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NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 402650, 7094]*) (*NotebookOutlinePosition[ 403914, 7130]*) (* CellTagsIndexPosition[ 403870, 7126]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Symmetry Based Decomposition and its Application in Evolutionary Modelling \ System\ \>", "Title"], Cell[TextData[StyleBox["Michael Y. Bogatyrev", "Author"]], "Author"], Cell["\<\ Tula State University Lenin Avenue 92, Tula 300600 Russia mibo@klax.tula.ru\ \>", "TextAboutAuthor"], Cell[TextData[StyleBox["Vladimir Latov", "Author"]], "Author"], Cell["\<\ Tula State University Lenin Avenue, 92, Tula 300600 Russia vlatov@yandex.ru \ \>", "TextAboutAuthor"], Cell["Sergey Behterev", "Author"], Cell[TextData[StyleBox["Institute of Earth Phisics of RAS\nBolshaya \ Gruzinskaya, 10, Moscow 123995 Russia\ncalm@klax.tula.ru", FontSize->12]], "TextAboutAuthor"], Cell["\<\ A method for analyzing and decomposing systems having symmetries under \ finite groups of transformations is considered. This method is based on the \ theory of group representations and needs not to solve eigenvalue problem. \ The method is applyed to create decomposition for the model of genetic \ algorithm used in evolutionary modelling system. \ \>", "Abstract"], Cell[CellGroupData[{ Cell["Introduction", "Subsection"], Cell[TextData[{ "Symmetry is a fundamental property of any phenomenon or process and group \ theory is a universally recognized instrument for analyzing symmetry. \n\t\ Symmetry analysis in its most accomplished and rigorous issue is based on Lie \ groups theory applied to differential equations. There is the school of Group \ Analysis of Differential Equations founded by Ovsyannikov [1], Olver [2] \ currently being developed by modern research groups which also apply ", StyleBox["Mathematica", FontSlant->"Italic"], " [3]. The main research problem here is to find all symmetries allowable \ by given differential equations and presented by corresponding Lie groups. \ The knowledge of these symmetries can be used to simplify the analysis of the \ problems governed by the equations. Namely, the symmetry group can be used to \ find some possible new solutions of equations or in constructing an \ equivalent system of equations having reduced order. \n\tExcept abstract \ symmetries described by Lie groups there are so called visual symmetries \ described by finite or point groups containing geometrical symmetry \ operations of rotation, mirror reflection and their compositions. Visual \ symmetry is often identified from the geometrical symmetry of an object or a \ system. A system with visual symmetry has detectable identical elements such \ as identical channels in control system or identical values of model\ \[CloseCurlyQuote]s parameters and they constitute a structure which can be \ recognized as symmetrical one. A system or a model is symmetrical if it is \ invariant under a set of transformations which constitutes a group. \ Unfortunately there is no finite basis of invariants for discrete groups [4], \ therefore a problem of discovering model\[CloseCurlyQuote]s symmetries under \ discrete groups may be hard to solve. Visual symmetry often helps to assign \ symmetry group directly from the experiment. \nIn this work we consider the \ method of decomposition of symmetrical systems based on the fact that \ symmetry group is known.