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In mathematics, a translation plane is a particular kind of projective plane, as considered as a combinatorial object.^[1]

In a projective plane, $\scriptstyle p$ represents a point, and $\scriptstyle L$ represents a line. A central collineation with center $\scriptstyle p$ and axis $\scriptstyle L$ is a collineation fixing every point on $\scriptstyle L$ and every line through $\scriptstyle p$ . It is called an "elation" if $\scriptstyle p$ is on $\scriptstyle L$ , otherwise it is called a "homology". The central collineations with centre $\scriptstyle p$ and axis $\scriptstyle L$ form a group.^[2]

A projective plane $\scriptstyle \Pi$ is called a translation plane if there exists a line $\scriptstyle L$ such that the group of elations with axis $\scriptstyle L$ is transitive on the affine plane Π_l (the affine derivative of Π).

Relationship to spreads

Translation planes are related to spreads in finite projective spaces by the André/Bruck-Bose construction.^[3] A spread of $\scriptstyle PG(3,q)$ is a set of q² + 1 lines, with no two intersecting. Equivalently, it is a partition of the points of $\scriptstyle PG(3,q)$ into lines.

Given a spread $\scriptstyle S$ of $\scriptstyle PG(3,q)$ , the André/Bruck-Bose construction¹ produces a translation plane $\scriptstyle \pi (S)$ of order q² as follows: Embed $\scriptstyle PG(3,q)$ as a hyperplane of $\scriptstyle PG(4,q)$ . Define an incidence structure $\scriptstyle A(S)$ with "points," the points of $\scriptstyle PG(4,q)$ not on $\scriptstyle PG(3,q)$ and "lines" the planes of $\scriptstyle PG(4,q)$ meeting $\scriptstyle PG(3,q)$ in a line of $\scriptstyle S$ . Then $\scriptstyle A(S)$ is a translation affine plane of order q². Let $\scriptstyle \pi (S)$ be the projective completion of $\scriptstyle A(S)$ .^[4]^[5]

References

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External links

↑ Projective Planes On projective planes
↑ Geometry Translation Plane Retrieved on June 13, 2007
↑ Template:Cite web
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↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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[1] Projective Planes On projective planes

[2] Geometry Translation Plane Retrieved on June 13, 2007

[3] Template:Cite web

[4] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[5] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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[2]

[3]

