simplex 13860011550 25 t0,7{3,3,3,3,3,3,3,3,3} Heptellated 10-simplex 184801320 26 t1,7{3,3,3,3,3,3,3,3,3} Bihexicated 10-simplex 693004620 27 t2,7{3,3,3,3,3,3,3,3,3} Tripentellated 10-simplex 1386009240 28 t0,8{3,3,3,3,3,3,3,3,3} Octellated 10-simplex 5940495 29 t1,8{3,3,3,3,3,3,3,3,3} Biheptellated 10-simplex 277201980 30 t0,9{3,3,3,3,3,3,3,3,3} Ennecated 10-simplex 990110 31 t0,1,2,3,4,5,6,7,8,9{3,3,3,3,3,3,3,3,3} Omnitruncated 10-simplex 19958400039916800 The B10 family There are 1023 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Twelve cases are shown below: ten single-ring (rectified) forms, and two truncations. Bowers-style acronym names are given in parentheses for cross-referencing. # Graph Coxeter-Dynkin diagram Schläfli symbol Name Element counts 9-faces 8-faces 7-faces 6-faces 5-faces 4-faces Cells Faces Edges Vertices 1 t0{4,3,3,3,3,3,3,3,3} 10-cube (deker) 201809603360806413440153601152051201024 2 t0,1{4,3,3,3,3,3,3,3,3} Truncated 10-cube (tade) 51200 10240 3 t1{4,3,3,3,3,3,3,3,3} Rectified 10-cube (rade) 46080 5120 4 t2{4,3,3,3,3,3,3,3,3} Birectified 10-cube (brade) 184320 11520 5 t3{4,3,3,3,3,3,3,3,3} Trirectified 10-cube (trade) 322560 15360 6 t4{4,3,3,3,3,3,3,3,3} Quadrirectified 10-cube (terade) 322560 13440 7 t4{3,3,3,3,3,3,3,3,4} Quadrirectified 10-orthoplex (terake) 201600 8064 8 t3{3,3,3,3,3,3,3,4} Trirectified 10-orthoplex (trake) 80640 3360 9 t2{3,3,3,3,3,3,3,3,4} Birectified 10-orthoplex (brake) 20160 960 10 t1{3,3,3,3,3,3,3,3,4} Rectified 10-orthoplex (rake) 2880 180 11 t0,1{3,3,3,3,3,3,3,3,4} Truncated 10-orthoplex (take) 3060 360 12 t0{3,3,3,3,3,3,3,3,4} 10-orthoplex (ka) 102451201152015360134408064336096018020 The D10 family The D10 family has symmetry of order 1,857,945,600 (10 factorial × 29). This family has 3×256−1=767 Wythoffian uniform polytopes, generated by marking one or more nodes of the D10 Coxeter-Dynkin diagram. Of these, 511 (2×256−1) are repeated from the B10 family and 256 are unique to this family, with 2 listed below. Bowers-style acronym names are given in parentheses for cross-referencing. # Graph Coxeter-Dynkin diagram Schläfli symbol Name Element counts 9-faces 8-faces 7-faces 6-faces 5-faces 4-faces Cells Faces Edges Vertices 1 10-demicube (hede) 532530024000648001155841424641228806144011520512 2 Truncated 10-demicube (thede) 19584023040 Regular and uniform honeycombs There are four fundamental affine Coxeter groups that generate regular and uniform tessellations in 9-space: # Coxeter group Coxeter-Dynkin diagram 1${\tilde {A}}_{9}$[3[10]] 2${\tilde {B}}_{9}$[4,37,4] 3${\tilde {C}}_{9}$h[4,37,4] [4,36,31,1] 4${\tilde {D}}_{9}$q[4,37,4] [31,1,35,31,1] Regular and uniform tessellations include: • Regular 9-hypercubic honeycomb, with symbols {4,37,4}, • Uniform alternated 9-hypercubic honeycomb with symbols h{4,37,4}, Regular and uniform hyperbolic honeycombs There are no compact hyperbolic Coxeter groups of rank 10, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 3 paracompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 9-space as permutations of rings of the Coxeter diagrams. ${\bar {Q}}_{9}$ = [31,1,34,32,1]: ${\bar {S}}_{9}$ = [4,35,32,1]: $E_{10}$ or ${\bar {T}}_{9}$ = [36,2,1]: Three honeycombs from the $E_{10}$ family, generated by end-ringed Coxeter diagrams are: • 621 honeycomb: • 261 honeycomb: • 162 honeycomb: References 1. Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008. • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 • A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910 • H.S.M. Coxeter: • H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954 • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 • Klitzing, Richard. "10D uniform polytopes (polyxenna)". External links • Polytope names • Polytopes of Various Dimensions, Jonathan Bowers • Multi-dimensional Glossary • Glossary for hyperspace, George Olshevsky. Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square p-gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221 Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321 Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421 Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes and compounds	Wikipedia
Uniform 1 k2 polytope In geometry, 1k2 polytope is a uniform polytope in n-dimensions (n = k+4) constructed from the En Coxeter group. The family was named by their Coxeter symbol 1k2 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. It can be named by an extended Schläfli symbol {3,3k,2}. Family members The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5-demicube (demipenteract) in 5-dimensions, and the 4-simplex (5-cell) in 4-dimensions. Each polytope is constructed from 1k-1,2 and (n-1)-demicube facets. Each has a vertex figure of a {31,n-2,2} polytope is a birectified n-simplex, t2{3n}. The sequence ends with k=6 (n=10), as an infinite tessellation of 9-dimensional hyperbolic space. The complete family of 1k2 polytope polytopes are: 1. 5-cell: 102, (5 tetrahedral cells) 2. 112 polytope, (16 5-cell, and 10 16-cell facets) 3. 122 polytope, (54 demipenteract facets) 4. 132 polytope, (56 122 and 126 demihexeract facets) 5. 142 polytope, (240 132 and 2160 demihepteract facets) 6. 152 honeycomb, tessellates Euclidean 8-space (∞ 142 and ∞ demiocteract facets) 7. 162 honeycomb, tessellates hyperbolic 9-space (∞ 152 and ∞ demienneract facets) Elements Gosset 1k2 figures n 1k2 Petrie polygon projection Name Coxeter-Dynkin diagram Facets Elements 1k-1,2 (n-1)-demicube Vertices Edges Faces Cells 4-faces 5-faces 6-faces 7-faces 4 102 120 -- 5 110 5 10 10 5 5 112 121 16 120 10 111 16 80 160 120 26 6 122 122 27 112 27 121 72 720 2160 2160 702 54 7 132 132 56 122 126 131 576 10080 40320 50400 23688 4284 182 8 142 142 240 132 2160 141 17280 483840 2419200 3628800 2298240 725760 106080 2400 9 152 152 (8-space tessellation) ∞ 142 ∞ 151 ∞ 10 162 162 (9-space hyperbolic tessellation) ∞ 152 ∞ 161 ∞ See also • k21 polytope family • 2k1 polytope family References • Alicia Boole Stott Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910 • Stott, A. B. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11, 3-24, 1910. • Alicia Boole Stott, "Geometrical deduction of semiregular from regular polytopes and space fillings," Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, (eerste sectie), Vol. 11, No. 1, pp. 1–24 plus 3 plates, 1910. • Stott, A. B. 1910. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam • Schoute, P. H., Analytical treatment of the polytopes regularly derived from the regular polytopes, Ver. der Koninklijke Akad. van Wetenschappen te Amsterdam (eerstie sectie), vol 11.5, 1913. • H. S. M. Coxeter: Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin, 1940 • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 • H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Berlin, 1985 • H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part III, Mathematische Zeitschrift, Springer, Berlin, 1988 External links • PolyGloss v0.05: Gosset figures (Gossetododecatope) Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square p-gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221 Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321 Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421 Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes and compounds Fundamental convex regular and uniform honeycombs in dimensions 2–9 Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$ E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4 E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6 E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222 E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331 E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521 E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10 E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11 En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21	Wikipedia
Uniform 2 k1 polytope In geometry, 2k1 polytope is a uniform polytope in n dimensions (n = k+4) constructed from the En Coxeter group. The family was named by their Coxeter symbol as 2k1 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence. It can be named by an extended Schläfli symbol {3,3,3k,1}. Family members The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5-orthoplex (pentacross) in 5-dimensions, and the 4-simplex (5-cell) in 4-dimensions. Each polytope is constructed from (n-1)-simplex and 2k-1,1 (n-1)-polytope facets, each has a vertex figure as an (n-1)-demicube, {31,n-2,1}. The sequence ends with k=6 (n=10), as an infinite hyperbolic tessellation of 9-space. The complete family of 2k1 polytope polytopes are: 1. 5-cell: 201, (5 tetrahedra cells) 2. Pentacross: 211, (32 5-cell (201) facets) 3. 221, (72 5-simplex and 27 5-orthoplex (211) facets) 4. 231, (576 6-simplex and 56 221 facets) 5. 241, (17280 7-simplex and 240 231 facets) 6. 251, tessellates Euclidean 8-space (∞ 8-simplex and ∞ 241 facets) 7. 261, tessellates hyperbolic 9-space (∞ 9-simplex and ∞ 251 facets) Elements Gosset 2k1 figures n 2k1 Petrie polygon projection Name Coxeter-Dynkin diagram Facets Elements 2k-1,1 polytope (n-1)-simplex Vertices Edges Faces Cells 4-faces 5-faces 6-faces 7-faces 4 201 5-cell {32,0,1} -- 5 {33} 5 10 10 5 5 211 pentacross {32,1,1} 16 {32,0,1} 16 {34} 10 40 80 80 32 6 221 2 21 polytope {32,2,1} 27 {32,1,1} 72 {35} 27 216 720 1080 648 99 7 231 2 31 polytope {32,3,1} 56 {32,2,1} 576 {36} 126 2016 10080 20160 16128 4788 632 8 241 2 41 polytope {32,4,1} 240 {32,3,1} 17280 {37} 2160 69120 483840 1209600 1209600 544320 144960 17520 9 251 2 51 honeycomb (8-space tessellation) {32,5,1} ∞ {32,4,1} ∞ {38} ∞ 10 261 2 61 honeycomb (9-space tessellation) {32,6,1} ∞ {32,5,1} ∞ {39} ∞ See also • k21 polytope family • 1k2 polytope family References • Alicia Boole Stott Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910 • Stott, A. B. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11, 3-24, 1910. • Alicia Boole Stott, "Geometrical deduction of semiregular from regular polytopes and space fillings," Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, (eerste sectie), Vol. 11, No. 1, pp. 1–24 plus 3 plates, 1910. • Stott, A. B. 1910. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam • Schoute, P. H., Analytical treatment of the polytopes regularly derived from the regular polytopes, Ver. der Koninklijke Akad. van Wetenschappen te Amsterdam (eerstie sectie), vol 11.5, 1913. • H. S. M. Coxeter: Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin, 1940 • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 • H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Berlin, 1985 • H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part III, Mathematische Zeitschrift, Springer, Berlin, 1988 External links • PolyGloss v0.05: Gosset figures (Gossetoctotope) Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square p-gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221 Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321 Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421 Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes and compounds Fundamental convex regular and uniform honeycombs in dimensions 2–9 Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$ E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4 E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6 E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222 E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331 E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521 E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10 E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11 En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21	Wikipedia
Uniform 6-polytope In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes. Graphs of three regular and related uniform polytopes 6-simplex Truncated 6-simplex Rectified 6-simplex Cantellated 6-simplex Runcinated 6-simplex Stericated 6-simplex Pentellated 6-simplex 6-orthoplex Truncated 6-orthoplex Rectified 6-orthoplex Cantellated 6-orthoplex Runcinated 6-orthoplex Stericated 6-orthoplex Cantellated 6-cube Runcinated 6-cube Stericated 6-cube Pentellated 6-cube 6-cube Truncated 6-cube Rectified 6-cube 6-demicube Truncated 6-demicube Cantellated 6-demicube Runcinated 6-demicube Stericated 6-demicube 221 122 Truncated 221 Truncated 122 The complete set of convex uniform 6-polytopes has not been determined, but most can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter-Dynkin diagrams. Each combination of at least one ring on every connected group of nodes in the diagram produces a uniform 6-polytope. The simplest uniform polypeta are regular polytopes: the 6-simplex {3,3,3,3,3}, the 6-cube (hexeract) {4,3,3,3,3}, and the 6-orthoplex (hexacross) {3,3,3,3,4}. History of discovery • Regular polytopes: (convex faces) • 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions. • Convex semiregular polytopes: (Various definitions before Coxeter's uniform category) • 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular polytera) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.[1] • Convex uniform polytopes: • 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes. • Nonregular uniform star polytopes: (similar to the nonconvex uniform polyhedra) • Ongoing: Jonathan Bowers and other researchers search for other non-convex uniform 6-polytopes, with a current count of 41348 known uniform 6-polytopes outside infinite families (convex and non-convex), excluding the prisms of the uniform 5-polytopes. The list is not proven complete.[2][3] Uniform 6-polytopes by fundamental Coxeter groups Uniform 6-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams. There are four fundamental reflective symmetry groups which generate 153 unique uniform 6-polytopes. # Coxeter group Coxeter-Dynkin diagram 1A6[3,3,3,3,3] 2B6[3,3,3,3,4] 3D6[3,3,3,31,1] 4 E6 [32,2,1] [3,32,2] Coxeter-Dynkin diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence. Uniform prismatic families Uniform prism There are 6 categorical uniform prisms based on the uniform 5-polytopes. # Coxeter group Notes 1A5A1[3,3,3,3,2]Prism family based on 5-simplex 2B5A1[4,3,3,3,2]Prism family based on 5-cube 3aD5A1[32,1,1,2]Prism family based on 5-demicube # Coxeter group Notes 4A3I2(p)A1[3,3,2,p,2]Prism family based on tetrahedral-p-gonal duoprisms 5B3I2(p)A1[4,3,2,p,2]Prism family based on cubic-p-gonal duoprisms 6H3I2(p)A1[5,3,2,p,2]Prism family based on dodecahedral-p-gonal duoprisms Uniform duoprism There are 11 categorical uniform duoprismatic families of polytopes based on Cartesian products of lower-dimensional uniform polytopes. Five are formed as the product of a uniform 4-polytope with a regular polygon, and six are formed by the product of two uniform polyhedra: # Coxeter group Notes 1A4I2(p)[3,3,3,2,p]Family based on 5-cell-p-gonal duoprisms. 2B4I2(p)[4,3,3,2,p]Family based on tesseract-p-gonal duoprisms. 3F4I2(p)[3,4,3,2,p]Family based on 24-cell-p-gonal duoprisms. 4H4I2(p)[5,3,3,2,p]Family based on 120-cell-p-gonal duoprisms. 5D4I2(p)[31,1,1,2,p]Family based on demitesseract-p-gonal duoprisms. # Coxeter group Notes 6A32[3,3,2,3,3]Family based on tetrahedral duoprisms. 7A3B3[3,3,2,4,3]Family based on tetrahedral-cubic duoprisms. 8A3H3[3,3,2,5,3]Family based on tetrahedral-dodecahedral duoprisms. 9B32[4,3,2,4,3]Family based on cubic duoprisms. 10B3H3[4,3,2,5,3]Family based on cubic-dodecahedral duoprisms. 11H32[5,3,2,5,3]Family based on dodecahedral duoprisms. Uniform triaprism There is one infinite family of uniform triaprismatic families of polytopes constructed as a Cartesian products of three regular polygons. Each combination of at least one ring on every connected group produces a uniform prismatic 6-polytope. # Coxeter group Notes 1I2(p)I2(q)I2(r)[p,2,q,2,r]Family based on p,q,r-gonal triprisms Enumerating the convex uniform 6-polytopes • Simplex family: A6 [34] - • 35 uniform 6-polytopes as permutations of rings in the group diagram, including one regular: 1. {34} - 6-simplex - • Hypercube/orthoplex family: B6 [4,34] - • 63 uniform 6-polytopes as permutations of rings in the group diagram, including two regular forms: 1. {4,33} — 6-cube (hexeract) - 2. {33,4} — 6-orthoplex, (hexacross) - • Demihypercube D6 family: [33,1,1] - • 47 uniform 6-polytopes (16 unique) as permutations of rings in the group diagram, including: 1. {3,32,1}, 121 6-demicube (demihexeract) - ; also as h{4,33}, 2. {3,3,31,1}, 211 6-orthoplex - , a half symmetry form of . • E6 family: [33,1,1] - • 39 uniform 6-polytopes as permutations of rings in the group diagram, including: 1. {3,3,32,1}, 221 - 2. {3,32,2}, 122 - These fundamental families generate 153 nonprismatic convex uniform polypeta. In addition, there are 57 uniform 6-polytope constructions based on prisms of the uniform 5-polytopes: [3,3,3,3,2], [4,3,3,3,2], [32,1,1,2], excluding the penteract prism as a duplicate of the hexeract. In addition, there are infinitely many uniform 6-polytope based on: 1. Duoprism prism families: [3,3,2,p,2], [4,3,2,p,2], [5,3,2,p,2]. 2. Duoprism families: [3,3,3,2,p], [4,3,3,2,p], [5,3,3,2,p]. 3. Triaprism family: [p,2,q,2,r]. The A6 family Further information: list of A6 polytopes There are 32+4−1=35 forms, derived by marking one or more nodes of the Coxeter-Dynkin diagram. All 35 are enumerated below. They are named by Norman Johnson from the Wythoff construction operations upon regular 6-simplex (heptapeton). Bowers-style acronym names are given in parentheses for cross-referencing. The A6 family has symmetry of order 5040 (7 factorial). The coordinates of uniform 6-polytopes with 6-simplex symmetry can be generated as permutations of simple integers in 7-space, all in hyperplanes with normal vector (1,1,1,1,1,1,1). # Coxeter-Dynkin Johnson naming system Bowers name and (acronym) Base point Element counts 543210 1 6-simplex heptapeton (hop) (0,0,0,0,0,0,1) 7213535217 2 Rectified 6-simplex rectified heptapeton (ril) (0,0,0,0,0,1,1) 146314017510521 3 Truncated 6-simplex truncated heptapeton (til) (0,0,0,0,0,1,2) 146314017512642 4 Birectified 6-simplex birectified heptapeton (bril) (0,0,0,0,1,1,1) 148424535021035 5 Cantellated 6-simplex small rhombated heptapeton (sril) (0,0,0,0,1,1,2) 35210560805525105 6 Bitruncated 6-simplex bitruncated heptapeton (batal) (0,0,0,0,1,2,2) 1484245385315105 7 Cantitruncated 6-simplex great rhombated heptapeton (gril) (0,0,0,0,1,2,3) 35210560805630210 8 Runcinated 6-simplex small prismated heptapeton (spil) (0,0,0,1,1,1,2) 7045513301610840140 9 Bicantellated 6-simplex small birhombated heptapeton (sabril) (0,0,0,1,1,2,2) 7045512951610840140 10 Runcitruncated 6-simplex prismatotruncated heptapeton (patal) (0,0,0,1,1,2,3) 70560182028001890420 11 Tritruncated 6-simplex tetradecapeton (fe) (0,0,0,1,2,2,2) 1484280490420140 12 Runcicantellated 6-simplex prismatorhombated heptapeton (pril) (0,0,0,1,2,2,3) 70455129519601470420 13 Bicantitruncated 6-simplex great birhombated heptapeton (gabril) (0,0,0,1,2,3,3) 4932998015401260420 14 Runcicantitruncated 6-simplex great prismated heptapeton (gapil) (0,0,0,1,2,3,4) 70560182030102520840 15 Stericated 6-simplex small cellated heptapeton (scal) (0,0,1,1,1,1,2) 10570014701400630105 16 Biruncinated 6-simplex small biprismato-tetradecapeton (sibpof) (0,0,1,1,1,2,2) 84714210025201260210 17 Steritruncated 6-simplex cellitruncated heptapeton (catal) (0,0,1,1,1,2,3) 105945294037802100420 18 Stericantellated 6-simplex cellirhombated heptapeton (cral) (0,0,1,1,2,2,3) 1051050346550403150630 19 Biruncitruncated 6-simplex biprismatorhombated heptapeton (bapril) (0,0,1,1,2,3,3) 84714231035702520630 20 Stericantitruncated 6-simplex celligreatorhombated heptapeton (cagral) (0,0,1,1,2,3,4) 10511554410714050401260 21 Steriruncinated 6-simplex celliprismated heptapeton (copal) (0,0,1,2,2,2,3) 105700199526601680420 22 Steriruncitruncated 6-simplex celliprismatotruncated heptapeton (captal) (0,0,1,2,2,3,4) 1059453360567044101260 23 Steriruncicantellated 6-simplex celliprismatorhombated heptapeton (copril) (0,0,1,2,3,3,4) 10510503675588044101260 24 Biruncicantitruncated 6-simplex great biprismato-tetradecapeton (gibpof) (0,0,1,2,3,4,4) 847142520441037801260 25 Steriruncicantitruncated 6-simplex great cellated heptapeton (gacal) (0,0,1,2,3,4,5) 10511554620861075602520 26 Pentellated 6-simplex small teri-tetradecapeton (staff) (0,1,1,1,1,1,2) 12643463049021042 27 Pentitruncated 6-simplex teracellated heptapeton (tocal) (0,1,1,1,1,2,3) 12682617851820945210 28 Penticantellated 6-simplex teriprismated heptapeton (topal) (0,1,1,1,2,2,3) 1261246357043402310420 29 Penticantitruncated 6-simplex terigreatorhombated heptapeton (togral) (0,1,1,1,2,3,4) 1261351409553903360840 30 Pentiruncitruncated 6-simplex tericellirhombated heptapeton (tocral) (0,1,1,2,2,3,4) 12614915565861056701260 31 Pentiruncicantellated 6-simplex teriprismatorhombi-tetradecapeton (taporf) (0,1,1,2,3,3,4) 12615965250756050401260 32 Pentiruncicantitruncated 6-simplex terigreatoprismated heptapeton (tagopal) (0,1,1,2,3,4,5) 126170168251155088202520 33 Pentisteritruncated 6-simplex tericellitrunki-tetradecapeton (tactaf) (0,1,2,2,2,3,4) 1261176378052503360840 34 Pentistericantitruncated 6-simplex tericelligreatorhombated heptapeton (tacogral) (0,1,2,2,3,4,5) 126159665101134088202520 35 Omnitruncated 6-simplex great teri-tetradecapeton (gotaf) (0,1,2,3,4,5,6) 1261806840016800151205040 The B6 family Further information: list of B6 polytopes There are 63 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. The B6 family has symmetry of order 46080 (6 factorial x 26). They are named by Norman Johnson from the Wythoff construction operations upon the regular 6-cube and 6-orthoplex. Bowers names and acronym names are given for cross-referencing. # Coxeter-Dynkin diagram Schläfli symbol Names Element counts 543210 36 t0{3,3,3,3,4}6-orthoplex Hexacontatetrapeton (gee) 641922401606012 37 t1{3,3,3,3,4}Rectified 6-orthoplex Rectified hexacontatetrapeton (rag) 765761200112048060 38 t2{3,3,3,3,4}Birectified 6-orthoplex Birectified hexacontatetrapeton (brag) 76636216028801440160 39 t2{4,3,3,3,3}Birectified 6-cube Birectified hexeract (brox) 76636208032001920240 40 t1{4,3,3,3,3}Rectified 6-cube Rectified hexeract (rax) 7644411201520960192 41 t0{4,3,3,3,3}6-cube Hexeract (ax) 126016024019264 42 t0,1{3,3,3,3,4}Truncated 6-orthoplex Truncated hexacontatetrapeton (tag) 7657612001120540120 43 t0,2{3,3,3,3,4}Cantellated 6-orthoplex Small rhombated hexacontatetrapeton (srog) 1361656504064003360480 44 t1,2{3,3,3,3,4}Bitruncated 6-orthoplex Bitruncated hexacontatetrapeton (botag) 1920480 45 t0,3{3,3,3,3,4}Runcinated 6-orthoplex Small prismated hexacontatetrapeton (spog) 7200960 46 t1,3{3,3,3,3,4}Bicantellated 6-orthoplex Small birhombated hexacontatetrapeton (siborg) 86401440 47 t2,3{4,3,3,3,3}Tritruncated 6-cube Hexeractihexacontitetrapeton (xog) 3360960 48 t0,4{3,3,3,3,4}Stericated 6-orthoplex Small cellated hexacontatetrapeton (scag) 5760960 49 t1,4{4,3,3,3,3}Biruncinated 6-cube Small biprismato-hexeractihexacontitetrapeton (sobpoxog) 115201920 50 t1,3{4,3,3,3,3}Bicantellated 6-cube Small birhombated hexeract (saborx) 96001920 51 t1,2{4,3,3,3,3}Bitruncated 6-cube Bitruncated hexeract (botox) 2880960 52 t0,5{4,3,3,3,3}Pentellated 6-cube Small teri-hexeractihexacontitetrapeton (stoxog) 1920384 53 t0,4{4,3,3,3,3}Stericated 6-cube Small cellated hexeract (scox) 5760960 54 t0,3{4,3,3,3,3}Runcinated 6-cube Small prismated hexeract (spox) 76801280 55 t0,2{4,3,3,3,3}Cantellated 6-cube Small rhombated hexeract (srox) 4800960 56 t0,1{4,3,3,3,3}Truncated 6-cube Truncated hexeract (tox) 76444112015201152384 57 t0,1,2{3,3,3,3,4}Cantitruncated 6-orthoplex Great rhombated hexacontatetrapeton (grog) 3840960 58 t0,1,3{3,3,3,3,4}Runcitruncated 6-orthoplex Prismatotruncated hexacontatetrapeton (potag) 158402880 59 t0,2,3{3,3,3,3,4}Runcicantellated 6-orthoplex Prismatorhombated hexacontatetrapeton (prog) 115202880 60 t1,2,3{3,3,3,3,4}Bicantitruncated 6-orthoplex Great birhombated hexacontatetrapeton (gaborg) 100802880 61 t0,1,4{3,3,3,3,4}Steritruncated 6-orthoplex Cellitruncated hexacontatetrapeton (catog) 192003840 62 t0,2,4{3,3,3,3,4}Stericantellated 6-orthoplex Cellirhombated hexacontatetrapeton (crag) 288005760 63 t1,2,4{3,3,3,3,4}Biruncitruncated 6-orthoplex Biprismatotruncated hexacontatetrapeton (boprax) 230405760 64 t0,3,4{3,3,3,3,4}Steriruncinated 6-orthoplex Celliprismated hexacontatetrapeton (copog) 153603840 65 t1,2,4{4,3,3,3,3}Biruncitruncated 6-cube Biprismatotruncated hexeract (boprag) 230405760 66 t1,2,3{4,3,3,3,3}Bicantitruncated 6-cube Great birhombated hexeract (gaborx) 115203840 67 t0,1,5{3,3,3,3,4}Pentitruncated 6-orthoplex Teritruncated hexacontatetrapeton (tacox) 86401920 68 t0,2,5{3,3,3,3,4}Penticantellated 6-orthoplex Terirhombated hexacontatetrapeton (tapox) 211203840 69 t0,3,4{4,3,3,3,3}Steriruncinated 6-cube Celliprismated hexeract (copox) 153603840 70 t0,2,5{4,3,3,3,3}Penticantellated 6-cube Terirhombated hexeract (topag) 211203840 71 t0,2,4{4,3,3,3,3}Stericantellated 6-cube Cellirhombated hexeract (crax) 288005760 72 t0,2,3{4,3,3,3,3}Runcicantellated 6-cube Prismatorhombated hexeract (prox) 134403840 73 t0,1,5{4,3,3,3,3}Pentitruncated 6-cube Teritruncated hexeract (tacog) 86401920 74 t0,1,4{4,3,3,3,3}Steritruncated 6-cube Cellitruncated hexeract (catax) 192003840 75 t0,1,3{4,3,3,3,3}Runcitruncated 6-cube Prismatotruncated hexeract (potax) 172803840 76 t0,1,2{4,3,3,3,3}Cantitruncated 6-cube Great rhombated hexeract (grox) 57601920 77 t0,1,2,3{3,3,3,3,4}Runcicantitruncated 6-orthoplex Great prismated hexacontatetrapeton (gopog) 201605760 78 t0,1,2,4{3,3,3,3,4}Stericantitruncated 6-orthoplex Celligreatorhombated hexacontatetrapeton (cagorg) 4608011520 79 t0,1,3,4{3,3,3,3,4}Steriruncitruncated 6-orthoplex Celliprismatotruncated hexacontatetrapeton (captog) 4032011520 80 t0,2,3,4{3,3,3,3,4}Steriruncicantellated 6-orthoplex Celliprismatorhombated hexacontatetrapeton (coprag) 4032011520 81 t1,2,3,4{4,3,3,3,3}Biruncicantitruncated 6-cube Great biprismato-hexeractihexacontitetrapeton (gobpoxog) 3456011520 82 t0,1,2,5{3,3,3,3,4}Penticantitruncated 6-orthoplex Terigreatorhombated hexacontatetrapeton (togrig) 307207680 83 t0,1,3,5{3,3,3,3,4}Pentiruncitruncated 6-orthoplex Teriprismatotruncated hexacontatetrapeton (tocrax) 5184011520 84 t0,2,3,5{4,3,3,3,3}Pentiruncicantellated 6-cube Teriprismatorhombi-hexeractihexacontitetrapeton (tiprixog) 4608011520 85 t0,2,3,4{4,3,3,3,3}Steriruncicantellated 6-cube Celliprismatorhombated hexeract (coprix) 4032011520 86 t0,1,4,5{4,3,3,3,3}Pentisteritruncated 6-cube Tericelli-hexeractihexacontitetrapeton (tactaxog) 307207680 87 t0,1,3,5{4,3,3,3,3}Pentiruncitruncated 6-cube Teriprismatotruncated hexeract (tocrag) 5184011520 88 t0,1,3,4{4,3,3,3,3}Steriruncitruncated 6-cube Celliprismatotruncated hexeract (captix) 4032011520 89 t0,1,2,5{4,3,3,3,3}Penticantitruncated 6-cube Terigreatorhombated hexeract (togrix) 307207680 90 t0,1,2,4{4,3,3,3,3}Stericantitruncated 6-cube Celligreatorhombated hexeract (cagorx) 4608011520 91 t0,1,2,3{4,3,3,3,3}Runcicantitruncated 6-cube Great prismated hexeract (gippox) 230407680 92 t0,1,2,3,4{3,3,3,3,4}Steriruncicantitruncated 6-orthoplex Great cellated hexacontatetrapeton (gocog) 6912023040 93 t0,1,2,3,5{3,3,3,3,4}Pentiruncicantitruncated 6-orthoplex Terigreatoprismated hexacontatetrapeton (tagpog) 8064023040 94 t0,1,2,4,5{3,3,3,3,4}Pentistericantitruncated 6-orthoplex Tericelligreatorhombated hexacontatetrapeton (tecagorg) 8064023040 95 t0,1,2,4,5{4,3,3,3,3}Pentistericantitruncated 6-cube Tericelligreatorhombated hexeract (tocagrax) 8064023040 96 t0,1,2,3,5{4,3,3,3,3}Pentiruncicantitruncated 6-cube Terigreatoprismated hexeract (tagpox) 8064023040 97 t0,1,2,3,4{4,3,3,3,3}Steriruncicantitruncated 6-cube Great cellated hexeract (gocax) 6912023040 98 t0,1,2,3,4,5{4,3,3,3,3}Omnitruncated 6-cube Great teri-hexeractihexacontitetrapeton (gotaxog) 13824046080 The D6 family Further information: list of D6 polytopes The D6 family has symmetry of order 23040 (6 factorial x 25). This family has 3×16−1=47 Wythoffian uniform polytopes, generated by marking one or more nodes of the D6 Coxeter-Dynkin diagram. Of these, 31 (2×16−1) are repeated from the B6 family and 16 are unique to this family. The 16 unique forms are enumerated below. Bowers-style acronym names are given for cross-referencing. # Coxeter diagram Names Base point (Alternately signed) Element counts Circumrad 543210 99 = 6-demicube Hemihexeract (hax) (1,1,1,1,1,1)44252640640240320.8660254 100 = Cantic 6-cube Truncated hemihexeract (thax) (1,1,3,3,3,3)766362080320021604802.1794493 101 = Runcic 6-cube Small rhombated hemihexeract (sirhax) (1,1,1,3,3,3)38406401.9364916 102 = Steric 6-cube Small prismated hemihexeract (sophax) (1,1,1,1,3,3)33604801.6583123 103 = Pentic 6-cube Small cellated demihexeract (sochax) (1,1,1,1,1,3)14401921.3228756 104 = Runcicantic 6-cube Great rhombated hemihexeract (girhax) (1,1,3,5,5,5)576019203.2787192 105 = Stericantic 6-cube Prismatotruncated hemihexeract (pithax) (1,1,3,3,5,5)1296028802.95804 106 = Steriruncic 6-cube Prismatorhombated hemihexeract (prohax) (1,1,1,3,5,5)768019202.7838821 107 = Penticantic 6-cube Cellitruncated hemihexeract (cathix) (1,1,3,3,3,5)960019202.5980761 108 = Pentiruncic 6-cube Cellirhombated hemihexeract (crohax) (1,1,1,3,3,5)1056019202.3979158 109 = Pentisteric 6-cube Celliprismated hemihexeract (cophix) (1,1,1,1,3,5)52809602.1794496 110 = Steriruncicantic 6-cube Great prismated hemihexeract (gophax) (1,1,3,5,7,7)1728057604.0926762 111 = Pentiruncicantic 6-cube Celligreatorhombated hemihexeract (cagrohax) (1,1,3,5,5,7)2016057603.7080991 112 = Pentistericantic 6-cube Celliprismatotruncated hemihexeract (capthix) (1,1,3,3,5,7)2304057603.4278274 113 = Pentisteriruncic 6-cube Celliprismatorhombated hemihexeract (caprohax) (1,1,1,3,5,7)1536038403.2787192 114 = Pentisteriruncicantic 6-cube Great cellated hemihexeract (gochax) (1,1,3,5,7,9)34560115204.5552168 The E6 family Further information: list of E6 polytopes There are 39 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Bowers-style acronym names are given for cross-referencing. The E6 family has symmetry of order 51,840. # Coxeter diagram Names Element counts 5-faces 4-faces Cells Faces Edges Vertices 115221 Icosiheptaheptacontidipeton (jak) 99648108072021627 116Rectified 221 Rectified icosiheptaheptacontidipeton (rojak) 1261350432050402160216 117Truncated 221 Truncated icosiheptaheptacontidipeton (tojak) 1261350432050402376432 118Cantellated 221 Small rhombated icosiheptaheptacontidipeton (sirjak) 34239421512024480151202160 119Runcinated 221 Small demiprismated icosiheptaheptacontidipeton (shopjak) 3424662162001944086401080 120Demified icosiheptaheptacontidipeton (hejak)3422430720079203240432 121Bitruncated 221 Bitruncated icosiheptaheptacontidipeton (botajik) 2160 122Demirectified icosiheptaheptacontidipeton (harjak)1080 123Cantitruncated 221 Great rhombated icosiheptaheptacontidipeton (girjak) 4320 124Runcitruncated 221 Demiprismatotruncated icosiheptaheptacontidipeton (hopitjak) 4320 125Steritruncated 221 Cellitruncated icosiheptaheptacontidipeton (catjak) 2160 126Demitruncated icosiheptaheptacontidipeton (hotjak)2160 127Runcicantellated 221 Demiprismatorhombated icosiheptaheptacontidipeton (haprojak) 6480 128Small demirhombated icosiheptaheptacontidipeton (shorjak)4320 129Small prismated icosiheptaheptacontidipeton (spojak)4320 130Tritruncated icosiheptaheptacontidipeton (titajak)4320 131Runcicantitruncated 221 Great demiprismated icosiheptaheptacontidipeton (ghopjak) 12960 132Stericantitruncated 221 Celligreatorhombated icosiheptaheptacontidipeton (cograjik) 12960 133Great demirhombated icosiheptaheptacontidipeton (ghorjak)8640 134Prismatotruncated icosiheptaheptacontidipeton (potjak)12960 135Demicellitruncated icosiheptaheptacontidipeton (hictijik)8640 136Prismatorhombated icosiheptaheptacontidipeton (projak)12960 137Great prismated icosiheptaheptacontidipeton (gapjak)25920 138Demicelligreatorhombated icosiheptaheptacontidipeton (hocgarjik)25920 # Coxeter diagram Names Element counts 5-faces 4-faces Cells Faces Edges Vertices 139 = 122 Pentacontatetrapeton (mo) 547022160216072072 140 = Rectified 122 Rectified pentacontatetrapeton (ram) 12615666480108006480720 141 = Birectified 122 Birectified pentacontatetrapeton (barm) 12622861080019440129602160 142 = Trirectified 122 Trirectified pentacontatetrapeton (trim) 5584608864064802160270 143 = Truncated 122 Truncated pentacontatetrapeton (tim) 136801440 144 = Bitruncated 122 Bitruncated pentacontatetrapeton (bitem) 6480 145 = Tritruncated 122 Tritruncated pentacontatetrapeton (titam) 8640 146 = Cantellated 122 Small rhombated pentacontatetrapeton (sram) 6480 147 = Cantitruncated 122 Great rhombated pentacontatetrapeton (gram) 12960 148 = Runcinated 122 Small prismated pentacontatetrapeton (spam) 2160 149 = Bicantellated 122 Small birhombated pentacontatetrapeton (sabrim) 6480 150 = Bicantitruncated 122 Great birhombated pentacontatetrapeton (gabrim) 12960 151 = Runcitruncated 122 Prismatotruncated pentacontatetrapeton (patom) 12960 152 = Runcicantellated 122 Prismatorhombated pentacontatetrapeton (prom) 25920 153 = Omnitruncated 122 Great prismated pentacontatetrapeton (gopam) 51840 Triaprisms Uniform triaprisms, {p}×{q}×{r}, form an infinite class for all integers p,q,r>2. {4}×{4}×{4} makes a lower symmetry form of the 6-cube. The extended f-vector is (p,p,1)(q,q,1)(r,r,1)=(pqr,3pqr,3pqr+pq+pr+qr,3p(p+1),3p,1). Coxeter diagram Names Element counts 5-faces 4-faces Cells Faces Edges Vertices {p}×{q}×{r} [4]p+q+rpq+pr+qr+p+q+rpqr+2(pq+pr+qr)3pqr+pq+pr+qr3pqrpqr {p}×{p}×{p}3p3p(p+1)p2(p+6)3p2(p+1)3p3p3 {3}×{3}×{3} (trittip)93681998127 {4}×{4}×{4} = 6-cube126016024019264 Non-Wythoffian 6-polytopes In 6 dimensions and above, there are an infinite amount of non-Wythoffian convex uniform polytopes: the Cartesian product of the grand antiprism in 4 dimensions and any regular polygon in 2 dimensions. It is not yet proven whether or not there are more. Regular and uniform honeycombs There are four fundamental affine Coxeter groups and 27 prismatic groups that generate regular and uniform tessellations in 5-space: # Coxeter group Coxeter diagram Forms 1${\tilde {A}}_{5}$[3[6]]12 2${\tilde {C}}_{5}$[4,33,4]35 3${\tilde {B}}_{5}$[4,3,31,1] [4,33,4,1+] 47 (16 new) 4${\tilde {D}}_{5}$[31,1,3,31,1] [1+,4,33,4,1+] 20 (3 new) Regular and uniform honeycombs include: • ${\tilde {A}}_{5}$ There are 12 unique uniform honeycombs, including: • 5-simplex honeycomb • Truncated 5-simplex honeycomb • Omnitruncated 5-simplex honeycomb • ${\tilde {C}}_{5}$ There are 35 uniform honeycombs, including: • Regular hypercube honeycomb of Euclidean 5-space, the 5-cube honeycomb, with symbols {4,33,4}, = • ${\tilde {B}}_{5}$ There are 47 uniform honeycombs, 16 new, including: • The uniform alternated hypercube honeycomb, 5-demicubic honeycomb, with symbols h{4,33,4}, = = • ${\tilde {D}}_{5}$, [31,1,3,31,1]: There are 20 unique ringed permutations, and 3 new ones. Coxeter calls the first one a quarter 5-cubic honeycomb, with symbols q{4,33,4}, = . The other two new ones are = , = . Prismatic groups # Coxeter group Coxeter-Dynkin diagram 1${\tilde {A}}_{4}$x${\tilde {I}}_{1}$[3[5],2,∞] 2${\tilde {B}}_{4}$x${\tilde {I}}_{1}$[4,3,31,1,2,∞] 3${\tilde {C}}_{4}$x${\tilde {I}}_{1}$[4,3,3,4,2,∞] 4${\tilde {D}}_{4}$x${\tilde {I}}_{1}$[31,1,1,1,2,∞] 5${\tilde {F}}_{4}$x${\tilde {I}}_{1}$[3,4,3,3,2,∞] 6${\tilde {C}}_{3}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$[4,3,4,2,∞,2,∞] 7${\tilde {B}}_{3}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$[4,31,1,2,∞,2,∞] 8${\tilde {A}}_{3}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$[3[4],2,∞,2,∞] 9${\tilde {C}}_{2}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$[4,4,2,∞,2,∞,2,∞] 10${\tilde {H}}_{2}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$[6,3,2,∞,2,∞,2,∞] 11${\tilde {A}}_{2}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$[3[3],2,∞,2,∞,2,∞] 12${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$[∞,2,∞,2,∞,2,∞,2,∞] 13${\tilde {A}}_{2}$x${\tilde {A}}_{2}$x${\tilde {I}}_{1}$[3[3],2,3[3],2,∞] 14${\tilde {A}}_{2}$x${\tilde {B}}_{2}$x${\tilde {I}}_{1}$[3[3],2,4,4,2,∞] 15${\tilde {A}}_{2}$x${\tilde {G}}_{2}$x${\tilde {I}}_{1}$[3[3],2,6,3,2,∞] 16${\tilde {B}}_{2}$x${\tilde {B}}_{2}$x${\tilde {I}}_{1}$[4,4,2,4,4,2,∞] 17${\tilde {B}}_{2}$x${\tilde {G}}_{2}$x${\tilde {I}}_{1}$[4,4,2,6,3,2,∞] 18${\tilde {G}}_{2}$x${\tilde {G}}_{2}$x${\tilde {I}}_{1}$[6,3,2,6,3,2,∞] 19${\tilde {A}}_{3}$x${\tilde {A}}_{2}$[3[4],2,3[3]] 20${\tilde {B}}_{3}$x${\tilde {A}}_{2}$[4,31,1,2,3[3]] 21${\tilde {C}}_{3}$x${\tilde {A}}_{2}$[4,3,4,2,3[3]] 22${\tilde {A}}_{3}$x${\tilde {B}}_{2}$[3[4],2,4,4] 23${\tilde {B}}_{3}$x${\tilde {B}}_{2}$[4,31,1,2,4,4] 24${\tilde {C}}_{3}$x${\tilde {B}}_{2}$[4,3,4,2,4,4] 25${\tilde {A}}_{3}$x${\tilde {G}}_{2}$[3[4],2,6,3] 26${\tilde {B}}_{3}$x${\tilde {G}}_{2}$[4,31,1,2,6,3] 27${\tilde {C}}_{3}$x${\tilde {G}}_{2}$[4,3,4,2,6,3] Regular and uniform hyperbolic honeycombs There are no compact hyperbolic Coxeter groups of rank 6, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 12 paracompact hyperbolic Coxeter groups of rank 6, each generating uniform honeycombs in 5-space as permutations of rings of the Coxeter diagrams. Hyperbolic paracompact groups ${\bar {P}}_{5}$ = [3,3[5]]: ${\widehat {AU}}_{5}$ = [(3,3,3,3,3,4)]: ${\widehat {AR}}_{5}$ = [(3,3,4,3,3,4)]: ${\bar {S}}_{5}$ = [4,3,32,1]: ${\bar {O}}_{5}$ = [3,4,31,1]: ${\bar {N}}_{5}$ = [3,(3,4)1,1]: ${\bar {U}}_{5}$ = [3,3,3,4,3]: ${\bar {X}}_{5}$ = [3,3,4,3,3]: ${\bar {R}}_{5}$ = [3,4,3,3,4]: ${\bar {Q}}_{5}$ = [32,1,1,1]: ${\bar {M}}_{5}$ = [4,3,31,1,1]: ${\bar {L}}_{5}$ = [31,1,1,1,1]: Notes on the Wythoff construction for the uniform 6-polytopes Construction of the reflective 6-dimensional uniform polytopes are done through a Wythoff construction process, and represented through a Coxeter-Dynkin diagram, where each node represents a mirror. Nodes are ringed to imply which mirrors are active. The full set of uniform polytopes generated are based on the unique permutations of ringed nodes. Uniform 6-polytopes are named in relation to the regular polytopes in each family. Some families have two regular constructors and thus may have two ways of naming them. Here's the primary operators available for constructing and naming the uniform 6-polytopes. The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity. Operation Extended Schläfli symbol Coxeter- Dynkin diagram Description Parent t0{p,q,r,s,t} Any regular 6-polytope Rectified t1{p,q,r,s,t} The edges are fully truncated into single points. The 6-polytope now has the combined faces of the parent and dual. Birectified t2{p,q,r,s,t} Birectification reduces cells to their duals. Truncated t0,1{p,q,r,s,t} Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated 6-polytope. The 6-polytope has its original faces doubled in sides, and contains the faces of the dual. Bitruncated t1,2{p,q,r,s,t} Bitrunction transforms cells to their dual truncation. Tritruncated t2,3{p,q,r,s,t} Tritruncation transforms 4-faces to their dual truncation. Cantellated t0,2{p,q,r,s,t} In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms. Bicantellated t1,3{p,q,r,s,t} In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms. Runcinated t0,3{p,q,r,s,t} Runcination reduces cells and creates new cells at the vertices and edges. Biruncinated t1,4{p,q,r,s,t} Runcination reduces cells and creates new cells at the vertices and edges. Stericated t0,4{p,q,r,s,t} Sterication reduces 4-faces and creates new 4-faces at the vertices, edges, and faces in the gaps. Pentellated t0,5{p,q,r,s,t} Pentellation reduces 5-faces and creates new 5-faces at the vertices, edges, faces, and cells in the gaps. (expansion operation for polypeta) Omnitruncated t0,1,2,3,4,5{p,q,r,s,t} All five operators, truncation, cantellation, runcination, sterication, and pentellation are applied. See also • List of regular polytopes#Higher dimensions Notes 1. T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 2. Uniform Polypeta, Jonathan Bowers 3. Uniform polytope 4. "N,m,k-tip". References • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 • A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910 • H.S.M. Coxeter: • H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954 • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 • Klitzing, Richard. "6D uniform polytopes (polypeta)". • Klitzing, Richard. "Uniform polytopes truncation operators". External links • Polytope names • Polytopes of Various Dimensions, Jonathan Bowers • Multi-dimensional Glossary • Glossary for hyperspace, George Olshevsky. Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square p-gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221 Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321 Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421 Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes and compounds Fundamental convex regular and uniform honeycombs in dimensions 2–9 Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$ E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4 E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6 E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222 E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331 E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521 E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10 E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11 En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21	Wikipedia
Uniform integrability In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. Measure-theoretic definition Uniform integrability is an extension to the notion of a family of functions being dominated in $L_{1}$ which is central in dominated convergence. Several textbooks on real analysis and measure theory use the following definition:[1][2] Definition A: Let $(X,{\mathfrak {M}},\mu )$ be a positive measure space. A set $\Phi \subset L^{1}(\mu )$ is called uniformly integrable if $\sup _{f\in \Phi }\\|f\\|_{L_{1}(\mu )}<\infty $, and to each $\varepsilon >0$ there corresponds a $\delta >0$ such that $\int _{E}\|f\|\,d\mu <\varepsilon $ whenever $f\in \Phi $ and $\mu (E)<\delta .$ Definition A is rather restrictive for infinite measure spaces. A more general definition[3] of uniform integrability that works well in general measures spaces was introduced by G. A. Hunt. Definition H: Let $(X,{\mathfrak {M}},\mu )$ be a positive measure space. A set $\Phi \subset L^{1}(\mu )$ is called uniformly integrable if and only if $\inf _{g\in L_{+}^{1}(\mu )}\sup _{f\in \Phi }\int _{\{\|f\|>g\}}\|f\|\,d\mu =0$ where $L_{+}^{1}(\mu )=\{g\in L^{1}(\mu ):g\geq 0\}$. For finite measure spaces the following result[4] follows from Definition H: Theorem 1: If $(X,{\mathfrak {M}},\mu )$ is a (positive) finite measure space, then a set $\Phi \subset L^{1}(\mu )$ is uniformly integrable if and only if $\inf _{a\geq 0}\sup _{f\in \Phi }\int _{\{\|f\|>a\}}\|f\|\,d\mu =0$ Many textbooks in probability present Theorem 1 as the definition of uniform integrability in Probability spaces. When the space $(X,{\mathfrak {M}},\mu )$ is $\sigma $-finite, Definition H yields the following equivalency: Theorem 2: Let $(X,{\mathfrak {M}},\mu )$ be a $\sigma $-finite measure space, and $h\in L^{1}(\mu )$ be such that $h>0$ almost surely. A set $\Phi \subset L^{1}(\mu )$ is uniformly integrable if and only if $\sup _{f\in \Phi }\\|f\\|_{L_{1}(\mu )}<\infty $, and for any $\varepsilon >0$, there exits $\delta >0$ such that $\sup _{f\in \Phi }\int _{A}\|f\|\,d\mu <\varepsilon $ whenever $\int _{A}h\,d\mu <\delta $. In particular, the equivalence of Definitions A and H for finite measures follows immediately from Theorem 2; for this case, the statement in Definition A is obtained by taking $h\equiv 1$ in Theorem 2. Probability definition In the theory of probability, Definition A or the statement of Theorem 1 are often presented as definitions of uniform integrability using the notation expectation of random variables.,[5][6][7] that is, 1. A class ${\mathcal {C}}$ of random variables is called uniformly integrable if: • There exists a finite $M$ such that, for every $X$ in ${\mathcal {C}}$, $\operatorname {E} (\|X\|)\leq M$ and • For every $\varepsilon >0$ there exists $\delta >0$ such that, for every measurable $A$ such that $P(A)\leq \delta $ and every $X$ in ${\mathcal {C}}$, $\operatorname {E} (\|X\|I_{A})\leq \varepsilon $. or alternatively 2. A class ${\mathcal {C}}$ of random variables is called uniformly integrable (UI) if for every $\varepsilon >0$ there exists $K\in [0,\infty )$ such that $\operatorname {E} (\|X\|I_{\|X\|\geq K})\leq \varepsilon \ {\text{ for all }}X\in {\mathcal {C}}$, where $I_{\|X\|\geq K}$ is the indicator function $I_{\|X\|\geq K}={\begin{cases}1&{\text{if }}\|X\|\geq K,\\0&{\text{if }}\|X\|<K.\end{cases}}$. Tightness and uniform integrability One consequence of uniformly integrability of a class ${\mathcal {C}}$ of random variables is that family of laws or distributions $\{P\circ \|X\|^{-1}(\cdot ):X\in {\mathcal {C}}\}$ is tight. That is, for each $\delta >0$, there exists $a>0$ such that $P(\|X\|>a)\leq \delta $ for all $X\in {\mathcal {C}}$.[8] This however, does not mean that the family of measures ${\mathcal {V}}_{\mathcal {C}}:={\Big \{}\mu _{X}:A\mapsto \int _{A}\|X\|\,dP,\,X\in {\mathcal {C}}{\Big \}}$ is tight. (In any case, tightness would require a topology on $\Omega $ in order to be defined.) Uniform absolute continuity There is another notion of uniformity, slightly different than uniform integrability, which also has many applications in probability and measure theory, and which does not require random variables to have a finite integral[9] Definition: Suppose $(\Omega ,{\mathcal {F}},P)$ is a probability space. A classed ${\mathcal {C}}$ of random variables is uniformly absolutely continuous with respect to $P$ if for any $\varepsilon >0$, there is $\delta >0$ such that $E[\|X\|I_{A}]<\varepsilon $ whenever $P(A)<\delta $. It is equivalent to uniform integrability if the measure is finite and has no atoms. The term "uniform absolute continuity" is not standard, but is used by some authors.[10][11] Related corollaries The following results apply to the probabilistic definition.[12] • Definition 1 could be rewritten by taking the limits as $\lim _{K\to \infty }\sup _{X\in {\mathcal {C}}}\operatorname {E} (\|X\|\,I_{\|X\|\geq K})=0.$ • A non-UI sequence. Let $\Omega =[0,1]\subset \mathbb {R} $, and define $X_{n}(\omega )={\begin{cases}n,&\omega \in (0,1/n),\\0,&{\text{otherwise.}}\end{cases}}$ Clearly $X_{n}\in L^{1}$, and indeed $\operatorname {E} (\|X_{n}\|)=1\ ,$ for all n. However, $\operatorname {E} (\|X_{n}\|I_{\{\|X_{n}\|\geq K\}})=1\ {\text{ for all }}n\geq K,$ and comparing with definition 1, it is seen that the sequence is not uniformly integrable. • By using Definition 2 in the above example, it can be seen that the first clause is satisfied as $L^{1}$ norm of all $X_{n}$s are 1 i.e., bounded. But the second clause does not hold as given any $\delta $ positive, there is an interval $(0,1/n)$ with measure less than $\delta $ and $E[\|X_{m}\|:(0,1/n)]=1$ for all $m\geq n$. • If $X$ is a UI random variable, by splitting $\operatorname {E} (\|X\|)=\operatorname {E} (\|X\|I_{\{\|X\|\geq K\}})+\operatorname {E} (\|X\|I_{\{\|X\|<K\}})$ and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in $L^{1}$. • If any sequence of random variables $X_{n}$ is dominated by an integrable, non-negative $Y$: that is, for all ω and n, $\|X_{n}(\omega )\|\leq Y(\omega ),\ Y(\omega )\geq 0,\ \operatorname {E} (Y)<\infty ,$ then the class ${\mathcal {C}}$ of random variables $\{X_{n}\}$ is uniformly integrable. • A class of random variables bounded in $L^{p}$ ($p>1$) is uniformly integrable. Relevant theorems In the following we use the probabilistic framework, but regardless of the finiteness of the measure, by adding the boundedness condition on the chosen subset of $L^{1}(\mu )$. • Dunford–Pettis theorem[13][14] A class of random variables $X_{n}\subset L^{1}(\mu )$ is uniformly integrable if and only if it is relatively compact for the weak topology $\sigma (L^{1},L^{\infty })$. • de la Vallée-Poussin theorem[15][16] The family $\{X_{\alpha }\}_{\alpha \in \mathrm {A} }\subset L^{1}(\mu )$ is uniformly integrable if and only if there exists a non-negative increasing convex function $G(t)$ such that $\lim _{t\to \infty }{\frac {G(t)}{t}}=\infty {\text{ and }}\sup _{\alpha }\operatorname {E} (G(\|X_{\alpha }\|))<\infty .$ Relation to convergence of random variables Main article: Convergence of random variables A sequence $\{X_{n}\}$ converges to $X$ in the $L_{1}$ norm if and only if it converges in measure to $X$ and it is uniformly integrable. In probability terms, a sequence of random variables converging in probability also converge in the mean if and only if they are uniformly integrable.[17] This is a generalization of Lebesgue's dominated convergence theorem, see Vitali convergence theorem. Citations 1. Rudin, Walter (1987). Real and Complex Analysis (3 ed.). Singapore: McGraw–Hill Book Co. p. 133. ISBN 0-07-054234-1. 2. Royden, H.L. & Fitzpatrick, P.M. (2010). Real Analysis (4 ed.). Boston: Prentice Hall. p. 93. ISBN 978-0-13-143747-0. 3. Hunt, G. A. (1966). Martingales et Processus de Markov. Paris: Dunod. p. 254. 4. Klenke, A. (2008). Probability Theory: A Comprehensive Course. Berlin: Springer Verlag. pp. 134–137. ISBN 978-1-84800-047-6. 5. Williams, David (1997). Probability with Martingales (Repr. ed.). Cambridge: Cambridge Univ. Press. pp. 126–132. ISBN 978-0-521-40605-5. 6. Gut, Allan (2005). Probability: A Graduate Course. Springer. pp. 214–218. ISBN 0-387-22833-0. 7. Bass, Richard F. (2011). Stochastic Processes. Cambridge: Cambridge University Press. pp. 356–357. ISBN 978-1-107-00800-7. 8. Gut 2005, p. 236. 9. Bass 2011, p. 356. sfn error: no target: CITEREFBass2011 (help) 10. Benedetto, J. J. (1976). Real Variable and Integration. Stuttgart: B. G. Teubner. p. 89. ISBN 3-519-02209-5. 11. Burrill, C. W. (1972). Measure, Integration, and Probability. McGraw-Hill. p. 180. ISBN 0-07-009223-0. 12. Gut 2005, pp. 215–216. 13. Dunford, Nelson (1938). "Uniformity in linear spaces". Transactions of the American Mathematical Society. 44 (2): 305–356. doi:10.1090/S0002-9947-1938-1501971-X. ISSN 0002-9947. 14. Dunford, Nelson (1939). "A mean ergodic theorem". Duke Mathematical Journal. 5 (3): 635–646. doi:10.1215/S0012-7094-39-00552-1. ISSN 0012-7094. 15. Meyer, P.A. (1966). Probability and Potentials, Blaisdell Publishing Co, N. Y. (p.19, Theorem T22). 16. Poussin, C. De La Vallee (1915). "Sur L'Integrale de Lebesgue". Transactions of the American Mathematical Society. 16 (4): 435–501. doi:10.2307/1988879. hdl:10338.dmlcz/127627. JSTOR 1988879. 17. Bogachev, Vladimir I. (2007). Measure Theory Volume I. Berlin Heidelberg: Springer-Verlag. p. 268. doi:10.1007/978-3-540-34514-5_4. ISBN 978-3-540-34513-8. References • Shiryaev, A.N. (1995). Probability (2 ed.). New York: Springer-Verlag. pp. 187–188. ISBN 978-0-387-94549-1. • Diestel, J. and Uhl, J. (1977). Vector measures, Mathematical Surveys 15, American Mathematical Society, Providence, RI ISBN 978-0-8218-1515-1	Wikipedia
Uniform absolute-convergence In mathematics, uniform absolute-convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is changed. Motivation A convergent series of numbers can often be reordered in such a way that the new series diverges. This is not possible for series of nonnegative numbers, however, so the notion of absolute-convergence precludes this phenomenon. When dealing with uniformly convergent series of functions, the same phenomenon occurs: the series can potentially be reordered into a non-uniformly convergent series, or a series which does not even converge pointwise. This is impossible for series of nonnegative functions, so the notion of uniform absolute-convergence can be used to rule out these possibilities. Definition Given a set X and functions $f_{n}:X\to \mathbb {C} $ (or to any normed vector space), the series $\sum _{n=0}^{\infty }f_{n}(x)$ is called uniformly absolutely-convergent if the series of nonnegative functions $\sum _{n=0}^{\infty }\|f_{n}(x)\|$ is uniformly convergent.[1] Distinctions A series can be uniformly convergent and absolutely convergent without being uniformly absolutely-convergent. For example, if ƒn(x) = xn/n on the open interval (−1,0), then the series Σfn(x) converges uniformly by comparison of the partial sums to those of Σ(−1)n/n, and the series Σ\|fn(x)\| converges absolutely at each point by the geometric series test, but Σ\|fn(x)\| does not converge uniformly. Intuitively, this is because the absolute-convergence gets slower and slower as x approaches −1, where convergence holds but absolute convergence fails. Generalizations If a series of functions is uniformly absolutely-convergent on some neighborhood of each point of a topological space, it is locally uniformly absolutely-convergent. If a series is uniformly absolutely-convergent on all compact subsets of a topological space, it is compactly (uniformly) absolutely-convergent. If the topological space is locally compact, these notions are equivalent. Properties • If a series of functions into C (or any Banach space) is uniformly absolutely-convergent, then it is uniformly convergent. • Uniform absolute-convergence is independent of the ordering of a series. This is because, for a series of nonnegative functions, uniform convergence is equivalent to the property that, for any ε > 0, there are finitely many terms of the series such that excluding these terms results in a series with total sum less than the constant function ε, and this property does not refer to the ordering. See also • Modes of convergence (annotated index) References 1. Kiyosi Itō (1987). Encyclopedic Dictionary of Mathematics, MIT Press.	Wikipedia
Uniform algebra In functional analysis, a uniform algebra A on a compact Hausdorff topological space X is a closed (with respect to the uniform norm) subalgebra of the C-algebra C(X) (the continuous complex-valued functions on X) with the following properties:[1] the constant functions are contained in A for every x, y $\in $ X there is f$\in $A with f(x)$\neq $f(y). This is called separating the points of X. As a closed subalgebra of the commutative Banach algebra C(X) a uniform algebra is itself a unital commutative Banach algebra (when equipped with the uniform norm). Hence, it is, (by definition) a Banach function algebra. A uniform algebra A on X is said to be natural if the maximal ideals of A are precisely the ideals $M_{x}$ of functions vanishing at a point x in X. Abstract characterization If A is a unital commutative Banach algebra such that $\|\|a^{2}\|\|=\|\|a\|\|^{2}$ for all a in A, then there is a compact Hausdorff X such that A is isomorphic as a Banach algebra to a uniform algebra on X. This result follows from the spectral radius formula and the Gelfand representation. Notes 1. (Gamelin 2005, p. 25) References • Gamelin, Theodore W. (2005). Uniform Algebras. ISBN 978-0-8218-4049-8. • Gorin, E.A. (2001) [1994], "Uniform algebra", Encyclopedia of Mathematics, EMS Press Functional analysis (topics – glossary) Spaces • Banach • Besov • Fréchet • Hilbert • Hölder • Nuclear • Orlicz • Schwartz • Sobolev • Topological vector Properties • Barrelled • Complete • Dual (Algebraic/Topological) • Locally convex • Reflexive • Reparable Theorems • Hahn–Banach • Riesz representation • Closed graph • Uniform boundedness principle • Kakutani fixed-point • Krein–Milman • Min–max • Gelfand–Naimark • Banach–Alaoglu Operators • Adjoint • Bounded • Compact • Hilbert–Schmidt • Normal • Nuclear • Trace class • Transpose • Unbounded • Unitary Algebras • Banach algebra • C-algebra • Spectrum of a C-algebra • Operator algebra • Group algebra of a locally compact group • Von Neumann algebra Open problems • Invariant subspace problem • Mahler's conjecture Applications • Hardy space • Spectral theory of ordinary differential equations • Heat kernel • Index theorem • Calculus of variations • Functional calculus • Integral operator • Jones polynomial • Topological quantum field theory • Noncommutative geometry • Riemann hypothesis • Distribution (or Generalized functions) Advanced topics • Approximation property • Balanced set • Choquet theory • Weak topology • Banach–Mazur distance • Tomita–Takesaki theory • Mathematics portal • Category • Commons Spectral theory and -algebras Basic concepts • Involution/-algebra • Banach algebra • B-algebra • C-algebra • Noncommutative topology • Projection-valued measure • Spectrum • Spectrum of a C-algebra • Spectral radius • Operator space Main results • Gelfand–Mazur theorem • Gelfand–Naimark theorem • Gelfand representation • Polar decomposition • Singular value decomposition • Spectral theorem • Spectral theory of normal C-algebras Special Elements/Operators • Isospectral • Normal operator • Hermitian/Self-adjoint operator • Unitary operator • Unit Spectrum • Krein–Rutman theorem • Normal eigenvalue • Spectrum of a C-algebra • Spectral radius • Spectral asymmetry • Spectral gap Decomposition • Decomposition of a spectrum • Continuous • Point • Residual • Approximate point • Compression • Direct integral • Discrete • Spectral abscissa Spectral Theorem • Borel functional calculus • Min-max theorem • Positive operator-valued measure • Projection-valued measure • Riesz projector • Rigged Hilbert space • Spectral theorem • Spectral theory of compact operators • Spectral theory of normal C-algebras Special algebras • Amenable Banach algebra • With an Approximate identity • Banach function algebra • Disk algebra • Nuclear C-algebra • Uniform algebra • Von Neumann algebra • Tomita–Takesaki theory Finite-Dimensional • Alon–Boppana bound • Bauer–Fike theorem • Numerical range • Schur–Horn theorem Generalizations • Dirac spectrum • Essential spectrum • Pseudospectrum • Structure space (Shilov boundary) Miscellaneous • Abstract index group • Banach algebra cohomology • Cohen–Hewitt factorization theorem • Extensions of symmetric operators • Fredholm theory • Limiting absorption principle • Schröder–Bernstein theorems for operator algebras • Sherman–Takeda theorem • Unbounded operator Examples • Wiener algebra Applications • Almost Mathieu operator • Corona theorem • Hearing the shape of a drum (Dirichlet eigenvalue) • Heat kernel • Kuznetsov trace formula • Lax pair • Proto-value function • Ramanujan graph • Rayleigh–Faber–Krahn inequality • Spectral geometry • Spectral method • Spectral theory of ordinary differential equations • Sturm–Liouville theory • Superstrong approximation • Transfer operator • Transform theory • Weyl law • Wiener–Khinchin theorem Authority control: National • Israel • United States	Wikipedia
Antiprism In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies (not mirror images) of an n-sided polygon, connected by an alternating band of 2n triangles. They are represented by the Conway notation An. Set of uniform n-gonal antiprisms Uniform hexagonal antiprism (n = 6) Typeuniform in the sense of semiregular polyhedron Faces2 regular n-gons 2n equilateral triangles Edges4n Vertices2n Vertex configuration3.3.3.n Schläfli symbol{ }⊗{n} [1] s{2,2n} sr{2,n} Conway notationAn Coxeter diagram Symmetry groupDnd, [2+,2n], (2n), order 4n Rotation groupDn, [2,n]+, (22n), order 2n Dual polyhedronconvex dual-uniform n-gonal trapezohedron Propertiesconvex, vertex-transitive, regular polygon faces, congruent & coaxial bases Net Net of uniform enneagonal antiprism (n = 9) Antiprisms are a subclass of prismatoids, and are a (degenerate) type of snub polyhedron. Antiprisms are similar to prisms, except that the bases are twisted relatively to each other, and that the side faces (connecting the bases) are 2n triangles, rather than n quadrilaterals. The dual polyhedron of an n-gonal antiprism is an n-gonal trapezohedron. History At the intersection of modern-day graph theory and coding theory, the triangulation of a set of points have interested mathematicians since Isaac Newton, who fruitlessly sought a mathematical proof of the kissing number problem in 1694.[2] The existence of antiprisms was discussed, and their name was coined by Johannes Kepler, though it is possible that they were previously known to Archimedes, as they satisfy the same conditions on faces and on vertices as the Archimedean solids. According to Ericson and Zinoviev, Harold Scott MacDonald Coxeter wrote at length on the topic,[2] and was among the first to apply the mathematics of Victor Schlegel to this field. Knowledge in this field is "quite incomplete" and "was obtained fairly recently", i.e. in the 20th century. For example, as of 2001 it had been proven for only a limited number of non-trivial cases that the n-gonal antiprism is the mathematically optimal arrangement of 2n points in the sense of maximizing the minimum Euclidean distance between any two points on the set: in 1943 by László Fejes Tóth for 4 and 6 points (digonal and trigonal antiprisms, which are Platonic solids); in 1951 by Kurt Schütte and Bartel Leendert van der Waerden for 8 points (tetragonal antiprism, which is not a cube).[2] The chemical structure of binary compounds has been remarked to be in the family of antiprisms;[3] especially those of the family of boron hydrides (in 1975) and carboranes because they are isoelectronic. This is a mathematically real conclusion reached by studies of X-ray diffraction patterns,[4] and stems from the 1971 work of Kenneth Wade,[5] the nominative source for Wade's rules of polyhedral skeletal electron pair theory. Rare-earth metals such as the lanthanides form antiprismatic compounds with some of the halides or some of the iodides. The study of crystallography is useful here.[6] Some lanthanides, when arranged in peculiar antiprismatic structures with chlorine and water, can form molecule-based magnets.[7] Right antiprism For an antiprism with regular n-gon bases, one usually considers the case where these two copies are twisted by an angle of 180/n degrees. The axis of a regular polygon is the line perpendicular to the polygon plane and lying in the polygon centre. For an antiprism with congruent regular n-gon bases, twisted by an angle of 180/n degrees, more regularity is obtained if the bases have the same axis: are coaxial; i.e. (for non-coplanar bases): if the line connecting the base centers is perpendicular to the base planes. Then the antiprism is called a right antiprism, and its 2n side faces are isosceles triangles. Uniform antiprism A uniform n-antiprism has two congruent regular n-gons as base faces, and 2n equilateral triangles as side faces. Uniform antiprisms form an infinite class of vertex-transitive polyhedra, as do uniform prisms. For n = 2, we have the regular tetrahedron as a digonal antiprism (degenerate antiprism); for n = 3, the regular octahedron as a triangular antiprism (non-degenerate antiprism). Family of uniform n-gonal antiprisms Antiprism name Digonal antiprism (Trigonal) Triangular antiprism (Tetragonal) Square antiprism Pentagonal antiprism Hexagonal antiprism Heptagonal antiprism Octagonal antiprism Enneagonal antiprism Decagonal antiprism Hendecagonal antiprism Dodecagonal antiprism ... Apeirogonal antiprism Polyhedron image ... Spherical tiling image Plane tiling image Vertex config. 2.3.3.3 3.3.3.3 4.3.3.3 5.3.3.3 6.3.3.3 7.3.3.3 8.3.3.3 9.3.3.3 10.3.3.3 11.3.3.3 12.3.3.3 ... ∞.3.3.3 Schlegel diagrams A3 A4 A5 A6 A7 A8 Cartesian coordinates Cartesian coordinates for the vertices of a right n-antiprism (i.e. with regular n-gon bases and 2n isosceles triangle side faces) are: $\left(\cos {\frac {k\pi }{n}},\sin {\frac {k\pi }{n}},(-1)^{k}h\right)$ where 0 ≤ k ≤ 2n – 1; if the n-antiprism is uniform (i.e. if the triangles are equilateral), then: $2h^{2}=\cos {\frac {\pi }{n}}-\cos {\frac {2\pi }{n}}.$ Volume and surface area Let a be the edge-length of a uniform n-gonal antiprism; then the volume is: $V={\frac {n~{\sqrt {4\cos ^{2}{\frac {\pi }{2n}}-1}}\sin {\frac {3\pi }{2n}}}{12\sin ^{2}{\frac {\pi }{n}}}}~a^{3},$ and the surface area is: $A={\frac {n}{2}}\left(\cot {\frac {\pi }{n}}+{\sqrt {3}}\right)a^{2}.$ Related polyhedra There are an infinite set of truncated antiprisms, including a lower-symmetry form of the truncated octahedron (truncated triangular antiprism). These can be alternated to create snub antiprisms, two of which are Johnson solids, and the snub triangular antiprism is a lower symmetry form of the regular icosahedron. Antiprisms ... s{2,4} s{2,6} s{2,8} s{2,10} s{2,2n} Truncated antiprisms ... ts{2,4} ts{2,6} ts{2,8} ts{2,10} ts{2,2n} Snub antiprisms J84 Icosahedron J85 Irregular faces... ... ss{2,4} ss{2,6} ss{2,8} ss{2,10} ss{2,2n} Four-dimensional antiprisms can be defined as having two dual polyhedra as parallel opposite faces, so that each three-dimensional face between them comes from two dual parts of the polyhedra: a vertex and a dual polygon, or two dual edges. Every three-dimensional polyhedron is combinatorially equivalent to one of the two opposite faces of a four-dimensional antiprism, constructed from its canonical polyhedron and its polar dual.[8] However, there exist four-dimensional polyhedra that cannot be combined with their duals to form five-dimensional antiprisms.[9] Symmetry The symmetry group of a right n-antiprism (i.e. with regular bases and isosceles side faces) is Dnd = Dnv of order 4n, except in the cases of: • n = 2: the regular tetrahedron, which has the larger symmetry group Td of order 24 = 3×(4×2), which has three versions of D2d as subgroups; • n = 3: the regular octahedron, which has the larger symmetry group Oh of order 48 = 4×(4×3), which has four versions of D3d as subgroups. The symmetry group contains inversion if and only if n is odd. The rotation group is Dn of order 2n, except in the cases of: • n = 2: the regular tetrahedron, which has the larger rotation group T of order 12 = 3×(2×2), which has three versions of D2 as subgroups; • n = 3: the regular octahedron, which has the larger rotation group O of order 24 = 4×(2×3), which has four versions of D3 as subgroups. Note: The right n-antiprisms have congruent regular n-gon bases and congruent isosceles triangle side faces, thus have the same (dihedral) symmetry group as the uniform n-antiprism, for n ≥ 4. Star antiprism 5/2-antiprism 5/3-antiprism 9/2-antiprism 9/4-antiprism 9/5-antiprism Further information: Prismatic uniform polyhedron Uniform star antiprisms are named by their star polygon bases, {p/q}, and exist in prograde and in retrograde (crossed) solutions. Crossed forms have intersecting vertex figures, and are denoted by "inverted" fractions: p/(p – q) instead of p/q; example: 5/3 instead of 5/2. A right star antiprism has two congruent coaxial regular convex or star polygon base faces, and 2n isosceles triangle side faces. Any star antiprism with regular convex or star polygon bases can be made a right star antiprism (by translating and/or twisting one of its bases, if necessary). In the retrograde forms but not in the prograde forms, the triangles joining the convex or star bases intersect the axis of rotational symmetry. Thus: • Retrograde star antiprisms with regular convex polygon bases cannot have all equal edge lengths, so cannot be uniform. "Exception": a retrograde star antiprism with equilateral triangle bases (vertex configuration: 3.3/2.3.3) can be uniform; but then, it has the appearance of an equilateral triangle: it is a degenerate star polyhedron. • Similarly, some retrograde star antiprisms with regular star polygon bases cannot have all equal edge lengths, so cannot be uniform. Example: a retrograde star antiprism with regular star 7/5-gon bases (vertex configuration: 3.3.3.7/5) cannot be uniform. Also, star antiprism compounds with regular star p/q-gon bases can be constructed if p and q have common factors. Example: a star 10/4-antiprism is the compound of two star 5/2-antiprisms. Star p/q-antiprisms by symmetry, for p ≤ 12 Symmetry group Uniform stars Right stars D4d [2+,8] (24) 3.3/2.3.4 D5h [2,5] (225) 3.3.3.5/2 3.3/2.3.5 D5d [2+,10] (25) 3.3.3.5/3 D6d [2+,12] (26) 3.3/2.3.6 D7h [2,7] (227) 3.3.3.7/2 3.3.3.7/4 D7d [2+,14] (27) 3.3.3.7/3 D8d [2+,16] (28) 3.3.3.8/3 3.3.3.8/5 D9h [2,9] (229) 3.3.3.9/2 3.3.3.9/4 D9d [2+,18] (29) 3.3.3.9/5 D10d [2+,20] (210) 3.3.3.10/3 D11h [2,11] (2.2.11) 3.3.3.11/2 3.3.3.11/4 3.3.3.11/6 D11d [2+,22] (211) 3.3.3.11/3 3.3.3.11/5 3.3.3.11/7 D12d [2+,24] (212) 3.3.3.12/5 3.3.3.12/7 ... ... See also • Apeirogonal antiprism • Grand antiprism – a four-dimensional polytope • One World Trade Center, a building consisting primarily of an elongated square antiprism • Skew polygon References 1. N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.3 Pyramids, Prisms, and Antiprisms, Figure 11.3c 2. Ericson, Thomas; Zinoviev, Victor (2001). "Codes in dimension n = 3". Codes on Euclidean Spheres. North-Holland Mathematical Library. Vol. 63. pp. 67–106. doi:10.1016/S0924-6509(01)80048-9. ISBN 9780444503299. 3. Beall, Herbert; Gaines, Donald F. (2003). "Boron Hydrides". Encyclopedia of Physical Science and Technology. pp. 301–316. doi:10.1016/B0-12-227410-5/00073-9. ISBN 9780122274107. 4. “Boron Hydride Chemistry” (E. L. Muetterties, ed.), Academic Press, New York 5. Wade, K. (1971). "The structural significance of the number of skeletal bonding electron-pairs in carboranes, the higher boranes and borane anions, and various transition-metal carbonyl cluster compounds". J. Chem. Soc. D. 1971 (15): 792–793. doi:10.1039/C29710000792. 6. Meyer, Gerd (2014). "Symbiosis of Intermetallic and Salt". Including Actinides. Handbook on the Physics and Chemistry of Rare Earths. Vol. 45. pp. 111–178. doi:10.1016/B978-0-444-63256-2.00264-3. ISBN 9780444632562. 7. Bartolomé, Elena; Arauzo, Ana; Luzón, Javier; Bartolomé, Juan; Bartolomé, Fernando (2017). Magnetic Relaxation of Lanthanide-Based Molecular Magnets. Handbook of Magnetic Materials. Vol. 26. pp. 1–289. doi:10.1016/bs.hmm.2017.09.002. ISBN 9780444639271. 8. Grünbaum, Branko (2005). "Are prisms and antiprisms really boring? (Part 3)" (PDF). Geombinatorics. 15 (2): 69–78. MR 2298896. 9. Dobbins, Michael Gene (2017). "Antiprismlessness, or: reducing combinatorial equivalence to projective equivalence in realizability problems for polytopes". Discrete & Computational Geometry. 57 (4): 966–984. doi:10.1007/s00454-017-9874-y. MR 3639611. • Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 2: Archimedean polyhedra, prisms and antiprisms • Media related to Antiprisms at Wikimedia Commons • Weisstein, Eric W. "Antiprism". MathWorld. • Nonconvex Prisms and Antiprisms • Paper models of prisms and antiprisms Convex polyhedra Platonic solids (regular) • tetrahedron • cube • octahedron • dodecahedron • icosahedron Archimedean solids (semiregular or uniform) • truncated tetrahedron • cuboctahedron • truncated cube • truncated octahedron • rhombicuboctahedron • truncated cuboctahedron • snub cube • icosidodecahedron • truncated dodecahedron • truncated icosahedron • rhombicosidodecahedron • truncated icosidodecahedron • snub dodecahedron Catalan solids (duals of Archimedean) • triakis tetrahedron • rhombic dodecahedron • triakis octahedron • tetrakis hexahedron • deltoidal icositetrahedron • disdyakis dodecahedron • pentagonal icositetrahedron • rhombic triacontahedron • triakis icosahedron • pentakis dodecahedron • deltoidal hexecontahedron • disdyakis triacontahedron • pentagonal hexecontahedron Dihedral regular • dihedron • hosohedron Dihedral uniform • prisms • antiprisms duals: • bipyramids • trapezohedra Dihedral others • pyramids • truncated trapezohedra • gyroelongated bipyramid • cupola • bicupola • frustum • bifrustum • rotunda • birotunda • prismatoid • scutoid Degenerate polyhedra are in italics.	Wikipedia
Uniform antiprismatic prism In 4-dimensional geometry, a uniform antiprismatic prism or antiduoprism is a uniform 4-polytope with two uniform antiprism cells in two parallel 3-space hyperplanes, connected by uniform prisms cells between pairs of faces. The symmetry of a p-gonal antiprismatic prism is [2p,2+,2], order 8p. Set of uniform antiprismatic prisms TypePrismatic uniform 4-polytope Schläfli symbols{2,p}×{} Coxeter diagram Cells2 p-gonal antiprisms, 2 p-gonal prisms and 2p triangular prisms Faces4p {3}, 4p {4} and 4 {p} Edges10p Vertices4p Vertex figure Trapezoidal pyramid Symmetry group[2p,2+,2], order 8p [(p,2)+,2], order 4p Propertiesconvex if the base is convex A p-gonal antiprismatic prism or p-gonal antiduoprism has 2 p-gonal antiprism, 2 p-gonal prism, and 2p triangular prism cells. It has 4p equilateral triangle, 4p square and 4 regular p-gon faces. It has 10p edges, and 4p vertices. Example 15-gonal antiprismatic prism Schlegel diagram Net Convex uniform antiprismatic prisms There is an infinite series of convex uniform antiprismatic prisms, starting with the digonal antiprismatic prism is a tetrahedral prism, with two of the tetrahedral cells degenerated into squares. The triangular antiprismatic prism is the first nondegenerate form, which is also an octahedral prism. The remainder are unique uniform 4-polytopes. Convex p-gonal antiprismatic prisms Name s{2,2}×{} s{2,3}×{} s{2,4}×{} s{2,5}×{} s{2,6}×{} s{2,7}×{} s{2,8}×{} s{2,p}×{} Coxeter diagram Image Vertex figure Cells 2 s{2,2} (2) {2}×{}={4} 4 {3}×{} 2 s{2,3} 2 {3}×{} 6 {3}×{} 2 s{2,4} 2 {4}×{} 8 {3}×{} 2 s{2,5} 2 {5}×{} 10 {3}×{} 2 s{2,6} 2 {6}×{} 12 {3}×{} 2 s{2,7} 2 {7}×{} 14 {3}×{} 2 s{2,8} 2 {8}×{} 16 {3}×{} 2 s{2,p} 2 {p}×{} 2p {3}×{} Net Star antiprismatic prisms There are also star forms following the set of star antiprisms, starting with the pentagram {5/2}: Name Coxeter diagram Cells Image Net Pentagrammic antiprismatic prism 5/2 antiduoprism 2 pentagrammic antiprisms 2 pentagrammic prisms 10 triangular prisms Pentagrammic crossed antiprismatic prism 5/3 antiduoprism 2 pentagrammic crossed antiprisms 2 pentagrammic prisms 10 triangular prisms ... Square antiprismatic prism Square antiprismatic prism TypePrismatic uniform 4-polytope Schläfli symbols{2,4}x{} Coxeter-Dynkin Cells2 (3.3.3.4) 8 (3.4.4) 2 4.4.4 Faces16 {3}, 20 {4} Edges40 Vertices16 Vertex figure Trapezoidal pyramid Symmetry group[(4,2)+,2], order 16 [8,2+,2], order 32 Propertiesconvex A square antiprismatic prism or square antiduoprism is a convex uniform 4-polytope. It is formed as two parallel square antiprisms connected by cubes and triangular prisms. The symmetry of a square antiprismatic prism is [8,2+,2], order 32. It has 16 triangle, 16 square and 4 square faces. It has 40 edges, and 16 vertices. Square antiprismatic prism Schlegel diagram Net Pentagonal antiprismatic prism Pentagonal antiprismatic prism TypePrismatic uniform 4-polytope Schläfli symbols{2,5}x{} Coxeter-Dynkin Cells2 (3.3.3.5) 10 (3.4.4) 2 (4.4.5) Faces20 {3}, 20 {4}, 4 {5} Edges50 Vertices20 Vertex figure Trapezoidal pyramid Symmetry group[(5,2)+,2], order 20 [10,2+,2], order 40 Propertiesconvex A pentagonal antiprismatic prism or pentagonal antiduoprism is a convex uniform 4-polytope. It is formed as two parallel pentagonal antiprisms connected by cubes and triangular prisms. The symmetry of a pentagonal antiprismatic prism is [10,2+,2], order 40. It has 20 triangle, 20 square and 4 pentagonal faces. It has 50 edges, and 20 vertices. Pentagonal antiprismatic prism Schlegel diagram Net Hexagonal antiprismatic prism Hexagonal antiprismatic prism TypePrismatic uniform 4-polytope Schläfli symbols{2,6}x{} Coxeter-Dynkin Cells2 (3.3.3.6) 12 (3.4.4) 2 (4.4.6) Faces24 {3}, 24 {4}, 4 {6} Edges60 Vertices24 Vertex figure Trapezoidal pyramid Symmetry group[(2,6)+,2], order 24 [12,2+,2], order 48 Propertiesconvex A hexagonal antiprismatic prism or hexagonal antiduoprism is a convex uniform 4-polytope. It is formed as two parallel hexagonal antiprisms connected by cubes and triangular prisms. The symmetry of a hexagonal antiprismatic prism is [12,2+,2], order 48. It has 24 triangle, 24 square and 4 hexagon faces. It has 60 edges, and 24 vertices. Hexagonal antiprismatic prism Schlegel diagram Net Heptagonal antiprismatic prism Heptagonal antiprismatic prism TypePrismatic uniform 4-polytope Schläfli symbols{2,7}×{} Coxeter-Dynkin Cells2 (3.3.3.7) 14 (3.4.4) 2 (4.4.7) Faces28 {3}, 28 {4}, 4 {7} Edges70 Vertices28 Vertex figure Trapezoidal pyramid Symmetry group[(7,2)+,2], order 28 [14,2+,2], order 56 Propertiesconvex A heptagonal antiprismatic prism or heptagonal antiduoprism is a convex uniform 4-polytope. It is formed as two parallel heptagonal antiprisms connected by cubes and triangular prisms. The symmetry of a heptagonal antiprismatic prism is [14,2+,2], order 56. It has 28 triangle, 28 square and 4 heptagonal faces. It has 70 edges, and 28 vertices. Heptagonal antiprismatic prism Schlegel diagram Net Octagonal antiprismatic prism Octagonal antiprismatic prism TypePrismatic uniform 4-polytope Schläfli symbols{2,8}×{} Coxeter-Dynkin Cells2 (3.3.3.8) 16 (3.4.4) 2 (4.4.8) Faces32 {3}, 32 {4}, 4 {8} Edges80 Vertices32 Vertex figure Trapezoidal pyramid Symmetry group[(8,2)+,2], order 32 [16,2+,2], order 64 Propertiesconvex A octagonal antiprismatic prism or octagonal antiduoprism is a convex uniform 4-polytope (four-dimensional polytope). It is formed as two parallel octagonal antiprisms connected by cubes and triangular prisms. The symmetry of an octagonal antiprismatic prism is [16,2+,2], order 64. It has 32 triangle, 32 square and 4 octagonal faces. It has 80 edges, and 32 vertices. Octagonal antiprismatic prism Schlegel diagram Net See also • Duoprism References • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26) • Norman Johnson Uniform Polytopes, Manuscript (1991) External links • 6. Convex uniform prismatic polychora, George Olshevsky.	Wikipedia
Minimax approximation algorithm A minimax approximation algorithm (or L∞ approximation or uniform approximation) is a method to find an approximation of a mathematical function that minimizes maximum error.[1][2] For example, given a function $f$ defined on the interval $[a,b]$ and a degree bound $n$, a minimax polynomial approximation algorithm will find a polynomial $p$ of degree at most $n$ to minimize $\max _{a\leq x\leq b}\|f(x)-p(x)\|.$[3] Polynomial approximations The Weierstrass approximation theorem states that every continuous function defined on a closed interval [a,b] can be uniformly approximated as closely as desired by a polynomial function.[2] For practical work it is often desirable to minimize the maximum absolute or relative error of a polynomial fit for any given number of terms in an effort to reduce computational expense of repeated evaluation. Polynomial expansions such as the Taylor series expansion are often convenient for theoretical work but less useful for practical applications. Truncated Chebyshev series, however, closely approximate the minimax polynomial. One popular minimax approximation algorithm is the Remez algorithm. References 1. Muller, Jean-Michel; Brisebarre, Nicolas; de Dinechin, Florent; Jeannerod, Claude-Pierre; Lefèvre, Vincent; Melquiond, Guillaume; Revol, Nathalie; Stehlé, Damien; Torres, Serge (2010). Handbook of Floating-Point Arithmetic (1 ed.). Birkhäuser. p. 376. doi:10.1007/978-0-8176-4705-6. ISBN 978-0-8176-4704-9. LCCN 2009939668. 2. Phillips, George M. (2003). "Best Approximation". Interpolation and Approximation by Polynomials. CMS Books in Mathematics. Springer. pp. 49–11. doi:10.1007/0-387-21682-0_2. ISBN 0-387-00215-4. 3. Powell, M. J. D. (1981). "7: The theory of minimax approximation". Approximation Theory and Methods. Cambridge University Press. ISBN 0521295149. External links • Minimax approximation algorithm at MathWorld	Wikipedia
Arithmetic dynamics Arithmetic dynamics[1] is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex dynamics, the study of the iteration of self-maps of the complex plane or other complex algebraic varieties. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures. Global arithmetic dynamics is the study of analogues of classical diophantine geometry in the setting of discrete dynamical systems, while local arithmetic dynamics, also called p-adic or nonarchimedean dynamics, is an analogue of complex dynamics in which one replaces the complex numbers C by a p-adic field such as Qp or Cp and studies chaotic behavior and the Fatou and Julia sets. The following table describes a rough correspondence between Diophantine equations, especially abelian varieties, and dynamical systems: Diophantine equationsDynamical systems Rational and integer points on a variety Rational and integer points in an orbit Points of finite order on an abelian variety Preperiodic points of a rational function Definitions and notation from discrete dynamics Let S be a set and let F : S → S be a map from S to itself. The iterate of F with itself n times is denoted $F^{(n)}=F\circ F\circ \cdots \circ F.$ A point P ∈ S is periodic if F(n)(P) = P for some n ≥ 1. The point is preperiodic if F(k)(P) is periodic for some k ≥ 1. The (forward) orbit of P is the set $O_{F}(P)=\left\{P,F(P),F^{(2)}(P),F^{(3)}(P),\cdots \right\}.$ Thus P is preperiodic if and only if its orbit OF(P) is finite. Number theoretic properties of preperiodic points See also: Uniform boundedness conjecture for torsion points and Uniform boundedness conjecture for rational points Let F(x) be a rational function of degree at least two with coefficients in Q. A theorem of Douglas Northcott[2] says that F has only finitely many Q-rational preperiodic points, i.e., F has only finitely many preperiodic points in P1(Q). The uniform boundedness conjecture for preperiodic points[3] of Patrick Morton and Joseph Silverman says that the number of preperiodic points of F in P1(Q) is bounded by a constant that depends only on the degree of F. More generally, let F : PN → PN be a morphism of degree at least two defined over a number field K. Northcott's theorem says that F has only finitely many preperiodic points in PN(K), and the general Uniform Boundedness Conjecture says that the number of preperiodic points in PN(K) may be bounded solely in terms of N, the degree of F, and the degree of K over Q. The Uniform Boundedness Conjecture is not known even for quadratic polynomials Fc(x) = x2 + c over the rational numbers Q. It is known in this case that Fc(x) cannot have periodic points of period four,[4] five,[5] or six,[6] although the result for period six is contingent on the validity of the conjecture of Birch and Swinnerton-Dyer. Bjorn Poonen has conjectured that Fc(x) cannot have rational periodic points of any period strictly larger than three.[7] Integer points in orbits The orbit of a rational map may contain infinitely many integers. For example, if F(x) is a polynomial with integer coefficients and if a is an integer, then it is clear that the entire orbit OF(a) consists of integers. Similarly, if F(x) is a rational map and some iterate F(n)(x) is a polynomial with integer coefficients, then every n-th entry in the orbit is an integer. An example of this phenomenon is the map F(x) = x−d, whose second iterate is a polynomial. It turns out that this is the only way that an orbit can contain infinitely many integers. Theorem.[8] Let F(x) ∈ Q(x) be a rational function of degree at least two, and assume that no iterate[9] of F is a polynomial. Let a ∈ Q. Then the orbit OF(a) contains only finitely many integers. Dynamically defined points lying on subvarieties There are general conjectures due to Shouwu Zhang[10] and others concerning subvarieties that contain infinitely many periodic points or that intersect an orbit in infinitely many points. These are dynamical analogues of, respectively, the Manin–Mumford conjecture, proven by Michel Raynaud, and the Mordell–Lang conjecture, proven by Gerd Faltings. The following conjectures illustrate the general theory in the case that the subvariety is a curve. Conjecture. Let F : PN → PN be a morphism and let C ⊂ PN be an irreducible algebraic curve. Suppose that there is a point P ∈ PN such that C contains infinitely many points in the orbit OF(P). Then C is periodic for F in the sense that there is some iterate F(k) of F that maps C to itself. p-adic dynamics The field of p-adic (or nonarchimedean) dynamics is the study of classical dynamical questions over a field K that is complete with respect to a nonarchimedean absolute value. Examples of such fields are the field of p-adic rationals Qp and the completion of its algebraic closure Cp. The metric on K and the standard definition of equicontinuity leads to the usual definition of the Fatou and Julia sets of a rational map F(x) ∈ K(x). There are many similarities between the complex and the nonarchimedean theories, but also many differences. A striking difference is that in the nonarchimedean setting, the Fatou set is always nonempty, but the Julia set may be empty. This is the reverse of what is true over the complex numbers. Nonarchimedean dynamics has been extended to Berkovich space,[11] which is a compact connected space that contains the totally disconnected non-locally compact field Cp. Generalizations There are natural generalizations of arithmetic dynamics in which Q and Qp are replaced by number fields and their p-adic completions. Another natural generalization is to replace self-maps of P1 or PN with self-maps (morphisms) V → V of other affine or projective varieties. Other areas in which number theory and dynamics interact There are many other problems of a number theoretic nature that appear in the setting of dynamical systems, including: • dynamics over finite fields. • dynamics over function fields such as C(x). • iteration of formal and p-adic power series. • dynamics on Lie groups. • arithmetic properties of dynamically defined moduli spaces. • equidistribution[12] and invariant measures, especially on p-adic spaces. • dynamics on Drinfeld modules. • number-theoretic iteration problems that are not described by rational maps on varieties, for example, the Collatz problem. • symbolic codings of dynamical systems based on explicit arithmetic expansions of real numbers.[13] The Arithmetic Dynamics Reference List gives an extensive list of articles and books covering a wide range of arithmetical dynamical topics. See also • Arithmetic geometry • Arithmetic topology • Combinatorics and dynamical systems Notes and references 1. Silverman, Joseph H. (2007). The Arithmetic of Dynamical Systems. Graduate Texts in Mathematics. Vol. 241. New York: Springer. doi:10.1007/978-0-387-69904-2. ISBN 978-0-387-69903-5. MR 2316407. 2. Northcott, Douglas Geoffrey (1950). "Periodic points on an algebraic variety". Annals of Mathematics. 51 (1): 167–177. doi:10.2307/1969504. JSTOR 1969504. MR 0034607. 3. Morton, Patrick; Silverman, Joseph H. (1994). "Rational periodic points of rational functions". International Mathematics Research Notices. 1994 (2): 97–110. doi:10.1155/S1073792894000127. MR 1264933. 4. Morton, Patrick (1992). "Arithmetic properties of periodic points of quadratic maps". Acta Arithmetica. 62 (4): 343–372. doi:10.4064/aa-62-4-343-372. MR 1199627. 5. Flynn, Eugene V.; Poonen, Bjorn; Schaefer, Edward F. (1997). "Cycles of quadratic polynomials and rational points on a genus-2 curve". Duke Mathematical Journal. 90 (3): 435–463. arXiv:math/9508211. doi:10.1215/S0012-7094-97-09011-6. MR 1480542. S2CID 15169450. 6. Stoll, Michael (2008). "Rational 6-cycles under iteration of quadratic polynomials". LMS Journal of Computation and Mathematics. 11: 367–380. arXiv:0803.2836. Bibcode:2008arXiv0803.2836S. doi:10.1112/S1461157000000644. MR 2465796. S2CID 14082110. 7. Poonen, Bjorn (1998). "The classification of rational preperiodic points of quadratic polynomials over Q: a refined conjecture". Mathematische Zeitschrift. 228 (1): 11–29. doi:10.1007/PL00004405. MR 1617987. S2CID 118160396. 8. Silverman, Joseph H. (1993). "Integer points, Diophantine approximation, and iteration of rational maps". Duke Mathematical Journal. 71 (3): 793–829. doi:10.1215/S0012-7094-93-07129-3. MR 1240603. 9. An elementary theorem says that if F(x) ∈ C(x) and if some iterate of F is a polynomial, then already the second iterate is a polynomial. 10. Zhang, Shou-Wu (2006). "Distributions in algebraic dynamics". In Yau, Shing Tung (ed.). Differential Geometry: A Tribute to Professor S.-S. Chern. Surveys in Differential Geometry. Vol. 10. Somerville, MA: International Press. pp. 381–430. doi:10.4310/SDG.2005.v10.n1.a9. ISBN 978-1-57146-116-2. MR 2408228. 11. Rumely, Robert; Baker, Matthew (2010). Potential theory and dynamics on the Berkovich projective line. Mathematical Surveys and Monographs. Vol. 159. Providence, RI: American Mathematical Society. arXiv:math/0407433. doi:10.1090/surv/159. ISBN 978-0-8218-4924-8. MR 2599526. 12. Granville, Andrew; Rudnick, Zeév, eds. (2007). Equidistribution in number theory, an introduction. NATO Science Series II: Mathematics, Physics and Chemistry. Vol. 237. Dordrecht: Springer Netherlands. doi:10.1007/978-1-4020-5404-4. ISBN 978-1-4020-5403-7. MR 2290490. 13. Sidorov, Nikita (2003). "Arithmetic dynamics". In Bezuglyi, Sergey; Kolyada, Sergiy (eds.). Topics in dynamics and ergodic theory. Survey papers and mini-courses presented at the international conference and US-Ukrainian workshop on dynamical systems and ergodic theory, Katsiveli, Ukraine, August 21–30, 2000. Lond. Math. Soc. Lect. Note Ser. Vol. 310. Cambridge: Cambridge University Press. pp. 145–189. doi:10.1017/CBO9780511546716.010. ISBN 0-521-53365-1. MR 2052279. S2CID 15482676. Zbl 1051.37007. Further reading • Lecture Notes on Arithmetic Dynamics Arizona Winter School, March 13–17, 2010, Joseph H. Silverman • Chapter 15 of A first course in dynamics: with a panorama of recent developments, Boris Hasselblatt, A. B. Katok, Cambridge University Press, 2003, ISBN 978-0-521-58750-1 External links • The Arithmetic of Dynamical Systems home page • Arithmetic dynamics bibliography • Analysis and dynamics on the Berkovich projective line • Book review of Joseph H. Silverman's "The Arithmetic of Dynamical Systems", reviewed by Robert L. Benedetto Number theory Fields • Algebraic number theory (class field theory, non-abelian class field theory, Iwasawa theory, Iwasawa–Tate theory, Kummer theory) • Analytic number theory (analytic theory of L-functions, probabilistic number theory, sieve theory) • Geometric number theory • Computational number theory • Transcendental number theory • Diophantine geometry (Arakelov theory, Hodge–Arakelov theory) • Arithmetic combinatorics (additive number theory) • Arithmetic geometry (anabelian geometry, P-adic Hodge theory) • Arithmetic topology • Arithmetic dynamics Key concepts • Numbers • Natural numbers • Prime numbers • Rational numbers • Irrational numbers • Algebraic numbers • Transcendental numbers • P-adic numbers (P-adic analysis) • Arithmetic • Modular arithmetic • Chinese remainder theorem • Arithmetic functions Advanced concepts • Quadratic forms • Modular forms • L-functions • Diophantine equations • Diophantine approximation • Continued fractions • Category • List of topics • List of recreational topics • Wikibook • Wikiversity	Wikipedia
Uniform boundedness conjecture for rational points In arithmetic geometry, the uniform boundedness conjecture for rational points asserts that for a given number field $K$ and a positive integer $g\geq 2$ that there exists a number $N(K,g)$ depending only on $K$ and $g$ such that for any algebraic curve $C$ defined over $K$ having genus equal to $g$ has at most $N(K,g)$ $K$-rational points. This is a refinement of Faltings's theorem, which asserts that the set of $K$-rational points $C(K)$ is necessarily finite. For other uniform boundedness conjectures, see Uniform boundedness conjecture. Progress The first significant progress towards the conjecture was due to Caporaso, Harris, and Mazur.[1] They proved that the conjecture holds if one assumes the Bombieri–Lang conjecture. Mazur's Conjecture B A variant of the conjecture, due to Mazur, asserts that there should be a number $N(K,g,r)$ such that for any algebraic curve $C$ defined over $K$ having genus $g$ and whose Jacobian variety $J_{C}$ has Mordell–Weil rank over $K$ equal to $r$, the number of $K$-rational points of $C$ is at most $N(K,g,r)$. This variant of the conjecture is known as Mazur's Conjecture B. Michael Stoll proved that Mazur's Conjecture B holds for hyperelliptic curves with the additional hypothesis that $r\leq g-3$.[2] Stoll's result was further refined by Katz, Rabinoff, and Zureick-Brown in 2015.[3] Both of these works rely on Chabauty's method. Mazur's Conjecture B was resolved by Dimitrov, Gao, and Habegger in a preprint in 2020 which has since appeared in the Annals of Mathematics using the earlier work of Gao and Habegger on the geometric Bogomolov conjecture instead of Chabauty's method.[4] References 1. Caporaso, Lucia; Harris, Joe; Mazur, Barry (1997). "Uniformity of rational points". Journal of the American Mathematical Society. 10 (1): 1–35. doi:10.1090/S0894-0347-97-00195-1. 2. Stoll, Michael (2019). "Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank". Journal of the European Mathematical Society. 21 (3): 923–956. doi:10.4171/JEMS/857. 3. Katz, Eric; Rabinoff, Joseph; Zureick-Brown, David (2016). "Uniform bounds for the number of rational points on curves of small Mordell–Weil rank". Duke Mathematical Journal. 165 (16): 3189–3240. arXiv:1504.00694. doi:10.1215/00127094-3673558. S2CID 42267487. 4. Dimitrov, Vessilin; Gao, Ziyang; Habegger, Philipp (2021). "Uniformity in Mordell–Lang for curves" (PDF). Annals of Mathematics. 194: 237–298. doi:10.4007/annals.2021.194.1.4. S2CID 210932420.	Wikipedia
Compact convergence In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology. Definition Let $(X,{\mathcal {T}})$ be a topological space and $(Y,d_{Y})$ be a metric space. A sequence of functions $f_{n}:X\to Y$, $n\in \mathbb {N} ,$ is said to converge compactly as $n\to \infty $ to some function $f:X\to Y$ if, for every compact set $K\subseteq X$, $f_{n}\|_{K}\to f\|_{K}$ uniformly on $K$ as $n\to \infty $. This means that for all compact $K\subseteq X$, $\lim _{n\to \infty }\sup _{x\in K}d_{Y}\left(f_{n}(x),f(x)\right)=0.$ Examples • If $X=(0,1)\subseteq \mathbb {R} $ and $Y=\mathbb {R} $ with their usual topologies, with $f_{n}(x):=x^{n}$, then $f_{n}$ converges compactly to the constant function with value 0, but not uniformly. • If $X=(0,1]$, $Y=\mathbb {R} $ and $f_{n}(x)=x^{n}$, then $f_{n}$ converges pointwise to the function that is zero on $(0,1)$ and one at $1$, but the sequence does not converge compactly. • A very powerful tool for showing compact convergence is the Arzelà–Ascoli theorem. There are several versions of this theorem, roughly speaking it states that every sequence of equicontinuous and uniformly bounded maps has a subsequence that converges compactly to some continuous map. Properties • If $f_{n}\to f$ uniformly, then $f_{n}\to f$ compactly. • If $(X,{\mathcal {T}})$ is a compact space and $f_{n}\to f$ compactly, then $f_{n}\to f$ uniformly. • If $(X,{\mathcal {T}})$ is a locally compact space, then $f_{n}\to f$ compactly if and only if $f_{n}\to f$ locally uniformly. • If $(X,{\mathcal {T}})$ is a compactly generated space, $f_{n}\to f$ compactly, and each $f_{n}$ is continuous, then $f$ is continuous. See also • Modes of convergence (annotated index) • Montel's theorem References • R. Remmert Theory of complex functions (1991 Springer) p. 95	Wikipedia