\nAnother key idea of the work is that the symmetry \ in a system or in a problem to solve induces some symmetries in the method \ which is applied to solve a problem or to analyze a system. In other words, \ certain principle of conservation symmetry exists and can be applied in \ practice. " }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Symmetry Based Decomposition. ", "Subsection"], Cell["Consider a model of a system in the form ", "Text"], Cell[TextData[{ StyleBox[" y", FontWeight->"Bold"], StyleBox["(t)", FontSlant->"Italic"], " = \[ScriptCapitalA](", StyleBox["d/dt", FontSlant->"Italic"], ", ", StyleBox["a", FontWeight->"Bold"], ") ", StyleBox["x", FontWeight->"Bold"], StyleBox["(t),", FontSlant->"Italic"] }], "NumberedEquation"], Cell[TextData[{ "where ", StyleBox["x", FontWeight->"Bold"], StyleBox["(t) ", FontSlant->"Italic"], "\[Element] ", Cell[BoxData[ FormBox[ SuperscriptBox[ FormBox["R", "TraditionalForm"], "n"], TraditionalForm]]], " \[Dash] input vector, ", StyleBox["y", FontWeight->"Bold"], StyleBox["(t) ", FontSlant->"Italic"], "\[Element] ", Cell[BoxData[ FormBox[ SuperscriptBox[ FormBox["R", "TraditionalForm"], "m"], TraditionalForm]]], " - output vector, ", Cell[BoxData[ FormBox[ SuperscriptBox[ FormBox["R", "TraditionalForm"], "n"], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SuperscriptBox[ FormBox["R", "TraditionalForm"], "m"], TraditionalForm]]], " \[Dash] corresponding linear vector spaces, \[ScriptCapitalA](", StyleBox["d/dt", FontSlant->"Italic"], ", ", StyleBox["a", FontWeight->"Bold"], ") - differential operator having vector of parameters ", StyleBox["a", FontWeight->"Bold"], " \[Element] ", Cell[BoxData[ FormBox[ SuperscriptBox[ FormBox["R", "TraditionalForm"], "p"], TraditionalForm]]], ". Corresponding initial conditions are assumed for concrete form of \ operator \[ScriptCapitalA]. 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If a model (1) has symmetry (2) then it can be decomposed \ to the set of independent sub-models of reduced dimensions. This \ decomposition is based on the decision of the problem of reducing group \ representation and Wigner\[CloseCurlyQuote]s theorem which is based on \ Schur's lemmas [5]. If a matrix commutes with matrices of reducible group ", StyleBox["representation", FontFamily->"Times New Roman"], ", then all these matrices have matched block-diagonal structure in \ special", StyleBox[" ", FontFamily->"Times New Roman", FontSize->10], "canonical basis. " }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(j = \ 1\)\%\(\(q\)\(\[CirclePlus]\)\)E\_\(s\_j\)\)]], Cell[BoxData[ \(TraditionalForm\`\(\[CircleTimes]\[Tau]\_j\)\)]], "(g)", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(i = \ 1\)\%\(\(q\)\(\[CirclePlus]\)\)A\_i\)]], Cell[BoxData[ \(TraditionalForm\`\(\[CircleTimes]E\_\(l\_i\)\)\)]], " = ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(i = \ 1\)\%\(\(q\)\(\[CirclePlus]\)\)A\_i\)]], Cell[BoxData[ \(TraditionalForm\`\(\[CircleTimes]E\_\(l\_i\)\)\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(j = \ 1\)\%\(\(q\)\(\[CirclePlus]\)\)E\_\(t\_j\)\)]], Cell[BoxData[ \(TraditionalForm\`\(\[CircleTimes]\[Tau]\_j\)\)]], "(g)" }], "NumberedEquation", TextAlignment->Left, TextJustification->0], Cell[TextData[{ "\nHere ", Cell[BoxData[ \(TraditionalForm\`\[CirclePlus]\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\[CircleTimes]\)]], " - Cartesian sum and product, ", StyleBox[" q", FontSlant->"Italic"], " - number of non-equivalent ", StyleBox["irreducible representations", FontSlant->"Italic"], " of group ", StyleBox["G", FontWeight->"Bold", FontSlant->"Italic"], ", \[Tau]", StyleBox["j", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], "(", StyleBox["g", FontSlant->"Italic"], ") - their square matrices of ", StyleBox["l", FontSlant->"Italic"], StyleBox["j", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox[" ", FontSlant->"Italic"], "dimensions, ", Cell[BoxData[ \(TraditionalForm\`s\_j\)]], ", ", Cell[BoxData[ \(TraditionalForm\`t\_j\)]], StyleBox[" ", FontSlant->"Italic"], "- repetition factors of occurrence irreducible representation ", Cell[BoxData[ \(TraditionalForm\`\[Tau]\_j\)]], "(", StyleBox["g", FontSlant->"Italic"], ") in reducible representations ", StyleBox["S", FontWeight->"Bold"], " and ", StyleBox["\:0422, ", FontWeight->"Bold"], Cell[BoxData[ \(TraditionalForm\`E\_\(s\_j\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`E\_\(l\_j\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`E\_\(t\_j\)\)]], "- unit matrices of corresponding dimensions, ", StyleBox[Cell[BoxData[ \(TraditionalForm\`s\_j\)], "DisplayFormula"], "DisplayFormula"], StyleBox["=", "DisplayFormula"], StyleBox[Cell[BoxData[ \(TraditionalForm\`1\/n\)], "DisplayFormula"], "DisplayFormula"], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[Sum]\+g\( \[CurlyKappa]\_j\%*\)(g)\)], "DisplayFormula"], "DisplayFormula"], StyleBox[Cell[BoxData[ \(TraditionalForm\`\(\[CurlyKappa]\_S\)(g)\)], "DisplayFormula"], "DisplayFormula"], StyleBox[",", "DisplayFormula"], StyleBox[Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \ \)\(t\_j\)\)\)], "DisplayFormula"], "DisplayFormula"], StyleBox["=", "DisplayFormula"], StyleBox[Cell[BoxData[ \(TraditionalForm\`1\/n\)], "DisplayFormula"], "DisplayFormula"], StyleBox[Cell[BoxData[ \(TraditionalForm\`\[Sum]\+g\( \[CurlyKappa]\_j\%*\)(g)\)], "DisplayFormula"], "DisplayFormula"], StyleBox[Cell[BoxData[ \(TraditionalForm\`\(\[CurlyKappa]\_T\)(g)\)], "DisplayFormula"], "DisplayFormula"], StyleBox[",", "DisplayFormula"], "\n", StyleBox["where ", TextAlignment->Left, TextJustification->0], Cell[BoxData[ \(TraditionalForm\`\(\[CurlyKappa]\_j\)(g)\)]], StyleBox[" - ", TextAlignment->Left, TextJustification->0], StyleBox["character", TextAlignment->Left, TextJustification->0, FontSlant->"Italic"], StyleBox[" of ", TextAlignment->Left, TextJustification->0], StyleBox["j", TextAlignment->Left, TextJustification->0, FontSlant->"Italic"], StyleBox["-th irreducible representation calculated as ", TextAlignment->Left, TextJustification->0], Cell[BoxData[ \(TraditionalForm\`\(\[CurlyKappa]\_j\)(g)\)]], "=", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(i = 1\)\%\(l\_j\)\), SubsuperscriptBox["\[Tau]", StyleBox["ii", FontSlant->"Italic"], \((j)\)]}], TraditionalForm]]], " (g)", StyleBox[", ", TextAlignment->Left, TextJustification->0], Cell[BoxData[ \(TraditionalForm\`\(\[CurlyKappa]\_S\)(g)\)]], " ", StyleBox["and ", TextAlignment->Left, TextJustification->0], Cell[BoxData[ \(TraditionalForm\`\(\[CurlyKappa]\_T\)(g)\)]], StyleBox[" are characters ", TextAlignment->Left, TextJustification->0], StyleBox["(traces)", TextAlignment->Left, TextJustification->0, FontFamily->"TimesNewRoman"], StyleBox[" of representations ", TextAlignment->Left, TextJustification->0], StyleBox["S", TextAlignment->Left, TextJustification->0, FontWeight->"Bold"], StyleBox[" and ", TextAlignment->Left, TextJustification->0], StyleBox["\:0422 ", TextAlignment->Left, TextJustification->0, FontWeight->"Bold"], StyleBox["correspondingly, the sign * means complex conjugation, summing is \ doing over all elements of group ", TextAlignment->Left, TextJustification->0], StyleBox["G", TextAlignment->Left, TextJustification->0, FontWeight->"Bold", FontSlant->"Italic"], StyleBox[". ", TextAlignment->Left, TextJustification->0], StyleBox["There exist coordinate transformations ", TextAlignment->Left, TextJustification->0, FontFamily->"Times New Roman"], StyleBox[Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["x", FontSlant->"Plain"], "^"], TraditionalForm]], "DisplayFormula", FontWeight->"Bold"], "DisplayFormula"], StyleBox[" = Mx", "DisplayFormula", FontWeight->"Bold"], StyleBox[", ", "DisplayFormula"], StyleBox[Cell[BoxData[ FormBox[ OverscriptBox[ StyleBox["y", FontWeight->"Bold", FontSlant->"Plain"], "^"], TraditionalForm]], "DisplayFormula"], "DisplayFormula"], StyleBox[" = ", "DisplayFormula"], StyleBox["Ny", "DisplayFormula", FontWeight->"Bold"], StyleBox[", which transform initial model to equivalent model in canonical \ basis so that the model has block-diagonal structure of equivalent matrix: ", TextAlignment->Left, TextJustification->0] }], "Text", TextAlignment->Left, TextJustification->0], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{\(\[ScriptCapitalA]\&^\), " ", "=", " ", RowBox[{ StyleBox["diag", FontSlant->"Italic"], " ", \({\[ScriptCapitalA]\&^\_i\[CircleTimes]\(E\_l\)\_\(\_j\)}\)}\ ]}], TraditionalForm]]], ", ", StyleBox["i", FontSlant->"Italic"], " = 1, 2, ... ", StyleBox["q", FontSlant->"Italic"], StyleBox[". ", TextAlignment->Left, TextJustification->0] }], "NumberedEquation", TextAlignment->Left, TextJustification->0], Cell[TextData[{ StyleBox["Decomposition (3) concerns the structure of the matrix of \ operator ", TextAlignment->Left, TextJustification->0], StyleBox["A", TextAlignment->Left, TextJustification->0, FontWeight->"Bold"], StyleBox[" and therefore it is universal as when as ", TextAlignment->Left, TextJustification->0], StyleBox["A", TextAlignment->Left, TextJustification->0, FontWeight->"Bold"], StyleBox[" is numerical matrix [6].", TextAlignment->Left, TextJustification->0], "\nKnowing the group ", StyleBox["G", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" ", FontWeight->"Bold"], "and parameters", StyleBox[" ", FontWeight->"Bold"], "of its representations is enough to find how many blocks will be in \ decomposed matrix and what their dimensions are. Parameters", StyleBox[" ", FontWeight->"Bold"], "of reducible and irreducible representations of mostly important groups \ are well known and can be found in the Internet [7]. \n\tSimply, we only name \ a kind of symmetry in the model (as a name of corresponding group) and then \ get model\[CloseCurlyQuote]s decomposition.\n\tGroup theoretic approach is \ very general and in some sense here it serves as foundation of well known \ technique of decomposition based on solving eigenvalue problem. As usual, one \ could not hope to have very concrete result by applying very general method. \ Here we have it since we have additional concrete information: instead of \ knowing eigenvalues we know symmetry group parameters." }], "Text", TextAlignment->Left, TextJustification->0] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Method realization.", FontFamily->"Arial", FontSize->13, FontWeight->"Bold"]], "Subsection"], Cell[TextData[{ "As it is followed from (2) symmetry group is incarnated by its \ representations. According to Cayley's group theorem ", StyleBox[" ", FontVariations->{"CompatibilityType"->"Superscript"}], "[5]", StyleBox[" ", FontColor->RGBColor[0, 0, 1]], "every finite group of order ", StyleBox["n", FontSlant->"Italic"], " can be represented as a subgroup of permutation group on ", StyleBox["n", FontSlant->"Italic"], " symbols. Therefore matrices ", StyleBox["S", FontWeight->"Bold"], "(", StyleBox["g", FontSlant->"Italic"], "), ", StyleBox["T", FontWeight->"Bold"], "(", StyleBox["g", FontSlant->"Italic"], ") can be always realized as ", StyleBox["permutation matrices", FontSlant->"Italic"], ". Using permutation representation is of fundamental importance for the \ purpose of applying symmetry based decomposition in evolutionary modeling as \ it is shown below. \n\tTo realize all the objects needed in the method such \ as groups, group representations, permutations", ",", " existing and newly created ", StyleBox["Mathematica", FontSlant->"Italic"], " packages were applied. We used several techniques of storing and \ processing large [0, 1] permutation matrices and realized them in our \ packages.