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-In [[mathematics]], the '''Bernoulli scheme''' or '''Bernoulli shift''' is a generalization of the [[Bernoulli process]] to more than two possible outcomes.<ref>P. Shields, ''The theory of Bernoulli shifts'' , Univ. Chicago Press  (1973)</ref><ref>Michael S. Keane, "Ergodic theory and subshifts of finite type", (1991), appearing as Chapter 2 in ''Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces'', Tim Bedford, Michael Keane and Caroline Series, Eds. Oxford University Press, Oxford (1991). ISBN 0-19-853390-X</ref> Bernoulli schemes are important in the study of [[dynamical system]]s, as most such systems (such as [[Axiom A system]]s) exhibit a [[repellor]] that is the product of the [[Cantor set]] and a [[smooth manifold]], and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift.<ref>Pierre Gaspard, ''Chaos, scattering and statistical mechanics''(1998), Cambridge University press</ref> This is essentially the [[Markov partition]]. The term ''shift'' is in reference to the [[shift operator]], which may be used to study Bernoulli schemes.  The [[Ornstein isomorphism theorem]]<ref>{{springer|author=D.S. Ornstein|title=Ornstein isomorphism theorem|id=Ornstein_isomorphism_theorem&oldid=17997}}</ref> shows that Bernoulli shifts are isomorphic when their [[Kolmogorov entropy|entropy]] is equal. Finite [[stationary stochastic process]]es are isomorphic to the Bernoulli shift; in this sense, Bernoulli shifts are [[universal property|universal]].
+In [[mathematics]], a '''translation plane''' is a particular kind of [[projective plane]], as considered as a combinatorial object.<ref>Projective Planes [http://www.maths.qmul.ac.uk/~pjc/pps/pps2.pdf On projective planes]</ref>
-==Definition==
+In a projective plane, <math>\scriptstyle p</math> represents a point, and <math>\scriptstyle L</math> represents a line. A central [[collineation]] with center <math>\scriptstyle p</math> and axis <math>\scriptstyle L</math> is a collineation fixing every point on <math>\scriptstyle L</math> and every line through <math>\scriptstyle p</math>. It is called an "elation" if <math>\scriptstyle p</math> is on <math>\scriptstyle L</math>, otherwise it is called a "homology". The central collineations with centre <math>\scriptstyle p</math> and axis <math>\scriptstyle L</math> form a group.<ref>Geometry [http://www.math.uni-kiel.de/geometrie/klein/math/geometry/translation.html Translation Plane] Retrieved on June 13, 2007</ref>
-A Bernoulli scheme is a [[discrete-time]] [[stochastic process]] where each [[statistical independence|independent]] [[random variable]] may take on one of ''N'' distinct possible values, with the outcome ''i'' occurring with probability <math>p_i</math>, with ''i''&nbsp;=&nbsp;1,&nbsp;...,&nbsp;''N'', and
-:<math>\sum_{i=1}^N p_i = 1. </math>
+A projective plane <math>\scriptstyle \Pi</math> is called a translation plane if there exists a line <math>\scriptstyle L</math> such that the group of elations with axis <math>\scriptstyle L</math> is transitive on the affine plane Π<sub>l</sub> (the [[Affine geometry|affine]] derivative of Π).
-The [[sample space]] is usually denoted as
+== Relationship to spreads ==
+Translation planes are related to spreads in finite projective spaces by the André/Bruck-Bose construction.<ref>{{cite web|url=http://www-ma4.upc.es/~simeon/bblpsympspread.pdf|title=Symplectice Spreads|last=Ball|first=Simeon|author2=John Bamberg |author3=Michel Lavrauw |author4=Tim Penttila |date=2003-09-15|publisher=[[Polytechnic University of Catalonia]]|accessdate=2008-10-08}}</ref> A spread of <math>\scriptstyle PG(3, q) </math> is a set of ''q''<sup>2</sup>&nbsp;+&nbsp;1 lines, with no two intersecting. Equivalently, it is a partition of the points of <math>\scriptstyle PG(3, q) </math> into lines.
-:<math>X=\{1,\ldots,N \}^\mathbb{Z}</math>
+Given a spread <math>\scriptstyle S</math> of <math>\scriptstyle PG(3, q) </math>, the André/Bruck-Bose construction<sup>1</sup> produces a translation plane <math>\scriptstyle \pi(S)</math> of order ''q''<sup>2</sup> as follows:  Embed <math>\scriptstyle PG(3, q) </math> as a hyperplane of <math>\scriptstyle PG(4, q) </math>. Define an incidence structure <math>\scriptstyle A(S)</math> with "points," the points of <math>\scriptstyle PG(4, q) </math> not on <math>\scriptstyle PG(3, q) </math> and "lines" the planes of <math>\scriptstyle PG(4, q) </math> meeting <math>\scriptstyle PG(3, q) </math> in a line of <math>\scriptstyle S</math>. Then <math>\scriptstyle A(S)</math> is a translation affine plane of order ''q''<sup>2</sup>. Let <math>\scriptstyle \pi(S)</math> be the projective completion of <math>\scriptstyle A(S)</math>.<ref>{{cite book