Uniform convergence In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions $(f_{n})$ converges uniformly to a limiting function $f$ on a set $E$ as the function domain if, given any arbitrarily small positive number $\epsilon $, a number $N$ can be found such that each of the functions $f_{N},f_{N+1},f_{N+2},\ldots $ differs from $f$ by no more than $\epsilon $ at every point $x$ in $E$. Described in an informal way, if $f_{n}$ converges to $f$ uniformly, then the rate at which $f_{n}(x)$ approaches $f(x)$ is "uniform" throughout its domain in the following sense: in order to show that $f_{n}(x)$ uniformly falls within a certain distance $\epsilon $ of $f(x)$, we do not need to know the value of $x\in E$ in question — there can be found a single value of $N=N(\epsilon )$ independent of $x$, such that choosing $n\geq N$ will ensure that $f_{n}(x)$ is within $\epsilon $ of $f(x)$ for all $x\in E$. In contrast, pointwise convergence of $f_{n}$ to $f$ merely guarantees that for any $x\in E$ given in advance, we can find $N=N(\epsilon ,x)$ (i.e., $N$ can depend on the value of $x$) such that, for that particular $x$, $f_{n}(x)$ falls within $\epsilon $ of $f(x)$ whenever $n\geq N$ (a different $x$ requiring a different $N$ for pointwise convergence). The difference between uniform convergence and pointwise convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning. The concept, which was first formalized by Karl Weierstrass, is important because several properties of the functions $f_{n}$, such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit $f$ if the convergence is uniform, but not necessarily if the convergence is not uniform. History In 1821 Augustin-Louis Cauchy published a proof that a convergent sum of continuous functions is always continuous, to which Niels Henrik Abel in 1826 found purported counterexamples in the context of Fourier series, arguing that Cauchy's proof had to be incorrect. Completely standard notions of convergence did not exist at the time, and Cauchy handled convergence using infinitesimal methods. When put into the modern language, what Cauchy proved is that a uniformly convergent sequence of continuous functions has a continuous limit. The failure of a merely pointwise-convergent limit of continuous functions to converge to a continuous function illustrates the importance of distinguishing between different types of convergence when handling sequences of functions.[1] The term uniform convergence was probably first used by Christoph Gudermann, in an 1838 paper on elliptic functions, where he employed the phrase "convergence in a uniform way" when the "mode of convergence" of a series $ \sum _{n=1}^{\infty }f_{n}(x,\phi ,\psi )$ is independent of the variables $\phi $ and $\psi .$ While he thought it a "remarkable fact" when a series converged in this way, he did not give a formal definition, nor use the property in any of his proofs.[2] Later Gudermann's pupil Karl Weierstrass, who attended his course on elliptic functions in 1839–1840, coined the term gleichmäßig konvergent (German: uniformly convergent) which he used in his 1841 paper Zur Theorie der Potenzreihen, published in 1894. Independently, similar concepts were articulated by Philipp Ludwig von Seidel[3] and George Gabriel Stokes. G. H. Hardy compares the three definitions in his paper "Sir George Stokes and the concept of uniform convergence" and remarks: "Weierstrass's discovery was the earliest, and he alone fully realized its far-reaching importance as one of the fundamental ideas of analysis." Under the influence of Weierstrass and Bernhard Riemann this concept and related questions were intensely studied at the end of the 19th century by Hermann Hankel, Paul du Bois-Reymond, Ulisse Dini, Cesare Arzelà and others. Definition We first define uniform convergence for real-valued functions, although the concept is readily generalized to functions mapping to metric spaces and, more generally, uniform spaces (see below). Suppose $E$ is a set and $(f_{n})_{n\in \mathbb {N} }$ is a sequence of real-valued functions on it. We say the sequence $(f_{n})_{n\in \mathbb {N} }$ is uniformly convergent on $E$ with limit $f:E\to \mathbb {R} $ if for every $\epsilon >0,$ there exists a natural number $N$ such that for all $n\geq N$ and for all $x\in E$ $\|f_{n}(x)-f(x)\|<\epsilon .$ The notation for uniform convergence of $f_{n}$ to $f$ is not quite standardized and different authors have used a variety of symbols, including (in roughly decreasing order of popularity): $f_{n}\rightrightarrows f,\quad {\underset {n\to \infty }{\mathrm {unif\ lim} }}f_{n}=f,\quad f_{n}{\overset {\mathrm {unif.} }{\longrightarrow }}f,\quad f=u-\lim _{n\to \infty }f_{n}.$ Frequently, no special symbol is used, and authors simply write $f_{n}\to f\quad \mathrm {uniformly} $ to indicate that convergence is uniform. (In contrast, the expression $f_{n}\to f$ on $E$ without an adverb is taken to mean pointwise convergence on $E$: for all $x\in E$, $f_{n}(x)\to f(x)$ as $n\to \infty $.) Since $\mathbb {R} $ is a complete metric space, the Cauchy criterion can be used to give an equivalent alternative formulation for uniform convergence: $(f_{n})_{n\in \mathbb {N} }$ converges uniformly on $E$ (in the previous sense) if and only if for every $\epsilon >0$, there exists a natural number $N$ such that $x\in E,m,n\geq N\implies \|f_{m}(x)-f_{n}(x)\|<\epsilon $. In yet another equivalent formulation, if we define $d_{n}=\sup _{x\in E}\|f_{n}(x)-f(x)\|,$ then $f_{n}$ converges to $f$ uniformly if and only if $d_{n}\to 0$ as $n\to \infty $. Thus, we can characterize uniform convergence of $(f_{n})_{n\in \mathbb {N} }$ on $E$ as (simple) convergence of $(f_{n})_{n\in \mathbb {N} }$ in the function space $\mathbb {R} ^{E}$ with respect to the uniform metric (also called the supremum metric), defined by $d(f,g)=\sup _{x\in E}\|f(x)-g(x)\|.$ Symbolically, $f_{n}\rightrightarrows f\iff d(f_{n},f)\to 0$. The sequence $(f_{n})_{n\in \mathbb {N} }$ is said to be locally uniformly convergent with limit $f$ if $E$ is a metric space and for every $x\in E$, there exists an $r>0$ such that $(f_{n})$ converges uniformly on $B(x,r)\cap E.$ It is clear that uniform convergence implies local uniform convergence, which implies pointwise convergence. Notes Intuitively, a sequence of functions $f_{n}$ converges uniformly to $f$ if, given an arbitrarily small $\epsilon >0$, we can find an $N\in \mathbb {N} $ so that the functions $f_{n}$ with $n>N$ all fall within a "tube" of width $2\epsilon $ centered around $f$ (i.e., between $f(x)-\epsilon $ and $f(x)+\epsilon $) for the entire domain of the function. Note that interchanging the order of quantifiers in the definition of uniform convergence by moving "for all $x\in E$" in front of "there exists a natural number $N$" results in a definition of pointwise convergence of the sequence. To make this difference explicit, in the case of uniform convergence, $N=N(\epsilon )$ can only depend on $\epsilon $, and the choice of $N$ has to work for all $x\in E$, for a specific value of $\epsilon $ that is given. In contrast, in the case of pointwise convergence, $N=N(\epsilon ,x)$ may depend on both $\epsilon $ and $x$, and the choice of $N$ only has to work for the specific values of $\epsilon $ and $x$ that are given. Thus uniform convergence implies pointwise convergence, however the converse is not true, as the example in the section below illustrates. Generalizations One may straightforwardly extend the concept to functions E → M, where (M, d) is a metric space, by replacing $\|f_{n}(x)-f(x)\|$ with $d(f_{n}(x),f(x))$. The most general setting is the uniform convergence of nets of functions E → X, where X is a uniform space. We say that the net $(f_{\alpha })$ converges uniformly with limit f : E → X if and only if for every entourage V in X, there exists an $\alpha _{0}$, such that for every x in E and every $\alpha \geq \alpha _{0}$, $(f_{\alpha }(x),f(x))$ is in V. In this situation, uniform limit of continuous functions remains continuous. Definition in a hyperreal setting Uniform convergence admits a simplified definition in a hyperreal setting. Thus, a sequence $f_{n}$ converges to f uniformly if for all x in the domain of $f^{}$ and all infinite n, $f_{n}^{}(x)$ is infinitely close to $f^{*}(x)$ (see microcontinuity for a similar definition of uniform continuity). Examples For $x\in [0,1)$, a basic example of uniform convergence can be illustrated as follows: the sequence $(1/2)^{x+n}$ converges uniformly, while $x^{n}$ does not. Specifically, assume $\epsilon =1/4$. Each function $(1/2)^{x+n}$ is less than or equal to $1/4$ when $n\geq 2$, regardless of the value of $x$. On the other hand, $x^{n}$ is only less than or equal to $1/4$ at ever increasing values of $n$ when values of $x$ are selected closer and closer to 1 (explained more in depth further below). Given a topological space X, we can equip the space of bounded real or complex-valued functions over X with the uniform norm topology, with the uniform metric defined by $d(f,g)=\\|f-g\\|_{\infty }=\sup _{x\in X}\|f(x)-g(x)\|.$ Then uniform convergence simply means convergence in the uniform norm topology: $\lim _{n\to \infty }\\|f_{n}-f\\|_{\infty }=0$. The sequence of functions $(f_{n})$ ${\begin{cases}f_{n}:[0,1]\to [0,1]\\f_{n}(x)=x^{n}\end{cases}}$ is a classic example of a sequence of functions that converges to a function $f$ pointwise but not uniformly. To show this, we first observe that the pointwise limit of $(f_{n})$ as $n\to \infty $ is the function $f$, given by $f(x)=\lim _{n\to \infty }f_{n}(x)={\begin{cases}0,&x\in [0,1);\\1,&x=1.\end{cases}}$ Pointwise convergence: Convergence is trivial for $x=0$ and $x=1$, since $f_{n}(0)=f(0)=0$ and $f_{n}(1)=f(1)=1$, for all $n$. For $x\in (0,1)$ and given $\epsilon >0$, we can ensure that $\|f_{n}(x)-f(x)\|<\epsilon $ whenever $n\geq N$ by choosing $N=\lceil \log \epsilon /\log x\rceil $ (here the upper square brackets indicate rounding up, see ceiling function). Hence, $f_{n}\to f$ pointwise for all $x\in [0,1]$. Note that the choice of $N$ depends on the value of $\epsilon $ and $x$. Moreover, for a fixed choice of $\epsilon $, $N$ (which cannot be defined to be smaller) grows without bound as $x$ approaches 1. These observations preclude the possibility of uniform convergence. Non-uniformity of convergence: The convergence is not uniform, because we can find an $\epsilon >0$ so that no matter how large we choose $N,$ there will be values of $x\in [0,1]$ and $n\geq N$ such that $\|f_{n}(x)-f(x)\|\geq \epsilon .$ To see this, first observe that regardless of how large $n$ becomes, there is always an $x_{0}\in [0,1)$ such that $f_{n}(x_{0})=1/2.$ Thus, if we choose $\epsilon =1/4,$ we can never find an $N$ such that $\|f_{n}(x)-f(x)\|<\epsilon $ for all $x\in [0,1]$ and $n\geq N$. Explicitly, whatever candidate we choose for $N$, consider the value of $f_{N}$ at $x_{0}=(1/2)^{1/N}$. Since $\left\|f_{N}(x_{0})-f(x_{0})\right\|=\left\|\left[\left({\frac {1}{2}}\right)^{\frac {1}{N}}\right]^{N}-0\right\|={\frac {1}{2}}>{\frac {1}{4}}=\epsilon ,$ the candidate fails because we have found an example of an $x\in [0,1]$ that "escaped" our attempt to "confine" each $f_{n}\ (n\geq N)$ to within $\epsilon $ of $f$ for all $x\in [0,1]$. In fact, it is easy to see that $\lim _{n\to \infty }\\|f_{n}-f\\|_{\infty }=1,$ contrary to the requirement that $\\|f_{n}-f\\|_{\infty }\to 0$ if $f_{n}\rightrightarrows f$. In this example one can easily see that pointwise convergence does not preserve differentiability or continuity. While each function of the sequence is smooth, that is to say that for all n, $f_{n}\in C^{\infty }([0,1])$, the limit $\lim _{n\to \infty }f_{n}$ is not even continuous. Exponential function The series expansion of the exponential function can be shown to be uniformly convergent on any bounded subset $S\subset \mathbb {C} $ using the Weierstrass M-test. Theorem (Weierstrass M-test). Let $(f_{n})$ be a sequence of functions $f_{n}:E\to \mathbb {C} $ and let $M_{n}$ be a sequence of positive real numbers such that $\|f_{n}(x)\|\leq M_{n}$ for all $x\in E$ and $n=1,2,3,\ldots $ If $ \sum _{n}M_{n}$ converges, then $ \sum _{n}f_{n}$ converges absolutely and uniformly on $E$. The complex exponential function can be expressed as the series: $\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.$ Any bounded subset is a subset of some disc $D_{R}$ of radius $R,$ centered on the origin in the complex plane. The Weierstrass M-test requires us to find an upper bound $M_{n}$ on the terms of the series, with $M_{n}$ independent of the position in the disc: $\left\|{\frac {z^{n}}{n!}}\right\|\leq M_{n},\forall z\in D_{R}.$ To do this, we notice $\left\|{\frac {z^{n}}{n!}}\right\|\leq {\frac {\|z\|^{n}}{n!}}\leq {\frac {R^{n}}{n!}}$ and take $M_{n}={\tfrac {R^{n}}{n!}}.$ If $\sum _{n=0}^{\infty }M_{n}$ is convergent, then the M-test asserts that the original series is uniformly convergent. The ratio test can be used here: $\lim _{n\to \infty }{\frac {M_{n+1}}{M_{n}}}=\lim _{n\to \infty }{\frac {R^{n+1}}{R^{n}}}{\frac {n!}{(n+1)!}}=\lim _{n\to \infty }{\frac {R}{n+1}}=0$ which means the series over $M_{n}$ is convergent. Thus the original series converges uniformly for all $z\in D_{R},$ and since $S\subset D_{R}$, the series is also uniformly convergent on $S.$ Properties • Every uniformly convergent sequence is locally uniformly convergent. • Every locally uniformly convergent sequence is compactly convergent. • For locally compact spaces local uniform convergence and compact convergence coincide. • A sequence of continuous functions on metric spaces, with the image metric space being complete, is uniformly convergent if and only if it is uniformly Cauchy. • If $S$ is a compact interval (or in general a compact topological space), and $(f_{n})$ is a monotone increasing sequence (meaning $f_{n}(x)\leq f_{n+1}(x)$ for all n and x) of continuous functions with a pointwise limit $f$ which is also continuous, then the convergence is necessarily uniform (Dini's theorem). Uniform convergence is also guaranteed if $S$ is a compact interval and $(f_{n})$ is an equicontinuous sequence that converges pointwise. Applications To continuity Main article: Uniform limit theorem If $E$ and $M$ are topological spaces, then it makes sense to talk about the continuity of the functions $f_{n},f:E\to M$. If we further assume that $M$ is a metric space, then (uniform) convergence of the $f_{n}$ to $f$ is also well defined. The following result states that continuity is preserved by uniform convergence: Uniform limit theorem — Suppose $E$ is a topological space, $M$ is a metric space, and $(f_{n})$ is a sequence of continuous functions $f_{n}:E\to M$. If $f_{n}\rightrightarrows f$ on $E$, then $f$ is also continuous. This theorem is proved by the "ε/3 trick", and is the archetypal example of this trick: to prove a given inequality (ε), one uses the definitions of continuity and uniform convergence to produce 3 inequalities (ε/3), and then combines them via the triangle inequality to produce the desired inequality. This theorem is an important one in the history of real and Fourier analysis, since many 18th century mathematicians had the intuitive understanding that a sequence of continuous functions always converges to a continuous function. The image above shows a counterexample, and many discontinuous functions could, in fact, be written as a Fourier series of continuous functions. The erroneous claim that the pointwise limit of a sequence of continuous functions is continuous (originally stated in terms of convergent series of continuous functions) is infamously known as "Cauchy's wrong theorem". The uniform limit theorem shows that a stronger form of convergence, uniform convergence, is needed to ensure the preservation of continuity in the limit function. More precisely, this theorem states that the uniform limit of uniformly continuous functions is uniformly continuous; for a locally compact space, continuity is equivalent to local uniform continuity, and thus the uniform limit of continuous functions is continuous. To differentiability If $S$ is an interval and all the functions $f_{n}$ are differentiable and converge to a limit $f$, it is often desirable to determine the derivative function $f'$ by taking the limit of the sequence $f'_{n}$. This is however in general not possible: even if the convergence is uniform, the limit function need not be differentiable (not even if the sequence consists of everywhere-analytic functions, see Weierstrass function), and even if it is differentiable, the derivative of the limit function need not be equal to the limit of the derivatives. Consider for instance $f_{n}(x)=n^{-1/2}{\sin(nx)}$ with uniform limit $f_{n}\rightrightarrows f\equiv 0$. Clearly, $f'$ is also identically zero. However, the derivatives of the sequence of functions are given by $f'_{n}(x)=n^{1/2}\cos nx,$ and the sequence $f'_{n}$ does not converge to $f',$ or even to any function at all. In order to ensure a connection between the limit of a sequence of differentiable functions and the limit of the sequence of derivatives, the uniform convergence of the sequence of derivatives plus the convergence of the sequence of functions at at least one point is required:[4] If $(f_{n})$ is a sequence of differentiable functions on $[a,b]$ such that $\lim _{n\to \infty }f_{n}(x_{0})$ exists (and is finite) for some $x_{0}\in [a,b]$ and the sequence $(f'_{n})$ converges uniformly on $[a,b]$, then $f_{n}$ converges uniformly to a function $f$ on $[a,b]$, and $f'(x)=\lim _{n\to \infty }f'_{n}(x)$ for $x\in [a,b]$. To integrability Similarly, one often wants to exchange integrals and limit processes. For the Riemann integral, this can be done if uniform convergence is assumed: If $(f_{n})_{n=1}^{\infty }$ is a sequence of Riemann integrable functions defined on a compact interval $I$ which uniformly converge with limit $f$, then $f$ is Riemann integrable and its integral can be computed as the limit of the integrals of the $f_{n}$: $\int _{I}f=\lim _{n\to \infty }\int _{I}f_{n}.$ In fact, for a uniformly convergent family of bounded functions on an interval, the upper and lower Riemann integrals converge to the upper and lower Riemann integrals of the limit function. This follows because, for n sufficiently large, the graph of $f_{n}$ is within ε of the graph of f, and so the upper sum and lower sum of $f_{n}$ are each within $\varepsilon \|I\|$ of the value of the upper and lower sums of $f$, respectively. Much stronger theorems in this respect, which require not much more than pointwise convergence, can be obtained if one abandons the Riemann integral and uses the Lebesgue integral instead. To analyticity Using Morera's Theorem, one can show that if a sequence of analytic functions converges uniformly in a region S of the complex plane, then the limit is analytic in S. This example demonstrates that complex functions are more well-behaved than real functions, since the uniform limit of analytic functions on a real interval need not even be differentiable (see Weierstrass function). To series We say that $ \sum _{n=1}^{\infty }f_{n}$ converges: 1. pointwise on E if and only if the sequence of partial sums $s_{n}(x)=\sum _{j=1}^{n}f_{j}(x)$ converges for every $x\in E$. 2. uniformly on E if and only if sn converges uniformly as $n\to \infty $. 3. absolutely on E if and only if $ \sum _{n=1}^{\infty }\|f_{n}\|$ converges for every $x\in E$. With this definition comes the following result: Let x0 be contained in the set E and each fn be continuous at x0. If $ f=\sum _{n=1}^{\infty }f_{n}$ converges uniformly on E then f is continuous at x0 in E. Suppose that $E=[a,b]$ and each fn is integrable on E. If $ \sum _{n=1}^{\infty }f_{n}$ converges uniformly on E then f is integrable on E and the series of integrals of fn is equal to integral of the series of fn. Almost uniform convergence If the domain of the functions is a measure space E then the related notion of almost uniform convergence can be defined. We say a sequence of functions $(f_{n})$ converges almost uniformly on E if for every $\delta >0$ there exists a measurable set $E_{\delta }$ with measure less than $\delta $ such that the sequence of functions $(f_{n})$ converges uniformly on $E\setminus E_{\delta }$. In other words, almost uniform convergence means there are sets of arbitrarily small measure for which the sequence of functions converges uniformly on their complement. Note that almost uniform convergence of a sequence does not mean that the sequence converges uniformly almost everywhere as might be inferred from the name. However, Egorov's theorem does guarantee that on a finite measure space, a sequence of functions that converges almost everywhere also converges almost uniformly on the same set. Almost uniform convergence implies almost everywhere convergence and convergence in measure. See also • Uniform convergence in probability • Modes of convergence (annotated index) • Dini's theorem • Arzelà–Ascoli theorem Notes 1. Sørensen, Henrik Kragh (2005). "Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem". Historia Mathematica. 32 (4): 453–480. doi:10.1016/j.hm.2004.11.010. 2. Jahnke, Hans Niels (2003). "6.7 The Foundation of Analysis in the 19th Century: Weierstrass". A history of analysis. AMS Bookstore. p. 184. ISBN 978-0-8218-2623-2. 3. Lakatos, Imre (1976). Proofs and Refutations. Cambridge University Press. pp. 141. ISBN 978-0-521-21078-2. 4. Rudin, Walter (1976). Principles of Mathematical Analysis 3rd edition, Theorem 7.17. McGraw-Hill: New York. References • Konrad Knopp, Theory and Application of Infinite Series; Blackie and Son, London, 1954, reprinted by Dover Publications, ISBN 0-486-66165-2. • G. H. Hardy, Sir George Stokes and the concept of uniform convergence; Proceedings of the Cambridge Philosophical Society, 19, pp. 148–156 (1918) • Bourbaki; Elements of Mathematics: General Topology. Chapters 5–10 (paperback); ISBN 0-387-19374-X • Walter Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw–Hill, 1976. • Gerald Folland, Real Analysis: Modern Techniques and Their Applications, Second Edition, John Wiley & Sons, Inc., 1999, ISBN 0-471-31716-0. • William Wade, An Introduction to Analysis, 3rd ed., Pearson, 2005 External links • "Uniform convergence", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Graphic examples of uniform convergence of Fourier series from the University of Colorado Sequences and series Integer sequences Basic • Arithmetic progression • Geometric progression • Harmonic progression • Square number • Cubic number • Factorial • Powers of two • Powers of three • Powers of 10 Advanced (list) • Complete sequence • Fibonacci sequence • Figurate number • Heptagonal number • Hexagonal number • Lucas number • Pell number • Pentagonal number • Polygonal number • Triangular number Properties of sequences • Cauchy sequence • Monotonic function • Periodic sequence Properties of series Series • Alternating • Convergent • Divergent • Telescoping Convergence • Absolute • Conditional • Uniform Explicit series Convergent • 1/2 − 1/4 + 1/8 − 1/16 + ⋯ • 1/2 + 1/4 + 1/8 + 1/16 + ⋯ • 1/4 + 1/16 + 1/64 + 1/256 + ⋯ • 1 + 1/2s + 1/3s + ... (Riemann zeta function) Divergent • 1 + 1 + 1 + 1 + ⋯ • 1 − 1 + 1 − 1 + ⋯ (Grandi's series) • 1 + 2 + 3 + 4 + ⋯ • 1 − 2 + 3 − 4 + ⋯ • 1 + 2 + 4 + 8 + ⋯ • 1 − 2 + 4 − 8 + ⋯ • Infinite arithmetic series • 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) • 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) • 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes) Kinds of series • Taylor series • Power series • Formal power series • Laurent series • Puiseux series • Dirichlet series • Trigonometric series • Fourier series • Generating series Hypergeometric series • Generalized hypergeometric series • Hypergeometric function of a matrix argument • Lauricella hypergeometric series • Modular hypergeometric series • Riemann's differential equation • Theta hypergeometric series • Category	Wikipedia
Uniform convergence in probability Uniform convergence in probability is a form of convergence in probability in statistical asymptotic theory and probability theory. It means that, under certain conditions, the empirical frequencies of all events in a certain event-family converge to their theoretical probabilities. Uniform convergence in probability has applications to statistics as well as machine learning as part of statistical learning theory. For other uses, see uniform convergence. The law of large numbers says that, for each single event $A$, its empirical frequency in a sequence of independent trials converges (with high probability) to its theoretical probability. In many application however, the need arises to judge simultaneously the probabilities of events of an entire class $S$ from one and the same sample. Moreover it, is required that the relative frequency of the events converge to the probability uniformly over the entire class of events $S$ [1] The Uniform Convergence Theorem gives a sufficient condition for this convergence to hold. Roughly, if the event-family is sufficiently simple (its VC dimension is sufficiently small) then uniform convergence holds. Definitions For a class of predicates $H$ defined on a set $X$ and a set of samples $x=(x_{1},x_{2},\dots ,x_{m})$, where $x_{i}\in X$, the empirical frequency of $h\in H$ on $x$ is ${\widehat {Q}}_{x}(h)={\frac {1}{m}}\|\{i:1\leq i\leq m,h(x_{i})=1\}\|.$ The theoretical probability of $h\in H$ is defined as $Q_{P}(h)=P\{y\in X:h(y)=1\}.$ The Uniform Convergence Theorem states, roughly, that if $H$ is "simple" and we draw samples independently (with replacement) from $X$ according to any distribution $P$, then with high probability, the empirical frequency will be close to its expected value, which is the theoretical probability. Here "simple" means that the Vapnik–Chervonenkis dimension of the class $H$ is small relative to the size of the sample. In other words, a sufficiently simple collection of functions behaves roughly the same on a small random sample as it does on the distribution as a whole. The Uniform Convergence Theorem was first proved by Vapnik and Chervonenkis[1] using the concept of growth function. Uniform convergence theorem The statement of the uniform convergence theorem is as follows:[2] If $H$ is a set of $\{0,1\}$-valued functions defined on a set $X$ and $P$ is a probability distribution on $X$ then for $\varepsilon >0$ and $m$ a positive integer, we have: $P^{m}\{\|Q_{P}(h)-{\widehat {Q_{x}}}(h)\|\geq \varepsilon {\text{ for some }}h\in H\}\leq 4\Pi _{H}(2m)e^{-\varepsilon ^{2}m/8}.$ where, for any $x\in X^{m},$, $Q_{P}(h)=P\{(y\in X:h(y)=1\},$ ${\widehat {Q}}_{x}(h)={\frac {1}{m}}\|\{i:1\leq i\leq m,h(x_{i})=1\}\|$ and $\|x\|=m$. $P^{m}$ indicates that the probability is taken over $x$ consisting of $m$ i.i.d. draws from the distribution $P$. $\Pi _{H}$ is defined as: For any $\{0,1\}$-valued functions $H$ over $X$ and $D\subseteq X$, $\Pi _{H}(D)=\{h\cap D:h\in H\}.$ And for any natural number $m$, the shattering number $\Pi _{H}(m)$ is defined as: $\Pi _{H}(m)=\max \|\{h\cap D:\|D\|=m,h\in H\}\|.$ From the point of Learning Theory one can consider $H$ to be the Concept/Hypothesis class defined over the instance set $X$. Before getting into the details of the proof of the theorem we will state Sauer's Lemma which we will need in our proof. Sauer–Shelah lemma The Sauer–Shelah lemma[3] relates the shattering number $\Pi _{h}(m)$ to the VC Dimension. Lemma: $\Pi _{H}(m)\leq \left({\frac {em}{d}}\right)^{d}$, where $d$ is the VC Dimension of the concept class $H$. Corollary: $\Pi _{H}(m)\leq m^{d}$. Proof of uniform convergence theorem [1] and [2] are the sources of the proof below. Before we get into the details of the proof of the Uniform Convergence Theorem we will present a high level overview of the proof. 1. Symmetrization: We transform the problem of analyzing $\|Q_{P}(h)-{\widehat {Q}}_{x}(h)\|\geq \varepsilon $ into the problem of analyzing $\|{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)\|\geq \varepsilon /2$, where $r$ and $s$ are i.i.d samples of size $m$ drawn according to the distribution $P$. One can view $r$ as the original randomly drawn sample of length $m$, while $s$ may be thought as the testing sample which is used to estimate $Q_{P}(h)$. 2. Permutation: Since $r$ and $s$ are picked identically and independently, so swapping elements between them will not change the probability distribution on $r$ and $s$. So, we will try to bound the probability of $\|{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)\|\geq \varepsilon /2$ for some $h\in H$ by considering the effect of a specific collection of permutations of the joint sample $x=r\|\|s$. Specifically, we consider permutations $\sigma (x)$ which swap $x_{i}$ and $x_{m+i}$ in some subset of ${1,2,...,m}$. The symbol $r\|\|s$ means the concatenation of $r$ and $s$. 3. Reduction to a finite class: We can now restrict the function class $H$ to a fixed joint sample and hence, if $H$ has finite VC Dimension, it reduces to the problem to one involving a finite function class. We present the technical details of the proof. Symmetrization Lemma: Let $V=\{x\in X^{m}:\|Q_{P}(h)-{\widehat {Q}}_{x}(h)\|\geq \varepsilon {\text{ for some }}h\in H\}$ and $R=\{(r,s)\in X^{m}\times X^{m}:\|{\widehat {Q_{r}}}(h)-{\widehat {Q}}_{s}(h)\|\geq \varepsilon /2{\text{ for some }}h\in H\}.$ Then for $m\geq {\frac {2}{\varepsilon ^{2}}}$, $P^{m}(V)\leq 2P^{2m}(R)$. Proof: By the triangle inequality, if $\|Q_{P}(h)-{\widehat {Q}}_{r}(h)\|\geq \varepsilon $ and $\|Q_{P}(h)-{\widehat {Q}}_{s}(h)\|\leq \varepsilon /2$ then $\|{\widehat {Q}}_{r}(h)-{\widehat {Q}}_{s}(h)\|\geq \varepsilon /2$. Therefore, ${\begin{aligned}&P^{2m}(R)\\[5pt]\geq {}&P^{2m}\{\exists h\in H,\|Q_{P}(h)-{\widehat {Q}}_{r}(h)\|\geq \varepsilon {\text{ and }}\|Q_{P}(h)-{\widehat {Q}}_{s}(h)\|\leq \varepsilon /2\}\\[5pt]={}&\int _{V}P^{m}\{s:\exists h\in H,\|Q_{P}(h)-{\widehat {Q}}_{r}(h)\|\geq \varepsilon {\text{ and }}\|Q_{P}(h)-{\widehat {Q}}_{s}(h)\|\leq \varepsilon /2\}\,dP^{m}(r)\\[5pt]={}&A\end{aligned}}$ since $r$ and $s$ are independent. Now for $r\in V$ fix an $h\in H$ such that $\|Q_{P}(h)-{\widehat {Q}}_{r}(h)\|\geq \varepsilon $. For this $h$, we shall show that $P^{m}\left\{\|Q_{P}(h)-{\widehat {Q}}_{s}(h)\|\leq {\frac {\varepsilon }{2}}\right\}\geq {\frac {1}{2}}.$ Thus for any $r\in V$, $A\geq {\frac {P^{m}(V)}{2}}$ and hence $P^{2m}(R)\geq {\frac {P^{m}(V)}{2}}$. And hence we perform the first step of our high level idea. Notice, $m\cdot {\widehat {Q}}_{s}(h)$ is a binomial random variable with expectation $m\cdot Q_{P}(h)$ and variance $m\cdot Q_{P}(h)(1-Q_{P}(h))$. By Chebyshev's inequality we get $P^{m}\left\{\|Q_{P}(h)-{\widehat {Q_{s}(h)}}\|>{\frac {\varepsilon }{2}}\right\}\leq {\frac {m\cdot Q_{P}(h)(1-Q_{P}(h))}{(\varepsilon m/2)^{2}}}\leq {\frac {1}{\varepsilon ^{2}m}}\leq {\frac {1}{2}}$ for the mentioned bound on $m$. Here we use the fact that $x(1-x)\leq 1/4$ for $x$. Permutations Let $\Gamma _{m}$ be the set of all permutations of $\{1,2,3,\dots ,2m\}$ that swaps $i$ and $m+i$ $\forall i$ in some subset of $\{1,2,3,\ldots ,2m\}$. Lemma: Let $R$ be any subset of $X^{2m}$ and $P$ any probability distribution on $X$. Then, $P^{2m}(R)=E[\Pr[\sigma (x)\in R]]\leq \max _{x\in X^{2m}}(\Pr[\sigma (x)\in R]),$ where the expectation is over $x$ chosen according to $P^{2m}$, and the probability is over $\sigma $ chosen uniformly from $\Gamma _{m}$. Proof: For any $\sigma \in \Gamma _{m},$ $P^{2m}(R)=P^{2m}\{x:\sigma (x)\in R\}$ (since coordinate permutations preserve the product distribution $P^{2m}$.) ${\begin{aligned}\therefore P^{2m}(R)={}&\int _{X^{2m}}1_{R}(x)\,dP^{2m}(x)\\[5pt]={}&{\frac {1}{\|\Gamma _{m}\|}}\sum _{\sigma \in \Gamma _{m}}\int _{X^{2m}}1_{R}(\sigma (x))\,dP^{2m}(x)\\[5pt]={}&\int _{X^{2m}}{\frac {1}{\|\Gamma _{m}\|}}\sum _{\sigma \in \Gamma _{m}}1_{R}(\sigma (x))\,dP^{2m}(x)\\[5pt]&{\text{(because }}\|\Gamma _{m}\|{\text{ is finite)}}\\[5pt]={}&\int _{X^{2m}}\Pr[\sigma (x)\in R]\,dP^{2m}(x)\quad {\text{(the expectation)}}\\[5pt]\leq {}&\max _{x\in X^{2m}}(\Pr[\sigma (x)\in R]).\end{aligned}}$ The maximum is guaranteed to exist since there is only a finite set of values that probability under a random permutation can take. Reduction to a finite class Lemma: Basing on the previous lemma, $\max _{x\in X^{2m}}(\Pr[\sigma (x)\in R])\leq 4\Pi _{H}(2m)e^{-\varepsilon ^{2}m/8}$. Proof: Let us define $x=(x_{1},x_{2},\ldots ,x_{2m})$ and $t=\|H\|_{x}\|$ which is at most $\Pi _{H}(2m)$. This means there are functions $h_{1},h_{2},\ldots ,h_{t}\in H$ such that for any $h\in H,\exists i$ between $1$ and $t$ with $h_{i}(x_{k})=h(x_{k})$ for $1\leq k\leq 2m.$ We see that $\sigma (x)\in R$ iff for some $h$ in $H$ satisfies, $\|{\frac {1}{m}}\|\{1\leq i\leq m:h(x_{\sigma _{i}})=1\}\|-{\frac {1}{m}}\|\{m+1\leq i\leq 2m:h(x_{\sigma _{i}})=1\}\|\|\geq {\frac {\varepsilon }{2}}$. Hence if we define $w_{i}^{j}=1$ if $h_{j}(x_{i})=1$ and $w_{i}^{j}=0$ otherwise. For $1\leq i\leq m$ and $1\leq j\leq t$, we have that $\sigma (x)\in R$ iff for some $j$ in ${1,\ldots ,t}$ satisfies $\|{\frac {1}{m}}\left(\sum _{i}w_{\sigma (i)}^{j}-\sum _{i}w_{\sigma (m+i)}^{j}\right)\|\geq {\frac {\varepsilon }{2}}$. By union bound we get $\Pr[\sigma (x)\in R]\leq t\cdot \max \left(\Pr[\|{\frac {1}{m}}\left(\sum _{i}w_{\sigma _{i}}^{j}-\sum _{i}w_{\sigma _{m+i}}^{j}\right)\|\geq {\frac {\varepsilon }{2}}]\right)$ $\leq \Pi _{H}(2m)\cdot \max \left(\Pr \left[\left\|{\frac {1}{m}}\left(\sum _{i}w_{\sigma _{i}}^{j}-\sum _{i}w_{\sigma _{m+i}}^{j}\right)\right\|\geq {\frac {\varepsilon }{2}}\right]\right).$ Since, the distribution over the permutations $\sigma $ is uniform for each $i$, so $w_{\sigma _{i}}^{j}-w_{\sigma _{m+i}}^{j}$ equals $\pm \|w_{i}^{j}-w_{m+i}^{j}\|$, with equal probability. Thus, $\Pr \left[\left\|{\frac {1}{m}}\left(\sum _{i}\left(w_{\sigma _{i}}^{j}-w_{\sigma _{m+i}}^{j}\right)\right)\right\|\geq {\frac {\varepsilon }{2}}\right]=\Pr \left[\left\|{\frac {1}{m}}\left(\sum _{i}\|w_{i}^{j}-w_{m+i}^{j}\|\beta _{i}\right)\right\|\geq {\frac {\varepsilon }{2}}\right],$ where the probability on the right is over $\beta _{i}$ and both the possibilities are equally likely. By Hoeffding's inequality, this is at most $2e^{-m\varepsilon ^{2}/8}$. Finally, combining all the three parts of the proof we get the Uniform Convergence Theorem. References 1. Vapnik, V. N.; Chervonenkis, A. Ya. (1971). "On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities". Theory of Probability & Its Applications. 16 (2): 264. doi:10.1137/1116025. This is an English translation, by B. Seckler, of the Russian paper: "On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities". Dokl. Akad. Nauk. 181 (4): 781. 1968. The translation was reproduced as: Vapnik, V. N.; Chervonenkis, A. Ya. (2015). "On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities". Measures of Complexity. p. 11. doi:10.1007/978-3-319-21852-6_3. ISBN 978-3-319-21851-9. 2. Martin Anthony Peter, l. Bartlett. Neural Network Learning: Theoretical Foundations, pages 46–50. First Edition, 1999. Cambridge University Press ISBN 0-521-57353-X 3. Sham Kakade and Ambuj Tewari, CMSC 35900 (Spring 2008) Learning Theory, Lecture 11	Wikipedia
Convex uniform honeycomb In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. Twenty-eight such honeycombs are known: • the familiar cubic honeycomb and 7 truncations thereof; • the alternated cubic honeycomb and 4 truncations thereof; • 10 prismatic forms based on the uniform plane tilings (11 if including the cubic honeycomb); • 5 modifications of some of the above by elongation and/or gyration. They can be considered the three-dimensional analogue to the uniform tilings of the plane. The Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra. History • 1900: Thorold Gosset enumerated the list of semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions, including one regular cubic honeycomb, and two semiregular forms with tetrahedra and octahedra. • 1905: Alfredo Andreini enumerated 25 of these tessellations. • 1991: Norman Johnson's manuscript Uniform Polytopes identified the list of 28.[1] • 1994: Branko Grünbaum, in his paper Uniform tilings of 3-space, also independently enumerated all 28, after discovering errors in Andreini's publication. He found the 1905 paper, which listed 25, had 1 wrong, and 4 being missing. Grünbaum states in this paper that Norman Johnson deserves priority for achieving the same enumeration in 1991. He also mentions that I. Alexeyev of Russia had contacted him regarding a putative enumeration of these forms, but that Grünbaum was unable to verify this at the time. • 2006: George Olshevsky, in his manuscript Uniform Panoploid Tetracombs, along with repeating the derived list of 11 convex uniform tilings, and 28 convex uniform honeycombs, expands a further derived list of 143 convex uniform tetracombs (Honeycombs of uniform 4-polytopes in 4-space).[2][1] Only 14 of the convex uniform polyhedra appear in these patterns: • three of the five Platonic solids (the tetrahedron, cube, and octahedron), • six of the thirteen Archimedean solids (the ones with reflective tetrahedral or octahedral symmetry), and • five of the infinite family of prisms (the 3-, 4-, 6-, 8-, and 12-gonal ones; the 4-gonal prism duplicates the cube). The icosahedron, snub cube, and square antiprism appear in some alternations, but those honeycombs cannot be realised with all edges unit length. Names This set can be called the regular and semiregular honeycombs. It has been called the Archimedean honeycombs by analogy with the convex uniform (non-regular) polyhedra, commonly called Archimedean solids. Recently Conway has suggested naming the set as the Architectonic tessellations and the dual honeycombs as the Catoptric tessellations. The individual honeycombs are listed with names given to them by Norman Johnson. (Some of the terms used below are defined in Uniform 4-polytope#Geometric derivations for 46 nonprismatic Wythoffian uniform 4-polytopes) For cross-referencing, they are given with list indices from Andreini (1-22), Williams(1-2,9-19), Johnson (11-19, 21–25, 31–34, 41–49, 51–52, 61–65), and Grünbaum(1-28). Coxeter uses δ4 for a cubic honeycomb, hδ4 for an alternated cubic honeycomb, qδ4 for a quarter cubic honeycomb, with subscripts for other forms based on the ring patterns of the Coxeter diagram. Compact Euclidean uniform tessellations (by their infinite Coxeter group families) The fundamental infinite Coxeter groups for 3-space are: 1. The ${\tilde {C}}_{3}$, [4,3,4], cubic, (8 unique forms plus one alternation) 2. The ${\tilde {B}}_{3}$, [4,31,1], alternated cubic, (11 forms, 3 new) 3. The ${\tilde {A}}_{3}$ cyclic group, [(3,3,3,3)] or [3[4]], (5 forms, one new) There is a correspondence between all three families. Removing one mirror from ${\tilde {C}}_{3}$ produces ${\tilde {B}}_{3}$, and removing one mirror from ${\tilde {B}}_{3}$ produces ${\tilde {A}}_{3}$. This allows multiple constructions of the same honeycombs. If cells are colored based on unique positions within each Wythoff construction, these different symmetries can be shown. In addition there are 5 special honeycombs which don't have pure reflectional symmetry and are constructed from reflectional forms with elongation and gyration operations. The total unique honeycombs above are 18. The prismatic stacks from infinite Coxeter groups for 3-space are: 1. The ${\tilde {C}}_{2}$×${\tilde {I}}_{1}$, [4,4,2,∞] prismatic group, (2 new forms) 2. The ${\tilde {G}}_{2}$×${\tilde {I}}_{1}$, [6,3,2,∞] prismatic group, (7 unique forms) 3. The ${\tilde {A}}_{2}$×${\tilde {I}}_{1}$, [(3,3,3),2,∞] prismatic group, (No new forms) 4. The ${\tilde {I}}_{1}$×${\tilde {I}}_{1}$×${\tilde {I}}_{1}$, [∞,2,∞,2,∞] prismatic group, (These all become a cubic honeycomb) In addition there is one special elongated form of the triangular prismatic honeycomb. The total unique prismatic honeycombs above (excluding the cubic counted previously) are 10. Combining these counts, 18 and 10 gives us the total 28 uniform honeycombs. The C̃3, [4,3,4] group (cubic) The regular cubic honeycomb, represented by Schläfli symbol {4,3,4}, offers seven unique derived uniform honeycombs via truncation operations. (One redundant form, the runcinated cubic honeycomb, is included for completeness though identical to the cubic honeycomb.) The reflectional symmetry is the affine Coxeter group [4,3,4]. There are four index 2 subgroups that generate alternations: [1+,4,3,4], [(4,3,4,2+)], [4,3+,4], and [4,3,4]+, with the first two generated repeated forms, and the last two are nonuniform. C3 honeycombs Space group Fibrifold Extended symmetry Extended diagram Order Honeycombs Pm3m (221) 4−:2 [4,3,4] ×1 1, 2, 3, 4, 5, 6 Fm3m (225) 2−:2 [1+,4,3,4] ↔ [4,31,1] ↔ Half 7, 11, 12, 13 I43m (217) 4o:2 [[(4,3,4,2+)]] Half × 2 (7), Fd3m (227) 2+:2 [[1+,4,3,4,1+]] ↔ [[3[4]]] ↔ Quarter × 2 10, Im3m (229) 8o:2 [[4,3,4]] ×2 (1), 8, 9 [4,3,4], space group Pm3m (221) Reference Indices Honeycomb name Coxeter diagram and Schläfli symbol Cell counts/vertex and positions in cubic honeycomb Frames (Perspective) Vertex figure Dual cell (0) (1) (2) (3) Alt Solids (Partial) J11,15 A1 W1 G22 δ4 cubic (chon) t0{4,3,4} {4,3,4} (8) (4.4.4) octahedron Cube, J12,32 A15 W14 G7 O1 rectified cubic (rich) t1{4,3,4} r{4,3,4} (2) (3.3.3.3) (4) (3.4.3.4) cuboid Square bipyramid J13 A14 W15 G8 t1δ4 O15 truncated cubic (tich) t0,1{4,3,4} t{4,3,4} (1) (3.3.3.3) (4) (3.8.8) square pyramid Isosceles square pyramid J14 A17 W12 G9 t0,2δ4 O14 cantellated cubic (srich) t0,2{4,3,4} rr{4,3,4} (1) (3.4.3.4) (2) (4.4.4) (2) (3.4.4.4) oblique triangular prism Triangular bipyramid J17 A18 W13 G25 t0,1,2δ4 O17 cantitruncated cubic (grich) t0,1,2{4,3,4} tr{4,3,4} (1) (4.6.6) (1) (4.4.4) (2) (4.6.8) irregular tetrahedron Triangular pyramidille J18 A19 W19 G20 t0,1,3δ4 O19 runcitruncated cubic (prich) t0,1,3{4,3,4} (1) (3.4.4.4) (1) (4.4.4) (2) (4.4.8) (1) (3.8.8) oblique trapezoidal pyramid Square quarter pyramidille J21,31,51 A2 W9 G1 hδ4 O21 alternated cubic (octet) h{4,3,4} (8) (3.3.3) (6) (3.3.3.3) cuboctahedron Dodecahedrille J22,34 A21 W17 G10 h2δ4 O25 Cantic cubic (tatoh) ↔ (1) (3.4.3.4) (2) (3.6.6) (2) (4.6.6) rectangular pyramid Half oblate octahedrille J23 A16 W11 G5 h3δ4 O26 Runcic cubic (sratoh) ↔ (1) (4.4.4) (1) (3.3.3) (3) (3.4.4.4) tapered triangular prism Quarter cubille J24 A20 W16 G21 h2,3δ4 O28 Runcicantic cubic (gratoh) ↔ (1) (3.8.8) (1) (3.6.6) (2) (4.6.8) Irregular tetrahedron Half pyramidille Nonuniformb snub rectified cubic (serch) sr{4,3,4} (1) (3.3.3.3.3) (1) (3.3.3) (2) (3.3.3.3.4) (4) (3.3.3) Irr. tridiminished icosahedron Nonuniform Cantic snub cubic (casch) 2s0{4,3,4} (1) (3.3.3.3.3) (2) (3.4.4.4) (3) (3.4.4) Nonuniform Runcicantic snub cubic (rusch) (1) (3.4.3.4) (2) (4.4.4) (1) (3.3.3) (1) (3.6.6) (3) Tricup Nonuniform Runcic cantitruncated cubic (esch) sr3{4,3,4} (1) (3.3.3.3.4) (1) (4.4.4) (1) (4.4.4) (1) (3.4.4.4) (3) (3.4.4) [[4,3,4]] honeycombs, space group Im3m (229) Reference Indices Honeycomb name Coxeter diagram and Schläfli symbol Cell counts/vertex and positions in cubic honeycomb Solids (Partial) Frames (Perspective) Vertex figure Dual cell (0,3) (1,2) Alt J11,15 A1 W1 G22 δ4 O1 runcinated cubic (same as regular cubic) (chon) t0,3{4,3,4} (2) (4.4.4) (6) (4.4.4) octahedron Cube J16 A3 W2 G28 t1,2δ4 O16 bitruncated cubic (batch) t1,2{4,3,4} 2t{4,3,4} (4) (4.6.6) (disphenoid) Oblate tetrahedrille J19 A22 W18 G27 t0,1,2,3δ4 O20 omnitruncated cubic (gippich) t0,1,2,3{4,3,4} (2) (4.6.8) (2) (4.4.8) irregular tetrahedron Eighth pyramidille J21,31,51 A2 W9 G1 hδ4 O27 Quarter cubic honeycomb (batatoh) ht0ht3{4,3,4} (2) (3.3.3) (6) (3.6.6) elongated triangular antiprism Oblate cubille J21,31,51 A2 W9 G1 hδ4 O21 Alternated runcinated cubic (octet) (same as alternated cubic) ht0,3{4,3,4} (2) (3.3.3) (6) (3.3.3) (6) (3.3.3.3) cuboctahedron Nonuniform Biorthosnub cubic honeycomb (gabreth) 2s0,3{(4,2,4,3)} (2) (4.6.6) (2) (4.4.4) (2) (4.4.6) Nonuniforma Alternated bitruncated cubic (bisch) h2t{4,3,4} (4) (3.3.3.3.3) (4) (3.3.3) Nonuniform Cantic bisnub cubic (cabisch) 2s0,3{4,3,4} (2) (3.4.4.4) (2) (4.4.4) (2) (4.4.4) Nonuniformc Alternated omnitruncated cubic (snich) ht0,1,2,3{4,3,4} (2) (3.3.3.3.4) (2) (3.3.3.4) (4) (3.3.3) B̃3, [4,31,1] group The ${\tilde {B}}_{3}$, [4,3] group offers 11 derived forms via truncation operations, four being unique uniform honeycombs. There are 3 index 2 subgroups that generate alternations: [1+,4,31,1], [4,(31,1)+], and [4,31,1]+. The first generates repeated honeycomb, and the last two are nonuniform but included for completeness. The honeycombs from this group are called alternated cubic because the first form can be seen as a cubic honeycomb with alternate vertices removed, reducing cubic cells to tetrahedra and creating octahedron cells in the gaps. Nodes are indexed left to right as 0,1,0',3 with 0' being below and interchangeable with 0. The alternate cubic names given are based on this ordering. B3 honeycombs Space group Fibrifold Extended symmetry Extended diagram Order Honeycombs Fm3m (225) 2−:2 [4,31,1] ↔ [4,3,4,1+] ↔ ×1 1, 2, 3, 4 Fm3m (225) 2−:2 <[1+,4,31,1]> ↔ <[3[4]]> ↔ ×2 (1), (3) Pm3m (221) 4−:2 <[4,31,1]> ×2 5, 6, 7, (6), 9, 10, 11 [4,31,1] uniform honeycombs, space group Fm3m (225) Referenced indices Honeycomb name Coxeter diagrams Cells by location (and count around each vertex) Solids (Partial) Frames (Perspective) vertex figure (0) (1) (0') (3) J21,31,51 A2 W9 G1 hδ4 O21 Alternated cubic (octet) ↔ (6) (3.3.3.3) (8) (3.3.3) cuboctahedron J22,34 A21 W17 G10 h2δ4 O25 Cantic cubic (tatoh) ↔ (1) (3.4.3.4) (2) (4.6.6) (2) (3.6.6) rectangular pyramid J23 A16 W11 G5 h3δ4 O26 Runcic cubic (sratoh) ↔ (1) cube (3) (3.4.4.4) (1) (3.3.3) tapered triangular prism J24 A20 W16 G21 h2,3δ4 O28 Runcicantic cubic (gratoh) ↔ (1) (3.8.8) (2) (4.6.8) (1) (3.6.6) Irregular tetrahedron <[4,31,1]> uniform honeycombs, space group Pm3m (221) Referenced indices Honeycomb name Coxeter diagrams ↔ Cells by location (and count around each vertex) Solids (Partial) Frames (Perspective) vertex figure (0,0') (1) (3) Alt J11,15 A1 W1 G22 δ4 O1 Cubic (chon) ↔ (8) (4.4.4) octahedron J12,32 A15 W14 G7 t1δ4 O15 Rectified cubic (rich) ↔ (4) (3.4.3.4) (2) (3.3.3.3) cuboid Rectified cubic (rich) ↔ (2) (3.3.3.3) (4) (3.4.3.4) cuboid J13 A14 W15 G8 t0,1δ4 O14 Truncated cubic (tich) ↔ (4) (3.8.8) (1) (3.3.3.3) square pyramid J14 A17 W12 G9 t0,2δ4 O17 Cantellated cubic (srich) ↔ (2) (3.4.4.4) (2) (4.4.4) (1) (3.4.3.4) obilique triangular prism J16 A3 W2 G28 t0,2δ4 O16 Bitruncated cubic (batch) ↔ (2) (4.6.6) (2) (4.6.6) isosceles tetrahedron J17 A18 W13 G25 t0,1,2δ4 O18 Cantitruncated cubic (grich) ↔ (2) (4.6.8) (1) (4.4.4) (1) (4.6.6) irregular tetrahedron J21,31,51 A2 W9 G1 hδ4 O21 Alternated cubic (octet) ↔ (8) (3.3.3) (6) (3.3.3.3) cuboctahedron J22,34 A21 W17 G10 h2δ4 O25 Cantic cubic (tatoh) ↔ (2) (3.6.6) (1) (3.4.3.4) (2) (4.6.6) rectangular pyramid Nonuniforma Alternated bitruncated cubic (bisch) ↔ (2) (3.3.3.3.3) (2) (3.3.3.3.3) (4) (3.3.3) Nonuniformb Alternated cantitruncated cubic (serch) ↔ (2) (3.3.3.3.4) (1) (3.3.3) (1) (3.3.3.3.3) (4) (3.3.3) Irr. tridiminished icosahedron Ã3, [3[4]] group There are 5 forms[3] constructed from the ${\tilde {A}}_{3}$, [3[4]] Coxeter group, of which only the quarter cubic honeycomb is unique. There is one index 2 subgroup [3[4]]+ which generates the snub form, which is not uniform, but included for completeness. A3 honeycombs Space group Fibrifold Square symmetry Extended symmetry Extended diagram Extended group Honeycomb diagrams F43m (216) 1o:2 a1 [3[4]] ${\tilde {A}}_{3}$ (None) Fm3m (225) 2−:2 d2 <[3[4]]> ↔ [4,31,1] ↔ ${\tilde {A}}_{3}$×21 ↔ ${\tilde {B}}_{3}$ 1, 2 Fd3m (227) 2+:2 g2 [[3[4]]] or [2+[3[4]]] ↔ ${\tilde {A}}_{3}$×22 3 Pm3m (221) 4−:2 d4 <2[3[4]]> ↔ [4,3,4] ↔ ${\tilde {A}}_{3}$×41 ↔ ${\tilde {C}}_{3}$ 4 I3 (204) 8−o r8 [4[3[4]]]+ ↔ [[4,3+,4]] ↔ ½${\tilde {A}}_{3}$×8 ↔ ½${\tilde {C}}_{3}$×2 (*) Im3m (229) 8o:2 [4[3[4]]] ↔ [[4,3,4]] ${\tilde {A}}_{3}$×8 ↔ ${\tilde {C}}_{3}$×2 5 [[3[4]]] uniform honeycombs, space group Fd3m (227) Referenced indices Honeycomb name Coxeter diagrams Cells by location (and count around each vertex) Solids (Partial) Frames (Perspective) vertex figure (0,1) (2,3) J25,33 A13 W10 G6 qδ4 O27 quarter cubic (batatoh) ↔ q{4,3,4} (2) (3.3.3) (6) (3.6.6) triangular antiprism <[3[4]]> ↔ [4,31,1] uniform honeycombs, space group Fm3m (225) Referenced indices Honeycomb name Coxeter diagrams ↔ Cells by location (and count around each vertex) Solids (Partial) Frames (Perspective) vertex figure 0 (1,3) 2 J21,31,51 A2 W9 G1 hδ4 O21 alternated cubic (octet) ↔ ↔ h{4,3,4} (8) (3.3.3) (6) (3.3.3.3) cuboctahedron J22,34 A21 W17 G10 h2δ4 O25 cantic cubic (tatoh) ↔ ↔ h2{4,3,4} (2) (3.6.6) (1) (3.4.3.4) (2) (4.6.6) Rectangular pyramid [2[3[4]]] ↔ [4,3,4] uniform honeycombs, space group Pm3m (221) Referenced indices Honeycomb name Coxeter diagrams ↔ Cells by location (and count around each vertex) Solids (Partial) Frames (Perspective) vertex figure (0,2) (1,3) J12,32 A15 W14 G7 t1δ4 O1 rectified cubic (rich) ↔ ↔ ↔ r{4,3,4} (2) (3.4.3.4) (1) (3.3.3.3) cuboid [4[3[4]]] ↔ [[4,3,4]] uniform honeycombs, space group Im3m (229) Referenced indices Honeycomb name Coxeter diagrams ↔ ↔ Cells by location (and count around each vertex) Solids (Partial) Frames (Perspective) vertex figure (0,1,2,3) Alt J16 A3 W2 G28 t1,2δ4 O16 bitruncated cubic (batch) ↔ ↔ 2t{4,3,4} (4) (4.6.6) isosceles tetrahedron Nonuniforma Alternated cantitruncated cubic (bisch) ↔ ↔ h2t{4,3,4} (4) (3.3.3.3.3) (4) (3.3.3) Nonwythoffian forms (gyrated and elongated) Three more uniform honeycombs are generated by breaking one or another of the above honeycombs where its faces form a continuous plane, then rotating alternate layers by 60 or 90 degrees (gyration) and/or inserting a layer of prisms (elongation). The elongated and gyroelongated alternated cubic tilings have the same vertex figure, but are not alike. In the elongated form, each prism meets a tetrahedron at one triangular end and an octahedron at the other. In the gyroelongated form, prisms that meet tetrahedra at both ends alternate with prisms that meet octahedra at both ends. The gyroelongated triangular prismatic tiling has the same vertex figure as one of the plain prismatic tilings; the two may be derived from the gyrated and plain triangular prismatic tilings, respectively, by inserting layers of cubes. Referenced indices symbol Honeycomb name cell types (# at each vertex) Solids (Partial) Frames (Perspective) vertex figure J52 A2' G2 O22 h{4,3,4}:g gyrated alternated cubic (gytoh) tetrahedron (8) octahedron (6) triangular orthobicupola J61 A? G3 O24 h{4,3,4}:ge gyroelongated alternated cubic (gyetoh) triangular prism (6) tetrahedron (4) octahedron (3) J62 A? G4 O23 h{4,3,4}:e elongated alternated cubic (etoh) triangular prism (6) tetrahedron (4) octahedron (3) J63 A? G12 O12 {3,6}:g × {∞} gyrated triangular prismatic (gytoph) triangular prism (12) J64 A? G15 O13 {3,6}:ge × {∞} gyroelongated triangular prismatic (gyetaph) triangular prism (6) cube (4) Prismatic stacks Eleven prismatic tilings are obtained by stacking the eleven uniform plane tilings, shown below, in parallel layers. (One of these honeycombs is the cubic, shown above.) The vertex figure of each is an irregular bipyramid whose faces are isosceles triangles. The C̃2×Ĩ1(∞), [4,4,2,∞], prismatic group There are only 3 unique honeycombs from the square tiling, but all 6 tiling truncations are listed below for completeness, and tiling images are shown by colors corresponding to each form. Indices Coxeter-Dynkin and Schläfli symbols Honeycomb name Plane tiling Solids (Partial) Tiling J11,15 A1 G22 {4,4}×{∞} Cubic (Square prismatic) (chon) (4.4.4.4) r{4,4}×{∞} rr{4,4}×{∞} J45 A6 G24 t{4,4}×{∞} Truncated/Bitruncated square prismatic (tassiph) (4.8.8) tr{4,4}×{∞} J44 A11 G14 sr{4,4}×{∞} Snub square prismatic (sassiph) (3.3.4.3.4) Nonuniform ht0,1,2,3{4,4,2,∞} The G̃2xĨ1(∞), [6,3,2,∞] prismatic group Indices Coxeter-Dynkin and Schläfli symbols Honeycomb name Plane tiling Solids (Partial) Tiling J41 A4 G11 {3,6} × {∞} Triangular prismatic (tiph) (36) J42 A5 G26 {6,3} × {∞} Hexagonal prismatic (hiph) (63) t{3,6} × {∞} J43 A8 G18 r{6,3} × {∞} Trihexagonal prismatic (thiph) (3.6.3.6) J46 A7 G19 t{6,3} × {∞} Truncated hexagonal prismatic (thaph) (3.12.12) J47 A9 G16 rr{6,3} × {∞} Rhombi-trihexagonal prismatic (srothaph) (3.4.6.4) J48 A12 G17 sr{6,3} × {∞} Snub hexagonal prismatic (snathaph) (3.3.3.3.6) J49 A10 G23 tr{6,3} × {∞} truncated trihexagonal prismatic (grothaph) (4.6.12) J65 A11' G13 {3,6}:e × {∞} elongated triangular prismatic (etoph) (3.3.3.4.4) J52 A2' G2 h3t{3,6,2,∞} gyrated tetrahedral-octahedral (gytoh) (36) s2r{3,6,2,∞} Nonuniform ht0,1,2,3{3,6,2,∞} Enumeration of Wythoff forms All nonprismatic Wythoff constructions by Coxeter groups are given below, along with their alternations. Uniform solutions are indexed with Branko Grünbaum's listing. Green backgrounds are shown on repeated honeycombs, with the relations are expressed in the extended symmetry diagrams. Coxeter group Extended symmetry Honeycombs Chiral extended symmetry Alternation honeycombs [4,3,4] [4,3,4] 6 7 \| 8 9 \| 25 \| 20 [1+,4,3+,4,1+](2) b [2+[4,3,4]] = (1) 22 [2+[(4,3+,4,2+)]](1) 6 [2+[4,3,4]] 1 28 [2+[(4,3+,4,2+)]](1) a [2+[4,3,4]] 2 27 [2+[4,3,4]]+(1) c [4,31,1] [4,31,1] 4 7 \| 10 \| 28 [1[4,31,1]]=[4,3,4] = (7) 22 \| 7 \| 22 \| 7 \| 9 \| 28 \| 25 [1[1+,4,31,1]]+(2) 6 \| a [1[4,31,1]]+ =[4,3,4]+ (1) b [3[4]] [3[4]] (none) [2+[3[4]]] 1 6 [1[3[4]]]=[4,31,1] = (2) 10 [2[3[4]]]=[4,3,4] = (1) 7 [(2+,4)[3[4]]]=[2+[4,3,4]] = (1) 28 [(2+,4)[3[4]]]+ = [2+[4,3,4]]+ (1)a Examples All 28 of these tessellations are found in crystal arrangements. The alternated cubic honeycomb is of special importance since its vertices form a cubic close-packing of spheres. The space-filling truss of packed octahedra and tetrahedra was apparently first discovered by Alexander Graham Bell and independently re-discovered by Buckminster Fuller (who called it the octet truss and patented it in the 1940s). . Octet trusses are now among the most common types of truss used in construction. Frieze forms If cells are allowed to be uniform tilings, more uniform honeycombs can be defined: Families: • ${\tilde {C}}_{2}$×$A_{1}$: [4,4,2] Cubic slab honeycombs (3 forms) • ${\tilde {G}}_{2}$×$A_{1}$: [6,3,2] Tri-hexagonal slab honeycombs (8 forms) • ${\tilde {A}}_{2}$×$A_{1}$: [(3,3,3),2] Triangular slab honeycombs (No new forms) • ${\tilde {I}}_{1}$×$A_{1}$×$A_{1}$: [∞,2,2] = Cubic column honeycombs (1 form) • $I_{2}(p)$×${\tilde {I}}_{1}$: [p,2,∞] Polygonal column honeycombs (analogous to duoprisms: these look like a single infinite tower of p-gonal prisms, with the remaining space filled with apeirogonal prisms) • ${\tilde {I}}_{1}$×${\tilde {I}}_{1}$×$A_{1}$: [∞,2,∞,2] = [4,4,2] - = (Same as cubic slab honeycomb family) Examples (partially drawn) Cubic slab honeycomb Alternated hexagonal slab honeycomb Trihexagonal slab honeycomb (4) 43: cube (1) 44: square tiling (4) 33: tetrahedron (3) 34: octahedron (1) 36: triangular tiling (2) 3.4.4: triangular prism (2) 4.4.6: hexagonal prism (1) (3.6)2: trihexagonal tiling The first two forms shown above are semiregular (uniform with only regular facets), and were listed by Thorold Gosset in 1900 respectively as the 3-ic semi-check and tetroctahedric semi-check.[4] Scaliform honeycomb A scaliform honeycomb is vertex-transitive, like a uniform honeycomb, with regular polygon faces while cells and higher elements are only required to be orbiforms, equilateral, with their vertices lying on hyperspheres. For 3D honeycombs, this allows a subset of Johnson solids along with the uniform polyhedra. Some scaliforms can be generated by an alternation process, leaving, for example, pyramid and cupola gaps.[5] Euclidean honeycomb scaliforms Frieze slabs Prismatic stacks s3{2,6,3}, s3{2,4,4}, s{2,4,4}, 3s4{4,4,2,∞}, (1) 3.4.3.4: triangular cupola (2) 3.4.6: triangular cupola (1) 3.3.3.3: octahedron (1) 3.6.3.6: trihexagonal tiling (1) 3.4.4.4: square cupola (2) 3.4.8: square cupola (1) 3.3.3: tetrahedron (1) 4.8.8: truncated square tiling (1) 3.3.3.3: square pyramid (4) 3.3.4: square pyramid (4) 3.3.3: tetrahedron (1) 4.4.4.4: square tiling (1) 3.3.3.3: square pyramid (4) 3.3.4: square pyramid (4) 3.3.3: tetrahedron (4) 4.4.4: cube Hyperbolic forms Main article: Uniform honeycombs in hyperbolic space There are 9 Coxeter group families of compact uniform honeycombs in hyperbolic 3-space, generated as Wythoff constructions, and represented by ring permutations of the Coxeter-Dynkin diagrams for each family. From these 9 families, there are a total of 76 unique honeycombs generated: • [3,5,3] : - 9 forms • [5,3,4] : - 15 forms • [5,3,5] : - 9 forms • [5,31,1] : - 11 forms (7 overlap with [5,3,4] family, 4 are unique) • [(4,3,3,3)] : - 9 forms • [(4,3,4,3)] : - 6 forms • [(5,3,3,3)] : - 9 forms • [(5,3,4,3)] : - 9 forms • [(5,3,5,3)] : - 6 forms Several non-Wythoffian forms outside the list of 76 are known; it is not known how many there are. Paracompact hyperbolic forms Main article: Paracompact uniform honeycombs There are also 23 paracompact Coxeter groups of rank 4. These families can produce uniform honeycombs with unbounded facets or vertex figure, including ideal vertices at infinity: Simplectic hyperbolic paracompact group summary Type Coxeter groups Unique honeycomb count Linear graphs \| \| \| \| \| 4×15+6+8+8 = 82 Tridental graphs \| 4+4+0 = 8 Cyclic graphs \| \| \| \| \| \| \| 4×9+5+1+4+1+0 = 47 Loop-n-tail graphs \| \| 4+4+4+2 = 14 References 1. Sloane, N. J. A. (ed.). "Sequence A242941 (Convex uniform tessellations in dimension n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2. George Olshevsky, (2006, Uniform Panoploid Tetracombs, Manuscript (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) 3. , A000029 6-1 cases, skipping one with zero marks 4. Gosset, Thorold (1900). "On the regular and semi-regular figures in space of n dimensions". Messenger of Mathematics. 29: 43–48. 5. "Polytope-tree". • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292–298, includes all the nonprismatic forms) • Branko Grünbaum, (1994) Uniform tilings of 3-space. Geombinatorics 4, 49 - 56. • Norman Johnson (1991) Uniform Polytopes, Manuscript • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Chapter 5: Polyhedra packing and space filling) • Critchlow, Keith (1970). Order in Space: A design source book. Viking Press. ISBN 0-500-34033-1. • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings) • A. Andreini, (1905) Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 75–129. PDF • D. M. Y. Sommerville, (1930) An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, . 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes • Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 5. Joining polyhedra • Crystallography of Quasicrystals: Concepts, Methods and Structures by Walter Steurer, Sofia Deloudi (2009), p. 54-55. 12 packings of 2 or more uniform polyhedra with cubic symmetry External links Wikimedia Commons has media related to Uniform tilings of Euclidean 3-space. • Weisstein, Eric W. "Honeycomb". MathWorld. • Uniform Honeycombs in 3-Space VRML models • Elementary Honeycombs Vertex transitive space filling honeycombs with non-uniform cells. • Uniform partitions of 3-space, their relatives and embedding, 1999 • The Uniform Polyhedra • Virtual Reality Polyhedra The Encyclopedia of Polyhedra • octet truss animation • Review: A. F. Wells, Three-dimensional nets and polyhedra, H. S. M. Coxeter (Source: Bull. Amer. Math. Soc. Volume 84, Number 3 (1978), 466-470.) • Klitzing, Richard. "3D Euclidean tesselations". • (sequence A242941 in the OEIS) Fundamental convex regular and uniform honeycombs in dimensions 2–9 Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$ E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4 E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6 E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222 E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331 E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521 E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10 E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11 En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21	Wikipedia
Uniform 4-polytope In geometry, a uniform 4-polytope (or uniform polychoron)[1] is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons. There are 47 non-prismatic convex uniform 4-polytopes. There are two infinite sets of convex prismatic forms, along with 17 cases arising as prisms of the convex uniform polyhedra. There are also an unknown number of non-convex star forms. History of discovery • Convex Regular polytopes: • 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 6 regular polytopes in 4 dimensions and only 3 in 5 or more dimensions. • Regular star 4-polytopes (star polyhedron cells and/or vertex figures) • 1852: Ludwig Schläfli also found 4 of the 10 regular star 4-polytopes, discounting 6 with cells or vertex figures {5/2,5} and {5,5/2}. • 1883: Edmund Hess completed the list of 10 of the nonconvex regular 4-polytopes, in his book (in German) Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder . • Convex semiregular polytopes: (Various definitions before Coxeter's uniform category) • 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions. In four dimensions, this gives the rectified 5-cell, the rectified 600-cell, and the snub 24-cell.[2] • 1910: Alicia Boole Stott, in her publication Geometrical deduction of semiregular from regular polytopes and space fillings, expanded the definition by also allowing Archimedean solid and prism cells. This construction enumerated 45 semiregular 4-polytopes, corresponding to the nonprismatic forms listed below. The snub 24-cell and grand antiprism were missing from her list.[3] • 1911: Pieter Hendrik Schoute published Analytic treatment of the polytopes regularly derived from the regular polytopes, followed Boole-Stott's notations, enumerating the convex uniform polytopes by symmetry based on 5-cell, 8-cell/16-cell, and 24-cell. • 1912: E. L. Elte independently expanded on Gosset's list with the publication The Semiregular Polytopes of the Hyperspaces, polytopes with one or two types of semiregular facets.[4] • Convex uniform polytopes: • 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes. • Convex uniform 4-polytopes: • 1965: The complete list of convex forms was finally enumerated by John Horton Conway and Michael Guy, in their publication Four-Dimensional Archimedean Polytopes, established by computer analysis, adding only one non-Wythoffian convex 4-polytope, the grand antiprism. • 1966 Norman Johnson completes his Ph.D. dissertation The Theory of Uniform Polytopes and Honeycombs under advisor Coxeter, completes the basic theory of uniform polytopes for dimensions 4 and higher. • 1986 Coxeter published a paper Regular and Semi-Regular Polytopes II which included analysis of the unique snub 24-cell structure, and the symmetry of the anomalous grand antiprism. • 1998[5]-2000: The 4-polytopes were systematically named by Norman Johnson, and given by George Olshevsky's online indexed enumeration (used as a basis for this listing). Johnson named the 4-polytopes as polychora, like polyhedra for 3-polytopes, from the Greek roots poly ("many") and choros ("room" or "space").[6] The names of the uniform polychora started with the 6 regular polychora with prefixes based on rings in the Coxeter diagrams; truncation t0,1, cantellation, t0,2, runcination t0,3, with single ringed forms called rectified, and bi,tri-prefixes added when the first ring was on the second or third nodes.[7][8] • 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope. Möller reproduced Johnson's naming system in his listing.[9] • 2008: The Symmetries of Things[10] was published by John H. Conway and contains the first print-published listing of the convex uniform 4-polytopes and higher dimensional polytopes by Coxeter group family, with general vertex figure diagrams for each ringed Coxeter diagram permutation—snub, grand antiprism, and duoprisms—which he called proprisms for product prisms. He used his own ijk-ambo naming scheme for the indexed ring permutations beyond truncation and bitruncation, and all of Johnson's names were included in the book index. • Nonregular uniform star 4-polytopes: (similar to the nonconvex uniform polyhedra) • 1966: Johnson describes three nonconvex uniform antiprisms in 4-space in his dissertation.[11] • 1990-2006: In a collaborative search, up to 2005 a total of 1845 uniform 4-polytopes (convex and nonconvex) had been identified by Jonathan Bowers and George Olshevsky,[12] with an additional four discovered in 2006 for a total of 1849. The count includes the 74 prisms of the 75 non-prismatic uniform polyhedra (since that is a finite set – the cubic prism is excluded as it duplicates the tesseract), but not the infinite categories of duoprisms or prisms of antiprisms.[13] • 2020-2023: 342 new polychora were found, bringing up the total number of known uniform 4-polytopes to 2191. The list has not been proven complete.[13][14] Regular 4-polytopes Regular 4-polytopes are a subset of the uniform 4-polytopes, which satisfy additional requirements. Regular 4-polytopes can be expressed with Schläfli symbol {p,q,r} have cells of type {p,q}, faces of type {p}, edge figures {r}, and vertex figures {q,r}. The existence of a regular 4-polytope {p,q,r} is constrained by the existence of the regular polyhedra {p,q} which becomes cells, and {q,r} which becomes the vertex figure. Existence as a finite 4-polytope is dependent upon an inequality:[15] $\sin \left({\frac {\pi }{p}}\right)\sin \left({\frac {\pi }{r}}\right)>\cos \left({\frac {\pi }{q}}\right).$ The 16 regular 4-polytopes, with the property that all cells, faces, edges, and vertices are congruent: • 6 regular convex 4-polytopes: 5-cell {3,3,3}, 8-cell {4,3,3}, 16-cell {3,3,4}, 24-cell {3,4,3}, 120-cell {5,3,3}, and 600-cell {3,3,5}. • 10 regular star 4-polytopes: icosahedral 120-cell {3,5,5/2}, small stellated 120-cell {5/2,5,3}, great 120-cell {5,5/2,5}, grand 120-cell {5,3,5/2}, great stellated 120-cell {5/2,3,5}, grand stellated 120-cell {5/2,5,5/2}, great grand 120-cell {5,5/2,3}, great icosahedral 120-cell {3,5/2,5}, grand 600-cell {3,3,5/2}, and great grand stellated 120-cell {5/2,3,3}. Convex uniform 4-polytopes Symmetry of uniform 4-polytopes in four dimensions Main article: Point groups in four dimensions Orthogonal subgroups The 24 mirrors of F4 can be decomposed into 2 orthogonal D4 groups: 1. = (12 mirrors) 2. = (12 mirrors) The 10 mirrors of B3×A1 can be decomposed into orthogonal groups, 4A1 and D3: 1. = (3+1 mirrors) 2. = (6 mirrors) There are 5 fundamental mirror symmetry point group families in 4-dimensions: A4 = , B4 = , D4 = , F4 = , H4 = .[7] There are also 3 prismatic groups A3A1 = , B3A1 = , H3A1 = , and duoprismatic groups: I2(p)×I2(q) = . Each group defined by a Goursat tetrahedron fundamental domain bounded by mirror planes. Each reflective uniform 4-polytope can be constructed in one or more reflective point group in 4 dimensions by a Wythoff construction, represented by rings around permutations of nodes in a Coxeter diagram. Mirror hyperplanes can be grouped, as seen by colored nodes, separated by even-branches. Symmetry groups of the form [a,b,a], have an extended symmetry, [[a,b,a]], doubling the symmetry order. This includes [3,3,3], [3,4,3], and [p,2,p]. Uniform polytopes in these group with symmetric rings contain this extended symmetry. If all mirrors of a given color are unringed (inactive) in a given uniform polytope, it will have a lower symmetry construction by removing all of the inactive mirrors. If all the nodes of a given color are ringed (active), an alternation operation can generate a new 4-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not generally adjustable to create uniform solutions. Weyl group Conway Quaternion Abstract structure Order Coxeter diagram Coxeter notation Commutator subgroup Coxeter number (h) Mirrors m=2h Irreducible A4 +1/60[I×I].21S5120[3,3,3][3,3,3]+510 D4 ±1/3[T×T].21/2.2S4192[31,1,1][31,1,1]+612 B4 ±1/6[O×O].22S4 = S2≀S4384[4,3,3]8412 F4 ±1/2[O×O].233.2S41152[3,4,3][3+,4,3+]121212 H4 ±[I×I].22.(A5×A5).214400[5,3,3][5,3,3]+3060 Prismatic groups A3A1 +1/24[O×O].23S4×D148[3,3,2] = [3,3]×[ ][3,3]+-61 B3A1 ±1/24[O×O].2S4×D196[4,3,2] = [4,3]×[ ]-361 H3A1 ±1/60[I×I].2A5×D1240[5,3,2] = [5,3]×[ ][5,3]+-151 Duoprismatic groups (Use 2p,2q for even integers) I2(p)I2(q) ±1/2[D2p×D2q]Dp×Dq4pq[p,2,q] = [p]×[q][p+,2,q+]-p q I2(2p)I2(q) ±1/2[D4p×D2q]D2p×Dq8pq[2p,2,q] = [2p]×[q]-p p q I2(2p)I2(2q) ±1/2[D4p×D4q]D2p×D2q16pq[2p,2,2q] = [2p]×[2q]-p p q q Enumeration There are 64 convex uniform 4-polytopes, including the 6 regular convex 4-polytopes, and excluding the infinite sets of the duoprisms and the antiprismatic prisms. • 5 are polyhedral prisms based on the Platonic solids (1 overlap with regular since a cubic hyperprism is a tesseract) • 13 are polyhedral prisms based on the Archimedean solids • 9 are in the self-dual regular A4 [3,3,3] group (5-cell) family. • 9 are in the self-dual regular F4 [3,4,3] group (24-cell) family. (Excluding snub 24-cell) • 15 are in the regular B4 [3,3,4] group (tesseract/16-cell) family (3 overlap with 24-cell family) • 15 are in the regular H4 [3,3,5] group (120-cell/600-cell) family. • 1 special snub form in the [3,4,3] group (24-cell) family. • 1 special non-Wythoffian 4-polytope, the grand antiprism. • TOTAL: 68 − 4 = 64 These 64 uniform 4-polytopes are indexed below by George Olshevsky. Repeated symmetry forms are indexed in brackets. In addition to the 64 above, there are 2 infinite prismatic sets that generate all of the remaining convex forms: • Set of uniform antiprismatic prisms - sr{p,2}×{ } - Polyhedral prisms of two antiprisms. • Set of uniform duoprisms - {p}×{q} - A Cartesian product of two polygons. The A4 family Further information: A4 polytope The 5-cell has diploid pentachoric [3,3,3] symmetry,[7] of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way. Facets (cells) are given, grouped in their Coxeter diagram locations by removing specified nodes. [3,3,3] uniform polytopes # Name Bowers name (and acronym) Vertex figure Coxeter diagram and Schläfli symbols Cell counts by location Element counts Pos. 3 (5) Pos. 2 (10) Pos. 1 (10) Pos. 0 (5) Cells Faces Edges Vertices 1 5-cell Pentachoron[7] (pen) {3,3,3} (4) (3.3.3) 5 10 10 5 2 rectified 5-cell Rectified pentachoron (rap) r{3,3,3} (3) (3.3.3.3) (2) (3.3.3) 10 30 30 10 3 truncated 5-cell Truncated pentachoron (tip) t{3,3,3} (3) (3.6.6) (1) (3.3.3) 10 30 40 20 4 cantellated 5-cell Small rhombated pentachoron (srip) rr{3,3,3} (2) (3.4.3.4) (2) (3.4.4) (1) (3.3.3.3) 20 80 90 30 7 cantitruncated 5-cell Great rhombated pentachoron (grip) tr{3,3,3} (2) (4.6.6) (1) (3.4.4) (1) (3.6.6) 20 80 120 60 8 runcitruncated 5-cell Prismatorhombated pentachoron (prip) t0,1,3{3,3,3} (1) (3.6.6) (2) (4.4.6) (1) (3.4.4) (1) (3.4.3.4) 30 120 150 60 [[3,3,3]] uniform polytopes # Name Bowers name (and acronym) Vertex figure Coxeter diagram and Schläfli symbols Cell counts by location Element counts Pos. 3-0 (10) Pos. 1-2 (20) Alt Cells Faces Edges Vertices 5 runcinated 5-cell Small prismatodecachoron (spid) t0,3{3,3,3} (2) (3.3.3) (6) (3.4.4) 30 70 60 20 6 bitruncated 5-cell Decachoron (deca) 2t{3,3,3} (4) (3.6.6) 10 40 60 30 9 omnitruncated 5-cell Great prismatodecachoron (gippid) t0,1,2,3{3,3,3} (2) (4.6.6) (2) (4.4.6) 30 150 240 120 Nonuniform omnisnub 5-cell Snub decachoron (snad) Snub pentachoron (snip)[16] ht0,1,2,3{3,3,3} (2) (3.3.3.3.3) (2) (3.3.3.3) (4) (3.3.3) 90 300 270 60 The three uniform 4-polytopes forms marked with an asterisk, , have the higher extended pentachoric symmetry, of order 240, [[3,3,3]] because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual. There is one small index subgroup [3,3,3]+, order 60, or its doubling [[3,3,3]]+, order 120, defining an omnisnub 5-cell which is listed for completeness, but is not uniform. The B4 family Further information: B4 polytope This family has diploid hexadecachoric symmetry,[7] [4,3,3], of order 24×16=384: 4!=24 permutations of the four axes, 24=16 for reflection in each axis. There are 3 small index subgroups, with the first two generate uniform 4-polytopes which are also repeated in other families, [1+,4,3,3], [4,(3,3)+], and [4,3,3]+, all order 192. Tesseract truncations # Name (Bowers name and acronym) Vertex figure Coxeter diagram and Schläfli symbols Cell counts by location Element counts Pos. 3 (8) Pos. 2 (24) Pos. 1 (32) Pos. 0 (16) Cells Faces Edges Vertices 10 tesseract or 8-cell Tesseract (tes) {4,3,3} (4) (4.4.4) 8 24 32 16 11 Rectified tesseract (rit) r{4,3,3} (3) (3.4.3.4) (2) (3.3.3) 24 88 96 32 13 Truncated tesseract (tat) t{4,3,3} (3) (3.8.8) (1) (3.3.3) 24 88 128 64 14 Cantellated tesseract Small rhombated tesseract (srit) rr{4,3,3} (2) (3.4.4.4) (2) (3.4.4) (1) (3.3.3.3) 56 248 288 96 15 Runcinated tesseract (also runcinated 16-cell) Small disprismatotesseractihexadecachoron (sidpith) t0,3{4,3,3} (1) (4.4.4) (3) (4.4.4) (3) (3.4.4) (1) (3.3.3) 80 208 192 64 16 Bitruncated tesseract (also bitruncated 16-cell) Tesseractihexadecachoron (tah) 2t{4,3,3} (2) (4.6.6) (2) (3.6.6) 24 120 192 96 18 Cantitruncated tesseract Great rhombated tesseract (grit) tr{4,3,3} (2) (4.6.8) (1) (3.4.4) (1) (3.6.6) 56 248 384 192 19 Runcitruncated tesseract Prismatorhombated hexadecachoron (proh) t0,1,3{4,3,3} (1) (3.8.8) (2) (4.4.8) (1) (3.4.4) (1) (3.4.3.4) 80 368 480 192 21 Omnitruncated tesseract (also omnitruncated 16-cell) Great disprismatotesseractihexadecachoron (gidpith) t0,1,2,3{3,3,4} (1) (4.6.8) (1) (4.4.8) (1) (4.4.6) (1) (4.6.6) 80 464 768 384 Related half tesseract, [1+,4,3,3] uniform 4-polytopes # Name (Bowers style acronym) Vertex figure Coxeter diagram and Schläfli symbols Cell counts by location Element counts Pos. 3 (8) Pos. 2 (24) Pos. 1 (32) Pos. 0 (16) Alt Cells Faces Edges Vertices 12 Half tesseract Demitesseract = 16-cell (hex) = h{4,3,3}={3,3,4} (4) (3.3.3) (4) (3.3.3) 16 32 24 8 [17] Cantic tesseract = Truncated 16-cell (thex) = h2{4,3,3}=t{4,3,3} (4) (6.6.3) (1) (3.3.3.3) 24 96 120 48 [11] Runcic tesseract = Rectified tesseract (rit) = h3{4,3,3}=r{4,3,3} (3) (3.4.3.4) (2) (3.3.3) 24 88 96 32 [16] Runcicantic tesseract = Bitruncated tesseract (tah) = h2,3{4,3,3}=2t{4,3,3} (2) (3.4.3.4) (2) (3.6.6) 24 120 192 96 [11] = Rectified tesseract (rat) = h1{4,3,3}=r{4,3,3} 24 88 96 32 [16] = Bitruncated tesseract (tah) = h1,2{4,3,3}=2t{4,3,3} 24 120 192 96 [23] = Rectified 24-cell (rico) = h1,3{4,3,3}=rr{3,3,4} 48 240 288 96 [24] = Truncated 24-cell (tico) = h1,2,3{4,3,3}=tr{3,3,4} 48 240 384 192 # Name (Bowers style acronym) Vertex figure Coxeter diagram and Schläfli symbols Cell counts by location Element counts Pos. 3 (8) Pos. 2 (24) Pos. 1 (32) Pos. 0 (16) Alt Cells Faces Edges Vertices Nonuniform omnisnub tesseract Snub tesseract (snet)[17] (Or omnisnub 16-cell) ht0,1,2,3{4,3,3} (1) (3.3.3.3.4) (1) (3.3.3.4) (1) (3.3.3.3) (1) (3.3.3.3.3) (4) (3.3.3) 272 944 864 192 16-cell truncations # Name (Bowers name and acronym) Vertex figure Coxeter diagram and Schläfli symbols Cell counts by location Element counts Pos. 3 (8) Pos. 2 (24) Pos. 1 (32) Pos. 0 (16) Alt Cells Faces Edges Vertices [12] 16-cell Hexadecachoron[7] (hex) {3,3,4} (8) (3.3.3) 16 32 24 8 [22] Rectified 16-cell (Same as 24-cell) (ico) = r{3,3,4} (2) (3.3.3.3) (4) (3.3.3.3) 24 96 96 24 17 Truncated 16-cell Truncated hexadecachoron (thex) t{3,3,4} (1) (3.3.3.3) (4) (3.6.6) 24 96 120 48 [23] Cantellated 16-cell (Same as rectified 24-cell) (rico) = rr{3,3,4} (1) (3.4.3.4) (2) (4.4.4) (2) (3.4.3.4) 48 240 288 96 [15] Runcinated 16-cell (also runcinated tesseract) (sidpith) t0,3{3,3,4} (1) (4.4.4) (3) (4.4.4) (3) (3.4.4) (1) (3.3.3) 80 208 192 64 [16] Bitruncated 16-cell (also bitruncated tesseract) (tah) 2t{3,3,4} (2) (4.6.6) (2) (3.6.6) 24 120 192 96 [24] Cantitruncated 16-cell (Same as truncated 24-cell) (tico) = tr{3,3,4} (1) (4.6.6) (1) (4.4.4) (2) (4.6.6) 48 240 384 192 20 Runcitruncated 16-cell Prismatorhombated tesseract (prit) t0,1,3{3,3,4} (1) (3.4.4.4) (1) (4.4.4) (2) (4.4.6) (1) (3.6.6) 80 368 480 192 [21] Omnitruncated 16-cell (also omnitruncated tesseract) (gidpith) t0,1,2,3{3,3,4} (1) (4.6.8) (1) (4.4.8) (1) (4.4.6) (1) (4.6.6) 80 464 768 384 [31] alternated cantitruncated 16-cell (Same as the snub 24-cell) (sadi) sr{3,3,4} (1) (3.3.3.3.3) (1) (3.3.3) (2) (3.3.3.3.3) (4) (3.3.3) 144 480 432 96 Nonuniform Runcic snub rectified 16-cell Pyritosnub tesseract (pysnet) sr3{3,3,4} (1) (3.4.4.4) (2) (3.4.4) (1) (4.4.4) (1) (3.3.3.3.3) (2) (3.4.4) 176 656 672 192 () Just as rectifying the tetrahedron produces the octahedron, rectifying the 16-cell produces the 24-cell, the regular member of the following family. The snub 24-cell is repeat to this family for completeness. It is an alternation of the cantitruncated 16-cell or truncated 24-cell, with the half symmetry group [(3,3)+,4]. The truncated octahedral cells become icosahedra. The cubes becomes tetrahedra, and 96 new tetrahedra are created in the gaps from the removed vertices. The F4 family Further information: F4 polytope This family has diploid icositetrachoric symmetry,[7] [3,4,3], of order 24×48=1152: the 48 symmetries of the octahedron for each of the 24 cells. There are 3 small index subgroups, with the first two isomorphic pairs generating uniform 4-polytopes which are also repeated in other families, [3+,4,3], [3,4,3+], and [3,4,3]+, all order 576. [3,4,3] uniform 4-polytopes # Name Vertex figure Coxeter diagram and Schläfli symbols Cell counts by location Element counts Pos. 3 (24) Pos. 2 (96) Pos. 1 (96) Pos. 0 (24) Cells Faces Edges Vertices 22 24-cell (Same as rectified 16-cell) Icositetrachoron[7] (ico) {3,4,3} (6) (3.3.3.3) 24 96 96 24 23 rectified 24-cell (Same as cantellated 16-cell) Rectified icositetrachoron (rico) r{3,4,3} (3) (3.4.3.4) (2) (4.4.4) 48 240 288 96 24 truncated 24-cell (Same as cantitruncated 16-cell) Truncated icositetrachoron (tico) t{3,4,3} (3) (4.6.6) (1) (4.4.4) 48 240 384 192 25 cantellated 24-cell Small rhombated icositetrachoron (srico) rr{3,4,3} (2) (3.4.4.4) (2) (3.4.4) (1) (3.4.3.4) 144 720 864 288 28 cantitruncated 24-cell Great rhombated icositetrachoron (grico) tr{3,4,3} (2) (4.6.8) (1) (3.4.4) (1) (3.8.8) 144 720 1152 576 29 runcitruncated 24-cell Prismatorhombated icositetrachoron (prico) t0,1,3{3,4,3} (1) (4.6.6) (2) (4.4.6) (1) (3.4.4) (1) (3.4.4.4) 240 1104 1440 576 [3+,4,3] uniform 4-polytopes # Name Vertex figure Coxeter diagram and Schläfli symbols Cell counts by location Element counts Pos. 3 (24) Pos. 2 (96) Pos. 1 (96) Pos. 0 (24) Alt Cells Faces Edges Vertices 31 †snub 24-cell Snub disicositetrachoron (sadi) s{3,4,3} (3) (3.3.3.3.3) (1) (3.3.3) (4) (3.3.3) 144 480 432 96 Nonuniform runcic snub 24-cell Prismatorhombisnub icositetrachoron (prissi) s3{3,4,3} (1) (3.3.3.3.3) (2) (3.4.4) (1) (3.6.6) (3) Tricup 240 960 1008 288 [25] cantic snub 24-cell (Same as cantellated 24-cell) (srico) s2{3,4,3} (2) (3.4.4.4) (1) (3.4.3.4) (2) (3.4.4) 144 720 864 288 [29] runcicantic snub 24-cell (Same as runcitruncated 24-cell) (prico) s2,3{3,4,3} (1) (4.6.6) (1) (3.4.4) (1) (3.4.4.4) (2) (4.4.6) 240 1104 1440 576 (†) The snub 24-cell here, despite its common name, is not analogous to the snub cube; rather, it is derived by an alternation of the truncated 24-cell. Its symmetry number is only 576, (the ionic diminished icositetrachoric group, [3+,4,3]). Like the 5-cell, the 24-cell is self-dual, and so the following three forms have twice as many symmetries, bringing their total to 2304 (extended icositetrachoric symmetry [[3,4,3]]). [[3,4,3]] uniform 4-polytopes # Name Vertex figure Coxeter diagram and Schläfli symbols Cell counts by location Element counts Pos. 3-0 (48) Pos. 2-1 (192) Cells Faces Edges Vertices 26 runcinated 24-cell Small prismatotetracontoctachoron (spic) t0,3{3,4,3} (2) (3.3.3.3) (6) (3.4.4) 240 672 576 144 27 bitruncated 24-cell Tetracontoctachoron (cont) 2t{3,4,3} (4) (3.8.8) 48 336 576 288 30 omnitruncated 24-cell Great prismatotetracontoctachoron (gippic) t0,1,2,3{3,4,3} (2) (4.6.8) (2) (4.4.6) 240 1392 2304 1152 [[3,4,3]]+ isogonal 4-polytope # Name Vertex figure Coxeter diagram and Schläfli symbols Cell counts by location Element counts Pos. 3-0 (48) Pos. 2-1 (192) Alt Cells Faces Edges Vertices Nonuniform omnisnub 24-cell Snub tetracontoctachoron (snoc) Snub icositetrachoron (sni)[18] ht0,1,2,3{3,4,3} (2) (3.3.3.3.4) (2) (3.3.3.3) (4) (3.3.3) 816 2832 2592 576 The H4 family Further information: H4 polytope This family has diploid hexacosichoric symmetry,[7] [5,3,3], of order 120×120=24×600=14400: 120 for each of the 120 dodecahedra, or 24 for each of the 600 tetrahedra. There is one small index subgroups [5,3,3]+, all order 7200. 120-cell truncations # Name (Bowers name and acronym) Vertex figure Coxeter diagram and Schläfli symbols Cell counts by location Element counts Pos. 3 (120) Pos. 2 (720) Pos. 1 (1200) Pos. 0 (600) Alt Cells Faces Edges Vertices 32 120-cell (hecatonicosachoron or dodecacontachoron)[7] Hecatonicosachoron (hi) {5,3,3} (4) (5.5.5) 120 720 1200 600 33 rectified 120-cell Rectified hecatonicosachoron (rahi) r{5,3,3} (3) (3.5.3.5) (2) (3.3.3) 720 3120 3600 1200 36 truncated 120-cell Truncated hecatonicosachoron (thi) t{5,3,3} (3) (3.10.10) (1) (3.3.3) 720 3120 4800 2400 37 cantellated 120-cell Small rhombated hecatonicosachoron (srahi) rr{5,3,3} (2) (3.4.5.4) (2) (3.4.4) (1) (3.3.3.3) 1920 9120 10800 3600 38 runcinated 120-cell (also runcinated 600-cell) Small disprismatohexacosihecatonicosachoron (sidpixhi) t0,3{5,3,3} (1) (5.5.5) (3) (4.4.5) (3) (3.4.4) (1) (3.3.3) 2640 7440 7200 2400 39 bitruncated 120-cell (also bitruncated 600-cell) Hexacosihecatonicosachoron (xhi) 2t{5,3,3} (2) (5.6.6) (2) (3.6.6) 720 4320 7200 3600 42 cantitruncated 120-cell Great rhombated hecatonicosachoron (grahi) tr{5,3,3} (2) (4.6.10) (1) (3.4.4) (1) (3.6.6) 1920 9120 14400 7200 43 runcitruncated 120-cell Prismatorhombated hexacosichoron (prix) t0,1,3{5,3,3} (1) (3.10.10) (2) (4.4.10) (1) (3.4.4) (1) (3.4.3.4) 2640 13440 18000 7200 46 omnitruncated 120-cell (also omnitruncated 600-cell) Great disprismatohexacosihecatonicosachoron (gidpixhi) t0,1,2,3{5,3,3} (1) (4.6.10) (1) (4.4.10) (1) (4.4.6) (1) (4.6.6) 2640 17040 28800 14400 Nonuniform omnisnub 120-cell Snub hecatonicosachoron (snahi)[19] (Same as the omnisnub 600-cell) ht0,1,2,3{5,3,3} (1) (3.3.3.3.5) (1) (3.3.3.5) (1) (3.3.3.3) (1) (3.3.3.3.3) (4) (3.3.3) 9840 35040 32400 7200 600-cell truncations # Name (Bowers style acronym) Vertex figure Coxeter diagram and Schläfli symbols Symmetry Cell counts by location Element counts Pos. 3 (120) Pos. 2 (720) Pos. 1 (1200) Pos. 0 (600) Cells Faces Edges Vertices 35 600-cell Hexacosichoron[7] (ex) {3,3,5} [5,3,3] order 14400 (20) (3.3.3) 600 1200 720 120 [47] 20-diminished 600-cell = Grand antiprism (gap) Nonwythoffian construction [[10,2+,10]] order 400 Index 36 (2) (3.3.3.5) (12) (3.3.3) 320 720 500 100 [31] 24-diminished 600-cell = Snub 24-cell (sadi) Nonwythoffian construction [3+,4,3] order 576 index 25 (3) (3.3.3.3.3) (5) (3.3.3) 144 480 432 96 Nonuniform bi-24-diminished 600-cell Bi-icositetradiminished hexacosichoron (bidex) Nonwythoffian construction order 144 index 100 (6) tdi 48 192 216 72 34 rectified 600-cell Rectified hexacosichoron (rox) r{3,3,5} [5,3,3] (2) (3.3.3.3.3) (5) (3.3.3.3) 720 3600 3600 720 Nonuniform 120-diminished rectified 600-cell Swirlprismatodiminished rectified hexacosichoron (spidrox) Nonwythoffian construction order 1200 index 12 (2) 3.3.3.5 (2) 4.4.5 (5) P4 840 2640 2400 600 41 truncated 600-cell Truncated hexacosichoron (tex) t{3,3,5} [5,3,3] (1) (3.3.3.3.3) (5) (3.6.6) 720 3600 4320 1440 40 cantellated 600-cell Small rhombated hexacosichoron (srix) rr{3,3,5} [5,3,3] (1) (3.5.3.5) (2) (4.4.5) (1) (3.4.3.4) 1440 8640 10800 3600 [38] runcinated 600-cell (also runcinated 120-cell) (sidpixhi) t0,3{3,3,5} [5,3,3] (1) (5.5.5) (3) (4.4.5) (3) (3.4.4) (1) (3.3.3) 2640 7440 7200 2400 [39] bitruncated 600-cell (also bitruncated 120-cell) (xhi) 2t{3,3,5} [5,3,3] (2) (5.6.6) (2) (3.6.6) 720 4320 7200 3600 45 cantitruncated 600-cell Great rhombated hexacosichoron (grix) tr{3,3,5} [5,3,3] (1) (5.6.6) (1) (4.4.5) (2) (4.6.6) 1440 8640 14400 7200 44 runcitruncated 600-cell Prismatorhombated hecatonicosachoron (prahi) t0,1,3{3,3,5} [5,3,3] (1) (3.4.5.4) (1) (4.4.5) (2) (4.4.6) (1) (3.6.6) 2640 13440 18000 7200 [46] omnitruncated 600-cell (also omnitruncated 120-cell) (gidpixhi) t0,1,2,3{3,3,5} [5,3,3] (1) (4.6.10) (1) (4.4.10) (1) (4.4.6) (1) (4.6.6) 2640 17040 28800 14400 The D4 family Further information: D4 polytope This demitesseract family, [31,1,1], introduces no new uniform 4-polytopes, but it is worthy to repeat these alternative constructions. This family has order 12×16=192: 4!/2=12 permutations of the four axes, half as alternated, 24=16 for reflection in each axis. There is one small index subgroups that generating uniform 4-polytopes, [31,1,1]+, order 96. [31,1,1] uniform 4-polytopes # Name (Bowers style acronym) Vertex figure Coxeter diagram = = Cell counts by location Element counts Pos. 0 (8) Pos. 2 (24) Pos. 1 (8) Pos. 3 (8) Pos. Alt (96) 3 2 1 0 [12] demitesseract half tesseract (Same as 16-cell) (hex) = h{4,3,3} (4) (3.3.3) (4) (3.3.3) 16 32 24 8 [17] cantic tesseract (Same as truncated 16-cell) (thex) = h2{4,3,3} (1) (3.3.3.3) (2) (3.6.6) (2) (3.6.6) 24 96 120 48 [11] runcic tesseract (Same as rectified tesseract) (rit) = h3{4,3,3} (1) (3.3.3) (1) (3.3.3) (3) (3.4.3.4) 24 88 96 32 [16] runcicantic tesseract (Same as bitruncated tesseract) (tah) = h2,3{4,3,3} (1) (3.6.6) (1) (3.6.6) (2) (4.6.6) 24 96 96 24 When the 3 bifurcated branch nodes are identically ringed, the symmetry can be increased by 6, as [3[31,1,1]] = [3,4,3], and thus these polytopes are repeated from the 24-cell family. [3[31,1,1]] uniform 4-polytopes # Name (Bowers style acronym) Vertex figure Coxeter diagram = = Cell counts by location Element counts Pos. 0,1,3 (24) Pos. 2 (24) Pos. Alt (96) 3 2 1 0 [22] rectified 16-cell (Same as 24-cell) (ico) = = = {31,1,1} = r{3,3,4} = {3,4,3} (6) (3.3.3.3) 48 240 288 96 [23] cantellated 16-cell (Same as rectified 24-cell) (rico) = = = r{31,1,1} = rr{3,3,4} = r{3,4,3} (3) (3.4.3.4) (2) (4.4.4) 24 120 192 96 [24] cantitruncated 16-cell (Same as truncated 24-cell) (tico) = = = t{31,1,1} = tr{3,3,4} = t{3,4,3} (3) (4.6.6) (1) (4.4.4) 48 240 384 192 [31] snub 24-cell (sadi) = = = s{31,1,1} = sr{3,3,4} = s{3,4,3} (3) (3.3.3.3.3) (1) (3.3.3) (4) (3.3.3) 144 480 432 96 Here again the snub 24-cell, with the symmetry group [31,1,1]+ this time, represents an alternated truncation of the truncated 24-cell creating 96 new tetrahedra at the position of the deleted vertices. In contrast to its appearance within former groups as partly snubbed 4-polytope, only within this symmetry group it has the full analogy to the Kepler snubs, i.e. the snub cube and the snub dodecahedron. The grand antiprism There is one non-Wythoffian uniform convex 4-polytope, known as the grand antiprism, consisting of 20 pentagonal antiprisms forming two perpendicular rings joined by 300 tetrahedra. It is loosely analogous to the three-dimensional antiprisms, which consist of two parallel polygons joined by a band of triangles. Unlike them, however, the grand antiprism is not a member of an infinite family of uniform polytopes. Its symmetry is the ionic diminished Coxeter group, [[10,2+,10]], order 400. # Name (Bowers style acronym) Picture Vertex figure Coxeter diagram and Schläfli symbols Cells by type Element counts Net Cells Faces Edges Vertices 47 grand antiprism (gap) No symbol 300 (3.3.3) 20 (3.3.3.5) 320 20 {5} 700 {3} 500 100 Prismatic uniform 4-polytopes A prismatic polytope is a Cartesian product of two polytopes of lower dimension; familiar examples are the 3-dimensional prisms, which are products of a polygon and a line segment. The prismatic uniform 4-polytopes consist of two infinite families: • Polyhedral prisms: products of a line segment and a uniform polyhedron. This family is infinite because it includes prisms built on 3-dimensional prisms and antiprisms. • Duoprisms: products of two polygons. Convex polyhedral prisms The most obvious family of prismatic 4-polytopes is the polyhedral prisms, i.e. products of a polyhedron with a line segment. The cells of such a 4-polytopes are two identical uniform polyhedra lying in parallel hyperplanes (the base cells) and a layer of prisms joining them (the lateral cells). This family includes prisms for the 75 nonprismatic uniform polyhedra (of which 18 are convex; one of these, the cube-prism, is listed above as the tesseract). There are 18 convex polyhedral prisms created from 5 Platonic solids and 13 Archimedean solids as well as for the infinite families of three-dimensional prisms and antiprisms. The symmetry number of a polyhedral prism is twice that of the base polyhedron. Tetrahedral prisms: A3 × A1 This prismatic tetrahedral symmetry is [3,3,2], order 48. There are two index 2 subgroups, [(3,3)+,2] and [3,3,2]+, but the second doesn't generate a uniform 4-polytope. [3,3,2] uniform 4-polytopes # Name (Bowers style acronym) Picture Vertex figure Coxeter diagram and Schläfli symbols Cells by type Element counts Net Cells Faces Edges Vertices 48 Tetrahedral prism (tepe) {3,3}×{ } t0,3{3,3,2} 2 3.3.3 4 3.4.4 6 8 {3} 6 {4} 16 8 49 Truncated tetrahedral prism (tuttip) t{3,3}×{ } t0,1,3{3,3,2} 2 3.6.6 4 3.4.4 4 4.4.6 10 8 {3} 18 {4} 8 {6} 48 24 [[3,3],2] uniform 4-polytopes # Name (Bowers style acronym) Picture Vertex figure Coxeter diagram and Schläfli symbols Cells by type Element counts Net Cells Faces Edges Vertices [51] Rectified tetrahedral prism (Same as octahedral prism) (ope) r{3,3}×{ } t1,3{3,3,2} 2 3.3.3.3 4 3.4.4 6 16 {3} 12 {4} 30 12 [50] Cantellated tetrahedral prism (Same as cuboctahedral prism) (cope) rr{3,3}×{ } t0,2,3{3,3,2} 2 3.4.3.4 8 3.4.4 6 4.4.4 16 16 {3} 36 {4} 60 24 [54] Cantitruncated tetrahedral prism (Same as truncated octahedral prism) (tope) tr{3,3}×{ } t0,1,2,3{3,3,2} 2 4.6.6 8 6.4.4 6 4.4.4 16 48 {4} 16 {6} 96 48 [59] Snub tetrahedral prism (Same as icosahedral prism) (ipe) sr{3,3}×{ } 2 3.3.3.3.3 20 3.4.4 22 40 {3} 30 {4} 72 24 Nonuniform omnisnub tetrahedral antiprism Pyritohedral icosahedral antiprism (pikap) $s\left\{{\begin{array}{l}3\\3\\2\end{array}}\right\}$ 2 3.3.3.3.3 8 3.3.3.3 6+24 3.3.3 40 16+96 {3} 96 24 Octahedral prisms: B3 × A1 This prismatic octahedral family symmetry is [4,3,2], order 96. There are 6 subgroups of index 2, order 48 that are expressed in alternated 4-polytopes below. Symmetries are [(4,3)+,2], [1+,4,3,2], [4,3,2+], [4,3+,2], [4,(3,2)+], and [4,3,2]+. # Name (Bowers style acronym) Picture Vertex figure Coxeter diagram and Schläfli symbols Cells by type Element counts Net Cells Faces Edges Vertices [10] Cubic prism (Same as tesseract) (Same as 4-4 duoprism) (tes) {4,3}×{ } t0,3{4,3,2} 2 4.4.4 6 4.4.4 824 {4}3216 50 Cuboctahedral prism (Same as cantellated tetrahedral prism) (cope) r{4,3}×{ } t1,3{4,3,2} 2 3.4.3.4 8 3.4.4 6 4.4.4 1616 {3} 36 {4} 6024 51 Octahedral prism (Same as rectified tetrahedral prism) (Same as triangular antiprismatic prism) (ope) {3,4}×{ } t2,3{4,3,2} 2 3.3.3.3 8 3.4.4 1016 {3} 12 {4} 3012 52 Rhombicuboctahedral prism (sircope) rr{4,3}×{ } t0,2,3{4,3,2} 2 3.4.4.4 8 3.4.4 18 4.4.4 2816 {3} 84 {4} 12048 53 Truncated cubic prism (ticcup) t{4,3}×{ } t0,1,3{4,3,2} 2 3.8.8 8 3.4.4 6 4.4.8 1616 {3} 36 {4} 12 {8} 9648 54 Truncated octahedral prism (Same as cantitruncated tetrahedral prism) (tope) t{3,4}×{ } t1,2,3{4,3,2} 2 4.6.6 6 4.4.4 8 4.4.6 1648 {4} 16 {6} 9648 55 Truncated cuboctahedral prism (gircope) tr{4,3}×{ } t0,1,2,3{4,3,2} 2 4.6.8 12 4.4.4 8 4.4.6 6 4.4.8 2896 {4} 16 {6} 12 {8} 19296 56 Snub cubic prism (sniccup) sr{4,3}×{ } 2 3.3.3.3.4 32 3.4.4 6 4.4.4 4064 {3} 72 {4} 14448 [48] Tetrahedral prism (tepe) h{4,3}×{ } 2 3.3.3 4 3.4.4 68 {3} 6 {4} 168 [49] Truncated tetrahedral prism (tuttip) h2{4,3}×{ } 2 3.3.6 4 3.4.4 4 4.4.6 68 {3} 6 {4} 168 [50] Cuboctahedral prism (cope) rr{3,3}×{ } 2 3.4.3.4 8 3.4.4 6 4.4.4 1616 {3} 36 {4} 6024 [52] Rhombicuboctahedral prism (sircope) s2{3,4}×{ } 2 3.4.4.4 8 3.4.4 18 4.4.4 2816 {3} 84 {4} 12048 [54] Truncated octahedral prism (tope) tr{3,3}×{ } 2 4.6.6 6 4.4.4 8 4.4.6 1648 {4} 16 {6} 9648 [59] Icosahedral prism (ipe) s{3,4}×{ } 2 3.3.3.3.3 20 3.4.4 2240 {3} 30 {4} 7224 [12] 16-cell (hex) s{2,4,3} 2+6+8 3.3.3.3 1632 {3}248 Nonuniform Omnisnub tetrahedral antiprism = Pyritohedral icosahedral antiprism (pikap) sr{2,3,4} 2 3.3.3.3.3 8 3.3.3.3 6+24 3.3.3 4016+96 {3}9624 Nonuniform Edge-snub octahedral hosochoron Pyritosnub alterprism (pysna) sr3{2,3,4} 2 3.4.4.4 6 4.4.4 8 3.3.3.3 24 3.4.4 4016+48 {3} 12+12+24+24 {4} 14448 Nonuniform Omnisnub cubic antiprism Snub cubic antiprism (sniccap) $s\left\{{\begin{array}{l}4\\3\\2\end{array}}\right\}$ 2 3.3.3.3.4 12+48 3.3.3 8 3.3.3.3 6 3.3.3.4 7616+192 {3} 12 {4} 19248 Nonuniform Runcic snub cubic hosochoron Truncated tetrahedral alterprism (tuta) s3{2,4,3} 2 3.6.6 6 3.3.3 8 triangular cupola 16526024 Icosahedral prisms: H3 × A1 This prismatic icosahedral symmetry is [5,3,2], order 240. There are two index 2 subgroups, [(5,3)+,2] and [5,3,2]+, but the second doesn't generate a uniform polychoron. # Name (Bowers name and acronym) Picture Vertex figure Coxeter diagram and Schläfli symbols Cells by type Element counts Net Cells Faces Edges Vertices 57 Dodecahedral prism (dope) {5,3}×{ } t0,3{5,3,2} 2 5.5.5 12 4.4.5 14 30 {4} 24 {5} 80 40 58 Icosidodecahedral prism (iddip) r{5,3}×{ } t1,3{5,3,2} 2 3.5.3.5 20 3.4.4 12 4.4.5 34 40 {3} 60 {4} 24 {5} 150 60 59 Icosahedral prism (same as snub tetrahedral prism) (ipe) {3,5}×{ } t2,3{5,3,2} 2 3.3.3.3.3 20 3.4.4 22 40 {3} 30 {4} 72 24 60 Truncated dodecahedral prism (tiddip) t{5,3}×{ } t0,1,3{5,3,2} 2 3.10.10 20 3.4.4 12 4.4.10 34 40 {3} 90 {4} 24 {10} 240 120 61 Rhombicosidodecahedral prism (sriddip) rr{5,3}×{ } t0,2,3{5,3,2} 2 3.4.5.4 20 3.4.4 30 4.4.4 12 4.4.5 64 40 {3} 180 {4} 24 {5} 300 120 62 Truncated icosahedral prism (tipe) t{3,5}×{ } t1,2,3{5,3,2} 2 5.6.6 12 4.4.5 20 4.4.6 34 90 {4} 24 {5} 40 {6} 240 120 63 Truncated icosidodecahedral prism (griddip) tr{5,3}×{ } t0,1,2,3{5,3,2} 2 4.6.10 30 4.4.4 20 4.4.6 12 4.4.10 64 240 {4} 40 {6} 24 {10} 480 240 64 Snub dodecahedral prism (sniddip) sr{5,3}×{ } 2 3.3.3.3.5 80 3.4.4 12 4.4.5 94 160 {3} 150 {4} 24 {5} 360 120 Nonuniform Omnisnub dodecahedral antiprism Snub dodecahedral antiprism (sniddap) $s\left\{{\begin{array}{l}5\\3\\2\end{array}}\right\}$ 2 3.3.3.3.5 30+120 3.3.3 20 3.3.3.3 12 3.3.3.5 18420+240 {3} 24 {5} 220120 Duoprisms: [p] × [q] The second is the infinite family of uniform duoprisms, products of two regular polygons. A duoprism's Coxeter-Dynkin diagram is . Its vertex figure is a disphenoid tetrahedron, . This family overlaps with the first: when one of the two "factor" polygons is a square, the product is equivalent to a hyperprism whose base is a three-dimensional prism. The symmetry number of a duoprism whose factors are a p-gon and a q-gon (a "p,q-duoprism") is 4pq if p≠q; if the factors are both p-gons, the symmetry number is 8p2. The tesseract can also be considered a 4,4-duoprism. The extended f-vector of {p}×{q} is (p,p,1)(q,q,1) = (pq,2pq,pq+p+q,p+q). • Cells: p q-gonal prisms, q p-gonal prisms • Faces: pq squares, p q-gons, q p-gons • Edges: 2pq • Vertices: pq There is no uniform analogue in four dimensions to the infinite family of three-dimensional antiprisms. Infinite set of p-q duoprism - - p q-gonal prisms, q p-gonal prisms: Name Coxeter graph Cells Images Net 3-3 duoprism (triddip) 3+3 triangular prisms 3-4 duoprism (tisdip) 3 cubes 4 triangular prisms 4-4 duoprism (tes) (same as tesseract) 4+4 cubes 3-5 duoprism (trapedip) 3 pentagonal prisms 5 triangular prisms 4-5 duoprism (squipdip) 4 pentagonal prisms 5 cubes 5-5 duoprism (pedip) 5+5 pentagonal prisms 3-6 duoprism (thiddip) 3 hexagonal prisms 6 triangular prisms 4-6 duoprism (shiddip) 4 hexagonal prisms 6 cubes 5-6 duoprism (phiddip) 5 hexagonal prisms 6 pentagonal prisms 6-6 duoprism (hiddip) 6+6 hexagonal prisms 3-3 3-4 3-5 3-6 3-7 3-8 4-3 4-4 4-5 4-6 4-7 4-8 5-3 5-4 5-5 5-6 5-7 5-8 6-3 6-4 6-5 6-6 6-7 6-8 7-3 7-4 7-5 7-6 7-7 7-8 8-3 8-4 8-5 8-6 8-7 8-8 Alternations are possible. = gives the family of duoantiprisms, but they generally cannot be made uniform. p=q=2 is the only convex case that can be made uniform, giving the regular 16-cell. p=5, q=5/3 is the only nonconvex case that can be made uniform, giving the so-called great duoantiprism. gives the p-2q-gonal prismantiprismoid (an edge-alternation of the 2p-4q duoprism), but this cannot be made uniform in any cases.[20] Polygonal prismatic prisms: [p] × [ ] × [ ] The infinite set of uniform prismatic prisms overlaps with the 4-p duoprisms: (p≥3) - - p cubes and 4 p-gonal prisms - (All are the same as 4-p duoprism) The second polytope in the series is a lower symmetry of the regular tesseract, {4}×{4}. Convex p-gonal prismatic prisms Name {3}×{4} {4}×{4} {5}×{4} {6}×{4} {7}×{4} {8}×{4} {p}×{4} Coxeter diagrams Image Cells 3 {4}×{} 4 {3}×{} 4 {4}×{} 4 {4}×{} 5 {4}×{} 4 {5}×{} 6 {4}×{} 4 {6}×{} 7 {4}×{} 4 {7}×{} 8 {4}×{} 4 {8}×{} p {4}×{} 4 {p}×{} Net Polygonal antiprismatic prisms: [p] × [ ] × [ ] The infinite sets of uniform antiprismatic prisms are constructed from two parallel uniform antiprisms): (p≥2) - - 2 p-gonal antiprisms, connected by 2 p-gonal prisms and 2p triangular prisms. Convex p-gonal antiprismatic prisms Name s{2,2}×{} s{2,3}×{} s{2,4}×{} s{2,5}×{} s{2,6}×{} s{2,7}×{} s{2,8}×{} s{2,p}×{} Coxeter diagram Image Vertex figure Cells 2 s{2,2} (2) {2}×{}={4} 4 {3}×{} 2 s{2,3} 2 {3}×{} 6 {3}×{} 2 s{2,4} 2 {4}×{} 8 {3}×{} 2 s{2,5} 2 {5}×{} 10 {3}×{} 2 s{2,6} 2 {6}×{} 12 {3}×{} 2 s{2,7} 2 {7}×{} 14 {3}×{} 2 s{2,8} 2 {8}×{} 16 {3}×{} 2 s{2,p} 2 {p}×{} 2p {3}×{} Net A p-gonal antiprismatic prism has 4p triangle, 4p square and 4 p-gon faces. It has 10p edges, and 4p vertices. Nonuniform alternations Coxeter showed only two uniform solutions for rank 4 Coxeter groups with all rings alternated (shown with empty circle nodes). The first is , s{21,1,1} which represented an index 24 subgroup (symmetry [2,2,2]+, order 8) form of the demitesseract, , h{4,3,3} (symmetry [1+,4,3,3] = [31,1,1], order 192). The second is , s{31,1,1}, which is an index 6 subgroup (symmetry [31,1,1]+, order 96) form of the snub 24-cell, , s{3,4,3}, (symmetry [3+,4,3], order 576). Other alternations, such as , as an alternation from the omnitruncated tesseract , can not be made uniform as solving for equal edge lengths are in general overdetermined (there are six equations but only four variables). Such nonuniform alternated figures can be constructed as vertex-transitive 4-polytopes by the removal of one of two half sets of the vertices of the full ringed figure, but will have unequal edge lengths. Just like uniform alternations, they will have half of the symmetry of uniform figure, like [4,3,3]+, order 192, is the symmetry of the alternated omnitruncated tesseract.[21] Wythoff constructions with alternations produce vertex-transitive figures that can be made equilateral, but not uniform because the alternated gaps (around the removed vertices) create cells that are not regular or semiregular. A proposed name for such figures is scaliform polytopes.[22] This category allows a subset of Johnson solids as cells, for example triangular cupola. Each vertex configuration within a Johnson solid must exist within the vertex figure. For example, a square pyramid has two vertex configurations: 3.3.4 around the base, and 3.3.3.3 at the apex. The nets and vertex figures of the four convex equilateral cases are given below, along with a list of cells around each vertex. Four convex vertex-transitive equilateral 4-polytopes with nonuniform cells Coxeter diagram s3{2,4,3}, s3{3,4,3}, Others Relation 24 of 48 vertices of rhombicuboctahedral prism 288 of 576 vertices of runcitruncated 24-cell 72 of 120 vertices of 600-cell 600 of 720 vertices of rectified 600-cell Projection Two rings of pyramids Net runcic snub cubic hosochoron[23][24] runcic snub 24-cell[25][26] [27][28][29] [30][31] Cells Vertex figure (1) 3.4.3.4: triangular cupola (2) 3.4.6: triangular cupola (1) 3.3.3: tetrahedron (1) 3.6.6: truncated tetrahedron (1) 3.4.3.4: triangular cupola (2) 3.4.6: triangular cupola (2) 3.4.4: triangular prism (1) 3.6.6: truncated tetrahedron (1) 3.3.3.3.3: icosahedron (2) 3.3.3.5: tridiminished icosahedron (4) 3.5.5: tridiminished icosahedron (1) 3.3.3.3: square pyramid (4) 3.3.4: square pyramid (2) 4.4.5: pentagonal prism (2) 3.3.3.5 pentagonal antiprism Geometric derivations for 46 nonprismatic Wythoffian uniform polychora The 46 Wythoffian 4-polytopes include the six convex regular 4-polytopes. The other forty can be derived from the regular polychora by geometric operations which preserve most or all of their symmetries, and therefore may be classified by the symmetry groups that they have in common. Summary chart of truncation operations Example locations of kaleidoscopic generator point on fundamental domain. The geometric operations that derive the 40 uniform 4-polytopes from the regular 4-polytopes are truncating operations. A 4-polytope may be truncated at the vertices, edges or faces, leading to addition of cells corresponding to those elements, as shown in the columns of the tables below. The Coxeter-Dynkin diagram shows the four mirrors of the Wythoffian kaleidoscope as nodes, and the edges between the nodes are labeled by an integer showing the angle between the mirrors (π/n radians or 180/n degrees). Circled nodes show which mirrors are active for each form; a mirror is active with respect to a vertex that does not lie on it. Operation Schläfli symbol Symmetry Coxeter diagram Description Parent t0{p,q,r} [p,q,r] Original regular form {p,q,r} Rectification t1{p,q,r} Truncation operation applied until the original edges are degenerated into points. Birectification (Rectified dual) t2{p,q,r} Face are fully truncated to points. Same as rectified dual. Trirectification (dual) t3{p,q,r} Cells are truncated to points. Regular dual {r,q,p} Truncation t0,1{p,q,r} Each vertex is cut off so that the middle of each original edge remains. Where the vertex was, there appears a new cell, the parent's vertex figure. Each original cell is likewise truncated. Bitruncation t1,2{p,q,r} A truncation between a rectified form and the dual rectified form. Tritruncation t2,3{p,q,r} Truncated dual {r,q,p}. Cantellation t0,2{p,q,r} A truncation applied to edges and vertices and defines a progression between the regular and dual rectified form. Bicantellation t1,3{p,q,r} Cantellated dual {r,q,p}. Runcination (or expansion) t0,3{p,q,r} A truncation applied to the cells, faces and edges; defines a progression between a regular form and the dual. Cantitruncation t0,1,2{p,q,r} Both the cantellation and truncation operations applied together. Bicantitruncation t1,2,3{p,q,r} Cantitruncated dual {r,q,p}. Runcitruncation t0,1,3{p,q,r} Both the runcination and truncation operations applied together. Runcicantellation t0,1,3{p,q,r} Runcitruncated dual {r,q,p}. Omnitruncation (runcicantitruncation) t0,1,2,3{p,q,r} Application of all three operators. Half h{2p,3,q} [1+,2p,3,q] =[(3,p,3),q] Alternation of , same as Cantic h2{2p,3,q} Same as Runcic h3{2p,3,q} Same as Runcicantic h2,3{2p,3,q} Same as Quarter q{2p,3,2q} [1+,2p,3,2q,1+] Same as Snub s{p,2q,r} [p+,2q,r] Alternated truncation Cantic snub s2{p,2q,r} Cantellated alternated truncation Runcic snub s3{p,2q,r} Runcinated alternated truncation Runcicantic snub s2,3{p,2q,r} Runcicantellated alternated truncation Snub rectified sr{p,q,2r} [(p,q)+,2r] Alternated truncated rectification ht0,3{2p,q,2r} [(2p,q,2r,2+)] Alternated runcination Bisnub 2s{2p,q,2r} [2p,q+,2r] Alternated bitruncation Omnisnub ht0,1,2,3{p,q,r} [p,q,r]+ Alternated omnitruncation See also convex uniform honeycombs, some of which illustrate these operations as applied to the regular cubic honeycomb. If two polytopes are duals of each other (such as the tesseract and 16-cell, or the 120-cell and 600-cell), then bitruncating, runcinating or omnitruncating either produces the same figure as the same operation to the other. Thus where only the participle appears in the table it should be understood to apply to either parent. Summary of constructions by extended symmetry The 46 uniform polychora constructed from the A4, B4, F4, H4 symmetry are given in this table by their full extended symmetry and Coxeter diagrams. The D4 symmetry is also included, though it only creates duplicates. Alternations are grouped by their chiral symmetry. All alternations are given, although the snub 24-cell, with its 3 constructions from different families is the only one that is uniform. Counts in parenthesis are either repeats or nonuniform. The Coxeter diagrams are given with subscript indices 1 through 46. The 3-3 and 4-4 duoprismatic family is included, the second for its relation to the B4 family. Coxeter group Extended symmetry Polychora Chiral extended symmetry Alternation honeycombs [3,3,3] [3,3,3] (order 120) 6 (2) \| (3) (4) \| (7) \| (8) [2+[3,3,3]] (order 240) 3 (6) \| (9) [2+[3,3,3]]+ (order 120) (1) (−) [3,31,1] [3,31,1] (order 192) 0 (none) [1[3,31,1]]=[4,3,3] = (order 384) (4) (17) \| (11) \| (16) [3[31,1,1]]=[3,4,3] = (order 1152) (3) (23) \| (24) [3[3,31,1]]+ =[3,4,3]+ (order 576) (1) (31) (= ) (−) [4,3,3] [3[1+,4,3,3]]=[3,4,3] = (order 1152) (3)(22) \| (23) \| (24) [4,3,3] (order 384) 12 (10) \| (11) \| (12) \| (13) \| (14) (15) \| (16) \| (17) \| (18) \| (19) (20) \| (21) [1+,4,3,3]+ (order 96) (2) (12) (= ) (31) (−) [4,3,3]+ (order 192) (1) (−) [3,4,3] [3,4,3] (order 1152) 6 (23) \| (24) (25) \| (28) \| (29) [2+[3+,4,3+]] (order 576) 1 (31) [2+[3,4,3]] (order 2304) 3 (27) \| (30) [2+[3,4,3]]+ (order 1152) (1) (−) [5,3,3] [5,3,3] (order 14400) 15 (33) \| (34) \| (35) \| (36) (37) \| (38) \| (39) \| (40) \| (41) (42) \| (43) \| (44) \| (45) \| (46) [5,3,3]+ (order 7200) (1) (−) [3,2,3] [3,2,3] (order 36) 0 (none) [3,2,3]+ (order 18) 0 (none) [2+[3,2,3]] (order 72) 0 [2+[3,2,3]]+ (order 36) 0 (none) [[3],2,3]=[6,2,3] = (order 72) 1 [1[3,2,3]]=[[3],2,3]+=[6,2,3]+ (order 36) (1) [(2+,4)[3,2,3]]=[2+[6,2,6]] = (order 288) 1 [(2+,4)[3,2,3]]+=[2+[6,2,6]]+ (order 144) (1) [4,2,4] [4,2,4] (order 64) 0 (none) [4,2,4]+ (order 32) 0 (none) [2+[4,2,4]] (order 128) 0 (none) [2+[(4,2+,4,2+)]] (order 64) 0 (none) [(3,3)[4,2,4]]=[4,3,3] = (order 384) (1) (10) [(3,3)[4,2*,4]]+=[4,3,3]+ (order 192) (1) (12) [[4],2,4]=[8,2,4] = (order 128) (1) [1[4,2,4]]=[[4],2,4]+=[8,2,4]+ (order 64) (1) [(2+,4)[4,2,4]]=[2+[8,2,8]] = (order 512) (1) [(2+,4)[4,2,4]]+=[2+[8,2,8]]+ (order 256) (1) Uniform star polychora Other than the aforementioned infinite duoprism and antiprism prism families, which have infinitely many nonconvex members, many uniform star polychora have been discovered. In 1852, Ludwig Schläfli discovered four regular star polychora: {5,3,5/2}, {5/2,3,5}, {3,3,5/2}, and {5/2,3,3}. In 1883, Edmund Hess found the other six: {3,5,5/2}, {5/2,5,3}, {5,5/2,5}, {5/2,5,5/2}, {5,5/2,3}, and {3,5/2,5}. Norman Johnson described three uniform antiprism-like star polychora in his doctoral dissertation of 1966: they are based on the three ditrigonal polyhedra sharing the edges and vertices of the regular dodecahedron. Many more have been found since then by other researchers, including Jonathan Bowers and George Olshevsky, creating a total count of 2125 known uniform star polychora at present (not counting the infinite set of duoprisms based on star polygons). There is currently no proof of the set's completeness. See also • Finite regular skew polyhedra of 4-space • Convex uniform honeycomb - related infinite 4-polytopes in Euclidean 3-space. • Convex uniform honeycombs in hyperbolic space - related infinite 4-polytopes in Hyperbolic 3-space. • Paracompact uniform honeycombs References 1. N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.1 Polytopes and Honeycombs, p.224 2. T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 3. "Archived copy" (PDF). Archived from the original (PDF) on 2009-12-29. Retrieved 2010-08-13.{{cite web}}: CS1 maint: archived copy as title (link) 4. Elte (1912) 5. https://web.archive.org/web/19981206035238/http://members.aol.com/Polycell/uniform.html December 6, 1998 oldest archive 6. The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes By David Darling, (2004) ASIN: B00SB4TU58 7. Johnson (2015), Chapter 11, section 11.5 Spherical Coxeter groups, 11.5.5 full polychoric groups 8. Uniform Polytopes in Four Dimensions, George Olshevsky. 9. Möller, Marco (2004). Vierdimensionale Archimedische Polytope (PDF) (Doctoral thesis) (in German). University of Hamburg. 10. Conway (2008) 11. Multidimensional Glossary, George Olshevsky 12. https://www.mit.edu/~hlb/Associahedron/program.pdf Convex and Abstract Polytopes workshop (2005), N.Johnson — "Uniform Polychora" abstract 13. "Uniform Polychora". www.polytope.net. Retrieved February 20, 2020. 14. Polytope Wiki 15. Coxeter, Regular polytopes, 7.7 Schlaefli's criterion eq 7.78, p.135 16. "S3s3s3s". 17. "S3s3s4s". 18. "S3s4s3s". 19. "S3s3s5s". 20. sns2s2mx, Richard Klitzing 21. H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) p. 582-588 2.7 The four-dimensional analogues of the snub cube 22. "Polytope-tree". 23. "tuta". 24. Category S1: Simple Scaliforms tutcup 25. "Prissi". 26. Category S3: Special Scaliforms prissi 27. http://bendwavy.org/klitzing/incmats/bidex.htm 28. Category S3: Special Scaliforms bidex 29. The Bi-icositetradiminished 600-cell 30. http://bendwavy.org/klitzing/incmats/spidrox.htm 31. Category S4: Scaliform Swirlprisms spidrox • A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910 • B. Grünbaum Convex Polytopes, New York ; London : Springer, c2003. ISBN 0-387-00424-6. Second edition prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler. • Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen, ISBN 1-4181-7968-X • H.S.M. Coxeter: • H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londen, 1954 • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] • H.S.M. Coxeter and W. O. J. Moser. Generators and Relations for Discrete Groups 4th ed, Springer-Verlag. New York. 1980 p. 92, p. 122. • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26) • John H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965 • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966 • N.W. Johnson: Geometries and Transformations, (2015) Chapter 11: Finite symmetry groups • Richard Klitzing, Snubs, alternated facetings, and Stott-Coxeter-Dynkin diagrams, Symmetry: Culture and Science, Vol. 21, No.4, 329-344, (2010) • Schoute, Pieter Hendrik (1911), "Analytic treatment of the polytopes regularly derived from the regular polytopes", Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, 11 (3): 87 pp Googlebook, 370-381 External links • Convex uniform 4-polytopes • Uniform, convex polytopes in four dimensions, Marco Möller (in German). Includes alternative names for these figures, including those from Jonathan Bowers, George Olshevsky, and Norman Johnson. • Regular and semi-regular convex polytopes a short historical overview • Java3D Applets with sources • Nonconvex uniform 4-polytopes • Uniform polychora by Jonathan Bowers • Stella4D Stella (software) produces interactive views of known uniform polychora including the 64 convex forms and the infinite prismatic families. • Klitzing, Richard. "4D uniform polytopes". • 4D-Polytopes and Their Dual Polytopes of the Coxeter Group W(A4) Represented by Quaternions International Journal of Geometric Methods in Modern Physics,Vol. 9, No. 4 (2012) Mehmet Koca, Nazife Ozdes Koca, Mudhahir Al-Ajmi (2012) Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square p-gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221 Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321 Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421 Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes and compounds	Wikipedia
Dijkstra's algorithm Dijkstra's algorithm (/ˈdaɪkstrəz/ DYKE-strəz) is an algorithm for finding the shortest paths between nodes in a weighted graph, which may represent, for example, road networks. It was conceived by computer scientist Edsger W. Dijkstra in 1956 and published three years later.[4][5][6] Dijkstra's algorithm Dijkstra's algorithm to find the shortest path between a and b. It picks the unvisited vertex with the lowest distance, calculates the distance through it to each unvisited neighbor, and updates the neighbor's distance if smaller. Mark visited (set to red) when done with neighbors. ClassSearch algorithm Greedy algorithm Dynamic programming[1] Data structureGraph Usually used with priority queue or heap for optimization[2][3] Worst-case performance$\Theta (\|E\|+\|V\|\log \|V\|)$[3] The algorithm exists in many variants. Dijkstra's original algorithm found the shortest path between two given nodes,[6] but a more common variant fixes a single node as the "source" node and finds shortest paths from the source to all other nodes in the graph, producing a shortest-path tree. For a given source node in the graph, the algorithm finds the shortest path between that node and every other.[7]: 196–206 It can also be used for finding the shortest paths from a single node to a single destination node by stopping the algorithm once the shortest path to the destination node has been determined. For example, if the nodes of the graph represent cities and costs of edge paths represent driving distances between pairs of cities connected by a direct road (for simplicity, ignore red lights, stop signs, toll roads and other obstructions), then Dijkstra's algorithm can be used to find the shortest route between one city and all other cities. A widely used application of shortest path algorithms is network routing protocols, most notably IS-IS (Intermediate System to Intermediate System) and OSPF (Open Shortest Path First). It is also employed as a subroutine in other algorithms such as Johnson's. The Dijkstra algorithm uses labels that are positive integers or real numbers, which are totally ordered. It can be generalized to use any labels that are partially ordered, provided the subsequent labels (a subsequent label is produced when traversing an edge) are monotonically non-decreasing. This generalization is called the generic Dijkstra shortest-path algorithm.[8][9] Dijkstra's algorithm uses a data structure for storing and querying partial solutions sorted by distance from the start. Dijkstra's original algorithm does not use a min-priority queue and runs in time $\Theta (\|V\|^{2})$(where $\|V\|$ is the number of nodes).[10] The idea of this algorithm is also given in Leyzorek et al. 1957. Fredman & Tarjan 1984 propose using a Fibonacci heap min-priority queue to optimize the running time complexity to $\Theta (\|E\|+\|V\|\log \|V\|)$. This is asymptotically the fastest known single-source shortest-path algorithm for arbitrary directed graphs with unbounded non-negative weights. However, specialized cases (such as bounded/integer weights, directed acyclic graphs etc.) can indeed be improved further as detailed in Specialized variants. Additionally, if preprocessing is allowed algorithms such as contraction hierarchies can be up to seven orders of magnitude faster. In some fields, artificial intelligence in particular, Dijkstra's algorithm or a variant of it is known as uniform cost search and formulated as an instance of the more general idea of best-first search.[11] History What is the shortest way to travel from Rotterdam to Groningen, in general: from given city to given city. It is the algorithm for the shortest path, which I designed in about twenty minutes. One morning I was shopping in Amsterdam with my young fiancée, and tired, we sat down on the café terrace to drink a cup of coffee and I was just thinking about whether I could do this, and I then designed the algorithm for the shortest path. As I said, it was a twenty-minute invention. In fact, it was published in '59, three years later. The publication is still readable, it is, in fact, quite nice. One of the reasons that it is so nice was that I designed it without pencil and paper. I learned later that one of the advantages of designing without pencil and paper is that you are almost forced to avoid all avoidable complexities. Eventually, that algorithm became to my great amazement, one of the cornerstones of my fame. — Edsger Dijkstra, in an interview with Philip L. Frana, Communications of the ACM, 2001[5] Dijkstra thought about the shortest path problem when working at the Mathematical Center in Amsterdam in 1956 as a programmer to demonstrate the capabilities of a new computer called ARMAC.[12] His objective was to choose both a problem and a solution (that would be produced by computer) that non-computing people could understand. He designed the shortest path algorithm and later implemented it for ARMAC for a slightly simplified transportation map of 64 cities in the Netherlands (64, so that 6 bits would be sufficient to encode the city number).[5] A year later, he came across another problem from hardware engineers working on the institute's next computer: minimize the amount of wire needed to connect the pins on the back panel of the machine. As a solution, he re-discovered the algorithm known as Prim's minimal spanning tree algorithm (known earlier to Jarník, and also rediscovered by Prim).[13][14] Dijkstra published the algorithm in 1959, two years after Prim and 29 years after Jarník.[15][16] Algorithm Let the node at which we are starting be called the initial node. Let the distance of node Y be the distance from the initial node to Y. Dijkstra's algorithm will initially start with infinite distances and will try to improve them step by step. 1. Mark all nodes unvisited. Create a set of all the unvisited nodes called the unvisited set. 2. Assign to every node a tentative distance value: set it to zero for our initial node and to infinity for all other nodes. During the run of the algorithm, the tentative distance of a node v is the length of the shortest path discovered so far between the node v and the starting node. Since initially no path is known to any other vertex than the source itself (which is a path of length zero), all other tentative distances are initially set to infinity. Set the initial node as current.[17] 3. For the current node, consider all of its unvisited neighbors and calculate their tentative distances through the current node. Compare the newly calculated tentative distance to the one currently assigned to the neighbor and assign it the smaller one. For example, if the current node A is marked with a distance of 6, and the edge connecting it with a neighbor B has length 2, then the distance to B through A will be 6 + 2 = 8. If B was previously marked with a distance greater than 8 then change it to 8. Otherwise, the current value will be kept. 4. When we are done considering all of the unvisited neighbors of the current node, mark the current node as visited and remove it from the unvisited set. A visited node will never be checked again (this is valid and optimal in connection with the behavior in step 6.: that the next nodes to visit will always be in the order of 'smallest distance from initial node first' so any visits after would have a greater distance). 5. If the destination node has been marked visited (when planning a route between two specific nodes) or if the smallest tentative distance among the nodes in the unvisited set is infinity (when planning a complete traversal; occurs when there is no connection between the initial node and remaining unvisited nodes), then stop. The algorithm has finished. 6. Otherwise, select the unvisited node that is marked with the smallest tentative distance, set it as the new current node, and go back to step 3. When planning a route, it is actually not necessary to wait until the destination node is "visited" as above: the algorithm can stop once the destination node has the smallest tentative distance among all "unvisited" nodes (and thus could be selected as the next "current"). Description Suppose you would like to find the shortest path between two intersections on a city map: a starting point and a destination. Dijkstra's algorithm initially marks the distance (from the starting point) to every other intersection on the map with infinity. This is done not to imply that there is an infinite distance, but to note that those intersections have not been visited yet. Some variants of this method leave the intersections' distances unlabeled. Now select the current intersection at each iteration. For the first iteration, the current intersection will be the starting point, and the distance to it (the intersection's label) will be zero. For subsequent iterations (after the first), the current intersection will be a closest unvisited intersection to the starting point (this will be easy to find). From the current intersection, update the distance to every unvisited intersection that is directly connected to it. This is done by determining the sum of the distance between an unvisited intersection and the value of the current intersection and then relabeling the unvisited intersection with this value (the sum) if it is less than the unvisited intersection's current value. In effect, the intersection is relabeled if the path to it through the current intersection is shorter than the previously known paths. To facilitate shortest path identification, in pencil, mark the road with an arrow pointing to the relabeled intersection if you label/relabel it, and erase all others pointing to it. After you have updated the distances to each neighboring intersection, mark the current intersection as visited and select an unvisited intersection with minimal distance (from the starting point) – or the lowest label—as the current intersection. Intersections marked as visited are labeled with the shortest path from the starting point to it and will not be revisited or returned to. Continue this process of updating the neighboring intersections with the shortest distances, marking the current intersection as visited, and moving onto a closest unvisited intersection until you have marked the destination as visited. Once you have marked the destination as visited (as is the case with any visited intersection), you have determined the shortest path to it from the starting point and can trace your way back following the arrows in reverse. In the algorithm's implementations, this is usually done (after the algorithm has reached the destination node) by following the nodes' parents from the destination node up to the starting node; that's why we also keep track of each node's parent. This algorithm makes no attempt of direct "exploration" towards the destination as one might expect. Rather, the sole consideration in determining the next "current" intersection is its distance from the starting point. This algorithm therefore expands outward from the starting point, interactively considering every node that is closer in terms of shortest path distance until it reaches the destination. When understood in this way, it is clear how the algorithm necessarily finds the shortest path. However, it may also reveal one of the algorithm's weaknesses: its relative slowness in some topologies. Pseudocode In the following pseudocode algorithm, dist is an array that contains the current distances from the source to other vertices, i.e. dist[u] is the current distance from the source to the vertex u. The prev array contains pointers to previous-hop nodes on the shortest path from source to the given vertex (equivalently, it is the next-hop on the path from the given vertex to the source). The code u ← vertex in Q with min dist[u], searches for the vertex u in the vertex set Q that has the least dist[u] value. Graph.Edges(u, v) returns the length of the edge joining (i.e. the distance between) the two neighbor-nodes u and v. The variable alt on line 14 is the length of the path from the root node to the neighbor node v if it were to go through u. If this path is shorter than the current shortest path recorded for v, that current path is replaced with this alt path.[7] 1 function Dijkstra(Graph, source): 2 3 for each vertex v in Graph.Vertices: 4 dist[v] ← INFINITY 5 prev[v] ← UNDEFINED 6 add v to Q 7 dist[source] ← 0 8 9 while Q is not empty: 10 u ← vertex in Q with min dist[u] 11 remove u from Q 12 13 for each neighbor v of u still in Q: 14 alt ← dist[u] + Graph.Edges(u, v) 15 if alt < dist[v]: 16 dist[v] ← alt 17 prev[v] ← u 18 19 return dist[], prev[] If we are only interested in a shortest path between vertices source and target, we can terminate the search after line 10 if u = target. Now we can read the shortest path from source to target by reverse iteration: 1 S ← empty sequence 2 u ← target 3 if prev[u] is defined or u = source: // Do something only if the vertex is reachable 4 while u is defined: // Construct the shortest path with a stack S 5 insert u at the beginning of S // Push the vertex onto the stack 6 u ← prev[u] // Traverse from target to source Now sequence S is the list of vertices constituting one of the shortest paths from source to target, or the empty sequence if no path exists. A more general problem would be to find all the shortest paths between source and target (there might be several different ones of the same length). Then instead of storing only a single node in each entry of prev[] we would store all nodes satisfying the relaxation condition. For example, if both r and source connect to target and both of them lie on different shortest paths through target (because the edge cost is the same in both cases), then we would add both r and source to prev[target]. When the algorithm completes, prev[] data structure will actually describe a graph that is a subset of the original graph with some edges removed. Its key property will be that if the algorithm was run with some starting node, then every path from that node to any other node in the new graph will be the shortest path between those nodes in the original graph, and all paths of that length from the original graph will be present in the new graph. Then to actually find all these shortest paths between two given nodes we would use a path finding algorithm on the new graph, such as depth-first search. Using a priority queue A min-priority queue is an abstract data type that provides 3 basic operations: add_with_priority(), decrease_priority() and extract_min(). As mentioned earlier, using such a data structure can lead to faster computing times than using a basic queue. Notably, Fibonacci heap[18] or Brodal queue offer optimal implementations for those 3 operations. As the algorithm is slightly different, it is mentioned here, in pseudocode as well: 1 function Dijkstra(Graph, source): 2 dist[source] ← 0 // Initialization 3 4 create vertex priority queue Q 5 6 for each vertex v in Graph.Vertices: 7 if v ≠ source 8 dist[v] ← INFINITY // Unknown distance from source to v 9 prev[v] ← UNDEFINED // Predecessor of v 10 11 Q.add_with_priority(v, dist[v]) 12 13 14 while Q is not empty: // The main loop 15 u ← Q.extract_min() // Remove and return best vertex 16 for each neighbor v of u: // Go through all v neighbors of u 17 alt ← dist[u] + Graph.Edges(u, v) 18 if alt < dist[v]: 19 dist[v] ← alt 20 prev[v] ← u 21 Q.decrease_priority(v, alt) 22 23 return dist, prev Instead of filling the priority queue with all nodes in the initialization phase, it is also possible to initialize it to contain only source; then, inside the if alt < dist[v] block, the decrease_priority() becomes an add_with_priority() operation if the node is not already in the queue.[7]: 198 Yet another alternative is to add nodes unconditionally to the priority queue and to instead check after extraction that it isn't revisiting, or that no shorter connection was found yet. This can be done by additionally extracting the associated priority p from the queue and only processing further if p == dist[u] inside the while Q is not empty loop. [19] These alternatives can use entirely array-based priority queues without decrease-key functionality, which have been found to achieve even faster computing times in practice. However, the difference in performance was found to be narrower for denser graphs.[20] Proof of correctness Proof of Dijkstra's algorithm is constructed by induction on the number of visited nodes. Invariant hypothesis: For each visited node v, dist[v] is the shortest distance from source to v, and for each unvisited node u, dist[u] is the shortest distance from source to u when traveling via visited nodes only, or infinity if no such path exists. (Note: we do not assume dist[u] is the actual shortest distance for unvisited nodes, while dist[v] is the actual shortest distance) The base case is when there is just one visited node, namely the initial node source, in which case the hypothesis is trivial. Next, assume the hypothesis for k-1 visited nodes. Next, we choose u to be the next visited node according to the algorithm. We claim that dist[u] is the shortest distance from source to u. To prove that claim, we will proceed with a proof by contradiction. If there were a shorter path, then there can be two cases, either the shortest path contains another unvisited node or not. In the first case, let w be the first unvisited node on the shortest path. By the induction hypothesis, the shortest path from source to u and w through visited node only has cost dist[u] and dist[w] respectively. That means the cost of going from source to u through w has the cost of at least dist[w] + the minimal cost of going from w to u. As the edge costs are positive, the minimal cost of going from w to u is a positive number. Also dist[u] < dist[w] because the algorithm picked u instead of w. Now we arrived at a contradiction that dist[u] < dist[w] yet dist[w] + a positive number < dist[u]. In the second case, let w be the last but one node on the shortest path. That means dist[w] + Graph.Edges[w,u] < dist[u]. That is a contradiction because by the time w is visited, it should have set dist[u] to at most dist[w] + Graph.Edges[w,u]. For all other visited nodes v, the induction hypothesis told us dist[v] is the shortest distance from source already, and the algorithm step is not changing that. After processing u it will still be true that for each unvisited node w, dist[w] will be the shortest distance from source to w using visited nodes only, because if there were a shorter path that doesn't go by u we would have found it previously, and if there were a shorter path using u we would have updated it when processing u. After all nodes are visited, the shortest path from source to any node v consists only of visited nodes, therefore dist[v] is the shortest distance. Running time Bounds of the running time of Dijkstra's algorithm on a graph with edges E and vertices V can be expressed as a function of the number of edges, denoted $\|E\|$, and the number of vertices, denoted $\|V\|$, using big-O notation. The complexity bound depends mainly on the data structure used to represent the set Q. In the following, upper bounds can be simplified because $\|E\|$ is $O(\|V\|^{2})$ for any graph, but that simplification disregards the fact that in some problems, other upper bounds on $\|E\|$ may hold. For any data structure for the vertex set Q, the running time is in[2] $\Theta (\|E\|\cdot T_{\mathrm {dk} }+\|V\|\cdot T_{\mathrm {em} }),$ where $T_{\mathrm {dk} }$ and $T_{\mathrm {em} }$ are the complexities of the decrease-key and extract-minimum operations in Q, respectively. The simplest version of Dijkstra's algorithm stores the vertex set Q as a linked list or array, and edges as an adjacency list or matrix. In this case, extract-minimum is simply a linear search through all vertices in Q, so the running time is $\Theta (\|E\|+\|V\|^{2})=\Theta (\|V\|^{2})$. For sparse graphs, that is, graphs with far fewer than $\|V\|^{2}$ edges, Dijkstra's algorithm can be implemented more efficiently by storing the graph in the form of adjacency lists and using a self-balancing binary search tree, binary heap, pairing heap, or Fibonacci heap as a priority queue to implement extracting minimum efficiently. To perform decrease-key steps in a binary heap efficiently, it is necessary to use an auxiliary data structure that maps each vertex to its position in the heap, and to keep this structure up to date as the priority queue Q changes. With a self-balancing binary search tree or binary heap, the algorithm requires $\Theta ((\|E\|+\|V\|)\log \|V\|)$ time in the worst case (where $\log $ denotes the binary logarithm $\log _{2}$); for connected graphs this time bound can be simplified to $\Theta (\|E\|\log \|V\|)$. The Fibonacci heap improves this to $\Theta (\|E\|+\|V\|\log \|V\|).$ When using binary heaps, the average case time complexity is lower than the worst-case: assuming edge costs are drawn independently from a common probability distribution, the expected number of decrease-key operations is bounded by $\Theta (\|V\|\log(\|E\|/\|V\|))$, giving a total running time of[7]: 199–200 $O\left(\|E\|+\|V\|\log {\frac {\|E\|}{\|V\|}}\log \|V\|\right).$ Practical optimizations and infinite graphs In common presentations of Dijkstra's algorithm, initially all nodes are entered into the priority queue. This is, however, not necessary: the algorithm can start with a priority queue that contains only one item, and insert new items as they are discovered (instead of doing a decrease-key, check whether the key is in the queue; if it is, decrease its key, otherwise insert it).[7]: 198 This variant has the same worst-case bounds as the common variant, but maintains a smaller priority queue in practice, speeding up the queue operations.[11] Moreover, not inserting all nodes in a graph makes it possible to extend the algorithm to find the shortest path from a single source to the closest of a set of target nodes on infinite graphs or those too large to represent in memory. The resulting algorithm is called uniform-cost search (UCS) in the artificial intelligence literature[11][21][22] and can be expressed in pseudocode as procedure uniform_cost_search(start) is node ← start frontier ← priority queue containing node only expanded ← empty set do if frontier is empty then return failure node ← frontier.pop() if node is a goal state then return solution(node) expanded.add(node) for each of node's neighbors n do if n is not in expanded and not in frontier then frontier.add(n) else if n is in frontier with higher cost replace existing node with n The complexity of this algorithm can be expressed in an alternative way for very large graphs: when C* is the length of the shortest path from the start node to any node satisfying the "goal" predicate, each edge has cost at least ε, and the number of neighbors per node is bounded by b, then the algorithm's worst-case time and space complexity are both in O(b1+⌊C* ⁄ ε⌋).[21] Further optimizations of Dijkstra's algorithm for the single-target case include bidirectional variants, goal-directed variants such as the A* algorithm (see § Related problems and algorithms), graph pruning to determine which nodes are likely to form the middle segment of shortest paths (reach-based routing), and hierarchical decompositions of the input graph that reduce s–t routing to connecting s and t to their respective "transit nodes" followed by shortest-path computation between these transit nodes using a "highway".[23] Combinations of such techniques may be needed for optimal practical performance on specific problems.[24] Specialized variants When arc weights are small integers (bounded by a parameter $C$), specialized queues which take advantage of this fact can be used to speed up Dijkstra's algorithm. The first algorithm of this type was Dial's algorithm (Dial 1969) for graphs with positive integer edge weights, which uses a bucket queue to obtain a running time $O(\|E\|+\|V\|C)$. The use of a Van Emde Boas tree as the priority queue brings the complexity to $O(\|E\|\log \log C)$ (Ahuja et al. 1990). Another interesting variant based on a combination of a new radix heap and the well-known Fibonacci heap runs in time $O(\|E\|+\|V\|{\sqrt {\log C}})$ (Ahuja et al. 1990). Finally, the best algorithms in this special case are as follows. The algorithm given by (Thorup 2000) runs in $O(\|E\|\log \log \|V\|)$ time and the algorithm given by (Raman 1997) runs in $O(\|E\|+\|V\|\min\{(\log \|V\|)^{1/3+\varepsilon },(\log C)^{1/4+\varepsilon }\})$ time. Related problems and algorithms The functionality of Dijkstra's original algorithm can be extended with a variety of modifications. For example, sometimes it is desirable to present solutions which are less than mathematically optimal. To obtain a ranked list of less-than-optimal solutions, the optimal solution is first calculated. A single edge appearing in the optimal solution is removed from the graph, and the optimum solution to this new graph is calculated. Each edge of the original solution is suppressed in turn and a new shortest-path calculated. The secondary solutions are then ranked and presented after the first optimal solution. Dijkstra's algorithm is usually the working principle behind link-state routing protocols, OSPF and IS-IS being the most common ones. Unlike Dijkstra's algorithm, the Bellman–Ford algorithm can be used on graphs with negative edge weights, as long as the graph contains no negative cycle reachable from the source vertex s. The presence of such cycles means there is no shortest path, since the total weight becomes lower each time the cycle is traversed. (This statement assumes that a "path" is allowed to repeat vertices. In graph theory that is normally not allowed. In theoretical computer science it often is allowed.) It is possible to adapt Dijkstra's algorithm to handle negative weight edges by combining it with the Bellman-Ford algorithm (to remove negative edges and detect negative cycles); such an algorithm is called Johnson's algorithm. The A* algorithm is a generalization of Dijkstra's algorithm that cuts down on the size of the subgraph that must be explored, if additional information is available that provides a lower bound on the "distance" to the target. The process that underlies Dijkstra's algorithm is similar to the greedy process used in Prim's algorithm. Prim's purpose is to find a minimum spanning tree that connects all nodes in the graph; Dijkstra is concerned with only two nodes. Prim's does not evaluate the total weight of the path from the starting node, only the individual edges. Breadth-first search can be viewed as a special-case of Dijkstra's algorithm on unweighted graphs, where the priority queue degenerates into a FIFO queue. The fast marching method can be viewed as a continuous version of Dijkstra's algorithm which computes the geodesic distance on a triangle mesh. Dynamic programming perspective From a dynamic programming point of view, Dijkstra's algorithm is a successive approximation scheme that solves the dynamic programming functional equation for the shortest path problem by the Reaching method.[25][26][27] In fact, Dijkstra's explanation of the logic behind the algorithm,[28] namely Problem 2. Find the path of minimum total length between two given nodes $P$ and $Q$. We use the fact that, if $R$ is a node on the minimal path from $P$ to $Q$, knowledge of the latter implies the knowledge of the minimal path from $P$ to $R$. is a paraphrasing of Bellman's famous Principle of Optimality in the context of the shortest path problem. Applications Least-cost paths are calculated for instance to establish tracks of electricity lines or oil pipelines. The algorithm has also been used to calculate optimal long-distance footpaths in Ethiopia and contrast them with the situation on the ground.[29] See also • A* search algorithm • Bellman–Ford algorithm • Euclidean shortest path • Floyd–Warshall algorithm • Johnson's algorithm • Longest path problem • Parallel all-pairs shortest path algorithm Notes 1. Controversial, see Moshe Sniedovich (2006). "Dijkstra's algorithm revisited: the dynamic programming connexion". Control and Cybernetics. 35: 599–620. and below part. 2. Cormen et al. 2001 3. Fredman & Tarjan 1987 4. Richards, Hamilton. "Edsger Wybe Dijkstra". A.M. Turing Award. Association for Computing Machinery. Retrieved 16 October 2017. At the Mathematical Centre a major project was building the ARMAC computer. For its official inauguration in 1956, Dijkstra devised a program to solve a problem interesting to a nontechnical audience: Given a network of roads connecting cities, what is the shortest route between two designated cities? 5. Frana, Phil (August 2010). "An Interview with Edsger W. Dijkstra". Communications of the ACM. 53 (8): 41–47. doi:10.1145/1787234.1787249. 6. Dijkstra, E. W. (1959). "A note on two problems in connexion with graphs" (PDF). Numerische Mathematik. 1: 269–271. doi:10.1007/BF01386390. S2CID 123284777. 7. Mehlhorn, Kurt; Sanders, Peter (2008). "Chapter 10. Shortest Paths" (PDF). Algorithms and Data Structures: The Basic Toolbox. Springer. doi:10.1007/978-3-540-77978-0. ISBN 978-3-540-77977-3. 8. Szcześniak, Ireneusz; Jajszczyk, Andrzej; Woźna-Szcześniak, Bożena (2019). "Generic Dijkstra for optical networks". Journal of Optical Communications and Networking. 11 (11): 568–577. arXiv:1810.04481. doi:10.1364/JOCN.11.000568. S2CID 52958911. 9. Szcześniak, Ireneusz; Woźna-Szcześniak, Bożena (2023), "Generic Dijkstra: Correctness and tractability", NOMS 2023-2023 IEEE/IFIP Network Operations and Management Symposium, pp. 1–7, arXiv:2204.13547, doi:10.1109/NOMS56928.2023.10154322, ISBN 978-1-6654-7716-1, S2CID 248427020 10. Schrijver, Alexander (2012). "On the history of the shortest path problem" (PDF). Documenta Mathematica. 11. Felner, Ariel (2011). Position Paper: Dijkstra's Algorithm versus Uniform Cost Search or a Case Against Dijkstra's Algorithm. Proc. 4th Int'l Symp. on Combinatorial Search. In a route-finding problem, Felner finds that the queue can be a factor 500–600 smaller, taking some 40% of the running time. 12. "ARMAC". Unsung Heroes in Dutch Computing History. 2007. Archived from the original on 13 November 2013. 13. Dijkstra, Edsger W., Reflections on "A note on two problems in connexion with graphs (PDF) 14. Tarjan, Robert Endre (1983), Data Structures and Network Algorithms, CBMS_NSF Regional Conference Series in Applied Mathematics, vol. 44, Society for Industrial and Applied Mathematics, p. 75, The third classical minimum spanning tree algorithm was discovered by Jarník and rediscovered by Prim and Dikstra; it is commonly known as Prim's algorithm. 15. Prim, R.C. (1957). "Shortest connection networks and some generalizations" (PDF). Bell System Technical Journal. 36 (6): 1389–1401. Bibcode:1957BSTJ...36.1389P. doi:10.1002/j.1538-7305.1957.tb01515.x. Archived from the original (PDF) on 18 July 2017. Retrieved 18 July 2017. 16. V. Jarník: O jistém problému minimálním [About a certain minimal problem], Práce Moravské Přírodovědecké Společnosti, 6, 1930, pp. 57–63. (in Czech) 17. Gass, Saul; Fu, Michael (2013). "Dijkstra's Algorithm". In Gass, Saul I; Fu, Michael C (eds.). Encyclopedia of Operations Research and Management Science. Vol. 1. Springer. doi:10.1007/978-1-4419-1153-7. ISBN 978-1-4419-1137-7 – via Springer Link. 18. Fredman & Tarjan 1984. 19. Observe that p < dist[u] cannot ever hold because of the update dist[v] ← alt when updating the queue. See https://cs.stackexchange.com/questions/118388/dijkstra-without-decrease-key for discussion. 20. Chen, M.; Chowdhury, R. A.; Ramachandran, V.; Roche, D. L.; Tong, L. (2007). Priority Queues and Dijkstra's Algorithm – UTCS Technical Report TR-07-54 – 12 October 2007 (PDF). Austin, Texas: The University of Texas at Austin, Department of Computer Sciences. 21. Russell, Stuart; Norvig, Peter (2009) [1995]. Artificial Intelligence: A Modern Approach (3rd ed.). Prentice Hall. pp. 75, 81. ISBN 978-0-13-604259-4. 22. Sometimes also least-cost-first search: Nau, Dana S. (1983). "Expert computer systems" (PDF). Computer. IEEE. 16 (2): 63–85. doi:10.1109/mc.1983.1654302. S2CID 7301753. 23. Wagner, Dorothea; Willhalm, Thomas (2007). Speed-up techniques for shortest-path computations. STACS. pp. 23–36. 24. Bauer, Reinhard; Delling, Daniel; Sanders, Peter; Schieferdecker, Dennis; Schultes, Dominik; Wagner, Dorothea (2010). "Combining hierarchical and goal-directed speed-up techniques for Dijkstra's algorithm". J. Experimental Algorithmics. 15: 2.1. doi:10.1145/1671970.1671976. S2CID 1661292. 25. Sniedovich, M. (2006). "Dijkstra's algorithm revisited: the dynamic programming connexion" (PDF). Journal of Control and Cybernetics. 35 (3): 599–620. Online version of the paper with interactive computational modules. 26. Denardo, E.V. (2003). Dynamic Programming: Models and Applications. Mineola, NY: Dover Publications. ISBN 978-0-486-42810-9. 27. Sniedovich, M. (2010). Dynamic Programming: Foundations and Principles. Francis & Taylor. ISBN 978-0-8247-4099-3. 28. Dijkstra 1959, p. 270 29. Nyssen, J., Tesfaalem Ghebreyohannes, Hailemariam Meaza, Dondeyne, S., 2020. Exploration of a medieval African map (Aksum, Ethiopia) – How do historical maps fit with topography? In: De Ryck, M., Nyssen, J., Van Acker, K., Van Roy, W., Liber Amicorum: Philippe De Maeyer In Kaart. Wachtebeke (Belgium): University Press: 165-178. References • Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). "Section 24.3: Dijkstra's algorithm". Introduction to Algorithms (Second ed.). MIT Press and McGraw–Hill. pp. 595–601. ISBN 0-262-03293-7. • Dial, Robert B. (1969). "Algorithm 360: Shortest-path forest with topological ordering [H]". Communications of the ACM. 12 (11): 632–633. doi:10.1145/363269.363610. S2CID 6754003. • Fredman, Michael Lawrence; Tarjan, Robert E. (1984). Fibonacci heaps and their uses in improved network optimization algorithms. 25th Annual Symposium on Foundations of Computer Science. IEEE. pp. 338–346. doi:10.1109/SFCS.1984.715934. • Fredman, Michael Lawrence; Tarjan, Robert E. (1987). "Fibonacci heaps and their uses in improved network optimization algorithms". Journal of the Association for Computing Machinery. 34 (3): 596–615. doi:10.1145/28869.28874. S2CID 7904683. • Zhan, F. Benjamin; Noon, Charles E. (February 1998). "Shortest Path Algorithms: An Evaluation Using Real Road Networks". Transportation Science. 32 (1): 65–73. doi:10.1287/trsc.32.1.65. S2CID 14986297. • Leyzorek, M.; Gray, R. S.; Johnson, A. A.; Ladew, W. C.; Meaker, Jr., S. R.; Petry, R. M.; Seitz, R. N. (1957). Investigation of Model Techniques – First Annual Report – 6 June 1956 – 1 July 1957 – A Study of Model Techniques for Communication Systems. Cleveland, Ohio: Case Institute of Technology. • Knuth, D.E. (1977). "A Generalization of Dijkstra's Algorithm". Information Processing Letters. 6 (1): 1–5. doi:10.1016/0020-0190(77)90002-3. • Ahuja, Ravindra K.; Mehlhorn, Kurt; Orlin, James B.; Tarjan, Robert E. (April 1990). "Faster Algorithms for the Shortest Path Problem" (PDF). Journal of the ACM. 37 (2): 213–223. doi:10.1145/77600.77615. hdl:1721.1/47994. S2CID 5499589. • Raman, Rajeev (1997). "Recent results on the single-source shortest paths problem". SIGACT News. 28 (2): 81–87. doi:10.1145/261342.261352. S2CID 18031586. • Thorup, Mikkel (2000). "On RAM priority Queues". SIAM Journal on Computing. 30 (1): 86–109. doi:10.1137/S0097539795288246. S2CID 5221089. • Thorup, Mikkel (1999). "Undirected single-source shortest paths with positive integer weights in linear time". Journal of the ACM. 46 (3): 362–394. doi:10.1145/316542.316548. S2CID 207654795. External links Wikimedia Commons has media related to Dijkstra's algorithm. • Oral history interview with Edsger W. Dijkstra, Charles Babbage Institute, University of Minnesota, Minneapolis • Implementation of Dijkstra's algorithm using TDD, Robert Cecil Martin, The Clean Code Blog Edsger Dijkstra Works • A Primer of ALGOL 60 Programming (book) • Structured Programming (book) • A Discipline of Programming (book) • A Method of Programming (book) • Predicate Calculus and Program Semantics (book) • Selected Writings on Computing: A Personal Perspective (book) • A Note on Two Problems in Connexion with Graphs • Cooperating Sequential Processes • Solution of a Problem in Concurrent Programming Control • The Structure of the 'THE'-Multiprogramming System • Go To Statement Considered Harmful • Notes on Structured Programming • The Humble Programmer • Programming Considered as a Human Activity • How Do We Tell Truths That Might Hurt? • On the Role of Scientific Thought • Self-stabilizing Systems in Spite of Distributed Control • On the Cruelty of Really Teaching Computer Science • Selected papers • EWD manuscripts Main research areas • Theoretical computing science • Software engineering • Systems science • Algorithm design • Concurrent computing • Distributed computing • Formal methods • Programming methodology • Programming language research • Program design and development • Software architecture • Philosophy of computer programming and computing science Related people • Shlomi Dolev • Per Brinch Hansen • Tony Hoare • Ole-Johan Dahl • Leslie Lamport • David Parnas • Jaap Zonneveld • Carel S. Scholten • Adriaan van Wijngaarden • Niklaus Wirth • Wikiquote Graph and tree traversal algorithms • α–β pruning • A* • IDA* • LPA* • SMA* • Best-first search • Beam search • Bidirectional search • Breadth-first search • Lexicographic • Parallel • B* • Depth-first search • Iterative Deepening • D* • Fringe search • Jump point search • Monte Carlo tree search • SSS* Shortest path • Bellman–Ford • Dijkstra's • Floyd–Warshall • Johnson's • Shortest path faster • Yen's Minimum spanning tree • Borůvka's • Kruskal's • Prim's • Reverse-delete List of graph search algorithms Optimization: Algorithms, methods, and heuristics Unconstrained nonlinear Functions • Golden-section search • Interpolation methods • Line search • Nelder–Mead method • Successive parabolic interpolation Gradients Convergence • Trust region • Wolfe conditions Quasi–Newton • Berndt–Hall–Hall–Hausman • Broyden–Fletcher–Goldfarb–Shanno and L-BFGS • Davidon–Fletcher–Powell • Symmetric rank-one (SR1) Other methods • Conjugate gradient • Gauss–Newton • Gradient • Mirror • Levenberg–Marquardt • Powell's dog leg method • Truncated Newton Hessians • Newton's method Constrained nonlinear General • Barrier methods • Penalty methods Differentiable • Augmented Lagrangian methods • Sequential quadratic programming • Successive linear programming Convex optimization Convex minimization • Cutting-plane method • Reduced gradient (Frank–Wolfe) • Subgradient method Linear and quadratic Interior point • Affine scaling • Ellipsoid algorithm of Khachiyan • Projective algorithm of Karmarkar Basis-exchange • Simplex algorithm of Dantzig • Revised simplex algorithm • Criss-cross algorithm • Principal pivoting algorithm of Lemke Combinatorial Paradigms • Approximation algorithm • Dynamic programming • Greedy algorithm • Integer programming • Branch and bound/cut Graph algorithms Minimum spanning tree • Borůvka • Prim • Kruskal Shortest path • Bellman–Ford • SPFA • Dijkstra • Floyd–Warshall Network flows • Dinic • Edmonds–Karp • Ford–Fulkerson • Push–relabel maximum flow Metaheuristics • Evolutionary algorithm • Hill climbing • Local search • Parallel metaheuristics • Simulated annealing • Spiral optimization algorithm • Tabu search • Software	Wikipedia
Uniform field theory Uniform field theory is a formula for determining the effective electrical resistance of a parallel wire system. By calculating the mean square field acting throughout a section of coil, formulae are obtained for the effective resistances of single- and multi-layer solenoidal coils of either solid or stranded wire.[1] See also • Mathematical methods in electronics References 1. E. F. Armstrong and T. P. Hilditch, A study of catalytic actions at solid surfaces, Nature, 107, pages 573-575 (30 June 1921).	Wikipedia
Kernel (statistics) The term kernel is used in statistical analysis to refer to a window function. The term "kernel" has several distinct meanings in different branches of statistics. Bayesian statistics In statistics, especially in Bayesian statistics, the kernel of a probability density function (pdf) or probability mass function (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the variables in the domain are omitted. Note that such factors may well be functions of the parameters of the pdf or pmf. These factors form part of the normalization factor of the probability distribution, and are unnecessary in many situations. For example, in pseudo-random number sampling, most sampling algorithms ignore the normalization factor. In addition, in Bayesian analysis of conjugate prior distributions, the normalization factors are generally ignored during the calculations, and only the kernel considered. At the end, the form of the kernel is examined, and if it matches a known distribution, the normalization factor can be reinstated. Otherwise, it may be unnecessary (for example, if the distribution only needs to be sampled from). For many distributions, the kernel can be written in closed form, but not the normalization constant. An example is the normal distribution. Its probability density function is $p(x\|\mu ,\sigma ^{2})={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}$ and the associated kernel is $p(x\|\mu ,\sigma ^{2})\propto e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}$ Note that the factor in front of the exponential has been omitted, even though it contains the parameter $\sigma ^{2}$ , because it is not a function of the domain variable $x$ . Pattern analysis The kernel of a reproducing kernel Hilbert space is used in the suite of techniques known as kernel methods to perform tasks such as statistical classification, regression analysis, and cluster analysis on data in an implicit space. This usage is particularly common in machine learning. Nonparametric statistics Further information: Kernel smoothing In nonparametric statistics, a kernel is a weighting function used in non-parametric estimation techniques. Kernels are used in kernel density estimation to estimate random variables' density functions, or in kernel regression to estimate the conditional expectation of a random variable. Kernels are also used in time-series, in the use of the periodogram to estimate the spectral density where they are known as window functions. An additional use is in the estimation of a time-varying intensity for a point process where window functions (kernels) are convolved with time-series data. Commonly, kernel widths must also be specified when running a non-parametric estimation. Definition Further information: Integral kernel A kernel is a non-negative real-valued integrable function K. For most applications, it is desirable to define the function to satisfy two additional requirements: • Normalization: $\int _{-\infty }^{+\infty }K(u)\,du=1\,;$ • Symmetry: $K(-u)=K(u){\mbox{ for all values of }}u\,.$ The first requirement ensures that the method of kernel density estimation results in a probability density function. The second requirement ensures that the average of the corresponding distribution is equal to that of the sample used. If K is a kernel, then so is the function K* defined by K*(u) = λK(λu), where λ > 0. This can be used to select a scale that is appropriate for the data. Kernel functions in common use Several types of kernel functions are commonly used: uniform, triangle, Epanechnikov,[1] quartic (biweight), tricube,[2] triweight, Gaussian, quadratic[3] and cosine. In the table below, if $K$ is given with a bounded support, then $K(u)=0$ for values of u lying outside the support. Kernel Functions, K(u) $\textstyle \int u^{2}K(u)du$ $\textstyle \int K(u)^{2}du$ Efficiency[4] relative to the Epanechnikov kernel Uniform ("rectangular window") $K(u)={\frac {1}{2}}$ Support: $\|u\|\leq 1$ "Boxcar function" ${\frac {1}{3}}$ ${\frac {1}{2}}$ 92.9% Triangular $K(u)=(1-\|u\|)$ Support: $\|u\|\leq 1$ ${\frac {1}{6}}$ ${\frac {2}{3}}$ 98.6% Epanechnikov (parabolic) $K(u)={\frac {3}{4}}(1-u^{2})$ Support: $\|u\|\leq 1$ ${\frac {1}{5}}$ ${\frac {3}{5}}$ 100% Quartic (biweight) $K(u)={\frac {15}{16}}(1-u^{2})^{2}$ Support: $\|u\|\leq 1$ ${\frac {1}{7}}$ ${\frac {5}{7}}$ 99.4% Triweight $K(u)={\frac {35}{32}}(1-u^{2})^{3}$ Support: $\|u\|\leq 1$ ${\frac {1}{9}}$ ${\frac {350}{429}}$ 98.7% Tricube $K(u)={\frac {70}{81}}(1-{\left\|u\right\|}^{3})^{3}$ Support: $\|u\|\leq 1$ ${\frac {35}{243}}$ ${\frac {175}{247}}$ 99.8% Gaussian $K(u)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}u^{2}}$ $1\,$ ${\frac {1}{2{\sqrt {\pi }}}}$ 95.1% Cosine $K(u)={\frac {\pi }{4}}\cos \left({\frac {\pi }{2}}u\right)$ Support: $\|u\|\leq 1$ $1-{\frac {8}{\pi ^{2}}}$ ${\frac {\pi ^{2}}{16}}$ 99.9% Logistic $K(u)={\frac {1}{e^{u}+2+e^{-u}}}$ ${\frac {\pi ^{2}}{3}}$ ${\frac {1}{6}}$ 88.7% Sigmoid function $K(u)={\frac {2}{\pi }}{\frac {1}{e^{u}+e^{-u}}}$ ${\frac {\pi ^{2}}{4}}$ ${\frac {2}{\pi ^{2}}}$ 84.3% Silverman kernel[5] $K(u)={\frac {1}{2}}e^{-{\frac {\|u\|}{\sqrt {2}}}}\cdot \sin \left({\frac {\|u\|}{\sqrt {2}}}+{\frac {\pi }{4}}\right)$ $0$ ${\frac {3{\sqrt {2}}}{16}}$ not applicable See also • Kernel density estimation • Kernel smoother • Stochastic kernel • Positive-definite kernel • Density estimation • Multivariate kernel density estimation References 1. Named for Epanechnikov, V. A. (1969). "Non-Parametric Estimation of a Multivariate Probability Density". Theory Probab. Appl. 14 (1): 153–158. doi:10.1137/1114019. 2. Altman, N. S. (1992). "An introduction to kernel and nearest neighbor nonparametric regression". The American Statistician. 46 (3): 175–185. doi:10.1080/00031305.1992.10475879. hdl:1813/31637. 3. Cleveland, W. S.; Devlin, S. J. (1988). "Locally weighted regression: An approach to regression analysis by local fitting". Journal of the American Statistical Association. 83 (403): 596–610. doi:10.1080/01621459.1988.10478639. 4. Efficiency is defined as ${\sqrt {\int u^{2}K(u)\,du}}\int K(u)^{2}\,du$. 5. Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman and Hall, London. • Li, Qi; Racine, Jeffrey S. (2007). Nonparametric Econometrics: Theory and Practice. Princeton University Press. ISBN 978-0-691-12161-1. • Zucchini, Walter. "APPLIED SMOOTHING TECHNIQUES Part 1: Kernel Density Estimation" (PDF). Retrieved 6 September 2018. • Comaniciu, D; Meer, P (2002). "Mean shift: A robust approach toward feature space analysis". IEEE Transactions on Pattern Analysis and Machine Intelligence. 24 (5): 603–619. CiteSeerX 10.1.1.76.8968. doi:10.1109/34.1000236.	Wikipedia
Uniform limit theorem In mathematics, the uniform limit theorem states that the uniform limit of any sequence of continuous functions is continuous. Statement More precisely, let X be a topological space, let Y be a metric space, and let ƒn : X → Y be a sequence of functions converging uniformly to a function ƒ : X → Y. According to the uniform limit theorem, if each of the functions ƒn is continuous, then the limit ƒ must be continuous as well. This theorem does not hold if uniform convergence is replaced by pointwise convergence. For example, let ƒn : [0, 1] → R be the sequence of functions ƒn(x) = xn. Then each function ƒn is continuous, but the sequence converges pointwise to the discontinuous function ƒ that is zero on [0, 1) but has ƒ(1) = 1. Another example is shown in the adjacent image. In terms of function spaces, the uniform limit theorem says that the space C(X, Y) of all continuous functions from a topological space X to a metric space Y is a closed subset of YX under the uniform metric. In the case where Y is complete, it follows that C(X, Y) is itself a complete metric space. In particular, if Y is a Banach space, then C(X, Y) is itself a Banach space under the uniform norm. The uniform limit theorem also holds if continuity is replaced by uniform continuity. That is, if X and Y are metric spaces and ƒn : X → Y is a sequence of uniformly continuous functions converging uniformly to a function ƒ, then ƒ must be uniformly continuous. Proof In order to prove the continuity of f, we have to show that for every ε > 0, there exists a neighbourhood U of any point x of X such that: $d_{Y}(f(x),f(y))<\varepsilon ,\qquad \forall y\in U$ Consider an arbitrary ε > 0. Since the sequence of functions (fn) converges uniformly to f by hypothesis, there exists a natural number N such that: $d_{Y}(f_{N}(t),f(t))<{\frac {\varepsilon }{3}},\qquad \forall t\in X$ Moreover, since fN is continuous on X by hypothesis, for every x there exists a neighbourhood U such that: $d_{Y}(f_{N}(x),f_{N}(y))<{\frac {\varepsilon }{3}},\qquad \forall y\in U$ In the final step, we apply the triangle inequality in the following way: ${\begin{aligned}d_{Y}(f(x),f(y))&\leq d_{Y}(f(x),f_{N}(x))+d_{Y}(f_{N}(x),f_{N}(y))+d_{Y}(f_{N}(y),f(y))\\&<{\frac {\varepsilon }{3}}+{\frac {\varepsilon }{3}}+{\frac {\varepsilon }{3}}=\varepsilon ,\qquad \forall y\in U\end{aligned}}$ Hence, we have shown that the first inequality in the proof holds, so by definition f is continuous everywhere on X. Uniform limit theorem in complex analysis There are also variants of the uniform limit theorem that are used in complex analysis, albeit with modified assumptions. Theorem.[1] Let $\Omega $ be an open and connected subset of the complex numbers. Suppose that $(f_{n})_{n=1}^{\infty }$ is a sequence of holomorphic functions $f_{n}:\Omega \to \mathbb {C} $ that converges uniformly to a function $f:\Omega \to \mathbb {C} $ on every compact subset of $\Omega $. Then $f$ is holomorphic in $\Omega $, and moreover, the sequence of derivatives $(f'_{n})_{n=1}^{\infty }$ converges uniformly to $f'$ on every compact subset of $\Omega $. Theorem.[2] Let $\Omega $ be an open and connected subset of the complex numbers. Suppose that $(f_{n})_{n=1}^{\infty }$ is a sequence of univalent[3] functions $f_{n}:\Omega \to \mathbb {C} $ that converges uniformly to a function $f:\Omega \to \mathbb {C} $. Then $f$ is holomorphic, and moreover, $f$ is either univalent or constant in $\Omega $. Notes 1. E.M.Stein, R.Shakarachi (2003), pp.53-54. 2. E.C.Titchmarsh (1939), p.200. 3. Univalent means holomorphic and injective. References • James Munkres (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2. • E. M. Stein, R. Shakarachi (2003). Complex Analysis (Princeton Lectures in Analysis, No. 2), Princeton University Press, pp.53-54. • E. C. Titchmarsh (1939). The Theory of Functions, 2002 Reprint, Oxford Science Publications.	Wikipedia
Uniform matroid In mathematics, a uniform matroid is a matroid in which the independent sets are exactly the sets containing at most r elements, for some fixed integer r. An alternative definition is that every permutation of the elements is a symmetry. Definition The uniform matroid $U{}_{n}^{r}$ is defined over a set of $n$ elements. A subset of the elements is independent if and only if it contains at most $r$ elements. A subset is a basis if it has exactly $r$ elements, and it is a circuit if it has exactly $r+1$ elements. The rank of a subset $S$ is $\min(\|S\|,r)$ and the rank of the matroid is $r$.[1][2] A matroid of rank $r$ is uniform if and only if all of its circuits have exactly $r+1$ elements.[3] The matroid $U{}_{n}^{2}$ is called the $n$-point line. Duality and minors The dual matroid of the uniform matroid $U{}_{n}^{r}$ is another uniform matroid $U{}_{n}^{n-r}$. A uniform matroid is self-dual if and only if $r=n/2$.[4] Every minor of a uniform matroid is uniform. Restricting a uniform matroid $U{}_{n}^{r}$ by one element (as long as $r<n$) produces the matroid $U{}_{n-1}^{r}$ and contracting it by one element (as long as $r>0$) produces the matroid $U{}_{n-1}^{r-1}$.[5] Realization The uniform matroid $U{}_{n}^{r}$ may be represented as the matroid of affinely independent subsets of $n$ points in general position in $r$-dimensional Euclidean space, or as the matroid of linearly independent subsets of $n$ vectors in general position in an $(r+1)$-dimensional real vector space. Every uniform matroid may also be realized in projective spaces and vector spaces over all sufficiently large finite fields.[6] However, the field must be large enough to include enough independent vectors. For instance, the $n$-point line $U{}_{n}^{2}$ can be realized only over finite fields of $n-1$ or more elements (because otherwise the projective line over that field would have fewer than $n$ points): $U{}_{4}^{2}$ is not a binary matroid, $U{}_{5}^{2}$ is not a ternary matroid, etc. For this reason, uniform matroids play an important role in Rota's conjecture concerning the forbidden minor characterization of the matroids that can be realized over finite fields.[7] Algorithms The problem of finding the minimum-weight basis of a weighted uniform matroid is well-studied in computer science as the selection problem. It may be solved in linear time.[8] Any algorithm that tests whether a given matroid is uniform, given access to the matroid via an independence oracle, must perform an exponential number of oracle queries, and therefore cannot take polynomial time.[9] Related matroids Unless $r\in \{0,n\}$, a uniform matroid $U{}_{n}^{r}$ is connected: it is not the direct sum of two smaller matroids.[10] The direct sum of a family of uniform matroids (not necessarily all with the same parameters) is called a partition matroid. Every uniform matroid is a paving matroid,[11] a transversal matroid[12] and a strict gammoid.[6] Not every uniform matroid is graphic, and the uniform matroids provide the smallest example of a non-graphic matroid, $U{}_{4}^{2}$. The uniform matroid $U{}_{n}^{1}$ is the graphic matroid of an $n$-edge dipole graph, and the dual uniform matroid $U{}_{n}^{n-1}$ is the graphic matroid of its dual graph, the $n$-edge cycle graph. $U{}_{n}^{0}$ is the graphic matroid of a graph with $n$ self-loops, and $U{}_{n}^{n}$ is the graphic matroid of an $n$-edge forest. Other than these examples, every uniform matroid $U{}_{n}^{r}$ with $1<r<n-1$ contains $U{}_{4}^{2}$ as a minor and therefore is not graphic.[13] The $n$-point line provides an example of a Sylvester matroid, a matroid in which every line contains three or more points.[14] See also • K-set (geometry) References 1. Oxley, James G. (2006), "Example 1.2.7", Matroid Theory, Oxford Graduate Texts in Mathematics, vol. 3, Oxford University Press, p. 19, ISBN 9780199202508. For the rank function, see p. 26. 2. Welsh, D. J. A. (2010), Matroid Theory, Courier Dover Publications, p. 10, ISBN 9780486474397. 3. Oxley (2006), p. 27. 4. Oxley (2006), pp. 77 & 111. 5. Oxley (2006), pp. 106–107 & 111. 6. Oxley (2006), p. 100. 7. Oxley (2006), pp. 202–206. 8. Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001), "Chapter 9: Medians and Order Statistics", Introduction to Algorithms (2nd ed.), MIT Press and McGraw-Hill, pp. 183–196, ISBN 0-262-03293-7. 9. Jensen, Per M.; Korte, Bernhard (1982), "Complexity of matroid property algorithms", SIAM Journal on Computing, 11 (1): 184–190, doi:10.1137/0211014, MR 0646772. 10. Oxley (2006), p. 126. 11. Oxley (2006, p. 26). 12. Oxley (2006), pp. 48–49. 13. Welsh (2010), p. 30. 14. Welsh (2010), p. 297.	Wikipedia
Uniform norm In mathematical analysis, the uniform norm (or sup norm) assigns to real- or complex-valued bounded functions $f$ defined on a set $S$ the non-negative number $\\|f\\|_{\infty }=\\|f\\|_{\infty ,S}=\sup \left\{\,\|f(s)\|:s\in S\,\right\}.$ This article is about the function space norm. For the finite-dimensional vector space distance, see Chebyshev distance. For the uniformity norm in additive combinatorics, see Gowers norm. This norm is also called the supremum norm, the Chebyshev norm, the infinity norm, or, when the supremum is in fact the maximum, the max norm. The name "uniform norm" derives from the fact that a sequence of functions $\left\{f_{n}\right\}$ converges to $f$ under the metric derived from the uniform norm if and only if $f_{n}$ converges to $f$ uniformly.[1] If $f$ is a continuous function on a closed and bounded interval, or more generally a compact set, then it is bounded and the supremum in the above definition is attained by the Weierstrass extreme value theorem, so we can replace the supremum by the maximum. In this case, the norm is also called the maximum norm. In particular, if $x$ is some vector such that $x=\left(x_{1},x_{2},\ldots ,x_{n}\right)$ in finite dimensional coordinate space, it takes the form: $\\|x\\|_{\infty }:=\max \left(\left\|x_{1}\right\|,\ldots ,\left\|x_{n}\right\|\right).$ Metric and topology The metric generated by this norm is called the Chebyshev metric, after Pafnuty Chebyshev, who was first to systematically study it. If we allow unbounded functions, this formula does not yield a norm or metric in a strict sense, although the obtained so-called extended metric still allows one to define a topology on the function space in question. The binary function $d(f,g)=\\|f-g\\|_{\infty }$ is then a metric on the space of all bounded functions (and, obviously, any of its subsets) on a particular domain. A sequence $\left\{f_{n}:n=1,2,3,\ldots \right\}$ converges uniformly to a function $f$ if and only if $\lim _{n\rightarrow \infty }\left\\|f_{n}-f\right\\|_{\infty }=0.\,$ We can define closed sets and closures of sets with respect to this metric topology; closed sets in the uniform norm are sometimes called uniformly closed and closures uniform closures. The uniform closure of a set of functions A is the space of all functions that can be approximated by a sequence of uniformly-converging functions on $A.$ For instance, one restatement of the Stone–Weierstrass theorem is that the set of all continuous functions on $[a,b]$ is the uniform closure of the set of polynomials on $[a,b].$ For complex continuous functions over a compact space, this turns it into a C* algebra. Properties The set of vectors whose infinity norm is a given constant, $c,$ forms the surface of a hypercube with edge length $2c.$ The reason for the subscript "$\infty $" is that whenever $f$ is continuous $\lim _{p\to \infty }\\|f\\|_{p}=\\|f\\|_{\infty },$ where $\\|f\\|_{p}=\left(\int _{D}\|f\|^{p}\,d\mu \right)^{1/p}$ where $D$ is the domain of $f$ and the integral amounts to a sum if $D$ is a discrete set (see p-norm). See also • L-infinity – Space of bounded sequences • Uniform continuity – Uniform restraint of the change in functions • Uniform space – Topological space with a notion of uniform properties • Chebyshev distance – Mathematical metric References 1. Rudin, Walter (1964). Principles of Mathematical Analysis. New York: McGraw-Hill. pp. 151. ISBN 0-07-054235-X. Lp spaces Basic concepts • Banach & Hilbert spaces • Lp spaces • Measure • Lebesgue • Measure space • Measurable space/function • Minkowski distance • Sequence spaces L1 spaces • Integrable function • Lebesgue integration • Taxicab geometry L2 spaces • Bessel's • Cauchy–Schwarz • Euclidean distance • Hilbert space • Parseval's identity • Polarization identity • Pythagorean theorem • Square-integrable function $L^{\infty }$ spaces • Bounded function • Chebyshev distance • Infimum and supremum • Essential • Uniform norm Maps • Almost everywhere • Convergence almost everywhere • Convergence in measure • Function space • Integral transform • Locally integrable function • Measurable function • Symmetric decreasing rearrangement Inequalities • Babenko–Beckner • Chebyshev's • Clarkson's • Hanner's • Hausdorff–Young • Hölder's • Markov's • Minkowski • Young's convolution Results • Marcinkiewicz interpolation theorem • Plancherel theorem • Riemann–Lebesgue • Riesz–Fischer theorem • Riesz–Thorin theorem For Lebesgue measure • Isoperimetric inequality • Brunn–Minkowski theorem • Milman's reverse • Minkowski–Steiner formula • Prékopa–Leindler inequality • Vitale's random Brunn–Minkowski inequality Applications & related • Bochner space • Fourier analysis • Lorentz space • Probability theory • Quasinorm • Real analysis • Sobolev space • -algebra • C-algebra • Von Neumann Banach space topics Types of Banach spaces • Asplund • Banach • list • Banach lattice • Grothendieck • Hilbert • Inner product space • Polarization identity • (Polynomially) Reflexive • Riesz • L-semi-inner product • (B • Strictly • Uniformly) convex • Uniformly smooth • (Injective • Projective) Tensor product (of Hilbert spaces) Banach spaces are: • Barrelled • Complete • F-space • Fréchet • tame • Locally convex • Seminorms/Minkowski functionals • Mackey • Metrizable • Normed • norm • Quasinormed • Stereotype Function space Topologies • Banach–Mazur compactum • Dual • Dual space • Dual norm • Operator • Ultraweak • Weak • polar • operator • Strong • polar • operator • Ultrastrong • Uniform convergence Linear operators • Adjoint • Bilinear • form • operator • sesquilinear • (Un)Bounded • Closed • Compact • on Hilbert spaces • (Dis)Continuous • Densely defined • Fredholm • kernel • operator • Hilbert–Schmidt • Functionals • positive • Pseudo-monotone • Normal • Nuclear • Self-adjoint • Strictly singular • Trace class • Transpose • Unitary Operator theory • Banach algebras • C-algebras • Operator space • Spectrum • C-algebra • radius • Spectral theory • of ODEs • Spectral theorem • Polar decomposition • Singular value decomposition Theorems • Anderson–Kadec • Banach–Alaoglu • Banach–Mazur • Banach–Saks • Banach–Schauder (open mapping) • Banach–Steinhaus (Uniform boundedness) • Bessel's inequality • Cauchy–Schwarz inequality • Closed graph • Closed range • Eberlein–Šmulian • Freudenthal spectral • Gelfand–Mazur • Gelfand–Naimark • Goldstine • Hahn–Banach • hyperplane separation • Kakutani fixed-point • Krein–Milman • Lomonosov's invariant subspace • Mackey–Arens • Mazur's lemma • M. Riesz extension • Parseval's identity • Riesz's lemma • Riesz representation • Robinson-Ursescu • Schauder fixed-point Analysis • Abstract Wiener space • Banach manifold • bundle • Bochner space • Convex series • Differentiation in Fréchet spaces • Derivatives • Fréchet • Gateaux • functional • holomorphic • quasi • Integrals • Bochner • Dunford • Gelfand–Pettis • regulated • Paley–Wiener • weak • Functional calculus • Borel • continuous • holomorphic • Measures • Lebesgue • Projection-valued • Vector • Weakly / Strongly measurable function Types of sets • Absolutely convex • Absorbing • Affine • Balanced/Circled • Bounded • Convex • Convex cone (subset) • Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) • Linear cone (subset) • Radial • Radially convex/Star-shaped • Symmetric • Zonotope Subsets / set operations • Affine hull • (Relative) Algebraic interior (core) • Bounding points • Convex hull • Extreme point • Interior • Linear span • Minkowski addition • Polar • (Quasi) Relative interior Examples • Absolute continuity AC • $ba(\Sigma )$ • c space • Banach coordinate BK • Besov $B_{p,q}^{s}(\mathbb {R} )$ • Birnbaum–Orlicz • Bounded variation BV • Bs space • Continuous C(K) with K compact Hausdorff • Hardy Hp • Hilbert H • Morrey–Campanato $L^{\lambda ,p}(\Omega )$ • ℓp • $\ell ^{\infty }$ • Lp • $L^{\infty }$ • weighted • Schwartz $S\left(\mathbb {R} ^{n}\right)$ • Segal–Bargmann F • Sequence space • Sobolev Wk,p • Sobolev inequality • Triebel–Lizorkin • Wiener amalgam $W(X,L^{p})$ Applications • Differential operator • Finite element method • Mathematical formulation of quantum mechanics • Ordinary Differential Equations (ODEs) • Validated numerics Functional analysis (topics – glossary) Spaces • Banach • Besov • Fréchet • Hilbert • Hölder • Nuclear • Orlicz • Schwartz • Sobolev • Topological vector Properties • Barrelled • Complete • Dual (Algebraic/Topological) • Locally convex • Reflexive • Reparable Theorems • Hahn–Banach • Riesz representation • Closed graph • Uniform boundedness principle • Kakutani fixed-point • Krein–Milman • Min–max • Gelfand–Naimark • Banach–Alaoglu Operators • Adjoint • Bounded • Compact • Hilbert–Schmidt • Normal • Nuclear • Trace class • Transpose • Unbounded • Unitary Algebras • Banach algebra • C-algebra • Spectrum of a C-algebra • Operator algebra • Group algebra of a locally compact group • Von Neumann algebra Open problems • Invariant subspace problem • Mahler's conjecture Applications • Hardy space • Spectral theory of ordinary differential equations • Heat kernel • Index theorem • Calculus of variations • Functional calculus • Integral operator • Jones polynomial • Topological quantum field theory • Noncommutative geometry • Riemann hypothesis • Distribution (or Generalized functions) Advanced topics • Approximation property • Balanced set • Choquet theory • Weak topology • Banach–Mazur distance • Tomita–Takesaki theory • Mathematics portal • Category • Commons	Wikipedia
Uniform module In abstract algebra, a module is called a uniform module if the intersection of any two nonzero submodules is nonzero. This is equivalent to saying that every nonzero submodule of M is an essential submodule. A ring may be called a right (left) uniform ring if it is uniform as a right (left) module over itself. Alfred Goldie used the notion of uniform modules to construct a measure of dimension for modules, now known as the uniform dimension (or Goldie dimension) of a module. Uniform dimension generalizes some, but not all, aspects of the notion of the dimension of a vector space. Finite uniform dimension was a key assumption for several theorems by Goldie, including Goldie's theorem, which characterizes which rings are right orders in a semisimple ring. Modules of finite uniform dimension generalize both Artinian modules and Noetherian modules. In the literature, uniform dimension is also referred to as simply the dimension of a module or the rank of a module. Uniform dimension should not be confused with the related notion, also due to Goldie, of the reduced rank of a module. Properties and examples of uniform modules Being a uniform module is not usually preserved by direct products or quotient modules. The direct sum of two nonzero uniform modules always contains two submodules with intersection zero, namely the two original summand modules. If N1 and N2 are proper submodules of a uniform module M and neither submodule contains the other, then $M/(N_{1}\cap N_{2})$ fails to be uniform, as $N_{1}/(N_{1}\cap N_{2})\cap N_{2}/(N_{1}\cap N_{2})=\{0\}.$ Uniserial modules are uniform, and uniform modules are necessarily directly indecomposable. Any commutative domain is a uniform ring, since if a and b are nonzero elements of two ideals, then the product ab is a nonzero element in the intersection of the ideals. Uniform dimension of a module The following theorem makes it possible to define a dimension on modules using uniform submodules. It is a module version of a vector space theorem: Theorem: If Ui and Vj are members of a finite collection of uniform submodules of a module M such that $\oplus _{i=1}^{n}U_{i}$ and $\oplus _{i=1}^{m}V_{i}$ are both essential submodules of M, then n = m. The uniform dimension of a module M, denoted u.dim(M), is defined to be n if there exists a finite set of uniform submodules Ui such that $\oplus _{i=1}^{n}U_{i}$ is an essential submodule of M. The preceding theorem ensures that this n is well defined. If no such finite set of submodules exists, then u.dim(M) is defined to be ∞. When speaking of the uniform dimension of a ring, it is necessary to specify whether u.dim(RR) or rather u.dim(RR) is being measured. It is possible to have two different uniform dimensions on the opposite sides of a ring. If N is a submodule of M, then u.dim(N) ≤ u.dim(M) with equality exactly when N is an essential submodule of M. In particular, M and its injective hull E(M) always have the same uniform dimension. It is also true that u.dim(M) = n if and only if E(M) is a direct sum of n indecomposable injective modules. It can be shown that u.dim(M) = ∞ if and only if M contains an infinite direct sum of nonzero submodules. Thus if M is either Noetherian or Artinian, M has finite uniform dimension. If M has finite composition length k, then u.dim(M) ≤ k with equality exactly when M is a semisimple module. (Lam 1999) A standard result is that a right Noetherian domain is a right Ore domain. In fact, we can recover this result from another theorem attributed to Goldie, which states that the following three conditions are equivalent for a domain D: • D is right Ore • u.dim(DD) = 1 • u.dim(DD) < ∞ Hollow modules and co-uniform dimension The dual notion of a uniform module is that of a hollow module: a module M is said to be hollow if, when N1 and N2 are submodules of M such that $N_{1}+N_{2}=M$, then either N1 = M or N2 = M. Equivalently, one could also say that every proper submodule of M is a superfluous submodule. These modules also admit an analogue of uniform dimension, called co-uniform dimension, corank, hollow dimension or dual Goldie dimension. Studies of hollow modules and co-uniform dimension were conducted in (Fleury 1974) harv error: no target: CITEREFFleury1974 (help), (Reiter 1981) harv error: no target: CITEREFReiter1981 (help), (Takeuchi 1976) harv error: no target: CITEREFTakeuchi1976 (help), (Varadarajan 1979) harv error: no target: CITEREFVaradarajan1979 (help) and (Miyashita 1966) harv error: no target: CITEREFMiyashita1966 (help). The reader is cautioned that Fleury explored distinct ways of dualizing Goldie dimension. Varadarajan, Takeuchi and Reiter's versions of hollow dimension are arguably the more natural ones. Grzeszczuk and Puczylowski in (Grzeszczuk & Puczylowski 1984) harv error: no target: CITEREFGrzeszczukPuczylowski1984 (help) gave a definition of uniform dimension for modular lattices such that the hollow dimension of a module was the uniform dimension of its dual lattice of submodules. It is always the case that a finitely cogenerated module has finite uniform dimension. This raises the question: does a finitely generated module have finite hollow dimension? The answer turns out to be no: it was shown in (Sarath & Varadarajan 1979) harv error: no target: CITEREFSarathVaradarajan1979 (help) that if a module M has finite hollow dimension, then M/J(M) is a semisimple, Artinian module. There are many rings with unity for which R/J(R) is not semisimple Artinian, and given such a ring R, R itself is finitely generated but has infinite hollow dimension. Sarath and Varadarajan showed later, that M/J(M) being semisimple Artinian is also sufficient for M to have finite hollow dimension provided J(M) is a superfluous submodule of M.[1] This shows that the rings R with finite hollow dimension either as a left or right R-module are precisely the semilocal rings. An additional corollary of Varadarajan's result is that RR has finite hollow dimension exactly when RR does. This contrasts the finite uniform dimension case, since it is known a ring can have finite uniform dimension on one side and infinite uniform dimension on the other. Textbooks • Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0525-8, ISBN 978-0-387-98428-5, MR 1653294 Primary sources 1. The same result can be found in (Reiter 1981) harv error: no target: CITEREFReiter1981 (help) and (Hanna & Shamsuddin 1984) harv error: no target: CITEREFHannaShamsuddin1984 (help) • Fleury, Patrick (1974), "A note on dualizing Goldie dimension", Canadian Mathematical Bulletin, 17 (4): 511–517, doi:10.4153/cmb-1974-090-0 • Goldie, A. W. (1958), "The structure of prime rings under ascending chain conditions", Proc. London Math. Soc., Series 3, 8 (4): 589–608, doi:10.1112/plms/s3-8.4.589, ISSN 0024-6115, MR 0103206 • Goldie, A. W. (1960), "Semi-prime rings with maximum condition", Proc. London Math. Soc., Series 3, 10: 201–220, doi:10.1112/plms/s3-10.1.201, ISSN 0024-6115, MR 0111766 • Grezeszcuk, P; Puczylowski, E (1984), "On Goldie and dual Goldie dimension", Journal of Pure and Applied Algebra, 31 (1–3): 47–55, doi:10.1016/0022-4049(84)90075-6 • Hanna, A.; Shamsuddin, A. (1984), Duality in the category of modules: Applications, Reinhard Fischer, ISBN 978-3889270177 • Miyashita, Y. (1966), "Quasi-projective modules, perfect modules, and a theorem for modular lattices", J. Fac. Sci. Hokkaido Ser. I, 19: 86–110, MR 0213390 • Reiter, E. (1981), "A dual to the Goldie ascending chain condition on direct sums of submodules", Bull. Calcutta Math. Soc., 73: 55–63 • Sarath B.; Varadarajan, K. (1979), "Dual Goldie dimension II", Communications in Algebra, 7 (17): 1885–1899, doi:10.1080/00927877908822434 • Takeuchi, T. (1976), "On cofinite-dimensional modules.", Hokkaido Mathematical Journal, 5 (1): 1–43, doi:10.14492/hokmj/1381758746, ISSN 0385-4035, MR 0213390 • Varadarajan, K. (1979), "Dual Goldie dimension", Comm. Algebra, 7 (6): 565–610, doi:10.1080/00927877908822364, ISSN 0092-7872, MR 0524269	Wikipedia
Uniform polyhedron compound In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts transitively on the compound's vertices. The uniform polyhedron compounds were first enumerated by John Skilling in 1976, with a proof that the enumeration is complete. The following table lists them according to his numbering. The prismatic compounds of {p/q}-gonal prisms (UC20 and UC21) exist only when p/q > 2, and when p and q are coprime. The prismatic compounds of {p/q}-gonal antiprisms (UC22, UC23, UC24 and UC25) exist only when p/q > 3/2, and when p and q are coprime. Furthermore, when p/q = 2, the antiprisms degenerate into tetrahedra with digonal bases. Compound Bowers acronym Picture Polyhedral count Polyhedral type Faces Edges Vertices Notes Symmetry group Subgroup restricting to one constituent UC01 sis 6 tetrahedra 24{3} 36 24 Rotational freedom Td S4 UC02 dis 12 tetrahedra 48{3} 72 48 Rotational freedom Oh S4 UC03 snu 6 tetrahedra 24{3} 36 24 Oh D2d UC04 so 2 tetrahedra 8{3} 12 8 Regular Oh Td UC05 ki 5 tetrahedra 20{3} 30 20 Regular I T UC06 e 10 tetrahedra 40{3} 60 20 Regular 2 polyhedra per vertex Ih T UC07 risdoh 6 cubes (12+24){4} 72 48 Rotational freedom Oh C4h UC08 rah 3 cubes (6+12){4} 36 24 Oh D4h UC09 rhom 5 cubes 30{4} 60 20 Regular 2 polyhedra per vertex Ih Th UC10 dissit 4 octahedra (8+24){3} 48 24 Rotational freedom Th S6 UC11 daso 8 octahedra (16+48){3} 96 48 Rotational freedom Oh S6 UC12 sno 4 octahedra (8+24){3} 48 24 Oh D3d UC13 addasi 20 octahedra (40+120){3} 240 120 Rotational freedom Ih S6 UC14 dasi 20 octahedra (40+120){3} 240 60 2 polyhedra per vertex Ih S6 UC15 gissi 10 octahedra (20+60){3} 120 60 Ih D3d UC16 si 10 octahedra (20+60){3} 120 60 Ih D3d UC17 se 5 octahedra 40{3} 60 30 Regular Ih Th UC18 hirki 5 tetrahemihexahedra 20{3} 15{4} 60 30 I T UC19 sapisseri 20 tetrahemihexahedra (20+60){3} 60{4} 240 60 2 polyhedra per vertex I C3 UC20 - 2n (2n ≥ 2) p/q-gonal prisms 4n{p/q} 2np{4} 6np 4np Rotational freedom Dnph Cph UC21 - n (n ≥ 2) p/q-gonal prisms 2n{p/q} np{4} 3np 2np Dnph Dph UC22 - 2n (2n ≥ 2) (q odd) p/q-gonal antiprisms (q odd) 4n{p/q} (if p/q ≠ 2) 4np{3} 8np 4np Rotational freedom Dnpd (if n odd) Dnph (if n even) S2p UC23 - n (n ≥ 2) p/q-gonal antiprisms (q odd) 2n{p/q} (if p/q ≠ 2) 2np{3} 4np 2np Dnpd (if n odd) Dnph (if n even) Dpd UC24 - 2n (2n ≥ 2) p/q-gonal antiprisms (q even) 4n{p/q} (if p/q ≠ 2) 4np{3} 8np 4np Rotational freedom Dnph Cph UC25 - n (n ≥ 2) p/q-gonal antiprisms (q even) 2n{p/q} (if p/q ≠ 2) 2np{3} 4np 2np Dnph Dph UC26 gadsid 12 pentagonal antiprisms 120{3} 24{5} 240 120 Rotational freedom Ih S10 UC27 gassid 6 pentagonal antiprisms 60{3} 12{5} 120 60 Ih D5d UC28 gidasid 12 pentagrammic crossed antiprisms 120{3} 24{5/2} 240 120 Rotational freedom Ih S10 UC29 gissed 6 pentagrammic crossed antiprisms 60{3} 125 120 60 Ih D5d UC30 ro 4 triangular prisms 8{3} 12{4} 36 24 O D3 UC31 dro 8 triangular prisms 16{3} 24{4} 72 48 Oh D3 UC32 kri 10 triangular prisms 20{3} 30{4} 90 60 I D3 UC33 dri 20 triangular prisms 40{3} 60{4} 180 60 2 polyhedra per vertex Ih D3 UC34 kred 6 pentagonal prisms 30{4} 12{5} 90 60 I D5 UC35 dird 12 pentagonal prisms 60{4} 24{5} 180 60 2 polyhedra per vertex Ih D5 UC36 gikrid 6 pentagrammic prisms 30{4} 12{5/2} 90 60 I D5 UC37 giddird 12 pentagrammic prisms 60{4} 24{5/2} 180 60 2 polyhedra per vertex Ih D5 UC38 griso 4 hexagonal prisms 24{4} 8{6} 72 48 Oh D3d UC39 rosi 10 hexagonal prisms 60{4} 20{6} 180 120 Ih D3d UC40 rassid 6 decagonal prisms 60{4} 12{10} 180 120 Ih D5d UC41 grassid 6 decagrammic prisms 60{4} 12{10/3} 180 120 Ih D5d UC42 gassic 3 square antiprisms 24{3} 6{4} 48 24 O D4 UC43 gidsac 6 square antiprisms 48{3} 12{4} 96 48 Oh D4 UC44 sassid 6 pentagrammic antiprisms 60{3} 12{5/2} 120 60 I D5 UC45 sadsid 12 pentagrammic antiprisms 120{3} 24{5/2} 240 120 Ih D5 UC46 siddo 2 icosahedra (16+24){3} 60 24 Oh Th UC47 sne 5 icosahedra (40+60){3} 150 60 Ih Th UC48 presipsido 2 great dodecahedra 24{5} 60 24 Oh Th UC49 presipsi 5 great dodecahedra 60{5} 150 60 Ih Th UC50 passipsido 2 small stellated dodecahedra 24{5/2} 60 24 Oh Th UC51 passipsi 5 small stellated dodecahedra 60{5/2} 150 60 Ih Th UC52 sirsido 2 great icosahedra (16+24){3} 60 24 Oh Th UC53 sirsei 5 great icosahedra (40+60){3} 150 60 Ih Th UC54 tisso 2 truncated tetrahedra 8{3} 8{6} 36 24 Oh Td UC55 taki 5 truncated tetrahedra 20{3} 20{6} 90 60 I T UC56 te 10 truncated tetrahedra 40{3} 40{6} 180 120 Ih T UC57 tar 5 truncated cubes 40{3} 30{8} 180 120 Ih Th UC58 quitar 5 stellated truncated hexahedra 40{3} 30{8/3} 180 120 Ih Th UC59 arie 5 cuboctahedra 40{3} 30{4} 120 60 Ih Th UC60 gari 5 cubohemioctahedra 30{4} 20{6} 120 60 Ih Th UC61 iddei 5 octahemioctahedra 40{3} 20{6} 120 60 Ih Th UC62 rasseri 5 rhombicuboctahedra 40{3} (30+60){4} 240 120 Ih Th UC63 rasher 5 small rhombihexahedra 60{4} 30{8} 240 120 Ih Th UC64 rahrie 5 small cubicuboctahedra 40{3} 30{4} 30{8} 240 120 Ih Th UC65 raquahri 5 great cubicuboctahedra 40{3} 30{4} 30{8/3} 240 120 Ih Th UC66 rasquahr 5 great rhombihexahedra 60{4} 30{8/3} 240 120 Ih Th UC67 rosaqri 5 nonconvex great rhombicuboctahedra 40{3} (30+60){4} 240 120 Ih Th UC68 disco 2 snub cubes (16+48){3} 12{4} 120 48 Oh O UC69 dissid 2 snub dodecahedra (40+120){3} 24{5} 300 120 Ih I UC70 giddasid 2 great snub icosidodecahedra (40+120){3} 24{5/2} 300 120 Ih I UC71 gidsid 2 great inverted snub icosidodecahedra (40+120){3} 24{5/2} 300 120 Ih I UC72 gidrissid 2 great retrosnub icosidodecahedra (40+120){3} 24{5/2} 300 120 Ih I UC73 disdid 2 snub dodecadodecahedra 120{3} 24{5} 24{5/2} 300 120 Ih I UC74 idisdid 2 inverted snub dodecadodecahedra 120{3} 24{5} 24{5/2} 300 120 Ih I UC75 desided 2 snub icosidodecadodecahedra (40+120){3} 24{5} 24{5/2} 360 120 Ih I References • Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79: 447–457, doi:10.1017/S0305004100052440, MR 0397554. External links • http://www.interocitors.com/polyhedra/UCs/ShortNames.html - Bowers style acronyms for uniform polyhedron compounds	Wikipedia