+  | last =André  | first =Johannes   | authorlink =  | title = Über nicht-Dessarguessche Ebenen mit transitiver Translationsgruppe  | publisher =  | year =1954  | location =  | pages =156–186  | url =  | doi =   | id =  }}</ref><ref>{{cite book
-as a short-hand for
+   | last =Bruck  | first = R. H. | authorlink = Richard Bruck|author2=R. C. Bose   | title = The Construction of Translation Planes from Projective Spaces  | publisher =  | year =1964  | location =  | pages = 85–102  | url =  | doi =  | id =  }}</ref>
-:<math>X=\{ x=(\ldots,x_{-1},x_0,x_1,\ldots) :
-x_k \in \{1,\ldots,N\} \; \forall k \in \mathbb{Z} \}.</math>
-The associated [[measure (mathematics)|measure]] is called the '''Bernoulli measure'''<ref>Achim Klenke, ''Probability Theory'' (2006) Springer-Verlag, ISBN 978-1-848000-047-6 doi:10.1007/978-1-848000-048-3</ref>
-:<math>\mu = \{p_1,\ldots,p_N\}^\mathbb{Z}</math>
-The [[sigma-algebra|&sigma;-algebra]] <math>\mathcal{A}</math> on ''X'' is the product sigma algebra; that is, it is the (countable) [[direct product]] of the σ-algebras of the finite set {1,&nbsp;...,&nbsp;''N''}.  Thus, the triplet
-:<math>(X,\mathcal{A},\mu)</math>
-is a [[measure space]].  The elements of <math>\mathcal{A}</math> are commonly called [[cylinder set]]s. Given a cylinder set <math>[x_0, x_1,\ldots,x_n]</math>, its measure is
-:<math>\mu\left([x_0, x_1,\ldots,x_n]\right)=
-\prod_{i=0}^n p_{x_i}</math>
-The equivalent expression, using the notation of probability theory, is
-:<math>\mu\left([x_0, x_1,\ldots,x_n]\right)=
-\mathrm{Pr}(X_0=x_0, X_1=x_1, \ldots, X_n=x_n)</math>
-for the random variables <math>\{X_k\}</math>
-The Bernoulli scheme, as any stochastic process, may be viewed as a [[dynamical system]] by endowing it with the [[shift operator]] ''T'' where
-:<math>Tx_k = x_{k+1}.</math>
-Since the outcomes are independent, the shift preserves the measure, and thus ''T'' is a [[measure-preserving transformation]]. The quadruplet
-:<math>(X,\mathcal{A},\mu, T)</math>
-is a [[measure-preserving dynamical system]], and is called a '''Bernoulli scheme''' or a '''Bernoulli shift'''. It is often denoted by
-:<math>BS(p)=BS(p_1,\ldots,p_N).</math>
-The ''N'' = 2 Bernoulli scheme is called a [[Bernoulli process]].  The Bernoulli shift can be understood as a special case of the [[Markov shift]], where all entries in the [[adjacency matrix]] are one, the corresponding graph thus being a [[clique (graph theory)|clique]].
-==Generalizations==
-Most of the properties of the Bernoulli scheme follow from the countable [[direct product]], rather than from the finite base space.  Thus, one may take the base space to be any [[standard probability space]] <math>(Y,\mathcal{B},\nu)</math>, and define the Bernoulli scheme as
-:<math>(X, \mathcal{A}, \mu)=(Y,\mathcal{B},\nu)^\mathbb{Z}</math>
-This works because the countable direct product of a standard probability space is again a standard probability space.
-As a further generalization, one may replace the in integers <math>\mathbb{Z}</math> by a [[countable]] [[discrete group]] <math>G</math>, so that
-:<math>(X, \mathcal{A}, \mu)=(Y,\mathcal{B},\nu)^G</math>
-For this last case, the shift operator is replaced by the [[group action]]
-:<math>gx(f)=x(g^{-1}f)</math>
-for group elements <math>f,g\in G</math> and <math>x\in Y^G</math> understood as a function <math>x:G\to Y</math> (any direct product <math>Y^G</math> can be understood to be the set of functions <math>[G\to Y]</math>, as this is the [[exponential object]]). The measure <math>\mu</math> is taken as the [[Haar measure]], which is invariant under the group action:
-:<math>\mu(gx)=\mu(x). \, </math>
-These generalizations are also commonly called Bernoulli schemes, as they still share most properties with the finite case.
-==Properties==
-[[Ya. Sinai]] demonstrated that the [[Kolmogorov entropy]] of a Bernoulli scheme is given by<ref>Ya.G. Sinai, (1959) "On the Notion of Entropy of a Dynamical System", ''Doklady of Russian Academy of Sciences'' '''124''', pp. 768–771.</ref><ref>Ya. G. Sinai, (2007) "[http://web.math.princeton.edu/facultypapers/Sinai/MetricEntropy2.pdf Metric Entropy of Dynamical System]"</ref>