Uniform polyhedron In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also face- and edge-transitive), quasi-regular (if also edge-transitive but not face-transitive), or semi-regular (if neither edge- nor face-transitive). The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra. There are two infinite classes of uniform polyhedra, together with 75 other polyhedra: • Infinite classes: • prisms, • antiprisms. • Convex exceptional: • 5 Platonic solids: regular convex polyhedra, • 13 Archimedean solids: 2 quasiregular and 11 semiregular convex polyhedra. • Star (nonconvex) exceptional: • 4 Kepler–Poinsot polyhedra: regular nonconvex polyhedra, • 53 uniform star polyhedra: 14 quasiregular and 39 semiregular. Hence 5 + 13 + 4 + 53 = 75. There are also many degenerate uniform polyhedra with pairs of edges that coincide, including one found by John Skilling called the great disnub dirhombidodecahedron (Skilling's figure). Dual polyhedra to uniform polyhedra are face-transitive (isohedral) and have regular vertex figures, and are generally classified in parallel with their dual (uniform) polyhedron. The dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid. The concept of uniform polyhedron is a special case of the concept of uniform polytope, which also applies to shapes in higher-dimensional (or lower-dimensional) space. Definition The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others continues to afflict all the work on this topic (including that of the present author). It arises from the fact that the traditional usage of the term “regular polyhedra” was, and is, contrary to syntax and to logic: the words seem to imply that we are dealing, among the objects we call “polyhedra”, with those special ones that deserve to be called “regular”. But at each stage— Euclid, Kepler, Poinsot, Hess, Brückner,…—the writers failed to define what are the “polyhedra” among which they are finding the “regular” ones. (Branko Grünbaum 1994) Coxeter, Longuet-Higgins & Miller (1954) define uniform polyhedra to be vertex-transitive polyhedra with regular faces. They define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property. By a polygon they implicitly mean a polygon in 3-dimensional Euclidean space; these are allowed to be non-convex and to intersect each other. There are some generalizations of the concept of a uniform polyhedron. If the connectedness assumption is dropped, then we get uniform compounds, which can be split as a union of polyhedra, such as the compound of 5 cubes. If we drop the condition that the realization of the polyhedron is non-degenerate, then we get the so-called degenerate uniform polyhedra. These require a more general definition of polyhedra. Grünbaum (1994) gave a rather complicated definition of a polyhedron, while McMullen & Schulte (2002) gave a simpler and more general definition of a polyhedron: in their terminology, a polyhedron is a 2-dimensional abstract polytope with a non-degenerate 3-dimensional realization. Here an abstract polytope is a poset of its "faces" satisfying various condition, a realization is a function from its vertices to some space, and the realization is called non-degenerate if any two distinct faces of the abstract polytope have distinct realizations. Some of the ways they can be degenerate are as follows: • Hidden faces. Some polyhedra have faces that are hidden, in the sense that no points of their interior can be seen from the outside. These are usually not counted as uniform polyhedra. • Degenerate compounds. Some polyhedra have multiple edges and their faces are the faces of two or more polyhedra, though these are not compounds in the previous sense since the polyhedra share edges. • Double covers. There are some non-orientable polyhedra that have double covers satisfying the definition of a uniform polyhedron. There double covers have doubled faces, edges and vertices. They are usually not counted as uniform polyhedra. • Double faces. There are several polyhedra with doubled faces produced by Wythoff's construction. Most authors do not allow doubled faces and remove them as part of the construction. • Double edges. Skilling's figure has the property that it has double edges (as in the degenerate uniform polyhedra) but its faces cannot be written as a union of two uniform polyhedra. History Regular convex polyhedra • The Platonic solids date back to the classical Greeks and were studied by the Pythagoreans, Plato (c. 424 – 348 BC), Theaetetus (c. 417 BC – 369 BC), Timaeus of Locri (ca. 420–380 BC) and Euclid (fl. 300 BC). The Etruscans discovered the regular dodecahedron before 500 BC.[1] Nonregular uniform convex polyhedra • The cuboctahedron was known by Plato. • Archimedes (287 BC – 212 BC) discovered all of the 13 Archimedean solids. His original book on the subject was lost, but Pappus of Alexandria (c. 290 – c. 350 AD) mentioned Archimedes listed 13 polyhedra. • Piero della Francesca (1415 – 1492) rediscovered the five truncations of the Platonic solids: truncated tetrahedron, truncated octahedron, truncated cube, truncated dodecahedron, and truncated icosahedron, and included illustrations and calculations of their metric properties in his book De quinque corporibus regularibus. He also discussed the cuboctahedron in a different book.[2] • Luca Pacioli plagiarized Francesca's work in De divina proportione in 1509, adding the rhombicuboctahedron, calling it a icosihexahedron for its 26 faces, which was drawn by Leonardo da Vinci. • Johannes Kepler (1571–1630) was the first to publish the complete list of Archimedean solids, in 1619. He also identified the infinite families of uniform prisms and antiprisms. Regular star polyhedra • Kepler (1619) discovered two of the regular Kepler–Poinsot polyhedra, the small stellated dodecahedron and great stellated dodecahedron. • Louis Poinsot (1809) discovered the other two, the great dodecahedron and great icosahedron. • The set of four were proven complete by Augustin-Louis Cauchy in 1813, and named by Arthur Cayley in 1859. Other 53 nonregular star polyhedra • Of the remaining 53, Edmund Hess (1878) discovered two, Albert Badoureau (1881) discovered 36 more, and Pitsch (1881) independently discovered 18, of which 3 had not previously been discovered. Together these gave 41 polyhedra. • The geometer H.S.M. Coxeter discovered the remaining twelve in collaboration with J. C. P. Miller (1930–1932) but did not publish. M.S. Longuet-Higgins and H.C. Longuet-Higgins independently discovered eleven of these. Lesavre and Mercier rediscovered five of them in 1947. • Coxeter, Longuet-Higgins & Miller (1954) published the list of uniform polyhedra. • Sopov (1970) proved their conjecture that the list was complete. • In 1974, Magnus Wenninger published his book Polyhedron models, which lists all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by Norman Johnson. • Skilling (1975) independently proved the completeness, and showed that if the definition of uniform polyhedron is relaxed to allow edges to coincide then there is just one extra possibility (the great disnub dirhombidodecahedron). • In 1987, Edmond Bonan drew all the uniform polyhedra and their duals in 3D, with a Turbo Pascal program called Polyca. Most of them were shown during the International Stereoscopic Union Congress held in 1993, at the Congress Theatre, Eastbourne, England; and again in 2005 at the Kursaal of Besançon, France.[3] • In 1993, Zvi Har'El (1949–2008)[4] produced a complete kaleidoscopic construction of the uniform polyhedra and duals with a computer program called Kaleido, and summarized in a paper Uniform Solution for Uniform Polyhedra, counting figures 1-80.[5] • Also in 1993, R. Mäder ported this Kaleido solution to Mathematica with a slightly different indexing system.[6] • In 2002 Peter W. Messer discovered a minimal set of closed-form expressions for determining the main combinatorial and metrical quantities of any uniform polyhedron (and its dual) given only its Wythoff symbol.[7] Uniform star polyhedra The 57 nonprismatic nonconvex forms, with exception of the great dirhombicosidodecahedron, are compiled by Wythoff constructions within Schwarz triangles. Convex forms by Wythoff construction The convex uniform polyhedra can be named by Wythoff construction operations on the regular form. In more detail the convex uniform polyhedron are given below by their Wythoff construction within each symmetry group. Within the Wythoff construction, there are repetitions created by lower symmetry forms. The cube is a regular polyhedron, and a square prism. The octahedron is a regular polyhedron, and a triangular antiprism. The octahedron is also a rectified tetrahedron. Many polyhedra are repeated from different construction sources, and are colored differently. The Wythoff construction applies equally to uniform polyhedra and uniform tilings on the surface of a sphere, so images of both are given. The spherical tilings including the set of hosohedrons and dihedrons which are degenerate polyhedra. These symmetry groups are formed from the reflectional point groups in three dimensions, each represented by a fundamental triangle (p q r), where p > 1, q > 1, r > 1 and 1/p + 1/q + 1/r < 1. • Tetrahedral symmetry (3 3 2) – order 24 • Octahedral symmetry (4 3 2) – order 48 • Icosahedral symmetry (5 3 2) – order 120 • Dihedral symmetry (n 2 2), for n = 3,4,5,... – order 4n The remaining nonreflective forms are constructed by alternation operations applied to the polyhedra with an even number of sides. Along with the prisms and their dihedral symmetry, the spherical Wythoff construction process adds two regular classes which become degenerate as polyhedra : the dihedra and the hosohedra, the first having only two faces, and the second only two vertices. The truncation of the regular hosohedra creates the prisms. Below the convex uniform polyhedra are indexed 1–18 for the nonprismatic forms as they are presented in the tables by symmetry form. For the infinite set of prismatic forms, they are indexed in four families: 1. Hosohedra H2... (only as spherical tilings) 2. Dihedra D2... (only as spherical tilings) 3. Prisms P3... (truncated hosohedra) 4. Antiprisms A3... (snub prisms) Summary tables Johnson name Parent Truncated Rectified Bitruncated (tr. dual) Birectified (dual) Cantellated Omnitruncated (cantitruncated) Snub Coxeter diagram Extended Schläfli symbol ${\begin{Bmatrix}p,q\end{Bmatrix}}$ $t{\begin{Bmatrix}p,q\end{Bmatrix}}$ ${\begin{Bmatrix}p\\q\end{Bmatrix}}$ $t{\begin{Bmatrix}q,p\end{Bmatrix}}$ ${\begin{Bmatrix}q,p\end{Bmatrix}}$ $r{\begin{Bmatrix}p\\q\end{Bmatrix}}$ $t{\begin{Bmatrix}p\\q\end{Bmatrix}}$ $s{\begin{Bmatrix}p\\q\end{Bmatrix}}$ {p,q} t{p,q} r{p,q} 2t{p,q} 2r{p,q} rr{p,q} tr{p,q} sr{p,q} t0{p,q} t0,1{p,q} t1{p,q} t1,2{p,q} t2{p,q} t0,2{p,q} t0,1,2{p,q} ht0,1,2{p,q} Wythoff symbol (p q 2) q \| p 2 2 q \| p 2 \| p q 2 p \| q p \| q 2 p q \| 2 p q 2 \| \| p q 2 Vertex figure pq q.2p.2p (p.q)2 p.2q.2q qp p.4.q.4 4.2p.2q 3.3.p.3.q Tetrahedral (3 3 2) 3.3.3 3.6.6 3.3.3.3 3.6.6 3.3.3 3.4.3.4 4.6.6 3.3.3.3.3 Octahedral (4 3 2) 4.4.4 3.8.8 3.4.3.4 4.6.6 3.3.3.3 3.4.4.4 4.6.8 3.3.3.3.4 Icosahedral (5 3 2) 5.5.5 3.10.10 3.5.3.5 5.6.6 3.3.3.3.3 3.4.5.4 4.6.10 3.3.3.3.5 And a sampling of dihedral symmetries: (The sphere is not cut, only the tiling is cut.) (On a sphere, an edge is the arc of the great circle, the shortest way, between its two vertices. Hence, a digon whose vertices are not polar-opposite is flat: it looks like an edge.) (p 2 2) Parent Truncated Rectified Bitruncated (tr. dual) Birectified (dual) Cantellated Omnitruncated (cantitruncated) Snub Coxeter diagram Extended Schläfli symbol ${\begin{Bmatrix}p,2\end{Bmatrix}}$ $t{\begin{Bmatrix}p,2\end{Bmatrix}}$ ${\begin{Bmatrix}p\\2\end{Bmatrix}}$ $t{\begin{Bmatrix}2,p\end{Bmatrix}}$ ${\begin{Bmatrix}2,p\end{Bmatrix}}$ $r{\begin{Bmatrix}p\\2\end{Bmatrix}}$ $t{\begin{Bmatrix}p\\2\end{Bmatrix}}$ $s{\begin{Bmatrix}p\\2\end{Bmatrix}}$ {p,2} t{p,2} r{p,2} 2t{p,2} 2r{p,2} rr{p,2} tr{p,2} sr{p,2} t0{p,2} t0,1{p,2} t1{p,2} t1,2{p,2} t2{p,2} t0,2{p,2} t0,1,2{p,2} ht0,1,2{p,2} Wythoff symbol 2 \| p 2 2 2 \| p 2 \| p 2 2 p \| 2 p \| 2 2 p 2 \| 2 p 2 2 \| \| p 2 2 Vertex figure p2 2.2p.2p p.2.p.2 p.4.4 2p p.4.2.4 4.2p.4 3.3.3.p Dihedral (2 2 2) {2,2} 2.4.4 2.2.2.2 4.4.2 2.2 2.4.2.4 4.4.4 3.3.3.2 Dihedral (3 2 2) 3.3 2.6.6 2.3.2.3 4.4.3 2.2.2 2.4.3.4 4.4.6 3.3.3.3 Dihedral (4 2 2) 4.4 2.8.8 2.4.2.4 4.4.4 2.2.2.2 2.4.4.4 4.4.8 3.3.3.4 Dihedral (5 2 2) 5.5 2.10.10 2.5.2.5 4.4.5 2.2.2.2.2 2.4.5.4 4.4.10 3.3.3.5 Dihedral (6 2 2) 6.6 2.12.12 2.6.2.6 4.4.6 2.2.2.2.2.2 2.4.6.4 4.4.12 3.3.3.6 (3 3 2) Td tetrahedral symmetry The tetrahedral symmetry of the sphere generates 5 uniform polyhedra, and a 6th form by a snub operation. The tetrahedral symmetry is represented by a fundamental triangle with one vertex with two mirrors, and two vertices with three mirrors, represented by the symbol (3 3 2). It can also be represented by the Coxeter group A2 or [3,3], as well as a Coxeter diagram: . There are 24 triangles, visible in the faces of the tetrakis hexahedron, and in the alternately colored triangles on a sphere: # Name Graph A3 Graph A2 Picture Tiling Vertex figure Coxeter and Schläfli symbols Face counts by position Element counts Pos. 2 [3] (4) Pos. 1 [2] (6) Pos. 0 [3] (4) Faces Edges Vertices 1 Tetrahedron {3,3} {3} 4 6 4 [1] Birectified tetrahedron (same as tetrahedron) t2{3,3}={3,3} {3} 4 6 4 2 Rectified tetrahedron Tetratetrahedron (same as octahedron) t1{3,3}=r{3,3} {3} {3} 8 12 6 3 Truncated tetrahedron t0,1{3,3}=t{3,3} {6} {3} 8 18 12 [3] Bitruncated tetrahedron (same as truncated tetrahedron) t1,2{3,3}=t{3,3} {3} {6} 8 18 12 4 Cantellated tetrahedron Rhombitetratetrahedron (same as cuboctahedron) t0,2{3,3}=rr{3,3} {3} {4} {3} 14 24 12 5 Omnitruncated tetrahedron Truncated tetratetrahedron (same as truncated octahedron) t0,1,2{3,3}=tr{3,3} {6} {4} {6} 14 36 24 6 Snub tetratetrahedron (same as icosahedron) sr{3,3} {3} 2 {3} {3} 20 30 12 (4 3 2) Oh octahedral symmetry The octahedral symmetry of the sphere generates 7 uniform polyhedra, and a 7 more by alternation. Six of these forms are repeated from the tetrahedral symmetry table above. The octahedral symmetry is represented by a fundamental triangle (4 3 2) counting the mirrors at each vertex. It can also be represented by the Coxeter group B2 or [4,3], as well as a Coxeter diagram: . There are 48 triangles, visible in the faces of the disdyakis dodecahedron, and in the alternately colored triangles on a sphere: # Name Graph B3 Graph B2 Picture Tiling Vertex figure Coxeter and Schläfli symbols Face counts by position Element counts Pos. 2 [4] (6) Pos. 1 [2] (12) Pos. 0 [3] (8) Faces Edges Vertices 7 Cube {4,3} {4} 6 12 8 [2] Octahedron {3,4} {3} 8 12 6 [4] Rectified cube Rectified octahedron (Cuboctahedron) {4,3} {4} {3} 14 24 12 8 Truncated cube t0,1{4,3}=t{4,3} {8} {3} 14 36 24 [5] Truncated octahedron t0,1{3,4}=t{3,4} {4} {6} 14 36 24 9 Cantellated cube Cantellated octahedron Rhombicuboctahedron t0,2{4,3}=rr{4,3} {4} {4} {3} 26 48 24 10 Omnitruncated cube Omnitruncated octahedron Truncated cuboctahedron t0,1,2{4,3}=tr{4,3} {8} {4} {6} 26 72 48 [6] Snub octahedron (same as Icosahedron) = s{3,4}=sr{3,3} {3} {3} 20 30 12 [1] Half cube (same as Tetrahedron) = h{4,3}={3,3} 1/2 {3} 4 6 4 [2] Cantic cube (same as Truncated tetrahedron) = h2{4,3}=t{3,3} 1/2 {6} 1/2 {3} 8 18 12 [4] (same as Cuboctahedron) = rr{3,3} 14 24 12 [5] (same as Truncated octahedron) = tr{3,3} 14 36 24 [9] Cantic snub octahedron (same as Rhombicuboctahedron) s2{3,4}=rr{3,4} 26 48 24 11 Snub cuboctahedron sr{4,3} {4} 2 {3} {3} 38 60 24 (5 3 2) Ih icosahedral symmetry The icosahedral symmetry of the sphere generates 7 uniform polyhedra, and a 1 more by alternation. Only one is repeated from the tetrahedral and octahedral symmetry table above. The icosahedral symmetry is represented by a fundamental triangle (5 3 2) counting the mirrors at each vertex. It can also be represented by the Coxeter group G2 or [5,3], as well as a Coxeter diagram: . There are 120 triangles, visible in the faces of the disdyakis triacontahedron, and in the alternately colored triangles on a sphere: # Name Graph (A2) [6] Graph (H3) [10] Picture Tiling Vertex figure Coxeter and Schläfli symbols Face counts by position Element counts Pos. 2 [5] (12) Pos. 1 [2] (30) Pos. 0 [3] (20) Faces Edges Vertices 12 Dodecahedron {5,3} {5} 12 30 20 [6] Icosahedron {3,5} {3} 20 30 12 13 Rectified dodecahedron Rectified icosahedron Icosidodecahedron t1{5,3}=r{5,3} {5} {3} 32 60 30 14 Truncated dodecahedron t0,1{5,3}=t{5,3} {10} {3} 32 90 60 15 Truncated icosahedron t0,1{3,5}=t{3,5} {5} {6} 32 90 60 16 Cantellated dodecahedron Cantellated icosahedron Rhombicosidodecahedron t0,2{5,3}=rr{5,3} {5} {4} {3} 62 120 60 17 Omnitruncated dodecahedron Omnitruncated icosahedron Truncated icosidodecahedron t0,1,2{5,3}=tr{5,3} {10} {4} {6} 62 180 120 18 Snub icosidodecahedron sr{5,3} {5} 2 {3} {3} 92 150 60 (p 2 2) Prismatic [p,2], I2(p) family (Dph dihedral symmetry) Main article: Prismatic uniform polyhedron The dihedral symmetry of the sphere generates two infinite sets of uniform polyhedra, prisms and antiprisms, and two more infinite set of degenerate polyhedra, the hosohedra and dihedra which exist as tilings on the sphere. The dihedral symmetry is represented by a fundamental triangle (p 2 2) counting the mirrors at each vertex. It can also be represented by the Coxeter group I2(p) or [n,2], as well as a prismatic Coxeter diagram: . Below are the first five dihedral symmetries: D2 ... D6. The dihedral symmetry Dp has order 4n, represented the faces of a bipyramid, and on the sphere as an equator line on the longitude, and n equally-spaced lines of longitude. (2 2 2) Dihedral symmetry There are 8 fundamental triangles, visible in the faces of the square bipyramid (Octahedron) and alternately colored triangles on a sphere: # Name Picture Tiling Vertex figure Coxeter and Schläfli symbols Face counts by position Element counts Pos. 2 [2] (2) Pos. 1 [2] (2) Pos. 0 [2] (2) Faces Edges Vertices D2 H2 Digonal dihedron, digonal hosohedron {2,2} {2} 2 2 2 D4 Truncated digonal dihedron (same as square dihedron) t{2,2}={4,2} {4} 2 4 4 P4 [7] Omnitruncated digonal dihedron (same as cube) t0,1,2{2,2}=tr{2,2} {4} {4} {4} 6 12 8 A2 [1] Snub digonal dihedron (same as tetrahedron) sr{2,2} 2 {3} 4 6 4 (3 2 2) D3h dihedral symmetry There are 12 fundamental triangles, visible in the faces of the hexagonal bipyramid and alternately colored triangles on a sphere: # Name Picture Tiling Vertex figure Coxeter and Schläfli symbols Face counts by position Element counts Pos. 2 [3] (2) Pos. 1 [2] (3) Pos. 0 [2] (3) Faces Edges Vertices D3 Trigonal dihedron {3,2} {3} 2 3 3 H3 Trigonal hosohedron {2,3} {2} 3 3 2 D6 Truncated trigonal dihedron (same as hexagonal dihedron) t{3,2} {6} 2 6 6 P3 Truncated trigonal hosohedron (Triangular prism) t{2,3} {3} {4} 5 9 6 P6 Omnitruncated trigonal dihedron (Hexagonal prism) t0,1,2{2,3}=tr{2,3} {6} {4} {4} 8 18 12 A3 [2] Snub trigonal dihedron (same as Triangular antiprism) (same as octahedron) sr{2,3} {3} 2 {3} 8 12 6 P3 Cantic snub trigonal dihedron (Triangular prism) s2{2,3}=t{2,3} 5 9 6 (4 2 2) D4h dihedral symmetry There are 16 fundamental triangles, visible in the faces of the octagonal bipyramid and alternately colored triangles on a sphere: # Name Picture Tiling Vertex figure Coxeter and Schläfli symbols Face counts by position Element counts Pos. 2 [4] (2) Pos. 1 [2] (4) Pos. 0 [2] (4) Faces Edges Vertices D4 square dihedron {4,2} {4} 2 4 4 H4 square hosohedron {2,4} {2} 4 4 2 D8 Truncated square dihedron (same as octagonal dihedron) t{4,2} {8} 2 8 8 P4 [7] Truncated square hosohedron (Cube) t{2,4} {4} {4} 6 12 8 D8 Omnitruncated square dihedron (Octagonal prism) t0,1,2{2,4}=tr{2,4} {8} {4} {4} 10 24 16 A4 Snub square dihedron (Square antiprism) sr{2,4} {4} 2 {3} 10 16 8 P4 [7] Cantic snub square dihedron (Cube) s2{4,2}=t{2,4} 6 12 8 A2 [1] Snub square hosohedron (Digonal antiprism) (Tetrahedron) s{2,4}=sr{2,2} 4 6 4 (5 2 2) D5h dihedral symmetry There are 20 fundamental triangles, visible in the faces of the decagonal bipyramid and alternately colored triangles on a sphere: # Name Picture Tiling Vertex figure Coxeter and Schläfli symbols Face counts by position Element counts Pos. 2 [5] (2) Pos. 1 [2] (5) Pos. 0 [2] (5) Faces Edges Vertices D5 Pentagonal dihedron {5,2} {5} 2 5 5 H5 Pentagonal hosohedron {2,5} {2} 5 5 2 D10 Truncated pentagonal dihedron (same as decagonal dihedron) t{5,2} {10} 2 10 10 P5 Truncated pentagonal hosohedron (same as pentagonal prism) t{2,5} {5} {4} 7 15 10 P10 Omnitruncated pentagonal dihedron (Decagonal prism) t0,1,2{2,5}=tr{2,5} {10} {4} {4} 12 30 20 A5 Snub pentagonal dihedron (Pentagonal antiprism) sr{2,5} {5} 2 {3} 12 20 10 P5 Cantic snub pentagonal dihedron (Pentagonal prism) s2{5,2}=t{2,5} 7 15 10 (6 2 2) D6h dihedral symmetry There are 24 fundamental triangles, visible in the faces of the dodecagonal bipyramid and alternately colored triangles on a sphere. # Name Picture Tiling Vertex figure Coxeter and Schläfli symbols Face counts by position Element counts Pos. 2 [6] (2) Pos. 1 [2] (6) Pos. 0 [2] (6) Faces Edges Vertices D6 Hexagonal dihedron {6,2} {6} 2 6 6 H6 Hexagonal hosohedron {2,6} {2} 6 6 2 D12 Truncated hexagonal dihedron (same as dodecagonal dihedron) t{6,2} {12} 2 12 12 H6 Truncated hexagonal hosohedron (same as hexagonal prism) t{2,6} {6} {4} 8 18 12 P12 Omnitruncated hexagonal dihedron (Dodecagonal prism) t0,1,2{2,6}=tr{2,6} {12} {4} {4} 14 36 24 A6 Snub hexagonal dihedron (Hexagonal antiprism) sr{2,6} {6} 2 {3} 14 24 12 P3 Cantic hexagonal dihedron (Triangular prism) = h2{6,2}=t{2,3} 5 9 6 P6 Cantic snub hexagonal dihedron (Hexagonal prism) s2{6,2}=t{2,6} 8 18 12 A3 [2] Snub hexagonal hosohedron (same as Triangular antiprism) (same as octahedron) s{2,6}=sr{2,3} 8 12 6 Wythoff construction operators Operation Symbol Coxeter diagram Description Parent {p,q} t0{p,q} Any regular polyhedron or tiling Rectified (r) r{p,q} t1{p,q} The edges are fully truncated into single points. The polyhedron now has the combined faces of the parent and dual. Polyhedra are named by the number of sides of the two regular forms: {p,q} and {q,p}, like cuboctahedron for r{4,3} between a cube and octahedron. Birectified (2r) (also dual) 2r{p,q} t2{p,q} The birectified (dual) is a further truncation so that the original faces are reduced to points. New faces are formed under each parent vertex. The number of edges is unchanged and are rotated 90 degrees. A birectification can be seen as the dual. Truncated (t) t{p,q} t0,1{p,q} Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated polyhedron. The polyhedron has its original faces doubled in sides, and contains the faces of the dual. Bitruncated (2t) (also truncated dual) 2t{p,q} t1,2{p,q} A bitruncation can be seen as the truncation of the dual. A bitruncated cube is a truncated octahedron. Cantellated (rr) (Also expanded) rr{p,q} In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms. A cantellated polyhedron is named as a rhombi-r{p,q}, like rhombicuboctahedron for rr{4,3}. Cantitruncated (tr) (Also omnitruncated) tr{p,q} t0,1,2{p,q} The truncation and cantellation operations are applied together to create an omnitruncated form which has the parent's faces doubled in sides, the dual's faces doubled in sides, and squares where the original edges existed. Alternation operations Operation Symbol Coxeter diagram Description Snub rectified (sr) sr{p,q} The alternated cantitruncated. All the original faces end up with half as many sides, and the squares degenerate into edges. Since the omnitruncated forms have 3 faces/vertex, new triangles are formed. Usually these alternated faceting forms are slightly deformed thereafter in order to end again as uniform polyhedra. The possibility of the latter variation depends on the degree of freedom. Snub (s) s{p,2q} Alternated truncation Cantic snub (s2) s2{p,2q} Alternated cantellation (hrr) hrr{2p,2q} Only possible in uniform tilings (infinite polyhedra), alternation of For example, Half (h) h{2p,q} Alternation of , same as Cantic (h2) h2{2p,q} Same as Half rectified (hr) hr{2p,2q} Only possible in uniform tilings (infinite polyhedra), alternation of , same as or For example, = or Quarter (q) q{2p,2q} Only possible in uniform tilings (infinite polyhedra), same as For example, = or See also • Polyhedron • Regular polyhedron • Quasiregular polyhedron • Semiregular polyhedron • List of uniform polyhedra • List of uniform polyhedra by vertex figure • List of uniform polyhedra by Wythoff symbol • List of uniform polyhedra by Schwarz triangle • List of Johnson solids • List of Wenninger polyhedron models • Polyhedron model • Uniform tiling • Uniform tilings in hyperbolic plane • Pseudo-uniform polyhedron • List of shapes Notes 1. Regular Polytopes, p.13 2. Piero della Francesca's Polyhedra 3. Edmond Bonan, "Polyèdres Eastbourne 1993", Stéréo-Club Français 1993, 4. Dr. Zvi Har’El (December 14, 1949 – February 2, 2008) and International Jules Verne Studies - A Tribute 5. Har'el, Zvi (1993). "Uniform Solution for Uniform Polyhedra" (PDF). Geometriae Dedicata. 47: 57–110. doi:10.1007/BF01263494. Zvi Har’El, Kaleido software, Images, dual images 6. Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48-57, 1993. 7. Messer, Peter W. (2002). "Closed-Form Expressions for Uniform Polyhedra and Their Duals". Discrete & Computational Geometry. 27 (3): 353–375. doi:10.1007/s00454-001-0078-2. References • Brückner, M. Vielecke und vielflache. Theorie und geschichte.. Leipzig, Germany: Teubner, 1900. • Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954). "Uniform polyhedra" (PDF). Philosophical Transactions of the Royal Society A. 246 (916): 401–450. Bibcode:1954RSPTA.246..401C. doi:10.1098/rsta.1954.0003. ISSN 0080-4614. JSTOR 91532. MR 0062446. S2CID 202575183. • Grünbaum, B. (1994), "Polyhedra with Hollow Faces", in Tibor Bisztriczky; Peter McMullen; Rolf Schneider; et al. (eds.), Proceedings of the NATO Advanced Study Institute on Polytopes: Abstract, Convex and Computational, Springer, pp. 43–70, doi:10.1007/978-94-011-0924-6_3, ISBN 978-94-010-4398-4 • McMullen, Peter; Schulte, Egon (2002), Abstract Regular Polytopes, Cambridge University Press • Skilling, J. (1975). "The complete set of uniform polyhedra". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences. 278 (1278): 111–135. Bibcode:1975RSPTA.278..111S. doi:10.1098/rsta.1975.0022. ISSN 0080-4614. JSTOR 74475. MR 0365333. S2CID 122634260. • Sopov, S. P. (1970). "A proof of the completeness on the list of elementary homogeneous polyhedra". Ukrainskiui Geometricheskiui Sbornik (8): 139–156. MR 0326550. • Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 978-0-521-09859-5. External links • Weisstein, Eric W. "Uniform Polyhedron". MathWorld. • Uniform Solution for Uniform Polyhedra • The Uniform Polyhedra • Virtual Polyhedra Uniform Polyhedra • Uniform polyhedron gallery • Uniform Polyhedron -- from Wolfram MathWorld Has a visual chart of all 75 Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square p-gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221 Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321 Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421 Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes and compounds	Wikipedia
Focused proof In mathematical logic, focused proofs are a family of analytic proofs that arise through goal-directed proof-search, and are a topic of study in structural proof theory and reductive logic. They form the most general definition of goal-directed proof-search—in which someone chooses a formula and performs hereditary reductions until the result meets some condition. The extremal case where reduction only terminates when axioms are reached forms the sub-family of uniform proofs.[1] A sequent calculus is said to have the focusing property when focused proofs are complete for some terminating condition. For System LK, System LJ, and System LL, uniform proofs are focused proofs where all the atoms are assigned negative polarity.[2] Many other sequent calculi has been shown to have the focusing property, notably the nested sequent calculi of both the classical and intuitionistic variants of the modal logics in the S5 cube.[3][4] Uniform proofs In the sequent calculus for an intuitionistic logic, the uniform proofs can be characterised as those in which the upward reading performs all right rules before the left rules. Typically, uniform proofs are not complete for the logic i.e., not all sequents or formulas admit a uniform proof, so one considers fragments where they are complete e.g., the hereditary Harrop fragment of Intuitionistic logic. Due to the deterministic behaviour, uniform proof-search has been used as the control mechanism defining the programming language paradigm of logic programming.[1] Occasionally, uniform proof-search is implemented in a variant of the sequent calculus for the given logic where context management is automatic thereby increasing the fragment for which one can define a logic programming langue.[5] Focused proofs The focusing principle was originally classified through the disambiguation between synchronous and asynchronous connective in Linear Logic i.e., connectives that interact with the context and those that do not, as consequence of research on logic programming. They are now an increasingly important example of control in reductive logic, and can drastically improve proof-search procedures in industry. The essential idea of focusing is to identify and coalesce the non-deterministic choices in a proof, so that a proof can be seen as an alternation of negative phases ( where invertible rules are applied eagerly), and positive phases (where applications of the other rules are confined and controlled).[3] Polarisation According to the rules in the sequent calculus, formulas are canonically put into one of two classes called positive and negative e.g., in LK and LJ the formula $\phi \lor \psi $ is positive. The only freedom is over atoms are assigned a polarity freely. For negative formulas provability is invariant under the application of a right rule; and, dually, for a positive formulas provability is invariant under the application of a left rule. In either case one can safely apply rules in any order to hereditary sub-formulas of the same polarity. In the case of a right rule applied to a positive formula, or a left rule applied to a negative formula, one may result in invalid sequents e.g., in LK and LJ there is no proof of the sequent $B\lor A\implies A\lor B$ beginning with a right rule. A calculus admits the focusing principle if when an original reduct was provable then the hereditary reducts of the same polarity are also provable. That is, one can commit to focusing on decomposing a formula and its sub-formulas of the same polarity without loss of completeness. Focused system A sequent calculus is often shown to have the focusing property by working in a related calculus where polarity explicitly controls which rules apply. Proofs in such systems are in focused, unfocused, or neutral phases, where the first two are characterised by hereditary decomposition; and the latter by forcing a choice of focus. One of the most important operational behaviours a procedure can undergo is backtracking i.e., returning to an earlier stage in the computation where a choice was made. In focused systems for classical and Intuitionistic logic, the use of backtracking can be simulated by pseudo-contraction. Let $\uparrow $ and $\downarrow $ denote change of polarity, the former making a formula negative, and the latter positive; and call a formula with an arrow neutral. Recall that $\lor $ is positive, and consider the neutral polarized sequent $\downarrow \uparrow \phi \lor \psi \implies \uparrow \phi \lor \psi $, which is interpreted as the actual sequent $\phi \lor \psi \implies \phi \lor \psi $. For neutral sequents such as this, the focused system forces on to make an explicit choice of which formula to focus on, denoted by $\langle \,\rangle $. To perform a proof-search the best thing is to choose the left formula, since $\lor $ is positive, indeed (as discussed above) in some cases there are no proofs where the focus is on the right formula. To overcome this, some focused calculi create a backtracking point such that focusing on the right yields $\downarrow \uparrow \phi \lor \psi \implies \langle \phi \lor \psi \rangle ,\uparrow \phi \lor \psi $, which is still as $\phi \lor \psi \implies \phi \lor \psi $. The second formula on the right can be removed only when the focused phase has finished, but if proof-search gets stuck before this happens the sequent may remove the focused component thereby returning to the choice e.g., $\downarrow \uparrow B\lor A\implies \langle A\rangle ,\uparrow A\lor B$ must be taken to $\downarrow \uparrow B\lor A\implies \uparrow A\lor B$ as no other reductive inference can be made. This is a pseudo-contraction since it has the syntactic form of a contraction on the right, but the actual formula doesn't exist i.e., in the interpretation of the proof in the focused system the sequent has only one formula on the right. References 1. Miller, Dale; Nadathur, Gopalan; Pfenning, Frank; Scedrov, Andre (1991-03-14). "Uniform proofs as a foundation for logic programming". Annals of Pure and Applied Logic. 51 (1): 125–157. doi:10.1016/0168-0072(91)90068-W. ISSN 0168-0072. 2. Liang, Chuck; Miller, Dale (2009-11-01). "Focusing and polarization in linear, intuitionistic, and classical logics". Theoretical Computer Science. Abstract Interpretation and Logic Programming: In honor of professor Giorgio Levi. 410 (46): 4747–4768. doi:10.1016/j.tcs.2009.07.041. ISSN 0304-3975. 3. Chaudhuri, Kaustuv; Marin, Sonia; Straßburger, Lutz (2016), Jacobs, Bart; Löding, Christof (eds.), "Focused and Synthetic Nested Sequents", Foundations of Software Science and Computation Structures, Berlin, Heidelberg: Springer Berlin Heidelberg, vol. 9634, pp. 390–407, doi:10.1007/978-3-662-49630-5_23, ISBN 978-3-662-49629-9 4. Chaudhuri, Kaustuv; Marin, Sonia; Straßburger, Lutz (2016). Modular Focused Proof Systems for Intuitionistic Modal Logics. Leibniz International Proceedings in Informatics (LIPIcs). Vol. 52. Marc Herbstritt. pp. 16:1–16:18. doi:10.4230/LIPICS.FSCD.2016.16. ISBN 9783959770101. 5. Armelín, Pablo A.; Pym, David J. (2001), "Bunched Logic Programming", Automated Reasoning, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 289–304, doi:10.1007/3-540-45744-5_21, ISBN 978-3-540-42254-9	Wikipedia
Uniform property In the mathematical field of topology a uniform property or uniform invariant is a property of a uniform space which is invariant under uniform isomorphisms. Since uniform spaces come as topological spaces and uniform isomorphisms are homeomorphisms, every topological property of a uniform space is also a uniform property. This article is (mostly) concerned with uniform properties that are not topological properties. Uniform properties • Separated. A uniform space X is separated if the intersection of all entourages is equal to the diagonal in X × X. This is actually just a topological property, and equivalent to the condition that the underlying topological space is Hausdorff (or simply T0 since every uniform space is completely regular). • Complete. A uniform space X is complete if every Cauchy net in X converges (i.e. has a limit point in X). • Totally bounded (or Precompact). A uniform space X is totally bounded if for each entourage E ⊂ X × X there is a finite cover {Ui} of X such that Ui × Ui is contained in E for all i. Equivalently, X is totally bounded if for each entourage E there exists a finite subset {xi} of X such that X is the union of all E[xi]. In terms of uniform covers, X is totally bounded if every uniform cover has a finite subcover. • Compact. A uniform space is compact if it is complete and totally bounded. Despite the definition given here, compactness is a topological property and so admits a purely topological description (every open cover has a finite subcover). • Uniformly connected. A uniform space X is uniformly connected if every uniformly continuous function from X to a discrete uniform space is constant. • Uniformly disconnected. A uniform space X is uniformly disconnected if it is not uniformly connected. See also • Topological property References • James, I. M. (1990). Introduction to Uniform Spaces. Cambridge, UK: Cambridge University Press. ISBN 0-521-38620-9. • Willard, Stephen (1970). General Topology. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6.	Wikipedia
Scaling (geometry) In affine geometry, uniform scaling (or isotropic scaling[1]) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions. The result of uniform scaling is similar (in the geometric sense) to the original. A scale factor of 1 is normally allowed, so that congruent shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a photograph, or when creating a scale model of a building, car, airplane, etc. More general is scaling with a separate scale factor for each axis direction. Non-uniform scaling (anisotropic scaling) is obtained when at least one of the scaling factors is different from the others; a special case is directional scaling or stretching (in one direction). Non-uniform scaling changes the shape of the object; e.g. a square may change into a rectangle, or into a parallelogram if the sides of the square are not parallel to the scaling axes (the angles between lines parallel to the axes are preserved, but not all angles). It occurs, for example, when a faraway billboard is viewed from an oblique angle, or when the shadow of a flat object falls on a surface that is not parallel to it. When the scale factor is larger than 1, (uniform or non-uniform) scaling is sometimes also called dilation or enlargement. When the scale factor is a positive number smaller than 1, scaling is sometimes also called contraction or reduction. In the most general sense, a scaling includes the case in which the directions of scaling are not perpendicular. It also includes the case in which one or more scale factors are equal to zero (projection), and the case of one or more negative scale factors (a directional scaling by -1 is equivalent to a reflection). Scaling is a linear transformation, and a special case of homothetic transformation (scaling about a point). In most cases, the homothetic transformations are non-linear transformations. Uniform scaling A scale factor is usually a decimal which scales, or multiplies, some quantity. In the equation y = Cx, C is the scale factor for x. C is also the coefficient of x, and may be called the constant of proportionality of y to x. For example, doubling distances corresponds to a scale factor of two for distance, while cutting a cake in half results in pieces with a scale factor for volume of one half. The basic equation for it is image over preimage. In the field of measurements, the scale factor of an instrument is sometimes referred to as sensitivity. The ratio of any two corresponding lengths in two similar geometric figures is also called a scale. Matrix representation A scaling can be represented by a scaling matrix. To scale an object by a vector v = (vx, vy, vz), each point p = (px, py, pz) would need to be multiplied with this scaling matrix: $S_{v}={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}.$ As shown below, the multiplication will give the expected result: $S_{v}p={\begin{bmatrix}v_{x}&0&0\\0&v_{y}&0\\0&0&v_{z}\\\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\end{bmatrix}}.$ Such a scaling changes the diameter of an object by a factor between the scale factors, the area by a factor between the smallest and the largest product of two scale factors, and the volume by the product of all three. The scaling is uniform if and only if the scaling factors are equal (vx = vy = vz). If all except one of the scale factors are equal to 1, we have directional scaling. In the case where vx = vy = vz = k, scaling increases the area of any surface by a factor of k2 and the volume of any solid object by a factor of k3. Scaling in arbitrary dimensions In $n$-dimensional space $\mathbb {R} ^{n}$, uniform scaling by a factor $v$ is accomplished by scalar multiplication with $v$, that is, multiplying each coordinate of each point by $v$. As a special case of linear transformation, it can be achieved also by multiplying each point (viewed as a column vector) with a diagonal matrix whose entries on the diagonal are all equal to $v$, namely $vI$ . Non-uniform scaling is accomplished by multiplication with any symmetric matrix. The eigenvalues of the matrix are the scale factors, and the corresponding eigenvectors are the axes along which each scale factor applies. A special case is a diagonal matrix, with arbitrary numbers $v_{1},v_{2},\ldots v_{n}$ along the diagonal: the axes of scaling are then the coordinate axes, and the transformation scales along each axis $i$ by the factor $v_{i}$. In uniform scaling with a non-zero scale factor, all non-zero vectors retain their direction (as seen from the origin), or all have the direction reversed, depending on the sign of the scaling factor. In non-uniform scaling only the vectors that belong to an eigenspace will retain their direction. A vector that is the sum of two or more non-zero vectors belonging to different eigenspaces will be tilted towards the eigenspace with largest eigenvalue. Using homogeneous coordinates In projective geometry, often used in computer graphics, points are represented using homogeneous coordinates. To scale an object by a vector v = (vx, vy, vz), each homogeneous coordinate vector p = (px, py, pz, 1) would need to be multiplied with this projective transformation matrix: $S_{v}={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}.$ As shown below, the multiplication will give the expected result: $S_{v}p={\begin{bmatrix}v_{x}&0&0&0\\0&v_{y}&0&0\\0&0&v_{z}&0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}v_{x}p_{x}\\v_{y}p_{y}\\v_{z}p_{z}\\1\end{bmatrix}}.$ Since the last component of a homogeneous coordinate can be viewed as the denominator of the other three components, a uniform scaling by a common factor s (uniform scaling) can be accomplished by using this scaling matrix: $S_{v}={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}.$ For each vector p = (px, py, pz, 1) we would have $S_{v}p={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&{\frac {1}{s}}\end{bmatrix}}{\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\1\end{bmatrix}}={\begin{bmatrix}p_{x}\\p_{y}\\p_{z}\\{\frac {1}{s}}\end{bmatrix}},$ which would be equivalent to ${\begin{bmatrix}sp_{x}\\sp_{y}\\sp_{z}\\1\end{bmatrix}}.$ Function dilation and contraction Given a point $P(x,y)$, the dilation associates it with the point $P'(x',y')$ through the equations ${\begin{cases}x'=mx\\y'=ny\end{cases}}$ for $m,n\in \mathbb {R} ^{+}$. Therefore, given a function $y=f(x)$, the equation of the dilated function is $y=nf\left({\frac {x}{m}}\right).$ Particular cases If $n=1$, the transformation is horizontal; when $m>1$, it is a dilation, when $m<1$, it is a contraction. If $m=1$, the transformation is vertical; when $n>1$ it is a dilation, when $n<1$, it is a contraction. If $m=1/n$ or $n=1/m$, the transformation is a squeeze mapping. See also • 2D_computer_graphics#Scaling • Digital zoom • Dilation (metric space) • Homogeneous function • Homothetic transformation • Orthogonal coordinates • Scalar (mathematics) • Scale (disambiguation) • Scale (ratio) • Scale (map) • Scale factor (computer science) • Scale factor (cosmology) • Scales of scale models • Scaling in statistical estimation • Scaling in gravity • Squeeze mapping • Transformation matrix • Image scaling Footnotes 1. Durand; Cutler. "Transformations" (PowerPoint). Massachusetts Institute of Technology. Retrieved 12 September 2008. External links Wikimedia Commons has media related to Scaling (geometry). • Understanding 2D Scaling and Understanding 3D Scaling by Roger Germundsson, The Wolfram Demonstrations Project. • Scale Factor Calculator Computer graphics Vector graphics • Diffusion curve • Pixel 2D graphics 2.5D • Isometric graphics • Mode 7 • Parallax scrolling • Ray casting • Skybox • Alpha compositing • Layers • Text-to-image 3D graphics • 3D projection • 3D rendering • (Image-based • Spectral • Unbiased) • Aliasing • Anisotropic filtering • Cel shading • Lighting • Global illumination • Hidden-surface determination • Polygon mesh • (Triangle mesh) • Shading • Deferred • Surface triangulation • Wire-frame model Concepts • Affine transformation • Back-face culling • Clipping • Collision detection • Planar projection • Rendering • Rotation • Scaling • Shadow mapping • Shadow volume • Shear matrix • Translation Algorithms • List of computer graphics algorithms	Wikipedia
List of uniform polyhedra by Schwarz triangle There are many relationships among the uniform polyhedra. The Wythoff construction is able to construct almost all of the uniform polyhedra from the acute and obtuse Schwarz triangles. The numbers that can be used for the sides of a non-dihedral acute or obtuse Schwarz triangle that does not necessarily lead to only degenerate uniform polyhedra are 2, 3, 3/2, 4, 4/3, 5, 5/2, 5/3, and 5/4 (but numbers with numerator 4 and those with numerator 5 may not occur together). (4/2 can also be used, but only leads to degenerate uniform polyhedra as 4 and 2 have a common factor.) There are 44 such Schwarz triangles (5 with tetrahedral symmetry, 7 with octahedral symmetry and 32 with icosahedral symmetry), which, together with the infinite family of dihedral Schwarz triangles, can form almost all of the non-degenerate uniform polyhedra. Many degenerate uniform polyhedra, with completely coincident vertices, edges, or faces, may also be generated by the Wythoff construction, and those that arise from Schwarz triangles not using 4/2 are also given in the tables below along with their non-degenerate counterparts. Reflex Schwarz triangles have not been included, as they simply create duplicates or degenerates; however, a few are mentioned outside the tables due to their application to three of the snub polyhedra. There are a few non-Wythoffian uniform polyhedra, which no Schwarz triangles can generate; however, most of them can be generated using the Wythoff construction as double covers (the non-Wythoffian polyhedron is covered twice instead of once) or with several additional coinciding faces that must be discarded to leave no more than two faces at every edge (see Omnitruncated polyhedron#Other even-sided nonconvex polyhedra). Such polyhedra are marked by an asterisk in this list. The only uniform polyhedra which still fail to be generated by the Wythoff construction are the great dirhombicosidodecahedron and the great disnub dirhombidodecahedron. Each tiling of Schwarz triangles on a sphere may cover the sphere only once, or it may instead wind round the sphere a whole number of times, crossing itself in the process. The number of times the tiling winds round the sphere is the density of the tiling, and is denoted μ. Jonathan Bowers' short names for the polyhedra, known as Bowers acronyms, are used instead of the full names for the polyhedra to save space.[1] The Maeder index is also given. Except for the dihedral Schwarz triangles, the Schwarz triangles are ordered by their densities. The analogous cases of Euclidean tilings are also listed, and those of hyperbolic tilings briefly and incompletely discussed. Möbius and Schwarz triangles There are 4 spherical triangles with angles π/p, π/q, π/r, where (p q r) are integers: (Coxeter, "Uniform polyhedra", 1954) 1. (2 2 r) - Dihedral 2. (2 3 3) - Tetrahedral 3. (2 3 4) - Octahedral 4. (2 3 5) - Icosahedral These are called Möbius triangles. In addition Schwarz triangles consider (p q r) which are rational numbers. Each of these can be classified in one of the 4 sets above. Density (μ) Dihedral Tetrahedral Octahedral Icosahedral d(2 2 n/d) 1(2 3 3)(2 3 4)(2 3 5) 2(3/2 3 3)(3/2 4 4)(3/2 5 5), (5/2 3 3) 3(2 3/2 3)(2 5/2 5) 4(3 4/3 4)(3 5/3 5) 5(2 3/2 3/2)(2 3/2 4) 6(3/2 3/2 3/2)(5/2 5/2 5/2), (3/2 3 5), (5/4 5 5) 7(2 3 4/3)(2 3 5/2) 8(3/2 5/2 5) 9(2 5/3 5) 10(3 5/3 5/2), (3 5/4 5) 11(2 3/2 4/3)(2 3/2 5) 13(2 3 5/3) 14(3/2 4/3 4/3)(3/2 5/2 5/2), (3 3 5/4) 16(3 5/4 5/2) 17(2 3/2 5/2) 18(3/2 3 5/3), (5/3 5/3 5/2) 19(2 3 5/4) 21(2 5/4 5/2) 22(3/2 3/2 5/2) 23(2 3/2 5/3) 26(3/2 5/3 5/3) 27(2 5/4 5/3) 29(2 3/2 5/4) 32(3/2 5/4 5/3) 34(3/2 3/2 5/4) 38(3/2 5/4 5/4) 42(5/4 5/4 5/4) Although a polyhedron usually has the same density as the Schwarz triangle it is generated from, this is not always the case. Firstly, polyhedra that have faces passing through the centre of the model (including the hemipolyhedra, great dirhombicosidodecahedron, and great disnub dirhombidodecahedron) do not have a well-defined density. Secondly, the distortion necessary to recover uniformity when changing a spherical polyhedron to its planar counterpart can push faces through the centre of the polyhedron and back out the other side, changing the density. This happens in the following cases: • The great truncated cuboctahedron, 2 3 4/3 \|. While the Schwarz triangle (2 3 4/3) has density 7, recovering uniformity pushes the eight hexagons through the centre, yielding density \|7 − 8\| = 1, the same as that of the colunar Schwarz triangle (2 3 4) that shares the same great circles. • The truncated dodecadodecahedron, 2 5/3 5 \|. While the Schwarz triangle (2 5/3 5) has density 9, recovering uniformity pushes the twelve decagons through the centre, yielding density \|9 − 12\| = 3, the same as that of the colunar Schwarz triangle (2 5/2 5) that shares the same great circles. • Three snub polyhedra: the great icosahedron \| 2 3/2 3/2, the small retrosnub icosicosidodecahedron \| 3/2 3/2 5/2, and the great retrosnub icosidodecahedron \| 2 3/2 5/3. Here the vertex figures have been distorted into pentagrams or hexagrams rather than pentagons or hexagons, pushing all the snub triangles through the centre and yielding densities of \|5 − 12\| = 7, \|22 − 60\| = 38, and \|23 − 60\| = 37 respectively. These densities are the same as those of colunar reflex-angled Schwarz triangles that are not included above. Thus the great icosahedron may be considered to come from (2/3 3 3) or (2 3 3/4), the small retrosnub icosicosidodecahedron from (3 3 5/8) or (3 3/4 5/3), and the great retrosnub icosidodecahedron from (2/3 3 5/2), (2 3/4 5/3), or (2 3 5/7). (Coxeter, "Uniform polyhedra", 1954) Summary table There are seven generator points with each set of p,q,r (and a few special forms): General Right triangle (r=2) Description Wythoff symbol Vertex configuration Coxeter diagram Wythoff symbol Vertex configuration Schläfli symbol Coxeter diagram regular and quasiregular q \| p r (p.r)q q \| p 2 pq {p,q} p \| q r (q.r)p p \| q 2 qp {q,p} r \| p q (q.p)r 2 \| p q (q.p)2 t1{p,q} truncated and expanded q r \| p q.2p.r.2p q 2 \| p q.2p.2p t0,1{p,q} p r \| q p.2q.r.2q p 2 \| q p. 2q.2q t0,1{q,p} p q \| r 2r.q.2r.p p q \| 2 4.q.4.p t0,2{p,q} even-faced p q r \| 2r.2q.2p p q 2 \| 4.2q.2p t0,1,2{p,q} p q r s \| 2p.2q.-2p.-2q - p 2 r s \| 2p.4.-2p.4/3 - snub \| p q r 3.r.3.q.3.p \| p q 2 3.3.q.3.p sr{p,q} \| p q r s (4.p.4.q.4.r.4.s)/2 - - - - There are four special cases: • p q r s \| – This is a mixture of p q r \| and p q s \|. Both the symbols p q r \| and p q s \| generate a common base polyhedron with some extra faces. The notation p q r s \| then represents the base polyhedron, made up of the faces common to both p q r \| and p q s \|. • \| p q r – Snub forms (alternated) are given this otherwise unused symbol. • \| p q r s – A unique snub form for U75 that isn't Wythoff-constructible using triangular fundamental domains. Four numbers are included in this Wythoff symbol as this polyhedron has a tetragonal spherical fundamental domain. • \| (p) q (r) s – A unique snub form for Skilling's figure that isn't Wythoff-constructible. This conversion table from Wythoff symbol to vertex configuration fails for the exceptional five polyhedra listed above whose densities do not match the densities of their generating Schwarz triangle tessellations. In these cases the vertex figure is highly distorted to achieve uniformity with flat faces: in the first two cases it is an obtuse triangle instead of an acute triangle, and in the last three it is a pentagram or hexagram instead of a pentagon or hexagon, winding around the centre twice. This results in some faces being pushed right through the polyhedron when compared with the topologically equivalent forms without the vertex figure distortion and coming out retrograde on the other side.[2] In the tables below, red backgrounds mark degenerate polyhedra. Green backgrounds mark the convex uniform polyhedra. Dihedral (prismatic) In dihedral Schwarz triangles, two of the numbers are 2, and the third may be any rational number strictly greater than 1. 1. (2 2 n/d) – degenerate if gcd(n, d) > 1. Many of the polyhedra with dihedral symmetry have digon faces that make them degenerate polyhedra (e.g. dihedra and hosohedra). Columns of the table that only give degenerate uniform polyhedra are not included: special degenerate cases (only in the (2 2 2) Schwarz triangle) are marked with a large cross. Uniform crossed antiprisms with a base {p} where p < 3/2 cannot exist as their vertex figures would violate the triangular inequality; these are also marked with a large cross. The 3/2-crossed antiprism (trirp) is degenerate, being flat in Euclidean space, and is also marked with a large cross. The Schwarz triangles (2 2 n/d) are listed here only when gcd(n, d) = 1, as they otherwise result in only degenerate uniform polyhedra. The list below gives all possible cases where n ≤ 6. (p q r) q r \| p q.2p.r.2p p r \| q p. 2q.r.2q p q r \| 2r.2q.2p \| p q r 3.r.3.q.3.p (2 2 2) (μ=1) X X 4.4.4 cube 4-p 3.3.3 tet 2-ap (2 2 3) (μ=1) 4.3.4 trip 3-p 4.3.4 trip 3-p 6.4.4 hip 6-p 3.3.3.3 oct 3-ap (2 2 3/2) (μ=2) 4.3.4 trip 3-p 4.3.4 trip 3-p 6/2.4.4 2trip 6/2-p X (2 2 4) (μ=1) 4.4.4 cube 4-p 4.4.4 cube 4-p 8.4.4 op 8-p 3.4.3.3 squap 4-ap (2 2 4/3) (μ=3) 4.4.4 cube 4-p 4.4.4 cube 4-p 8/3.4.4 stop 8/3-p X (2 2 5) (μ=1) 4.5.4 pip 5-p 4.5.4 pip 5-p 10.4.4 dip 10-p 3.5.3.3 pap 5-ap (2 2 5/2) (μ=2) 4.5/2.4 stip 5/2-p 4.5/2.4 stip 5/2-p 10/2.4.4 2pip 10/2-p 3.5/2.3.3 stap 5/2-ap (2 2 5/3) (μ=3) 4.5/2.4 stip 5/2-p 4.5/2.4 stip 5/2-p 10/3.4.4 stiddip 10/3-p 3.5/3.3.3 starp 5/3-ap (2 2 5/4) (μ=4) 4.5.4 pip 5-p 4.5.4 pip 5-p 10/4.4.4 2stip 10/4-p X (2 2 6) (μ=1) 4.6.4 hip 6-p 4.6.4 hip 6-p 12.4.4 twip 12-p 3.6.3.3 hap 6-ap (2 2 6/5) (μ=5) 4.6.4 hip 6-p 4.6.4 hip 6-p 12/5.4.4 stwip 12/5-p X (2 2 n) (μ=1) 4.n.4 n-p 4.n.4 n-p 2n.4.4 2n-p 3.n.3.3 n-ap (2 2 n/d) (μ=d) 4.n/d.4 n/d-p 4.n/d.4 n/d-p 2n/d.4.4 2n/d-p 3.n/d.3.3 n/d-ap Tetrahedral In tetrahedral Schwarz triangles, the maximum numerator allowed is 3. # (p q r) q \| p r (p.r)q p \| q r (q.r)p r \| p q (q.p)r q r \| p q.2p.r.2p p r \| q p. 2q.r.2q p q \| r 2r.q.2r.p p q r \| 2r.2q.2p \| p q r 3.r.3.q.3.p 1 (3 3 2) (μ=1) 3.3.3 tet U1 3.3.3 tet U1 3.3.3.3 oct U5 3.6.6 tut U2 3.6.6 tut U2 4.3.4.3 co U7 4.6.6 toe U8 3.3.3.3.3 ike U22 2 (3 3 3/2) (μ=2) (3.3.3.3.3.3)/2 2tet – (3.3.3.3.3.3)/2 2tet – (3.3.3.3.3.3)/2 2tet – 3.6.3/2.6 oho U3 3.6.3/2.6 oho U3 2(6/2.3.6/2.3) 2oct – 2(6/2.6.6) 2tut – 2(3.3/2.3.3.3.3) 2oct+8{3} – 3 (3 2 3/2) (μ=3) 3.3.3.3 oct U5 3.3.3 tet U1 3.3.3 tet U1 3.6.6 tut U2 2(3/2.4.3.4) 2thah U4* 3(3.6/2.6/2) 3tet – 2(6/2.4.6) cho+4{6/2} U15* 3(3.3.3) 3tet – 4 (2 3/2 3/2) (μ=5) 3.3.3 tet U1 3.3.3.3 oct U5 3.3.3 tet U1 3.4.3.4 co U7 3(6/2.3.6/2) 3tet – 3(6/2.3.6/2) 3tet – 4(6/2.6/2.4) 2oct+6{4} – (3.3.3.3.3)/2 gike U53 5 (3/2 3/2 3/2) (μ=6) (3.3.3.3.3.3)/2 2tet – (3.3.3.3.3.3)/2 2tet – (3.3.3.3.3.3)/2 2tet – 2(6/2.3.6/2.3) 2oct – 2(6/2.3.6/2.3) 2oct – 2(6/2.3.6/2.3) 2oct – 6(6/2.6/2.6/2) 6tet – ? Octahedral In octahedral Schwarz triangles, the maximum numerator allowed is 4. There also exist octahedral Schwarz triangles which use 4/2 as a number, but these only lead to degenerate uniform polyhedra as 4 and 2 have a common factor. # (p q r) q \| p r (p.r)q p \| q r (q.r)p r \| p q (q.p)r q r \| p q.2p.r.2p p r \| q p. 2q.r.2q p q \| r 2r.q.2r.p p q r \| 2r.2q.2p \| p q r 3.r.3.q.3.p 1 (4 3 2) (μ=1) 4.4.4 cube U6 3.3.3.3 oct U5 3.4.3.4 co U7 3.8.8 tic U9 4.6.6 toe U8 4.3.4.4 sirco U10 4.6.8 girco U11 3.3.3.3.4 snic U12 2 (4 4 3/2) (μ=2) (3/2.4)4 oct+6{4} – (3/2.4)4 oct+6{4} – (4.4.4.4.4.4)/2 2cube – 3/2.8.4.8 socco U13 3/2.8.4.8 socco U13 2(6/2.4.6/2.4) 2co – 2(6/2.8.8) 2tic – ? 3 (4 3 4/3) (μ=4) (4.4.4.4.4.4)/2 2cube – (3/2.4)4 oct+6{4} – (3/2.4)4 oct+6{4} – 3/2.8.4.8 socco U13 2(4/3.6.4.6) 2cho U15* 3.8/3.4.8/3 gocco U14 6.8.8/3 cotco U16 ? 4 (4 2 3/2) (μ=5) 3.4.3.4 co U7 3.3.3.3 oct U5 4.4.4 cube U6 3.8.8 tic U9 4.4.3/2.4 querco U17 4(4.6/2.6/2) 2oct+6{4} – 2(4.6/2.8) sroh+8{6/2} U18* ? 5 (3 2 4/3) (μ=7) 3.4.3.4 co U7 4.4.4 cube U6 3.3.3.3 oct U5 4.6.6 toe U8 4.4.3/2.4 querco U17 3.8/3.8/3 quith U19 4.6/5.8/3 quitco U20 ? 6 (2 3/2 4/3) (μ=11) 4.4.4 cube U6 3.4.3.4 co U7 3.3.3.3 oct U5 4.3.4.4 sirco U10 4(4.6/2.6/2) 2oct+6{4} – 3.8/3.8/3 quith U19 2(4.6/2.8/3) groh+8{6/2} U21* ? 7 (3/2 4/3 4/3) (μ=14) (3/2.4)4 = (3.4)4/3 oct+6{4} – (4.4.4.4.4.4)/2 2cube – (3/2.4)4 = (3.4)4/3 oct+6{4} – 2(6/2.4.6/2.4) 2co – 3.8/3.4.8/3 gocco U14 3.8/3.4.8/3 gocco U14 2(6/2.8/3.8/3) 2quith – ? Icosahedral In icosahedral Schwarz triangles, the maximum numerator allowed is 5. Additionally, the numerator 4 cannot be used at all in icosahedral Schwarz triangles, although numerators 2 and 3 are allowed. (If 4 and 5 could occur together in some Schwarz triangle, they would have to do so in some Möbius triangle as well; but this is impossible as (2 4 5) is a hyperbolic triangle, not a spherical one.) # (p q r) q \| p r (p.r)q p \| q r (q.r)p r \| p q (q.p)r q r \| p q.2p.r.2p p r \| q p. 2q.r.2q p q \| r 2r.q.2r.p p q r \| 2r.2q.2p \| p q r 3.r.3.q.3.p 1 (5 3 2) (μ=1) 5.5.5 doe U23 3.3.3.3.3 ike U22 3.5.3.5 id U24 3.10.10 tid U26 5.6.6 ti U25 4.3.4.5 srid U27 4.6.10 grid U28 3.3.3.3.5 snid U29 2 (3 3 5/2) (μ=2) 3.5/2.3.5/2.3.5/2 sidtid U30 3.5/2.3.5/2.3.5/2 sidtid U30 (310)/2 2ike – 3.6.5/2.6 siid U31 3.6.5/2.6 siid U31 2(10/2.3.10/2.3) 2id – 2(10/2.6.6) 2ti – 3.5/2.3.3.3.3 seside U32 3 (5 5 3/2) (μ=2) (5.3/2)5 cid – (5.3/2)5 cid – (5.5.5.5.5.5)/2 2doe – 5.10.3/2.10 saddid U33 5.10.3/2.10 saddid U33 2(6/2.5.6/2.5) 2id – 2(6/2.10.10) 2tid – 2(3.3/2.3.5.3.5) 2id+40{3} – 4 (5 5/2 2) (μ=3) (5.5.5.5.5)/2 gad U35 5/2.5/2.5/2.5/2.5/2 sissid U34 5/2.5.5/2.5 did U36 5/2.10.10 tigid U37 5.10/2.10/2 3doe – 4.5/2.4.5 raded U38 2(4.10/2.10) sird+12{10/2} U39* 3.3.5/2.3.5 siddid U40 5 (5 3 5/3) (μ=4) 5.5/3.5.5/3.5.5/3 ditdid U41 (3.5/3)5 gacid – (3.5)5/3 cid – 3.10.5/3.10 sidditdid U43 5.6.5/3.6 ided U44 10/3.3.10/3.5 gidditdid U42 10/3.6.10 idtid U45 3.5/3.3.3.3.5 sided U46 6 (5/2 5/2 5/2) (μ=6) (5/2)10/2 2sissid – (5/2)10/2 2sissid – (5/2)10/2 2sissid – 2(5/2.10/2)2 2did – 2(5/2.10/2)2 2did – 2(5/2.10/2)2 2did – 6(10/2.10/2.10/2) 6doe – 3(3.5/2.3.5/2.3.5/2) 3sidtid – 7 (5 3 3/2) (μ=6) (3.5.3.5.3.5)/2 gidtid U47 (310)/4 2gike – (3.5.3.5.3.5)/2 gidtid U47 2(3.10.3/2.10) 2seihid U49* 5.6.3/2.6 giid U48 5(6/2.3.6/2.5) 3ike+gad – 2(6.6/2.10) siddy+20{6/2} U50* 5(3.3.3.3.3.5)/2 5ike+gad – 8 (5 5 5/4) (μ=6) (510)/4 2gad – (510)/4 2gad – (510)/4 2gad – 2(5.10.5/4.10) 2sidhid U51* 2(5.10.5/4.10) 2sidhid U51* 10/4.5.10/4.5 2did – 2(10/4.10.10) 2tigid – 3(3.5.3.5.3.5) 3cid – 9 (3 5/2 2) (μ=7) (3.3.3.3.3)/2 gike U53 5/2.5/2.5/2 gissid U52 5/2.3.5/2.3 gid U54 5/2.6.6 tiggy U55 3.10/2.10/2 2gad+ike – 3(4.5/2.4.3) sicdatrid – 4.10/2.6 ri+12{10/2} U56* 3.3.5/2.3.3 gosid U57 10 (5 5/2 3/2) (μ=8) (5.3/2)5 cid – (5/3.3)5 gacid – 5.5/3.5.5/3.5.5/3 ditdid U41 5/3.10.3.10 sidditdid U43 5(5.10/2.3.10/2) ike+3gad – 3(6/2.5/2.6/2.5) sidtid+gidtid – 4(6/2.10/2.10) id+seihid+sidhid – ? (3\|3 5/2) + (3/2\|3 5) 11 (5 2 5/3) (μ=9) 5.5/2.5.5/2 did U36 5/2.5/2.5/2.5/2.5/2 sissid U34 (5.5.5.5.5)/2 gad U35 5/2.10.10 tigid U37 3(5.4.5/3.4) cadditradid – 10/3.5.5 quit sissid U58 10/3.4.10/9 quitdid U59 3.5/3.3.3.5 isdid U60 12 (3 5/2 5/3) (μ=10) (3.5/3)5 gacid – (5/2)6/2 2gissid – (5/2.3)5/3 gacid – 2(5/2.6.5/3.6) 2sidhei U62* 3(3.10/2.5/3.10/2) ditdid+gidtid – 10/3.5/2.10/3.3 gaddid U61 10/3.10/2.6 giddy+12{10/2} U63* 3.5/3.3.5/2.3.3 gisdid U64 13 (5 3 5/4) (μ=10) (5.5.5.5.5.5)/2 2doe – (3/2.5)5 cid – (3.5)5/3 cid – 3/2.10.5.10 saddid U33 2(5.6.5/4.6) 2gidhei U65* 3(10/4.3.10/4.5) sidtid+ditdid – 2(10/4.6.10) siddy+12{10/4} U50* ? 14 (5 2 3/2) (μ=11) 5.3.5.3 id U24 3.3.3.3.3 ike U22 5.5.5 doe U23 3.10.10 tid U26 3(5/4.4.3/2.4) gicdatrid – 5(5.6/2.6/2) 2ike+gad – 2(6/2.4.10) sird+20{6/2} U39* 5(3.3.3.5.3)/2 4ike+gad – 15 (3 2 5/3) (μ=13) 3.5/2.3.5/2 gid U54 5/2.5/2.5/2 gissid U52 (3.3.3.3.3)/2 gike U53 5/2.6.6 tiggy U55 3.4.5/3.4 qrid U67 10/3.10/3.3 quit gissid U66 10/3.4.6 gaquatid U68 3.5/3.3.3.3 gisid U69 16 (5/2 5/2 3/2) (μ=14) (5/3.3)5 gacid – (5/3.3)5 gacid – (5/2)6/2 2gissid – 3(5/3.10/2.3.10/2) ditdid+gidtid – 3(5/3.10/2.3.10/2) ditdid+gidtid – 2(6/2.5/2.6/2.5/2) 2gid – 10(6/2.10/2.10/2) 2ike+4gad – ? 17 (3 3 5/4) (μ=14) (3.5.3.5.3.5)/2 gidtid U47 (3.5.3.5.3.5)/2 gidtid U47 (3)10/4 2gike – 3/2.6.5.6 giid U48 3/2.6.5.6 giid U48 2(10/4.3.10/4.3) 2gid – 2(10/4.6.6) 2tiggy – ? 18 (3 5/2 5/4) (μ=16) (3/2.5)5 cid – 5/3.5.5/3.5.5/3.5 ditdid U41 (5/2.3)5/3 gacid – 5/3.6.5.6 ided U44 5(3/2.10/2.5.10/2) ike+3gad – 5(10/4.5/2.10/4.3) 3sissid+gike – 4(10/4.10/2.6) did+sidhei+gidhei – ? 19 (5/2 2 3/2) (μ=17) 3.5/2.3.5/2 gid U54 (3.3.3.3.3)/2 gike U53 5/2.5/2.5/2 gissid U52 5(10/2.3.10/2) 2gad+ike – 5/3.4.3.4 qrid U67 5(6/2.6/2.5/2) 2gike+sissid – 6(6/2.4.10/2) 2gidtid+rhom – ? 20 (5/2 5/3 5/3) (μ=18) (5/2)10/2 2sissid – (5/2)10/2 2sissid – (5/2)10/2 2sissid – 2(5/2.10/2)2 2did – 2(5/2.10/3.5/3.10/3) 2gidhid U70* 2(5/2.10/3.5/3.10/3) 2gidhid U70* 2(10/3.10/3.10/2) 2quitsissid – ? 21 (3 5/3 3/2) (μ=18) (310)/2 2ike – 5/2.3.5/2.3.5/2.3 sidtid U30 5/2.3.5/2.3.5/2.3 sidtid U30 5/2.6.3.6 siid U31 2(3.10/3.3/2.10/3) 2geihid U71* 5(6/2.5/3.6/2.3) sissid+3gike – 2(6/2.10/3.6) giddy+20{6/2} U63* ? 22 (3 2 5/4) (μ=19) 3.5.3.5 id U24 5.5.5 doe U23 3.3.3.3.3 ike U22 5.6.6 ti U25 3(3/2.4.5/4.4) gicdatrid – 5(10/4.10/4.3) 2sissid+gike – 2(10/4.4.6) ri+12{10/4} U56* ? 23 (5/2 2 5/4) (μ=21) 5/2.5.5/2.5 did U36 (5.5.5.5.5)/2 gad U35 5/2.5/2.5/2.5/2.5/2 sissid U34 3(10/2.5.10/2) 3doe – 3(5/3.4.5.4) cadditradid – 3(10/4.5/2.10/4) 3gissid – 6(10/4.4.10/2) 2ditdid+rhom – ? 24 (5/2 3/2 3/2) (μ=22) 5/2.3.5/2.3.5/2.3 sidtid U30 (310)/2 2ike – 5/2.3.5/2.3.5/2.3 sidtid U30 2(3.10/2.3.10/2) 2id – 5(5/3.6/2.3.6/2) sissid+3gike – 5(5/3.6/2.3.6/2) sissid+3gike – 10(6/2.6/2.10/2) 4ike+2gad – (3.3.3.3.3.5/2)/2 sirsid U72 25 (2 5/3 3/2) (μ=23) (3.3.3.3.3)/2 gike U53 5/2.3.5/2.3 gid U54 5/2.5/2.5/2 gissid U52 3(5/2.4.3.4) sicdatrid – 10/3.3.10/3 quit gissid U66 5(6/2.5/2.6/2) 2gike+sissid – 2(6/2.10/3.4) gird+20{6/2} U73* (3.3.3.5/2.3)/2 girsid U74 26 (5/3 5/3 3/2) (μ=26) (5/2.3)5/3 gacid – (5/2.3)5/3 gacid – (5/2)6/2 2gissid – 5/2.10/3.3.10/3 gaddid U61 5/2.10/3.3.10/3 gaddid U61 2(6/2.5/2.6/2.5/2) 2gid – 2(6/2.10/3.10/3) 2quitgissid – ? 27 (2 5/3 5/4) (μ=27) (5.5.5.5.5)/2 gad U35 5/2.5.5/2.5 did U36 5/2.5/2.5/2.5/2.5/2 sissid U34 5/2.4.5.4 raded U38 10/3.5.10/3 quit sissid U58 3(10/4.5/2.10/4) 3gissid – 2(10/4.10/3.4) gird+12{10/4} U73* ? 28 (2 3/2 5/4) (μ=29) 5.5.5 doe U23 3.5.3.5 id U24 3.3.3.3.3 ike U22 3.4.5.4 srid U27 2(6/2.5.6/2) 2ike+gad – 5(10/4.3.10/4) 2sissid+gike – 6(10/4.6/2.4/3) 2sidtid+rhom – ? 29 (5/3 3/2 5/4) (μ=32) 5/3.5.5/3.5.5/3.5 ditdid U41 (3.5)5/3 cid – (3.5/2)5/3 gacid – 3.10/3.5.10/3 gidditdid U42 3(5/2.6/2.5.6/2) sidtid+gidtid – 5(10/4.3.10/4.5/2) 3sissid+gike – 4(10/4.6/2.10/3) gid+geihid+gidhid – ? 30 (3/2 3/2 5/4) (μ=34) (3.5.3.5.3.5)/2 gidtid U47 (3.5.3.5.3.5)/2 gidtid U47 (3)10/4 2gike – 5(3.6/2.5.6/2) 3ike+gad – 5(3.6/2.5.6/2) 3ike+gad – 2(10/4.3.10/4.3) 2gid – 10(10/4.6/2.6/2) 2sissid+4gike – ? 31 (3/2 5/4 5/4) (μ=38) (3.5)5/3 cid – (5.5.5.5.5.5)/2 2doe – (3.5)5/3 cid – 2(5.6/2.5.6/2) 2id – 3(3.10/4.5/4.10/4) sidtid+ditdid – 3(3.10/4.5/4.10/4) sidtid+ditdid – 10(10/4.10/4.6/2) 4sissid+2gike – 5(3.3.3.5/4.3.5/4) 4ike+2gad – 32 (5/4 5/4 5/4) (μ=42) (5)10/4 2gad – (5)10/4 2gad – (5)10/4 2gad – 2(5.10/4.5.10/4) 2did – 2(5.10/4.5.10/4) 2did – 2(5.10/4.5.10/4) 2did – 6(10/4.10/4.10/4) 2gissid – 3(3/2.5.3/2.5.3/2.5) 3cid – Non-Wythoffian Hemi forms Apart from the octahemioctahedron, the hemipolyhedra are generated as double coverings by the Wythoff construction.[3] 3/2.4.3.4 thah U4 hemi(3 3/2 \| 2) 4/3.6.4.6 cho U15 hemi(4 4/3 \| 3) 5/4.10.5.10 sidhid U51 hemi(5 5/4 \| 5) 5/2.6.5/3.6 sidhei U62 hemi(5/2 5/3 \| 3) 5/2.10/3.5/3.10/3 gidhid U70 hemi(5/2 5/3 \| 5/3) 3/2.6.3.6 oho U3 hemi(?) 3/2.10.3.10 seihid U49 hemi(3 3/2 \| 5) 5.6.5/4.6 gidhei U65 hemi(5 5/4 \| 3) 3.10/3.3/2.10/3 geihid U71 hemi(3 3/2 \| 5/3) Reduced forms These polyhedra are generated with extra faces by the Wythoff construction. Wythoff Polyhedron Extra faces Wythoff Polyhedron Extra faces Wythoff Polyhedron Extra faces 3 2 3/2 \| 4.6.4/3.6 cho U15 4{6/2} 4 2 3/2 \| 4.8.4/3.8/7 sroh U18 8{6/2} 2 3/2 4/3 \| 4.8/3.4/3.8/5 groh U21 8{6/2} 5 5/2 2 \| 4.10.4/3.10/9 sird U39 12{10/2} 5 3 3/2 \| 10.6.10/9.6/5 siddy U50 20{6/2} 3 5/2 2 \| 6.4.6/5.4/3 ri U56 12{10/2} 5 5/2 3/2 \| 3/2.10.3.10 seihid U49 id + sidhid 5 5/2 3/2 \| 5/4.10.5.10 sidhid U51 id + seihid 5 3 5/4 \| 10.6.10/9.6/5 siddy U50 12{10/4} 3 5/2 5/3 \| 6.10/3.6/5.10/7 giddy U63 12{10/2} 5 2 3/2 \| 4.10/3.4/3.10/9 sird U39 20{6/2} 3 5/2 5/4 \| 5.6.5/4.6 gidhei U65 did + sidhei 3 5/2 5/4 \| 5/2.6.5/3.6 sidhei U62 did + gidhei 3 5/3 3/2 \| 6.10/3.6/5.10/7 giddy U63 20{6/2} 3 2 5/4 \| 6.4.6/5.4/3 ri U56 12{10/4} 2 5/3 3/2 \| 4.10/3.4/3.10/7 gird U73 20{6/2} 5/3 3/2 5/4 \| 3.10/3.3/2.10/3 geihid U71 gid + gidhid 5/3 3/2 5/4 \| 5/2.10/3.5/3.10/3 gidhid U70 gid + geihid 2 5/3 5/4 \| 4.10/3.4/3.10/7 gird U73 12{10/4} The tetrahemihexahedron (thah, U4) is also a reduced version of the {3/2}-cupola (retrograde triangular cupola, ratricu) by {6/2}. As such it may also be called the crossed triangular cuploid. Many cases above are derived from degenerate omnitruncated polyhedra p q r \|. In these cases, two distinct degenerate cases p q r \| and p q s \| can be generated from the same p and q; the result has faces {2p}'s, {2q}'s, and coinciding {2r}'s or {2s}'s respectively. These both yield the same nondegenerate uniform polyhedra when the coinciding faces are discarded, which Coxeter symbolised p q r s \|. These cases are listed below: 4.6.4/3.6 cho U15 2 3 3/2 3/2 \| 4.8.4/3.8/7 sroh U18 2 3 3/2 4/2 \| 4.10.4/3.10/9 sird U39 2 3 3/2 5/2 \| 6.10/3.6/5.10/7 giddy U63 3 5/3 3/2 5/2 \| 6.4.6/5.4/3 ri U56 2 3 5/4 5/2 \| 4.8/3.4/3.8/5 groh U21 2 4/3 3/2 4/2 \| 4.10/3.4/3.10/7 gird U73 2 5/3 3/2 5/4 \| 10.6.10/9.6/5 siddy U50 3 5 3/2 5/4 \| In the small and great rhombihexahedra, the fraction 4/2 is used despite it not being in lowest terms. While 2 4 2 \| and 2 4/3 2 \| represent a single octagonal or octagrammic prism respectively, 2 4 4/2 \| and 2 4/3 4/2 \| represent three such prisms, which share some of their square faces (precisely those doubled up to produce {8/2}'s). These {8/2}'s appear with fourfold and not twofold rotational symmetry, justifying the use of 4/2 instead of 2.[2] Other forms These two uniform polyhedra cannot be generated at all by the Wythoff construction. This is the set of uniform polyhedra commonly described as the "non-Wythoffians". Instead of the triangular fundamental domains of the Wythoffian uniform polyhedra, these two polyhedra have tetragonal fundamental domains. Skilling's figure is not given an index in Maeder's list due to it being an exotic uniform polyhedron, with ridges (edges in the 3D case) completely coincident. This is also true of some of the degenerate polyhedron included in the above list, such as the small complex icosidodecahedron. This interpretation of edges being coincident allows these figures to have two faces per edge: not doubling the edges would give them 4, 6, 8, 10 or 12 faces meeting at an edge, figures that are usually excluded as uniform polyhedra. Skilling's figure has 4 faces meeting at some edges. (p q r s) \| p q r s (4.p. 4.q.4.r.4.s)/2 \| (p) q (r) s (p3.4.q.4.r3.4.s.4)/2 (3/2 5/3 3 5/2) (4.3/2.4.5/3.4.3.4.5/2)/2 gidrid U75 (3/23.4.5/3.4.33.4.5/2.4)/2 gidisdrid Skilling Vertex figure of \| 3 5/3 5/2 Great snub dodecicosidodecahedron Great dirhombicosidodecahedron Vertex figure of \| 3/2 5/3 3 5/2 Great disnub dirhombidodecahedron Compound of twenty octahedra Compound of twenty tetrahemihexahedra Vertex figure of \|(3/2) 5/3 (3) 5/2 Both of these special polyhedra may be derived from the great snub dodecicosidodecahedron, \| 3 5/3 5/2 (U64). This is a chiral snub polyhedron, but its pentagrams appear in coplanar pairs. Combining one copy of this polyhedron with its enantiomorph, the pentagrams coincide and may be removed. As the edges of this polyhedron's vertex figure include three sides of a square, with the fourth side being contributed by its enantiomorph, we see that the resulting polyhedron is in fact the compound of twenty octahedra. Each of these octahedra contain one pair of parallel faces that stem from a fully symmetric triangle of \| 3 5/3 5/2, while the other three come from the original \| 3 5/3 5/2's snub triangles. Additionally, each octahedron can be replaced by the tetrahemihexahedron with the same edges and vertices. Taking the fully symmetric triangles in the octahedra, the original coinciding pentagrams in the great snub dodecicosidodecahedra, and the equatorial squares of the tetrahemihexahedra together yields the great dirhombicosidodecahedron (Miller's monster).[2] Taking the snub triangles of the octahedra instead yields the great disnub dirhombidodecahedron (Skilling's figure).[4] Euclidean tilings The only plane triangles that tile the plane once over are (3 3 3), (4 2 4), and (3 2 6): they are respectively the equilateral triangle, the 45-45-90 right isosceles triangle, and the 30-60-90 right triangle. It follows that any plane triangle tiling the plane multiple times must be built up from multiple copies of one of these. The only possibility is the 30-30-120 obtuse isosceles triangle (3/2 6 6) = (6 2 3) + (2 6 3) tiling the plane twice over. Each triangle counts twice with opposite orientations, with a branch point at the 120° vertices.[5] The tiling {∞,2} made from two apeirogons is not accepted, because its faces meet at more than one edge. Here ∞' denotes the retrograde counterpart to ∞. The degenerate named forms are: • chatit: compound of 3 hexagonal tilings + triangular tiling • chata: compound of 3 hexagonal tilings + triangular tiling + double covers of apeirogons along all edge sequences • cha: compound of 3 hexagonal tilings + double covers of apeirogons along all edge sequences • cosa: square tiling + double covers of apeirogons along all edge sequences (p q r) q \| p r (p.r)q p \| q r (q.r)p r \| p q (q.p)r q r \| p q.2p.r.2p p r \| q p.2q.r.2q p q \| r 2r.q.2r.p p q r \| 2r.2q.2p \| p q r 3.r.3.q.3.p (6 3 2) 6.6.6 hexat 3.3.3.3.3.3 trat 3.6.3.6 that 3.12.12 toxat 6.6.6 hexat 4.3.4.6 srothat 4.6.12 grothat 3.3.3.3.6 snathat (4 4 2) 4.4.4.4 squat 4.4.4.4 squat 4.4.4.4 squat 4.8.8 tosquat 4.8.8 tosquat 4.4.4.4 squat 4.8.8 tosquat 3.3.4.3.4 snasquat (3 3 3) 3.3.3.3.3.3 trat 3.3.3.3.3.3 trat 3.3.3.3.3.3 trat 3.6.3.6 that 3.6.3.6 that 3.6.3.6 that 6.6.6 hexat 3.3.3.3.3.3 trat (∞ 2 2) — — — — 4.4.∞ azip 4.4.∞ azip 4.4.∞ azip 3.3.3.∞ azap (3/2 3/2 3) 3.3.3.3.3.3 trat 3.3.3.3.3.3 trat 3.3.3.3.3.3 trat ∞-covered {3} ∞-covered {3} 3.6.3.6 that [degenerate] ? (4 4/3 2) 4.4.4.4 squat 4.4.4.4 squat 4.4.4.4 squat 4.8.8 tosquat 4.8/5.8/5 quitsquat ∞-covered {4} 4.8/3.8/7 qrasquit ? (4/3 4/3 2) 4.4.4.4 squat 4.4.4.4 squat 4.4.4.4 squat 4.8/5.8/5 quitsquat 4.8/5.8/5 quitsquat 4.4.4.4 squat 4.8/5.8/5 quitsquat 3.3.4/3.3.4/3 rasisquat (3/2 6 2) 3.3.3.3.3.3 trat 6.6.6 hexat 3.6.3.6 that [degenerate] 3.12.12 toxat 3/2.4.6/5.4 qrothat [degenerate] ? (3 6/5 2) 3.3.3.3.3.3 trat 6.6.6 hexat 3.6.3.6 that 6.6.6 hexat 3/2.12/5.12/5 quothat 3/2.4.6/5.4 qrothat 4.6/5.12/5 quitothit ? (3/2 6/5 2) 3.3.3.3.3.3 trat 6.6.6 hexat 3.6.3.6 that [degenerate] 3/2.12/5.12/5 quothat 3.4.6.4 srothat [degenerate] ? (3/2 6 6) (3/2.6)6 chatit (6.6.6.6.6.6)/2 2hexat (3/2.6)6 chatit [degenerate] 3/2.12.6.12 shothat 3/2.12.6.12 shothat [degenerate] ? (3 6 6/5) (3/2.6)6 chatit (6.6.6.6.6.6)/2 2hexat (3/2.6)6 chatit ∞-covered {6} 3/2.12.6.12 shothat 3.12/5.6/5.12/5 ghothat 6.12/5.12/11 thotithit ? (3/2 6/5 6/5) (3/2.6)6 chatit (6.6.6.6.6.6)/2 2hexat (3/2.6)6 chatit [degenerate] 3.12/5.6/5.12/5 ghothat 3.12/5.6/5.12/5 ghothat [degenerate] ? (3 3/2 ∞) (3.∞)3/2 = (3/2.∞)3 ditatha (3.∞)3/2 = (3/2.∞)3 ditatha — 6.3/2.6.∞ chata [degenerate] 3.∞.3/2.∞ tha [degenerate] ? (3 3 ∞') (3.∞)3/2 = (3/2.∞)3 ditatha (3.∞)3/2 = (3/2.∞)3 ditatha — 6.3/2.6.∞ chata 6.3/2.6.∞ chata [degenerate] [degenerate] ? (3/2 3/2 ∞') (3.∞)3/2 = (3/2.∞)3 ditatha (3.∞)3/2 = (3/2.∞)3 ditatha — [degenerate] [degenerate] [degenerate] [degenerate] ? (4 4/3 ∞) (4.∞)4/3 cosa (4.∞)4/3 cosa — 8.4/3.8.∞ gossa 8/3.4.8/3.∞ sossa 4.∞.4/3.∞ sha 8.8/3.∞ satsa 3.4.3.4/3.3.∞ snassa (4 4 ∞') (4.∞)4/3 cosa (4.∞)4/3 cosa — 8.4/3.8.∞ gossa 8.4/3.8.∞ gossa [degenerate] [degenerate] ? (4/3 4/3 ∞') (4.∞)4/3 cosa (4.∞)4/3 cosa — 8/3.4.8/3.∞ sossa 8/3.4.8/3.∞ sossa [degenerate] [degenerate] ? (6 6/5 ∞) (6.∞)6/5 cha (6.∞)6/5 cha — 6/5.12.∞.12 ghaha 6.12/5.∞.12/5 shaha 6.∞.6/5.∞ 2hoha 12.12/5.∞ hatha ? (6 6 ∞') (6.∞)6/5 cha (6.∞)6/5 cha — 6/5.12.∞.12 ghaha 6/5.12.∞.12 ghaha [degenerate] [degenerate] ? (6/5 6/5 ∞') (6.∞)6/5 cha (6.∞)6/5 cha — 6.12/5.∞.12/5 shaha 6.12/5.∞.12/5 shaha [degenerate] [degenerate] ? The tiling 6 6/5 \| ∞ is generated as a double cover by Wythoff's construction: 6.∞.6/5.∞ hoha hemi(6 6/5 \| ∞) Also there are a few tilings with the mixed symbol p q r s \|: 4.12.4/3.12/11 sraht 2 6 3/2 3 \| 4.12/5.4/3.12/7 graht 2 6/5 3/2 3 \| 8/3.8.8/5.8/7 sost 4/3 4 2 ∞ \| 12/5.12.12/7.12/11 huht 6/5 6 3 ∞ \| There are also some non-Wythoffian tilings: 3.3.3.4.4 etrat 3.3.3.4/3.4/3 retrat The set of uniform tilings of the plane is not proved to be complete, unlike the set of uniform polyhedra. The tilings above represent all found by Coxeter, Longuet-Higgins, and Miller in their 1954 paper on uniform polyhedra. They conjectured that the lists were complete: this was proven by Sopov in 1970 for the uniform polyhedra, but has not been proven for the uniform tilings. Indeed Branko Grünbaum, J. C. P. Miller, and G. C. Shephard list fifteen more non-Wythoffian uniform tilings in Uniform Tilings with Hollow Tiles (1981):[6] 4.8.8/3.4/3.∞ rorisassa 4.8/3.8.4/3.∞ rosassa 4.8.4/3.8.4/3.∞ rarsisresa (2 forms) 4.8/3.4.8/3.4/3.∞ rassersa (2 forms) 3/2.∞.3/2.∞.3/2.4.4 3/2.∞.3/2.∞.3/2.4/3.4/3 3/2.∞.3/2.∞.3/2.12/5.6.12/5 3/2.∞.3/2.∞.3/2.12/7.6/5.12/7 3/2.∞.3/2.∞.3/2.12.6/5.12 3/2.∞.3/2.∞.3/2.12/11.6/5.12/11 3/2.∞.3/2.4/3.4/3.3/2.4/3.4/3 3/2.∞.3/2.4.4.3/2.4.4 3/2.∞.3/2.4.4.3/2.4/3.4/3 There are two tilings each for the vertex figures 4.8.4/3.8.4/3.∞ and 4.8/3.4.8/3.4/3.∞; they use the same sets of vertices and edges, but have a different set of squares. There exists also a third tiling for each of these two vertex figure that is only pseudo-uniform (all vertices look alike, but they come in two symmetry orbits). Hence, for Euclidean tilings, the vertex configuration does not uniquely determine the tiling.[6] In the pictures below, the included squares with horizontal and vertical edges are marked with a central dot. A single square has edges highlighted.[6] • Uniform (wallpaper group p4m) • Uniform (wallpaper group p4g) • Pseudo-uniform Grünbaum, Miller, and Shephard also list 33 uniform tilings using zigzags (skew apeirogons) as faces, ten of which are families that have a free parameter (the angle of the zigzag). In eight cases this parameter is continuous; in two, it is discrete.[6] Hyperbolic tilings The set of triangles tiling the hyperbolic plane is infinite. Moreover in hyperbolic space the fundamental domain does not have to be a simplex. Consequently a full listing of the uniform tilings of the hyperbolic plane cannot be given. Even when restricted to convex tiles, it is possible to find multiple tilings with the same vertex configuration: see for example Snub order-6 square tiling#Related polyhedra and tiling.[7] A few small convex cases (not involving ideal faces or vertices) have been given below: (p q r) q \| p r (p.r)q p \| q r (q.r)p r \| p q (q.p)r q r \| p q.2p.r.2p p r \| q p.2q.r.2q p q \| r 2r.q.2r.p p q r \| 2r.2q.2p \| p q r 3.r.3.q.3.p (7 3 2) 7.7.7 heat 3.3.3.3.3.3.3 hetrat 3.7.3.7 thet 3.14.14 theat 6.6.7 thetrat 4.3.4.7 srothet 4.6.14 grothet 3.3.3.3.7 snathet (8 3 2) 8.8.8 ocat 3.3.3.3.3.3.3.3 otrat 3.8.3.8 toct 3.16.16 tocat 6.6.8 totrat 4.3.4.8 srotoct 4.6.16 grotoct 3.3.3.3.8 snatoct (5 4 2) 5.5.5.5 peat 4.4.4.4.4 pesquat 4.5.4.5 tepet 4.10.10 topeat 5.8.8 topesquat 4.4.4.5 srotepet 4.8.10 grotepet 3.3.4.3.5 stepet (6 4 2) 6.6.6.6 shexat 4.4.4.4.4.4 hisquat 4.6.4.6 tehat 4.12.12 toshexat 6.8.8 thisquat 4.4.4.6 srotehat 4.8.12 grotehat 3.3.4.3.6 snatehat (5 5 2) 5.5.5.5.5 pepat 5.5.5.5.5 pepat 5.5.5.5 peat 5.10.10 topepat 5.10.10 topepat 4.5.4.5 tepet 4.10.10 topeat 3.3.5.3.5 spepat (6 6 2) 6.6.6.6.6.6 hihat 6.6.6.6.6.6 hihat 6.6.6.6 shexat 6.12.12 thihat 6.12.12 thihat 4.6.4.6 tehat 4.12.12 toshexat 3.3.6.3.6 shihat (4 3 3) 3.4.3.4.3.4 dittitecat 3.3.3.3.3.3.3.3 otrat 3.4.3.4.3.4 dittitecat 3.8.3.8 toct 6.3.6.4 sittitetrat 6.3.6.4 sittitetrat 6.6.8 totrat 3.3.3.3.3.4 stititet (4 4 3) 3.4.3.4.3.4.3.4 ditetetrat 3.4.3.4.3.4.3.4 ditetetrat 4.4.4.4.4.4 hisquat 4.8.3.8 sittiteteat 4.8.3.8 sittiteteat 6.4.6.4 tehat 6.8.8 thisquat 3.3.3.4.3.4 stitetet (4 4 4) 4.4.4.4.4.4.4.4 osquat 4.4.4.4.4.4.4.4 osquat 4.4.4.4.4.4 osquat 4.8.4.8 teoct 4.8.4.8 teoct 4.8.4.8 teoct 8.8.8 ocat 3.4.3.4.3.4 dittitecat References 1. The Bowers acronyms for the uniform polyhedra are given in R. Klitzing, Axial-Symmetrical Edge-Facetings of Uniform Polyhedra, Symmetry: Culture and Science Vol. 13, No. 3-4, 241-258, 2002 2. Coxeter, 1954 3. Explicitly stated for the tetrahemihexahedron in Coxeter et al. 1954, pp. 415–6 4. Skilling, 1974 5. Coxeter, Regular Polytopes, p. 114 6. Grünbaum, Branko; Miller, J. C. P.; Shephard, G. C. (1981). "Uniform Tilings with Hollow Tiles". In Davis, Chandler; Grünbaum, Branko; Sherk, F. A. (eds.). The Geometric Vein: The Coxeter Festschrift. Springer. pp. 17–64. ISBN 978-1-4612-5650-2. 7. Semi-regular tilings of the hyperbolic plane, Basudeb Datta and Subhojoy Gupta • Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954). "Uniform polyhedra". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences. The Royal Society. 246 (916): 401–450. doi:10.1098/rsta.1954.0003. ISSN 0080-4614. JSTOR 91532. MR 0062446. S2CID 202575183. • Skilling, J. (1974). "The complete set of uniform polyhedra". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences. The Royal Society. 278 (1278): 111–135. doi:10.1098/rsta.1975.0022. ISSN 1364-503X. S2CID 122634260. Richard Klitzing: Polyhedra by • point-group symmetry • complexity • Schwarz triangles part 1, part 2 • Euclidean tessellations and honeycombs • Hyperbolic tessellations and honeycombs The tables are based on those presented by Klitzing at his site. External links Jim McNeill: • Tessellations of the Plane Zvi Har'El: • Uniform Solution for Uniform Polyhedra Hironori Sakamoto: • Uniform Tessellations on the Euclid plane	Wikipedia
Uniform tiling symmetry mutations In geometry, a symmetry mutation is a mapping of fundamental domains between two symmetry groups.[1] They are compactly expressed in orbifold notation. These mutations can occur from spherical tilings to Euclidean tilings to hyperbolic tilings. Hyperbolic tilings can also be divided between compact, paracompact and divergent cases. Example n32 symmetry mutations Spherical tilings (n = 3..5) 332 432 532 Euclidean plane tiling (n = 6) 632 Hyperbolic plane tilings (n = 7...∞) 732 832 ... ∞32 The uniform tilings are the simplest application of these mutations, although more complex patterns can be expressed within a fundamental domain. This article expressed progressive sequences of uniform tilings within symmetry families. Mutations of orbifolds Orbifolds with the same structure can be mutated between different symmetry classes, including across curvature domains from spherical, to Euclidean to hyperbolic. This table shows mutation classes.[1] This table is not complete for possible hyperbolic orbifolds. Orbifold Spherical Euclidean Hyperbolic o - o - pp 22, 33 ... ∞∞ - pp 22, 33 ... ∞∞ - p* 2, 3 ... ∞* - p× 2×, 3× ... ∞× - - × - × - ×× - ×× - ppp 222 333 444 ... pp* - 22* 33* ... pp× - 22× 33×, 44× ... pqq 222, 322 ... , 233 244 255 ..., 433 ... pqr 234, 235 236 237 ..., 245 ... pq* - - 23, 24 ... pq× - - 23×, 24× ... pq 22, 23 ... 33, 42 52 53 ..., 43, 44 ..., 34, 35 ... p* - - 2 ... p× - - 2× ... pppp - 2222 3333 ... pppq - - 2223... ppqq - - 2233 ppp - - 222 ... pqr - 222 322 ..., 232 ... ppp 222 333 444 ... pqq p22, 233 244 255 ..., 344... pqr 234, 235 236 237..., 245..., 345 ... pppp - - 2222 pqrs - 2222 2223... ppppp - - 22222 ... ... n22 symmetry Regular tilings Family of regular hosohedra · n22 symmetry mutations of regular hosohedral tilings: nn Space SphericalEuclidean Tiling name (Monogonal) Henagonal hosohedron Digonal hosohedron (Triangular) Trigonal hosohedron (Tetragonal) Square hosohedron Pentagonal hosohedron Hexagonal hosohedron Heptagonal hosohedron Octagonal hosohedron Enneagonal hosohedron Decagonal hosohedron Hendecagonal hosohedron Dodecagonal hosohedron ... Apeirogonal hosohedron Tiling image ... Schläfli symbol {2,1}{2,2}{2,3}{2,4}{2,5}{2,6}{2,7}{2,8}{2,9}{2,10}{2,11}{2,12}...{2,∞} Coxeter diagram ... Faces and edges 123456789101112...∞ Vertices 2...2 Vertex config. 22.223242526272829210211212...2∞ Family of regular dihedra · n22 symmetry mutations of regular dihedral tilings: nn Space SphericalEuclidean Tiling name (Hengonal) Monogonal dihedron Digonal dihedron (Triangular) Trigonal dihedron (Tetragonal) Square dihedron Pentagonal dihedron Hexagonal dihedron ... Apeirogonal dihedron Tiling image ... Schläfli symbol {1,2}{2,2}{3,2}{4,2}{5,2}{6,2}...{∞,2} Coxeter diagram ... Faces 2 {1}2 {2}2 {3}2 {4}2 {5}2 {6}...2 {∞} Edges and vertices 123456...∞ Vertex config. 1.12.23.34.45.56.6...∞.∞ Prism tilings n22 symmetry mutations of uniform prisms: n.4.4 Space Spherical Euclidean Tiling Config. 3.4.4 4.4.4 5.4.4 6.4.4 7.4.4 8.4.4 9.4.4 10.4.4 11.4.4 12.4.4 ...∞.4.4 Antiprism tilings n22 symmetry mutations of antiprism tilings: Vn.3.3.3 Space Spherical Euclidean Tiling Config. 2.3.3.3 3.3.3.3 4.3.3.3 5.3.3.3 6.3.3.3 7.3.3.3 8.3.3.3 ...∞.3.3.3 n32 symmetry Regular tilings n32 symmetry mutation of regular tilings: {3,n} Spherical Euclid. Compact hyper. Paraco. Noncompact hyperbolic 3.3 33 34 35 36 37 38 3∞ 312i 39i 36i 33i n32 symmetry mutation of regular tilings: {n,3} Spherical Euclidean Compact hyperb. Paraco. Noncompact hyperbolic {2,3} {3,3} {4,3} {5,3} {6,3} {7,3} {8,3} {∞,3} {12i,3} {9i,3} {6i,3} {3i,3} Truncated tilings n32 symmetry mutation of truncated tilings: t{n,3} Symmetry n32 [n,3] Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic 232 [2,3] 332 [3,3] 432 [4,3] 532 [5,3] 632 [6,3] 732 [7,3] 832 [8,3]... ∞32 [∞,3] [12i,3] [9i,3] [6i,3] Truncated figures Symbol t{2,3} t{3,3} t{4,3} t{5,3} t{6,3} t{7,3} t{8,3} t{∞,3} t{12i,3} t{9i,3} t{6i,3} Triakis figures Config. V3.4.4 V3.6.6 V3.8.8 V3.10.10 V3.12.12 V3.14.14 V3.16.16 V3.∞.∞ n32 symmetry mutation of truncated tilings: n.6.6 Sym. n42 [n,3] Spherical Euclid. Compact Parac. Noncompact hyperbolic 232 [2,3] 332 [3,3] 432 [4,3] 532 [5,3] 632 [6,3] 732 [7,3] 832 [8,3]... ∞32 [∞,3] [12i,3] [9i,3] [6i,3] Truncated figures Config. 2.6.6 3.6.6 4.6.6 5.6.6 6.6.6 7.6.6 8.6.6 ∞.6.6 12i.6.6 9i.6.6 6i.6.6 n-kis figures Config. V2.6.6 V3.6.6 V4.6.6 V5.6.6 V6.6.6 V7.6.6 V8.6.6 V∞.6.6 V12i.6.6 V9i.6.6 V6i.6.6 Quasiregular tilings Quasiregular tilings: (3.n)2 Sym. n32 [n,3] Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic 332 [3,3] Td 432 [4,3] Oh 532 [5,3] Ih 632 [6,3] p6m 732 [7,3] 832 [8,3]... ∞32 [∞,3] [12i,3] [9i,3] [6i,3] Figure Figure Vertex (3.3)2 (3.4)2 (3.5)2 (3.6)2 (3.7)2 (3.8)2 (3.∞)2 (3.12i)2 (3.9i)2 (3.6i)2 Schläfli r{3,3} r{3,4} r{3,5} r{3,6} r{3,7} r{3,8} r{3,∞} r{3,12i} r{3,9i} r{3,6i} Coxeter Dual uniform figures Dual conf. V(3.3)2 V(3.4)2 V(3.5)2 V(3.6)2 V(3.7)2 V(3.8)2 V(3.∞)2 Symmetry mutations of dual quasiregular tilings: V(3.n)2 n32 Spherical Euclidean Hyperbolic 332 432 532 632 732 832... ∞32 Tiling Conf. V(3.3)2 V(3.4)2 V(3.5)2 V(3.6)2 V(3.7)2 V(3.8)2 V(3.∞)2 Expanded tilings n42 symmetry mutation of expanded tilings: 3.4.n.4 Symmetry n32 [n,3] Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic 232 [2,3] 332 [3,3] 432 [4,3] 532 [5,3] 632 [6,3] 732 [7,3] 832 [8,3]... ∞32 [∞,3] [12i,3] [9i,3] [6i,3] Figure Config. 3.4.2.4 3.4.3.4 3.4.4.4 3.4.5.4 3.4.6.4 3.4.7.4 3.4.8.4 3.4.∞.4 3.4.12i.4 3.4.9i.4 3.4.6i.4 n32 symmetry mutation of dual expanded tilings: V3.4.n.4 Symmetry n32 [n,3] Spherical Euclid. Compact hyperb. Paraco. 232 [2,3] 332 [3,3] 432 [4,3] 532 [5,3] 632 [6,3] 732 [7,3] 832 [8,3]... ∞32 [∞,3] Figure Config. V3.4.2.4 V3.4.3.4 V3.4.4.4 V3.4.5.4 V3.4.6.4 V3.4.7.4 V3.4.8.4 V3.4.∞.4 Omnitruncated tilings n32 symmetry mutation of omnitruncated tilings: 4.6.2n Sym. n32 [n,3] Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic 232 [2,3] 332 [3,3] 432 [4,3] 532 [5,3] 632 [6,3] 732 [7,3] 832 [8,3] ∞32 [∞,3] [12i,3] [9i,3] [6i,3] [3i,3] Figures Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞ 4.6.24i 4.6.18i 4.6.12i 4.6.6i Duals Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i Snub tilings n32 symmetry mutations of snub tilings: 3.3.3.3.n Symmetry n32 Spherical Euclidean Compact hyperbolic Paracomp. 232 332 432 532 632 732 832 ∞32 Snub figures Config. 3.3.3.3.2 3.3.3.3.3 3.3.3.3.4 3.3.3.3.5 3.3.3.3.6 3.3.3.3.7 3.3.3.3.8 3.3.3.3.∞ Gyro figures Config. V3.3.3.3.2 V3.3.3.3.3 V3.3.3.3.4 V3.3.3.3.5 V3.3.3.3.6 V3.3.3.3.7 V3.3.3.3.8 V3.3.3.3.∞ n42 symmetry Regular tilings n42 symmetry mutation of regular tilings: {4,n} Spherical Euclidean Compact hyperbolic Paracompact {4,3} {4,4} {4,5} {4,6} {4,7} {4,8}... {4,∞} n42 symmetry mutation of regular tilings: {n,4} Spherical Euclidean Hyperbolic tilings 24 34 44 54 64 74 84 ...∞4 Quasiregular tilings n42 symmetry mutations of quasiregular tilings: (4.n)2 Symmetry 4n2 [n,4] Spherical Euclidean Compact hyperbolic Paracompact Noncompact 342 [3,4] 442 [4,4] 542 [5,4] 642 [6,4] 742 [7,4] 842 [8,4]... ∞42 [∞,4] [ni,4] Figures Config. (4.3)2 (4.4)2 (4.5)2 (4.6)2 (4.7)2 (4.8)2 (4.∞)2 (4.ni)2 n42 symmetry mutations of quasiregular dual tilings: V(4.n)2 Symmetry 4n2 [n,4] Spherical Euclidean Compact hyperbolic Paracompact Noncompact 342 [3,4] 442 [4,4] 542 [5,4] 642 [6,4] 742 [7,4] 842 [8,4]... ∞42 [∞,4] [iπ/λ,4] Tiling Conf. V4.3.4.3 V4.4.4.4 V4.5.4.5 V4.6.4.6 V4.7.4.7 V4.8.4.8 V4.∞.4.∞ V4.∞.4.∞ Truncated tilings n42 symmetry mutation of truncated tilings: 4.2n.2n Symmetry n42 [n,4] Spherical Euclidean Compact hyperbolic Paracomp. 242 [2,4] 342 [3,4] 442 [4,4] 542 [5,4] 642 [6,4] 742 [7,4] 842 [8,4]... ∞42 [∞,4] Truncated figures Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4.∞.∞ n-kis figures Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10 V4.12.12 V4.14.14 V4.16.16 V4.∞.∞ n42 symmetry mutation of truncated tilings: n.8.8 Symmetry n42 [n,4] Spherical Euclidean Compact hyperbolic Paracompact 242 [2,4] 342 [3,4] 442 [4,4] 542 [5,4] 642 [6,4] 742 [7,4] 842 [8,4]... ∞42 [∞,4] Truncated figures Config. 2.8.8 3.8.8 4.8.8 5.8.8 6.8.8 7.8.8 8.8.8 ∞.8.8 n-kis figures Config. V2.8.8 V3.8.8 V4.8.8 V5.8.8 V6.8.8 V7.8.8 V8.8.8 V∞.8.8 Expanded tilings n42 symmetry mutation of expanded tilings: n.4.4.4 Symmetry [n,4], (n42) Spherical Euclidean Compact hyperbolic Paracomp. 342 [3,4] 442 [4,4] 542 [5,4] 642 [6,4] 742 [7,4] 842 [8,4] ∞42 [∞,4] Expanded figures Config. 3.4.4.4 4.4.4.4 5.4.4.4 6.4.4.4 7.4.4.4 8.4.4.4 ∞.4.4.4 Rhombic figures config. V3.4.4.4 V4.4.4.4 V5.4.4.4 V6.4.4.4 V7.4.4.4 V8.4.4.4 V∞.4.4.4 Omnitruncated tilings n42 symmetry mutation of omnitruncated tilings: 4.8.2n Symmetry n42 [n,4] Spherical Euclidean Compact hyperbolic Paracomp. 242 [2,4] 342 [3,4] 442 [4,4] 542 [5,4] 642 [6,4] 742 [7,4] 842 [8,4]... ∞42 [∞,4] Omnitruncated figure 4.8.4 4.8.6 4.8.8 4.8.10 4.8.12 4.8.14 4.8.16 4.8.∞ Omnitruncated duals V4.8.4 V4.8.6 V4.8.8 V4.8.10 V4.8.12 V4.8.14 V4.8.16 V4.8.∞ Snub tilings 4n2 symmetry mutations of snub tilings: 3.3.4.3.n Symmetry 4n2 Spherical Euclidean Compact hyperbolic Paracomp. 242 342 442 542 642 742 842 ∞42 Snub figures Config. 3.3.4.3.2 3.3.4.3.3 3.3.4.3.4 3.3.4.3.5 3.3.4.3.6 3.3.4.3.7 3.3.4.3.8 3.3.4.3.∞ Gyro figures Config. V3.3.4.3.2 V3.3.4.3.3 V3.3.4.3.4 V3.3.4.3.5 V3.3.4.3.6 V3.3.4.3.7 V3.3.4.3.8 V3.3.4.3.∞ n52 symmetry Regular tilings n52 symmetry mutation of truncated tilings: 5n Sphere Hyperbolic plane {5,3} {5,4} {5,5} {5,6} {5,7} {5,8} ...{5,∞} n62 symmetry Regular tilings n62 symmetry mutation of regular tilings: {6,n} Spherical Euclidean Hyperbolic tilings {6,2} {6,3} {6,4} {6,5} {6,6} {6,7} {6,8} ... {6,∞} *n82 symmetry Regular tilings n82 symmetry mutations of regular tilings: 8n Space Spherical Compact hyperbolic Paracompact Tiling Config. 8.8 83 84 85 86 87 88 ...8∞ References 1. Two Dimensional symmetry Mutations by Daniel Huson Sources • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 • From hyperbolic 2-space to Euclidean 3-space: Tilings and patterns via topology Stephen Hyde	Wikipedia
Uniform tree In mathematics, a uniform tree is a locally finite tree which is the universal cover of a finite graph. Equivalently, the full automorphism group G=Aut(X) of the tree, which is a locally compact topological group, is unimodular and G\X is finite. Also equivalent is the existence of a uniform X-lattice in G. Sources • Bass, Hyman; Lubotzky, Alexander (2001), Tree Lattices, Progress in Mathematics, vol. 176, Birkhäuser, ISBN 0-8176-4120-3	Wikipedia
Uniformization theorem In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces. Since every Riemann surface has a universal cover which is a simply connected Riemann surface, the uniformization theorem leads to a classification of Riemann surfaces into three types: those that have the Riemann sphere as universal cover ("elliptic"), those with the plane as universal cover ("parabolic") and those with the unit disk as universal cover ("hyperbolic"). It further follows that every Riemann surface admits a Riemannian metric of constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case. The uniformization theorem also yields a similar classification of closed orientable Riemannian 2-manifolds into elliptic/parabolic/hyperbolic cases. Each such manifold has a conformally equivalent Riemannian metric with constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case. History Felix Klein (1883) and Henri Poincaré (1882) conjectured the uniformization theorem for (the Riemann surfaces of) algebraic curves. Henri Poincaré (1883) extended this to arbitrary multivalued analytic functions and gave informal arguments in its favor. The first rigorous proofs of the general uniformization theorem were given by Poincaré (1907) and Paul Koebe (1907a, 1907b, 1907c). Paul Koebe later gave several more proofs and generalizations. The history is described in Gray (1994); a complete account of uniformization up to the 1907 papers of Koebe and Poincaré is given with detailed proofs in de Saint-Gervais (2016) (the Bourbaki-type pseudonym of the group of fifteen mathematicians who jointly produced this publication). Classification of connected Riemann surfaces Every Riemann surface is the quotient of free, proper and holomorphic action of a discrete group on its universal covering and this universal covering, being a simply connected Riemann surface, is holomorphically isomorphic (one also says: "conformally equivalent" or "biholomorphic") to one of the following: 1. the Riemann sphere 2. the complex plane 3. the unit disk in the complex plane. For compact Riemann surfaces, those with universal cover the unit disk are precisely the hyperbolic surfaces of genus greater than 1, all with non-abelian fundamental group; those with universal cover the complex plane are the Riemann surfaces of genus 1, namely the complex tori or elliptic curves with fundamental group Z2; and those with universal cover the Riemann sphere are those of genus zero, namely the Riemann sphere itself, with trivial fundamental group. Classification of closed oriented Riemannian 2-manifolds On an oriented 2-manifold, a Riemannian metric induces a complex structure using the passage to isothermal coordinates. If the Riemannian metric is given locally as $ds^{2}=E\,dx^{2}+2F\,dx\,dy+G\,dy^{2},$ then in the complex coordinate z = x + iy, it takes the form $ds^{2}=\lambda \|dz+\mu \,d{\overline {z}}\|^{2},$ where $\lambda ={\frac {1}{4}}\left(E+G+2{\sqrt {EG-F^{2}}}\right),\ \ \mu ={\frac {1}{4\lambda }}(E-G+2iF),$ so that λ and μ are smooth with λ > 0 and \|μ\| < 1. In isothermal coordinates (u, v) the metric should take the form $ds^{2}=\rho (du^{2}+dv^{2})$ with ρ > 0 smooth. The complex coordinate w = u + i v satisfies $\rho \,\|dw\|^{2}=\rho \|w_{z}\|^{2}\left\|dz+{w_{\overline {z}} \over w_{z}}\,d{\overline {z}}\right\|^{2},$ so that the coordinates (u, v) will be isothermal locally provided the Beltrami equation ${\partial w \over \partial {\overline {z}}}=\mu {\partial w \over \partial z}$ has a locally diffeomorphic solution, i.e. a solution with non-vanishing Jacobian. These conditions can be phrased equivalently in terms of the exterior derivative and the Hodge star operator ∗.[1] u and v will be isothermal coordinates if ∗du = dv, where ∗ is defined on differentials by ∗(p dx + q dy) = −q dx + p dy. Let ∆ = ∗d∗d be the Laplace–Beltrami operator. By standard elliptic theory, u can be chosen to be harmonic near a given point, i.e. Δ u = 0, with du non-vanishing. By the Poincaré lemma dv = ∗du has a local solution v exactly when d(∗du) = 0. This condition is equivalent to Δ u = 0, so can always be solved locally. Since du is non-zero and the square of the Hodge star operator is −1 on 1-forms, du and dv must be linearly independent, so that u and v give local isothermal coordinates. The existence of isothermal coordinates can be proved by other methods, for example using the general theory of the Beltrami equation, as in Ahlfors (2006), or by direct elementary methods, as in Chern (1955) and Jost (2006). From this correspondence with compact Riemann surfaces, a classification of closed orientable Riemannian 2-manifolds follows. Each such is conformally equivalent to a unique closed 2-manifold of constant curvature, so a quotient of one of the following by a free action of a discrete subgroup of an isometry group: 1. the sphere (curvature +1) 2. the Euclidean plane (curvature 0) 3. the hyperbolic plane (curvature −1). • genus 0 • genus 1 • genus 2 • genus 3 The first case gives the 2-sphere, the unique 2-manifold with constant positive curvature and hence positive Euler characteristic (equal to 2). The second gives all flat 2-manifolds, i.e. the tori, which have Euler characteristic 0. The third case covers all 2-manifolds of constant negative curvature, i.e. the hyperbolic 2-manifolds all of which have negative Euler characteristic. The classification is consistent with the Gauss–Bonnet theorem, which implies that for a closed surface with constant curvature, the sign of that curvature must match the sign of the Euler characteristic. The Euler characteristic is equal to 2 – 2g, where g is the genus of the 2-manifold, i.e. the number of "holes". Methods of proof Many classical proofs of the uniformization theorem rely on constructing a real-valued harmonic function on the simply connected Riemann surface, possibly with a singularity at one or two points and often corresponding to a form of Green's function. Four methods of constructing the harmonic function are widely employed: the Perron method; the Schwarz alternating method; Dirichlet's principle; and Weyl's method of orthogonal projection. In the context of closed Riemannian 2-manifolds, several modern proofs invoke nonlinear differential equations on the space of conformally equivalent metrics. These include the Beltrami equation from Teichmüller theory and an equivalent formulation in terms of harmonic maps; Liouville's equation, already studied by Poincaré; and Ricci flow along with other nonlinear flows. Rado's theorem shows that every Riemann surface is automatically second-countable. Although Rado's theorem is often used in proofs of the uniformization theorem, some proofs have been formulated so that Rado's theorem becomes a consequence. Second countability is automatic for compact Riemann surfaces. Hilbert space methods See also: Planar Riemann surface § Uniformization theorem In 1913 Hermann Weyl published his classic textbook "Die Idee der Riemannschen Fläche" based on his Göttingen lectures from 1911 to 1912. It was the first book to present the theory of Riemann surfaces in a modern setting and through its three editions has remained influential. Dedicated to Felix Klein, the first edition incorporated Hilbert's treatment of the Dirichlet problem using Hilbert space techniques; Brouwer's contributions to topology; and Koebe's proof of the uniformization theorem and its subsequent improvements. Much later Weyl (1940) developed his method of orthogonal projection which gave a streamlined approach to the Dirichlet problem, also based on Hilbert space; that theory, which included Weyl's lemma on elliptic regularity, was related to Hodge's theory of harmonic integrals; and both theories were subsumed into the modern theory of elliptic operators and L2 Sobolev spaces. In the third edition of his book from 1955, translated into English in Weyl (1964), Weyl adopted the modern definition of differential manifold, in preference to triangulations, but decided not to make use of his method of orthogonal projection. Springer (1957) followed Weyl's account of the uniformisation theorem, but used the method of orthogonal projection to treat the Dirichlet problem. Kodaira (2007) describes the approach in Weyl's book and also how to shorten it using the method of orthogonal projection. A related account can be found in Donaldson (2011). Nonlinear flows See also: Ricci flow § Relationship to uniformization and geometrization Richard S. Hamilton showed that the normalized Ricci flow on a closed surface uniformizes the metric (i.e., the flow converges to a constant curvature metric). However, his proof relied on the uniformization theorem. The missing step involved Ricci flow on the 2-sphere: a method for avoiding an appeal to the uniformization theorem (for genus 0) was provided by Chen, Lu & Tian (2006);[2] a short self-contained account of Ricci flow on the 2-sphere was given in Andrews & Bryan (2010). Generalizations Koebe proved the general uniformization theorem that if a Riemann surface is homeomorphic to an open subset of the complex sphere (or equivalently if every Jordan curve separates it), then it is conformally equivalent to an open subset of the complex sphere. In 3 dimensions, there are 8 geometries, called the eight Thurston geometries. Not every 3-manifold admits a geometry, but Thurston's geometrization conjecture proved by Grigori Perelman states that every 3-manifold can be cut into pieces that are geometrizable. The simultaneous uniformization theorem of Lipman Bers shows that it is possible to simultaneously uniformize two compact Riemann surfaces of the same genus >1 with the same quasi-Fuchsian group. The measurable Riemann mapping theorem shows more generally that the map to an open subset of the complex sphere in the uniformization theorem can be chosen to be a quasiconformal map with any given bounded measurable Beltrami coefficient. See also • p-adic uniformization theorem Notes 1. DeTurck & Kazdan 1981; Taylor 1996a, pp. 377–378 2. Brendle 2010 References Historic references • Schwarz, H. A. (1870), "Über einen Grenzübergang durch alternierendes Verfahren", Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich, 15: 272–286, JFM 02.0214.02. • Klein, Felix (1883), "Neue Beiträge zur Riemann'schen Functionentheorie", Mathematische Annalen, 21 (2): 141–218, doi:10.1007/BF01442920, ISSN 0025-5831, JFM 15.0351.01, S2CID 120465625 • Koebe, P. (1907a), "Über die Uniformisierung reeller analytischer Kurven", Göttinger Nachrichten: 177–190, JFM 38.0453.01 • Koebe, P. (1907b), "Über die Uniformisierung beliebiger analytischer Kurven", Göttinger Nachrichten: 191–210, JFM 38.0454.01 • Koebe, P. (1907c), "Über die Uniformisierung beliebiger analytischer Kurven (Zweite Mitteilung)", Göttinger Nachrichten: 633–669, JFM 38.0455.02 • Koebe, Paul (1910a), "Über die Uniformisierung beliebiger analytischer Kurven", Journal für die Reine und Angewandte Mathematik, 138: 192–253, doi:10.1515/crll.1910.138.192, S2CID 120198686 • Koebe, Paul (1910b), "Über die Hilbertsche Uniformlsierungsmethode" (PDF), Göttinger Nachrichten: 61–65 • Poincaré, H. (1882), "Mémoire sur les fonctions fuchsiennes", Acta Mathematica, 1: 193–294, doi:10.1007/BF02592135, ISSN 0001-5962, JFM 15.0342.01 • Poincaré, Henri (1883), "Sur un théorème de la théorie générale des fonctions", Bulletin de la Société Mathématique de France, 11: 112–125, doi:10.24033/bsmf.261, ISSN 0037-9484, JFM 15.0348.01 • Poincaré, Henri (1907), "Sur l'uniformisation des fonctions analytiques", Acta Mathematica, 31: 1–63, doi:10.1007/BF02415442, ISSN 0001-5962, JFM 38.0452.02 • Hilbert, David (1909), "Zur Theorie der konformen Abbildung" (PDF), Göttinger Nachrichten: 314–323 • Perron, O. (1923), "Eine neue Behandlung der ersten Randwertaufgabe für Δu=0", Mathematische Zeitschrift, 18 (1): 42–54, doi:10.1007/BF01192395, ISSN 0025-5874, S2CID 122843531 • Weyl, Hermann (1913), Die Idee der Riemannschen Fläche (1997 reprint of the 1913 German original), Teubner, ISBN 978-3-8154-2096-6 • Weyl, Hermann (1940), "The method of orthogonal projections in potential theory", Duke Math. J., 7: 411–444, doi:10.1215/s0012-7094-40-00725-6 Historical surveys • Abikoff, William (1981), "The uniformization theorem", Amer. Math. Monthly, 88 (8): 574–592, doi:10.2307/2320507, JSTOR 2320507 • Gray, Jeremy (1994), "On the history of the Riemann mapping theorem" (PDF), Rendiconti del Circolo Matematico di Palermo. Serie II. Supplemento (34): 47–94, MR 1295591 • Bottazzini, Umberto; Gray, Jeremy (2013), Hidden Harmony—Geometric Fantasies: The Rise of Complex Function Theory, Sources and Studies in the History of Mathematics and Physical Sciences, Springer, ISBN 978-1461457251 • de Saint-Gervais, Henri Paul (2016), Uniformization of Riemann Surfaces: revisiting a hundred-year-old theorem, translated by Robert G. Burns, European Mathematical Society, doi:10.4171/145, ISBN 978-3-03719-145-3, translation of French text (prepared in 2007 during centenary of 1907 papers of Koebe and Poincaré) Harmonic functions Perron's method • Heins, M. (1949), "The conformal mapping of simply-connected Riemann surfaces", Ann. of Math., 50 (3): 686–690, doi:10.2307/1969555, JSTOR 1969555 • Heins, M. (1951), "Interior mapping of an orientable surface into S2", Proc. Amer. Math. Soc., 2 (6): 951–952, doi:10.1090/s0002-9939-1951-0045221-4 • Heins, M. (1957), "The conformal mapping of simply-connected Riemann surfaces. II", Nagoya Math. J., 12: 139–143, doi:10.1017/s002776300002198x • Pfluger, Albert (1957), Theorie der Riemannschen Flächen, Springer • Ahlfors, Lars V. (2010), Conformal invariants: topics in geometric function theory, AMS Chelsea Publishing, ISBN 978-0-8218-5270-5 • Beardon, A. F. (1984), "A primer on Riemann surfaces", London Mathematical Society Lecture Note Series, Cambridge University Press, 78, ISBN 978-0521271042 • Forster, Otto (1991), Lectures on Riemann surfaces, Graduate Texts in Mathematics, vol. 81, translated by Bruce Gilligan, Springer, ISBN 978-0-387-90617-1 • Farkas, Hershel M.; Kra, Irwin (1980), Riemann surfaces (2nd ed.), Springer, ISBN 978-0-387-90465-8 • Gamelin, Theodore W. (2001), Complex analysis, Undergraduate Texts in Mathematics, Springer, ISBN 978-0-387-95069-3 • Hubbard, John H. (2006), Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1. Teichmüller theory, Matrix Editions, ISBN 978-0971576629 • Schlag, Wilhelm (2014), A course in complex analysis and Riemann surfaces., Graduate Studies in Mathematics, vol. 154, American Mathematical Society, ISBN 978-0-8218-9847-5 Schwarz's alternating method • Nevanlinna, Rolf (1953), Uniformisierung, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, vol. 64, Springer, doi:10.1007/978-3-642-52801-9, ISBN 978-3-642-52802-6 • Behnke, Heinrich; Sommer, Friedrich (1965), Theorie der analytischen Funktionen einer komplexen Veränderlichen, Die Grundlehren der mathematischen Wissenschaften, vol. 77 (3rd ed.), Springer • Freitag, Eberhard (2011), Complex analysis. 2. Riemann surfaces, several complex variables, abelian functions, higher modular functions, Springer, ISBN 978-3-642-20553-8 Dirichlet principle • Weyl, Hermann (1964), The concept of a Riemann surface, translated by Gerald R. MacLane, Addison-Wesley, MR 0069903 • Courant, Richard (1977), Dirichlet's principle, conformal mapping, and minimal surfaces, Springer, ISBN 978-0-387-90246-3 • Siegel, C. L. (1988), Topics in complex function theory. Vol. I. Elliptic functions and uniformization theory, translated by A. Shenitzer; D. Solitar, Wiley, ISBN 978-0471608448 Weyl's method of orthogonal projection • Springer, George (1957), Introduction to Riemann surfaces, Addison-Wesley, MR 0092855 • Kodaira, Kunihiko (2007), Complex analysis, Cambridge Studies in Advanced Mathematics, vol. 107, Cambridge University Press, ISBN 9780521809375 • Donaldson, Simon (2011), Riemann surfaces, Oxford Graduate Texts in Mathematics, vol. 22, Oxford University Press, ISBN 978-0-19-960674-0 Sario operators • Sario, Leo (1952), "A linear operator method on arbitrary Riemann surfaces", Trans. Amer. Math. Soc., 72 (2): 281–295, doi:10.1090/s0002-9947-1952-0046442-2 • Ahlfors, Lars V.; Sario, Leo (1960), Riemann surfaces, Princeton Mathematical Series, vol. 26, Princeton University Press Nonlinear differential equations Beltrami's equation • Ahlfors, Lars V. (2006), Lectures on quasiconformal mappings, University Lecture Series, vol. 38 (2nd ed.), American Mathematical Society, ISBN 978-0-8218-3644-6 • Ahlfors, Lars V.; Bers, Lipman (1960), "Riemann's mapping theorem for variable metrics", Ann. of Math., 72 (2): 385–404, doi:10.2307/1970141, JSTOR 1970141 • Bers, Lipman (1960), "Simultaneous uniformization", Bull. Amer. Math. Soc., 66 (2): 94–97, doi:10.1090/s0002-9904-1960-10413-2 • Bers, Lipman (1961), "Uniformization by Beltrami equations", Comm. Pure Appl. Math., 14 (3): 215–228, doi:10.1002/cpa.3160140304 • Bers, Lipman (1972), "Uniformization, moduli, and Kleinian groups", The Bulletin of the London Mathematical Society, 4 (3): 257–300, doi:10.1112/blms/4.3.257, ISSN 0024-6093, MR 0348097 Harmonic maps • Jost, Jürgen (2006), Compact Riemann surfaces: an introduction to contemporary mathematics (3rd ed.), Springer, ISBN 978-3-540-33065-3 Liouville's equation • Berger, Melvyn S. (1971), "Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds", Journal of Differential Geometry, 5 (3–4): 325–332, doi:10.4310/jdg/1214429996 • Berger, Melvyn S. (1977), Nonlinearity and functional analysis, Academic Press, ISBN 978-0-12-090350-4 • Taylor, Michael E. (2011), Partial differential equations III. Nonlinear equations, Applied Mathematical Sciences, vol. 117 (2nd ed.), Springer, ISBN 978-1-4419-7048-0 Flows on Riemannian metrics • Hamilton, Richard S. (1988), "The Ricci flow on surfaces", Mathematics and general relativity (Santa Cruz, CA, 1986), Contemp. Math., vol. 71, American Mathematical Society, pp. 237–262 • Chow, Bennett (1991), "The Ricci flow on the 2-sphere", J. Differential Geom., 33 (2): 325–334, doi:10.4310/jdg/1214446319 • Osgood, B.; Phillips, R.; Sarnak, P. (1988), "Extremals of determinants of Laplacians", J. Funct. Anal., 80: 148–211, CiteSeerX 10.1.1.486.558, doi:10.1016/0022-1236(88)90070-5 • Chrusciel, P. (1991), "Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation", Communications in Mathematical Physics, 137 (2): 289–313, Bibcode:1991CMaPh.137..289C, CiteSeerX 10.1.1.459.9029, doi:10.1007/bf02431882, S2CID 53641998 • Chang, Shu-Cheng (2000), "Global existence and convergence of solutions of Calabi flow on surfaces of genus h ≥ 2", J. Math. Kyoto Univ., 40 (2): 363–377, doi:10.1215/kjm/1250517718 • Brendle, Simon (2010), Ricci flow and the sphere theorem, Graduate Studies in Mathematics, vol. 111, American Mathematical Society, ISBN 978-0-8218-4938-5 • Chen, Xiuxiong; Lu, Peng; Tian, Gang (2006), "A note on uniformization of Riemann surfaces by Ricci flow", Proceedings of the American Mathematical Society, 134 (11): 3391–3393, doi:10.1090/S0002-9939-06-08360-2, ISSN 0002-9939, MR 2231924 • Andrews, Ben; Bryan, Paul (2010), "Curvature bounds by isoperimetric comparison for normalized Ricci flow on the two-sphere", Calc. Var. Partial Differential Equations, 39 (3–4): 419–428, arXiv:0908.3606, doi:10.1007/s00526-010-0315-5, S2CID 1095459 • Mazzeo, Rafe; Taylor, Michael (2002), "Curvature and uniformization", Israel Journal of Mathematics, 130: 323–346, arXiv:math/0105016, doi:10.1007/bf02764082, S2CID 7192529 • Struwe, Michael (2002), "Curvature flows on surfaces", Ann. Sc. Norm. Super. Pisa Cl. Sci., 1: 247–274 General references • Chern, Shiing-shen (1955), "An elementary proof of the existence of isothermal parameters on a surface", Proc. Amer. Math. Soc., 6 (5): 771–782, doi:10.2307/2032933, JSTOR 2032933 • DeTurck, Dennis M.; Kazdan, Jerry L. (1981), "Some regularity theorems in Riemannian geometry", Annales Scientifiques de l'École Normale Supérieure, Série 4, 14 (3): 249–260, doi:10.24033/asens.1405, ISSN 0012-9593, MR 0644518. • Gusevskii, N.A. (2001) [1994], "Uniformization", Encyclopedia of Mathematics, EMS Press • Krushkal, S. L.; Apanasov, B. N.; Gusevskiĭ, N. A. (1986) [1981], Kleinian groups and uniformization in examples and problems, Translations of Mathematical Monographs, vol. 62, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4516-5, MR 0647770 • Taylor, Michael E. (1996a), Partial Differential Equations I: Basic Theory, Springer, pp. 376–378, ISBN 978-0-387-94654-2 • Taylor, Michael E. (1996b), Partial Differential Equations II:Qualitative studies of linear equations, Springer, ISBN 978-0-387-94651-1 • Bers, Lipman; John, Fritz; Schechter, Martin (1979), Partial differential equations (reprint of the 1964 original), Lectures in Applied Mathematics, vol. 3A, American Mathematical Society, ISBN 978-0-8218-0049-2 • Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley, ISBN 978-0-471-05059-9 • Warner, Frank W. (1983), Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer, doi:10.1007/978-1-4757-1799-0, ISBN 978-0-387-90894-6 External links • Conformal Transformation: from Circle to Square. Manifolds (Glossary) Basic concepts • Topological manifold • Atlas • Differentiable/Smooth manifold • Differential structure • Smooth atlas • Submanifold • Riemannian manifold • Smooth map • Submersion • Pushforward • Tangent space • Differential form • Vector field Main results (list) • Atiyah–Singer index • Darboux's • De Rham's • Frobenius • Generalized Stokes • Hopf–Rinow • Noether's • Sard's • Whitney embedding Maps • Curve • Diffeomorphism • Local • Geodesic • Exponential map • in Lie theory • Foliation • Immersion • Integral curve • Lie derivative • Section • Submersion Types of manifolds • Closed • (Almost) Complex • (Almost) Contact • Fibered • Finsler • Flat • G-structure • Hadamard • Hermitian • Hyperbolic • Kähler • Kenmotsu • Lie group • Lie algebra • Manifold with boundary • Oriented • Parallelizable • Poisson • Prime • Quaternionic • Hypercomplex • (Pseudo−, Sub−) Riemannian • Rizza • (Almost) Symplectic • Tame Tensors Vectors • Distribution • Lie bracket • Pushforward • Tangent space • bundle • Torsion • Vector field • Vector flow Covectors • Closed/Exact • Covariant derivative • Cotangent space • bundle • De Rham cohomology • Differential form • Vector-valued • Exterior derivative • Interior product • Pullback • Ricci curvature • flow • Riemann curvature tensor • Tensor field • density • Volume form • Wedge product Bundles • Adjoint • Affine • Associated • Cotangent • Dual • Fiber • (Co) Fibration • Jet • Lie algebra • (Stable) Normal • Principal • Spinor • Subbundle • Tangent • Tensor • Vector Connections • Affine • Cartan • Ehresmann • Form • Generalized • Koszul • Levi-Civita • Principal • Vector • Parallel transport Related • Classification of manifolds • Gauge theory • History • Morse theory • Moving frame • Singularity theory Generalizations • Banach manifold • Diffeology • Diffiety • Fréchet manifold • K-theory • Orbifold • Secondary calculus • over commutative algebras • Sheaf • Stratifold • Supermanifold • Stratified space	Wikipedia
Gowers norm In mathematics, in the field of additive combinatorics, a Gowers norm or uniformity norm is a class of norms on functions on a finite group or group-like object which quantify the amount of structure present, or conversely, the amount of randomness.[1] They are used in the study of arithmetic progressions in the group. They are named after Timothy Gowers, who introduced it in his work on Szemerédi's theorem.[2] "Uniformity norm" redirects here. For the function field norm, see uniform norm. For uniformity in topology, see uniform space. Definition Let $f$ be a complex-valued function on a finite abelian group $G$ and let $J$ denote complex conjugation. The Gowers $d$-norm is $\Vert f\Vert _{U^{d}(G)}^{2^{d}}=\sum _{x,h_{1},\ldots ,h_{d}\in G}\prod _{\omega _{1},\ldots ,\omega _{d}\in \{0,1\}}J^{\omega _{1}+\cdots +\omega _{d}}f\left({x+h_{1}\omega _{1}+\cdots +h_{d}\omega _{d}}\right)\ .$ Gowers norms are also defined for complex-valued functions f on a segment $[N]={0,1,2,...,N-1}$, where N is a positive integer. In this context, the uniformity norm is given as $\Vert f\Vert _{U^{d}[N]}=\Vert {\tilde {f}}\Vert _{U^{d}(\mathbb {Z} /{\tilde {N}}\mathbb {Z} )}/\Vert 1_{[N]}\Vert _{U^{d}(\mathbb {Z} /{\tilde {N}}\mathbb {Z} )}$, where ${\tilde {N}}$ is a large integer, $1_{[N]}$ denotes the indicator function of [N], and ${\tilde {f}}(x)$ is equal to $f(x)$ for $x\in [N]$ and $0$ for all other $x$. This definition does not depend on ${\tilde {N}}$, as long as ${\tilde {N}}>2^{d}N$. Inverse conjectures An inverse conjecture for these norms is a statement asserting that if a bounded function f has a large Gowers d-norm then f correlates with a polynomial phase of degree d − 1 or other object with polynomial behaviour (e.g. a (d − 1)-step nilsequence). The precise statement depends on the Gowers norm under consideration. The Inverse Conjecture for vector spaces over a finite field $\mathbb {F} $ asserts that for any $\delta >0$ there exists a constant $c>0$ such that for any finite-dimensional vector space V over $\mathbb {F} $ and any complex-valued function $f$ on $V$, bounded by 1, such that $\Vert f\Vert _{U^{d}[V]}\geq \delta $, there exists a polynomial sequence $P\colon V\to \mathbb {R} /\mathbb {Z} $ such that $\left\|{\frac {1}{\|V\|}}\sum _{x\in V}f(x)e\left(-P(x)\right)\right\|\geq c,$ where $e(x):=e^{2\pi ix}$. This conjecture was proved to be true by Bergelson, Tao, and Ziegler.[3][4][5] The Inverse Conjecture for Gowers $U^{d}[N]$ norm asserts that for any $\delta >0$, a finite collection of (d − 1)-step nilmanifolds ${\mathcal {M}}_{\delta }$ and constants $c,C$ can be found, so that the following is true. If $N$ is a positive integer and $f\colon [N]\to \mathbb {C} $ is bounded in absolute value by 1 and $\Vert f\Vert _{U^{d}[N]}\geq \delta $, then there exists a nilmanifold $G/\Gamma \in {\mathcal {M}}_{\delta }$ and a nilsequence $F(g^{n}x)$ where $g\in G,\ x\in G/\Gamma $ and $F\colon G/\Gamma \to \mathbb {C} $ bounded by 1 in absolute value and with Lipschitz constant bounded by $C$ such that: $\left\|{\frac {1}{N}}\sum _{n=0}^{N-1}f(n){\overline {F(g^{n}x}})\right\|\geq c.$ This conjecture was proved to be true by Green, Tao, and Ziegler.[6][7] It should be stressed that the appearance of nilsequences in the above statement is necessary. The statement is no longer true if we only consider polynomial phases. References 1. Hartnett, Kevin. "Mathematicians Catch a Pattern by Figuring Out How to Avoid It". Quanta Magazine. Retrieved 2019-11-26. 2. Gowers, Timothy (2001). "A new proof of Szemerédi's theorem". Geom. Funct. Anal. 11 (3): 465–588. doi:10.1007/s00039-001-0332-9. MR 1844079. 3. Bergelson, Vitaly; Tao, Terence; Ziegler, Tamar (2010). "An inverse theorem for the uniformity seminorms associated with the action of $\mathbb {F} _{p}^{\infty }$". Geom. Funct. Anal. 19 (6): 1539–1596. arXiv:0901.2602. doi:10.1007/s00039-010-0051-1. MR 2594614. 4. Tao, Terence; Ziegler, Tamar (2010). "The inverse conjecture for the Gowers norm over finite fields via the correspondence principle". Analysis & PDE. 3 (1): 1–20. arXiv:0810.5527. doi:10.2140/apde.2010.3.1. MR 2663409. 5. Tao, Terence; Ziegler, Tamar (2011). "The Inverse Conjecture for the Gowers Norm over Finite Fields in Low Characteristic". Annals of Combinatorics. 16: 121–188. arXiv:1101.1469. doi:10.1007/s00026-011-0124-3. MR 2948765. 6. Green, Ben; Tao, Terence; Ziegler, Tamar (2011). "An inverse theorem for the Gowers $U^{s+1}[N]$-norm". Electron. Res. Announc. Math. Sci. 18: 69–90. arXiv:1006.0205. doi:10.3934/era.2011.18.69. MR 2817840. 7. Green, Ben; Tao, Terence; Ziegler, Tamar (2012). "An inverse theorem for the Gowers $U^{s+1}[N]$-norm". Annals of Mathematics. 176 (2): 1231–1372. arXiv:1009.3998. doi:10.4007/annals.2012.176.2.11. MR 2950773. • Tao, Terence (2012). Higher order Fourier analysis. Graduate Studies in Mathematics. Vol. 142. Providence, RI: American Mathematical Society. ISBN 978-0-8218-8986-2. MR 2931680. Zbl 1277.11010.	Wikipedia
Uniformizable space In mathematics, a topological space X is uniformizable if there exists a uniform structure on X that induces the topology of X. Equivalently, X is uniformizable if and only if it is homeomorphic to a uniform space (equipped with the topology induced by the uniform structure). Any (pseudo)metrizable space is uniformizable since the (pseudo)metric uniformity induces the (pseudo)metric topology. The converse fails: There are uniformizable spaces that are not (pseudo)metrizable. However, it is true that the topology of a uniformizable space can always be induced by a family of pseudometrics; indeed, this is because any uniformity on a set X can be defined by a family of pseudometrics. Showing that a space is uniformizable is much simpler than showing it is metrizable. In fact, uniformizability is equivalent to a common separation axiom: A topological space is uniformizable if and only if it is completely regular. Induced uniformity One way to construct a uniform structure on a topological space X is to take the initial uniformity on X induced by C(X), the family of real-valued continuous functions on X. This is the coarsest uniformity on X for which all such functions are uniformly continuous. A subbase for this uniformity is given by the set of all entourages $D_{f,\varepsilon }=\{(x,y)\in X\times X:\|f(x)-f(y)\|<\varepsilon \}$ where f ∈ C(X) and ε > 0. The uniform topology generated by the above uniformity is the initial topology induced by the family C(X). In general, this topology will be coarser than the given topology on X. The two topologies will coincide if and only if X is completely regular. Fine uniformity Given a uniformizable space X there is a finest uniformity on X compatible with the topology of X called the fine uniformity or universal uniformity. A uniform space is said to be fine if it has the fine uniformity generated by its uniform topology. The fine uniformity is characterized by the universal property: any continuous function f from a fine space X to a uniform space Y is uniformly continuous. This implies that the functor F : CReg → Uni that assigns to any completely regular space X the fine uniformity on X is left adjoint to the forgetful functor sending a uniform space to its underlying completely regular space. Explicitly, the fine uniformity on a completely regular space X is generated by all open neighborhoods D of the diagonal in X × X (with the product topology) such that there exists a sequence D1, D2, … of open neighborhoods of the diagonal with D = D1 and $D_{n}\circ D_{n}\subseteq D_{n-1}$. The uniformity on a completely regular space X induced by C(X) (see the previous section) is not always the fine uniformity. References • Willard, Stephen (1970). General Topology. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6.	Wikipedia
Uniformization (probability theory) In probability theory, uniformization method, (also known as Jensen's method[1] or the randomization method[2]) is a method to compute transient solutions of finite state continuous-time Markov chains, by approximating the process by a discrete-time Markov chain.[2] The original chain is scaled by the fastest transition rate γ, so that transitions occur at the same rate in every state, hence the name. The method is simple to program and efficiently calculates an approximation to the transient distribution at a single point in time (near zero).[1] The method was first introduced by Winfried Grassmann in 1977.[3][4][5] Method description For a continuous-time Markov chain with transition rate matrix Q, the uniformized discrete-time Markov chain has probability transition matrix $P:=(p_{ij})_{i,j}$, which is defined by[1][6][7] $p_{ij}={\begin{cases}q_{ij}/\gamma &{\text{ if }}i\neq j\\1-\sum _{j\neq i}q_{ij}/\gamma &{\text{ if }}i=j\end{cases}}$ with γ, the uniform rate parameter, chosen such that $\gamma \geq \max _{i}\|q_{ii}\|.$ In matrix notation: $P=I+{\frac {1}{\gamma }}Q.$ For a starting distribution π(0), the distribution at time t, π(t) is computed by[1] $\pi (t)=\sum _{n=0}^{\infty }\pi (0)P^{n}{\frac {(\gamma t)^{n}}{n!}}e^{-\gamma t}.$ This representation shows that a continuous-time Markov chain can be described by a discrete Markov chain with transition matrix P as defined above where jumps occur according to a Poisson process with intensity γt. In practice this series is terminated after finitely many terms. Implementation Pseudocode for the algorithm is included in Appendix A of Reibman and Trivedi's 1988 paper.[8] Using a parallel version of the algorithm, chains with state spaces of larger than 107 have been analysed.[9] Limitations Reibman and Trivedi state that "uniformization is the method of choice for typical problems," though they note that for stiff problems some tailored algorithms are likely to perform better.[8] External links • Matlab implementation Notes 1. Stewart, William J. (2009). Probability, Markov chains, queues, and simulation: the mathematical basis of performance modeling. Princeton University Press. p. 361. ISBN 0-691-14062-6. 2. Ibe, Oliver C. (2009). Markov processes for stochastic modeling. Academic Press. p. 98. ISBN 0-12-374451-2. 3. Gross, D.; Miller, D. R. (1984). "The Randomization Technique as a Modeling Tool and Solution Procedure for Transient Markov Processes". Operations Research. 32 (2): 343–361. doi:10.1287/opre.32.2.343. 4. Grassmann, W. K. (1977). "Transient solutions in markovian queueing systems". Computers & Operations Research. 4: 47–00. doi:10.1016/0305-0548(77)90007-7. 5. Grassmann, W. K. (1977). "Transient solutions in Markovian queues". European Journal of Operational Research. 1 (6): 396–402. doi:10.1016/0377-2217(77)90049-2. 6. Cassandras, Christos G.; Lafortune, Stéphane (2008). Introduction to discrete event systems. Springer. ISBN 0-387-33332-0. 7. Ross, Sheldon M. (2007). Introduction to probability models. Academic Press. ISBN 0-12-598062-0. 8. Reibman, A.; Trivedi, K. (1988). "Numerical transient analysis of markov models" (PDF). Computers & Operations Research. 15: 19. doi:10.1016/0305-0548(88)90026-3. 9. Dingle, N.; Harrison, P. G.; Knottenbelt, W. J. (2004). "Uniformization and hypergraph partitioning for the distributed computation of response time densities in very large Markov models". Journal of Parallel and Distributed Computing. 64 (8): 908–920. doi:10.1016/j.jpdc.2004.03.017. hdl:10044/1/5771.	Wikipedia
Uniformization (set theory) In set theory, a branch of mathematics, the axiom of uniformization is a weak form of the axiom of choice. It states that if $R$ is a subset of $X\times Y$, where $X$ and $Y$ are Polish spaces, then there is a subset $f$ of $R$ that is a partial function from $X$ to $Y$, and whose domain (the set of all $x$ such that $f(x)$ exists) equals $\{x\in X\mid \exists y\in Y:(x,y)\in R\}\,$ Such a function is called a uniformizing function for $R$, or a uniformization of $R$. To see the relationship with the axiom of choice, observe that $R$ can be thought of as associating, to each element of $X$, a subset of $Y$. A uniformization of $R$ then picks exactly one element from each such subset, whenever the subset is non-empty. Thus, allowing arbitrary sets X and Y (rather than just Polish spaces) would make the axiom of uniformization equivalent to the axiom of choice. A pointclass ${\boldsymbol {\Gamma }}$ is said to have the uniformization property if every relation $R$ in ${\boldsymbol {\Gamma }}$ can be uniformized by a partial function in ${\boldsymbol {\Gamma }}$. The uniformization property is implied by the scale property, at least for adequate pointclasses of a certain form. It follows from ZFC alone that ${\boldsymbol {\Pi }}_{1}^{1}$ and ${\boldsymbol {\Sigma }}_{2}^{1}$ have the uniformization property. It follows from the existence of sufficient large cardinals that • ${\boldsymbol {\Pi }}_{2n+1}^{1}$ and ${\boldsymbol {\Sigma }}_{2n+2}^{1}$ have the uniformization property for every natural number $n$. • Therefore, the collection of projective sets has the uniformization property. • Every relation in L(R) can be uniformized, but not necessarily by a function in L(R). In fact, L(R) does not have the uniformization property (equivalently, L(R) does not satisfy the axiom of uniformization). • (Note: it's trivial that every relation in L(R) can be uniformized in V, assuming V satisfies the axiom of choice. The point is that every such relation can be uniformized in some transitive inner model of V in which the axiom of determinacy holds.) References • Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.	Wikipedia
Discrete valuation ring In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions: 1. R is a local principal ideal domain, and not a field. 2. R is a valuation ring with a value group isomorphic to the integers under addition. 3. R is a local Dedekind domain and not a field. 4. R is a Noetherian local domain whose maximal ideal is principal, and not a field.[1] 5. R is an integrally closed Noetherian local ring with Krull dimension one. 6. R is a principal ideal domain with a unique non-zero prime ideal. 7. R is a principal ideal domain with a unique irreducible element (up to multiplication by units). 8. R is a unique factorization domain with a unique irreducible element (up to multiplication by units). 9. R is Noetherian, not a field, and every nonzero fractional ideal of R is irreducible in the sense that it cannot be written as a finite intersection of fractional ideals properly containing it. 10. There is some discrete valuation ν on the field of fractions K of R such that R = {0} $\cup $ {x $\in $ K : ν(x) ≥ 0}. Examples Localization of Dedekind rings Let $\mathbb {Z} _{(2)}:=\{z/n\mid z,n\in \mathbb {Z} ,\,\,n{\text{ is odd}}\}$. Then, the field of fractions of $\mathbb {Z} _{(2)}$ is $\mathbb {Q} $. For any nonzero element $r$ of $\mathbb {Q} $, we can apply unique factorization to the numerator and denominator of r to write r as 2k z/n where z, n, and k are integers with z and n odd. In this case, we define ν(r)=k. Then $\mathbb {Z} _{(2)}$ is the discrete valuation ring corresponding to ν. The maximal ideal of $\mathbb {Z} _{(2)}$ is the principal ideal generated by 2, i.e. $2\mathbb {Z} _{(2)}$, and the "unique" irreducible element (up to units) is 2 (this is also known as a uniformizing parameter). Note that $\mathbb {Z} _{(2)}$ is the localization of the Dedekind domain $\mathbb {Z} $ at the prime ideal generated by 2. More generally, any localization of a Dedekind domain at a non-zero prime ideal is a discrete valuation ring; in practice, this is frequently how discrete valuation rings arise. In particular, we can define rings $\mathbb {Z} _{(p)}:=\left.\left\{{\frac {z}{n}}\,\right\|z,n\in \mathbb {Z} ,p\nmid n\right\}$ for any prime p in complete analogy. p-adic integers The ring $\mathbb {Z} _{p}$ of p-adic integers is a DVR, for any prime $p$. Here $p$ is an irreducible element; the valuation assigns to each $p$-adic integer $x$ the largest integer $k$ such that $p^{k}$ divides $x$. Formal power series Another important example of a DVR is the ring of formal power series $R=k[[T]]$ in one variable $T$ over some field $k$. The "unique" irreducible element is $T$, the maximal ideal of $R$ is the principal ideal generated by $T$, and the valuation $\nu $ assigns to each power series the index (i.e. degree) of the first non-zero coefficient. If we restrict ourselves to real or complex coefficients, we can consider the ring of power series in one variable that converge in a neighborhood of 0 (with the neighborhood depending on the power series). This is a discrete valuation ring. This is useful for building intuition with the Valuative criterion of properness. Ring in function field For an example more geometrical in nature, take the ring R = {f/g : f, g polynomials in R[X] and g(0) ≠ 0}, considered as a subring of the field of rational functions R(X) in the variable X. R can be identified with the ring of all real-valued rational functions defined (i.e. finite) in a neighborhood of 0 on the real axis (with the neighborhood depending on the function). It is a discrete valuation ring; the "unique" irreducible element is X and the valuation assigns to each function f the order (possibly 0) of the zero of f at 0. This example provides the template for studying general algebraic curves near non-singular points, the algebraic curve in this case being the real line. Henselian trait For a DVR $R$ it is common to write the fraction field as $K={\text{Frac}}(R)$ and $\kappa =R/{\mathfrak {m}}$ the residue field. These correspond to the generic and closed points of $S={\text{Spec}}(R).$ For example, the closed point of ${\text{Spec}}(\mathbb {Z} _{p})$ is $\mathbb {F} _{p}$ and the generic point is $\mathbb {Q} _{p}$. Sometimes this is denoted as $\eta \to S\leftarrow s$ where $\eta $ is the generic point and $s$ is the closed point . Localization of a point on a curve Given an algebraic curve $(X,{\mathcal {O}}_{X})$, the local ring ${\mathcal {O}}_{X,{\mathfrak {p}}}$ at a smooth point ${\mathfrak {p}}$ is a discrete valuation ring, because it is a principal valuation ring. Note because the point ${\mathfrak {p}}$ is smooth, the completion of the local ring is isomorphic to the completion of the localization of $\mathbb {A} ^{1}$ at some point ${\mathfrak {q}}$. Uniformizing parameter Given a DVR R, any irreducible element of R is a generator for the unique maximal ideal of R and vice versa. Such an element is also called a uniformizing parameter of R (or a uniformizing element, a uniformizer, or a prime element). If we fix a uniformizing parameter t, then M=(t) is the unique maximal ideal of R, and every other non-zero ideal is a power of M, i.e. has the form (t k) for some k≥0. All the powers of t are distinct, and so are the powers of M. Every non-zero element x of R can be written in the form αt k with α a unit in R and k≥0, both uniquely determined by x. The valuation is given by ν(x) = kv(t). So to understand the ring completely, one needs to know the group of units of R and how the units interact additively with the powers of t. The function v also makes any discrete valuation ring into a Euclidean domain. Topology Every discrete valuation ring, being a local ring, carries a natural topology and is a topological ring. We can also give it a metric space structure where the distance between two elements x and y can be measured as follows: $\|x-y\|=2^{-\nu (x-y)}$ (or with any other fixed real number > 1 in place of 2). Intuitively: an element z is "small" and "close to 0" iff its valuation ν(z) is large. The function \|x-y\|, supplemented by \|0\|=0, is the restriction of an absolute value defined on the field of fractions of the discrete valuation ring. A DVR is compact if and only if it is complete and its residue field R/M is a finite field. Examples of complete DVRs include • the ring of p-adic integers and • the ring of formal power series over any field For a given DVR, one often passes to its completion, a complete DVR containing the given ring that is often easier to study. This completion procedure can be thought of in a geometrical way as passing from rational functions to power series, or from rational numbers to the reals. Returning to our examples: the ring of all formal power series in one variable with real coefficients is the completion of the ring of rational functions defined (i.e. finite) in a neighborhood of 0 on the real line; it is also the completion of the ring of all real power series that converge near 0. The completion of $\mathbb {Z} _{(p)}=\mathbb {Q} \cap \mathbb {Z} _{p}$ (which can be seen as the set of all rational numbers that are p-adic integers) is the ring of all p-adic integers Zp. See also • Category:Localization (mathematics) • Local ring • Ramification of local fields • Cohen ring • Valuation ring References 1. "ac.commutative algebra - Condition for a local ring whose maximal ideal is principal to be Noetherian". MathOverflow. • Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8 • Dummit, David S.; Foote, Richard M. (2004), Abstract algebra (3rd ed.), New York: John Wiley & Sons, ISBN 978-0-471-43334-7, MR 2286236 • Discrete valuation ring, The Encyclopaedia of Mathematics.	Wikipedia
Uniformly Cauchy sequence In mathematics, a sequence of functions $\{f_{n}\}$ from a set S to a metric space M is said to be uniformly Cauchy if: • For all $\varepsilon >0$, there exists $N>0$ such that for all $x\in S$: $d(f_{n}(x),f_{m}(x))<\varepsilon $ whenever $m,n>N$. Another way of saying this is that $d_{u}(f_{n},f_{m})\to 0$ as $m,n\to \infty $, where the uniform distance $d_{u}$ between two functions is defined by $d_{u}(f,g):=\sup _{x\in S}d(f(x),g(x)).$ Convergence criteria A sequence of functions {fn} from S to M is pointwise Cauchy if, for each x ∈ S, the sequence {fn(x)} is a Cauchy sequence in M. This is a weaker condition than being uniformly Cauchy. In general a sequence can be pointwise Cauchy and not pointwise convergent, or it can be uniformly Cauchy and not uniformly convergent. Nevertheless, if the metric space M is complete, then any pointwise Cauchy sequence converges pointwise to a function from S to M. Similarly, any uniformly Cauchy sequence will tend uniformly to such a function. The uniform Cauchy property is frequently used when the S is not just a set, but a topological space, and M is a complete metric space. The following theorem holds: • Let S be a topological space and M a complete metric space. Then any uniformly Cauchy sequence of continuous functions fn : S → M tends uniformly to a unique continuous function f : S → M. Generalization to uniform spaces A sequence of functions $\{f_{n}\}$ from a set S to a uniform space U is said to be uniformly Cauchy if: • For all $x\in S$ and for any entourage $\varepsilon $, there exists $N>0$ such that $d(f_{n}(x),f_{m}(x))<\varepsilon $ whenever $m,n>N$. See also • Modes of convergence (annotated index)	Wikipedia
Uniform continuity In mathematics, a real function $f$ of real numbers is said to be uniformly continuous if there is a positive real number $\delta $ such that function values over any function domain interval of the size $\delta $ are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number $\epsilon $, then there is a positive real number $\delta $ such that $\|f(x)-f(y)\|<\epsilon $ at any $x$ and $y$ in any function interval of the size $\delta $. The difference between uniform continuity and (ordinary) continuity is that, in uniform continuity there is a globally applicable $\delta $ (the size of a function domain interval over which function value differences are less than $\epsilon $) that depends on only $\varepsilon $, while in (ordinary) continuity there is a locally applicable $\delta $ that depends on the both $\varepsilon $ and $x$. So uniform continuity is a stronger continuity condition than continuity; a function that is uniformly continuous is continuous but a function that is continuous is not necessarily uniformly continuous. The concepts of uniform continuity and continuity can be expanded to functions defined between metric spaces. Continuous functions can fail to be uniformly continuous if they are unbounded on a bounded domain, such as $f(x)={\tfrac {1}{x}}$ on $(0,1)$, or if their slopes become unbounded on an infinite domain, such as $f(x)=x^{2}$ on the real (number) line. However, any Lipschitz map between metric spaces is uniformly continuous, in particular any isometry (distance-preserving map). Although continuity can be defined for functions between general topological spaces, defining uniform continuity requires more structure. The concept relies on comparing the sizes of neighbourhoods of distinct points, so it requires a metric space, or more generally a uniform space. Definition for functions on metric spaces For a function $f:X\to Y$ with metric spaces $(X,d_{1})$ and $(Y,d_{2})$, the following definitions of uniform continuity and (ordinary) continuity hold. Definition of uniform continuity • $f$ is called uniformly continuous if for every real number $\varepsilon >0$ there exists a real number $\delta >0$ such that for every $x,y\in X$ with $d_{1}(x,y)<\delta $, we have $d_{2}(f(x),f(y))<\varepsilon $. The set $\{y\in X:d_{1}(x,y)<\delta \}$ for each $x$ is a neighbourhood of $x$ and the set $\{x\in X:d_{1}(x,y)<\delta \}$ for each $y$ is a neighbourhood of $y$ by the definition of a neighbourhood in a metric space. • If $X$ and $Y$ are subsets of the real line, then $d_{1}$ and $d_{2}$ can be the standard one-dimensional Euclidean distance, yielding the following definition: for every real number $\varepsilon >0$ there exists a real number $\delta >0$ such that for every $x,y\in X$, $\|x-y\|<\delta \implies \|f(x)-f(y)\|<\varepsilon $ (where $A\implies B$ is a material conditional statement saying "if $A$, then $B$"). • Equivalently, $f$ is said to be uniformly continuous if $\forall \varepsilon >0\;\exists \delta >0\;\forall x\in X\;\forall y\in X:\,d_{1}(x,y)<\delta \,\Rightarrow \,d_{2}(f(x),f(y))<\varepsilon $. Here quantifications ($\forall \varepsilon >0$, $\exists \delta >0$, $\forall x\in X$, and $\forall y\in X$) are used. • Alternatively, $f$ is said to be uniformly continuous if there is a function of all positive real numbers $\varepsilon $, $\delta (\varepsilon )$ representing the maximum positive real number, such that for every $x,y\in X$ if $d_{1}(x,y)<\delta (\varepsilon )$ then $d_{2}(f(x),f(y))<\varepsilon $. $\delta (\varepsilon )$ is a monotonically non-decreasing function. Definition of (ordinary) continuity • $f$ is called continuous ${\underline {{\text{at }}x}}$ if for every real number $\varepsilon >0$ there exists a real number $\delta >0$ such that for every $y\in X$ with $d_{1}(x,y)<\delta $, we have $d_{2}(f(x),f(y))<\varepsilon $. The set $\{y\in X:d_{1}(x,y)<\delta \}$ is a neighbourhood of $x$. Thus, (ordinary) continuity is a local property of the function at the point $x$. • Equivalently, a function $f$ is said to be continuous if $\forall x\in X\;\forall \varepsilon >0\;\exists \delta >0\;\forall y\in X:\,d_{1}(x,y)<\delta \,\Rightarrow \,d_{2}(f(x),f(y))<\varepsilon $. • Alternatively, a function $f$ is said to be continuous if there is a function of all positive real numbers $\varepsilon $ and $x\in X$, $\delta (\varepsilon ,x)$ representing the maximum positive real number, such that at each $x$ if $y\in X$ satisfies $d_{1}(x,y)<\delta (\varepsilon ,x)$ then $d_{2}(f(x),f(y))<\varepsilon $. At every $x$, $\delta (\varepsilon ,x)$ is a monotonically non-decreasing function. Local continuity versus global uniform continuity In the definitions, the difference between uniform continuity and continuity is that, in uniform continuity there is a globally applicable $\delta $ (the size of a neighbourhood in $X$ over which values of the metric for function values in $Y$ are less than $\varepsilon $) that depends on only $\varepsilon $ while in continuity there is a locally applicable $\delta $ that depends on the both $\varepsilon $ and $x$. Continuity is a local property of a function — that is, a function $f$ is continuous, or not, at a particular point $x$ of the function domain $X$, and this can be determined by looking at only the values of the function in an arbitrarily small neighbourhood of that point. When we speak of a function being continuous on an interval, we mean that the function is continuous at every point of the interval. In contrast, uniform continuity is a global property of $f$, in the sense that the standard definition of uniform continuity refers to every point of $X$. On the other hand, it is possible to give a definition that is local in terms of the natural extension $f^{}$(the characteristics of which at nonstandard points are determined by the global properties of $f$), although it is not possible to give a local definition of uniform continuity for an arbitrary hyperreal-valued function, see below. A mathematical definition that a function $f$ is continuous on an interval $I$ and a definition that $f$ is uniformly continuous on $I$ are structurally similar as shown in the following. Continuity of a function $f:X\to Y$ for metric spaces $(X,d_{1})$ and $(Y,d_{2})$ at every point $x$ of an interval $I\subseteq X$ (i.e., continuity of $f$ on the interval $I$) is expressed by a formula starting with quantifications $\forall x\in I\;\forall \varepsilon >0\;\exists \delta >0\;\forall y\in I:\,d_{1}(x,y)<\delta \,\Rightarrow \,d_{2}(f(x),f(y))<\varepsilon $, (metrics $d_{1}(x,y)$ and $d_{2}(f(x),f(y))$ are $\|x-y\|$ and $\|f(x)-f(y)\|$ for $f:\mathbb {R} \to \mathbb {R} $ for the set of real numbers $\mathbb {R} $). For uniform continuity, the order of the first, second, and third quantifications ($\forall x\in I$, $\forall \varepsilon >0$, and $\exists \delta >0$) are rotated: $\forall \varepsilon >0\;\exists \delta >0\;\forall x\in I\;\forall y\in I:\,d_{1}(x,y)<\delta \,\Rightarrow \,d_{2}(f(x),f(y))<\varepsilon $. Thus for continuity on the interval, one takes an arbitrary point $x$ of the interval, and then there must exist a distance $\delta $, $\cdots \forall x\,\exists \delta \cdots ,$ while for uniform continuity, a single $\delta $ must work uniformly for all points $x$ of the interval, $\cdots \exists \delta \,\forall x\cdots .$ Properties Every uniformly continuous function is continuous, but the converse does not hold. Consider for instance the continuous function $f\colon \mathbb {R} \rightarrow \mathbb {R} ,x\mapsto x^{2}$ where $\mathbb {R} $ is the set of real numbers. Given a positive real number $\varepsilon $, uniform continuity requires the existence of a positive real number $\delta $ such that for all $x_{1},x_{2}\in \mathbb {R} $ with $\|x_{1}-x_{2}\|<\delta $, we have $\|f(x_{1})-f(x_{2})\|<\varepsilon $. But $f\left(x+\delta \right)-f(x)=2x\cdot \delta +\delta ^{2},$ and as $x$ goes to be a higher and higher value, $\delta $ needs to be lower and lower to satisfy $\|f(x+\beta )-f(x)\|<\varepsilon $ for positive real numbers $\beta <\delta $ and the given $\varepsilon $. This means that there is no specifiable (no matter how small it is) positive real number $\delta $ to satisfy the condition for $f$ to be uniformly continuous so $f$ is not uniformly continuous. Any absolutely continuous function (over a compact interval) is uniformly continuous. On the other hand, the Cantor function is uniformly continuous but not absolutely continuous. The image of a totally bounded subset under a uniformly continuous function is totally bounded. However, the image of a bounded subset of an arbitrary metric space under a uniformly continuous function need not be bounded: as a counterexample, consider the identity function from the integers endowed with the discrete metric to the integers endowed with the usual Euclidean metric. The Heine–Cantor theorem asserts that every continuous function on a compact set is uniformly continuous. In particular, if a function is continuous on a closed bounded interval of the real line, it is uniformly continuous on that interval. The Darboux integrability of continuous functions follows almost immediately from this theorem. If a real-valued function $f$ is continuous on $[0,\infty )$ and $\lim _{x\to \infty }f(x)$ exists (and is finite), then $f$ is uniformly continuous. In particular, every element of $C_{0}(\mathbb {R} )$, the space of continuous functions on $\mathbb {R} $ that vanish at infinity, is uniformly continuous. This is a generalization of the Heine-Cantor theorem mentioned above, since $C_{c}(\mathbb {R} )\subset C_{0}(\mathbb {R} )$. Examples and nonexamples Examples • Linear functions $x\mapsto ax+b$ are the simplest examples of uniformly continuous functions. • Any continuous function on the interval $[0,1]$ is also uniformly continuous, since $[0,1]$ is a compact set. • If a function is differentiable on an open interval and its derivative is bounded, then the function is uniformly continuous on that interval. • Every Lipschitz continuous map between two metric spaces is uniformly continuous. More generally, every Hölder continuous function is uniformly continuous. • The absolute value function is uniformly continuous, despite not being differentiable at $x=0$. This shows uniformly continuous functions are not always differentiable. • Despite being nowhere differentiable, the Weierstrass function is uniformly continuous. • Every member of a uniformly equicontinuous set of functions is uniformly continuous. Nonexamples • Functions that are unbounded on a bounded domain are not uniformly continuous. The tangent function is continuous on the interval $(-\pi /2,\pi /2)$ but is not uniformly continuous on that interval, as it goes to infinity as $x\to \pi /2$. • Functions whose derivative tends to infinity as $x$ grows large cannot be uniformly continuous. The exponential function $x\mapsto e^{x}$ is continuous everywhere on the real line but is not uniformly continuous on the line, since its derivative is $e^{x}$, and $e^{x}\to \infty $ as $x\to \infty $. Visualization For a uniformly continuous function, for every positive real number $\varepsilon >0$ there is a positive real number $\delta >0$ such that two function values $f(x)$ and $f(y)$ have the maximum distance $\varepsilon $ whenever $x$ and $y$ are within the maximum distance $\delta $. Thus at each point $(x,f(x))$ of the graph, if we draw a rectangle with a height slightly less than $2\varepsilon $ and width a slightly less than $2\delta $ around that point, then the graph lies completely within the height of the rectangle, i.e., the graph do not pass through the top or the bottom side of the rectangle. For functions that are not uniformly continuous, this isn't possible; for these functions, the graph might lie inside the height of the rectangle at some point on the graph but there is a point on the graph where the graph lies above or below the rectangle. (the graph penetrates the top or bottom side of the rectangle.) • For uniformly continuous functions, for each positive real number $\varepsilon >0$ there is a positive real number $\delta >0$ such that when we draw a rectangle around each point of the graph with a width slightly less than $2\delta $ and a height slightly less than $2\varepsilon $, the graph lies completely inside the height of the rectangle. • For functions that are not uniformly continuous, there is a positive real number $\varepsilon >0$ such that for every positive real number $\delta >0$ there is a point on the graph so that when we draw a rectangle with a height slightly less than $2\varepsilon $ and a width slightly less than $2\delta $ around that point, there is a function value directly above or below the rectangle. There might be a graph point where the graph is completely inside the height of the rectangle but this is not true for every point of the graph. History The first published definition of uniform continuity was by Heine in 1870, and in 1872 he published a proof that a continuous function on an open interval need not be uniformly continuous. The proofs are almost verbatim given by Dirichlet in his lectures on definite integrals in 1854. The definition of uniform continuity appears earlier in the work of Bolzano where he also proved that continuous functions on an open interval do not need to be uniformly continuous. In addition he also states that a continuous function on a closed interval is uniformly continuous, but he does not give a complete proof.[1] Other characterizations Non-standard analysis In non-standard analysis, a real-valued function $f$ of a real variable is microcontinuous at a point $a$ precisely if the difference $f^{}(a+\delta )-f^{}(a)$ is infinitesimal whenever $\delta $ is infinitesimal. Thus $f$ is continuous on a set $A$ in $\mathbb {R} $ precisely if $f^{}$ is microcontinuous at every real point $a\in A$. Uniform continuity can be expressed as the condition that (the natural extension of) $f$ is microcontinuous not only at real points in $A$, but at all points in its non-standard counterpart (natural extension) $^{}A$ in $^{}\mathbb {R} $. Note that there exist hyperreal-valued functions which meet this criterion but are not uniformly continuous, as well as uniformly continuous hyperreal-valued functions which do not meet this criterion, however, such functions cannot be expressed in the form $f^{*}$ for any real-valued function $f$. (see non-standard calculus for more details and examples). Cauchy continuity For a function between metric spaces, uniform continuity implies Cauchy continuity (Fitzpatrick 2006). More specifically, let $A$ be a subset of $\mathbb {R} ^{n}$. If a function $f:A\to \mathbb {R} ^{n}$ is uniformly continuous then for every pair of sequences $x_{n}$ and $y_{n}$ such that $\lim _{n\to \infty }\|x_{n}-y_{n}\|=0$ we have $\lim _{n\to \infty }\|f(x_{n})-f(y_{n})\|=0.$ Relations with the extension problem Let $X$ be a metric space, $S$ a subset of $X$, $R$ a complete metric space, and $f:S\rightarrow R$ a continuous function. A question to answer: When can $f$ be extended to a continuous function on all of $X$? If $S$ is closed in $X$, the answer is given by the Tietze extension theorem. So it is necessary and sufficient to extend $f$ to the closure of $S$ in $X$: that is, we may assume without loss of generality that $S$ is dense in $X$, and this has the further pleasant consequence that if the extension exists, it is unique. A sufficient condition for $f$ to extend to a continuous function $f:X\rightarrow R$ is that it is Cauchy-continuous, i.e., the image under $f$ of a Cauchy sequence remains Cauchy. If $X$ is complete (and thus the completion of $S$), then every continuous function from $X$ to a metric space $Y$ is Cauchy-continuous. Therefore when $X$ is complete, $f$ extends to a continuous function $f:X\rightarrow R$ if and only if $f$ is Cauchy-continuous. It is easy to see that every uniformly continuous function is Cauchy-continuous and thus extends to $X$. The converse does not hold, since the function $f:R\rightarrow R,x\mapsto x^{2}$ is, as seen above, not uniformly continuous, but it is continuous and thus Cauchy continuous. In general, for functions defined on unbounded spaces like $R$, uniform continuity is a rather strong condition. It is desirable to have a weaker condition from which to deduce extendability. For example, suppose $a>1$ is a real number. At the precalculus level, the function $f:x\mapsto a^{x}$ can be given a precise definition only for rational values of $x$ (assuming the existence of qth roots of positive real numbers, an application of the Intermediate Value Theorem). One would like to extend $f$ to a function defined on all of $R$. The identity $f(x+\delta )-f(x)=a^{x}\left(a^{\delta }-1\right)$ shows that $f$ is not uniformly continuous on the set $Q$ of all rational numbers; however for any bounded interval $I$ the restriction of $f$ to $Q\cap I$ is uniformly continuous, hence Cauchy-continuous, hence $f$ extends to a continuous function on $I$. But since this holds for every $I$, there is then a unique extension of $f$ to a continuous function on all of $R$. More generally, a continuous function $f:S\rightarrow R$ whose restriction to every bounded subset of $S$ is uniformly continuous is extendable to $X$, and the converse holds if $X$ is locally compact. A typical application of the extendability of a uniformly continuous function is the proof of the inverse Fourier transformation formula. We first prove that the formula is true for test functions, there are densely many of them. We then extend the inverse map to the whole space using the fact that linear map is continuous; thus, uniformly continuous. Generalization to topological vector spaces In the special case of two topological vector spaces $V$ and $W$, the notion of uniform continuity of a map $f:V\to W$ becomes: for any neighborhood $B$ of zero in $W$, there exists a neighborhood $A$ of zero in $V$ such that $v_{1}-v_{2}\in A$ implies $f(v_{1})-f(v_{2})\in B.$ For linear transformations $f:V\to W$, uniform continuity is equivalent to continuity. This fact is frequently used implicitly in functional analysis to extend a linear map off a dense subspace of a Banach space. Generalization to uniform spaces Just as the most natural and general setting for continuity is topological spaces, the most natural and general setting for the study of uniform continuity are the uniform spaces. A function $f:X\to Y$ between uniform spaces is called uniformly continuous if for every entourage $V$ in $Y$ there exists an entourage $U$ in $X$ such that for every $(x_{1},x_{2})$ in $U$ we have $(f(x_{1}),f(x_{2}))$ in $V$. In this setting, it is also true that uniformly continuous maps transform Cauchy sequences into Cauchy sequences. Each compact Hausdorff space possesses exactly one uniform structure compatible with the topology. A consequence is a generalization of the Heine-Cantor theorem: each continuous function from a compact Hausdorff space to a uniform space is uniformly continuous. See also • Contraction mapping – Function reducing distance between all points • Uniform isomorphism – Uniformly continuous homeomorphism References 1. Rusnock & Kerr-Lawson 2005. Further reading • Bourbaki, Nicolas. General Topology: Chapters 1–4 [Topologie Générale]. ISBN 0-387-19374-X. Chapter II is a comprehensive reference of uniform spaces. • Dieudonné, Jean (1960). Foundations of Modern Analysis. Academic Press. • Fitzpatrick, Patrick (2006). Advanced Calculus. Brooks/Cole. ISBN 0-534-92612-6. • Kelley, John L. (1955). General topology. Graduate Texts in Mathematics. Springer-Verlag. ISBN 0-387-90125-6. • Kudryavtsev, L.D. (2001) [1994], "Uniform continuity", Encyclopedia of Mathematics, EMS Press • Rudin, Walter (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. ISBN 978-0-07-054235-8. • Rusnock, P.; Kerr-Lawson, A. (2005), "Bolzano and uniform continuity", Historia Mathematica, 32 (3): 303–311, doi:10.1016/j.hm.2004.11.003	Wikipedia
Uniformly connected space In topology and related areas of mathematics a uniformly connected space or Cantor connected space is a uniform space U such that every uniformly continuous function from U to a discrete uniform space is constant. A uniform space U is called uniformly disconnected if it is not uniformly connected. Properties A compact uniform space is uniformly connected if and only if it is connected Examples • every connected space is uniformly connected • the rational numbers and the irrational numbers are disconnected but uniformly connected See also • connectedness References 1. Cantor, Georg Über Unendliche, lineare punktmannigfaltigkeiten, Mathematische Annalen. 21 (1883) 545-591.	Wikipedia
Uniformly disconnected space In mathematics, a uniformly disconnected space is a metric space $(X,d)$ for which there exists $\lambda >0$ such that no pair of distinct points $x,y\in X$ can be connected by a $\lambda $-chain. A $\lambda $-chain between $x$ and $y$ is a sequence of points $x=x_{0},x_{1},\ldots ,x_{n}=y$ in $X$ such that $d(x_{i},x_{i+1})\leq \lambda d(x,y),\forall i\in \{0,\ldots ,n\}$.[1] Properties Uniform disconnectedness is invariant under quasi-Möbius maps.[2] References 1. Heinonen, Juha (2001). Lectures on Analysis on Metric Spaces. Universitext. New York: Springer-Verlag. pp. x+140. ISBN 0-387-95104-0. 2. Heer, Loreno (2017-08-28). "Some Invariant Properties of Quasi-Möbius Maps". Analysis and Geometry in Metric Spaces. 5 (1): 69–77. arXiv:1603.07521. doi:10.1515/agms-2017-0004. ISSN 2299-3274.	Wikipedia
Delone set In the mathematical theory of metric spaces, ε-nets, ε-packings, ε-coverings, uniformly discrete sets, relatively dense sets, and Delone sets (named after Boris Delone) are several closely related definitions of well-spaced sets of points, and the packing radius and covering radius of these sets measure how well-spaced they are. These sets have applications in coding theory, approximation algorithms, and the theory of quasicrystals. Definitions If (M,d) is a metric space, and X is a subset of M, then the packing radius of X is half of the infimum of distances between distinct members of X. If the packing radius is r, then open balls of radius r centered at the points of X will all be disjoint from each other, and each open ball centered at one of the points of X with radius 2r will be disjoint from the rest of X. The covering radius of X is the infimum of the numbers r such that every point of M is within distance r of at least one point in X; that is, it is the smallest radius such that closed balls of that radius centered at the points of X have all of M as their union. An ε-packing is a set X of packing radius ≥ ε/2 (equivalently, minimum distance ≥ ε), an ε-covering is a set X of covering radius ≤ ε, and an ε-net is a set that is both an ε-packing and an ε-covering. A set is uniformly discrete if it has a nonzero packing radius, and relatively dense if it has a finite covering radius. A Delone set is a set that is both uniformly discrete and relatively dense; thus, every ε-net is Delone, but not vice versa.[1][2] Construction of ε-nets As the most restrictive of the definitions above, ε-nets are at least as difficult to construct as ε-packings, ε-coverings, and Delone sets. However, whenever the points of M have a well-ordering, transfinite induction shows that it is possible to construct an ε-net N, by including in N every point for which the infimum of distances to the set of earlier points in the ordering is at least ε. For finite sets of points in a Euclidean space of bounded dimension, each point may be tested in constant time by mapping it to a grid of cells of diameter ε, and using a hash table to test which nearby cells already contain points of N; thus, in this case, an ε-net can be constructed in linear time.[3][4] For more general finite or compact metric spaces, an alternative algorithm of Teo Gonzalez based on the farthest-first traversal can be used to construct a finite ε-net. This algorithm initializes the net N to be empty, and then repeatedly adds to N the farthest point in M from N, breaking ties arbitrarily and stopping when all points of M are within distance ε of N.[5] In spaces of bounded doubling dimension, Gonzalez' algorithm can be implemented in O(n log n) time for point sets with a polynomial ratio between their farthest and closest distances, and approximated in the same time bound for arbitrary point sets.[6] Applications Coding theory Main article: Hamming bound § Covering radius and packing radius In the theory of error-correcting codes, the metric space containing a block code C consists of strings of a fixed length, say n, taken over an alphabet of size q (can be thought of as vectors), with the Hamming metric. This space is denoted by $\scriptstyle {\mathcal {A}}_{q}^{n}$. The covering radius and packing radius of this metric space are related to the code's ability to correct errors. Approximation algorithms Har-Peled & Raichel (2013) describe an algorithmic paradigm that they call "net and prune" for designing approximation algorithms for certain types of geometric optimization problems defined on sets of points in Euclidean spaces. An algorithm of this type works by performing the following steps: 1. Choose a random point p from the point set, find its nearest neighbor q, and set ε to the distance between p and q. 2. Test whether ε is (approximately) bigger than or smaller than the optimal solution value (using a technique specific to the particular optimization problem being solved) • If it is bigger, remove from the input the points whose closest neighbor is farther than ε • If it is smaller, construct an ε-net N, and remove from the input the points that are not in N. In both cases, the expected number of remaining points decreases by a constant factor, so the time is dominated by the testing step. As they show, this paradigm can be used to construct fast approximation algorithms for k-center clustering, finding a pair of points with median distance, and several related problems. A hierarchical system of nets, called a net-tree, may be used in spaces of bounded doubling dimension to construct well-separated pair decompositions, geometric spanners, and approximate nearest neighbors.[6][7] Crystallography For points in Euclidean space, a set X is a Meyer set if it is relatively dense and its difference set X − X is uniformly discrete. Equivalently, X is a Meyer set if both X and X − X are Delone sets. Meyer sets are named after Yves Meyer, who introduced them (with a different but equivalent definition based on harmonic analysis) as a mathematical model for quasicrystals. They include the point sets of lattices, Penrose tilings, and the Minkowski sums of these sets with finite sets.[8] The Voronoi cells of symmetric Delone sets form space-filling polyhedra called plesiohedra.[9] See also • Danzer set References 1. Clarkson, Kenneth L. (2006), "Building triangulations using ε-nets", STOC'06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, New York: ACM, pp. 326–335, doi:10.1145/1132516.1132564, ISBN 1595931341, MR 2277158, S2CID 14132888. 2. Some sources use "ε-net" for what is here called an "ε-covering"; see, e.g. Sutherland, W. A. (1975), Introduction to metric and topological spaces, Oxford University Press, p. 110, ISBN 0-19-853161-3, Zbl 0304.54002. 3. Har-Peled, S. (2004), "Clustering motion", Discrete and Computational Geometry, 31 (4): 545–565, doi:10.1007/s00454-004-2822-7, MR 2053498. 4. Har-Peled, S.; Raichel, B. (2013), "Net and prune: A linear time algorithm for Euclidean distance problems", STOC'13: Proceedings of the 45th Annual ACM Symposium on Theory of Computing, pp. 605–614, arXiv:1409.7425. 5. Gonzalez, T. F. (1985), "Clustering to minimize the maximum intercluster distance", Theoretical Computer Science, 38 (2–3): 293–306, doi:10.1016/0304-3975(85)90224-5, MR 0807927. 6. Har-Peled, S.; Mendel, M. (2006), "Fast construction of nets in low-dimensional metrics, and their applications", SIAM Journal on Computing, 35 (5): 1148–1184, arXiv:cs/0409057, doi:10.1137/S0097539704446281, MR 2217141, S2CID 37346335. 7. Krauthgamer, Robert; Lee, James R. (2004), "Navigating nets: simple algorithms for proximity search", Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '04), Philadelphia, PA, USA: Society for Industrial and Applied Mathematics, pp. 798–807, ISBN 0-89871-558-X. 8. Moody, Robert V. (1997), "Meyer sets and their duals", The Mathematics of Long-Range Aperiodic Order (Waterloo, ON, 1995), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 489, Dordrecht: Kluwer Academic Publishers, pp. 403–441, MR 1460032. 9. Grünbaum, Branko; Shephard, G. C. (1980), "Tilings with congruent tiles", Bulletin of the American Mathematical Society, New Series, 3 (3): 951–973, doi:10.1090/S0273-0979-1980-14827-2, MR 0585178. External links • Delone set – Tilings Encyclopedia	Wikipedia
Uniformly distributed measure In mathematics — specifically, in geometric measure theory — a uniformly distributed measure on a metric space is one for which the measure of an open ball depends only on its radius and not on its centre. By convention, the measure is also required to be Borel regular, and to take positive and finite values on open balls of finite radius. Thus, if (X, d) is a metric space, a Borel regular measure μ on X is said to be uniformly distributed if $0<\mu (\mathbf {B} _{r}(x))=\mu (\mathbf {B} _{r}(y))<+\infty $ for all points x and y of X and all 0 < r < +∞, where $\mathbf {B} _{r}(x):=\{z\in X\|d(x,z)<r\}.$ Christensen's lemma As it turns out, uniformly distributed measures are very rigid objects. On any "decent" metric space, the uniformly distributed measures form a one-parameter linearly dependent family: Let μ and ν be uniformly distributed Borel regular measures on a separable metric space (X, d). Then there is a constant c such that μ = cν. References • Christensen, Jens Peter Reus (1970). "On some measures analogous to Haar measure". Mathematica Scandinavica. 26: 103–106. ISSN 0025-5521. MR0260979 • Mattila, Pertti (1995). Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability. Cambridge Studies in Advanced Mathematics No. 44. Cambridge: Cambridge University Press. pp. xii+343. ISBN 0-521-46576-1. MR1333890 (See chapter 3)	Wikipedia
Equidistributed sequence In mathematics, a sequence (s1, s2, s3, ...) of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that subinterval. Such sequences are studied in Diophantine approximation theory and have applications to Monte Carlo integration. Definition A sequence (s1, s2, s3, ...) of real numbers is said to be equidistributed on a non-degenerate interval [a, b] if for every subinterval [c, d ] of [a, b] we have $\lim _{n\to \infty }{\left\|\{\,s_{1},\dots ,s_{n}\,\}\cap [c,d]\right\| \over n}={d-c \over b-a}.$ (Here, the notation \|{s1,...,sn} ∩ [c, d ]\| denotes the number of elements, out of the first n elements of the sequence, that are between c and d.) For example, if a sequence is equidistributed in [0, 2], since the interval [0.5, 0.9] occupies 1/5 of the length of the interval [0, 2], as n becomes large, the proportion of the first n members of the sequence which fall between 0.5 and 0.9 must approach 1/5. Loosely speaking, one could say that each member of the sequence is equally likely to fall anywhere in its range. However, this is not to say that (sn) is a sequence of random variables; rather, it is a determinate sequence of real numbers. Discrepancy We define the discrepancy DN for a sequence (s1, s2, s3, ...) with respect to the interval [a, b] as $D_{N}=\sup _{a\leq c\leq d\leq b}\left\vert {\frac {\left\|\{\,s_{1},\dots ,s_{N}\,\}\cap [c,d]\right\|}{N}}-{\frac {d-c}{b-a}}\right\vert .$ A sequence is thus equidistributed if the discrepancy DN tends to zero as N tends to infinity. Equidistribution is a rather weak criterion to express the fact that a sequence fills the segment leaving no gaps. For example, the drawings of a random variable uniform over a segment will be equidistributed in the segment, but there will be large gaps compared to a sequence which first enumerates multiples of ε in the segment, for some small ε, in an appropriately chosen way, and then continues to do this for smaller and smaller values of ε. For stronger criteria and for constructions of sequences that are more evenly distributed, see low-discrepancy sequence. Riemann integral criterion for equidistribution Recall that if f is a function having a Riemann integral in the interval [a, b], then its integral is the limit of Riemann sums taken by sampling the function f in a set of points chosen from a fine partition of the interval. Therefore, if some sequence is equidistributed in [a, b], it is expected that this sequence can be used to calculate the integral of a Riemann-integrable function. This leads to the following criterion[1] for an equidistributed sequence: Suppose (s1, s2, s3, ...) is a sequence contained in the interval [a, b]. Then the following conditions are equivalent: 1. The sequence is equidistributed on [a, b]. 2. For every Riemann-integrable (complex-valued) function f : [a, b] → $\mathbb {C} $, the following limit holds: $\lim _{N\to \infty }{\frac {1}{N}}\sum _{n=1}^{N}f\left(s_{n}\right)={\frac {1}{b-a}}\int _{a}^{b}f(x)\,dx$ Proof First note that the definition of an equidistributed sequence is equivalent to the integral criterion whenever f is the indicator function of an interval: If f = 1[c, d], then the left hand side is the proportion of points of the sequence falling in the interval [c, d], and the right hand side is exactly $\textstyle {\frac {d-c}{b-a}}.$ This means 2 ⇒ 1 (since indicator functions are Riemann-integrable), and 1 ⇒ 2 for f being an indicator function of an interval. It remains to assume that the integral criterion holds for indicator functions and prove that it holds for general Riemann-integrable functions as well. Note that both sides of the integral criterion equation are linear in f, and therefore the criterion holds for linear combinations of interval indicators, that is, step functions. To show it holds for f being a general Riemann-integrable function, first assume f is real-valued. Then by using Darboux's definition of the integral, we have for every ε > 0 two step functions f1 and f2 such that f1 ≤ f ≤ f2 and $\textstyle \int _{a}^{b}(f_{2}(x)-f_{1}(x))\,dx\leq \varepsilon (b-a).$ Notice that: ${\frac {1}{b-a}}\int _{a}^{b}f_{1}(x)\,dx=\lim _{N\to \infty }{\frac {1}{N}}\sum _{n=1}^{N}f_{1}(s_{n})\leq \liminf _{N\to \infty }{\frac {1}{N}}\sum _{n=1}^{N}f(s_{n})$ ${\frac {1}{b-a}}\int _{a}^{b}f_{2}(x)\,dx=\lim _{N\to \infty }{\frac {1}{N}}\sum _{n=1}^{N}f_{2}(s_{n})\geq \limsup _{N\to \infty }{\frac {1}{N}}\sum _{n=1}^{N}f(s_{n})$ By subtracting, we see that the limit superior and limit inferior of $\textstyle {\frac {1}{N}}\sum _{n=1}^{N}f(s_{n})$ differ by at most ε. Since ε is arbitrary, we have the existence of the limit, and by Darboux's definition of the integral, it is the correct limit. Finally, for complex-valued Riemann-integrable functions, the result follows again from linearity, and from the fact that every such function can be written as f = u + vi, where u, v are real-valued and Riemann-integrable. ∎ This criterion leads to the idea of Monte-Carlo integration, where integrals are computed by sampling the function over a sequence of random variables equidistributed in the interval. It is not possible to generalize the integral criterion to a class of functions bigger than just the Riemann-integrable ones. For example, if the Lebesgue integral is considered and f is taken to be in L1, then this criterion fails. As a counterexample, take f to be the indicator function of some equidistributed sequence. Then in the criterion, the left hand side is always 1, whereas the right hand side is zero, because the sequence is countable, so f is zero almost everywhere. In fact, the de Bruijn–Post Theorem states the converse of the above criterion: If f is a function such that the criterion above holds for any equidistributed sequence in [a, b], then f is Riemann-integrable in [a, b].[2] Equidistribution modulo 1 A sequence (a1, a2, a3, ...) of real numbers is said to be equidistributed modulo 1 or uniformly distributed modulo 1 if the sequence of the fractional parts of an, denoted by (an) or by an − ⌊an⌋, is equidistributed in the interval [0, 1]. Examples • The equidistribution theorem: The sequence of all multiples of an irrational α, 0, α, 2α, 3α, 4α, ... is equidistributed modulo 1.[3] • More generally, if p is a polynomial with at least one coefficient other than the constant term irrational then the sequence p(n) is uniformly distributed modulo 1. This was proven by Weyl and is an application of van der Corput's difference theorem.[4] • The sequence log(n) is not uniformly distributed modulo 1.[3] This fact is related to Benford's law. • The sequence of all multiples of an irrational α by successive prime numbers, 2α, 3α, 5α, 7α, 11α, ... is equidistributed modulo 1. This is a famous theorem of analytic number theory, published by I. M. Vinogradov in 1948.[5] • The van der Corput sequence is equidistributed.[6] Weyl's criterion Weyl's criterion states that the sequence an is equidistributed modulo 1 if and only if for all non-zero integers ℓ, $\lim _{n\to \infty }{\frac {1}{n}}\sum _{j=1}^{n}e^{2\pi i\ell a_{j}}=0.$ The criterion is named after, and was first formulated by, Hermann Weyl.[7] It allows equidistribution questions to be reduced to bounds on exponential sums, a fundamental and general method. Sketch of proof If the sequence is equidistributed modulo 1, then we can apply the Riemann integral criterion (described above) on the function $\textstyle f(x)=e^{2\pi i\ell x},$ which has integral zero on the interval [0, 1]. This gives Weyl's criterion immediately. Conversely, suppose Weyl's criterion holds. Then the Riemann integral criterion holds for functions f as above, and by linearity of the criterion, it holds for f being any trigonometric polynomial. By the Stone–Weierstrass theorem and an approximation argument, this extends to any continuous function f. Finally, let f be the indicator function of an interval. It is possible to bound f from above and below by two continuous functions on the interval, whose integrals differ by an arbitrary ε. By an argument similar to the proof of the Riemann integral criterion, it is possible to extend the result to any interval indicator function f, thereby proving equidistribution modulo 1 of the given sequence. ∎ Generalizations • A quantitative form of Weyl's criterion is given by the Erdős–Turán inequality. • Weyl's criterion extends naturally to higher dimensions, assuming the natural generalization of the definition of equidistribution modulo 1: The sequence vn of vectors in Rk is equidistributed modulo 1 if and only if for any non-zero vector ℓ ∈ Zk, $\lim _{n\to \infty }{\frac {1}{n}}\sum _{j=0}^{n-1}e^{2\pi i\ell \cdot v_{j}}=0.$ Example of usage Weyl's criterion can be used to easily prove the equidistribution theorem, stating that the sequence of multiples 0, α, 2α, 3α, ... of some real number α is equidistributed modulo 1 if and only if α is irrational.[3] Suppose α is irrational and denote our sequence by aj = jα (where j starts from 0, to simplify the formula later). Let ℓ ≠ 0 be an integer. Since α is irrational, ℓα can never be an integer, so $ e^{2\pi i\ell \alpha }$ can never be 1. Using the formula for the sum of a finite geometric series, $\left\|\sum _{j=0}^{n-1}e^{2\pi i\ell j\alpha }\right\|=\left\|\sum _{j=0}^{n-1}\left(e^{2\pi i\ell \alpha }\right)^{j}\right\|=\left\|{\frac {1-e^{2\pi i\ell n\alpha }}{1-e^{2\pi i\ell \alpha }}}\right\|\leq {\frac {2}{\left\|1-e^{2\pi i\ell \alpha }\right\|}},$ a finite bound that does not depend on n. Therefore, after dividing by n and letting n tend to infinity, the left hand side tends to zero, and Weyl's criterion is satisfied. Conversely, notice that if α is rational then this sequence is not equidistributed modulo 1, because there are only a finite number of options for the fractional part of aj = jα. Complete uniform distribution A sequence $(a_{1},a_{2},\dots )$ of real numbers is said to be k-uniformly distributed mod 1 if not only the sequence of fractional parts $a_{n}':=a_{n}-[a_{n}]$ is uniformly distributed in $[0,1]$ but also the sequence $(b_{1},b_{2},\dots )$, where $b_{n}$ is defined as $b_{n}:=(a'_{n+1},\dots ,a'_{n+k})\in [0,1]^{k}$, is uniformly distributed in $[0,1]^{k}$. A sequence $(a_{1},a_{2},\dots )$ of real numbers is said to be completely uniformly distributed mod 1 it is $k$-uniformly distributed for each natural number $k\geq 1$. For example, the sequence $(\alpha ,2\alpha ,\dots )$ is uniformly distributed mod 1 (or 1-uniformly distributed) for any irrational number $\alpha $, but is never even 2-uniformly distributed. In contrast, the sequence $(\alpha ,\alpha ^{2},\alpha ^{3},\dots )$ is completely uniformly distributed for almost all $\alpha >1$ (i.e., for all $\alpha $ except for a set of measure 0). van der Corput's difference theorem A theorem of Johannes van der Corput[8] states that if for each h the sequence sn+h − sn is uniformly distributed modulo 1, then so is sn.[9][10][11] A van der Corput set is a set H of integers such that if for each h in H the sequence sn+h − sn is uniformly distributed modulo 1, then so is sn.[10][11] Metric theorems Metric theorems describe the behaviour of a parametrised sequence for almost all values of some parameter α: that is, for values of α not lying in some exceptional set of Lebesgue measure zero. • For any sequence of distinct integers bn, the sequence (bnα) is equidistributed mod 1 for almost all values of α.[12] • The sequence (α n) is equidistributed mod 1 for almost all values of α > 1.[13] It is not known whether the sequences (en ) or (π n ) are equidistributed mod 1. However it is known that the sequence (αn) is not equidistributed mod 1 if α is a PV number. Well-distributed sequence A sequence (s1, s2, s3, ...) of real numbers is said to be well-distributed on [a, b] if for any subinterval [c, d ] of [a, b] we have $\lim _{n\to \infty }{\left\|\{\,s_{k+1},\dots ,s_{k+n}\,\}\cap [c,d]\right\| \over n}={d-c \over b-a}$ uniformly in k. Clearly every well-distributed sequence is uniformly distributed, but the converse does not hold. The definition of well-distributed modulo 1 is analogous. Sequences equidistributed with respect to an arbitrary measure For an arbitrary probability measure space $(X,\mu )$, a sequence of points $(x_{n})$ is said to be equidistributed with respect to $\mu $ if the mean of point measures converges weakly to $\mu $:[14] ${\frac {\sum _{k=1}^{n}\delta _{x_{k}}}{n}}\Rightarrow \mu \ .$ In any Borel probability measure on a separable, metrizable space, there exists an equidistributed sequence with respect to the measure; indeed, this follows immediately from the fact that such a space is standard. The general phenomenon of equidistribution comes up a lot for dynamical systems associated with Lie groups, for example in Margulis' solution to the Oppenheim conjecture. See also • Equidistribution theorem • Low-discrepancy sequence • Erdős–Turán inequality Notes 1. Kuipers & Niederreiter (2006) pp. 2–3 2. http://math.uga.edu/~pete/udnotes.pdf, Theorem 8 3. Kuipers & Niederreiter (2006) p. 8 4. Kuipers & Niederreiter (2006) p. 27 5. Kuipers & Niederreiter (2006) p. 129 6. Kuipers & Niederreiter (2006) p. 127 7. Weyl, H. (September 1916). "Über die Gleichverteilung von Zahlen mod. Eins" [On the distribution of numbers modulo one] (PDF). Math. Ann. (in German). 77 (3): 313–352. doi:10.1007/BF01475864. S2CID 123470919. 8. van der Corput, J. (1931), "Diophantische Ungleichungen. I. Zur Gleichverteilung Modulo Eins", Acta Mathematica, Springer Netherlands, 56: 373–456, doi:10.1007/BF02545780, ISSN 0001-5962, JFM 57.0230.05, Zbl 0001.20102 9. Kuipers & Niederreiter (2006) p. 26 10. Montgomery (1994) p. 18 11. Montgomery, Hugh L. (2001). "Harmonic analysis as found in analytic number theory" (PDF). In Byrnes, James S. (ed.). Twentieth century harmonic analysis–a celebration. Proceedings of the NATO Advanced Study Institute, Il Ciocco, Italy, July 2–15, 2000. NATO Sci. Ser. II, Math. Phys. Chem. Vol. 33. Dordrecht: Kluwer Academic Publishers. pp. 271–293. doi:10.1007/978-94-010-0662-0_13. ISBN 978-0-7923-7169-4. Zbl 1001.11001. 12. See Bernstein, Felix (1911), "Über eine Anwendung der Mengenlehre auf ein aus der Theorie der säkularen Störungen herrührendes Problem", Mathematische Annalen, 71 (3): 417–439, doi:10.1007/BF01456856, S2CID 119558177. 13. Koksma, J. F. (1935), "Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins", Compositio Mathematica, 2: 250–258, JFM 61.0205.01, Zbl 0012.01401 14. Kuipers & Niederreiter (2006) p. 171 References • Kuipers, L.; Niederreiter, H. (2006) [1974]. Uniform Distribution of Sequences. Dover Publications. ISBN 0-486-45019-8. • Kuipers, L.; Niederreiter, H. (1974). Uniform Distribution of Sequences. John Wiley & Sons Inc. ISBN 0-471-51045-9. Zbl 0281.10001. • Montgomery, Hugh L. (1994). Ten lectures on the interface between analytic number theory and harmonic analysis. Regional Conference Series in Mathematics. Vol. 84. Providence, RI: American Mathematical Society. ISBN 0-8218-0737-4. Zbl 0814.11001. Further reading • Granville, Andrew; Rudnick, Zeév, eds. (2007). Equidistribution in number theory, an introduction. Proceedings of the NATO Advanced Study Institute on equidistribution in number theory, Montréal, Canada, July 11–22, 2005. NATO Science Series II: Mathematics, Physics and Chemistry. Vol. 237. Dordrecht: Springer-Verlag. ISBN 978-1-4020-5403-7. Zbl 1121.11004. • Tao, Terence (2012). Higher order Fourier analysis. Graduate Studies in Mathematics. Vol. 142. Providence, RI: American Mathematical Society. ISBN 978-0-8218-8986-2. Zbl 1277.11010. External links • Weisstein, Eric W. "Equidistributed Sequence". MathWorld. • Weisstein, Eric W. "Weyl's Criterion". MathWorld. • Weyl's Criterion at PlanetMath. • Lecture notes by Charles Walkden with proof of Weyl's Criterion	Wikipedia