-:<math>H = -\sum_{i=1}^N p_i \log p_i .</math>
-This may be seen as resulting from the general definition of the entropy of a [[Cartesian product]] of probability spaces, which follows from the [[asymptotic equipartition property]].  For the case of a general base space <math>(Y, \mathcal{B}, \nu)</math> (''i.e.'' a base space which is not countable), one typically considers the [[relative entropy]]. So, for example, if one has a countable [[partition of a set|partition]] <math>Y'\subset Y</math> of the base ''Y'', such that <math>\nu(Y')=1</math>, one may define the entropy as
-:<math>H_{Y'} = -\sum_{y'\in Y'} \nu(y') \log \nu(y') .</math>
-In general, this entropy will depend on the partition; however, for many [[dynamical system]]s, it is the case that the [[symbolic dynamics]] is independent of the partition (or rather, there are isomorphisms connecting the symbolic dynamics of different partitions, leaving the measure invariant), and so such systems can have a well-defined entropy independent of the partition.
-The [[Ornstein isomorphism theorem]] states that two Bernoulli schemes with the same entropy are [[isomorphism of dynamical systems|isomorphic]].<ref>Donald Ornstein, "Bernoulli shifts with the same entropy are isomorphic", ''Advances in Math.'' '''4''' (1970), pp.337–352</ref> The result is sharp<ref>Christopher Hoffman, "[http://www.ams.org/journals/tran/1999-351-10/S0002-9947-99-02446-0/ A K counterexample machine]", ''Trans. Amer. Math. Soc.'' '''351''' (1999), pp 4263–4280 </ref>, in that very similar, non-scheme systems, such as [[Kolmogorov automorphism]]s, do not have this property.
-The Ornstein isomorphism theorem is in fact considerably deeper: it provides a simple criterion by which many different [[measure-preserving dynamical system]]s can be judged to be isomorphic to Bernoulli schemes.   The result was surprising, as many systems previously believed to be unrelated proved to be isomorphic. These include all finite{{clarify|date=November 2010}} [[stationary stochastic process]]es, [[subshifts of finite type]], finite [[Markov chain]]s, [[Anosov flow]]s, and [[Sinai's billiards]]: these are all isomorphic to Bernoulli schemes.
-For the generalized case, the Ornstein isomorpism theorem still holds if the group ''G'' is a countably infinite [[amenable group]].
-<ref>D. Ornstein and B. Weiss. "Entropy and isomorphism theorems for actions of amenable groups." ''J. Analyse Math.'' '''48''' (1987), pp. 1–141. </ref><ref>Lewis Bowen (2011), "[http://arxiv.org/abs/1103.4424 Every countably infinite group is almost Ornstein]", ArXiv abs/1103.4424</ref>
-==Bernoulli automorphism==
-An invertible, [[measure-preserving transformation]] of a [[standard probability space]] (Lebesgue space) is called a '''Bernoulli automorphism''' if it [[isomorphism of dynamical systems|isomorphic]] to a Bernoulli shift.<ref>Peter Walters (1982) ''An Introduction to Ergodic Theory'', Springer-Verlag, ISBN 0-387-90599-5</ref>
-==See also==
-* [[Shift of finite type]]
-* [[Markov chain]]
-* [[Hidden Bernoulli model]]
 ==References==
-<references/>
+{{Reflist}}
-{{DEFAULTSORT:Bernoulli Scheme}}
+==Further reading==
-[[Category:Markov models]]
+* Mauro Biliotti, Vikram Jha, Norman L. Johnson (2001) ''Foundations of Translation Planes'', [[Marcel Dekker]] ISBN 0-8247-0609-9 .
-[[Category:Ergodic theory]]
-[[Category:Stochastic processes]]
-[[Category:Symbolic dynamics]]
+==External links==
+*[http://www.library.tuiasi.ro/ipm/vol13no34/pure.html  Foundations_of_Translation_Planes]
+*[http://www-math.ucdenver.edu/~wcherowi/courses/m6221/pglc3a.html Lecture Notes on Projective Geometry]
+*[http://mellinger.umwblogs.org/publications/ Publications of Keith Mellinger]
-[[fr:Décalage de Bernoulli (langage formel)]]
+{{DEFAULTSORT:Translation Plane}}
-[[ru:Символическая динамика]]
+[[Category:Projective geometry]]

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Revision as of 13:50, 14 August 2014

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Relationship to spreads

References

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Main Page: Difference between revisions

Revision as of 13:50, 14 August 2014

Relationship to spreads

References

Further reading

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