Equicontinuity In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable families, and thus sequences of functions. Equicontinuity appears in the formulation of Ascoli's theorem, which states that a subset of C(X), the space of continuous functions on a compact Hausdorff space X, is compact if and only if it is closed, pointwise bounded and equicontinuous. As a corollary, a sequence in C(X) is uniformly convergent if and only if it is equicontinuous and converges pointwise to a function (not necessarily continuous a-priori). In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions fn on either metric space or locally compact space[1] is continuous. If, in addition, fn are holomorphic, then the limit is also holomorphic. The uniform boundedness principle states that a pointwise bounded family of continuous linear operators between Banach spaces is equicontinuous.[2] Equicontinuity between metric spaces Let X and Y be two metric spaces, and F a family of functions from X to Y. We shall denote by d the respective metrics of these spaces. The family F is equicontinuous at a point x0 ∈ X if for every ε > 0, there exists a δ > 0 such that d(ƒ(x0), ƒ(x)) < ε for all ƒ ∈ F and all x such that d(x0, x) < δ. The family is pointwise equicontinuous if it is equicontinuous at each point of X.[3] The family F is uniformly equicontinuous if for every ε > 0, there exists a δ > 0 such that d(ƒ(x1), ƒ(x2)) < ε for all ƒ ∈ F and all x1, x2 ∈ X such that d(x1, x2) < δ.[4] For comparison, the statement 'all functions ƒ in F are continuous' means that for every ε > 0, every ƒ ∈ F, and every x0 ∈ X, there exists a δ > 0 such that d(ƒ(x0), ƒ(x)) < ε for all x ∈ X such that d(x0, x) < δ. • For continuity, δ may depend on ε, ƒ, and x0. • For uniform continuity, δ may depend on ε and ƒ. • For pointwise equicontinuity, δ may depend on ε and x0. • For uniform equicontinuity, δ may depend only on ε. More generally, when X is a topological space, a set F of functions from X to Y is said to be equicontinuous at x if for every ε > 0, x has a neighborhood Ux such that $d_{Y}(f(y),f(x))<\epsilon $ for all y ∈ Ux and ƒ ∈ F. This definition usually appears in the context of topological vector spaces. When X is compact, a set is uniformly equicontinuous if and only if it is equicontinuous at every point, for essentially the same reason as that uniform continuity and continuity coincide on compact spaces. Used on its own, the term "equicontinuity" may refer to either the pointwise or uniform notion, depending on the context. On a compact space, these notions coincide. Some basic properties follow immediately from the definition. Every finite set of continuous functions is equicontinuous. The closure of an equicontinuous set is again equicontinuous. Every member of a uniformly equicontinuous set of functions is uniformly continuous, and every finite set of uniformly continuous functions is uniformly equicontinuous. Examples • A set of functions with a common Lipschitz constant is (uniformly) equicontinuous. In particular, this is the case if the set consists of functions with derivatives bounded by the same constant. • Uniform boundedness principle gives a sufficient condition for a set of continuous linear operators to be equicontinuous. • A family of iterates of an analytic function is equicontinuous on the Fatou set.[5][6] Counterexamples • The sequence of functions fn(x) = arctan(nx), is not equicontinuous because the definition is violated at x0=0. Equicontinuity of maps valued in topological groups Suppose that T is a topological space and Y is an additive topological group (i.e. a group endowed with a topology making its operations continuous). Topological vector spaces are prominent examples of topological groups and every topological group has an associated canonical uniformity. Definition:[7] A family H of maps from T into Y is said to be equicontinuous at t ∈ T if for every neighborhood V of 0 in Y, there exists some neighborhood U of t in T such that h(U) ⊆ h(t) + V for every h ∈ H. We say that H is equicontinuous if it is equicontinuous at every point of T. Note that if H is equicontinuous at a point then every map in H is continuous at the point. Clearly, every finite set of continuous maps from T into Y is equicontinuous. Equicontinuous linear maps Because every topological vector space (TVS) is a topological group so the definition of an equicontinuous family of maps given for topological groups transfers to TVSs without change. Characterization of equicontinuous linear maps A family $H$ of maps of the form $X\to Y$ between two topological vector spaces is said to be equicontinuous at a point $x\in X$ if for every neighborhood $V$ of the origin in $Y$ there exists some neighborhood $U$ of the origin in $X$ such that $h(x+U)\subseteq h(x)+V$ for all $h\in H.$ If $H$ is a family of maps and $U$ is a set then let $H(U):=\bigcup _{h\in H}h(U).$ With notation, if $U$ and $V$ are sets then $h(U)\subseteq V$ for all $h\in H$ if and only if $H(U)\subseteq V.$ Let $X$ and $Y$ be topological vector spaces (TVSs) and $H$ be a family of linear operators from $X$ into $Y.$ Then the following are equivalent: 1. $H$ is equicontinuous; 2. $H$ is equicontinuous at every point of $X.$ 3. $H$ is equicontinuous at some point of $X.$ 4. $H$ is equicontinuous at the origin. • that is, for every neighborhood $V$ of the origin in $Y,$ there exists a neighborhood $U$ of the origin in $X$ such that $H(U)\subseteq V$ (or equivalently, $h(U)\subseteq V$for every $h\in H$). 5. [8] 6. for every neighborhood $V$ of the origin in $Y,$ $\bigcap _{h\in H}h^{-1}(V)$ is a neighborhood of the origin in $X.$ 7. the closure of $H$ in $L_{\sigma }(X;Y)$ is equicontinuous. • $L_{\sigma }(X;Y)$ denotes $L(X;Y)$endowed with the topology of point-wise convergence. 8. the balanced hull of $H$ is equicontinuous. while if $Y$ is locally convex then this list may be extended to include: 1. the convex hull of $H$ is equicontinuous.[9] 2. the convex balanced hull of $H$ is equicontinuous.[10][9] while if $X$ and $Y$ are locally convex then this list may be extended to include: 1. for every continuous seminorm $q$ on $Y,$ there exists a continuous seminorm $p$ on $X$ such that $q\circ h\leq p$ for all $h\in H.$ [9] • Here, $q\circ h\leq p$ means that $q(h(x))\leq p(x)$ for all $x\in X.$ while if $X$ is barreled and $Y$ is locally convex then this list may be extended to include: 1. $H$ is bounded in $L_{\sigma }(X;Y)$;[11] 2. $H$ is bounded in $L_{b}(X;Y).$ [11] • $L_{b}(X;Y)$ denotes $L(X;Y)$endowed with the topology of bounded convergence (that is, uniform convergence on bounded subsets of $X.$ while if $X$ and $Y$ are Banach spaces then this list may be extended to include: 1. $\sup\{\\|T\\|:T\in H\}<\infty $ (that is, $H$ is uniformly bounded in the operator norm). Characterization of equicontinuous linear functionals Let $X$ be a topological vector space (TVS) over the field $\mathbb {F} $ with continuous dual space $X^{\prime }.$ A family $H$ of linear functionals on $X$ is said to be equicontinuous at a point $x\in X$ if for every neighborhood $V$ of the origin in $\mathbb {F} $ there exists some neighborhood $U$ of the origin in $X$ such that $h(x+U)\subseteq h(x)+V$ for all $h\in H.$ For any subset $H\subseteq X^{\prime },$ the following are equivalent:[9] 1. $H$ is equicontinuous. 2. $H$ is equicontinuous at the origin. 3. $H$ is equicontinuous at some point of $X.$ 4. $H$ is contained in the polar of some neighborhood of the origin in $X$[10] 5. the (pre)polar of $H$ is a neighborhood of the origin in $X.$ 6. the weak* closure of $H$ in $X^{\prime }$ is equicontinuous. 7. the balanced hull of $H$ is equicontinuous. 8. the convex hull of $H$ is equicontinuous. 9. the convex balanced hull of $H$ is equicontinuous.[10] while if $X$ is normed then this list may be extended to include: 1. $H$ is a strongly bounded subset of $X^{\prime }.$[10] while if $X$ is a barreled space then this list may be extended to include: 1. $H$ is relatively compact in the weak* topology on $X^{\prime }.$[11] 2. $H$ is weak* bounded (that is, $H$ is $\sigma \left(X^{\prime },X\right)-$bounded in $X^{\prime }$).[11] 3. $H$ is bounded in the topology of bounded convergence (that is, $H$ is $b\left(X^{\prime },X\right)-$bounded in $X^{\prime }$).[11] Properties of equicontinuous linear maps The uniform boundedness principle (also known as the Banach–Steinhaus theorem) states that a set $H$ of linear maps between Banach spaces is equicontinuous if it is pointwise bounded; that is, $\sup _{h\in H}\\|h(x)\\|<\infty $ for each $x\in X.$ The result can be generalized to a case when $Y$ is locally convex and $X$ is a barreled space.[12] Properties of equicontinuous linear functionals Alaoglu's theorem implies that the weak-* closure of an equicontinuous subset of $X^{\prime }$ is weak-* compact; thus that every equicontinuous subset is weak-* relatively compact.[13][9] If $X$ is any locally convex TVS, then the family of all barrels in $X$ and the family of all subsets of $X^{\prime }$ that are convex, balanced, closed, and bounded in $X_{\sigma }^{\prime },$ correspond to each other by polarity (with respect to $\left\langle X,X^{\#}\right\rangle $).[14] It follows that a locally convex TVS $X$ is barreled if and only if every bounded subset of $X_{\sigma }^{\prime }$ is equicontinuous.[14] Theorem — Suppose that $X$ is a separable TVS. Then every closed equicontinuous subset of $X_{\sigma }^{\prime }$ is a compact metrizable space (under the subspace topology). If in addition $X$ is metrizable then $X_{\sigma }^{\prime }$ is separable.[14] Equicontinuity and uniform convergence Let X be a compact Hausdorff space, and equip C(X) with the uniform norm, thus making C(X) a Banach space, hence a metric space. Then Arzelà–Ascoli theorem states that a subset of C(X) is compact if and only if it is closed, uniformly bounded and equicontinuous. [15] This is analogous to the Heine–Borel theorem, which states that subsets of Rn are compact if and only if they are closed and bounded.[16] As a corollary, every uniformly bounded equicontinuous sequence in C(X) contains a subsequence that converges uniformly to a continuous function on X. In view of Arzelà–Ascoli theorem, a sequence in C(X) converges uniformly if and only if it is equicontinuous and converges pointwise. The hypothesis of the statement can be weakened a bit: a sequence in C(X) converges uniformly if it is equicontinuous and converges pointwise on a dense subset to some function on X (not assumed continuous). Proof Suppose fj is an equicontinuous sequence of continuous functions on a dense subset D of X. Let ε > 0 be given. By equicontinuity, for each z ∈ D, there exists a neighborhood Uz of z such that $\|f_{j}(x)-f_{j}(z)\|<\epsilon /3$ for all j and x ∈ Uz. By denseness and compactness, we can find a finite subset D′ ⊂ D such that X is the union of Uz over z ∈ D′. Since fj converges pointwise on D′, there exists N > 0 such that $\|f_{j}(z)-f_{k}(z)\|<\epsilon /3$ whenever z ∈ D′ and j, k > N. It follows that $\sup _{X}\|f_{j}-f_{k}\|<\epsilon $ for all j, k > N. In fact, if x ∈ X, then x ∈ Uz for some z ∈ D′ and so we get: $\|f_{j}(x)-f_{k}(x)\|\leq \|f_{j}(x)-f_{j}(z)\|+\|f_{j}(z)-f_{k}(z)\|+\|f_{k}(z)-f_{k}(x)\|<\epsilon $. Hence, fj is Cauchy in C(X) and thus converges by completeness. This weaker version is typically used to prove Arzelà–Ascoli theorem for separable compact spaces. Another consequence is that the limit of an equicontinuous pointwise convergent sequence of continuous functions on a metric space, or on a locally compact space, is continuous. (See below for an example.) In the above, the hypothesis of compactness of X cannot be relaxed. To see that, consider a compactly supported continuous function g on R with g(0) = 1, and consider the equicontinuous sequence of functions {ƒn} on R defined by ƒn(x) = g(x − n). Then, ƒn converges pointwise to 0 but does not converge uniformly to 0. This criterion for uniform convergence is often useful in real and complex analysis. Suppose we are given a sequence of continuous functions that converges pointwise on some open subset G of Rn. As noted above, it actually converges uniformly on a compact subset of G if it is equicontinuous on the compact set. In practice, showing the equicontinuity is often not so difficult. For example, if the sequence consists of differentiable functions or functions with some regularity (e.g., the functions are solutions of a differential equation), then the mean value theorem or some other kinds of estimates can be used to show the sequence is equicontinuous. It then follows that the limit of the sequence is continuous on every compact subset of G; thus, continuous on G. A similar argument can be made when the functions are holomorphic. One can use, for instance, Cauchy's estimate to show the equicontinuity (on a compact subset) and conclude that the limit is holomorphic. Note that the equicontinuity is essential here. For example, ƒn(x) = arctan n x converges to a multiple of the discontinuous sign function. Generalizations Equicontinuity in topological spaces The most general scenario in which equicontinuity can be defined is for topological spaces whereas uniform equicontinuity requires the filter of neighbourhoods of one point to be somehow comparable with the filter of neighbourhood of another point. The latter is most generally done via a uniform structure, giving a uniform space. Appropriate definitions in these cases are as follows: A set A of functions continuous between two topological spaces X and Y is topologically equicontinuous at the points x ∈ X and y ∈ Y if for any open set O about y, there are neighborhoods U of x and V of y such that for every f ∈ A, if the intersection of f[U] and V is nonempty, f[U] ⊆ O. Then A is said to be topologically equicontinuous at x ∈ X if it is topologically equicontinuous at x and y for each y ∈ Y. Finally, A is equicontinuous if it is equicontinuous at x for all points x ∈ X. A set A of continuous functions between two uniform spaces X and Y is uniformly equicontinuous if for every element W of the uniformity on Y, the set { (u,v) ∈ X × X: for all f ∈ A. (f(u),f(v)) ∈ W } is a member of the uniformity on X Introduction to uniform spaces Main article: Uniform space We now briefly describe the basic idea underlying uniformities. The uniformity 𝒱 is a non-empty collection of subsets of Y × Y where, among many other properties, every V ∈ 𝒱, V contains the diagonal of Y (i.e. {(y, y) ∈ Y}). Every element of 𝒱 is called an entourage. Uniformities generalize the idea (taken from metric spaces) of points that are "r-close" (for r > 0), meaning that their distance is < r. To clarify this, suppose that (Y, d) is a metric space (so the diagonal of Y is the set {(y, z) ∈ Y × Y : d(y, z) = 0}) For any r > 0, let Ur = {(y, z) ∈ Y × Y : d(y, z) < r} denote the set of all pairs of points that are r-close. Note that if we were to "forget" that d existed then, for any r > 0, we would still be able to determine whether or not two points of Y are r-close by using only the sets Ur. In this way, the sets Ur encapsulate all the information necessary to define things such as uniform continuity and uniform convergence without needing any metric. Axiomatizing the most basic properties of these sets leads to the definition of a uniformity. Indeed, the sets Ur generate the uniformity that is canonically associated with the metric space (Y, d). The benefit of this generalization is that we may now extend some important definitions that make sense for metric spaces (e.g. completeness) to a broader category of topological spaces. In particular, to topological groups and topological vector spaces. A weaker concept is that of even continuity A set A of continuous functions between two topological spaces X and Y is said to be evenly continuous at x ∈ X and y ∈ Y if given any open set O containing y there are neighborhoods U of x and V of y such that f[U] ⊆ O whenever f(x) ∈ V. It is evenly continuous at x if it is evenly continuous at x and y for every y ∈ Y, and evenly continuous if it is evenly continuous at x for every x ∈ X. Stochastic equicontinuity Stochastic equicontinuity is a version of equicontinuity used in the context of sequences of functions of random variables, and their convergence.[17] See also • Absolute continuity – Form of continuity for functions • Classification of discontinuities – Mathematical analysis of discontinuous points • Coarse function • Continuous function – Mathematical function with no sudden changes • Continuous function (set theory) – sequence of ordinals such that the values assumed at limit stages are the limits (limit suprema and limit infima) of all values at previous stagesPages displaying wikidata descriptions as a fallback • Continuous stochastic process – Stochastic process that is a continuous function of time or index parameter • Dini continuity • Direction-preserving function - an analogue of a continuous function in discrete spaces. • Microcontinuity – Mathematical term • Normal function – Function of ordinals in mathematics • Piecewise – Function defined by multiple sub-functions • Symmetrically continuous function • Uniform continuity – Uniform restraint of the change in functions Notes 1. More generally, on any compactly generated space; e.g., a first-countable space. 2. Rudin 1991, p. 44 §2.5. 3. Reed & Simon (1980), p. 29; Rudin (1987), p. 245 4. Reed & Simon (1980), p. 29 5. Alan F. Beardon, S. Axler, F.W. Gehring, K.A. Ribet : Iteration of Rational Functions: Complex Analytic Dynamical Systems. Springer, 2000; ISBN 0-387-95151-2, ISBN 978-0-387-95151-5; page 49 6. Joseph H. Silverman : The arithmetic of dynamical systems. Springer, 2007. ISBN 0-387-69903-1, ISBN 978-0-387-69903-5; page 22 7. Narici & Beckenstein 2011, pp. 133–136. 8. Rudin 1991, p. 44 Theorem 2.4. 9. Narici & Beckenstein 2011, pp. 225–273. 10. Trèves 2006, pp. 335–345. 11. Trèves 2006, pp. 346–350. 12. Schaefer 1966, Theorem 4.2. 13. Schaefer 1966, Corollary 4.3. 14. Schaefer & Wolff 1999, pp. 123–128. 15. Rudin 1991, p. 394 Appendix A5. 16. Rudin 1991, p. 18 Theorem 1.23. 17. de Jong, Robert M. (1993). "Stochastic Equicontinuity for Mixing Processes". Asymptotic Theory of Expanding Parameter Space Methods and Data Dependence in Econometrics. Amsterdam. pp. 53–72. ISBN 90-5170-227-2.{{cite book}}: CS1 maint: location missing publisher (link) References • "Equicontinuity", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Reed, Michael; Simon, Barry (1980), Functional Analysis (revised and enlarged ed.), Boston, MA: Academic Press, ISBN 978-0-12-585050-6. • Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. • Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. • Rudin, Walter (1987), Real and Complex Analysis (3rd ed.), New York: McGraw-Hill. • Schaefer, Helmut H. (1966), Topological vector spaces, New York: The Macmillan Company • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.	Wikipedia
Uniform isomorphism In the mathematical field of topology a uniform isomorphism or uniform homeomorphism is a special isomorphism between uniform spaces that respects uniform properties. Uniform spaces with uniform maps form a category. An isomorphism between uniform spaces is called a uniform isomorphism. Definition A function $f$ between two uniform spaces $X$ and $Y$ is called a uniform isomorphism if it satisfies the following properties • $f$ is a bijection • $f$ is uniformly continuous • the inverse function $f^{-1}$ is uniformly continuous In other words, a uniform isomorphism is a uniformly continuous bijection between uniform spaces whose inverse is also uniformly continuous. If a uniform isomorphism exists between two uniform spaces they are called uniformly isomorphic or uniformly equivalent. Uniform embeddings A uniform embedding is an injective uniformly continuous map $i:X\to Y$ between uniform spaces whose inverse $i^{-1}:i(X)\to X$ is also uniformly continuous, where the image $i(X)$ has the subspace uniformity inherited from $Y.$ Examples The uniform structures induced by equivalent norms on a vector space are uniformly isomorphic. See also • Homeomorphism – Mapping which preserves all topological properties of a given space — an isomorphism between topological spaces • Isometric isomorphism – Distance-preserving mathematical transformationPages displaying short descriptions of redirect targets — an isomorphism between metric spaces References • John L. Kelley, General topology, van Nostrand, 1955. P.181.	Wikipedia
Uniformly hyperfinite algebra In mathematics, particularly in the theory of C-algebras, a uniformly hyperfinite, or UHF, algebra is a C-algebra that can be written as the closure, in the norm topology, of an increasing union of finite-dimensional full matrix algebras. Definition A UHF C-algebra is the direct limit of an inductive system {An, φn} where each An is a finite-dimensional full matrix algebra and each φn : An → An+1 is a unital embedding. Suppressing the connecting maps, one can write $A={\overline {\cup _{n}A_{n}}}.$ Classification If $A_{n}\simeq M_{k_{n}}(\mathbb {C} ),$ then rkn = kn + 1 for some integer r and $\phi _{n}(a)=a\otimes I_{r},$ where Ir is the identity in the r × r matrices. The sequence ...kn\|kn + 1\|kn + 2... determines a formal product $\delta (A)=\prod _{p}p^{t_{p}}$ where each p is prime and tp = sup {m \| pm divides kn for some n}, possibly zero or infinite. The formal product δ(A) is said to be the supernatural number corresponding to A.[1] Glimm showed that the supernatural number is a complete invariant of UHF C-algebras.[2] In particular, there are uncountably many isomorphism classes of UHF C-algebras. If δ(A) is finite, then A is the full matrix algebra Mδ(A). A UHF algebra is said to be of infinite type if each tp in δ(A) is 0 or ∞. In the language of K-theory, each supernatural number $\delta (A)=\prod _{p}p^{t_{p}}$ specifies an additive subgroup of Q that is the rational numbers of the type n/m where m formally divides δ(A). This group is the K0 group of A. [1] CAR algebra One example of a UHF C-algebra is the CAR algebra. It is defined as follows: let H be a separable complex Hilbert space H with orthonormal basis fn and L(H) the bounded operators on H, consider a linear map $\alpha :H\rightarrow L(H)$ with the property that $\{\alpha (f_{n}),\alpha (f_{m})\}=0\quad {\mbox{and}}\quad \alpha (f_{n})^{}\alpha (f_{m})+\alpha (f_{m})\alpha (f_{n})^{}=\langle f_{m},f_{n}\rangle I.$ The CAR algebra is the C-algebra generated by $\{\alpha (f_{n})\}\;.$ The embedding $C^{}(\alpha (f_{1}),\cdots ,\alpha (f_{n}))\hookrightarrow C^{}(\alpha (f_{1}),\cdots ,\alpha (f_{n+1}))$ can be identified with the multiplicity 2 embedding $M_{2^{n}}\hookrightarrow M_{2^{n+1}}.$ Therefore, the CAR algebra has supernatural number 2∞.[3] This identification also yields that its K0 group is the dyadic rationals. References 1. Rørdam, M.; Larsen, F.; Laustsen, N.J. (2000). An Introduction to K-Theory for C-Algebras. Cambridge: Cambridge University Press. ISBN 0521789443. 2. Glimm, James G. (1 February 1960). "On a certain class of operator algebras" (PDF). Transactions of the American Mathematical Society. 95 (2): 318–340. doi:10.1090/S0002-9947-1960-0112057-5. Retrieved 2 March 2013. 3. Davidson, Kenneth (1997). C*-Algebras by Example. Fields Institute. pp. 166, 218–219, 234. ISBN 0-8218-0599-1.	Wikipedia
Recurrent word In mathematics, a recurrent word or sequence is an infinite word over a finite alphabet in which every factor occurs infinitely many times.[1][2][3] An infinite word is recurrent if and only if it is a sesquipower.[4][5] A uniformly recurrent word is a recurrent word in which for any given factor X in the sequence, there is some length nX (often much longer than the length of X) such that X appears in every block of length nX.[1][6][7] The terms minimal sequence[8] and almost periodic sequence (Muchnik, Semenov, Ushakov 2003) are also used. Examples • The easiest way to make a recurrent sequence is to form a periodic sequence, one where the sequence repeats entirely after a given number m of steps. Such a sequence is then uniformly recurrent and nX can be set to any multiple of m that is larger than twice the length of X. A recurrent sequence that is ultimately periodic is purely periodic.[2] • The Thue–Morse sequence is uniformly recurrent without being periodic, nor even eventually periodic (meaning periodic after some nonperiodic initial segment).[9] • All Sturmian words are uniformly recurrent.[10] References 1. Lothaire (2011) p. 30 2. Allouche & Shallit (2003) p.325 3. Pytheas Fogg (2002) p.2 4. Lothaire (2011) p. 141 5. Berstel et al (2009) p.133 6. Berthé & Rigo (2010) p.7 7. Allouche & Shallit (2003) p.328 8. Pytheas Fogg (2002) p.6 9. Lothaire (2011) p.31 10. Berthé & Rigo (2010) p.177 • Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015. • Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009). Combinatorics on words. Christoffel words and repetitions in words. CRM Monograph Series. Vol. 27. Providence, RI: American Mathematical Society. ISBN 978-0-8218-4480-9. Zbl 1161.68043. • Berthé, Valérie; Rigo, Michel, eds. (2010). Combinatorics, automata, and number theory. Encyclopedia of Mathematics and its Applications. Vol. 135. Cambridge: Cambridge University Press. ISBN 978-0-521-51597-9. Zbl 1197.68006. • Lothaire, M. (2011). Algebraic combinatorics on words. Encyclopedia of Mathematics and Its Applications. Vol. 90. With preface by Jean Berstel and Dominique Perrin (Reprint of the 2002 hardback ed.). Cambridge University Press. ISBN 978-0-521-18071-9. Zbl 1221.68183. • Pytheas Fogg, N. (2002). Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, Anne (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015. • An. Muchnik, A. Semenov, M. Ushakov, Almost periodic sequences, Theoret. Comput. Sci. vol.304 no.1-3 (2003), 1-33.	Wikipedia
Uniformly smooth space In mathematics, a uniformly smooth space is a normed vector space $X$ satisfying the property that for every $\epsilon >0$ there exists $\delta >0$ such that if $x,y\in X$ with $\\|x\\|=1$ and $\\|y\\|\leq \delta $ then $\\|x+y\\|+\\|x-y\\|\leq 2+\epsilon \\|y\\|.$ The modulus of smoothness of a normed space X is the function ρX defined for every t > 0 by the formula[1] $\rho _{X}(t)=\sup {\Bigl \{}{\frac {\\|x+y\\|+\\|x-y\\|}{2}}-1\,:\,\\|x\\|=1,\;\\|y\\|=t{\Bigr \}}.$ The triangle inequality yields that ρX(t ) ≤ t. The normed space X is uniformly smooth if and only if ρX(t ) / t tends to 0 as t tends to 0. Properties • Every uniformly smooth Banach space is reflexive.[2] • A Banach space $X$ is uniformly smooth if and only if its continuous dual $X^{}$ is uniformly convex (and vice versa, via reflexivity).[3] The moduli of convexity and smoothness are linked by $\rho _{X^{}}(t)=\sup\{t\varepsilon /2-\delta _{X}(\varepsilon ):\varepsilon \in [0,2]\},\quad t\geq 0,$ and the maximal convex function majorated by the modulus of convexity δX is given by[4] ${\tilde {\delta }}_{X}(\varepsilon )=\sup\{\varepsilon t/2-\rho _{X^{}}(t):t\geq 0\}.$ Furthermore,[5] $\delta _{X}(\varepsilon /2)\leq {\tilde {\delta }}_{X}(\varepsilon )\leq \delta _{X}(\varepsilon ),\quad \varepsilon \in [0,2].$ • A Banach space is uniformly smooth if and only if the limit $\lim _{t\to 0}{\frac {\\|x+ty\\|-\\|x\\|}{t}}$ exists uniformly for all $x,y\in S_{X}$ (where $S_{X}$ denotes the unit sphere of $X$). • When 1 < p < ∞, the Lp-spaces are uniformly smooth (and uniformly convex). Enflo proved[6] that the class of Banach spaces that admit an equivalent uniformly convex norm coincides with the class of super-reflexive Banach spaces, introduced by Robert C. James.[7] As a space is super-reflexive if and only if its dual is super-reflexive, it follows that the class of Banach spaces that admit an equivalent uniformly convex norm coincides with the class of spaces that admit an equivalent uniformly smooth norm. The Pisier renorming theorem[8] states that a super-reflexive space X admits an equivalent uniformly smooth norm for which the modulus of smoothness ρX satisfies, for some constant C and some p > 1 $\rho _{X}(t)\leq C\,t^{p},\quad t>0.$ It follows that every super-reflexive space Y admits an equivalent uniformly convex norm for which the modulus of convexity satisfies, for some constant c > 0 and some positive real q $\delta _{Y}(\varepsilon )\geq c\,\varepsilon ^{q},\quad \varepsilon \in [0,2].$ If a normed space admits two equivalent norms, one uniformly convex and one uniformly smooth, the Asplund averaging technique[9] produces another equivalent norm that is both uniformly convex and uniformly smooth. See also • Uniformly convex space Notes 1. see Definition 1.e.1, p. 59 in Lindenstrauss & Tzafriri (1979). 2. Proposition 1.e.3, p. 61 in Lindenstrauss & Tzafriri (1979). 3. Proposition 1.e.2, p. 61 in Lindenstrauss & Tzafriri (1979). 4. Proposition 1.e.6, p. 65 in Lindenstrauss & Tzafriri (1979). 5. Lemma 1.e.7 and 1.e.8, p. 66 in Lindenstrauss & Tzafriri (1979). 6. Enflo, Per (1973), "Banach spaces which can be given an equivalent uniformly convex norm", Israel Journal of Mathematics, 13 (3–4): 281–288, doi:10.1007/BF02762802 7. James, Robert C. (1972), "Super-reflexive Banach spaces", Canadian Journal of Mathematics, 24 (5): 896–904, doi:10.4153/CJM-1972-089-7 8. Pisier, Gilles (1975), "Martingales with values in uniformly convex spaces", Israel Journal of Mathematics, 20 (3–4): 326–350, doi:10.1007/BF02760337 9. Asplund, Edgar (1967), "Averaged norms", Israel Journal of Mathematics, 5 (4): 227–233, doi:10.1007/BF02771611 References • Diestel, Joseph (1984), Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, New York: Springer-Verlag, pp. xii+261, ISBN 0-387-90859-5 • Itô, Kiyosi (1993), Encyclopedic Dictionary of Mathematics, Volume 1, MIT Press, ISBN 0-262-59020-4 • Lindenstrauss, Joram; Tzafriri, Lior (1979), Classical Banach spaces. II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Berlin-New York: Springer-Verlag, pp. x+243, ISBN 3-540-08888-1. Banach space topics Types of Banach spaces • Asplund • Banach • list • Banach lattice • Grothendieck • Hilbert • Inner product space • Polarization identity • (Polynomially) Reflexive • Riesz • L-semi-inner product • (B • Strictly • Uniformly) convex • Uniformly smooth • (Injective • Projective) Tensor product (of Hilbert spaces) Banach spaces are: • Barrelled • Complete • F-space • Fréchet • tame • Locally convex • Seminorms/Minkowski functionals • Mackey • Metrizable • Normed • norm • Quasinormed • Stereotype Function space Topologies • Banach–Mazur compactum • Dual • Dual space • Dual norm • Operator • Ultraweak • Weak • polar • operator • Strong • polar • operator • Ultrastrong • Uniform convergence Linear operators • Adjoint • Bilinear • form • operator • sesquilinear • (Un)Bounded • Closed • Compact • on Hilbert spaces • (Dis)Continuous • Densely defined • Fredholm • kernel • operator • Hilbert–Schmidt • Functionals • positive • Pseudo-monotone • Normal • Nuclear • Self-adjoint • Strictly singular • Trace class • Transpose • Unitary Operator theory • Banach algebras • C-algebras • Operator space • Spectrum • C-algebra • radius • Spectral theory • of ODEs • Spectral theorem • Polar decomposition • Singular value decomposition Theorems • Anderson–Kadec • Banach–Alaoglu • Banach–Mazur • Banach–Saks • Banach–Schauder (open mapping) • Banach–Steinhaus (Uniform boundedness) • Bessel's inequality • Cauchy–Schwarz inequality • Closed graph • Closed range • Eberlein–Šmulian • Freudenthal spectral • Gelfand–Mazur • Gelfand–Naimark • Goldstine • Hahn–Banach • hyperplane separation • Kakutani fixed-point • Krein–Milman • Lomonosov's invariant subspace • Mackey–Arens • Mazur's lemma • M. Riesz extension • Parseval's identity • Riesz's lemma • Riesz representation • Robinson-Ursescu • Schauder fixed-point Analysis • Abstract Wiener space • Banach manifold • bundle • Bochner space • Convex series • Differentiation in Fréchet spaces • Derivatives • Fréchet • Gateaux • functional • holomorphic • quasi • Integrals • Bochner • Dunford • Gelfand–Pettis • regulated • Paley–Wiener • weak • Functional calculus • Borel • continuous • holomorphic • Measures • Lebesgue • Projection-valued • Vector • Weakly / Strongly measurable function Types of sets • Absolutely convex • Absorbing • Affine • Balanced/Circled • Bounded • Convex • Convex cone (subset) • Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) • Linear cone (subset) • Radial • Radially convex/Star-shaped • Symmetric • Zonotope Subsets / set operations • Affine hull • (Relative) Algebraic interior (core) • Bounding points • Convex hull • Extreme point • Interior • Linear span • Minkowski addition • Polar • (Quasi) Relative interior Examples • Absolute continuity AC • $ba(\Sigma )$ • c space • Banach coordinate BK • Besov $B_{p,q}^{s}(\mathbb {R} )$ • Birnbaum–Orlicz • Bounded variation BV • Bs space • Continuous C(K) with K compact Hausdorff • Hardy Hp • Hilbert H • Morrey–Campanato $L^{\lambda ,p}(\Omega )$ • ℓp • $\ell ^{\infty }$ • Lp • $L^{\infty }$ • weighted • Schwartz $S\left(\mathbb {R} ^{n}\right)$ • Segal–Bargmann F • Sequence space • Sobolev Wk,p • Sobolev inequality • Triebel–Lizorkin • Wiener amalgam $W(X,L^{p})$ Applications • Differential operator • Finite element method • Mathematical formulation of quantum mechanics • Ordinary Differential Equations (ODEs) • Validated numerics Functional analysis (topics – glossary) Spaces • Banach • Besov • Fréchet • Hilbert • Hölder • Nuclear • Orlicz • Schwartz • Sobolev • Topological vector Properties • Barrelled • Complete • Dual (Algebraic/Topological) • Locally convex • Reflexive • Reparable Theorems • Hahn–Banach • Riesz representation • Closed graph • Uniform boundedness principle • Kakutani fixed-point • Krein–Milman • Min–max • Gelfand–Naimark • Banach–Alaoglu Operators • Adjoint • Bounded • Compact • Hilbert–Schmidt • Normal • Nuclear • Trace class • Transpose • Unbounded • Unitary Algebras • Banach algebra • C-algebra • Spectrum of a C*-algebra • Operator algebra • Group algebra of a locally compact group • Von Neumann algebra Open problems • Invariant subspace problem • Mahler's conjecture Applications • Hardy space • Spectral theory of ordinary differential equations • Heat kernel • Index theorem • Calculus of variations • Functional calculus • Integral operator • Jones polynomial • Topological quantum field theory • Noncommutative geometry • Riemann hypothesis • Distribution (or Generalized functions) Advanced topics • Approximation property • Balanced set • Choquet theory • Weak topology • Banach–Mazur distance • Tomita–Takesaki theory • Mathematics portal • Category • Commons	Wikipedia
Unimodality In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object.[1] Unimodal probability distribution In statistics, a unimodal probability distribution or unimodal distribution is a probability distribution which has a single peak. The term "mode" in this context refers to any peak of the distribution, not just to the strict definition of mode which is usual in statistics. If there is a single mode, the distribution function is called "unimodal". If it has more modes it is "bimodal" (2), "trimodal" (3), etc., or in general, "multimodal".[2] Figure 1 illustrates normal distributions, which are unimodal. Other examples of unimodal distributions include Cauchy distribution, Student's t-distribution, chi-squared distribution and exponential distribution. Among discrete distributions, the binomial distribution and Poisson distribution can be seen as unimodal, though for some parameters they can have two adjacent values with the same probability. Figure 2 and Figure 3 illustrate bimodal distributions. Other definitions Other definitions of unimodality in distribution functions also exist. In continuous distributions, unimodality can be defined through the behavior of the cumulative distribution function (cdf).[3] If the cdf is convex for x < m and concave for x > m, then the distribution is unimodal, m being the mode. Note that under this definition the uniform distribution is unimodal,[4] as well as any other distribution in which the maximum distribution is achieved for a range of values, e.g. trapezoidal distribution. Usually this definition allows for a discontinuity at the mode; usually in a continuous distribution the probability of any single value is zero, while this definition allows for a non-zero probability, or an "atom of probability", at the mode. Criteria for unimodality can also be defined through the characteristic function of the distribution[3] or through its Laplace–Stieltjes transform.[5] Another way to define a unimodal discrete distribution is by the occurrence of sign changes in the sequence of differences of the probabilities.[6] A discrete distribution with a probability mass function, $\{p_{n}:n=\dots ,-1,0,1,\dots \}$, is called unimodal if the sequence $\dots ,p_{-2}-p_{-1},p_{-1}-p_{0},p_{0}-p_{1},p_{1}-p_{2},\dots $ has exactly one sign change (when zeroes don't count). Uses and results One reason for the importance of distribution unimodality is that it allows for several important results. Several inequalities are given below which are only valid for unimodal distributions. Thus, it is important to assess whether or not a given data set comes from a unimodal distribution. Several tests for unimodality are given in the article on multimodal distribution. Inequalities See also: Chebychev's inequality § Unimodal distributions Gauss's inequality A first important result is Gauss's inequality.[7] Gauss's inequality gives an upper bound on the probability that a value lies more than any given distance from its mode. This inequality depends on unimodality. Vysochanskiï–Petunin inequality A second is the Vysochanskiï–Petunin inequality,[8] a refinement of the Chebyshev inequality. The Chebyshev inequality guarantees that in any probability distribution, "nearly all" the values are "close to" the mean value. The Vysochanskiï–Petunin inequality refines this to even nearer values, provided that the distribution function is continuous and unimodal. Further results were shown by Sellke and Sellke.[9] Mode, median and mean Gauss also showed in 1823 that for a unimodal distribution[10] $\sigma \leq \omega \leq 2\sigma $ and $\|\nu -\mu \|\leq {\sqrt {\frac {3}{4}}}\omega ,$ where the median is ν, the mean is μ and ω is the root mean square deviation from the mode. It can be shown for a unimodal distribution that the median ν and the mean μ lie within (3/5)1/2 ≈ 0.7746 standard deviations of each other.[11] In symbols, ${\frac {\|\nu -\mu \|}{\sigma }}\leq {\sqrt {\frac {3}{5}}}$ where \| . \| is the absolute value. In 2020, Bernard, Kazzi, and Vanduffel generalized the previous inequality by deriving the maximum distance between the symmetric quantile average ${\frac {q_{\alpha }+q_{(1-\alpha )}}{2}}$ and the mean,[12] ${\frac {\left\|{\frac {q_{\alpha }+q_{(1-\alpha )}}{2}}-\mu \right\|}{\sigma }}\leq \left\{{\begin{array}{cl}{\frac {{\sqrt[{}]{{\frac {4}{9(1-\alpha )}}-1}}{\text{ }}+{\text{ }}{\sqrt[{}]{\frac {1-\alpha }{1/3+\alpha }}}}{2}}&{\text{for }}\alpha \in \left[{\frac {5}{6}},1\right)\!,\\{\frac {{\sqrt[{}]{\frac {3\alpha }{4-3\alpha }}}{\text{ }}+{\text{ }}{\sqrt[{}]{\frac {1-\alpha }{1/3+\alpha }}}}{2}}&{\text{for }}\alpha \in \left({\frac {1}{6}},{\frac {5}{6}}\right)\!,\\{\frac {{\sqrt[{}]{\frac {3\alpha }{4-3\alpha }}}{\text{ }}+{\text{ }}{\sqrt[{}]{{\frac {4}{9\alpha }}-1}}}{2}}&{\text{for }}\alpha \in \left(0,{\frac {1}{6}}\right]\!.\end{array}}\right.$ It is worth noting that the maximum distance is minimized at $\alpha =0.5$ (i.e., when the symmetric quantile average is equal to $q_{0.5}=\nu $), which indeed motivates the common choice of the median as a robust estimator for the mean. Moreover, when $\alpha =0.5$, the bound is equal to ${\sqrt {3/5}}$, which is the maximum distance between the median and the mean of a unimodal distribution. A similar relation holds between the median and the mode θ: they lie within 31/2 ≈ 1.732 standard deviations of each other: ${\frac {\|\nu -\theta \|}{\sigma }}\leq {\sqrt {3}}.$ It can also be shown that the mean and the mode lie within 31/2 of each other: ${\frac {\|\mu -\theta \|}{\sigma }}\leq {\sqrt {3}}.$ Skewness and kurtosis Rohatgi and Szekely claimed that the skewness and kurtosis of a unimodal distribution are related by the inequality:[13] $\gamma ^{2}-\kappa \leq {\frac {6}{5}}=1.2$ where κ is the kurtosis and γ is the skewness. Klaassen, Mokveld, and van Es showed that this only applies in certain settings, such as the set of unimodal distributions where the mode and mean coincide.[14] They derived a weaker inequality which applies to all unimodal distributions:[14] $\gamma ^{2}-\kappa \leq {\frac {186}{125}}=1.488$ This bound is sharp, as it is reached by the equal-weights mixture of the uniform distribution on [0,1] and the discrete distribution at {0}. Unimodal function As the term "modal" applies to data sets and probability distribution, and not in general to functions, the definitions above do not apply. The definition of "unimodal" was extended to functions of real numbers as well. A common definition is as follows: a function f(x) is a unimodal function if for some value m, it is monotonically increasing for x ≤ m and monotonically decreasing for x ≥ m. In that case, the maximum value of f(x) is f(m) and there are no other local maxima. Proving unimodality is often hard. One way consists in using the definition of that property, but it turns out to be suitable for simple functions only. A general method based on derivatives exists,[15] but it does not succeed for every function despite its simplicity. Examples of unimodal functions include quadratic polynomial functions with a negative quadratic coefficient, tent map functions, and more. The above is sometimes related to as strong unimodality, from the fact that the monotonicity implied is strong monotonicity. A function f(x) is a weakly unimodal function if there exists a value m for which it is weakly monotonically increasing for x ≤ m and weakly monotonically decreasing for x ≥ m. In that case, the maximum value f(m) can be reached for a continuous range of values of x. An example of a weakly unimodal function which is not strongly unimodal is every other row in Pascal's triangle. Depending on context, unimodal function may also refer to a function that has only one local minimum, rather than maximum.[16] For example, local unimodal sampling, a method for doing numerical optimization, is often demonstrated with such a function. It can be said that a unimodal function under this extension is a function with a single local extremum. One important property of unimodal functions is that the extremum can be found using search algorithms such as golden section search, ternary search or successive parabolic interpolation.[17] Other extensions A function f(x) is "S-unimodal" (often referred to as "S-unimodal map") if its Schwarzian derivative is negative for all $x\neq c$, where $c$ is the critical point.[18] In computational geometry if a function is unimodal it permits the design of efficient algorithms for finding the extrema of the function.[19] A more general definition, applicable to a function f(X) of a vector variable X is that f is unimodal if there is a one-to-one differentiable mapping X = G(Z) such that f(G(Z)) is convex. Usually one would want G(Z) to be continuously differentiable with nonsingular Jacobian matrix. Quasiconvex functions and quasiconcave functions extend the concept of unimodality to functions whose arguments belong to higher-dimensional Euclidean spaces. See also • Bimodal distribution References 1. Weisstein, Eric W. "Unimodal". MathWorld. 2. Weisstein, Eric W. "Mode". MathWorld. 3. A.Ya. Khinchin (1938). "On unimodal distributions". Trams. Res. Inst. Math. Mech. (in Russian). University of Tomsk. 2 (2): 1–7. 4. Ushakov, N.G. (2001) [1994], "Unimodal distribution", Encyclopedia of Mathematics, EMS Press 5. Vladimirovich Gnedenko and Victor Yu Korolev (1996). Random summation: limit theorems and applications. CRC-Press. ISBN 0-8493-2875-6. p. 31 6. Medgyessy, P. (March 1972). "On the unimodality of discrete distributions". Periodica Mathematica Hungarica. 2 (1–4): 245–257. doi:10.1007/bf02018665. S2CID 119817256. 7. Gauss, C. F. (1823). "Theoria Combinationis Observationum Erroribus Minimis Obnoxiae, Pars Prior". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. 5. 8. D. F. Vysochanskij, Y. I. Petunin (1980). "Justification of the 3σ rule for unimodal distributions". Theory of Probability and Mathematical Statistics. 21: 25–36. 9. Sellke, T.M.; Sellke, S.H. (1997). "Chebyshev inequalities for unimodal distributions". American Statistician. American Statistical Association. 51 (1): 34–40. doi:10.2307/2684690. JSTOR 2684690. 10. Gauss C.F. Theoria Combinationis Observationum Erroribus Minimis Obnoxiae. Pars Prior. Pars Posterior. Supplementum. Theory of the Combination of Observations Least Subject to Errors. Part One. Part Two. Supplement. 1995. Translated by G.W. Stewart. Classics in Applied Mathematics Series, Society for Industrial and Applied Mathematics, Philadelphia 11. Basu, S.; Dasgupta, A. (1997). "The Mean, Median, and Mode of Unimodal Distributions: A Characterization". Theory of Probability & Its Applications. 41 (2): 210–223. doi:10.1137/S0040585X97975447. 12. Bernard, Carole; Kazzi, Rodrigue; Vanduffel, Steven (2020). "Range Value-at-Risk bounds for unimodal distributions under partial information". Insurance: Mathematics and Economics. 94: 9–24. doi:10.1016/j.insmatheco.2020.05.013. 13. Rohatgi, Vijay K.; Székely, Gábor J. (1989). "Sharp inequalities between skewness and kurtosis". Statistics & Probability Letters. 8 (4): 297–299. doi:10.1016/0167-7152(89)90035-7. 14. Klaassen, Chris A.J.; Mokveld, Philip J.; Van Es, Bert (2000). "Squared skewness minus kurtosis bounded by 186/125 for unimodal distributions". Statistics & Probability Letters. 50 (2): 131–135. doi:10.1016/S0167-7152(00)00090-0. 15. "On the unimodality of METRIC Approximation subject to normally distributed demands" (PDF). Method in appendix D, Example in theorem 2 page 5. Retrieved 2013-08-28. 16. "Mathematical Programming Glossary". Retrieved 2020-03-29. 17. Demaine, Erik D.; Langerman, Stefan (2005). "Optimizing a 2D Function Satisfying Unimodality Properties". In Brodal, Gerth Stølting; Leonardi, Stefano (eds.). Algorithms – ESA 2005. Lecture Notes in Computer Science. Vol. 3669. Berlin, Heidelberg: Springer. pp. 887–898. doi:10.1007/11561071_78. ISBN 978-3-540-31951-1. 18. See e.g. John Guckenheimer; Stewart Johnson (July 1990). "Distortion of S-Unimodal Maps". Annals of Mathematics. Second Series. 132 (1): 71–130. doi:10.2307/1971501. JSTOR 1971501. 19. Godfried T. Toussaint (June 1984). "Complexity, convexity, and unimodality". International Journal of Computer and Information Sciences. 13 (3): 197–217. doi:10.1007/bf00979872. S2CID 11577312.	Wikipedia
Unimodular lattice In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in n-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamental domain for the lattice be 1. Not to be confused with modular lattice. The E8 lattice and the Leech lattice are two famous examples. Definitions • A lattice is a free abelian group of finite rank with a symmetric bilinear form (·, ·). • The lattice is integral if (·,·) takes integer values. • The dimension of a lattice is the same as its rank (as a Z-module). • The norm of a lattice element a is (a, a). • A lattice is positive definite if the norm of all nonzero elements is positive. • The determinant of a lattice is the determinant of the Gram matrix, a matrix with entries (ai, aj), where the elements ai form a basis for the lattice. • An integral lattice is unimodular if its determinant is 1 or −1. • A unimodular lattice is even or type II if all norms are even, otherwise odd or type I. • The minimum of a positive definite lattice is the lowest nonzero norm. • Lattices are often embedded in a real vector space with a symmetric bilinear form. The lattice is positive definite, Lorentzian, and so on if its vector space is. • The signature of a lattice is the signature of the form on the vector space. Examples The three most important examples of unimodular lattices are: • The lattice Z, in one dimension. • The E8 lattice, an even 8-dimensional lattice, • The Leech lattice, the 24-dimensional even unimodular lattice with no roots. Properties An integral lattice is unimodular if and only if its dual lattice is integral. Unimodular lattices are equal to their dual lattices, and for this reason, unimodular lattices are also known as self-dual. Given a pair (m,n) of nonnegative integers, an even unimodular lattice of signature (m,n) exists if and only if m−n is divisible by 8, but an odd unimodular lattice of signature (m,n) always exists. In particular, even unimodular definite lattices only exist in dimension divisible by 8. Examples in all admissible signatures are given by the IIm,n and Im,n constructions, respectively. The theta function of a unimodular positive definite lattice is a modular form whose weight is one half the rank. If the lattice is even, the form has level 1, and if the lattice is odd the form has Γ0(4) structure (i.e., it is a modular form of level 4). Due to the dimension bound on spaces of modular forms, the minimum norm of a nonzero vector of an even unimodular lattice is no greater than ⎣n/24⎦ + 1. An even unimodular lattice that achieves this bound is called extremal. Extremal even unimodular lattices are known in relevant dimensions up to 80,[1] and their non-existence has been proven for dimensions above 163,264.[2] Classification For indefinite lattices, the classification is easy to describe. Write Rm,n for the m + n dimensional vector space Rm+n with the inner product of (a1, ..., am+n) and (b1, ..., bm+n) given by $a_{1}b_{1}+\cdots +a_{m}b_{m}-a_{m+1}b_{m+1}-\cdots -a_{m+n}b_{m+n}.\,$ In Rm,n there is one odd indefinite unimodular lattice up to isomorphism, denoted by Im,n, which is given by all vectors (a1,...,am+n) in Rm,n with all the ai integers. There are no indefinite even unimodular lattices unless m − n is divisible by 8, in which case there is a unique example up to isomorphism, denoted by IIm,n. This is given by all vectors (a1,...,am+n) in Rm,n such that either all the ai are integers or they are all integers plus 1/2, and their sum is even. The lattice II8,0 is the same as the E8 lattice. Positive definite unimodular lattices have been classified up to dimension 25. There is a unique example In,0 in each dimension n less than 8, and two examples (I8,0 and II8,0) in dimension 8. The number of lattices increases moderately up to dimension 25 (where there are 665 of them), but beyond dimension 25 the Smith-Minkowski-Siegel mass formula implies that the number increases very rapidly with the dimension; for example, there are more than 80,000,000,000,000,000 in dimension 32. In some sense unimodular lattices up to dimension 9 are controlled by E8, and up to dimension 25 they are controlled by the Leech lattice, and this accounts for their unusually good behavior in these dimensions. For example, the Dynkin diagram of the norm-2 vectors of unimodular lattices in dimension up to 25 can be naturally identified with a configuration of vectors in the Leech lattice. The wild increase in numbers beyond 25 dimensions might be attributed to the fact that these lattices are no longer controlled by the Leech lattice. Even positive definite unimodular lattice exist only in dimensions divisible by 8. There is one in dimension 8 (the E8 lattice), two in dimension 16 (E82 and II16,0), and 24 in dimension 24, called the Niemeier lattices (examples: the Leech lattice, II24,0, II16,0 + II8,0, II8,03). Beyond 24 dimensions the number increases very rapidly; in 32 dimensions there are more than a billion of them. Unimodular lattices with no roots (vectors of norm 1 or 2) have been classified up to dimension 28. There are none of dimension less than 23 (other than the zero lattice!). There is one in dimension 23 (called the short Leech lattice), two in dimension 24 (the Leech lattice and the odd Leech lattice), and Bacher & Venkov (2001) showed that there are 0, 1, 3, 38 in dimensions 25, 26, 27, 28, respectively. Beyond this the number increases very rapidly; there are at least 8000 in dimension 29. In sufficiently high dimensions most unimodular lattices have no roots. The only non-zero example of even positive definite unimodular lattices with no roots in dimension less than 32 is the Leech lattice in dimension 24. In dimension 32 there are more than ten million examples, and above dimension 32 the number increases very rapidly. The following table from (King 2003) gives the numbers of (or lower bounds for) even or odd unimodular lattices in various dimensions, and shows the very rapid growth starting shortly after dimension 24. Dimension Odd lattices Odd lattices no roots Even lattices Even lattices no roots 00011 110 210 310 410 510 610 710 8101 (E8 lattice)0 920 1020 1120 1230 1330 1440 1550 16602 (E82, D16+)0 1790 18130 19160 20280 21400 22680 231171 (shorter Leech lattice) 242731 (odd Leech lattice)24 (Niemeier lattices)1 (Leech lattice) 256650 26≥ 23071 27≥ 141793 28≥ 32797238 29≥ 37938009≥ 8900 30≥ 20169641025≥ 82000000 31≥ 5000000000000≥ 800000000000 32≥ 80000000000000000≥ 10000000000000000≥ 1160000000≥ 10900000 Beyond 32 dimensions, the numbers increase even more rapidly. Applications The second cohomology group of a closed simply connected oriented topological 4-manifold is a unimodular lattice. Michael Freedman showed that this lattice almost determines the manifold: there is a unique such manifold for each even unimodular lattice, and exactly two for each odd unimodular lattice. In particular if we take the lattice to be 0, this implies the Poincaré conjecture for 4-dimensional topological manifolds. Donaldson's theorem states that if the manifold is smooth and the lattice is positive definite, then it must be a sum of copies of Z, so most of these manifolds have no smooth structure. One such example is the $E_{8}$ manifold. References 1. Nebe, Gabriele; Sloane, Neil. "Unimodular Lattices, Together With A Table of the Best Such Lattices". Online Catalogue of Lattices. Retrieved 2015-05-30. 2. Nebe, Gabriele (2013). "Boris Venkov's Theory of Lattices and Spherical Designs". In Wan, Wai Kiu; Fukshansky, Lenny; Schulze-Pillot, Rainer; et al. (eds.). Diophantine methods, lattices, and arithmetic theory of quadratic forms. Contemporary Mathematics. Vol. 587. Providence, RI: American Mathematical Society. pp. 1–19. arXiv:1201.1834. Bibcode:2012arXiv1201.1834N. MR 3074799. • Bacher, Roland; Venkov, Boris (2001), "Réseaux entiers unimodulaires sans racine en dimension 27 et 28" [Unimodular integral lattices without roots in dimensions 27 and 28], in Martinet, Jacques (ed.), Réseaux euclidiens, designs sphériques et formes modulaires [Euclidean lattices, spherical designs and modular forms], Monogr. Enseign. Math. (in French), vol. 37, Geneva: L'Enseignement Mathématique, pp. 212–267, ISBN 2-940264-02-3, MR 1878751, Zbl 1139.11319, archived from the original on 2007-09-28 • Conway, J.H.; Sloane, N.J.A. (1999), Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften, vol. 290, With contributions by Bannai, E.; Borcherds, R.E.; Leech, J.; Norton, S.P.; Odlyzko, A.M.; Parker, R.A.; Queen, L.; Venkov, B.B. (Third ed.), New York, NY: Springer-Verlag, ISBN 0-387-98585-9, MR 0662447, Zbl 0915.52003 • King, Oliver D. (2003), "A mass formula for unimodular lattices with no roots", Mathematics of Computation, 72 (242): 839–863, arXiv:math.NT/0012231, Bibcode:2003MaCom..72..839K, doi:10.1090/S0025-5718-02-01455-2, MR 1954971, S2CID 7766244, Zbl 1099.11035 • Milnor, John; Husemoller, Dale (1973), Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 73, New York-Heidelberg: Springer-Verlag, doi:10.1007/978-3-642-88330-9, ISBN 3-540-06009-X, MR 0506372, Zbl 0292.10016 • Serre, Jean-Pierre (1973), A Course in Arithmetic, Graduate Texts in Mathematics, vol. 7, Springer-Verlag, doi:10.1007/978-1-4684-9884-4, ISBN 0-387-90040-3, MR 0344216, Zbl 0256.12001 • Freedman, Michael H. (1982), "The topology of four-dimensional manifolds", J. Differential Geom., 17 (3): 357–453, doi:10.4310/jdg/1214437136 External links • Gabriele Nebe and Neil Sloane's catalogue of unimodular lattices. • OEIS sequence A005134 (Number of n-dimensional unimodular lattices)	Wikipedia
Coherent topology In topology, a coherent topology is a topology that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent with a family of subspaces if it is a topological union of those subspaces. It is also sometimes called the weak topology generated by the family of subspaces, a notion that is quite different from the notion of a weak topology generated by a set of maps.[1] Definition Let $X$ be a topological space and let $C=\left\{C_{\alpha }:\alpha \in A\right\}$ be a family of subsets of $X$ each having the subspace topology. (Typically $C$ will be a cover of $X$.) Then $X$ is said to be coherent with $C$ (or determined by $C$)[2] if the topology of $X$ is recovered as the one coming from the final topology coinduced by the inclusion maps $i_{\alpha }:C_{\alpha }\to X\qquad \alpha \in A.$ By definition, this is the finest topology on (the underlying set of) $X$ for which the inclusion maps are continuous. $X$ is coherent with $C$ if either of the following two equivalent conditions holds: • A subset $U$ is open in $X$ if and only if $U\cap C_{\alpha }$ is open in $C_{\alpha }$ for each $\alpha \in A.$ • A subset $U$ is closed in $X$ if and only if $U\cap C_{\alpha }$ is closed in $C_{\alpha }$ for each $\alpha \in A.$ Given a topological space $X$ and any family of subspaces $C$ there is a unique topology on (the underlying set of) $X$ that is coherent with $C.$ This topology will, in general, be finer than the given topology on $X.$ Examples • A topological space $X$ is coherent with every open cover of $X.$ More generally, $X$ is coherent with any family of subsets whose interiors cover $X.$ As examples of this, a weakly locally compact space is coherent with the family of its compact subspaces. And a locally connected space is coherent with the family of its connected subsets. • A topological space $X$ is coherent with every locally finite closed cover of $X.$ • A discrete space is coherent with every family of subspaces (including the empty family). • A topological space $X$ is coherent with a partition of $X$ if and only $X$ is homeomorphic to the disjoint union of the elements of the partition. • Finitely generated spaces are those determined by the family of all finite subspaces. • Compactly generated spaces are those determined by the family of all compact subspaces. • A CW complex $X$ is coherent with its family of $n$-skeletons $X_{n}.$ Topological union Let $\left\{X_{\alpha }:\alpha \in A\right\}$ be a family of (not necessarily disjoint) topological spaces such that the induced topologies agree on each intersection $X_{\alpha }\cap X_{\beta }.$ Assume further that $X_{\alpha }\cap X_{\beta }$ is closed in $X_{\alpha }$ for each $\alpha ,\beta \in A.$ Then the topological union $X$ is the set-theoretic union $X^{set}=\bigcup _{\alpha \in A}X_{\alpha }$ endowed with the final topology coinduced by the inclusion maps $i_{\alpha }:X_{\alpha }\to X^{set}$. The inclusion maps will then be topological embeddings and $X$ will be coherent with the subspaces $\left\{X_{\alpha }\right\}.$ Conversely, if $X$ is a topological space and is coherent with a family of subspaces $\left\{C_{\alpha }\right\}$ that cover $X,$ then $X$ is homeomorphic to the topological union of the family $\left\{C_{\alpha }\right\}.$ One can form the topological union of an arbitrary family of topological spaces as above, but if the topologies do not agree on the intersections then the inclusions will not necessarily be embeddings. One can also describe the topological union by means of the disjoint union. Specifically, if $X$ is a topological union of the family $\left\{X_{\alpha }\right\},$ then $X$ is homeomorphic to the quotient of the disjoint union of the family $\left\{X_{\alpha }\right\}$ by the equivalence relation $(x,\alpha )\sim (y,\beta )\Leftrightarrow x=y$ for all $\alpha ,\beta \in A.$; that is, $X\cong \coprod _{\alpha \in A}X_{\alpha }/\sim .$ If the spaces $\left\{X_{\alpha }\right\}$ are all disjoint then the topological union is just the disjoint union. Assume now that the set A is directed, in a way compatible with inclusion: $\alpha \leq \beta $ whenever $X_{\alpha }\subset X_{\beta }$. Then there is a unique map from $\varinjlim X_{\alpha }$ to $X,$ which is in fact a homeomorphism. Here $\varinjlim X_{\alpha }$ is the direct (inductive) limit (colimit) of $\left\{X_{\alpha }\right\}$ in the category Top. Properties Let $X$ be coherent with a family of subspaces $\left\{C_{\alpha }\right\}.$ A function $f:X\to Y$ from $X$ to a topological space $Y$ is continuous if and only if the restrictions $f{\big \vert }_{C_{\alpha }}:C_{\alpha }\to Y\,$ are continuous for each $\alpha \in A.$ This universal property characterizes coherent topologies in the sense that a space $X$ is coherent with $C$ if and only if this property holds for all spaces $Y$ and all functions $f:X\to Y.$ Let $X$ be determined by a cover $C=\left\{C_{\alpha }\right\}.$ Then • If $C$ is a refinement of a cover $D,$ then $X$ is determined by $D.$ In particular, if $C$ is a subcover of $D,$ $X$ is determined by $D.$ • If $D$ is a refinement of $C$ and each $C_{\alpha }$ is determined by the family of all $D_{\beta }$ contained in $C_{\alpha }$ then $X$ is determined by $D.$ • Let $Y$ be an open or closed subspace of $X,$ or more generally a locally closed subset of $X.$ Then $Y$ is determined by $\left\{Y\cap C_{\alpha }\right\}.$ • Let $f:X\to Y$ be a quotient map. Then $Y$ is determined by $\left\{f(C_{\alpha })\right\}.$ Let $f:X\to Y$ be a surjective map and suppose $Y$ is determined by $\left\{D_{\alpha }:\alpha \in A\right\}.$ For each $\alpha \in A$ let $ f_{\alpha }:f^{-1}(D_{\alpha })\to D_{\alpha }\,$be the restriction of $f$ to $f^{-1}(D_{\alpha }).$ Then • If $f$ is continuous and each $f_{\alpha }$ is a quotient map, then $f$ is a quotient map. • $f$ is a closed map (resp. open map) if and only if each $f_{\alpha }$ is closed (resp. open). Given a topological space $(X,\tau )$ and a family of subspaces $C=\{C_{\alpha }\}$ there is a unique topology $\tau _{C}$ on $X$ that is coherent with $C.$ The topology $\tau _{C}$ is finer than the original topology $\tau ,$ and strictly finer if $\tau $ was not coherent with $C.$ But the topologies $\tau $ and $\tau _{C}$ induce the same subspace topology on each of the $C_{\alpha }$ in the family $C.$ And the topology $\tau _{C}$ is always coherent with $C.$ As an example of this last construction, if $C$ is the collection of all compact subspaces of a topological space $(X,\tau ),$ the resulting topology $\tau _{C}$ defines the k-ification $kX$ of $X.$ The spaces $X$ and $kX$ have the same compact sets, with the same induced subspace topologies. And the k-ification $kX$ is compactly generated. See also • Final topology – Finest topology making some functions continuous Notes 1. Willard, p. 69 2. $X$ is also said to have the weak topology generated by $C.$ This is a potentially confusing name since the adjectives weak and strong are used with opposite meanings by different authors. In modern usage the term weak topology is synonymous with initial topology and strong topology is synonymous with final topology. It is the final topology that is being discussed here. References • Tanaka, Yoshio (2004). "Quotient Spaces and Decompositions". In K.P. Hart; J. Nagata; J.E. Vaughan (eds.). Encyclopedia of General Topology. Amsterdam: Elsevier Science. pp. 43–46. ISBN 0-444-50355-2. • Willard, Stephen (1970). General Topology. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6. (Dover edition).	Wikipedia
Axiom of union In axiomatic set theory, the axiom of union is one of the axioms of Zermelo–Fraenkel set theory. This axiom was introduced by Ernst Zermelo.[1] The axiom states that for each set x there is a set y whose elements are precisely the elements of the elements of x. Formal statement In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: $\forall A\,\exists B\,\forall c\,(c\in B\iff \exists D\,(c\in D\land D\in A)\,)$ or in words: Given any set A, there is a set B such that, for any element c, c is a member of B if and only if there is a set D such that c is a member of D and D is a member of A. or, more simply: For any set $A$, there is a set $\bigcup A\ $ which consists of just the elements of the elements of that set $A$. Relation to Pairing The axiom of union allows one to unpack a set of sets and thus create a flatter set. Together with the axiom of pairing, this implies that for any two sets, there is a set (called their union) that contains exactly the elements of the two sets. Relation to Replacement The axiom of replacement allows one to form many unions, such as the union of two sets. However, in its full generality, the axiom of union is independent from the rest of the ZFC-axioms: Replacement does not prove the existence of the union of a set of sets if the result contains an unbounded number of cardinalities. Together with the axiom schema of replacement, the axiom of union implies that one can form the union of a family of sets indexed by a set. Relation to Separation In the context of set theories which include the axiom of separation, the axiom of union is sometimes stated in a weaker form which only produces a superset of the union of a set. For example, Kunen[2] states the axiom as $\forall {\mathcal {F}}\,\exists A\,\forall Y\,\forall x[(x\in Y\land Y\in {\mathcal {F}})\Rightarrow x\in A].$ which is equivalent to $\forall {\mathcal {F}}\,\exists A\forall x[[\exists Y(x\in Y\land Y\in {\mathcal {F}})]\Rightarrow x\in A].$ Compared to the axiom stated at the top of this section, this variation asserts only one direction of the implication, rather than both directions. Relation to Intersection There is no corresponding axiom of intersection. If $A$ is a nonempty set containing $E$, it is possible to form the intersection $\bigcap A$ using the axiom schema of specification as $\bigcap A=\{c\in E:\forall D(D\in A\Rightarrow c\in D)\}$, so no separate axiom of intersection is necessary. (If A is the empty set, then trying to form the intersection of A as {c: for all D in A, c is in D} is not permitted by the axioms. Moreover, if such a set existed, then it would contain every set in the "universe", but the notion of a universal set is antithetical to Zermelo–Fraenkel set theory.) References 1. Ernst Zermelo, 1908, "Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen 65(2), pp. 261–281. English translation: Jean van Heijenoort, 1967, From Frege to Gödel: A Source Book in Mathematical Logic, pp. 199–215 ISBN 978-0-674-32449-7 2. Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. ISBN 0-444-86839-9. Further reading • Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition). • Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. ISBN 3-540-44085-2. Set theory Overview • Set (mathematics) Axioms • Adjunction • Choice • countable • dependent • global • Constructibility (V=L) • Determinacy • Extensionality • Infinity • Limitation of size • Pairing • Power set • Regularity • Union • Martin's axiom • Axiom schema • replacement • specification Operations • Cartesian product • Complement (i.e. set difference) • De Morgan's laws • Disjoint union • Identities • Intersection • Power set • Symmetric difference • Union • Concepts • Methods • Almost • Cardinality • Cardinal number (large) • Class • Constructible universe • Continuum hypothesis • Diagonal argument • Element • ordered pair • tuple • Family • Forcing • One-to-one correspondence • Ordinal number • Set-builder notation • Transfinite induction • Venn diagram Set types • Amorphous • Countable • Empty • Finite (hereditarily) • Filter • base • subbase • Ultrafilter • Fuzzy • Infinite (Dedekind-infinite) • Recursive • Singleton • Subset · Superset • Transitive • Uncountable • Universal Theories • Alternative • Axiomatic • Naive • Cantor's theorem • Zermelo • General • Principia Mathematica • New Foundations • Zermelo–Fraenkel • von Neumann–Bernays–Gödel • Morse–Kelley • Kripke–Platek • Tarski–Grothendieck • Paradoxes • Problems • Russell's paradox • Suslin's problem • Burali-Forti paradox Set theorists • Paul Bernays • Georg Cantor • Paul Cohen • Richard Dedekind • Abraham Fraenkel • Kurt Gödel • Thomas Jech • John von Neumann • Willard Quine • Bertrand Russell • Thoralf Skolem • Ernst Zermelo	Wikipedia
Unipotent representation In mathematics, a unipotent representation of a reductive group is a representation that has some similarities with unipotent conjugacy classes of groups. Informally, Langlands philosophy suggests that there should be a correspondence between representations of a reductive group and conjugacy classes of a Langlands dual group, and the unipotent representations should be roughly the ones corresponding to unipotent classes in the dual group. Unipotent representations are supposed to be the basic "building blocks" out of which one can construct all other representations in the following sense. Unipotent representations should form a small (preferably finite) set of irreducible representations for each reductive group, such that all irreducible representations can be obtained from unipotent representations of possibly smaller groups by some sort of systematic process, such as (cohomological or parabolic) induction. Finite fields Over finite fields, the unipotent representations are those that occur in the decomposition of the Deligne–Lusztig characters R1 T of the trivial representation 1 of a torus T . They were classified by Lusztig (1978, 1979). Some examples of unipotent representations over finite fields are the trivial 1-dimensional representation, the Steinberg representation, and θ10. Non-archimedean local fields Lusztig (1995) classified the unipotent characters over non-archimedean local fields. Archimedean local fields Vogan (1987) discusses several different possible definitions of unipotent representations of real Lie groups. See also • Deligne–Lusztig theory References • Barbasch, Dan (1991), "Unipotent representations for real reductive groups", in Satake, Ichirô (ed.), Proceedings of the International Congress of Mathematicians, Vol. II (Kyoto, 1990), Tokyo: Math. Soc. Japan, pp. 769–777, ISBN 978-4-431-70047-0, MR 1159263 • Lusztig, George (1979), "Unipotent representations of a finite Chevalley group of type E8", The Quarterly Journal of Mathematics, Second Series, 30 (3): 315–338, doi:10.1093/qmath/30.3.315, ISSN 0033-5606, MR 0545068 • Lusztig, George (1978), Representations of finite Chevalley groups, CBMS Regional Conference Series in Mathematics, vol. 39, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1689-9, MR 0518617 • Lusztig, George (1995), "Classification of unipotent representations of simple p-adic groups", International Mathematics Research Notices, 1995 (11): 517–589, arXiv:math/0111248, doi:10.1155/S1073792895000353, ISSN 1073-7928, MR 1369407 • Vogan, David A. (1987), Unitary representations of reductive Lie groups, Annals of Mathematics Studies, vol. 118, Princeton University Press, ISBN 978-0-691-08482-4	Wikipedia
Fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors.[3][4][5] For example, $1200=2^{4}\cdot 3^{1}\cdot 5^{2}=(2\cdot 2\cdot 2\cdot 2)\cdot 3\cdot (5\cdot 5)=5\cdot 2\cdot 5\cdot 2\cdot 3\cdot 2\cdot 2=\ldots $ Not to be confused with Fundamental theorem of algebra. The theorem says two things about this example: first, that 1200 can be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product. The requirement that the factors be prime is necessary: factorizations containing composite numbers may not be unique (for example, $12=2\cdot 6=3\cdot 4$). This theorem is one of the main reasons why 1 is not considered a prime number: if 1 were prime, then factorization into primes would not be unique; for example, $2=2\cdot 1=2\cdot 1\cdot 1=\ldots $ The theorem generalizes to other algebraic structures that are called unique factorization domains and include principal ideal domains, Euclidean domains, and polynomial rings over a field. However, the theorem does not hold for algebraic integers.[6] This failure of unique factorization is one of the reasons for the difficulty of the proof of Fermat's Last Theorem. The implicit use of unique factorization in rings of algebraic integers is behind the error of many of the numerous false proofs that have been written during the 358 years between Fermat's statement and Wiles's proof. History The fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of Euclid's Elements. If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers. — Euclid, Elements Book VII, Proposition 30 (In modern terminology: if a prime p divides the product ab, then p divides either a or b or both.) Proposition 30 is referred to as Euclid's lemma, and it is the key in the proof of the fundamental theorem of arithmetic. Any composite number is measured by some prime number. — Euclid, Elements Book VII, Proposition 31 (In modern terminology: every integer greater than one is divided evenly by some prime number.) Proposition 31 is proved directly by infinite descent. Any number either is prime or is measured by some prime number. — Euclid, Elements Book VII, Proposition 32 Proposition 32 is derived from proposition 31, and proves that the decomposition is possible. If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it. — Euclid, Elements Book IX, Proposition 14 (In modern terminology: a least common multiple of several prime numbers is not a multiple of any other prime number.) Book IX, proposition 14 is derived from Book VII, proposition 30, and proves partially that the decomposition is unique – a point critically noted by André Weil.[7] Indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case. While Euclid took the first step on the way to the existence of prime factorization, Kamāl al-Dīn al-Fārisī took the final step[8] and stated for the first time the fundamental theorem of arithmetic.[9] Article 16 of Gauss' Disquisitiones Arithmeticae is an early modern statement and proof employing modular arithmetic.[1] Applications Canonical representation of a positive integer Every positive integer n > 1 can be represented in exactly one way as a product of prime powers $n=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots p_{k}^{n_{k}}=\prod _{i=1}^{k}p_{i}^{n_{i}},$ where p1 < p2 < ... < pk are primes and the ni are positive integers. This representation is commonly extended to all positive integers, including 1, by the convention that the empty product is equal to 1 (the empty product corresponds to k = 0). This representation is called the canonical representation[10] of n, or the standard form[11][12] of n. For example, 999 = 33×37, 1000 = 23×53, 1001 = 7×11×13. Factors p0 = 1 may be inserted without changing the value of n (for example, 1000 = 23×30×53). In fact, any positive integer can be uniquely represented as an infinite product taken over all the positive prime numbers, as $n=2^{n_{1}}3^{n_{2}}5^{n_{3}}7^{n_{4}}\cdots =\prod _{i=1}^{\infty }p_{i}^{n_{i}},$ where a finite number of the ni are positive integers, and the others are zero. Allowing negative exponents provides a canonical form for positive rational numbers. Arithmetic operations The canonical representations of the product, greatest common divisor (GCD), and least common multiple (LCM) of two numbers a and b can be expressed simply in terms of the canonical representations of a and b themselves: ${\begin{alignedat}{2}a\cdot b&=2^{a_{1}+b_{1}}3^{a_{2}+b_{2}}5^{a_{3}+b_{3}}7^{a_{4}+b_{4}}\cdots &&=\prod p_{i}^{a_{i}+b_{i}},\\\gcd(a,b)&=2^{\min(a_{1},b_{1})}3^{\min(a_{2},b_{2})}5^{\min(a_{3},b_{3})}7^{\min(a_{4},b_{4})}\cdots &&=\prod p_{i}^{\min(a_{i},b_{i})},\\\operatorname {lcm} (a,b)&=2^{\max(a_{1},b_{1})}3^{\max(a_{2},b_{2})}5^{\max(a_{3},b_{3})}7^{\max(a_{4},b_{4})}\cdots &&=\prod p_{i}^{\max(a_{i},b_{i})}.\end{alignedat}}$ However, integer factorization, especially of large numbers, is much more difficult than computing products, GCDs, or LCMs. So these formulas have limited use in practice. Arithmetic functions Main article: Arithmetic function Many arithmetic functions are defined using the canonical representation. In particular, the values of additive and multiplicative functions are determined by their values on the powers of prime numbers. Proof The proof uses Euclid's lemma (Elements VII, 30): If a prime divides the product of two integers, then it must divide at least one of these integers. Existence It must be shown that every integer greater than 1 is either prime or a product of primes. First, 2 is prime. Then, by strong induction, assume this is true for all numbers greater than 1 and less than n. If n is prime, there is nothing more to prove. Otherwise, there are integers a and b, where n = a b, and 1 < a ≤ b < n. By the induction hypothesis, a = p1 p2 ⋅⋅⋅ pj and b = q1 q2 ⋅⋅⋅ qk are products of primes. But then n = a b = p1 p2 ⋅⋅⋅ pj q1 q2 ⋅⋅⋅ qk is a product of primes. Uniqueness Suppose, to the contrary, there is an integer that has two distinct prime factorizations. Let n be the least such integer and write n = p1 p2 ... pj = q1 q2 ... qk, where each pi and qi is prime. We see that p1 divides q1 q2 ... qk, so p1 divides some qi by Euclid's lemma. Without loss of generality, say p1 divides q1. Since p1 and q1 are both prime, it follows that p1 = q1. Returning to our factorizations of n, we may cancel these two factors to conclude that p2 ... pj = q2 ... qk. We now have two distinct prime factorizations of some integer strictly smaller than n, which contradicts the minimality of n. Uniqueness without Euclid's lemma The fundamental theorem of arithmetic can also be proved without using Euclid's lemma.[13] The proof that follows is inspired by Euclid's original version of the Euclidean algorithm. Assume that $s$ is the smallest positive integer which is the product of prime numbers in two different ways. Incidentally, this implies that $s$, if it exists, must be a composite number greater than $1$. Now, say ${\begin{aligned}s&=p_{1}p_{2}\cdots p_{m}\\&=q_{1}q_{2}\cdots q_{n}.\end{aligned}}$ Every $p_{i}$ must be distinct from every $q_{j}.$ Otherwise, if say $p_{i}=q_{j},$ then there would exist some positive integer $t=s/p_{i}=s/q_{j}$ that is smaller than s and has two distinct prime factorizations. One may also suppose that $p_{1}<q_{1},$ by exchanging the two factorizations, if needed. Setting $P=p_{2}\cdots p_{m}$ and $Q=q_{2}\cdots q_{n},$ one has $s=p_{1}P=q_{1}Q.$ Also, since $p_{1}<q_{1},$ one has $Q<P.$ It then follows that $s-p_{1}Q=(q_{1}-p_{1})Q=p_{1}(P-Q)<s.$ As the positive integers less than s have been supposed to have a unique prime factorization, $p_{1}$ must occur in the factorization of either $q_{1}-p_{1}$ or Q. The latter case is impossible, as Q, being smaller than s, must have a unique prime factorization, and $p_{1}$ differs from every $q_{j}.$ The former case is also impossible, as, if $p_{1}$ is a divisor of $q_{1}-p_{1},$ it must be also a divisor of $q_{1},$ which is impossible as $p_{1}$ and $q_{1}$ are distinct primes. Therefore, there cannot exist a smallest integer with more than a single distinct prime factorization. Every positive integer must either be a prime number itself, which would factor uniquely, or a composite that also factors uniquely into primes, or in the case of the integer $1$, not factor into any prime. Generalizations The first generalization of the theorem is found in Gauss's second monograph (1832) on biquadratic reciprocity. This paper introduced what is now called the ring of Gaussian integers, the set of all complex numbers a + bi where a and b are integers. It is now denoted by $\mathbb {Z} [i].$ He showed that this ring has the four units ±1 and ±i, that the non-zero, non-unit numbers fall into two classes, primes and composites, and that (except for order), the composites have unique factorization as a product of primes (up to the order and multiplication by units).[14] Similarly, in 1844 while working on cubic reciprocity, Eisenstein introduced the ring $\mathbb {Z} [\omega ]$, where $ \omega ={\frac {-1+{\sqrt {-3}}}{2}},$ $\omega ^{3}=1$ is a cube root of unity. This is the ring of Eisenstein integers, and he proved it has the six units $\pm 1,\pm \omega ,\pm \omega ^{2}$ and that it has unique factorization. However, it was also discovered that unique factorization does not always hold. An example is given by $\mathbb {Z} [{\sqrt {-5}}]$. In this ring one has[15] $6=2\cdot 3=\left(1+{\sqrt {-5}}\right)\left(1-{\sqrt {-5}}\right).$ Examples like this caused the notion of "prime" to be modified. In $\mathbb {Z} \left[{\sqrt {-5}}\right]$ it can be proven that if any of the factors above can be represented as a product, for example, 2 = ab, then one of a or b must be a unit. This is the traditional definition of "prime". It can also be proven that none of these factors obeys Euclid's lemma; for example, 2 divides neither (1 + √−5) nor (1 − √−5) even though it divides their product 6. In algebraic number theory 2 is called irreducible in $\mathbb {Z} \left[{\sqrt {-5}}\right]$ (only divisible by itself or a unit) but not prime in $\mathbb {Z} \left[{\sqrt {-5}}\right]$ (if it divides a product it must divide one of the factors). The mention of $\mathbb {Z} \left[{\sqrt {-5}}\right]$ is required because 2 is prime and irreducible in $\mathbb {Z} .$ Using these definitions it can be proven that in any integral domain a prime must be irreducible. Euclid's classical lemma can be rephrased as "in the ring of integers $\mathbb {Z} $ every irreducible is prime". This is also true in $\mathbb {Z} [i]$ and $\mathbb {Z} [\omega ],$ but not in $\mathbb {Z} [{\sqrt {-5}}].$ The rings in which factorization into irreducibles is essentially unique are called unique factorization domains. Important examples are polynomial rings over the integers or over a field, Euclidean domains and principal ideal domains. In 1843 Kummer introduced the concept of ideal number, which was developed further by Dedekind (1876) into the modern theory of ideals, special subsets of rings. Multiplication is defined for ideals, and the rings in which they have unique factorization are called Dedekind domains. There is a version of unique factorization for ordinals, though it requires some additional conditions to ensure uniqueness. See also • Integer factorization – Decomposition of a number into a product • Prime signature – Multiset of prime exponents in a prime factorization Notes 1. Gauss (1986, Art. 16) 2. Gauss (1986, Art. 131) 3. Long (1972, p. 44) 4. Pettofrezzo & Byrkit (1970, p. 53) 5. Hardy & Wright (2008, Thm 2) 6. In a ring of algebraic integers, the factorization into prime elements may be non unique, but one can recover a unique factorization if one factors into ideals. 7. Weil (2007, p. 5): "Even in Euclid, we fail to find a general statement about the uniqueness of the factorization of an integer into primes; surely he may have been aware of it, but all he has is a statement (Eucl.IX.I4) about the l.c.m. of any number of given primes." 8. A. Goksel Agargun and E. Mehmet Özkan. "A Historical Survey of the Fundamental Theorem of Arithmetic" (PDF). Historia Mathematica: 209. One could say that Euclid takes the first step on the way to the existence of prime factorization, and al-Farisi takes the final step by actually proving the existence of a finite prime factorization in his first proposition. 9. Rashed, Roshdi (2002-09-11). Encyclopedia of the History of Arabic Science. Routledge. p. 385. ISBN 9781134977246. The famous physicist and mathematician Kamal al-Din al-Farisi compiled a paper in which he set out deliberately to prove the theorem of Ibn Qurra in an algebraic way. This forced him to an understanding of the first arithmetical functions and to a full preparation which led him to state for the first time the fundamental theorem of arithmetic. 10. Long (1972, p. 45) 11. Pettofrezzo & Byrkit (1970, p. 55) 12. Hardy & Wright (2008, § 1.2) 13. Dawson, John W. (2015), Why Prove it Again? Alternative Proofs in Mathematical Practice., Springer, p. 45, ISBN 9783319173689 14. Gauss, BQ, §§ 31–34 15. Hardy & Wright (2008, § 14.6) References The Disquisitiones Arithmeticae has been translated from Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. • Gauss, Carl Friedrich (1986), Disquisitiones Arithemeticae (Second, corrected edition), translated by Clarke, Arthur A., New York: Springer, ISBN 978-0-387-96254-2 • Gauss, Carl Friedrich (1965), Untersuchungen über hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition) (in German), translated by Maser, H., New York: Chelsea, ISBN 0-8284-0191-8 The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76. Footnotes referencing these are of the form "Gauss, BQ, § n". Footnotes referencing the Disquisitiones Arithmeticae are of the form "Gauss, DA, Art. n". • Gauss, Carl Friedrich (1828), Theoria residuorum biquadraticorum, Commentatio prima, Göttingen: Comment. Soc. regiae sci, Göttingen 6 • Gauss, Carl Friedrich (1832), Theoria residuorum biquadraticorum, Commentatio secunda, Göttingen: Comment. Soc. regiae sci, Göttingen 7 These are in Gauss's Werke, Vol II, pp. 65–92 and 93–148; German translations are pp. 511–533 and 534–586 of the German edition of the Disquisitiones. • Euclid (1956), The thirteen books of the Elements, vol. 2 (Books III-IX), Translated by Thomas Little Heath (Second Edition Unabridged ed.), New York: Dover, ISBN 978-0-486-60089-5 • Hardy, G. H.; Wright, E. M. (2008) [1938], An Introduction to the Theory of Numbers, Revised by D. R. Heath-Brown and J. H. Silverman. Foreword by Andrew Wiles. (6th ed.), Oxford: Oxford University Press, ISBN 978-0-19-921986-5, MR 2445243, Zbl 1159.11001 • Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: D. C. Heath and Company, LCCN 77-171950. • Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: Prentice Hall, LCCN 77-81766. • Riesel, Hans (1994), Prime Numbers and Computer Methods for Factorization (second edition), Boston: Birkhäuser, ISBN 0-8176-3743-5 • Weil, André (2007) [1984], Number Theory: An Approach through History from Hammurapi to Legendre, Modern Birkhäuser Classics, Boston, MA: Birkhäuser, ISBN 978-0-817-64565-6 External links • Why isn’t the fundamental theorem of arithmetic obvious? • GCD and the Fundamental Theorem of Arithmetic at cut-the-knot. • PlanetMath: Proof of fundamental theorem of arithmetic • Fermat's Last Theorem Blog: Unique Factorization, a blog that covers the history of Fermat's Last Theorem from Diophantus of Alexandria to the proof by Andrew Wiles. • "Fundamental Theorem of Arithmetic" by Hector Zenil, Wolfram Demonstrations Project, 2007. • Grime, James, "1 and Prime Numbers", Numberphile, Brady Haran, archived from the original on 2021-12-11 Divisibility-based sets of integers Overview • Integer factorization • Divisor • Unitary divisor • Divisor function • Prime factor • Fundamental theorem of arithmetic Factorization forms • Prime • Composite • Semiprime • Pronic • Sphenic • Square-free • Powerful • Perfect power • Achilles • Smooth • Regular • Rough • Unusual Constrained divisor sums • Perfect • Almost perfect • Quasiperfect • Multiply perfect • Hemiperfect • Hyperperfect • Superperfect • Unitary perfect • Semiperfect • Practical • Erdős–Nicolas With many divisors • Abundant • Primitive abundant • Highly abundant • Superabundant • Colossally abundant • Highly composite • Superior highly composite • Weird Aliquot sequence-related • Untouchable • Amicable (Triple) • Sociable • Betrothed Base-dependent • Equidigital • Extravagant • Frugal • Harshad • Polydivisible • Smith Other sets • Arithmetic • Deficient • Friendly • Solitary • Sublime • Harmonic divisor • Descartes • Refactorable • Superperfect	Wikipedia
Ergodicity In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored into smaller components. Ergodic theory is the study of systems possessing ergodicity. Ergodic systems occur in a broad range of systems in physics and in geometry. This can be roughly understood to be due to a common phenomenon: the motion of particles, that is, geodesics on a hyperbolic manifold are divergent; when that manifold is compact, that is, of finite size, those orbits return to the same general area, eventually filling the entire space. Ergodic systems capture the common-sense, every-day notions of randomness, such that smoke might come to fill all of a smoke-filled room, or that a block of metal might eventually come to have the same temperature throughout, or that flips of a fair coin may come up heads and tails half the time. A stronger concept than ergodicity is that of mixing, which aims to mathematically describe the common-sense notions of mixing, such as mixing drinks or mixing cooking ingredients. The proper mathematical formulation of ergodicity is founded on the formal definitions of measure theory and dynamical systems, and rather specifically on the notion of a measure-preserving dynamical system. The origins of ergodicity lie in statistical physics, where Ludwig Boltzmann formulated the ergodic hypothesis. Informal explanation Ergodicity occurs in broad settings in physics and mathematics. All of these settings are unified by a common mathematical description, that of the measure-preserving dynamical system. Equivalently, ergodicity can be understood in terms of stochastic processes. They are one and the same, despite using dramatically different notation and language. Measure-preserving dynamical systems The mathematical definition of ergodicity aims to capture ordinary every-day ideas about randomness. This includes ideas about systems that move in such a way as to (eventually) fill up all of space, such as diffusion and Brownian motion, as well as common-sense notions of mixing, such as mixing paints, drinks, cooking ingredients, industrial process mixing, smoke in a smoke-filled room, the dust in Saturn's rings and so on. To provide a solid mathematical footing, descriptions of ergodic systems begin with the definition of a measure-preserving dynamical system. This is written as $(X,{\mathcal {A}},\mu ,T).$ The set $X$ is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, etc. The measure $\mu $ is understood to define the natural volume of the space $X$ and of its subspaces. The collection of subspaces is denoted by ${\mathcal {A}}$, and the size of any given subset $A\subset X$ is $\mu (A)$; the size is its volume. Naively, one could imagine ${\mathcal {A}}$ to be the power set of $X$; this doesn't quite work, as not all subsets of a space have a volume (famously, the Banach-Tarski paradox). Thus, conventionally, ${\mathcal {A}}$ consists of the measurable subsets—the subsets that do have a volume. It is always taken to be a Borel set—the collection of subsets that can be constructed by taking intersections, unions and set complements of open sets; these can always be taken to be measurable. The time evolution of the system is described by a map $T:X\to X$. Given some subset $A\subset X$, its map $T(A)$ will in general be a deformed version of $A$ – it is squashed or stretched, folded or cut into pieces. Mathematical examples include the baker's map and the horseshoe map, both inspired by bread-making. The set $T(A)$ must have the same volume as $A$; the squashing/stretching does not alter the volume of the space, only its distribution. Such a system is "measure-preserving" (area-preserving, volume-preserving). A formal difficulty arises when one tries to reconcile the volume of sets with the need to preserve their size under a map. The problem arises because, in general, several different points in the domain of a function can map to the same point in its range; that is, there may be $x\neq y$ with $T(x)=T(y)$. Worse, a single point $x\in X$ has no size. These difficulties can be avoided by working with the inverse map $T^{-1}:{\mathcal {A}}\to {\mathcal {A}}$; it will map any given subset $A\subset X$ to the parts that were assembled to make it: these parts are $T^{-1}(A)\in {\mathcal {A}}$. It has the important property of not losing track of where things came from. More strongly, it has the important property that any (measure-preserving) map ${\mathcal {A}}\to {\mathcal {A}}$ is the inverse of some map $X\to X$. The proper definition of a volume-preserving map is one for which $\mu (A)=\mu {\mathord {\left(T^{-1}(A)\right)}}$ because $T^{-1}(A)$ describes all the pieces-parts that $A$ came from. One is now interested in studying the time evolution of the system. If a set $A\in {\mathcal {A}}$ eventually comes to fill all of $X$ over a long period of time (that is, if $T^{n}(A)$ approaches all of $X$ for large $n$), the system is said to be ergodic. If every set $A$ behaves in this way, the system is a conservative system, placed in contrast to a dissipative system, where some subsets $A$ wander away, never to be returned to. An example would be water running downhill: once it's run down, it will never come back up again. The lake that forms at the bottom of this river can, however, become well-mixed. The ergodic decomposition theorem states that every ergodic system can be split into two parts: the conservative part, and the dissipative part. Mixing is a stronger statement than ergodicity. Mixing asks for this ergodic property to hold between any two sets $A,B$, and not just between some set $A$ and $X$. That is, given any two sets $A,B\in {\mathcal {A}}$, a system is said to be (topologically) mixing if there is an integer $N$ such that, for all $A,B$ and $n>N$, one has that $T^{n}(A)\cap B\neq \varnothing $. Here, $\cap $ denotes set intersection and $\varnothing $ is the empty set. Other notions of mixing include strong and weak mixing, which describe the notion that the mixed substances intermingle everywhere, in equal proportion. This can be non-trivial, as practical experience of trying to mix sticky, gooey substances shows. Ergodic Processes The above discussion appeals to a physical sense of a volume. The volume does not have to literally be some portion of 3D space; it can be some abstract volume. This is generally the case in statistical systems, where the volume (the measure) is given by the probability. The total volume corresponds to probability one. This correspondence works because the axioms of probability theory are identical to those of measure theory; these are the Kolmogorov axioms. The idea of a volume can be very abstract. Consider, for example, the set of all possible coin-flips: the set of infinite sequences of heads and tails. Assigning the volume of 1 to this space, it is clear that half of all such sequences start with heads, and half start with tails. One can slice up this volume in other ways: one can say "I don't care about the first $n-1$ coin-flips; but I want the $n$'th of them to be heads, and then I don't care about what comes after that". This can be written as the set $(,\cdots ,,h,,\cdots )$ where $$ is "don't care" and $h$ is "heads". The volume of this space is again one-half. The above is enough to build up a measure-preserving dynamical system, in its entirety. The sets of $h$ or $t$ occurring in the $n$'th place are called cylinder sets. The set of all possible intersections, unions and complements of the cylinder sets then form the Borel set ${\mathcal {A}}$ defined above. In formal terms, the cylinder sets form the base for a topology on the space $X$ of all possible infinite-length coin-flips. The measure $\mu $ has all of the common-sense properties one might hope for: the measure of a cylinder set with $h$ in the $m$'th position, and $t$ in the $k$'th position is obviously 1/4, and so on. These common-sense properties persist for set-complement and set-union: everything except for $h$ and $t$ in locations $m$ and $k$ obviously has the volume of 3/4. All together, these form the axioms of a sigma-additive measure; measure-preserving dynamical systems always use sigma-additive measures. For coin flips, this measure is called the Bernoulli measure. For the coin-flip process, the time-evolution operator $T$ is the shift operator that says "throw away the first coin-flip, and keep the rest". Formally, if $(x_{1},x_{2},\cdots )$ is a sequence of coin-flips, then $T(x_{1},x_{2},\cdots )=(x_{2},x_{3},\cdots )$. The measure is obviously shift-invariant: as long as we are talking about some set $A\in {\mathcal {A}}$ where the first coin-flip $x_{1}=$ is the "don't care" value, then the volume $\mu (A)$ does not change: $\mu (A)=\mu (T(A))$. In order to avoid talking about the first coin-flip, it is easier to define $T^{-1}$ as inserting a "don't care" value into the first position: $T^{-1}(x_{1},x_{2},\cdots )=(,x_{1},x_{2},\cdots )$. With this definition, one obviously has that $\mu {\mathord {\left(T^{-1}(A)\right)}}=\mu (A)$ with no constraints on $A$. This is again an example of why $T^{-1}$ is used in the formal definitions. The above development takes a random process, the Bernoulli process, and converts it to a measure-preserving dynamical system $(X,{\mathcal {A}},\mu ,T).$ The same conversion (equivalence, isomorphism) can be applied to any stochastic process. Thus, an informal definition of ergodicity is that a sequence is ergodic if it visits all of $X$; such sequences are "typical" for the process. Another is that its statistical properties can be deduced from a single, sufficiently long, random sample of the process (thus uniformly sampling all of $X$), or that any collection of random samples from a process must represent the average statistical properties of the entire process (that is, samples drawn uniformly from $X$ are representative of $X$ as a whole.) In the present example, a sequence of coin flips, where half are heads, and half are tails, is a "typical" sequence. There are several important points to be made about the Bernoulli process. If one writes 0 for tails and 1 for heads, one gets the set of all infinite strings of binary digits. These correspond to the base-two expansion of real numbers. Explicitly, given a sequence $(x_{1},x_{2},\cdots )$, the corresponding real number is $y=\sum _{n=1}^{\infty }{\frac {x_{n}}{2^{n}}}$ The statement that the Bernoulli process is ergodic is equivalent to the statement that the real numbers are uniformly distributed. The set of all such strings can be written in a variety of ways: $\{h,t\}^{\infty }=\{h,t\}^{\omega }=\{0,1\}^{\omega }=2^{\omega }=2^{\mathbb {N} }.$ This set is the Cantor set, sometimes called the Cantor space to avoid confusion with the Cantor function $C(x)=\sum _{n=1}^{\infty }{\frac {x_{n}}{3^{n}}}$ In the end, these are all "the same thing". The Cantor set plays key roles in many branches of mathematics. In recreational mathematics, it underpins the period-doubling fractals; in analysis, it appears in a vast variety of theorems. A key one for stochastic processes is the Wold decomposition, which states that any stationary process can be decomposed into a pair of uncorrelated processes, one deterministic, and the other being a moving average process. The Ornstein isomorphism theorem states that every stationary stochastic process is equivalent to a Bernoulli scheme (a Bernoulli process with an N-sided (and possibly unfair) gaming die). Other results include that every non-dissipative ergodic system is equivalent to the Markov odometer, sometimes called an "adding machine" because it looks like elementary-school addition, that is, taking a base-N digit sequence, adding one, and propagating the carry bits. The proof of equivalence is very abstract; understanding the result is not: by adding one at each time step, every possible state of the odometer is visited, until it rolls over, and starts again. Likewise, ergodic systems visit each state, uniformly, moving on to the next, until they have all been visited. Systems that generate (infinite) sequences of N letters are studied by means of symbolic dynamics. Important special cases include subshifts of finite type and sofic systems. History and etymology The term ergodic is commonly thought to derive from the Greek words ἔργον (ergon: "work") and ὁδός (hodos: "path", "way"), as chosen by Ludwig Boltzmann while he was working on a problem in statistical mechanics.[1] At the same time it is also claimed to be a derivation of ergomonode, coined by Boltzmann in a relatively obscure paper from 1884. The etymology appears to be contested in other ways as well.[2] The idea of ergodicity was born in the field of thermodynamics, where it was necessary to relate the individual states of gas molecules to the temperature of a gas as a whole and its time evolution thereof. In order to do this, it was necessary to state what exactly it means for gases to mix well together, so that thermodynamic equilibrium could be defined with mathematical rigor. Once the theory was well developed in physics, it was rapidly formalized and extended, so that ergodic theory has long been an independent area of mathematics in itself. As part of that progression, more than one slightly different definition of ergodicity and multitudes of interpretations of the concept in different fields coexist. For example, in classical physics the term implies that a system satisfies the ergodic hypothesis of thermodynamics,[3] the relevant state space being position and momentum space. In dynamical systems theory the state space is usually taken to be a more general phase space. On the other hand in coding theory the state space is often discrete in both time and state, with less concomitant structure. In all those fields the ideas of time average and ensemble average can also carry extra baggage as well—as is the case with the many possible thermodynamically relevant partition functions used to define ensemble averages in physics, back again. As such the measure theoretic formalization of the concept also serves as a unifying discipline. In 1913 Michel Plancherel proved the strict impossibility for ergodicity for a purely mechanical system. Ergodicity in physics and geometry A review of ergodicity in physics, and in geometry follows. In all cases, the notion of ergodicity is exactly the same as that for dynamical systems; there is no difference, except for outlook, notation, style of thinking and the journals where results are published. Physical systems can be split into three categories: classical mechanics, which describes machines with a finite number of moving parts, quantum mechanics, which describes the structure of atoms, and statistical mechanics, which describes gases, liquids, solids; this includes condensed matter physics. These presented below. In statistical mechanics This section reviews ergodicity in statistical mechanics. The above abstract definition of a volume is required as the appropriate setting for definitions of ergodicity in physics. Consider a container of liquid, or gas, or plasma, or other collection of atoms or particles. Each and every particle $x_{i}$ has a 3D position, and a 3D velocity, and is thus described by six numbers: a point in six-dimensional space $\mathbb {R} ^{6}.$ If there are $N$ of these particles in the system, a complete description requires $6N$ numbers. Any one system is just a single point in $\mathbb {R} ^{6N}.$ The physical system is not all of $\mathbb {R} ^{6N}$, of course; if it's a box of width, height and length $W\times H\times L$ then a point is in $\left(W\times H\times L\times \mathbb {R} ^{3}\right)^{N}.$ Nor can velocities be infinite: they are scaled by some probability measure, for example the Boltzmann–Gibbs measure for a gas. None-the-less, for $N$ close to the Avogadro number, this is obviously a very large space. This space is called the canonical ensemble. A physical system is said to be ergodic if any representative point of the system eventually comes to visit the entire volume of the system. For the above example, this implies that any given atom not only visits every part of the box $W\times H\times L$ with uniform probability, but it does so with every possible velocity, with probability given by the Boltzmann distribution for that velocity (so, uniform with respect to that measure). The ergodic hypothesis states that physical systems actually are ergodic. Multiple time scales are at work: gases and liquids appear to be ergodic over short time scales. Ergodicity in a solid can be viewed in terms of the vibrational modes or phonons, as obviously the atoms in a solid do not exchange locations. Glasses present a challenge to the ergodic hypothesis; time scales are assumed to be in the millions of years, but results are contentious. Spin glasses present particular difficulties. Formal mathematical proofs of ergodicity in statistical physics are hard to come by; most high-dimensional many-body systems are assumed to be ergodic, without mathematical proof. Exceptions include the dynamical billiards, which model billiard ball-type collisions of atoms in an ideal gas or plasma. The first hard-sphere ergodicity theorem was for Sinai's billiards, which considers two balls, one of them taken as being stationary, at the origin. As the second ball collides, it moves away; applying periodic boundary conditions, it then returns to collide again. By appeal to homogeneity, this return of the "second" ball can instead be taken to be "just some other atom" that has come into range, and is moving to collide with the atom at the origin (which can be taken to be just "any other atom".) This is one of the few formal proofs that exist; there are no equivalent statements e.g. for atoms in a liquid, interacting via van der Waals forces, even if it would be common sense to believe that such systems are ergodic (and mixing). More precise physical arguments can be made, though. Simple dynamical systems The formal study of ergodicity can be approached by examining fairly simple dynamical systems. Some of the primary ones are listed here. The irrational rotation of a circle is ergodic: the orbit of a point is such that eventually, every other point in the circle is visited. Such rotations are a special case of the interval exchange map. The beta expansions of a number are ergodic: beta expansions of a real number are done not in base-N, but in base-$\beta $ for some $\beta .$ The reflected version of the beta expansion is tent map; there are a variety of other ergodic maps of the unit interval. Moving to two dimensions, the arithmetic billiards with irrational angles are ergodic. One can also take a flat rectangle, squash it, cut it and reassemble it; this is the previously-mentioned baker's map. Its points can be described by the set of bi-infinite strings in two letters, that is, extending to both the left and right; as such, it looks like two copies of the Bernoulli process. If one deforms sideways during the squashing, one obtains Arnold's cat map. In most ways, the cat map is prototypical of any other similar transformation. In classical mechanics and geometry Ergodicity is a widespread phenomenon in the study of symplectic manifolds and Riemannian manifolds. Symplectic manifolds provide the generalized setting for classical mechanics, where the motion of a mechanical system is described by a geodesic. Riemannian manifolds are a special case: the cotangent bundle of a Riemannian manifold is always a symplectic manifold. In particular, the geodesics on a Riemannian manifold are given by the solution of the Hamilton–Jacobi equations. The geodesic flow of a flat torus following any irrational direction is ergodic; informally this means that when drawing a straight line in a square starting at any point, and with an irrational angle with respect to the sides, if every time one meets a side one starts over on the opposite side with the same angle, the line will eventually meet every subset of positive measure. More generally on any flat surface there are many ergodic directions for the geodesic flow. For non-flat surfaces, one has that the geodesic flow of any negatively curved compact Riemann surface is ergodic. A surface is "compact" in the sense that it has finite surface area. The geodesic flow is a generalization of the idea of moving in a "straight line" on a curved surface: such straight lines are geodesics. One of the earliest cases studied is Hadamard's billiards, which describes geodesics on the Bolza surface, topologically equivalent to a donut with two holes. Ergodicity can be demonstrated informally, if one has a sharpie and some reasonable example of a two-holed donut: starting anywhere, in any direction, one attempts to draw a straight line; rulers are useful for this. It doesn't take all that long to discover that one is not coming back to the starting point. (Of course, crooked drawing can also account for this; that's why we have proofs.) These results extend to higher dimensions. The geodesic flow for negatively curved compact Riemannian manifolds is ergodic. A classic example for this is the Anosov flow, which is the horocycle flow on a hyperbolic manifold. This can be seen to be a kind of Hopf fibration. Such flows commonly occur in classical mechanics, which is the study in physics of finite-dimensional moving machinery, e.g. the double pendulum and so-forth. Classical mechanics is constructed on symplectic manifolds. The flows on such systems can be deconstructed into stable and unstable manifolds; as a general rule, when this is possible, chaotic motion results. That this is generic can be seen by noting that the cotangent bundle of a Riemannian manifold is (always) a symplectic manifold; the geodesic flow is given by a solution to the Hamilton–Jacobi equations for this manifold. In terms of the canonical coordinates $(q,p)$ on the cotangent manifold, the Hamiltonian or energy is given by $H={\tfrac {1}{2}}\sum _{ij}g^{ij}(q)p_{i}p_{j}$ with $g^{ij}$ the (inverse of the) metric tensor and $p_{i}$ the momentum. The resemblance to the kinetic energy $E={\tfrac {1}{2}}mv^{2}$ of a point particle is hardly accidental; this is the whole point of calling such things "energy". In this sense, chaotic behavior with ergodic orbits is a more-or-less generic phenomenon in large tracts of geometry. Ergodicity results have been provided in translation surfaces, hyperbolic groups and systolic geometry. Techniques include the study of ergodic flows, the Hopf decomposition, and the Ambrose–Kakutani–Krengel–Kubo theorem. An important class of systems are the Axiom A systems. A number of both classification and "anti-classification" results have been obtained. The Ornstein isomorphism theorem applies here as well; again, it states that most of these systems are isomorphic to some Bernoulli scheme. This rather neatly ties these systems back into the definition of ergodicity given for a stochastic process, in the previous section. The anti-classification results state that there are more than a countably infinite number of inequivalent ergodic measure-preserving dynamical systems. This is perhaps not entirely a surprise, as one can use points in the Cantor set to construct similar-but-different systems. See measure-preserving dynamical system for a brief survey of some of the anti-classification results. In quantum mechanics As to quantum mechanics, there is no universal quantum definition of ergodocity or even chaos (see quantum chaos).[4] However, there is a quantum ergodicity theorem stating that the expectation value of an operator converges to the corresponding microcanonical classical average in the semiclassical limit $\hbar \rightarrow 0$. Nevertheless, the theorem does not imply that all eigenstates of the Hamiltionian whose classical counterpart is chaotic are features and random. For example, the quantum ergodicity theorem do not exclude the existence of non-ergodic states such as quantum scars. In addition to the conventional scarring,[5][6][7][8] there are two other types of quantum scarring, which further illustrate the weak-ergodicity breaking in quantum chaotic systems: perturbation-induced[9][10][11][12][13] and many-body quantum scars.[14] Definition for discrete-time systems Formal definition Let $(X,{\mathcal {B}})$ be a measurable space. If $T$ is a measurable function from $X$ to itself and $\mu $ a probability measure on $(X,{\mathcal {B}})$ then we say that $T$ is $\mu $-ergodic or $\mu $ is an ergodic measure for $T$ if $T$ preserves $\mu $ and the following condition holds: For any $A\in {\mathcal {B}}$ such that $T^{-1}(A)=A$ either $\mu (A)=0$ or $\mu (A)=1$. In other words there are no $T$-invariant subsets up to measure 0 (with respect to $\mu $). Recall that $T$ preserving $\mu $ (or $\mu $ being $T$-invariant) means that $\mu {\mathord {\left(T^{-1}(A)\right)}}=\mu (A)$ for all $A\in {\mathcal {B}}$ (see also measure-preserving dynamical system). Some authors[15] relax the requirement that $T$ preserves $\mu $ to the requirement that $T$ is a non-singular transformation, that means that it preserves the subsets of measure zero. Examples The simplest example is when $X$ is a finite set and $\mu $ the counting measure. Then a self-map of $X$ preserves $\mu $ if and only if it is a bijection, and it is ergodic if and only if $T$ has only one orbit (that is, for every $x,y\in X$ there exists $k\in \mathbb {N} $ such that $y=T^{k}(x)$). For example, if $X=\{1,2,\ldots ,n\}$ then the cycle $(1\,2\,\cdots \,n)$ is ergodic, but the permutation $(1\,2)(3\,4\,\cdots \,n)$ is not (it has the two invariant subsets $\{1,2\}$ and $\{3,4,\ldots ,n\}$). Equivalent formulations The definition given above admits the following immediate reformulations: • for every $A\in {\mathcal {B}}$ with $\mu {\mathord {\left(T^{-1}(A)\bigtriangleup A\right)}}=0$ we have $\mu (A)=0$ or $\mu (A)=1\,$ (where $\bigtriangleup $ denotes the symmetric difference); • for every $A\in {\mathcal {B}}$ with positive measure we have $ \mu {\mathord {\left(\bigcup _{n=1}^{\infty }T^{-n}(A)\right)}}=1$; • for every two sets $A,B\in {\mathcal {B}}$ of positive measure, there exists $n>0$ such that $\mu {\mathord {\left(\left(T^{-n}(A)\right)\cap B\right)}}>0$; • Every measurable function $f:X\to \mathbb {R} $ with $f\circ T=f$ is constant on a subset of full measure. Importantly for applications, the condition in the last characterisation can be restricted to square-integrable functions only: • If $f\in L^{2}(X,\mu )$ and $f\circ T=f$ then $f$ is constant almost everywhere. Bernoulli shifts and subshifts See also: Bernoulli shift Let $S$ be a finite set and $X=S^{\mathbb {Z} }$ with $\mu $ the product measure (each factor $S$ being endowed with its counting measure). Then the shift operator $T$ defined by $T\left((s_{k})_{k\in \mathbb {Z} })\right)=(s_{k+1})_{k\in \mathbb {Z} }$ is $\mu $-ergodic.[16] There are many more ergodic measures for the shift map $T$ on $X$. Periodic sequences give finitely supported measures. More interestingly, there are infinitely-supported ones which are subshifts of finite type. Irrational rotations Let $X$ be the unit circle $\{z\in \mathbb {C} ,\,\|z\|=1\}$, with its Lebesgue measure $\mu $. For any $\theta \in \mathbb {R} $ the rotation of $X$ of angle $\theta $ is given by $T_{\theta }(z)=e^{2i\pi \theta }z$. If $\theta \in \mathbb {Q} $ then $T_{\theta }$ is not ergodic for the Lebesgue measure as it has infinitely many finite orbits. On the other hand, if $\theta $ is irrational then $T_{\theta }$ is ergodic.[17] Arnold's cat map Let $X=\mathbb {R} ^{2}/\mathbb {Z} ^{2}$ be the 2-torus. Then any element $g\in \mathrm {SL} _{2}(\mathbb {Z} )$ defines a self-map of $X$ since $g\left(\mathbb {Z} ^{2}\right)=\mathbb {Z} ^{2}$. When $ g=\left({\begin{array}{cc}2&1\\1&1\end{array}}\right)$ one obtains the so-called Arnold's cat map, which is ergodic for the Lebesgue measure on the torus. Ergodic theorems Main article: Ergodic theory § Ergodic theorems If $\mu $ is a probability measure on a space $X$ which is ergodic for a transformation $T$ the pointwise ergodic theorem of G. Birkhoff states that for every measurable functions $f:X\to \mathbb {R} $ and for $\mu $-almost every point $x\in X$ the time average on the orbit of $x$ converges to the space average of $f$. Formally this means that $\lim _{k\to +\infty }\left({\frac {1}{k+1}}\sum _{i=0}^{k}f\left(T^{i}(x)\right)\right)=\int _{X}fd\mu .$ The mean ergodic theorem of J. von Neumann is a similar, weaker statement about averaged translates of square-integrable functions. Dense orbits An immediate consequence of the definition of ergodicity is that on a topological space $X$, and if ${\mathcal {B}}$ is the σ-algebra of Borel sets, if $T$ is $\mu $-ergodic then $\mu $-almost every orbit of $T$ is dense in the support of $\mu $. This is not an equivalence since for a transformation which is not uniquely ergodic, but for which there is an ergodic measure with full support $\mu _{0}$, for any other ergodic measure $\mu _{1}$ the measure $ {\frac {1}{2}}(\mu _{0}+\mu _{1})$ is not ergodic for $T$ but its orbits are dense in the support. Explicit examples can be constructed with shift-invariant measures.[18] Mixing Main article: Mixing (mathematics) A transformation $T$ of a probability measure space $(X,\mu )$ is said to be mixing for the measure $\mu $ if for any measurable sets $A,B\subset X$ the following holds: $\lim _{n\to +\infty }\mu \left(T^{-n}A\cap B\right)=\mu (A)\mu (B)$ It is immediate that a mixing transformation is also ergodic (taking $A$ to be a $T$-stable subset and $B$ its complement). The converse is not true, for example a rotation with irrational angle on the circle (which is ergodic per the examples above) is not mixing (for a sufficiently small interval its successive images will not intersect itself most of the time). Bernoulli shifts are mixing, and so is Arnold's cat map. This notion of mixing is sometimes called strong mixing, as opposed to weak mixing which means that $\lim _{n\to +\infty }{\frac {1}{n}}\sum _{k=1}^{n}\left\|\mu (T^{-n}A\cap B)-\mu (A)\mu (B)\right\|=0$ Proper ergodicity The transformation $T$ is said to be properly ergodic if it does not have an orbit of full measure. In the discrete case this means that the measure $\mu $ is not supported on a finite orbit of $T$. Definition for continuous-time dynamical systems The definition is essentially the same for continuous-time dynamical systems as for a single transformation. Let $(X,{\mathcal {B}})$ be a measurable space and for each $t\in \mathbb {R} _{+}$, then such a system is given by a family $T_{t}$ of measurable functions from $X$ to itself, so that for any $t,s\in \mathbb {R} _{+}$ the relation $T_{s+t}=T_{s}\circ T_{t}$ holds (usually it is also asked that the orbit map from $\mathbb {R} _{+}\times X\to X$ is also measurable). If $\mu $ is a probability measure on $(X,{\mathcal {B}})$ then we say that $T_{t}$ is $\mu $-ergodic or $\mu $ is an ergodic measure for $T$ if each $T_{t}$ preserves $\mu $ and the following condition holds: For any $A\in {\mathcal {B}}$, if for all $t\in \mathbb {R} _{+}$ we have $T_{t}^{-1}(A)\subset A$ then either $\mu (A)=0$ or $\mu (A)=1$. Examples As in the discrete case the simplest example is that of a transitive action, for instance the action on the circle given by $T_{t}(z)=e^{2i\pi t}z$ is ergodic for Lebesgue measure. An example with infinitely many orbits is given by the flow along an irrational slope on the torus: let $X=\mathbb {S} ^{1}\times \mathbb {S} ^{1}$ and $\alpha \in \mathbb {R} $. Let $T_{t}(z_{1},z_{2})=\left(e^{2i\pi t}z_{1},e^{2\alpha i\pi t}z_{2}\right)$; then if $\alpha \not \in \mathbb {Q} $ this is ergodic for the Lebesgue measure. Ergodic flows Further examples of ergodic flows are: • Billiards in convex Euclidean domains; • the geodesic flow of a negatively curved Riemannian manifold of finite volume is ergodic (for the normalised volume measure); • the horocycle flow on a hyperbolic manifold of finite volume is ergodic (for the normalised volume measure) Ergodicity in compact metric spaces If $X$ is a compact metric space it is naturally endowed with the σ-algebra of Borel sets. The additional structure coming from the topology then allows a much more detailed theory for ergodic transformations and measures on $X$. Functional analysis interpretation A very powerful alternate definition of ergodic measures can be given using the theory of Banach spaces. Radon measures on $X$ form a Banach space of which the set ${\mathcal {P}}(X)$ of probability measures on $X$ is a convex subset. Given a continuous transformation $T$ of $X$ the subset ${\mathcal {P}}(X)^{T}$ of $T$-invariant measures is a closed convex subset, and a measure is ergodic for $T$ if and only if it is an extreme point of this convex.[19] Existence of ergodic measures In the setting above it follows from the Banach-Alaoglu theorem that there always exists extremal points in ${\mathcal {P}}(X)^{T}$. Hence a transformation of a compact metric space always admits ergodic measures. Ergodic decomposition In general an invariant measure need not be ergodic, but as a consequence of Choquet theory it can always be expressed as the barycenter of a probability measure on the set of ergodic measures. This is referred to as the ergodic decomposition of the measure.[20] Example In the case of $X=\{1,\ldots ,n\}$ and $T=(1\,2)(3\,4\,\cdots \,n)$ the counting measure is not ergodic. The ergodic measures for $T$ are the uniform measures $\mu _{1},\mu _{2}$ supported on the subsets $\{1,2\}$ and $\{3,\ldots ,n\}$ and every $T$-invariant probability measure can be written in the form $t\mu _{1}+(1-t)\mu _{2}$ for some $t\in [0,1]$. In particular $ {\frac {2}{n}}\mu _{1}+{\frac {n-2}{n}}\mu _{2}$ is the ergodic decomposition of the counting measure. Continuous systems Everything in this section transfers verbatim to continuous actions of $\mathbb {R} $ or $\mathbb {R} _{+}$ on compact metric spaces. Unique ergodicity The transformation $T$ is said to be uniquely ergodic if there is a unique Borel probability measure $\mu $ on $X$ which is ergodic for $T$. In the examples considered above, irrational rotations of the circle are uniquely ergodic;[21] shift maps are not. Probabilistic interpretation: ergodic processes If $\left(X_{n}\right)_{n\geq 1}$ is a discrete-time stochastic process on a space $\Omega $, it is said to be ergodic if the joint distribution of the variables on $\Omega ^{\mathbb {N} }$ is invariant under the shift map $\left(x_{n}\right)_{n\geq 1}\mapsto \left(x_{n+1}\right)_{n\geq 1}$. This is a particular case of the notions discussed above. The simplest case is that of an independent and identically distributed process which corresponds to the shift map described above. Another important case is that of a Markov chain which is discussed in detail below. A similar interpretation holds for continuous-time stochastic processes though the construction of the measurable structure of the action is more complicated. Ergodicity of Markov chains The dynamical system associated with a Markov chain Let $S$ be a finite set. A Markov chain on $S$ is defined by a matrix $P\in [0,1]^{S\times S}$, where $P(s_{1},s_{2})$ is the transition probability from $s_{1}$ to $s_{2}$, so for every $s\in S$ we have $ \sum _{s'\in S}P(s,s')=1$. A stationary measure for $P$ is a probability measure $\nu $ on $S$ such that $\nu P=\nu $ ; that is $ \sum _{s'\in S}\nu (s')P(s',s)=\nu (s)$ for all $s\in S$. Using this data we can define a probability measure $\mu _{\nu }$ on the set $X=S^{\mathbb {Z} }$ with its product σ-algebra by giving the measures of the cylinders as follows: $\mu _{\nu }(\cdots \times S\times \{(s_{n},\ldots ,s_{m})\}\times S\times \cdots )=\nu (s_{n})P(s_{n},s_{n+1})\cdots P(s_{m-1},s_{m}).$ Stationarity of $\nu $ then means that the measure $\mu _{\nu }$ is invariant under the shift map $T\left(\left(s_{k}\right)_{k\in \mathbb {Z} })\right)=\left(s_{k+1}\right)_{k\in \mathbb {Z} }$. Criterion for ergodicity The measure $\mu _{\nu }$ is always ergodic for the shift map if the associated Markov chain is irreducible (any state can be reached with positive probability from any other state in a finite number of steps).[22] The hypotheses above imply that there is a unique stationary measure for the Markov chain. In terms of the matrix $P$ a sufficient condition for this is that 1 be a simple eigenvalue of the matrix $P$ and all other eigenvalues of $P$ (in $\mathbb {C} $) are of modulus <1. Note that in probability theory the Markov chain is called ergodic if in addition each state is aperiodic (the times where the return probability is positive are not multiples of a single integer >1). This is not necessary for the invariant measure to be ergodic; hence the notions of "ergodicity" for a Markov chain and the associated shift-invariant measure are different (the one for the chain is strictly stronger).[23] Moreover the criterion is an "if and only if" if all communicating classes in the chain are recurrent and we consider all stationary measures. Counting measure If $P(s,s')=1/\|S\|$ for all $s,s'\in S$ then the stationary measure is the counting measure, the measure $\mu _{P}$ is the product of counting measures. The Markov chain is ergodic, so the shift example from above is a special case of the criterion. Non-ergodic Markov chains Markov chains with recurring communicating classes are not irreducible are not ergodic, and this can be seen immediately as follows. If $S_{1}\subsetneq S$ are two distinct recurrent communicating classes there are nonzero stationary measures $\nu _{1},\nu _{2}$ supported on $S_{1},S_{2}$ respectively and the subsets $S_{1}^{\mathbb {Z} }$ and $S_{2}^{\mathbb {Z} }$ are both shift-invariant and of measure 1.2 for the invariant probability measure $ {\frac {1}{2}}(\nu _{1}+\nu _{2})$. A very simple example of that is the chain on $S=\{1,2\}$ given by the matrix $ \left({\begin{array}{cc}1&0\\0&1\end{array}}\right)$ (both states are stationary). A periodic chain The Markov chain on $S=\{1,2\}$ given by the matrix $ \left({\begin{array}{cc}0&1\\1&0\end{array}}\right)$ is irreducible but periodic. Thus it is not ergodic in the sense of Markov chain though the associated measure $\mu $ on $\{1,2\}^{\mathbb {Z} }$ is ergodic for the shift map. However the shift is not mixing for this measure, as for the sets $A=\cdots \times \{1,2\}\times 1\times \{1,2\}\times 1\times \{1,2\}\cdots $ and $B=\cdots \times \{1,2\}\times 2\times \{1,2\}\times 2\times \{1,2\}\cdots $ we have $ \mu (A)={\frac {1}{2}}=\mu (B)$ but $\mu \left(T^{-n}A\cap B\right)={\begin{cases}{\frac {1}{2}}{\text{ if }}n{\text{ is odd}}\\0{\text{ if }}n{\text{ is even.}}\end{cases}}$ Generalisations The definition of ergodicity also makes sense for group actions. The classical theory (for invertible transformations) corresponds to actions of $\mathbb {Z} $ or $\mathbb {R} $. For non-abelian groups there might not be invariant measures even on compact metric spaces. However the definition of ergodicity carries over unchanged if one replaces invariant measures by quasi-invariant measures. Important examples are the action of a semisimple Lie group (or a lattice therein) on its Furstenberg boundary. A measurable equivalence relation it is said to be ergodic if all saturated subsets are either null or conull. Notes 1. Walters 1982, §0.1, p. 2 2. Gallavotti, Giovanni (1995). "Ergodicity, ensembles, irreversibility in Boltzmann and beyond". Journal of Statistical Physics. 78 (5–6): 1571–1589. arXiv:chao-dyn/9403004. Bibcode:1995JSP....78.1571G. doi:10.1007/BF02180143. S2CID 17605281. 3. Feller, William (1 August 2008). An Introduction to Probability Theory and Its Applications (2nd ed.). Wiley India Pvt. Limited. p. 271. ISBN 978-81-265-1806-7. 4. Stöckmann, Hans-Jürgen (1999). Quantum Chaos: An Introduction. Cambridge: Cambridge University Press. doi:10.1017/cbo9780511524622. ISBN 978-0-521-02715-1. 5. Heller, Eric J. (1984-10-15). "Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits". Physical Review Letters. 53 (16): 1515–1518. Bibcode:1984PhRvL..53.1515H. doi:10.1103/PhysRevLett.53.1515. 6. Kaplan, L (1999-03-01). "Scars in quantum chaotic wavefunctions". Nonlinearity. 12 (2): R1–R40. doi:10.1088/0951-7715/12/2/009. ISSN 0951-7715. S2CID 250793219. 7. Kaplan, L.; Heller, E.J. (April 1998). "Linear and Nonlinear Theory of Eigenfunction Scars". Annals of Physics. 264 (2): 171–206. arXiv:chao-dyn/9809011. Bibcode:1998AnPhy.264..171K. doi:10.1006/aphy.1997.5773. S2CID 120635994. 8. Heller, Eric Johnson (2018). The semiclassical way to dynamics and spectroscopy. Princeton: Princeton University Press. ISBN 978-1-4008-9029-3. OCLC 1034625177. 9. Keski-Rahkonen, J.; Ruhanen, A.; Heller, E. J.; Räsänen, E. (2019-11-21). "Quantum Lissajous Scars". Physical Review Letters. 123 (21): 214101. arXiv:1911.09729. Bibcode:2019PhRvL.123u4101K. doi:10.1103/PhysRevLett.123.214101. PMID 31809168. S2CID 208248295. 10. Luukko, Perttu J. J.; Drury, Byron; Klales, Anna; Kaplan, Lev; Heller, Eric J.; Räsänen, Esa (2016-11-28). "Strong quantum scarring by local impurities". Scientific Reports. 6 (1): 37656. arXiv:1511.04198. Bibcode:2016NatSR...637656L. doi:10.1038/srep37656. ISSN 2045-2322. PMC 5124902. PMID 27892510. 11. Keski-Rahkonen, J.; Luukko, P. J. J.; Kaplan, L.; Heller, E. J.; Räsänen, E. (2017-09-20). "Controllable quantum scars in semiconductor quantum dots". Physical Review B. 96 (9): 094204. arXiv:1710.00585. Bibcode:2017PhRvB..96i4204K. doi:10.1103/PhysRevB.96.094204. S2CID 119083672. 12. Keski-Rahkonen, J; Luukko, P J J; Åberg, S; Räsänen, E (2019-01-21). "Effects of scarring on quantum chaos in disordered quantum wells". Journal of Physics: Condensed Matter. 31 (10): 105301. arXiv:1806.02598. Bibcode:2019JPCM...31j5301K. doi:10.1088/1361-648x/aaf9fb. ISSN 0953-8984. PMID 30566927. S2CID 51693305. 13. Keski-Rahkonen, Joonas (2020). Quantum Chaos in Disordered Two-Dimensional Nanostructures. Tampere University. ISBN 978-952-03-1699-0. 14. Turner, C. J.; Michailidis, A. A.; Abanin, D. A.; Serbyn, M.; Papić, Z. (July 2018). "Weak ergodicity breaking from quantum many-body scars". Nature Physics. 14 (7): 745–749. Bibcode:2018NatPh..14..745T. doi:10.1038/s41567-018-0137-5. ISSN 1745-2481. S2CID 256706206. 15. Aaronson, Jon (1997). "An introduction to infinite ergodic theory". American Mathematical Soc.: 21. ISBN 9780821804940. {{cite journal}}: Cite journal requires \|journal= (help) 16. Walters 1982, p. 32. 17. Walters 1982, p. 29. 18. "Example of a measure-preserving system with dense orbits that is not ergodic". MathOverflow. September 1, 2011. Retrieved May 16, 2020. 19. Walters 1982, p. 152. 20. Walters 1982, p. 153. 21. Walters 1982, p. 159. 22. Walters 1982, p. 42. 23. "Different uses of the word "ergodic"". MathOverflow. September 4, 2011. Retrieved May 16, 2020. References • Walters, Peter (1982). An Introduction to Ergodic Theory. Springer. ISBN 0-387-95152-0. • Brin, Michael; Garrett, Stuck (2002). Introduction to Dynamical Systems. Cambridge University Press. ISBN 0-521-80841-3. External links Look up ergodic in Wiktionary, the free dictionary. • Karma Dajani and Sjoerd Dirksin, "A Simple Introduction to Ergodic Theory"	Wikipedia
Unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units. "Unique factorization" redirects here. For the uniqueness of integer factorization, see fundamental theorem of arithmetic. Algebraic structures Group-like • Group • Semigroup / Monoid • Rack and quandle • Quasigroup and loop • Abelian group • Magma • Lie group Group theory Ring-like • Ring • Rng • Semiring • Near-ring • Commutative ring • Domain • Integral domain • Field • Division ring • Lie ring Ring theory Lattice-like • Lattice • Semilattice • Complemented lattice • Total order • Heyting algebra • Boolean algebra • Map of lattices • Lattice theory Module-like • Module • Group with operators • Vector space • Linear algebra Algebra-like • Algebra • Associative • Non-associative • Composition algebra • Lie algebra • Graded • Bialgebra • Hopf algebra Important examples of UFDs are the integers and polynomial rings in one or more variables with coefficients coming from the integers or from a field. Unique factorization domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields Definition Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero element x of R can be written as a product (an empty product if x is a unit) of irreducible elements pi of R and a unit u: x = u p1 p2 ⋅⋅⋅ pn with n ≥ 0 and this representation is unique in the following sense: If q1, ..., qm are irreducible elements of R and w is a unit such that x = w q1 q2 ⋅⋅⋅ qm with m ≥ 0, then m = n, and there exists a bijective map φ : {1, ..., n} → {1, ..., m} such that pi is associated to qφ(i) for i ∈ {1, ..., n}. The uniqueness part is usually hard to verify, which is why the following equivalent definition is useful: A unique factorization domain is an integral domain R in which every non-zero element can be written as a product of a unit and prime elements of R. Examples Most rings familiar from elementary mathematics are UFDs: • All principal ideal domains, hence all Euclidean domains, are UFDs. In particular, the integers (also see fundamental theorem of arithmetic), the Gaussian integers and the Eisenstein integers are UFDs. • If R is a UFD, then so is R[X], the ring of polynomials with coefficients in R. Unless R is a field, R[X] is not a principal ideal domain. By induction, a polynomial ring in any number of variables over any UFD (and in particular over a field or over the integers) is a UFD. • The formal power series ring K[[X1,...,Xn]] over a field K (or more generally over a regular UFD such as a PID) is a UFD. On the other hand, the formal power series ring over a UFD need not be a UFD, even if the UFD is local. For example, if R is the localization of k[x,y,z]/(x2 + y3 + z7) at the prime ideal (x,y,z) then R is a local ring that is a UFD, but the formal power series ring R[[X]] over R is not a UFD. • The Auslander–Buchsbaum theorem states that every regular local ring is a UFD. • $\mathbb {Z} \left[e^{\frac {2\pi i}{n}}\right]$ is a UFD for all integers 1 ≤ n ≤ 22, but not for n = 23. • Mori showed that if the completion of a Zariski ring, such as a Noetherian local ring, is a UFD, then the ring is a UFD.[1] The converse of this is not true: there are Noetherian local rings that are UFDs but whose completions are not. The question of when this happens is rather subtle: for example, for the localization of k[x,y,z]/(x2 + y3 + z5) at the prime ideal (x,y,z), both the local ring and its completion are UFDs, but in the apparently similar example of the localization of k[x,y,z]/(x2 + y3 + z7) at the prime ideal (x,y,z) the local ring is a UFD but its completion is not. • Let $R$ be a field of any characteristic other than 2. Klein and Nagata showed that the ring R[X1,...,Xn]/Q is a UFD whenever Q is a nonsingular quadratic form in the X's and n is at least 5. When n=4 the ring need not be a UFD. For example, $R[X,Y,Z,W]/(XY-ZW)$ is not a UFD, because the element $XY$ equals the element $ZW$ so that $XY$ and $ZW$ are two different factorizations of the same element into irreducibles. • The ring Q[x,y]/(x2 + 2y2 + 1) is a UFD, but the ring Q(i)[x,y]/(x2 + 2y2 + 1) is not. On the other hand, The ring Q[x,y]/(x2 + y2 – 1) is not a UFD, but the ring Q(i)[x,y]/(x2 + y2 – 1) is (Samuel 1964, p.35). Similarly the coordinate ring R[X,Y,Z]/(X2 + Y2 + Z2 − 1) of the 2-dimensional real sphere is a UFD, but the coordinate ring C[X,Y,Z]/(X2 + Y2 + Z2 − 1) of the complex sphere is not. • Suppose that the variables Xi are given weights wi, and F(X1,...,Xn) is a homogeneous polynomial of weight w. Then if c is coprime to w and R is a UFD and either every finitely generated projective module over R is free or c is 1 mod w, the ring R[X1,...,Xn,Z]/(Zc − F(X1,...,Xn)) is a UFD (Samuel 1964, p.31). Non-examples • The quadratic integer ring $\mathbb {Z} [{\sqrt {-5}}]$ of all complex numbers of the form $a+b{\sqrt {-5}}$, where a and b are integers, is not a UFD because 6 factors as both 2×3 and as $\left(1+{\sqrt {-5}}\right)\left(1-{\sqrt {-5}}\right)$. These truly are different factorizations, because the only units in this ring are 1 and −1; thus, none of 2, 3, $1+{\sqrt {-5}}$, and $1-{\sqrt {-5}}$ are associate. It is not hard to show that all four factors are irreducible as well, though this may not be obvious.[2] See also algebraic integer. • For a square-free positive integer d, the ring of integers of $\mathbb {Q} [{\sqrt {-d}}]$ will fail to be a UFD unless d is a Heegner number. • The ring of formal power series over the complex numbers is a UFD, but the subring of those that converge everywhere, in other words the ring of entire functions in a single complex variable, is not a UFD, since there exist entire functions with an infinity of zeros, and thus an infinity of irreducible factors, while a UFD factorization must be finite, e.g.: $\sin \pi z=\pi z\prod _{n=1}^{\infty }\left(1-{{z^{2}} \over {n^{2}}}\right).$ Properties Some concepts defined for integers can be generalized to UFDs: • In UFDs, every irreducible element is prime. (In any integral domain, every prime element is irreducible, but the converse does not always hold. For instance, the element $z\in K[x,y,z]/(z^{2}-xy)$ is irreducible, but not prime.) Note that this has a partial converse: a domain satisfying the ACCP is a UFD if and only if every irreducible element is prime. • Any two elements of a UFD have a greatest common divisor and a least common multiple. Here, a greatest common divisor of a and b is an element d which divides both a and b, and such that every other common divisor of a and b divides d. All greatest common divisors of a and b are associated. • Any UFD is integrally closed. In other words, if R is a UFD with quotient field K, and if an element k in K is a root of a monic polynomial with coefficients in R, then k is an element of R. • Let S be a multiplicatively closed subset of a UFD A. Then the localization $S^{-1}A$ is a UFD. A partial converse to this also holds; see below. Equivalent conditions for a ring to be a UFD A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal (a proof is given at the end). Also, a Dedekind domain is a UFD if and only if its ideal class group is trivial. In this case, it is in fact a principal ideal domain. In general, for an integral domain A, the following conditions are equivalent: 1. A is a UFD. 2. Every nonzero prime ideal of A contains a prime element. (Kaplansky) 3. A satisfies ascending chain condition on principal ideals (ACCP), and the localization S−1A is a UFD, where S is a multiplicatively closed subset of A generated by prime elements. (Nagata criterion) 4. A satisfies ACCP and every irreducible is prime. 5. A is atomic and every irreducible is prime. 6. A is a GCD domain satisfying ACCP. 7. A is a Schreier domain,[3] and atomic. 8. A is a pre-Schreier domain and atomic. 9. A has a divisor theory in which every divisor is principal. 10. A is a Krull domain in which every divisorial ideal is principal (in fact, this is the definition of UFD in Bourbaki.) 11. A is a Krull domain and every prime ideal of height 1 is principal.[4] In practice, (2) and (3) are the most useful conditions to check. For example, it follows immediately from (2) that a PID is a UFD, since every prime ideal is generated by a prime element in a PID. For another example, consider a Noetherian integral domain in which every height one prime ideal is principal. Since every prime ideal has finite height, it contains a height one prime ideal (induction on height) which is principal. By (2), the ring is a UFD. See also • Parafactorial local ring • Noncommutative unique factorization domain Citations 1. Bourbaki, 7.3, no 6, Proposition 4. 2. Artin, Michael (2011). Algebra. Prentice Hall. p. 360. ISBN 978-0-13-241377-0. 3. A Schreier domain is an integrally closed integral domain where, whenever x divides yz, x can be written as x = x1 x2 so that x1 divides y and x2 divides z. In particular, a GCD domain is a Schreier domain 4. Bourbaki, 7.3, no 2, Theorem 1. References • N. Bourbaki (1972). Commutative algebra. Paris, Hermann; Reading, Mass., Addison-Wesley Pub. Co. ISBN 9780201006445. • B. Hartley; T.O. Hawkes (1970). Rings, modules and linear algebra. Chapman and Hall. ISBN 0-412-09810-5. Chap. 4. • Chapter II.5 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001 • David Sharpe (1987). Rings and factorization. Cambridge University Press. ISBN 0-521-33718-6. • Samuel, Pierre (1964), Murthy, M. Pavman (ed.), Lectures on unique factorization domains, Tata Institute of Fundamental Research Lectures on Mathematics, vol. 30, Bombay: Tata Institute of Fundamental Research, MR 0214579 • Samuel, Pierre (1968). "Unique factorization". The American Mathematical Monthly. 75 (9): 945–952. doi:10.2307/2315529. ISSN 0002-9890. JSTOR 2315529. Authority control: National • Germany	Wikipedia
Unique games conjecture In computational complexity theory, the unique games conjecture (often referred to as UGC) is a conjecture made by Subhash Khot in 2002.[1][2][3] The conjecture postulates that the problem of determining the approximate value of a certain type of game, known as a unique game, has NP-hard computational complexity. It has broad applications in the theory of hardness of approximation. If the unique games conjecture is true and P ≠ NP,[4] then for many important problems it is not only impossible to get an exact solution in polynomial time (as postulated by the P versus NP problem), but also impossible to get a good polynomial-time approximation. The problems for which such an inapproximability result would hold include constraint satisfaction problems, which crop up in a wide variety of disciplines. Unsolved problem in computer science: Is the Unique Games Conjecture true? (more unsolved problems in computer science) The conjecture is unusual in that the academic world seems about evenly divided on whether it is true or not.[1] Formulations The unique games conjecture can be stated in a number of equivalent ways. Unique label cover The following formulation of the unique games conjecture is often used in hardness of approximation. The conjecture postulates the NP-hardness of the following promise problem known as label cover with unique constraints. For each edge, the colors on the two vertices are restricted to some particular ordered pairs. Unique constraints means that for each edge none of the ordered pairs have the same color for the same node. This means that an instance of label cover with unique constraints over an alphabet of size k can be represented as a directed graph together with a collection of permutations πe: [k] → [k], one for each edge e of the graph. An assignment to a label cover instance gives to each vertex of G a value in the set [k] = {1, 2, ... k}, often called “colours.” • An instance of unique label cover. The 4 vertices may be assigned the colors red, blue, and green while satisfying the constraints at each edge. • A solution to the unique label cover instance. Such instances are strongly constrained in the sense that the colour of a vertex uniquely defines the colours of its neighbours, and hence for its entire connected component. Thus, if the input instance admits a valid assignment, then such an assignment can be found efficiently by iterating over all colours of a single node. In particular, the problem of deciding if a given instance admits a satisfying assignment can be solved in polynomial time. • An instance of unique label cover that does not allow a satisfying assignment. • An assignment that satisfies all edges except the thick edge. Thus, this instance has value 3/4. The value of a unique label cover instance is the fraction of constraints that can be satisfied by any assignment. For satisfiable instances, this value is 1 and is easy to find. On the other hand, it seems to be very difficult to determine the value of an unsatisfiable game, even approximately. The unique games conjecture formalises this difficulty. More formally, the (c, s)-gap label-cover problem with unique constraints is the following promise problem (Lyes, Lno): • Lyes = {G: Some assignment satisfies at least a c-fraction of constraints in G} • Lno = {G: Every assignment satisfies at most an s-fraction of constraints in G} where G is an instance of the label cover problem with unique constraints. The unique games conjecture states that for every sufficiently small pair of constants ε, δ > 0, there exists a constant k such that the (1 − δ, ε)-gap label-cover problem with unique constraints over alphabet of size k is NP-hard. Instead of graphs, the label cover problem can be formulated in terms of linear equations. For example, suppose that we have a system of linear equations over the integers modulo 7: ${\begin{aligned}x_{1}&\equiv 2\cdot x_{2}{\pmod {7}},\\x_{2}&\equiv 4\cdot x_{5}{\pmod {7}},\\&{}\ \ \vdots \\x_{1}&\equiv 2\cdot x_{7}{\pmod {7}}.\end{aligned}}$ This is an instance of the label cover problem with unique constraints. For example, the first equation corresponds to the permutation π(1, 2) where π(1, 2)(x1) = 2x2 modulo 7. Two-prover proof systems A unique game is a special case of a two-prover one-round (2P1R) game. A two-prover one-round game has two players (also known as provers) and a referee. The referee sends each player a question drawn from a known probability distribution, and the players each have to send an answer. The answers come from a set of fixed size. The game is specified by a predicate that depends on the questions sent to the players and the answers provided by them. The players may decide on a strategy beforehand, although they cannot communicate with each other during the game. The players win if the predicate is satisfied by their questions and their answers. A two-prover one-round game is called a unique game if for every question and every answer by the first player, there is exactly one answer by the second player that results in a win for the players, and vice versa. The value of a game is the maximum winning probability for the players over all strategies. The unique games conjecture states that for every sufficiently small pair of constants ε, δ > 0, there exists a constant k such that the following promise problem (Lyes, Lno) is NP-hard: • Lyes = {G: the value of G is at least 1 − δ} • Lno = {G: the value of G is at most ε} where G is a unique game whose answers come from a set of size k. Probabilistically checkable proofs Alternatively, the unique games conjecture postulates the existence of a certain type of probabilistically checkable proof for problems in NP. A unique game can be viewed as a special kind of nonadaptive probabilistically checkable proof with query complexity 2, where for each pair of possible queries of the verifier and each possible answer to the first query, there is exactly one possible answer to the second query that makes the verifier accept, and vice versa. The unique games conjecture states that for every sufficiently small pair of constants $\varepsilon ,\delta >0$ there is a constant $K$ such that every problem in NP has a probabilistically checkable proof over an alphabet of size $K$ with completeness $1-\delta $, soundness $\varepsilon $, and randomness complexity $O(\log n)$ which is a unique game. Relevance Approximability results assuming P ≠ NP versus the UGC ProblemPoly.-time approx.NP hardnessUG hardness Max 2SAT$0.940\dots $[5]$0.954\dots +\varepsilon $[6]$0.9439\dots +\varepsilon $[7] Max cut$0.878...\dots $[8]${\tfrac {16}{17}}+\varepsilon \approx 0.941$[6]$0.878\dots +\varepsilon $[7] Min vertex cover$2$${\sqrt {2}}-\varepsilon $[9]$2-\varepsilon $[10] Feedback arc set$O(\log n\log \log n)$[11]$1.360\dots -\varepsilon $[12]All constants[13] Max acyclic subgraph${\tfrac {1}{2}}+\Omega (1/{\sqrt {\Delta }})$[14]${\tfrac {65}{66}}$[15]${\tfrac {1}{2}}+\varepsilon $[13] Betweenness${\tfrac {1}{3}}$${\tfrac {47}{48}}$[16]${\tfrac {1}{3}}+\varepsilon $[17] Some very natural, intrinsically interesting statements about things like voting and foams just popped out of studying the UGC.... Even if the UGC turns out to be false, it has inspired a lot of interesting math research. — Ryan O’Donnell, [1] The unique games conjecture was introduced by Subhash Khot in 2002 in order to make progress on certain questions in the theory of hardness of approximation. The truth of the unique games conjecture would imply the optimality of many known approximation algorithms (assuming P ≠ NP). For example, the approximation ratio achieved by the algorithm of Goemans and Williamson for approximating the maximum cut in a graph is optimal to within any additive constant assuming the unique games conjecture and P ≠ NP. A list of results that the unique games conjecture is known to imply is shown in the adjacent table together with the corresponding best results for the weaker assumption P ≠ NP. A constant of $c+\varepsilon $ or $c-\varepsilon $ means that the result holds for every constant (with respect to the problem size) strictly greater than or less than $c$, respectively. Discussion and alternatives Currently, there is no consensus regarding the truth of the unique games conjecture. Certain stronger forms of the conjecture have been disproved. A different form of the conjecture postulates that distinguishing the case when the value of a unique game is at least $1-\delta $ from the case when the value is at most $\varepsilon $ is impossible for polynomial-time algorithms (but perhaps not NP-hard). This form of the conjecture would still be useful for applications in hardness of approximation. The constant $\delta >0$ in the above formulations of the conjecture is necessary unless P = NP. If the uniqueness requirement is removed the corresponding statement is known to be true by the parallel repetition theorem, even when $\delta =0$. Marek Karpinski and Warren Schudy have constructed linear time approximation schemes for dense instances of unique games problem.[18] In 2008, Prasad Raghavendra has shown that if the unique games conjecture is true, then for every constraint satisfaction problem the best approximation ratio is given by a certain simple semidefinite programming instance, which is in particular polynomial.[19] In 2010, Prasad Raghavendra and David Steurer defined the Gap-Small-Set Expansion problem, and conjectured that it is NP-hard. This conjecture implies the unique games conjecture.[20] It has also been used to prove strong hardness of approximation results for finding complete bipartite subgraphs.[21] In 2010, Sanjeev Arora, Boaz Barak and David Steurer found a subexponential time approximation algorithm for the unique games problem.[22] In 2012, it was shown that distinguishing instances with value at most ${\tfrac {3}{8}}+\delta $ from instances with value at least ${\tfrac {1}{2}}$ is NP-hard.[23] In 2018, after a series of papers, a weaker version of the conjecture, called the 2-2 games conjecture, was proven. In a certain sense, this proves "a half" of the original conjecture.[24][25] This also improves the best known gap for unique label cover: it is NP-hard to distinguish instances with value at most $\delta $ from instances with value at least ${\tfrac {1}{2}}$.[26] Notes 1. Erica Klarreich (2011-10-06). "Approximately Hard: The Unique Games Conjecture". Simons Foundation. Retrieved 2012-10-29. 2. Dick Lipton (2010-05-05). "Unique Games: A Three Act Play". Gödel’s Lost Letter and P=NP. Retrieved 2012-10-29. 3. Khot, Subhash (2002), "On the power of unique 2-prover 1-round games", Proceedings of the thirty-fourth annual ACM symposium on Theory of computing, pp. 767–775, doi:10.1145/509907.510017, ISBN 1-58113-495-9, S2CID 207635974 4. The unique games conjecture is vacuously true if P = NP, as then every problem in NP would also be NP-hard. 5. Feige, Uriel; Goemans, Michel X. (1995), "Approximating the value of two prover proof systems, with applications to MAX 2SAT and MAX DICUT", Proc. 3rd Israel Symp. Theory of Computing and Systems, IEEE Computer Society Press, pp. 182–189 6. Håstad, Johan (1999), "Some Optimal Inapproximability Results", Journal of the ACM, 48 (4): 798–859, doi:10.1145/502090.502098, S2CID 5120748. 7. Khot, Subhash; Kindler, Guy; Mossel, Elchanan; O'Donnell, Ryan (2007), "Optimal inapproximability results for MAX-CUT and other two-variable CSPs?" (PDF), SIAM Journal on Computing, 37 (1): 319–357, doi:10.1137/S0097539705447372, S2CID 2090495 8. Goemans, Michel X.; Williamson, David P. (1995), "Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming", Journal of the ACM, 42 (6): 1115–1145, doi:10.1145/227683.227684, S2CID 15794408 9. Khot, Subhash; Minzer, Dor; Safra, Muli (2018), "Pseudorandom Sets in Grassmann Graph Have Near-Perfect Expansion", 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pp. 592–601, doi:10.1109/FOCS.2018.00062, ISBN 978-1-5386-4230-6, S2CID 3688775 10. Khot, Subhash; Regev, Oded (2003), "Vertex cover might be hard to approximate to within 2 − ε", IEEE Conference on Computational Complexity: 379– 11. Even, G.; Naor, J.; Schieber, B.; Sudan, M. (1998), "Approximating minimum feedback sets and multicuts in directed graphs", Algorithmica, 20 (2): 151–174, doi:10.1007/PL00009191, MR 1484534, S2CID 2437790 12. Dinur, Irit; Safra, Samuel (2005), "On the hardness of approximating minimum vertex cover" (PDF), Annals of Mathematics, 162 (1): 439–485, doi:10.4007/annals.2005.162.439, retrieved 2010-03-05. 13. Guruswami, Venkatesan; Manokaran, Rajsekar; Raghavendra, Prasad (2008), "Beating the Random Ordering is Hard: Inapproximability of Maximum Acyclic Subgraph", 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pp. 573–582, doi:10.1109/FOCS.2008.51, S2CID 8762205 14. Berger, Bonnie; Shor, Peter W. (1997), "Tight bounds for the maximum acyclic subgraph problem", Journal of Algorithms, 25 (1): 1–18, doi:10.1006/jagm.1997.0864, MR 1474592 15. Newman, A. (June 2000), Approximating the maximum acyclic subgraph (Master’s thesis), Massachusetts Institute of Technology, as cited by Guruswami, Manokaran & Raghavendra (2008) 16. Chor, Benny; Sudan, Madhu (1998), "A geometric approach to betweenness", SIAM Journal on Discrete Mathematics, 11 (4): 511–523 (electronic), doi:10.1137/S0895480195296221, MR 1640920. 17. Charikar, Moses; Guruswami, Venkatesan; Manokaran, Rajsekar (2009), "Every permutation CSP of arity 3 is approximation resistant", 24th Annual IEEE Conference on Computational Complexity, pp. 62–73, doi:10.1109/CCC.2009.29, MR 2932455, S2CID 257225. 18. Karpinski, Marek; Schudy, Warren (2009), "Linear time approximation schemes for the Gale-Berlekamp game and related minimization problems", Proceedings of the forty-first annual ACM symposium on Theory of computing, pp. 313–322, arXiv:0811.3244, doi:10.1145/1536414.1536458, ISBN 9781605585062, S2CID 6117694{{citation}}: CS1 maint: date and year (link) 19. Raghavendra, Prasad (2008), "Optimal algorithms and inapproximability results for every CSP?" (PDF), in Dwork, Cynthia (ed.), Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, {ACM}, pp. 245–254, doi:10.1145/1374376.1374414, S2CID 15075197 20. Raghavendra, Prasad; Steurer, David (2010), "Graph expansion and the unique games conjecture" (PDF), STOC'10—Proceedings of the 2010 ACM International Symposium on Theory of Computing, ACM, New York, pp. 755–764, doi:10.1145/1806689.1806792, MR 2743325, S2CID 1601199 21. Manurangsi, Pasin (2017), "Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis", in Chatzigiannakis, Ioannis; Indyk, Piotr; Kuhn, Fabian; Muscholl, Anca (eds.), 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), Leibniz International Proceedings in Informatics (LIPIcs), vol. 80, Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, pp. 79:1–79:14, doi:10.4230/LIPIcs.ICALP.2017.79, ISBN 978-3-95977-041-5 22. Arora, Sanjeev; Barak, Boaz; Steurer, David (2015), "Subexponential algorithms for unique games and related problems", Journal of the ACM, 62 (5): Art. 42, 25, doi:10.1145/2775105, MR 3424199, S2CID 622344. Previously announced at FOCS 2010. 23. O'Donnell, Ryan; Wright, John (2012), "A new point of NP-hardness for unique games", Proceedings of the 2012 ACM Symposium on Theory of Computing (STOC'12), New York: ACM, pp. 289–306, doi:10.1145/2213977.2214005, MR 2961512, S2CID 6737664. 24. Klarreich, Erica (April 24, 2018). "First Big Steps Toward Proving the Unique Games Conjecture". Quanta Magazine. 25. Barak, ~ Boaz (2018-01-10). "Unique Games Conjecture – halfway there?". Windows On Theory. Retrieved 2023-03-15. 26. Subhash, K.; Minzer, D.; Safra, M. (October 2018). "Pseudorandom Sets in Grassmann Graph Have Near-Perfect Expansion". 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS). pp. 592–601. doi:10.1109/FOCS.2018.00062. ISBN 978-1-5386-4230-6. S2CID 3688775. References • Khot, Subhash (2010), "On the Unique Games Conjecture", Proc. 25th IEEE Conference on Computational Complexity (PDF), pp. 99–121, doi:10.1109/CCC.2010.19. Computational hardness assumptions Number theoretic • Integer factorization • Phi-hiding • RSA problem • Strong RSA • Quadratic residuosity • Decisional composite residuosity • Higher residuosity Group theoretic • Discrete logarithm • Diffie-Hellman • Decisional Diffie–Hellman • Computational Diffie–Hellman Pairings • External Diffie–Hellman • Sub-group hiding • Decision linear Lattices • Shortest vector problem (gap) • Closest vector problem (gap) • Learning with errors • Ring learning with errors • Short integer solution Non-cryptographic • Exponential time hypothesis • Unique games conjecture • Planted clique conjecture	Wikipedia
Unique homomorphic extension theorem The unique homomorphic extension theorem is a result in mathematical logic which formalizes the intuition that the truth or falsity of a statement can be deduced from the truth values of its parts.[1][2][3] The lemma Let A be a non-empty set, X a subset of A, F a set of functions in A, and $X_{+}$ the inductive closure of X under F. Let be B any non-empty set and let G be the set of functions on B, such that there is a function $d:F\to G$ in G that maps with each function f of arity n in F the following function $d(f):B^{n}\to B$ in G (G cannot be a bijection). From this lemma we can now build the concept of unique homomorphic extension. The theorem If $X_{+}$ is a free set generated by X and F, for each function $h:X\to B$ there is a single function ${\hat {h}}:X_{+}\to B$ such that: $\forall x\in X,{\hat {h}}(x)=h(x);\qquad (1)$ For each function f of arity n > 0, for each $x_{1},\ldots ,x_{n}\in X_{+}^{n},$ ${\hat {h}}(f(x_{1},\ldots ,x_{n}))=g({\hat {h}}(x_{1}),\ldots ,{\hat {h}}(x_{n})),{\text{ where }}g=d(f)\qquad (2)$ Consequence The identities seen in (1) e (2) show that ${\hat {h}}$ is an homomorphism, specifically named the unique homomorphic extension of $h$. To prove the theorem, two requirements must be met: to prove that the extension (${\hat {h}}$) exists and is unique (assuring the lack of bijections). Proof of the theorem We must define a sequence of functions $h_{i}:X_{i}\to B$ inductively, satisfying conditions (1) and (2) restricted to $X_{i}$. For this, we define $h_{0}=h$, and given $h_{i}$ then $h_{i+1}$shall have the following graph: ${\{(f(x_{1},\ldots ,x_{n}),g(h_{i}(x_{1}),\ldots ,h_{i}(x_{n})))\mid (x_{1},\ldots ,x_{n})\in X_{i}^{n}-X_{i-1}^{n},f\in F\}}\cup {\operatorname {graph} (h_{i})}{\text{ with }}g=d(f)$ First we must be certain the graph actually has functionality, since $X_{+}$ is a free set, from the lemma we have $f(x_{1},\ldots ,x_{n})\in X_{i+1}-X_{i}$ when $(x_{1},\ldots ,x_{n})\in X_{i}^{n}-X_{i-1}^{n},(i\geq 0)$, so we only have to determine the functionality for the left side of the union. Knowing that the elements of G are functions(again, as defined by the lemma), the only instance where $(x,y)\in graph(h_{i})$ and $(x,z)\in graph(h_{i})$ for some $x\in X_{i+1}-X_{i}$ is possible is if we have $x=f(x_{1},\ldots ,x_{m})=f'(y_{1},\ldots ,y_{n})$ for some $(x_{1},\ldots ,x_{m})\in X_{i}^{m}-X_{i-1}^{m},(y_{1},\ldots ,y_{n})\in X_{i}^{n}-X_{i-1}^{n}$ and for some generators $f$ and ${f'}$ in $F$. Since $f(X_{+}^{m})$ and ${f'}(X_{+}^{n})$ are disjoint when $f\neq {f'},f(x_{1},\ldots ,x_{m})=f'(y_{1},\ldots ,Y_{n})$ this implies $f=f'$ and $m=n$. Being all $f\in F$ in $X_{+}^{n}$, we must have $x_{j}=y_{j},\forall j,1\leq j\leq n$. Then we have $y=z=g(x_{1},\ldots ,x_{n})$ with $g=d(f)$, displaying functionality. Before moving further we must make use of a new lemma that determines the rules for partial functions, it may be written as: (3)Be $(f_{n})_{n\geq 0}$ a sequence of partial functions $f_{n}:A\to B$ such that $f_{n}\subseteq f_{n+1},\forall n\geq 0$. Then, $g=(A,\bigcup graph(f_{n}),B)$ is a partial function. Using (3), ${\hat {h}}=\bigcup _{i\geq 0}h_{i}$ is a partial function. Since $dom({\hat {h}})=\bigcup dom(h_{i})=\bigcup X_{i}=X_{+}$ then ${\hat {h}}$ is total in $X_{+}$. Furthermore, it is clear from the definition of $h_{i}$ that ${\hat {h}}$ satisfies (1) and (2). To prove the uniqueness of ${\hat {h}}$, or any other function ${h'}$ that satisfies (1) and (2), it is enough to use a simple induction that shows ${\hat {h}}$ and ${h'}$ work for $X_{i},\forall i\geq 0$, and such is proved the Theorem of the Unique Homomorphic Extension. Example of a particular case We can use the theorem of unique homomorphic extension for calculating numeric expressions over whole numbers. First, we must define the following: $A=\Sigma ^{}$ where $\Sigma =\mathrm {Variables} \cup \{0,1,2,\ldots ,9\}\cup \{+,-,\}\cup \{(,)\},{\text{ where }}\|=\mathrm {Variables} \cup \{{0,\ldots ,9}\}$ Be $F=\{{f-,f+,f}\}$ $f:\Sigma ^{}\to \Sigma _{w\mapsto {-w}}^{}$ $f:\Sigma ^{}x\Sigma ^{}\to \Sigma _{w_{1},w_{2}\mapsto {w_{1}+w_{2}}}^{}$ $f:\Sigma ^{}x\Sigma ^{}\to \Sigma _{w_{1},w_{2}\mapsto {w_{1}w_{2}}}^{}$ Be $EXPR$ he inductive closure of $X$ under $F$ and be$B=\mathbb {Z} ,G={\{Soma(-.-),Mult(-,-),Menos(-)}\}$ Be $d:F\to G$ $d({f-})=menos$ $d({f+})=mais$ $d({f})=mult$ Then ${\hat {h}}:X_{+}\to \{{0,1}\}$ will be a function that calculates recursively the truth-value of a proposition, and in a way, will be an extension of the function $h:X\to \{{0,1}\}$that associates a truth-value to each atomic proposition, such that: (1)${\hat {h}}(\phi )=h(\phi )$ (2)${\hat {h}}({(\neg \phi )})=NAO({\hat {h}}(\psi ))$ (Negation) ${\hat {h}}({(\rho \land \theta )})=E({\hat {h}}(\rho ),{\hat {h}}(\theta ))$ (AND Operator) ${\hat {h}}({(\rho \lor \theta )})=OU({\hat {h}}(\rho ),{\hat {h}}(\theta ))$ (OR Operator) ${\hat {h}}({(\rho \to \theta )})=SE\,ENTAO({\hat {h}}(\rho ),{\hat {h}}(\theta ))$ (IF-THEN Operator) References 1. Gallier (2003), p. 25 2. Eiter, Thomas; Faber, Wolfgang; Trusczynksi, Miroslaw (2003-08-06). Logic Programming and Nonmonotonic Reasoning: 6th International Conference, LPNMR 2001, Vienna, Austria, September 17-19, 2001. Proceedings. Springer. p. 383. ISBN 9783540454021. 3. Bloch, Isabelle; Petrosino, Alfredo; Tettamanzi, Andrea G. B. (2006-02-15). Fuzzy Logic and Applications: 6th International Workshop, WILF 2005, Crema, Italy, September 15-17, 2005, Revised Selected Papers. Springer. ISBN 9783540325307. • Gallier, Jean (2003), Logic For Computer Science: Foundations of Automatic Theorem Proving (PDF), Philadelphia, retrieved 2017-10-25{{citation}}: CS1 maint: location missing publisher (link)	Wikipedia
Unique sink orientation In mathematics, a unique sink orientation is an orientation of the edges of a polytope such that, in every face of the polytope (including the whole polytope as one of the faces), there is exactly one vertex for which all adjoining edges are oriented inward (i.e. towards that vertex). If a polytope is given together with a linear objective function, and edges are oriented from vertices with smaller objective function values to vertices with larger objective values, the result is a unique sink orientation. Thus, unique sink orientations can be used to model linear programs as well as certain nonlinear programs such as the smallest circle problem. In hypercubes The problem of finding the sink in a unique sink orientation of a hypercube was formulated as an abstraction of linear complementarity problems by Stickney & Watson (1978) and it was termed "unique sink orientation" in 2001 (Szabó & Welzl 2001). It is possible for an algorithm to determine the unique sink of a d-dimensional hypercube in time cd for c < 2, substantially smaller than the 2d time required to examine all vertices. When the orientation has the additional property that the orientation forms a directed acyclic graph, which happens when unique sink orientations are used to model LP-type problems, it is possible to find the sink using a randomized algorithm in expected time exponential in the square root of d (Gärtner 2002). In simple polytopes A simple d-dimensional polytope is a polytope in which every vertex has exactly d incident edges. In a unique-sink orientation of a simple polytope, every subset of k incoming edges at a vertex v determines a k-dimensional face for which v is the unique sink. Therefore, the number of faces of all dimensions of the polytope (including the polytope itself, but not the empty set) can be computed by the sum of the number of subsets of incoming edges, $\sum _{v\in G(P)}2^{d_{\operatorname {in} }(v)}$ where G(P) is the graph of the polytope, and din(v) is the in-degree (number of incoming edges) of a vertex v in the given orientation (Kalai 1988). More generally, for any orientation of a simple polytope, the same sum counts the number of incident pairs of a face of the polytope and a sink of the face. And in an acyclic orientation, every face must have at least one sink. Therefore, an acyclic orientation is a unique sink orientation if and only if there is no other acyclic orientation with a smaller sum. Additionally, a k-regular subgraph of the given graph forms a face of the polytope if and only if its vertices form a lower set for at least one acyclic unique sink orientation. In this way, the face lattice of the polytope is uniquely determined from the graph (Kalai 1988). Based on this structure, the face lattices of simple polytopes can be reconstructed from their graphs in polynomial time using linear programming (Friedman 2009). References • Friedman, Eric J. (2009), "Finding a simple polytope from its graph in polynomial time", Discrete and Computational Geometry, 41 (2): 249–256, doi:10.1007/s00454-008-9121-7, MR 2471873. • Kalai, Gil (1988), "A simple way to tell a simple polytope from its graph", Journal of Combinatorial Theory, Series A, 49 (2): 381–383, doi:10.1016/0097-3165(88)90064-7, MR 0964396. • Matoušek, Jiří (2006), "The number of unique-sink orientations of the hypercube", Combinatorica, 26 (1): 91–99, CiteSeerX 10.1.1.5.491, doi:10.1007/s00493-006-0007-0, MR 2201286, S2CID 29950186. • Schurr, Ingo; Szabó, Tibor (2004), "Finding the sink takes some time: an almost quadratic lower bound for finding the sink of unique sink oriented cubes", Discrete & Computational Geometry, 31 (4): 627–642, doi:10.1007/s00454-003-0813-8, MR 2053502. • Stickney, Alan; Watson, Layne (1978), "Digraph models of Bard-type algorithms for the linear complementarity problem", Mathematics of Operations Research, 3 (4): 322–333, doi:10.1287/moor.3.4.322, MR 0509668. • Szabó, Tibor; Welzl, Emo (2001), "Unique sink orientations of cubes", 42nd IEEE Symposium on Foundations of Computer Science (Las Vegas, NV, 2001), Los Alamitos, CA: IEEE Computer Society, pp. 547–555, CiteSeerX 10.1.1.25.2115, doi:10.1109/SFCS.2001.959931, ISBN 978-0-7695-1116-0, MR 1948744, S2CID 6597643. • Gärtner, Bernd (2002), "The Random-Facet simplex algorithm on combinatorial cubes", Random Structures & Algorithms, 20 (3): 353–381, doi:10.1002/rsa.10034.	Wikipedia
Uniquely colorable graph In graph theory, a uniquely colorable graph is a k-chromatic graph that has only one possible (proper) k-coloring up to permutation of the colors. Equivalently, there is only one way to partition its vertices into k independent sets and there is no way to partition them into k − 1 independent sets. Examples A complete graph is uniquely colorable, because the only proper coloring is one that assigns each vertex a different color. Every k-tree is uniquely (k + 1)-colorable. The uniquely 4-colorable planar graphs are known to be exactly the Apollonian networks, that is, the planar 3-trees.[1] Every connected bipartite graph is uniquely 2-colorable. Its 2-coloring can be obtained by choosing a starting vertex arbitrarily, coloring the vertices at even distance from the starting vertex with one color, and coloring the vertices at odd distance from the starting vertex with the other color.[2] Properties Some properties of a uniquely k-colorable graph G with n vertices and m edges: 1. m ≥ (k - 1) n - k(k-1)/2.[3] Related concepts Minimal imperfection A minimal imperfect graph is a graph in which every subgraph is perfect. The deletion of any vertex from a minimal imperfect graph leaves a uniquely colorable subgraph. Unique edge colorability A uniquely edge-colorable graph is a k-edge-chromatic graph that has only one possible (proper) k-edge-coloring up to permutation of the colors. The only uniquely 2-edge-colorable graphs are the paths and the cycles. For any k, the stars K1,k are uniquely k-edge-colorable. Moreover, Wilson (1976) conjectured and Thomason (1978) proved that, when k ≥ 4, they are also the only members in this family. However, there exist uniquely 3-edge-colorable graphs that do not fit into this classification, such as the graph of the triangular pyramid. If a cubic graph is uniquely 3-edge-colorable, it must have exactly three Hamiltonian cycles, formed by the edges with two of its three colors, but some cubic graphs with only three Hamiltonian cycles are not uniquely 3-edge-colorable.[4] Every simple planar cubic graph that is uniquely 3-edge-colorable contains a triangle,[1] but W. T. Tutte (1976) observed that the generalized Petersen graph G(9,2) is non-planar, triangle-free, and uniquely 3-edge-colorable. For many years it was the only known such graph, and it had been conjectured to be the only such graph[5] but now infinitely many triangle-free non-planar cubic uniquely 3-edge-colorable graphs are known.[6] Unique total colorability A uniquely total colorable graph is a k-total-chromatic graph that has only one possible (proper) k-total-coloring up to permutation of the colors. Empty graphs, paths, and cycles of length divisible by 3 are uniquely total colorable graphs. Mahmoodian & Shokrollahi (1995) conjectured that they are also the only members in this family. Some properties of a uniquely k-total-colorable graph G with n vertices: 1. χ″(G) = Δ(G) + 1 unless G = K2.[7] 2. Δ(G) ≤ 2 δ(G).[7] 3. Δ(G) ≤ n/2 + 1.[8] Here χ″(G) is the total chromatic number; Δ(G) is the maximum degree; and δ(G) is the minimum degree. Notes 1. Fowler (1998). 2. Mahmoodian (1998). 3. Truszczyński (1984); Xu (1990). 4. Thomason (1982). 5. Bollobás (1978); Schwenk (1989). 6. belcastro & Haas (2015). 7. Akbari et al. (1997). 8. Akbari (2003). References • Akbari, S. (2003), "Two conjectures on uniquely totally colorable graphs", Discrete Mathematics, 266 (1–3): 41–45, doi:10.1016/S0012-365X(02)00797-5, MR 1991705. • Akbari, S.; Behzad, M.; Hajiabolhassan, H.; Mahmoodian, E. S. (1997), "Uniquely total colorable graphs", Graphs and Combinatorics, 13 (4): 305–314, doi:10.1016/S0012-365X(02)00797-5, MR 1485924. • belcastro, sarah-marie; Haas, Ruth (2015), "Triangle-free uniquely 3-edge colorable cubic graphs", Contributions to Discrete Mathematics, 10 (2): 39–44, arXiv:1508.06934, doi:10.11575/cdm.v10i2.62320, MR 3499076. • Bollobás, Béla (1978), Extremal Graph Theory, LMS Monographs, vol. 11, Academic Press, MR 0506522. • Fowler, Thomas (1998), Unique Coloring of Planar Graphs (PDF), Ph.D. thesis, Georgia Institute of Technology Mathematics Department. • Hillar, Christopher J.; Windfeldt, Troels (2008), "Algebraic characterization of uniquely vertex colorable graphs", Journal of Combinatorial Theory, Series B, 98 (2): 400–414, arXiv:math/0606565, doi:10.1016/j.jctb.2007.08.004, MR 2389606, S2CID 108304. • Mahmoodian, E. S. (1998), "Defining sets and uniqueness in graph colorings: a survey", Journal of Statistical Planning and Inference, 73 (1–2): 85–89, doi:10.1016/S0378-3758(98)00053-6, MR 1655213. • Mahmoodian, E. S.; Shokrollahi, M. A. (1995), "Open problems at the combinatorics workshop of AIMC25 (Tehran, 1994)", in C. J., Colbourn; E. S., Mahmoodian (eds.), Combinatorics Advances, Mathematics and its applications, vol. 329, Dordrecht; Boston; London: Kluwer Academic Publishers, pp. 321–324. • Schwenk, Allen J. (1989), "Enumeration of Hamiltonian cycles in certain generalized Petersen graphs", Journal of Combinatorial Theory, Series B, 47 (1): 53–59, doi:10.1016/0095-8956(89)90064-6, MR 1007713. • Thomason, A. G. (1978), "Hamiltonian cycles and uniquely edge colourable graphs", Advances in Graph Theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977), Annals of Discrete Mathematics, vol. 3, pp. 259–268, MR 0499124. • Thomason, Andrew (1982), "Cubic graphs with three Hamiltonian cycles are not always uniquely edge colorable", Journal of Graph Theory, 6 (2): 219–221, doi:10.1002/jgt.3190060218, MR 0655209. • Truszczyński, M. (1984), "Some results on uniquely colourable graphs", in Hajnal, A.; Lovász, L.; Sós, V. T. (eds.), Finite and Infinite Sets. Vol. I, II. Proceedings of the sixth Hungarian combinatorial colloquium held in Eger, July 6–11, 1981, Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, pp. 733–748, MR 0818274. • Tutte, William T. (1976), "Hamiltonian circuits", Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo I, Accad. Naz. Lincei, Rome, pp. 193–199. Atti dei Convegni Lincei, No. 17, MR 0480185. As cited by belcastro & Haas (2015). • Xu, Shao Ji (1990), "The size of uniquely colorable graphs", Journal of Combinatorial Theory, Series B, 50 (2): 319–320, doi:10.1016/0095-8956(90)90086-F, MR 1081235. • Wilson, R. J. (1976), "Problem 2", in Nash-Williams, C. St. J. A.; Sheehan, J. (eds.), Proc. British Comb. Conf. 1975, Winnipeg: Utilitas Math., p. 696. As cited by Thomason (1978). External links • Weisstein, Eric W., "Uniquely Colorable Graph", MathWorld	Wikipedia
Uniqueness case In mathematical finite group theory, the uniqueness case is one of the three possibilities for groups of characteristic 2 type given by the trichotomy theorem. The uniqueness case covers groups G of characteristic 2 type with e(G) ≥ 3 that have an almost strongly p-embedded maximal 2-local subgroup for all primes p whose 2-local p-rank is sufficiently large (usually at least 3). Aschbacher (1983a, 1983b) proved that there are no finite simple groups in the uniqueness case. References • Aschbacher, Michael (1983a), "The uniqueness case for finite groups. I", Annals of Mathematics, Second Series, 117 (2): 383–454, doi:10.2307/2007081, ISSN 0003-486X, MR 0690850 • Aschbacher, Michael (1983b), "The uniqueness case for finite groups. II", Annals of Mathematics, Second Series, 117 (3): 455–551, doi:10.2307/2007081, ISSN 0003-486X, JSTOR 2007034, MR 0690850 • Stroth, Gernot (1996), "The uniqueness case", in Arasu, K. T.; Dillon, J. F.; Harada, Koichiro; Sehgal, S.; Solomon., R. (eds.), Groups, difference sets, and the Monster (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., vol. 4, Berlin: de Gruyter, pp. 117–126, ISBN 978-3-11-014791-9, MR 1400413	Wikipedia
Nakayama algebra In algebra, a Nakayama algebra or generalized uniserial algebra is an algebra such that each left or right indecomposable projective module has a unique composition series. They were studied by Tadasi Nakayama (1940) who called them "generalized uni-serial rings". These algebras were further studied by Herbert Kupisch (1959) and later by Ichiro Murase (1963-64), by Kent Ralph Fuller (1968) and by Idun Reiten (1982). An example of a Nakayama algebra is k[x]/(xn) for k a field and n a positive integer. Current usage of uniserial differs slightly: an explanation of the difference appears here. References • Nakayama, Tadasi (1940), "Note on uni-serial and generalized uni-serial rings", Proc. Imp. Acad. Tokyo, 16: 285–289, MR 0003618 • Fuller, Kent Ralph (1968), "Generalized Uniserial Rings and their Kupisch Series", Math. Z., 106 (4): 248–260, doi:10.1007/BF01110273, S2CID 122522745 • Kupisch, Herbert (1959), "Beiträge zur Theorie nichthalbeinfacher Ringe mit Minimalbedingung", Crelle's Journal, 201: 100–112 • Murase, Ichiro (1964), "On the structure of generalized uniserial rings III.", Sci. Pap. Coll. Gen. Educ., Univ. Tokyo, 14: 11–25 • Reiten, Idun (1982), "The use of almost split sequences in the representation theory of Artin algebras", Representations of algebras (Puebla, 1980), Lecture Notes in Mathematics, vol. 944, Berlin, New York: Springer-Verlag, pp. 29–104, doi:10.1007/BFb0094057, ISBN 978-3-540-11577-9, MR 0672115	Wikipedia
Unisolvent functions In mathematics, a set of n functions f1, f2, ..., fn is unisolvent (meaning "uniquely solvable") on a domain Ω if the vectors ${\begin{bmatrix}f_{1}(x_{1})\\f_{1}(x_{2})\\\vdots \\f_{1}(x_{n})\end{bmatrix}},{\begin{bmatrix}f_{2}(x_{1})\\f_{2}(x_{2})\\\vdots \\f_{2}(x_{n})\end{bmatrix}},\dots ,{\begin{bmatrix}f_{n}(x_{1})\\f_{n}(x_{2})\\\vdots \\f_{n}(x_{n})\end{bmatrix}}$ are linearly independent for any choice of n distinct points x1, x2 ... xn in Ω. Equivalently, the collection is unisolvent if the matrix F with entries fi(xj) has nonzero determinant: det(F) ≠ 0 for any choice of distinct xj's in Ω. Unisolvency is a property of vector spaces, not just particular sets of functions. That is, a vector space of functions of dimension n is unisolvent if given any basis (equivalently, a linearly independent set of n functions), the basis is unisolvent (as a set of functions). This is because any two bases are related by an invertible matrix (the change of basis matrix), so one basis is unisolvent if and only if any other basis is unisolvent. Unisolvent systems of functions are widely used in interpolation since they guarantee a unique solution to the interpolation problem. The set of polynomials of degree at most $d$ (which form a vector space of dimension $d+1$) are unisolvent by the unisolvence theorem. Examples • 1, x, x2 is unisolvent on any interval by the unisolvence theorem • 1, x2 is unisolvent on [0, 1], but not unisolvent on [−1, 1] • 1, cos(x), cos(2x), ..., cos(nx), sin(x), sin(2x), ..., sin(nx) is unisolvent on [−π, π] • Unisolvent functions are used in linear inverse problems. Unisolvence in the finite element method When using "simple" functions to approximate an unknown function, such as in the finite element method, it is useful to consider a set of functionals $\{f_{i}\}_{i=1}^{n}$ that act on a finite dimensional vector space $V_{h}$ of functions, usually polynomials. Often, the functionals are given by evaluation at points in Euclidean space or some subset of it.[1][2] For example, let $V_{h}={\big \{}p(x)=\sum _{k=0}^{n}p_{k}x^{k}{\big \}}$ be the space of univariate polynomials of degree $m$ or less, and let $f_{k}(p):=f{\Big (}{\frac {k}{n}}{\Big )}$ for $0\leq i\leq n$ be defined by evaluation at $n+1$ equidistant points on the unit interval $[0,1]$. In this context, the unisolvence of $V_{h}$ with respect to $\{f_{k}\}_{k=1}^{n}$ means that $\{f_{k}\}_{k=1}^{n}$ is a basis for $V_{h}^{*}$, the dual space of $V_{h}$. Equivalently, and perhaps more intuitively, unisolvence here means that given any set of values $\{c_{k}\}_{k=1}^{n}$, there exists a unique polynomial $q(x)\in V_{h}$ such that $f_{k}(q)=q({\tfrac {k}{n}})=c_{k}$. Results of this type are widely applied in polynomial interpolation; given any function on $\phi \in C([0,1])$, by letting $c_{k}=\phi ({\tfrac {k}{n}})$, we can find a polynomial $q\in V_{h}$ that interpolates $\phi $ at each of the $n+1$ points: . $\phi ({\tfrac {k}{n}})=q({\tfrac {k}{n}}),\ \forall k\in \{0,1,..,n\}$ Dimensions Systems of unisolvent functions are much more common in 1 dimension than in higher dimensions. In dimension d = 2 and higher (Ω ⊂ Rd), the functions f1, f2, ..., fn cannot be unisolvent on Ω if there exists a single open set on which they are all continuous. To see this, consider moving points x1 and x2 along continuous paths in the open set until they have switched positions, such that x1 and x2 never intersect each other or any of the other xi. The determinant of the resulting system (with x1 and x2 swapped) is the negative of the determinant of the initial system. Since the functions fi are continuous, the intermediate value theorem implies that some intermediate configuration has determinant zero, hence the functions cannot be unisolvent. See also • Inverse problem References 1. Brenner, Susanne C.; Scott, L. Ridgway (2008). "The Mathematical Theory of Finite Element Methods". Texts in Applied Mathematics. doi:10.1007/978-0-387-75934-0. ISSN 0939-2475. 2. Ern, Alexandre; Guermond, Jean-Luc (2004). "Theory and Practice of Finite Elements". Applied Mathematical Sciences. doi:10.1007/978-1-4757-4355-5. ISSN 0066-5452. • Philip J. Davis: Interpolation and Approximation pp. 31–32	Wikipedia
Unisolvent point set In approximation theory, a finite collection of points $X\subset R^{n}$ is often called unisolvent for a space $W$ if any element $w\in W$ is uniquely determined by its values on $X$. $X$ is unisolvent for $\Pi _{n}^{m}$ (polynomials in n variables of degree at most m) if there exists a unique polynomial in $\Pi _{n}^{m}$ of lowest possible degree which interpolates the data $X$. Simple examples in $R$ would be the fact that two distinct points determine a line, three points determine a parabola, etc. It is clear that over $R$, any collection of k + 1 distinct points will uniquely determine a polynomial of lowest possible degree in $\Pi ^{k}$. See also • Padua points External links • Numerical Methods / Interpolation	Wikipedia
Unistochastic matrix In mathematics, a unistochastic matrix (also called unitary-stochastic) is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix. A square matrix B of size n is doubly stochastic (or bistochastic) if all its entries are non-negative real numbers and each of its rows and columns sum to 1. It is unistochastic if there exists a unitary matrix U such that $B_{ij}=\|U_{ij}\|^{2}{\text{ for }}i,j=1,\dots ,n.\,$ This definition is analogous to that for an orthostochastic matrix, which is a doubly stochastic matrix whose entries are the squares of the entries in some orthogonal matrix. Since all orthogonal matrices are necessarily unitary matrices, all orthostochastic matrices are also unistochastic. The converse, however, is not true. First, all 2-by-2 doubly stochastic matrices are both unistochastic and orthostochastic, but for larger n this is not the case. For example, take $n=3$ and consider the following doubly stochastic matrix: $B={\frac {1}{2}}{\begin{bmatrix}1&1&0\\0&1&1\\1&0&1\end{bmatrix}}.$ This matrix is not unistochastic, since any two vectors with moduli equal to the square root of the entries of two columns (or rows) of B cannot be made orthogonal by a suitable choice of phases. For $ n>2$, the set of orthostochastic matrices is a proper subset of the set of unistochastic matrices. • the set of unistochastic matrices contains all permutation matrices and its convex hull is the Birkhoff polytope of all doubly stochastic matrices • for $n\geq 3$ this set is not convex • for $n=3$ the set of triangle inequality on the moduli of the raw is a sufficient and necessary condition for the unistocasticity [1] • for $n=3$ the set of unistochastic matrices takes the form of a centrosymmetric matrix and unistochasticity of any bistochastic matrix B is implied by a non-negative value of its Jarlskog invariant[2] • for $n=3$ the relative volume of the set of unistochastic matrices with respect to the Birkhoff polytope of doubly stochastic matrices is [3] $8\pi ^{2}/105\approx 75.2\%$ • for $n=4$ explicit conditions for unistochasticity are not known yet, but there exists a numerical method to verify unistochasticity based on the algorithm by Haagerup [4] • The Schur-Horn theorem is equivalent to the following "weak convexity" property of the set ${\mathcal {U}}_{n}$ of unistochastic $n\times n$ matrices: for any vector $v\in \mathbb {R} ^{n}$ the set ${\mathcal {U}}_{n}v$ is the convex hull of the set of vectors obtained by all permutations of the entries of the vector $v$ (the permutation polytope generated by the vector $v$). • The set of $n\times n$ unistochastic matrices ${\mathcal {U}}_{n}\subset \mathbb {R} ^{(n-1)^{2}}$ has a nonempty interior. The unistochastic matrix corresponding to the unitary $n\times n$ matrix with the entries $U_{ij}=\delta _{ij}+{\frac {\theta -1}{n}}$, where $\|\theta \|=1$ and $\theta \neq \pm 1$, is an interior point of ${\mathcal {U}}_{n}$. References 1. Fedullo, A. (1992-12-01). "On the existence of a Hilbert-space model for finite-valued observables". Il Nuovo Cimento B. Springer. 107 (12): 1413–1426. doi:10.1007/BF02722852. ISSN 1826-9877. 2. Jarlskog, C. (1985-09-02). "Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Nonconservation". Physical Review Letters. American Physical Society (APS). 55 (10): 1039–1042. doi:10.1103/physrevlett.55.1039. ISSN 0031-9007. 3. Dunkl, Charles; Życzkowski, Karol (2009). "Volume of the set of unistochastic matrices of order 3 and the mean Jarlskog invariant". Journal of Mathematical Physics. AIP Publishing. 50 (12): 123521. arXiv:0909.0116. doi:10.1063/1.3272543. ISSN 0022-2488. 4. Rajchel, Grzegorz; Gąsiorowski, Adam; Życzkowski, Karol (2018-09-19). "Robust Hadamard Matrices, Unistochastic Rays in Birkhoff Polytope and Equi-Entangled Bases in Composite Spaces". Mathematics in Computer Science. Springer Science and Business Media LLC. 12 (4): 473–490. arXiv:1804.10715. doi:10.1007/s11786-018-0384-y. ISSN 1661-8270. • Bengtsson, Ingemar; Ericsson, Åsa; Kuś, Marek; Tadej, Wojciech; Życzkowski, Karol (2005), "Birkhoff's Polytope and Unistochastic Matrices, N = 3 and N = 4", Communications in Mathematical Physics, 259 (2): 307–324, arXiv:math/0402325, Bibcode:2005CMaPh.259..307B, doi:10.1007/s00220-005-1392-8. • Bengtsson, Ingemar (2004-03-11). "The importance of being unistochastic". arXiv:quant-ph/0403088. • Karabegov, Alexander (2008-06-14). "A mapping from the unitary to doubly stochastic matrices and symbols on a finite set". arXiv:0806.2357.	Wikipedia
Unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in ${\hat {\mathbf {v} }}$ (pronounced "v-hat"). Not to be confused with Vector of ones. The term direction vector, commonly denoted as d, is used to describe a unit vector being used to represent spatial direction and relative direction. 2D spatial directions are numerically equivalent to points on the unit circle and spatial directions in 3D are equivalent to a point on the unit sphere. The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e., $\mathbf {\hat {u}} ={\frac {\mathbf {u} }{\\|\mathbf {u} \\|}}$ where ‖u‖ is the norm (or length) of u.[1][2] The term normalized vector is sometimes used as a synonym for unit vector. Unit vectors are often chosen to form the basis of a vector space, and every vector in the space may be written as a linear combination of unit vectors. Orthogonal coordinates Cartesian coordinates Main article: Standard basis Unit vectors may be used to represent the axes of a Cartesian coordinate system. For instance, the standard unit vectors in the direction of the x, y, and z axes of a three dimensional Cartesian coordinate system are $\mathbf {\hat {i}} ={\begin{bmatrix}1\\0\\0\end{bmatrix}},\,\,\mathbf {\hat {j}} ={\begin{bmatrix}0\\1\\0\end{bmatrix}},\,\,\mathbf {\hat {k}} ={\begin{bmatrix}0\\0\\1\end{bmatrix}}$ They form a set of mutually orthogonal unit vectors, typically referred to as a standard basis in linear algebra. They are often denoted using common vector notation (e.g., i or ${\vec {\imath }}$) rather than standard unit vector notation (e.g., $\mathbf {\hat {\imath }} $). In most contexts it can be assumed that i, j, and k, (or ${\vec {\imath }},$ ${\vec {\jmath }},$ and ${\vec {k}}$) are versors of a 3-D Cartesian coordinate system. The notations $(\mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} )$, $(\mathbf {\hat {x}} _{1},\mathbf {\hat {x}} _{2},\mathbf {\hat {x}} _{3})$, $(\mathbf {\hat {e}} _{x},\mathbf {\hat {e}} _{y},\mathbf {\hat {e}} _{z})$, or $(\mathbf {\hat {e}} _{1},\mathbf {\hat {e}} _{2},\mathbf {\hat {e}} _{3})$, with or without hat, are also used,[1] particularly in contexts where i, j, k might lead to confusion with another quantity (for instance with index symbols such as i, j, k, which are used to identify an element of a set or array or sequence of variables). When a unit vector in space is expressed in Cartesian notation as a linear combination of i, j, k, its three scalar components can be referred to as direction cosines. The value of each component is equal to the cosine of the angle formed by the unit vector with the respective basis vector. This is one of the methods used to describe the orientation (angular position) of a straight line, segment of straight line, oriented axis, or segment of oriented axis (vector). Cylindrical coordinates See also: Jacobian matrix The three orthogonal unit vectors appropriate to cylindrical symmetry are: • ${\boldsymbol {\hat {\rho }}}$ (also designated $\mathbf {\hat {e}} $ or ${\boldsymbol {\hat {s}}}$), representing the direction along which the distance of the point from the axis of symmetry is measured; • ${\boldsymbol {\hat {\varphi }}}$, representing the direction of the motion that would be observed if the point were rotating counterclockwise about the symmetry axis; • $\mathbf {\hat {z}} $, representing the direction of the symmetry axis; They are related to the Cartesian basis ${\hat {x}}$, ${\hat {y}}$, ${\hat {z}}$ by: ${\boldsymbol {\hat {\rho }}}=\cos(\varphi )\mathbf {\hat {x}} +\sin(\varphi )\mathbf {\hat {y}} $ ${\boldsymbol {\hat {\varphi }}}=-\sin(\varphi )\mathbf {\hat {x}} +\cos(\varphi )\mathbf {\hat {y}} $ $\mathbf {\hat {z}} =\mathbf {\hat {z}} .$ The vectors ${\boldsymbol {\hat {\rho }}}$ and ${\boldsymbol {\hat {\varphi }}}$ are functions of $\varphi ,$ and are not constant in direction. When differentiating or integrating in cylindrical coordinates, these unit vectors themselves must also be operated on. The derivatives with respect to $\varphi $ are: ${\frac {\partial {\boldsymbol {\hat {\rho }}}}{\partial \varphi }}=-\sin \varphi \mathbf {\hat {x}} +\cos \varphi \mathbf {\hat {y}} ={\boldsymbol {\hat {\varphi }}}$ ${\frac {\partial {\boldsymbol {\hat {\varphi }}}}{\partial \varphi }}=-\cos \varphi \mathbf {\hat {x}} -\sin \varphi \mathbf {\hat {y}} =-{\boldsymbol {\hat {\rho }}}$ ${\frac {\partial \mathbf {\hat {z}} }{\partial \varphi }}=\mathbf {0} .$ Spherical coordinates The unit vectors appropriate to spherical symmetry are: $\mathbf {\hat {r}} $, the direction in which the radial distance from the origin increases; ${\boldsymbol {\hat {\varphi }}}$, the direction in which the angle in the x-y plane counterclockwise from the positive x-axis is increasing; and ${\boldsymbol {\hat {\theta }}}$, the direction in which the angle from the positive z axis is increasing. To minimize redundancy of representations, the polar angle $\theta $ is usually taken to lie between zero and 180 degrees. It is especially important to note the context of any ordered triplet written in spherical coordinates, as the roles of ${\boldsymbol {\hat {\varphi }}}$ and ${\boldsymbol {\hat {\theta }}}$ are often reversed. Here, the American "physics" convention[3] is used. This leaves the azimuthal angle $\varphi $ defined the same as in cylindrical coordinates. The Cartesian relations are: $\mathbf {\hat {r}} =\sin \theta \cos \varphi \mathbf {\hat {x}} +\sin \theta \sin \varphi \mathbf {\hat {y}} +\cos \theta \mathbf {\hat {z}} $ ${\boldsymbol {\hat {\theta }}}=\cos \theta \cos \varphi \mathbf {\hat {x}} +\cos \theta \sin \varphi \mathbf {\hat {y}} -\sin \theta \mathbf {\hat {z}} $ ${\boldsymbol {\hat {\varphi }}}=-\sin \varphi \mathbf {\hat {x}} +\cos \varphi \mathbf {\hat {y}} $ The spherical unit vectors depend on both $\varphi $ and $\theta $, and hence there are 5 possible non-zero derivatives. For a more complete description, see Jacobian matrix and determinant. The non-zero derivatives are: ${\frac {\partial \mathbf {\hat {r}} }{\partial \varphi }}=-\sin \theta \sin \varphi \mathbf {\hat {x}} +\sin \theta \cos \varphi \mathbf {\hat {y}} =\sin \theta {\boldsymbol {\hat {\varphi }}}$ ${\frac {\partial \mathbf {\hat {r}} }{\partial \theta }}=\cos \theta \cos \varphi \mathbf {\hat {x}} +\cos \theta \sin \varphi \mathbf {\hat {y}} -\sin \theta \mathbf {\hat {z}} ={\boldsymbol {\hat {\theta }}}$ ${\frac {\partial {\boldsymbol {\hat {\theta }}}}{\partial \varphi }}=-\cos \theta \sin \varphi \mathbf {\hat {x}} +\cos \theta \cos \varphi \mathbf {\hat {y}} =\cos \theta {\boldsymbol {\hat {\varphi }}}$ ${\frac {\partial {\boldsymbol {\hat {\theta }}}}{\partial \theta }}=-\sin \theta \cos \varphi \mathbf {\hat {x}} -\sin \theta \sin \varphi \mathbf {\hat {y}} -\cos \theta \mathbf {\hat {z}} =-\mathbf {\hat {r}} $ ${\frac {\partial {\boldsymbol {\hat {\varphi }}}}{\partial \varphi }}=-\cos \varphi \mathbf {\hat {x}} -\sin \varphi \mathbf {\hat {y}} =-\sin \theta \mathbf {\hat {r}} -\cos \theta {\boldsymbol {\hat {\theta }}}$ General unit vectors Main article: Orthogonal coordinates Common themes of unit vectors occur throughout physics and geometry:[4] Unit vector Nomenclature Diagram Tangent vector to a curve/flux line$\mathbf {\hat {t}} $ A normal vector $\mathbf {\hat {n}} $ to the plane containing and defined by the radial position vector $r\mathbf {\hat {r}} $ and angular tangential direction of rotation $\theta {\boldsymbol {\hat {\theta }}}$ is necessary so that the vector equations of angular motion hold. Normal to a surface tangent plane/plane containing radial position component and angular tangential component $\mathbf {\hat {n}} $ In terms of polar coordinates; $\mathbf {\hat {n}} =\mathbf {\hat {r}} \times {\boldsymbol {\hat {\theta }}}$ Binormal vector to tangent and normal $\mathbf {\hat {b}} =\mathbf {\hat {t}} \times \mathbf {\hat {n}} $[5] Parallel to some axis/line$\mathbf {\hat {e}} _{\parallel }$ One unit vector $\mathbf {\hat {e}} _{\parallel }$ aligned parallel to a principal direction (red line), and a perpendicular unit vector $\mathbf {\hat {e}} _{\bot }$ is in any radial direction relative to the principal line. Perpendicular to some axis/line in some radial direction $\mathbf {\hat {e}} _{\bot }$ Possible angular deviation relative to some axis/line $\mathbf {\hat {e}} _{\angle }$ Unit vector at acute deviation angle φ (including 0 or π/2 rad) relative to a principal direction. Curvilinear coordinates In general, a coordinate system may be uniquely specified using a number of linearly independent unit vectors $\mathbf {\hat {e}} _{n}$[1] (the actual number being equal to the degrees of freedom of the space). For ordinary 3-space, these vectors may be denoted $\mathbf {\hat {e}} _{1},\mathbf {\hat {e}} _{2},\mathbf {\hat {e}} _{3}$. It is nearly always convenient to define the system to be orthonormal and right-handed: $\mathbf {\hat {e}} _{i}\cdot \mathbf {\hat {e}} _{j}=\delta _{ij}$ $\mathbf {\hat {e}} _{i}\cdot (\mathbf {\hat {e}} _{j}\times \mathbf {\hat {e}} _{k})=\varepsilon _{ijk}$ where $\delta _{ij}$ is the Kronecker delta (which is 1 for i = j, and 0 otherwise) and $\varepsilon _{ijk}$ is the Levi-Civita symbol (which is 1 for permutations ordered as ijk, and −1 for permutations ordered as kji). Right versor A unit vector in $\mathbb {R} ^{3}$ was called a right versor by W. R. Hamilton, as he developed his quaternions $\mathbb {H} \subset \mathbb {R} ^{4}$. In fact, he was the originator of the term vector, as every quaternion $q=s+v$ has a scalar part s and a vector part v. If v is a unit vector in $\mathbb {R} ^{3}$, then the square of v in quaternions is –1. Thus by Euler's formula, $\exp(\theta v)=\operatorname {cis} \theta =\cos \theta +v\sin \theta $ is a versor in the 3-sphere. When θ is a right angle, the versor is a right versor: its scalar part is zero and its vector part v is a unit vector in $\mathbb {R} ^{3}$. See also Look up unit vector in Wiktionary, the free dictionary. • Cartesian coordinate system • Coordinate system • Curvilinear coordinates • Four-velocity • Jacobian matrix and determinant • Normal vector • Polar coordinate system • Standard basis • Unit interval • Unit square, cube, circle, sphere, and hyperbola • Vector notation • Vector of ones • Unit matrix Notes 1. Weisstein, Eric W. "Unit Vector". mathworld.wolfram.com. Retrieved 2020-08-19. 2. "Unit Vectors \| Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-08-19. 3. Tevian Dray and Corinne A. Manogue,Spherical Coordinates, College Math Journal 34, 168-169 (2003). 4. F. Ayres; E. Mendelson (2009). Calculus (Schaum's Outlines Series) (5th ed.). Mc Graw Hill. ISBN 978-0-07-150861-2. 5. M. R. Spiegel; S. Lipschutz; D. Spellman (2009). Vector Analysis (Schaum's Outlines Series) (2nd ed.). Mc Graw Hill. ISBN 978-0-07-161545-7. References • G. B. Arfken & H. J. Weber (2000). Mathematical Methods for Physicists (5th ed.). Academic Press. ISBN 0-12-059825-6. • Spiegel, Murray R. (1998). Schaum's Outlines: Mathematical Handbook of Formulas and Tables (2nd ed.). McGraw-Hill. ISBN 0-07-038203-4. • Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.	Wikipedia
Unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1.[1] Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as S1 because it is a one-dimensional unit n-sphere.[2][note 1] Trigonometry • Outline • History • Usage • Functions (inverse) • Generalized trigonometry Reference • Identities • Exact constants • Tables • Unit circle Laws and theorems • Sines • Cosines • Tangents • Cotangents • Pythagorean theorem Calculus • Trigonometric substitution • Integrals (inverse functions) • Derivatives If (x, y) is a point on the unit circle's circumference, then \|x\| and \|y\| are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation $x^{2}+y^{2}=1.$ Since x2 = (−x)2 for all x, and since the reflection of any point on the unit circle about the x- or y-axis is also on the unit circle, the above equation holds for all points (x, y) on the unit circle, not only those in the first quadrant. The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk. One may also use other notions of "distance" to define other "unit circles", such as the Riemannian circle; see the article on mathematical norms for additional examples. In the complex plane Main article: unit complex numbers In the complex plane, numbers of unit magnitude are called the unit complex numbers. This is the set of complex numbers z such that $\|z\|=1.$ When broken into real and imaginary components $z=x+iy,$ this condition is $\|z\|^{2}=z{\bar {z}}=x^{2}+y^{2}=1.$ The complex unit circle can be parametrized by angle measure $\theta $ from the positive real axis using the complex exponential function, $z=e^{i\theta }=\cos \theta +i\sin \theta .$ (See Euler's formula.) Under the complex multiplication operation, the unit complex numbers are group called the circle group, usually denoted $\mathbb {T} .$ In quantum mechanics, a unit complex number is called a phase factor. Trigonometric functions on the unit circle The trigonometric functions cosine and sine of angle θ may be defined on the unit circle as follows: If (x, y) is a point on the unit circle, and if the ray from the origin (0, 0) to (x, y) makes an angle θ from the positive x-axis, (where counterclockwise turning is positive), then $\cos \theta =x\quad {\text{and}}\quad \sin \theta =y.$ The equation x2 + y2 = 1 gives the relation $\cos ^{2}\theta +\sin ^{2}\theta =1.$ The unit circle also demonstrates that sine and cosine are periodic functions, with the identities $\cos \theta =\cos(2\pi k+\theta )$ $\sin \theta =\sin(2\pi k+\theta )$ for any integer k. Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OP from the origin O to a point P(x1,y1) on the unit circle such that an angle t with 0 < t < π/2 is formed with the positive arm of the x-axis. Now consider a point Q(x1,0) and line segments PQ ⊥ OQ. The result is a right triangle △OPQ with ∠QOP = t. Because PQ has length y1, OQ length x1, and OP has length 1 as a radius on the unit circle, sin(t) = y1 and cos(t) = x1. Having established these equivalences, take another radius OR from the origin to a point R(−x1,y1) on the circle such that the same angle t is formed with the negative arm of the x-axis. Now consider a point S(−x1,0) and line segments RS ⊥ OS. The result is a right triangle △ORS with ∠SOR = t. It can hence be seen that, because ∠ROQ = π − t, R is at (cos(π − t), sin(π − t)) in the same way that P is at (cos(t), sin(t)). The conclusion is that, since (−x1, y1) is the same as (cos(π − t), sin(π − t)) and (x1,y1) is the same as (cos(t),sin(t)), it is true that sin(t) = sin(π − t) and −cos(t) = cos(π − t). It may be inferred in a similar manner that tan(π − t) = −tan(t), since tan(t) = y1/x1 and tan(π − t) = y1/−x1. A simple demonstration of the above can be seen in the equality sin(π/4) = sin(3π/4) = 1/√2. When working with right triangles, sine, cosine, and other trigonometric functions only make sense for angle measures more than zero and less than π/2. However, when defined with the unit circle, these functions produce meaningful values for any real-valued angle measure – even those greater than 2π. In fact, all six standard trigonometric functions – sine, cosine, tangent, cotangent, secant, and cosecant, as well as archaic functions like versine and exsecant – can be defined geometrically in terms of a unit circle, as shown at right. Using the unit circle, the values of any trigonometric function for many angles other than those labeled can be easily calculated by hand using the angle sum and difference formulas. Complex dynamics Main article: Complex dynamics The Julia set of discrete nonlinear dynamical system with evolution function: $f_{0}(x)=x^{2}$ is a unit circle. It is a simplest case so it is widely used in the study of dynamical systems. See also • Angle measure • Pythagorean trigonometric identity • Riemannian circle • Radian • Unit disk • Unit sphere • Unit hyperbola • Unit square • Turn (angle) • z-transform • Smith chart Notes 1. Confusingly, in geometry a unit circle is often considered to be a 2-sphere—not a 1-sphere. The unit circle is "embedded" in a 2-dimensional plane that contains both height and width—hence why it is called a 2-sphere in geometry. However, the surface of the circle itself is one-dimensional, which is why topologists classify it as a 1-sphere. For further discussion, see the technical distinction between a circle and a disk.[2] References 1. Weisstein, Eric W. "Unit Circle". mathworld.wolfram.com. Retrieved 2020-05-05. 2. Weisstein, Eric W. "Hypersphere". mathworld.wolfram.com. Retrieved 2020-05-06.	Wikipedia
Unit fraction A unit fraction is a positive fraction with one as its numerator, 1/n. It is the multiplicative inverse (reciprocal) of the denominator of the fraction, which must be a positive natural number. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc. When an object is divided into equal parts, each part is a unit fraction of the whole. Multiplying two unit fractions produces another unit fraction, but other arithmetic operations do not preserve unit fractions. In modular arithmetic, unit fractions can be converted into equivalent whole numbers, allowing modular division to be transformed into multiplication. Every rational number can be represented as a sum of distinct unit fractions; these representations are called Egyptian fractions based on their use in ancient Egyptian mathematics. Many infinite sums of unit fractions are meaningful mathematically. In geometry, unit fractions can be used to characterize the curvature of triangle groups and the tangencies of Ford circles. Unit fractions are commonly used in fair division, and this familiar application is used in mathematics education as an early step toward the understanding of other fractions. Unit fractions are common in probability theory due to the principle of indifference. They also have applications in combinatorial optimization and in analyzing the pattern of frequencies in the hydrogen spectral series. Arithmetic The unit fractions are the rational numbers that can be written in the form ${\frac {1}{n}},$ where $n$ can be any positive natural number. They are thus the multiplicative inverses of the positive integers. When something is divided into $n$ equal parts, each part is a $1/n$ fraction of the whole.[1] Elementary arithmetic Multiplying any two unit fractions results in a product that is another unit fraction:[2] ${\frac {1}{x}}\times {\frac {1}{y}}={\frac {1}{xy}}.$ However, adding,[3] subtracting,[3] or dividing two unit fractions produces a result that is generally not a unit fraction: ${\frac {1}{x}}+{\frac {1}{y}}={\frac {x+y}{xy}}$ ${\frac {1}{x}}-{\frac {1}{y}}={\frac {y-x}{xy}}$ ${\frac {1}{x}}\div {\frac {1}{y}}={\frac {y}{x}}.$ As the last of these formulas shows, every fraction can be expressed as a quotient of two unit fractions.[4] Modular arithmetic In modular arithmetic, any unit fraction can be converted into an equivalent whole number using the extended Euclidean algorithm.[5][6] This conversion can be used to perform modular division: dividing by a number $x$, modulo $y$, can be performed by converting the unit fraction $1/x$ into an equivalent whole number modulo $y$, and then multiplying by that number.[7] In more detail, suppose that $x$ is relatively prime to $y$ (otherwise, division by $x$ is not defined modulo $y$). The extended Euclidean algorithm for the greatest common divisor can be used to find integers $a$ and $b$ such that Bézout's identity is satisfied: $\displaystyle ax+by=\gcd(x,y)=1.$ In modulo-$y$ arithmetic, the term $by$ can be eliminated as it is zero modulo $y$. This leaves $\displaystyle ax\equiv 1{\pmod {y}}.$ That is, $a$ is the modular inverse of $x$, the number that when multiplied by $x$ produces one. Equivalently,[5][6] $a\equiv {\frac {1}{x}}{\pmod {y}}.$ Thus division by $x$ (modulo $y$) can instead be performed by multiplying by the integer $a$.[7] Combinations Several constructions in mathematics involve combining multiple unit fractions together, often by adding them. Finite sums Any positive rational number can be written as the sum of distinct unit fractions, in multiple ways. For example, ${\frac {4}{5}}={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{20}}={\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{6}}+{\frac {1}{10}}.$ These sums are called Egyptian fractions, because the ancient Egyptian civilisations used them as notation for more general rational numbers. There is still interest today in analyzing the methods used by the ancients to choose among the possible representations for a fractional number, and to calculate with such representations.[8] The topic of Egyptian fractions has also seen interest in modern number theory; for instance, the Erdős–Graham conjecture[9] and the Erdős–Straus conjecture[10] concern sums of unit fractions, as does the definition of Ore's harmonic numbers.[11] In geometric group theory, triangle groups are classified into Euclidean, spherical, and hyperbolic cases according to whether an associated sum of unit fractions is equal to one, greater than one, or less than one respectively.[12] Infinite series Many well-known infinite series have terms that are unit fractions. These include: • The harmonic series, the sum of all positive unit fractions. This sum diverges, and its partial sums ${\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}$ closely approximate the natural logarithm of $n$ plus the Euler–Mascheroni constant.[13] Changing every other addition to a subtraction produces the alternating harmonic series, which sums to the natural logarithm of 2:[14] $\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots =\ln 2.$ • The Leibniz formula for π is[15] $1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots ={\frac {\pi }{4}}.$ • The Basel problem concerns the sum of the square unit fractions:[16] $1+{\frac {1}{4}}+{\frac {1}{9}}+{\frac {1}{16}}+\cdots ={\frac {\pi ^{2}}{6}}.$ Similarly, Apéry's constant is an irrational number, the sum of the cubed unit fractions.[17] • The binary geometric series is[18] $1+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+{\frac {1}{16}}+\cdots =2.$ Matrices A Hilbert matrix is a square matrix in which the elements on the $i$th antidiagonal all equal the unit fraction $1/i$. That is, it has elements $B_{i,j}={\frac {1}{i+j-1}}.$ For example, the matrix ${\begin{bmatrix}1&{\frac {1}{2}}&{\frac {1}{3}}\\{\frac {1}{2}}&{\frac {1}{3}}&{\frac {1}{4}}\\{\frac {1}{3}}&{\frac {1}{4}}&{\frac {1}{5}}\end{bmatrix}}$ is a Hilbert matrix. It has the unusual property that all elements in its inverse matrix are integers.[19] Similarly, Richardson (2001) defined a matrix whose elements are unit fractions whose denominators are Fibonacci numbers: $C_{i,j}={\frac {1}{F_{i+j-1}}},$ where $F_{i}$ denotes the $i$th Fibonacci number. He calls this matrix the Filbert matrix and it has the same property of having an integer inverse.[20] Adjacency and Ford circles Two fractions $a/b$ and $c/d$ (in lowest terms) are called adjacent if $ad-bc=\pm 1,$ which implies that they differ from each other by a unit fraction: $\left\|{\frac {1}{a}}-{\frac {1}{b}}\right\|={\frac {\|ad-bc\|}{bd}}={\frac {1}{bd}}.$ For instance, ${\tfrac {1}{2}}$ and ${\tfrac {3}{5}}$ are adjacent: $1\cdot 5-2\cdot 3=-1$ and ${\tfrac {3}{5}}-{\tfrac {1}{2}}={\tfrac {1}{10}}$. However, some pairs of fractions whose difference is a unit fraction are not adjacent in this sense: for instance, ${\tfrac {1}{3}}$ and ${\tfrac {2}{3}}$ differ by a unit fraction, but are not adjacent, because for them $ad-bc=3$.[21] This terminology comes from the study of Ford circles. These are a system of circles that are tangent to the number line at a given fraction and have the squared denominator of the fraction as their diameter. Fractions $a/b$ and $c/d$ are adjacent if and only if their Ford circles are tangent circles.[21] Applications Fair division and mathematics education In mathematics education, unit fractions are often introduced earlier than other kinds of fractions, because of the ease of explaining them visually as equal parts of a whole.[22][23] A common practical use of unit fractions is to divide food equally among a number of people, and exercises in performing this sort of fair division are a standard classroom example in teaching students to work with unit fractions.[24] Probability and statistics In a uniform distribution on a discrete space, all probabilities are equal unit fractions. Due to the principle of indifference, probabilities of this form arise frequently in statistical calculations.[25] Unequal probabilities related to unit fractions arise in Zipf's law. This states that, for many observed phenomena involving the selection of items from an ordered sequence, the probability that the $n$th item is selected is proportional to the unit fraction $1/n$.[26] Combinatorial optimization In the study of combinatorial optimization problems, bin packing problems involve an input sequence of items with fractional sizes, which must be placed into bins whose capacity (the total size of items placed into each bin) is one. Research into these problems has included the study of restricted bin packing problems where the item sizes are unit fractions.[27][28] One motivation for this is as a test case for more general bin packing methods. Another involves a form of pinwheel scheduling, in which a collection of messages of equal length must each be repeatedly broadcast on a limited number of communication channels, with each message having a maximum delay between the start times of its repeated broadcasts. An item whose delay is $k$ times the length of a message must occupy a fraction of at least $1/k$ of the time slots on the channel it is assigned to, so a solution to the scheduling problem can only come from a solution to the unit fraction bin packing problem with the channels as bins and the fractions $1/k$ as item sizes.[27] Even for bin packing problems with arbitrary item sizes, it can be helpful to round each item size up to the next larger unit fraction, and then apply a bin packing algorithm specialized for unit fraction sizes. In particular, the harmonic bin packing method does exactly this, and then packs each bin using items of only a single rounded unit fraction size.[28] Physics The energy levels of photons that can be absorbed or emitted by a hydrogen atom are, according to the Rydberg formula, proportional to the differences of two unit fractions. An explanation for this phenomenon is provided by the Bohr model, according to which the energy levels of electron orbitals in a hydrogen atom are inversely proportional to square unit fractions, and the energy of a photon is quantized to the difference between two levels.[29] Arthur Eddington argued that the fine-structure constant was a unit fraction. He initially thought it to be 1/136 and later changed his theory to 1/137. This contention has been falsified, given that current estimates of the fine structure constant are (to 6 significant digits) 1/137.036.[30] See also • 17-animal inheritance puzzle, a puzzle involving fair division into unit fractions • Submultiple, a number that produces a unit fraction when used as the numerator with a given denominator • Superparticular ratio, one plus a unit fraction, important in musical harmony References 1. Cavey, Laurie O.; Kinzel, Margaret T. (February 2014), "From whole numbers to invert and multiply", Teaching Children Mathematics, 20 (6): 374–383, doi:10.5951/teacchilmath.20.6.0374, JSTOR 10.5951/teacchilmath.20.6.0374 2. Solomon, Pearl Gold (2007), The Math We Need to Know and Do in Grades 6 9: Concepts, Skills, Standards, and Assessments, Corwin Press, p. 157, ISBN 9781412917261 3. Betz, William (1957), Algebra for Today, First Year, Ginn, p. 370 4. Humenberger, Hans (Fall 2014), "Egyptian fractions – representations as sums of unit fractions", Mathematics and Computer Education, 48 (3): 268–283, ProQuest 1622317875 5. Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001) [1990], "31.4 Solving modular linear equations", Introduction to Algorithms (2nd ed.), MIT Press and McGraw-Hill, pp. 869–872, ISBN 0-262-03293-7 6. Goodrich, Michael T.; Tamassia, Roberto (2015), "Section 24.2.2: Modular multiplicative inverses", Algorithm Design and Applications, Wiley, pp. 697–698, ISBN 978-1-118-33591-8 7. Brent, Richard P.; Zimmermann, Paul (2010), "2.5 Modular division and inversion", Modern Computer Arithmetic (PDF), Cambridge Monographs on Applied and Computational Mathematics, vol. 18, Cambridge University Press, pp. 65–68, arXiv:1004.4710, doi:10.1017/cbo9780511921698.001, ISBN 9781139492287 8. Guy, Richard K. (2004), "D11. Egyptian Fractions", Unsolved problems in number theory (3rd ed.), Springer-Verlag, pp. 252–262, ISBN 978-0-387-20860-2 9. Croot, Ernest S., III (2003), "On a coloring conjecture about unit fractions", Annals of Mathematics, 157 (2): 545–556, arXiv:math.NT/0311421, doi:10.4007/annals.2003.157.545, MR 1973054, S2CID 13514070{{citation}}: CS1 maint: multiple names: authors list (link) 10. Elsholtz, Christian; Tao, Terence (2013), "Counting the number of solutions to the Erdős–Straus equation on unit fractions" (PDF), Journal of the Australian Mathematical Society, 94 (1): 50–105, arXiv:1107.1010, doi:10.1017/S1446788712000468, MR 3101397, S2CID 17233943 11. Ore, Øystein (1948), "On the averages of the divisors of a number", The American Mathematical Monthly, 55 (10): 615–619, doi:10.2307/2305616, JSTOR 2305616 12. Magnus, Wilhelm (1974), Noneuclidean Tesselations and their Groups, Pure and Applied Mathematics, vol. 61, Academic Press, p. 65, ISBN 9780080873770, MR 0352287 13. Boas, R. P. Jr.; Wrench, J. W. Jr. (1971), "Partial sums of the harmonic series", The American Mathematical Monthly, 78 (8): 864–870, doi:10.1080/00029890.1971.11992881, JSTOR 2316476, MR 0289994 14. Freniche, Francisco J. (2010), "On Riemann's rearrangement theorem for the alternating harmonic series" (PDF), The American Mathematical Monthly, 117 (5): 442–448, doi:10.4169/000298910X485969, JSTOR 10.4169/000298910x485969, MR 2663251, S2CID 20575373 15. Roy, Ranjan (1990), "The discovery of the series formula for π by Leibniz, Gregory and Nilakantha" (PDF), Mathematics Magazine, 63 (5): 291–306, doi:10.1080/0025570X.1990.11977541 16. Ayoub, Raymond (1974), "Euler and the zeta function", The American Mathematical Monthly, 81 (10): 1067–86, doi:10.2307/2319041, JSTOR 2319041 17. van der Poorten, Alfred (1979), "A proof that Euler missed ... Apéry's proof of the irrationality of $\zeta (3)$" (PDF), The Mathematical Intelligencer, 1 (4): 195–203, doi:10.1007/BF03028234, S2CID 121589323, archived from the original (PDF) on 2011-07-06 18. Euler, Leonhard (September 1983), "From Elements of Algebra", Old Intelligencer, The Mathematical Intelligencer, 5 (3): 75–76, doi:10.1007/bf03026580, S2CID 122191726 19. Choi, Man Duen (1983), "Tricks or treats with the Hilbert matrix", The American Mathematical Monthly, 90 (5): 301–312, doi:10.2307/2975779, JSTOR 2975779, MR 0701570 20. Richardson, Thomas M. (2001), "The Filbert matrix" (PDF), Fibonacci Quarterly, 39 (3): 268–275, arXiv:math.RA/9905079, Bibcode:1999math......5079R 21. Ford, L. R. (1938), "Fractions", The American Mathematical Monthly, 45 (9): 586–601, doi:10.1080/00029890.1938.11990863, JSTOR 2302799, MR 1524411 22. Polkinghorne, Ada R. (May 1935), "Young-children and fractions", Childhood Education, 11 (8): 354–358, doi:10.1080/00094056.1935.10725374 23. Empson, Susan Baker; Jacobs, Victoria R.; Jessup, Naomi A.; Hewitt, Amy; Pynes, D'Anna; Krause, Gladys (April 2020), "Unit fractions as superheroes for instruction", The Mathematics Teacher, 113 (4): 278–286, doi:10.5951/mtlt.2018.0024, JSTOR 10.5951/mtlt.2018.0024, S2CID 216283105 24. Wilson, P. Holt; Edgington, Cynthia P.; Nguyen, Kenny H.; Pescosolido, Ryan C.; Confrey, Jere (November 2011), "Fractions: how to fair share", Mathematics Teaching in the Middle School, 17 (4): 230–236, doi:10.5951/mathteacmiddscho.17.4.0230, JSTOR 10.5951/mathteacmiddscho.17.4.0230 25. Welsh, Alan H. (1996), Aspects of Statistical Inference, Wiley Series in Probability and Statistics, vol. 246, John Wiley and Sons, p. 66, ISBN 978-0-471-11591-5 26. Saichev, Alexander; Malevergne, Yannick; Sornette, Didier (2009), Theory of Zipf's Law and Beyond, Lecture Notes in Economics and Mathematical Systems, vol. 632, Springer-Verlag, ISBN 978-3-642-02945-5 27. Bar-Noy, Amotz; Ladner, Richard E.; Tamir, Tami (2007), "Windows scheduling as a restricted version of bin packing", ACM Transactions on Algorithms, 3 (3): A28:1–A28:22, doi:10.1145/1273340.1273344, MR 2344019, S2CID 2461059 28. van Stee, Rob (June 2012), "SIGACT news online algorithms column 20: The power of harmony" (PDF), ACM SIGACT News, 43 (2): 127–136, doi:10.1145/2261417.2261440, S2CID 14805804 29. Yang, Fujia; Hamilton, Joseph H. (2009), Modern Atomic and Nuclear Physics, World Scientific, pp. 81–86, ISBN 978-981-283-678-6 30. Kilmister, Clive William (1994), Eddington's Search for a Fundamental Theory: A Key to the Universe, Cambridge University Press, ISBN 978-0-521-37165-0 Fractions and ratios Division and ratio • Dividend ÷ Divisor = Quotient Fraction • Numerator/Denominator = Quotient • Algebraic • Aspect • Binary • Continued • Decimal • Dyadic • Egyptian • Golden • Silver • Integer • Irreducible • Reduction • Just intonation • LCD • Musical interval • Paper size • Percentage • Unit	Wikipedia
Unit commitment problem in electrical power production The unit commitment problem (UC) in electrical power production is a large family of mathematical optimization problems where the production of a set of electrical generators is coordinated in order to achieve some common target, usually either matching the energy demand at minimum cost or maximizing revenue from electricity production. This is necessary because it is difficult to store electrical energy on a scale comparable with normal consumption; hence, each (substantial) variation in the consumption must be matched by a corresponding variation of the production. Coordinating generation units is a difficult task for a number of reasons: • the number of units can be large (hundreds or thousands); • there are several types of units, with significantly different energy production costs and constraints about how power can be produced; • generation is distributed across a vast geographical area (e.g., a country), and therefore the response of the electrical grid, itself a highly complex system, has to be taken into account: even if the production levels of all units are known, checking whether the load can be sustained and what the losses are requires highly complex power flow computations. Because the relevant details of the electrical system vary greatly worldwide, there are many variants of the UC problem, which are often very difficult to solve. This is also because, since some units require quite a long time (many hours) to start up or shut down, the decisions need be taken well in advance (usually, the day before), which implies that these problems have to be solved within tight time limits (several minutes to a few hours). UC is therefore one of the fundamental problems in power system management and simulation. It has been studied for many years,[1][2] and still is one of the most significant energy optimization problems. Recent surveys on the subject[3][4] count many hundreds of scientific articles devoted to the problem. Furthermore, several commercial products comprise specific modules for solving UC, such as MAON[5] and PLEXOS,[6] or are even entirely devoted to its solution.[7] Elements of unit commitment problems There are many different UC problems, as the electrical system is structured and governed differently across the world. Common elements are: • A time horizon along which the decisions have to be made, sampled at a finite number of time instants. This is usually one or two days, up to a week, where instants are usually hours or half-hours; less frequently, 15 or 5 minutes. Hence, time instants are typically between 24 and around 2000. • A set of generating units with the corresponding energy production cost and/or emission curves, and (complex) technical constraints. • A representation of the significant part of the grid network. • A (forecasted) load profile to be satisfied, i.e., the net amount of energy to be delivered to each node of the grid network at each time instant. • Possibly, a set of reliability constraints[8] ensuring that demand will be satisfied even if some unforeseen events occur. • Possibly, financial and/or regulatory conditions[9] (energy revenues, market operation constraints, financial instruments, ...). The decisions that have to be taken usually comprise: • commitment decisions: whether a unit is producing energy at any time instant; • production decisions: how much energy a unit is producing at any time instant; • network decisions: how much energy is flowing (and in which direction) on each branch of the transmission and/or distribution grid at any given time instant. While the above features are usually present, there are many combinations and many different cases. Among these we mention: • whether the units and the grid are all handled by a Monopolistic Operator (MO),[10] or a separate Transmission System Operator (TSO) manages the grid providing fair and not discriminatory access to Generating Companies (GenCos) that compete to satisfy the production on the (or, most often, several interconnected) energy market(s); • the different kinds of energy production units, such as thermal/nuclear ones, hydro-electric ones, and renewable sources (wind, solar, ...); • which units can be modulated, i.e., their produced energy can be decided by the operator (albeit subject to the technical constraints of the unit), as opposed to it being entirely dictated by external factors such as weather conditions; • the level of detail at which the working of the electrical grid must be considered, ranging from basically ignoring it to considering the possibility of dynamically opening (interrupting) a line in order to optimally change the energy routing on the grid.[11] Management objectives The objectives of UC depend on the aims of the actor for which it is solved. For a MO, this is basically to minimize energy production costs while satisfying the demand; reliability and emissions are usually treated as constraints. In a free-market regime, the aim is rather to maximize energy production profits, i.e., the difference between revenues (due to selling energy) and costs (due to producing it). If the GenCo is a price maker, i.e., it has sufficient size to influence market prices, it may in principle perform strategic bidding[12] in order to improve its profits. This means bidding its production at high cost so as to raise market prices, losing market share but retaining some because, essentially, there is not enough generation capacity. For some regions this may be due to the fact that there is not enough grid network capacity to import energy from nearby regions with available generation capacity.[13] While the electrical markets are highly regulated in order to, among other things, rule out such behavior, large producers can still benefit from simultaneously optimizing the bids of all their units to take into account their combined effect on market prices.[14] On the contrary, price takers can simply optimize each generator independently, as, not having a significant impact on prices, the corresponding decisions are not correlated.[15] Types of production units In the context of UC, generating units are usually classified as: • Thermal units, which include nuclear ones, that burn some sort of fuel to produce electricity. They are subject to numerous complex technical constraints, among which we mention minimum up/down time, ramp up/down rate, modulation/stability (a unit cannot change its production level too many times[16]), and start-up/shut-down ramp rate (when starting/stopping, a unit must follow a specific power curve which may depend on how long the plant has been offline/online[17]). Therefore, optimizing even a single unit is in principle already a complex problem which requires specific techniques.[18] • Hydro units, that generate energy by harvesting water potential energy, are often organized into systems of connected reservoirs called hydro valleys. Because water released by an upstream reservoir reaches the downstream one (after some time), and therefore becomes available to generate energy there, decisions on the optimal production must be taken for all units simultaneously, which makes the problem rather difficult even if no (or little) thermal production is involved,[19] even more so if the complete electrical system is considered.[20] Hydro units may include pumped-storage units, where energy can be spent to pump water uphill. This is the only current technology capable of storing enough (potential) energy to be significant at the typical level of the UC problem. Hydro units are subject to complex technical constraints. The amount of energy generated by turbining some amount of water is not constant, but it depends on the water head which in turn depends on previous decisions. The relationship is nonlinear and nonconvex, making the problem particularly difficult to solve.[21] • Renewable generation units, such as wind farms, solar plants, run-of-river hydro units (without a dedicated reservoir, and therefore whose production is dictated by the flowing water), and geothermal units. Most of these cannot be modulated, and several are also intermittent, i.e., their production is difficult to accurately forecast well in advance. In UC, these units do not really correspond to decisions, since they cannot be influenced. Rather, their production is considered fixed and added to that of the other sources. The substantial increase of intermittent renewable generation in recent years has significantly increased uncertainty in the net load (demand minus production that cannot be modulated), which has challenged the traditional view that the forecasted load in UC is accurate enough.[22] Electrical grid models There are three different ways in which the energy grid is represented within a UC: • In the single bus approximation the grid is ignored: demand is considered to be satisfied whenever total production equals total demand, irrespective of their geographical location. • In the DC approximation only Kirchhoff's current law is modeled; this corresponds to reactive power flow being neglected, the voltage angles differences being considered small, and the angle voltage profile being assumed constant; • In the full AC model the complete Kirchhoff laws are used: this results in highly nonlinear and nonconvex constraints in the model. When the full AC model is used, UC actually incorporates the optimal power flow problem, which is already a nonconvex nonlinear problem. Recently, the traditional "passive" view of the energy grid in UC has been challenged. In a fixed electrical network currents cannot be routed, their behavior being entirely dictated by nodal power injection: the only way to modify the network load is therefore to change nodal demand or production, for which there is limited scope. However, a somewhat counter-intuitive consequence of Kirchhoff laws is that interrupting a line (maybe even a congested one) causes a global re-routing of electrical energy and may therefore improve grid performances. This has led to defining the Optimal Transmission Switching problem,[11] whereby some of the lines of the grid can be dynamically opened and closed across the time horizon. Incorporating this feature in the UC problem makes it difficult to solve even with the DC approximation, even more so with the full AC model.[23] Uncertainty in unit commitment problems A troubling consequence of the fact that UC needs be solved well in advance to the actual operations is that the future state of the system is not known exactly, and therefore needs be estimated. This used to be a relatively minor problem when the uncertainty in the system was only due to variation of users' demand, which on aggregate can be forecasted quite effectively,[24][25] and occurrence of lines or generators faults, which can be dealt with by well established rules (spinning reserve). However, in recent years the production from intermittent renewable production sources has significantly increased. This has, in turn, very significantly increased the impact of uncertainty in the system, so that ignoring it (as traditionally done by taking average point estimates) risks significant cost increases.[22] This had made it necessary to resort to appropriate mathematical modeling techniques to properly take uncertainty into account, such as: • Robust optimization approaches; • Scenario optimization approaches; • Chance-constrained optimization approaches. The combination of the (already, many) traditional forms of UC problems with the several (old and) new forms of uncertainty gives rise to the even larger family of Uncertain Unit Commitment[4] (UUC) problems, which are currently at the frontier of applied and methodological research. Integrated Transmission and Distribution Models One of the major issues with the real-time unit commitment problem is the fact that the electricity demand of the transmission network is usually treated as a "load point" at each distribution system. The reality, however, is that each load point is a complex distribution network with its own sub-loads, generators, and DERs. By simplifying a distribution into load points can lead to extreme operational troubles of the whole power grid. Such problems include high pressure on the power transmission system and reverse power flow from the distribution systems towards the power transmission system. A newly pursued approach for more effectively solving the unit-commitment problem hence is born by Integrated Transmission and Distribution Systems.[26] In such models, the unit commitment problem of the Transmission Systems is usually combined with the Renewable Management Problem of the Distribution Systems by the means of bi-level programming tools. See also • Electricity market References 1. C.J. Baldwin, K.M. Dale, R.F. Dittrich. A study of the economic shutdown of generating units in daily dispatch. Transactions of the American Institute of Electrical Engineers Power Apparatus and Systems, Part III, 78(4):1272–1282, 1959. 2. J.F. Bard. Short-term scheduling of thermal-electric generators using Lagrangian relaxation. Operations Research 1338 36(5):765–766, 1988. 3. N.P. Padhy. Unit commitment – a bibliographical survey, IEEE Transactions On Power Systems 19(2):1196–1205, 2004. 4. M. Tahanan, W. van Ackooij, A. Frangioni, F. Lacalandra. Large-scale Unit Commitment under uncertainty, 4OR 13(2), 115–171, 2015. 5. Maon Model Handbook 6. PLEXOS® Integrated Energy Model 7. Power optimization 8. M. Shahidehpour, H. Yamin, and Z. Li. Market Operations in Electric Power Systems: Forecasting, Scheduling, and Risk Management, Wiley-IEEE Press, 2002. 9. C. Harris. Electricity markets: Pricing, structures and Economics, volume 565 of The Wiley Finance Series. John Wiley and Sons, 2011. 10. A.J. Conejo and F.J. Prieto. Mathematical programming and electricity markets, TOP 9(1):1–53, 2001. 11. E.B. Fisher, R.P. O'Neill, M.C. Ferris. Optimal transmission switching, IEEE Transactions on Power Systems 23(3):1346–1355, 2008. 12. A.K. David, F. Wen. Strategic bidding in competitive electricity markets: a literature survey In Proceedings IEEE PES Summer Meeting 4, 2168–2173, 2001. 13. T. Peng and K. Tomsovic. Congestion influence on bidding strategies in an electricity market, IEEE Transactions on Power Systems 18(3):1054–1061, August 2003. 14. A.J. Conejo, J. Contreras, J.M. Arroyo, S. de la Torre. Optimal response of an oligopolistic generating company to a competitive pool-based electric power market, IEEE Transactions on Power Systems 17(2):424–430, 2002. 15. J.M. Arroyo, A.J. Conejo. Optimal response of a thermal unit to an electricity spot market, IEEE Transactions on Power Systems 15(3):1098–1104, 2000. 16. J. Batut and A. Renaud. Daily scheduling with transmission constraints: A new class of algorithms, IEEE Transactions on Power Systems 7(3):982–989, 1992. 17. G. Morales-España, J.M. Latorre, A. Ramos. Tight and Compact MILP Formulation of Start-Up and Shut-Down Ramping in Unit Commitment, IEEE Transactions on Power Systems 28(2), 1288–1296, 2013. 18. A. Frangioni, C. Gentile. Solving Nonlinear Single-Unit Commitment Problems with Ramping Constraints, Operations Research 54(4), 767–775, 2006. 19. E.C. Finardi and E.L. Da Silva. Solving the hydro unit commitment problem via dual decomposition and sequential quadratic programming, IEEE Transactions on Power Systems 21(2):835–844, 2006. 20. F.Y.K. Takigawa, E.L. da Silva, E.C. Finardi, and R.N. Rodrigues. Solving the hydrothermal scheduling problem considering network constraints., Electric Power Systems Research 88:89–97, 2012. 21. A. Borghetti, C. D’Ambrosio, A. Lodi, S. Martello. A MILP approach for short-term hydro scheduling and unit commitment with head-dependent reservoir, IEEE Transactions on Power Systems 23(3):1115–1124, 2008. 22. A. Keyhani, M.N. Marwali, and M. Dai. Integration of Green and Renewable Energy in Electric Power Systems, Wiley, 2010. 23. K.W. Hedman, M.C. Ferris, R.P. O’Neill, E.B. Fisher, S.S. Oren. Co-optimization of generation unit commitment and transmission switching with n − 1 reliability, IEEE Transactions on Power Systems 25(2):1052–1063, 2010. 24. E.A. Feinberg, D. Genethliou. Load Forecasting, in Applied Mathematics for Restructured Electric Power Systems, J.H. Chow, F.F. Wu, and J. Momoh eds., Springer, 269–285, 2005 25. H. Hahn, S. Meyer-Nieberg, S. Pickl. Electric load forecasting methods: Tools for decision making, European Journal of Operational Research 199(3), 902–907, 2009 26. Fathabad, Abolhassan Mohammadi; Cheng, Jianqiang; Pan, Kai (2021-01-01), Ren, Jingzheng (ed.), "Chapter 5 - Integrated power transmission and distribution systems", Renewable-Energy-Driven Future, Academic Press, pp. 169–199, ISBN 978-0-12-820539-6, retrieved 2021-01-09 External links • A description of the role of unit commitment problems in the overall context of power system management can be found in the Energy Optimization Wiki developed by the COST TD1207 project.	Wikipedia
Unit disk graph In geometric graph theory, a unit disk graph is the intersection graph of a family of unit disks in the Euclidean plane. That is, it is a graph with one vertex for each disk in the family, and with an edge between two vertices whenever the corresponding vertices lie within a unit distance of each other. They are commonly formed from a Poisson point process, making them a simple example of a random structure. Definitions There are several possible definitions of the unit disk graph, equivalent to each other up to a choice of scale factor: • Unit disk graphs are the graph formed from a collection of points in the Euclidean plane, with a vertex for each point and an edge connecting each pair of points whose distance is below a fixed threshold. • Unit disk graphs are the intersection graphs of equal-radius circles, or of equal-radius disks. These graphs have a vertex for each circle or disk, and an edge connecting each pair of circles or disks that have a nonempty intersection. • Unit disk graphs may be formed in a different way from a collection of equal-radius circles, by connecting two circles with an edge whenever one circle contains the center of the other circle. Properties Every induced subgraph of a unit disk graph is also a unit disk graph. An example of a graph that is not a unit disk graph is the star $K_{1,6}$ with one central node connected to six leaves: if each of six unit disks touches a common unit disk, some two of the six disks must touch each other. Therefore, unit disk graphs cannot contain an induced $K_{1,6}$ subgraph.[1] Infinitely many other forbidden induced subgraphs are known.[2] The number of unit disk graphs on $n$ labeled vertices is within an exponential factor of $n^{2n}$.[3] This rapid growth implies that unit disk graphs do not have bounded twin-width.[4] Applications Beginning with the work of Huson & Sen (1995), unit disk graphs have been used in computer science to model the topology of ad hoc wireless communication networks. In this application, nodes are connected through a direct wireless connection without a base station. It is assumed that all nodes are homogeneous and equipped with omnidirectional antennas. Node locations are modelled as Euclidean points, and the area within which a signal from one node can be received by another node is modelled as a circle. If all nodes have transmitters of equal power, these circles are all equal. Random geometric graphs, formed as unit disk graphs with randomly generated disk centres, have also been used as a model of percolation and various other phenomena.[5] Computational complexity If one is given a collection of unit disks (or their centres) in a space of any fixed dimension, it is possible to construct the corresponding unit disk graph in linear time, by rounding the centres to nearby integer grid points, using a hash table to find all pairs of centres within constant distance of each other, and filtering the resulting list of pairs for the ones whose circles intersect. The ratio of the number of pairs considered by this algorithm to the number of edges in the eventual graph is a constant, giving the linear time bound. However, this constant grows exponentially as a function of the dimension.[6] It is NP-hard (more specifically, complete for the existential theory of the reals) to determine whether a graph, given without geometry, can be represented as a unit disk graph.[7] Additionally, it is provably impossible in polynomial time to output explicit coordinates of a unit disk graph representation: there exist unit disk graphs that require exponentially many bits of precision in any such representation.[8] However, many important and difficult graph optimization problems such as maximum independent set, graph coloring, and minimum dominating set can be approximated efficiently by using the geometric structure of these graphs,[9] and the maximum clique problem can be solved exactly for these graphs in polynomial time, given a disk representation.[10] Even if a disk representation is not known, and an abstract graph is given as input, it is possible in polynomial time to produce either a maximum clique or a proof that the graph is not a unit disk graph,[11] and to 3-approximate the optimum coloring by using a greedy coloring algorithm.[12] See also • Barrier resilience, an algorithmic problem of breaking cycles in unit disk graphs • Indifference graph, a one-dimensional analogue of the unit disk graphs • Penny graph, the unit disk graphs for which the disks can be tangent but not overlap (contact graph) • Coin graph, the contact graph of (not necessarily unit-sized) disks • Vietoris–Rips complex, a generalization of the unit disk graph that constructs higher-order topological spaces from unit distances in a metric space • Unit distance graph, a graph formed by connecting points that are at distance exactly one rather than (as here) at most a given threshold Notes 1. Dębski, Junosza-Szaniawski & Śleszyńska-Nowak (2020). 2. Atminas & Zamaraev (2018). 3. McDiarmid & Müller (2014). 4. Bonnet et al. (2022). 5. See, e.g., Dall & Christensen (2002). 6. Bentley, Stanat & Williams (1977). 7. Breu & Kirkpatrick (1998); Kang & Müller (2011). 8. McDiarmid & Mueller (2013). 9. Marathe et al. (1994); Matsui (2000). 10. Clark, Colbourn & Johnson (1990). 11. Raghavan & Spinrad (2003). 12. Gräf, Stumpf & Weißenfels (1998). References • Atminas, Aistis; Zamaraev, Viktor (2018), "On forbidden induced subgraphs for unit disk graphs", Discrete & Computational Geometry, 60 (1): 57–97, arXiv:1602.08148, doi:10.1007/s00454-018-9968-1, MR 3807349, S2CID 254025741 • Bentley, Jon L.; Stanat, Donald F.; Williams, E. Hollins, Jr. (1977), "The complexity of finding fixed-radius near neighbors", Information Processing Letters, 6 (6): 209–212, doi:10.1016/0020-0190(77)90070-9, MR 0489084{{citation}}: CS1 maint: multiple names: authors list (link). • Bonnet, Édouard; Geniet, Colin; Kim, Eun Jung; Thomassé, Stéphan; Watrigant, Rémi (2022), "Twin-width II: small classes", Combinatorial Theory, 2 (2): P10:1–P10:42, arXiv:2006.09877, doi:10.5070/C62257876, MR 4449818 • Breu, Heinz; Kirkpatrick, David G. (1998), "Unit disk graph recognition is NP-hard", Computational Geometry: Theory and Applications, 9 (1–2): 3–24, doi:10.1016/s0925-7721(97)00014-x. • Clark, Brent N.; Colbourn, Charles J.; Johnson, David S. (1990), "Unit disk graphs", Discrete Mathematics, 86 (1–3): 165–177, doi:10.1016/0012-365X(90)90358-O. • Dall, Jesper; Christensen, Michael (2002), "Random geometric graphs", Physical Review E, 66 (1): 016121, arXiv:cond-mat/0203026, Bibcode:2002PhRvE..66a6121D, doi:10.1103/PhysRevE.66.016121, PMID 12241440, S2CID 15193516. • Dębski, Michał; Junosza-Szaniawski, Konstanty; Śleszyńska-Nowak, Małgorzata (2020), "Strong chromatic index of $K_{1,t}$-free graphs", Discrete Applied Mathematics, 284: 53–60, doi:10.1016/j.dam.2020.03.024, MR 4115456, S2CID 216369782 • Gräf, A.; Stumpf, M.; Weißenfels, G. (1998), "On coloring unit disk graphs", Algorithmica, 20 (3): 277–293, doi:10.1007/PL00009196, MR 1489033, S2CID 36161020. • Huson, Mark L.; Sen, Arunabha (1995), "Broadcast scheduling algorithms for radio networks", Military Communications Conference, IEEE MILCOM '95, vol. 2, pp. 647–651, doi:10.1109/MILCOM.1995.483546, ISBN 0-7803-2489-7, S2CID 62039740. • Kang, Ross J.; Müller, Tobias (2011), "Sphere and dot product representations of graphs", Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry (SoCG'11), June 13–15, 2011, Paris, France, pp. 308–314. • Marathe, Madhav V.; Breu, Heinz; Hunt, III, Harry B.; Ravi, S. S.; Rosenkrantz, Daniel J. (1994), Geometry based heuristics for unit disk graphs, arXiv:math.CO/9409226, Bibcode:1994math......9226M. • Matsui, Tomomi (2000), "Approximation Algorithms for Maximum Independent Set Problems and Fractional Coloring Problems on Unit Disk Graphs", Discrete and Computational Geometry, Lecture Notes in Computer Science, vol. 1763, pp. 194–200, doi:10.1007/978-3-540-46515-7_16, ISBN 978-3-540-67181-7. • McDiarmid, Colin; Mueller, Tobias (2013), "Integer realizations of disk and segment graphs", Journal of Combinatorial Theory, Series B, 103 (1): 114–143, arXiv:1111.2931, Bibcode:2011arXiv1111.2931M, doi:10.1016/j.jctb.2012.09.004 • McDiarmid, Colin; Müller, Tobias (2014), "The number of disk graphs", European Journal of Combinatorics, 35: 413–431, doi:10.1016/j.ejc.2013.06.037, MR 3090514 • Raghavan, Vijay; Spinrad, Jeremy (2003), "Robust algorithms for restricted domains", Journal of Algorithms, 48 (1): 160–172, doi:10.1016/S0196-6774(03)00048-8, MR 2006100, S2CID 16327087.	Wikipedia
Unit distance graph In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points whenever the distance between them is exactly one. To distinguish these graphs from a broader definition that allows some non-adjacent pairs of vertices to be at distance one, they may also be called strict unit distance graphs or faithful unit distance graphs. As a hereditary family of graphs, they can be characterized by forbidden induced subgraphs. The unit distance graphs include the cactus graphs, the matchstick graphs and penny graphs, and the hypercube graphs. The generalized Petersen graphs are non-strict unit distance graphs. An unsolved problem of Paul Erdős asks how many edges a unit distance graph on $n$ vertices can have. The best known lower bound is slightly above linear in $n$—far from the upper bound, proportional to $n^{4/3}$. The number of colors required to color unit distance graphs is also unknown (the Hadwiger–Nelson problem): some unit distance graphs require five colors, and every unit distance graph can be colored with seven colors. For every algebraic number there is a unit distance graph with two vertices that must be that distance apart. According to the Beckman–Quarles theorem, the only plane transformations that preserve all unit distance graphs are the isometries. It is possible to construct a unit distance graph efficiently, given its points. Finding all unit distances has applications in pattern matching, where it can be a first step in finding congruent copies of larger patterns. However, determining whether a given graph can be represented as a unit distance graph is NP-hard, and more specifically complete for the existential theory of the reals. Definition The Petersen graph as a unit distance graph[1] The Möbius–Kantor graph as a subgraph of a unit distance graph The unit distance graph for a set of points in the plane is the undirected graph having those points as its vertices, with an edge between two vertices whenever their Euclidean distance is exactly one. An abstract graph is said to be a unit distance graph if it is possible to find distinct locations in the plane for its vertices, so that its edges have unit length and so that all non-adjacent pairs of vertices have non-unit distances. When this is possible, the abstract graph is isomorphic to the unit distance graph of the chosen locations. Alternatively, some sources use a broader definition, allowing non-adjacent pairs of vertices to be at unit distance. The resulting graphs are the subgraphs of the unit distance graphs (as defined here).[2] Where the terminology may be ambiguous, the graphs in which non-edges must be a non-unit distance apart may be called strict unit distance graphs[3] or faithful unit distance graphs.[2] The subgraphs of unit distance graphs are equivalently the graphs that can be drawn in the plane using only one edge length.[4] For brevity, this article refers to these as "non-strict unit distance graphs". Unit distance graphs should not be confused with unit disk graphs, which connect pairs of points when their distance is less than or equal to one, and are frequently used to model wireless communication networks.[5] Examples The complete graph on two vertices is a unit distance graph, as is the complete graph on three vertices (the triangle graph), but not the complete graph on four vertices.[3] Generalizing the triangle graph, every cycle graph is a unit distance graph, realized by a regular polygon.[4] Two finite unit distance graphs, connected at a single shared vertex, yield another unit distance graph, as one can be rotated with respect to the other to avoid undesired additional unit distances.[6] By thus connecting graphs, every finite tree or cactus graph may be realized as a unit distance graph.[7] The hypercube graph $Q_{4}$ has 16 vertices and 32 unit distances The Hamming graph $H(3,3)$ has 27 vertices and 81 unit distances Any Cartesian product of unit distance graphs produces another unit distance graph; however, the same is not true for some other common graph products. For instance, the strong product of graphs, applied to any two non-empty graphs, produces complete subgraphs with four vertices, which are not unit distance graphs. The Cartesian products of path graphs form grid graphs of any dimension, the Cartesian products of the complete graph on two vertices are the hypercube graphs,[8] and the Cartesian products of triangle graphs are the Hamming graphs $H(d,3)$.[9] Other specific graphs that are unit distance graphs include the Petersen graph,[10] the Heawood graph,[11] the wheel graph $W_{7}$ (the only wheel graph that is a unit distance graph),[3] and the Moser spindle and Golomb graph (small 4-chromatic unit distance graphs).[12] All generalized Petersen graphs, such as the Möbius–Kantor graph depicted, are non-strict unit distance graphs.[13] Matchstick graphs are a special case of unit distance graphs, in which no edges cross. Every matchstick graph is a planar graph,[14] but some otherwise-planar unit distance graphs (such as the Moser spindle) have a crossing in every representation as a unit distance graph. Additionally, in the context of unit distance graphs, the term 'planar' should be used with care, as some authors use it to refer to the plane in which the unit distances are defined, rather than to a prohibition on crossings.[3] The penny graphs are an even more special case of unit distance and matchstick graphs, in which every non-adjacent pair of vertices are more than one unit apart.[14] Properties Number of edges Unsolved problem in mathematics: How many unit distances can be determined by a set of $n$ points? (more unsolved problems in mathematics) Paul Erdős (1946) posed the problem of estimating how many pairs of points in a set of $n$ points could be at unit distance from each other. In graph-theoretic terms, the question asks how dense a unit distance graph can be, and Erdős's publication on this question was one of the first works in extremal graph theory.[15] The hypercube graphs and Hamming graphs provide a lower bound on the number of unit distances, proportional to $n\log n.$ By considering points in a square grid with carefully chosen spacing, Erdős found an improved lower bound of the form $n^{1+c/\log \log n}$ for a constant $c$, and offered $500 for a proof of whether the number of unit distances can also be bounded above by a function of this form.[16] The best known upper bound for this problem is ${\sqrt[{3}]{\frac {29n^{4}}{4}}}\approx 1.936n^{4/3}.$ This bound can be viewed as counting incidences between points and unit circles, and is closely related to the crossing number inequality and to the Szemerédi–Trotter theorem on incidences between points and lines.[17] For small values of $n$ (up to 14, as of 2022), the exact maximum number of possible edges is known. For $n=2,3,4,\dots $ these numbers of edges are:[18] 1, 3, 5, 7, 9, 12, 14, 18, 20, 23, 27, 30, 33, ... (sequence A186705 in the OEIS) Forbidden subgraphs If a given graph $G$ is not a non-strict unit distance graph, neither is any supergraph $H$ of $G$. A similar idea works for strict unit distance graphs, but using the concept of an induced subgraph, a subgraph formed from all edges between the pairs of vertices in a given subset of vertices. If $G$ is not a strict unit distance graph, then neither is any other $H$ that has $G$ as an induced subgraph. Because of these relations between whether a subgraph or its supergraph is a unit distance graph, it is possible to describe unit distance graphs by their forbidden subgraphs. These are the minimal graphs that are not unit distance graphs of the given type. They can be used to determine whether a given graph $G$ is a unit distance graph, of either type. $G$ is a non-strict unit distance graph, if and only if $G$ is not a supergraph of a forbidden graph for the non-strict unit distance graphs. $G$ is a strict unit distance graph, if and only if $G$ is not an induced supergraph of a forbidden graph for the strict unit distance graphs.[8] For both the non-strict and strict unit distance graphs, the forbidden graphs include both the complete graph $K_{4}$ and the complete bipartite graph $K_{2,3}$. For $K_{2,3}$, wherever the vertices on the two-vertex side of this graph are placed, there are at most two positions at unit distance from them to place the other three vertices, so it is impossible to place all three vertices at distinct points.[8] These are the only two forbidden graphs for the non-strict unit distance graphs on up to five vertices; there are six forbidden graphs on up to seven vertices[6] and 74 on graphs up to nine vertices. Because gluing two unit distance graphs (or subgraphs thereof) at a vertex produce strict (respectively non-strict) unit distance graphs, every forbidden graph is a biconnected graph, one that cannot be formed by this gluing process.[19] The wheel graph $W_{7}$ can be realized as a strict unit distance graph with six of its vertices forming a unit regular hexagon and the seventh at the center of the hexagon. Removing one of the edges from the center vertex produces a subgraph that still has unit-length edges, but which is not a strict unit distance graph. The regular-hexagon placement of its vertices is the only one way (up to congruence) to place the vertices at distinct locations such that adjacent vertices are a unit distance apart, and this placement also puts the two endpoints of the missing edge at unit distance. Thus, it is a forbidden graph for the strict unit distance graphs,[20] but not one of the six forbidden graphs for the non-strict unit distance graphs. Other examples of graphs that are non-strict unit distance graphs but not strict unit distance graphs include the graph formed by removing an outer edge from $W_{7}$, and the six-vertex graph formed from a triangular prism by removing an edge from one of its triangles.[19] Algebraic numbers and rigidity Main article: Beckman–Quarles theorem For every algebraic number $\alpha $, it is possible to construct a unit distance graph $G$ in which some pair of vertices are at distance $\alpha $ in all unit distance representations of $G$.[21] This result implies a finite version of the Beckman–Quarles theorem: for any two points $p$ and $q$ at distance $\alpha $ from each other, there exists a finite rigid unit distance graph containing $p$ and $q$ such that any transformation of the plane that preserves the unit distances in this graph also preserves the distance between $p$ and $q$.[22] The full Beckman–Quarles theorem states that the only transformations of the Euclidean plane (or a higher-dimensional Euclidean space) that preserve unit distances are the isometries. Equivalently, for the infinite unit distance graph generated by all the points in the plane, all graph automorphisms preserve all of the distances in the plane, not just the unit distances.[23] If $\alpha $ is an algebraic number of modulus 1 that is not a root of unity, then the integer combinations of powers of $\alpha $ form a finitely generated subgroup of the additive group of complex numbers whose unit distance graph has infinite degree. For instance, $\alpha $ can be chosen as one of the two complex roots of the polynomial $z^{4}-z^{3}-z^{2}-z+1$, producing an infinite-degree unit distance graph with four generators.[24] Coloring Main article: Hadwiger–Nelson problem Unsolved problem in mathematics: What is the largest possible chromatic number of a unit distance graph? (more unsolved problems in mathematics) The Hadwiger–Nelson problem concerns the chromatic number of unit distance graphs, and more specifically of the infinite unit distance graph formed from all points of the Euclidean plane. By the de Bruijn–Erdős theorem, which assumes the axiom of choice, this is equivalent to asking for the largest chromatic number of a finite unit distance graph. There exist unit distance graphs requiring five colors in any proper coloring,[25] and all unit distance graphs can be colored with at most seven colors.[26] Answering another question of Paul Erdős, it is possible for triangle-free unit distance graphs to require four colors.[27] Enumeration The number of strict unit distance graphs on $n\geq 4$ labeled vertices is at most[2] ${\binom {n(n-1)}{2n}}=O\left(2^{{\bigl (}4+o(1){\bigr )}n\log _{2}n}\right),$ as expressed using big O notation and little o notation. Generalization to higher dimensions The definition of a unit distance graph may naturally be generalized to any higher-dimensional Euclidean space. In three dimensions, unit distance graphs of $n$ points have at most $n^{3/2}\beta (n)$ edges, where $\beta $ is a very slowly growing function related to the inverse Ackermann function.[28] This result leads to a similar bound on the number of edges of three-dimensional relative neighborhood graphs.[29] In four or more dimensions, any complete bipartite graph is a unit distance graph, realized by placing the points on two perpendicular circles with a common center, so unit distance graphs can be dense graphs.[7] The enumeration formulas for unit distance graphs generalize to higher dimensions, and shows that in dimensions four or more the number of strict unit distance graphs is much larger than the number of subgraphs of unit distance graphs.[2] Any finite graph may be embedded as a unit distance graph in a sufficiently high dimension. Some graphs may need very different dimensions for embeddings as non-strict unit distance graphs and as strict unit distance graphs. For instance the $2n$-vertex crown graph may be embedded in four dimensions as a non-strict unit distance graph (that is, so that all its edges have unit length). However, it requires at least $n-2$ dimensions to be embedded as a strict unit distance graph, so that its edges are the only unit-distance pairs.[30] The dimension needed to realize any given graph as a strict unit graph is at most twice its maximum degree.[31] Computational complexity Constructing a unit distance graph from its points is an important step for other algorithms for finding congruent copies of some pattern in a larger point set. These algorithms use this construction to search for candidate positions where one of the distances in the pattern is present, and then use other methods to test the rest of the pattern for each candidate.[32] A method of Matoušek (1993) can be applied to this problem,[32] yielding an algorithm for finding a planar point set's unit distance graph in time $n^{4/3}2^{O(\log ^{}n)}$ where $\log ^{}$ is the slowly growing iterated logarithm function.[33] It is NP-hard—and more specifically, complete for the existential theory of the reals—to test whether a given graph is a (strict or non-strict) unit distance graph in the plane.[34] It is also NP-complete to determine whether a planar unit distance graph has a Hamiltonian cycle, even when the graph's vertices all have known integer coordinates.[35] References Notes 1. Griffiths (2019). 2. Alon & Kupavskii (2014). 3. Gervacio, Lim & Maehara (2008). 4. Carmi et al. (2008). 5. Huson & Sen (1995). 6. Chilakamarri & Mahoney (1995). 7. Erdős, Harary & Tutte (1965). 8. Horvat & Pisanski (2010). 9. Brouwer & Haemers (2012). 10. Erdős, Harary & Tutte (1965); Griffiths (2019) 11. Gerbracht (2009). 12. Soifer (2008), pp. 14–15, 19. 13. Žitnik, Horvat & Pisanski (2012). 14. Lavollée & Swanepoel (2022). 15. Szemerédi (2016). 16. Erdős (1990). 17. Spencer, Szemerédi & Trotter (1984); Clarkson et al. (1990); Pach & Tardos (2005); Ágoston & Pálvölgyi (2022) 18. Ágoston & Pálvölgyi (2022). 19. Globus & Parshall (2020). 20. Soifer (2008), p. 94. 21. Maehara (1991, 1992). 22. Tyszka (2000). 23. Beckman & Quarles (1953). 24. Radchenko (2021). 25. Langin (2018); de Grey (2018) 26. Soifer (2008), p. 17. 27. Wormald (1979); Chilakamarri (1995); O'Donnell (1995). 28. Clarkson et al. (1990). 29. Jaromczyk & Toussaint (1992). 30. Erdős & Simonovits (1980). 31. Maehara & Rödl (1990). 32. Braß (2002). 33. Matoušek (1993); see also Chan & Zheng (2022) for a closely related algorithm for listing point–line incidences in time $O(n^{4/3})$. 34. Schaefer (2013). 35. Itai, Papadimitriou & Szwarcfiter (1982). Sources • Ágoston, Péter; Pálvölgyi, Dömötör (April 2022), "An improved constant factor for the unit distance problem", Studia Scientiarum Mathematicarum Hungarica, Akademiai Kiado Zrt., 59 (1): 40–57, arXiv:2006.06285, doi:10.1556/012.2022.01517, S2CID 218479287 • Alon, Noga; Kupavskii, Andrey (2014), "Two notions of unit distance graphs" (PDF), Journal of Combinatorial Theory, Series A, 125: 1–17, doi:10.1016/j.jcta.2014.02.006, MR 3207464, S2CID 12043969 • Beckman, F. S.; Quarles, D. A., Jr. (1953), "On isometries of Euclidean spaces", Proceedings of the American Mathematical Society, 4 (5): 810–815, doi:10.2307/2032415, JSTOR 2032415, MR 0058193{{citation}}: CS1 maint: multiple names: authors list (link) • Braß, Peter (2002), "Combinatorial geometry problems in pattern recognition", Discrete & Computational Geometry, 28 (4): 495–510, doi:10.1007/s00454-002-2884-3, MR 1949897 • Brouwer, Andries E.; Haemers, Willem H. (2012), Spectra of Graphs, Universitext, New York: Springer, p. 178, doi:10.1007/978-1-4614-1939-6, ISBN 978-1-4614-1938-9, MR 2882891 • Carmi, Paz; Dujmović, Vida; Morin, Pat; Wood, David R. (2008), "Distinct distances in graph drawings", Electronic Journal of Combinatorics, 15 (1): Research Paper 107, arXiv:0804.3690, doi:10.37236/831, MR 2438579, S2CID 2955082 • Chan, Timothy M.; Zheng, Da Wei (2022), "Hopcroft's problem, log-star shaving, 2d fractional cascading, and decision trees", in Naor, Joseph (Seffi); Buchbinder, Niv (eds.), Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, Society for Industrial and Applied Mathematics, pp. 190–210, arXiv:2111.03744, doi:10.1137/1.9781611977073.10, S2CID 243847672 • Chilakamarri, Kiran B. (1995), "A 4-chromatic unit-distance graph with no triangles", Geombinatorics, 4 (3): 64–76, MR 1313386 • Chilakamarri, Kiran B.; Mahoney, Carolyn R. (1995), "Maximal and minimal forbidden unit-distance graphs in the plane", Bulletin of the Institute of Combinatorics and Its Applications, 13: 35–43, MR 1314500, as cited by Globus & Parshall (2020) • Clarkson, Kenneth L.; Edelsbrunner, Herbert; Guibas, Leonidas J.; Sharir, Micha; Welzl, Emo (1990), "Combinatorial complexity bounds for arrangements of curves and spheres", Discrete & Computational Geometry, 5 (2): 99–160, doi:10.1007/BF02187783, MR 1032370, S2CID 28143698 • de Grey, Aubrey D. N. J. (2018), "The chromatic number of the plane is at least 5", Geombinatorics, 28: 5–18, arXiv:1804.02385, MR 3820926 • Erdős, Paul (1946), "On sets of distances of $n$ points", American Mathematical Monthly, 53 (5): 248–250, doi:10.2307/2305092, JSTOR 2305092 • Erdős, Paul; Harary, Frank; Tutte, William T. (1965), "On the dimension of a graph" (PDF), Mathematika, 12 (2): 118–122, doi:10.1112/S0025579300005222, hdl:2027.42/152495, MR 0188096 • Erdős, Paul; Simonovits, Miklós (1980), "On the chromatic number of geometric graphs", Ars Combinatoria, 9: 229–246, as cited by Soifer (2008, p. 97) • Erdős, Paul (1990), "Some of my favourite unsolved problems", in Baker, A.; Bollobás, B.; Hajnal, A. (eds.), A tribute to Paul Erdős, Cambridge University Press, pp. 467–478, ISBN 0-521-38101-0, MR 1117038; see in particular p. 475 • Gerbracht, Eberhard H.-A. (2009), Eleven unit distance embeddings of the Heawood graph, arXiv:0912.5395, Bibcode:2009arXiv0912.5395G • Gervacio, Severino V.; Lim, Yvette F.; Maehara, Hiroshi (2008), "Planar unit-distance graphs having planar unit-distance complement", Discrete Mathematics, 308 (10): 1973–1984, doi:10.1016/j.disc.2007.04.050 • Globus, Aidan; Parshall, Hans (2020), "Small unit-distance graphs in the plane", Bulletin of the Institute of Combinatorics and Its Applications, 90: 107–138, arXiv:1905.07829, MR 4156400 • Griffiths, Martin (June 2019), "103.27 A property of a particular unit-distance graph", The Mathematical Gazette, 103 (557): 353–356, doi:10.1017/mag.2019.74, S2CID 233361952 • Horvat, Boris; Pisanski, Tomaž (2010), "Products of unit distance graphs", Discrete Mathematics, 310 (12): 1783–1792, doi:10.1016/j.disc.2009.11.035, MR 2610282 • Huson, Mark L.; Sen, Arunabha (1995), "Broadcast scheduling algorithms for radio networks", Military Communications Conference, IEEE MILCOM '95, vol. 2, pp. 647–651, doi:10.1109/MILCOM.1995.483546, ISBN 0-7803-2489-7, S2CID 62039740 • Itai, Alon; Papadimitriou, Christos H.; Szwarcfiter, Jayme Luiz (1982), "Hamilton paths in grid graphs", SIAM Journal on Computing, 11 (4): 676–686, CiteSeerX 10.1.1.383.1078, doi:10.1137/0211056, MR 0677661 • Jaromczyk, Jerzy W.; Toussaint, Godfried T. (1992), "Relative neighborhood graphs and their relatives", Proceedings of the IEEE, 80 (9): 1502–1517, doi:10.1109/5.163414 • Langin, Katie (April 18, 2018), "Amateur mathematician cracks decades-old math problem", Science • Lavollée, Jérémy; Swanepoel, Konrad J. (2022), "Bounding the number of edges of matchstick graphs", SIAM Journal on Discrete Mathematics, 36 (1): 777–785, arXiv:2108.07522, doi:10.1137/21M1441134, MR 4399020, S2CID 237142624 • Maehara, Hiroshi (1991), "Distances in a rigid unit-distance graph in the plane", Discrete Applied Mathematics, 31 (2): 193–200, doi:10.1016/0166-218X(91)90070-D • Maehara, Hiroshi (1992), "Extending a flexible unit-bar framework to a rigid one", Discrete Mathematics, 108 (1–3): 167–174, doi:10.1016/0012-365X(92)90671-2, MR 1189840 • Maehara, Hiroshi; Rödl, Vojtech (1990), "On the dimension to represent a graph by a unit distance graph", Graphs and Combinatorics, 6 (4): 365–367, doi:10.1007/BF01787703, S2CID 31148911 • Matoušek, Jiří (1993), "Range searching with efficient hierarchical cuttings", Discrete & Computational Geometry, 10 (2): 157–182, doi:10.1007/BF02573972, MR 1220545 • O'Donnell, Paul (1995), "A 40 vertex 4-chromatic triangle-free unit distance graph", Geombinatorics, 5 (1): 31–34, MR 1337155 • Pach, János; Tardos, Gábor (2005), "Forbidden patterns and unit distances", in Mitchell, Joseph S. B.; Rote, Günter (eds.), Proceedings of the 21st ACM Symposium on Computational Geometry, Pisa, Italy, June 6-8, 2005, Association for Computing Machinery, pp. 1–9, doi:10.1145/1064092.1064096, MR 2460341, S2CID 18752227 • Radchenko, Danylo (2021), "Unit distance graphs and algebraic integers", Discrete & Computational Geometry, 66 (1): 269–272, doi:10.1007/s00454-019-00152-4, hdl:21.11116/0000-0006-9CFD-E, MR 4270642, S2CID 119682489 • Schaefer, Marcus (2013), "Realizability of graphs and linkages", in Pach, János (ed.), Thirty Essays on Geometric Graph Theory, Springer, pp. 461–482, CiteSeerX 10.1.1.220.9651, doi:10.1007/978-1-4614-0110-0_24, ISBN 978-1-4614-0109-4 • Soifer, Alexander (2008), The Mathematical Coloring Book, Springer-Verlag, ISBN 978-0-387-74640-1 • Spencer, Joel; Szemerédi, Endre; Trotter, William T. (1984), "Unit distances in the Euclidean plane", in Bollobás, Béla (ed.), Graph Theory and Combinatorics, London: Academic Press, pp. 293–308, ISBN 978-0-12-111760-3, MR 0777185 • Szemerédi, Endre (2016), "Erdős's unit distance problem", in Nash, John Forbes, Jr.; Rassias, Michael Th. (eds.), Open Problems in Mathematics, Cham, Switzerland: Springer, pp. 459–477, doi:10.1007/978-3-319-32162-2_15, MR 3526946{{citation}}: CS1 maint: multiple names: editors list (link) • Tyszka, Apoloniusz (2000), "Discrete versions of the Beckman-Quarles theorem", Aequationes Mathematicae, 59 (1–2): 124–133, arXiv:math/9904047, doi:10.1007/PL00000119, MR 1741475, S2CID 14803182 • Wormald, Nicholas (1979), "A 4-chromatic graph with a special plane drawing", Journal of the Australian Mathematical Society, Series A, 28 (1): 1–8, doi:10.1017/S1446788700014865, MR 0541161, S2CID 124067465 • Žitnik, Arjana; Horvat, Boris; Pisanski, Tomaž (2012), "All generalized Petersen graphs are unit-distance graphs", Journal of the Korean Mathematical Society, 49 (3): 475–491, doi:10.4134/JKMS.2012.49.3.475, MR 2953031 External links • Venkatasubramanian, Suresh, "Problem 39: Distances among Point Sets in R2 and R3", The Open Problems Project • Weisstein, Eric W., "Unit-Distance Graph", MathWorld	Wikipedia
Unit doublet In mathematics, the unit doublet is the derivative of the Dirac delta function. It can be used to differentiate signals in electrical engineering:[1] If u1 is the unit doublet, then $(xu_{1})(t)={\frac {dx(t)}{dt}}$ where $$ is the convolution operator.[2] The function is zero for all values except zero, where its behaviour is interesting. Its integral over any interval enclosing zero is zero. However, the integral of its absolute value over any region enclosing zero goes to infinity. The function can be thought of as the limiting case of two rectangles, one in the second quadrant, and the other in the fourth. The length of each rectangle is k, whereas their breadth is 1/k2, where k tends to zero. References 1. "Signals and Systems Lecture #4" (PDF). Mit.edu. 16 September 2003. Archived from the original (PDF) on 19 February 2009. Retrieved 2 September 2009.	Wikipedia
Unit function In number theory, the unit function is a completely multiplicative function on the positive integers defined as: $\varepsilon (n)={\begin{cases}1,&{\mbox{if }}n=1\\0,&{\mbox{if }}n\neq 1\end{cases}}$ It is called the unit function because it is the identity element for Dirichlet convolution.[1] It may be described as the "indicator function of 1" within the set of positive integers. It is also written as u(n) (not to be confused with μ(n), which generally denotes the Möbius function). See also • Möbius inversion formula • Heaviside step function • Kronecker delta References 1. Estrada, Ricardo (1995), "Dirichlet convolution inverses and solution of integral equations", Journal of Integral Equations and Applications, 7 (2): 159–166, doi:10.1216/jiea/1181075867, MR 1355233.	Wikipedia
Unit (ring theory) In algebra, a unit or invertible element[lower-alpha 1] of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that $vu=uv=1,$ Not to be confused with Unit ring. where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u.[1][2] The set of units of R forms a group R× under multiplication, called the group of units or unit group of R.[lower-alpha 2] Other notations for the unit group are R∗, U(R), and E(R) (from the German term Einheit). Less commonly, the term unit is sometimes used to refer to the element 1 of the ring, in expressions like ring with a unit or unit ring, and also unit matrix. Because of this ambiguity, 1 is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng. Examples The multiplicative identity 1 and its additive inverse −1 are always units. More generally, any root of unity in a ring R is a unit: if rn = 1, then rn − 1 is a multiplicative inverse of r. In a nonzero ring, the element 0 is not a unit, so R× is not closed under addition. A nonzero ring R in which every nonzero element is a unit (that is, R× = R −{0}) is called a division ring (or a skew-field). A commutative division ring is called a field. For example, the unit group of the field of real numbers R is R − {0}. Integer ring In the ring of integers Z, the only units are 1 and −1. In the ring Z/nZ of integers modulo n, the units are the congruence classes (mod n) represented by integers coprime to n. They constitute the multiplicative group of integers modulo n. Ring of integers of a number field In the ring Z[√3] obtained by adjoining the quadratic integer √3 to Z, one has (2 + √3)(2 − √3) = 1, so 2 + √3 is a unit, and so are its powers, so Z[√3] has infinitely many units. More generally, for the ring of integers R in a number field F, Dirichlet's unit theorem states that R× is isomorphic to the group $\mathbf {Z} ^{n}\times \mu _{R}$ where $\mu _{R}$ is the (finite, cyclic) group of roots of unity in R and n, the rank of the unit group, is $n=r_{1}+r_{2}-1,$ where $r_{1},r_{2}$ are the number of real embeddings and the number of pairs of complex embeddings of F, respectively. This recovers the Z[√3] example: The unit group of (the ring of integers of) a real quadratic field is infinite of rank 1, since $r_{1}=2,r_{2}=0$. Polynomials and power series For a commutative ring R, the units of the polynomial ring R[x] are the polynomials $p(x)=a_{0}+a_{1}x+\dots +a_{n}x^{n}$ such that $a_{0}$ is a unit in R and the remaining coefficients $a_{1},\dots ,a_{n}$ are nilpotent, i.e., satisfy $a_{i}^{N}=0$ for some N.[4] In particular, if R is a domain (or more generally reduced), then the units of R[x] are the units of R. The units of the power series ring $R[[x]]$ are the power series $p(x)=\sum _{i=0}^{\infty }a_{i}x^{i}$ such that $a_{0}$ is a unit in R.[5] Matrix rings The unit group of the ring Mn(R) of n × n matrices over a ring R is the group GLn(R) of invertible matrices. For a commutative ring R, an element A of Mn(R) is invertible if and only if the determinant of A is invertible in R. In that case, A−1 can be given explicitly in terms of the adjugate matrix. In general For elements x and y in a ring R, if $1-xy$ is invertible, then $1-yx$ is invertible with inverse $1+y(1-xy)^{-1}x$;[6] this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series: $(1-yx)^{-1}=\sum _{n\geq 0}(yx)^{n}=1+y\left(\sum _{n\geq 0}(xy)^{n}\right)x=1+y(1-xy)^{-1}x.$ See Hua's identity for similar results. Group of units A commutative ring is a local ring if R − R× is a maximal ideal. As it turns out, if R − R× is an ideal, then it is necessarily a maximal ideal and R is local since a maximal ideal is disjoint from R×. If R is a finite field, then R× is a cyclic group of order $\|R\|-1$. Every ring homomorphism f : R → S induces a group homomorphism R× → S×, since f maps units to units. In fact, the formation of the unit group defines a functor from the category of rings to the category of groups. This functor has a left adjoint which is the integral group ring construction.[7] The group scheme $\operatorname {GL} _{1}$ is isomorphic to the multiplicative group scheme $\mathbb {G} _{m}$ over any base, so for any commutative ring R, the groups $\operatorname {GL} _{1}(R)$ and $\mathbb {G} _{m}(R)$ are canonically isomorphic to $U(R)$. Note that the functor $\mathbb {G} _{m}$ (that is, $R\mapsto U(R)$) is representable in the sense: $\mathbb {G} _{m}(R)\simeq \operatorname {Hom} (\mathbb {Z} [t,t^{-1}],R)$ for commutative rings R (this for instance follows from the aforementioned adjoint relation with the group ring construction). Explicitly this means that there is a natural bijection between the set of the ring homomorphisms $\mathbb {Z} [t,t^{-1}]\to R$ and the set of unit elements of R (in contrast, $\mathbb {Z} [t]$ represents the additive group $\mathbb {G} _{a}$, the forgetful functor from the category of commutative rings to the category of abelian groups). Associatedness Suppose that R is commutative. Elements r and s of R are called associate if there exists a unit u in R such that r = us; then write r ∼ s. In any ring, pairs of additive inverse elements[lower-alpha 3] x and −x are associate. For example, 6 and −6 are associate in Z. In general, ~ is an equivalence relation on R. Associatedness can also be described in terms of the action of R× on R via multiplication: Two elements of R are associate if they are in the same R×-orbit. In an integral domain, the set of associates of a given nonzero element has the same cardinality as R×. The equivalence relation ~ can be viewed as any one of Green's semigroup relations specialized to the multiplicative semigroup of a commutative ring R. See also • S-units • Localization of a ring and a module Notes 1. The use of "invertible element" without specifying the operation is not ambiguous in the case of rings, since all elements of a ring are invertible for addition. 2. The notation R×, introduced by André Weil, is commonly used in number theory, where unit groups arise frequently.[3] The symbol × is a reminder that the group operation is multiplication. Also, a superscript × is not frequently used in other contexts, whereas a superscript * often denotes dual. 3. x and −x are not necessarily distinct. For example, in the ring of integers modulo 6, one has 3 = −3 even though 1 ≠ −1. Citations 1. Dummit & Foote 2004. 2. Lang 2002. 3. Weil 1974. 4. Watkins (2007, Theorem 11.1) 5. Watkins (2007, Theorem 12.1) 6. Jacobson 2009, § 2.2. Exercise 4. 7. Exercise 10 in § 2.2. of Cohn, Paul M. (2003). Further algebra and applications (Revised ed. of Algebra, 2nd ed.). London: Springer-Verlag. ISBN 1-85233-667-6. Zbl 1006.00001. Sources • Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. • Jacobson, Nathan (2009). Basic Algebra 1 (2nd ed.). Dover. ISBN 978-0-486-47189-1. • Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X. • Watkins, John J. (2007), Topics in commutative ring theory, Princeton University Press, ISBN 978-0-691-12748-4, MR 2330411 • Weil, André (1974). Basic number theory. Grundlehren der mathematischen Wissenschaften. Vol. 144 (3rd ed.). Springer-Verlag. ISBN 978-3-540-58655-5.	Wikipedia
Unit hyperbola In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation $x^{2}-y^{2}=1.$ In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial length $r={\sqrt {x^{2}-y^{2}}}.$ Whereas the unit circle surrounds its center, the unit hyperbola requires the conjugate hyperbola $y^{2}-x^{2}=1$ to complement it in the plane. This pair of hyperbolas share the asymptotes y = x and y = −x. When the conjugate of the unit hyperbola is in use, the alternative radial length is $r={\sqrt {y^{2}-x^{2}}}.$ The unit hyperbola is a special case of the rectangular hyperbola, with a particular orientation, location, and scale. As such, its eccentricity equals ${\sqrt {2}}.$[1] The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry. A prominent instance is the depiction of spacetime as a pseudo-Euclidean space. There the asymptotes of the unit hyperbola form a light cone. Further, the attention to areas of hyperbolic sectors by Gregoire de Saint-Vincent led to the logarithm function and the modern parametrization of the hyperbola by sector areas. When the notions of conjugate hyperbolas and hyperbolic angles are understood, then the classical complex numbers, which are built around the unit circle, can be replaced with numbers built around the unit hyperbola. Asymptotes Main article: Asymptote Generally asymptotic lines to a curve are said to converge toward the curve. In algebraic geometry and the theory of algebraic curves there is a different approach to asymptotes. The curve is first interpreted in the projective plane using homogeneous coordinates. Then the asymptotes are lines that are tangent to the projective curve at a point at infinity, thus circumventing any need for a distance concept and convergence. In a common framework (x, y, z) are homogeneous coordinates with the line at infinity determined by the equation z = 0. For instance, C. G. Gibson wrote:[2] For the standard rectangular hyperbola $f=x^{2}-y^{2}-1$ in ℝ2, the corresponding projective curve is $F=x^{2}-y^{2}-z^{2},$ which meets z = 0 at the points P = (1 : 1 : 0) and Q = (1 : −1 : 0). Both P and Q are simple on F, with tangents x + y = 0, x − y = 0; thus we recover the familiar 'asymptotes' of elementary geometry. Minkowski diagram The Minkowski diagram is drawn in a spacetime plane where the spatial aspect has been restricted to a single dimension. The units of distance and time on such a plane are • units of 30 centimetres length and nanoseconds, or • astronomical units and intervals of 8 minutes and 20 seconds, or • light years and years. Each of these scales of coordinates results in photon connections of events along diagonal lines of slope plus or minus one. Five elements constitute the diagram Hermann Minkowski used to describe the relativity transformations: the unit hyperbola, its conjugate hyperbola, the axes of the hyperbola, a diameter of the unit hyperbola, and the conjugate diameter. The plane with the axes refers to a resting frame of reference. The diameter of the unit hyperbola represents a frame of reference in motion with rapidity a where tanh a = y/x and (x,y) is the endpoint of the diameter on the unit hyperbola. The conjugate diameter represents the spatial hyperplane of simultaneity corresponding to rapidity a. In this context the unit hyperbola is a calibration hyperbola[3][4] Commonly in relativity study the hyperbola with vertical axis is taken as primary: The arrow of time goes from the bottom to top of the figure — a convention adopted by Richard Feynman in his famous diagrams. Space is represented by planes perpendicular to the time axis. The here and now is a singularity in the middle.[5] The vertical time axis convention stems from Minkowski in 1908, and is also illustrated on page 48 of Eddington's The Nature of the Physical World (1928). Parametrization Main article: Hyperbolic angle A direct way to parameterizing the unit hyperbola starts with the hyperbola xy = 1 parameterized with the exponential function: $(e^{t},\ e^{-t}).$ This hyperbola is transformed into the unit hyperbola by a linear mapping having the matrix $A={\tfrac {1}{2}}{\begin{pmatrix}1&1\\1&-1\end{pmatrix}}\ :$ :} $(e^{t},\ e^{-t})\ A=({\frac {e^{t}+e^{-t}}{2}},\ {\frac {e^{t}-e^{-t}}{2}})=(\cosh t,\ \sinh t).$ This parameter t is the hyperbolic angle, which is the argument of the hyperbolic functions. One finds an early expression of the parametrized unit hyperbola in Elements of Dynamic (1878) by W. K. Clifford. He describes quasi-harmonic motion in a hyperbola as follows: The motion $\rho =\alpha \cosh(nt+\epsilon )+\beta \sinh(nt+\epsilon )$ has some curious analogies to elliptic harmonic motion. ... The acceleration ${\ddot {\rho }}=n^{2}\rho \ ;$ ;} thus it is always proportional to the distance from the centre, as in elliptic harmonic motion, but directed away from the centre.[6] As a particular conic, the hyperbola can be parametrized by the process of addition of points on a conic. The following description was given by Russian analysts: Fix a point E on the conic. Consider the points at which the straight line drawn through E parallel to AB intersects the conic a second time to be the sum of the points A and B. For the hyperbola $x^{2}-y^{2}=1$ with the fixed point E = (1,0) the sum of the points $(x_{1},\ y_{1})$ and $(x_{2},\ y_{2})$ is the point $(x_{1}x_{2}+y_{1}y_{2},\ y_{1}x_{2}+y_{2}x_{1})$ under the parametrization $x=\cosh \ t$ and $y=\sinh \ t$ this addition corresponds to the addition of the parameter t.[7] Complex plane algebra Main article: Split-complex number Whereas the unit circle is associated with complex numbers, the unit hyperbola is key to the split-complex number plane consisting of z = x + yj, where j 2 = +1. Then jz = y + xj, so the action of j on the plane is to swap the coordinates. In particular, this action swaps the unit hyperbola with its conjugate and swaps pairs of conjugate diameters of the hyperbolas. In terms of the hyperbolic angle parameter a, the unit hyperbola consists of points $\pm (\cosh a+j\sinh a)$, where j = (0,1). The right branch of the unit hyperbola corresponds to the positive coefficient. In fact, this branch is the image of the exponential map acting on the j-axis. Thus this branch is the curve $f(a)=\exp(aj).$ The slope of the curve at a is given by the derivative $f^{\prime }(a)=\sinh a+j\cosh a=jf(a).$ For any a, $f^{\prime }(a$) is hyperbolic-orthogonal to $f(a)$. This relation is analogous to the perpendicularity of exp(a i) and i exp(a i) when i2 = − 1. Since $\exp(aj)\exp(bj)=\exp((a+b)j)$, the branch is a group under multiplication. Unlike the circle group, this unit hyperbola group is not compact. Similar to the ordinary complex plane, a point not on the diagonals has a polar decomposition using the parametrization of the unit hyperbola and the alternative radial length. References 1. Eric Weisstein Rectangular hyperbola from Wolfram Mathworld 2. C.G. Gibson (1998) Elementary Geometry of Algebraic Curves, p 159, Cambridge University Press ISBN 0-521-64140-3 3. Anthony French (1968) Special Relativity, page 83, W. W. Norton & Company 4. W.G.V. Rosser (1964) Introduction to the Theory of Relativity, figure 6.4, page 256, London: Butterworths 5. A.P. French (1989) "Learning from the past; Looking to the future", acceptance speech for 1989 Oersted Medal, American Journal of Physics 57(7):587–92 6. William Kingdon Clifford (1878) Elements of Dynamic, pages 89 & 90, London: MacMillan & Co; on-line presentation by Cornell University Historical Mathematical Monographs 7. Viktor Prasolov & Yuri Solovyev (1997) Elliptic Functions and Elliptic Integrals, page one, Translations of Mathematical Monographs volume 170, American Mathematical Society • F. Reese Harvey (1990) Spinors and calibrations, Figure 4.33, page 70, Academic Press, ISBN 0-12-329650-1 .	Wikipedia
Unit cube A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long.[1][2] The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units.[3] Unit hypercube The term unit cube or unit hypercube is also used for hypercubes, or "cubes" in n-dimensional spaces, for values of n other than 3 and edge length 1.[1][2] Sometimes the term "unit cube" refers in specific to the set [0, 1]n of all n-tuples of numbers in the interval [0, 1].[1] The length of the longest diagonal of a unit hypercube of n dimensions is ${\sqrt {n}}$, the square root of n and the (Euclidean) length of the vector (1,1,1,....1,1) in n-dimensional space.[2] See also • Doubling the cube • K-cell • Robbins constant, the average distance between two random points in a unit cube • Tychonoff cube, an infinite-dimensional analogue of the unit cube • Unit square • Unit sphere References 1. Ball, Keith (2010), "High-dimensional geometry and its probabilistic analogues", in Gowers, Timothy (ed.), The Princeton Companion to Mathematics, Princeton University Press, pp. 670–680, ISBN 9781400830398. See in particular p. 671. 2. Gardner, Martin (2001), "Chapter 13: Hypercubes", The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems : Number Theory, Algebra, Geometry, Probability, Topology, Game Theory, Infinity, and Other Topics of Recreational Mathematics, W. W. Norton & Company, pp. 162–174, ISBN 9780393020236. 3. Geometry: Reteaching Masters, Holt Rinehart & Winston, 2001, p. 74, ISBN 9780030543289. External links • Weisstein, Eric W. "Unit cube". MathWorld.	Wikipedia

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