\nThe next specific decision is concerned with the problem of \ finding symmetry groups for given model which is actual to make the method \ general. We applied and tested some methods of generating subgroups for given \ permutation group - all of them or subgroups with certain properties." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Decomposition of genetic algorithms", FontFamily->"Arial", FontSize->13, FontWeight->"Bold"]], "Subsection"], Cell[TextData[{ "We applied symmetry based decomposition in the area of Evolutionary \ Modelling.\n\tEvolutionary modelling is a way to solve parametrical \ optimization problems when in the model (1) it is needed to find values of \ vector of parameters which form extremum of certain functional criteria ", StyleBox["F", FontWeight->"Bold"], "[", StyleBox["y", FontWeight->"Bold"], "(t)] and it is impossible to transform model (1) to any analytical model \ connecting ", StyleBox["a ", FontWeight->"Bold"], "and ", StyleBox["y", FontWeight->"Bold"], ". \n\tEvolutionary modelling is based on evolutionary algorithms and \ specifically ", StyleBox["genetic algorithms", FontSlant->"Italic"], ". Genetic algorithms become effective when there is no information about \ ", StyleBox["F", FontWeight->"Bold"], "[", StyleBox["y", FontWeight->"Bold"], "(t)] except an information about its very general properties. \n\tFor \ example, if ", StyleBox["F", FontWeight->"Bold"], " is multimodal, not smooth but has no infinite breaks then using genetic \ algorithm it is possible to find the global optima of ", StyleBox["F", FontWeight->"Bold"], ". \n\t", "It seems one of the first ", StyleBox["Mathematica", FontSlant->"Italic"], " notebooks devoted to genetic algorithms was the notebook by Bergsson. It \ contained very showy example of finding extremum of test function of Sin (\ \[Pi]", StyleBox["x Mod ", FontSlant->"Italic"], "(9x,1)). We are rendering this example using our modern version of genetic \ algorithm program.\n\tInitial population of decisions is shown on Figure 1. \ At the end of evolution they ", "cumulated at two extremas of the function that is ", "shown on Figure 2. 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Therefore appropriate model of genetic algorithm is needed for \ symmetry analysis. \t\n\tGenetic algorithms use random search and heuristics. \ As result, their modeling is not simple and there are several approaches in \ this area. Classical probability approach, approach based on Markov chains \ and algebraic approach are mostly known. That we name as algebraic approach \ is presented by the works of N. Radcliffe [8], M. Vose [9] and A. Wright [10] \ where vector - matrix model of genetic algorithm is developed and some group \ properties concerned genetic algorithms are investigated. Consider a model \ based on M. Vose notation [9]. \nA set of decisions generated by genetic \ algorithm is defined as a ", StyleBox["search space X. ", FontSlant->"Italic"], "A ", StyleBox["population", FontSlant->"Italic"], " of decisions for a genetic algorithm over length ", StyleBox["l", FontSlant->"Italic"], StyleBox[" ", FontFamily->"Times New Roman", FontSlant->"Italic"], "strings is usually interpreted as a multiset (set with repetitions) ", StyleBox["P", FontSlant->"Italic"], " ", StyleBox["X", FontSlant->"Italic"], " . Fitness function is a mapping from the search space to the space of \ positive numbers. ", StyleBox["Chromosom ", FontSlant->"Italic"], "is ", StyleBox["l", FontSlant->"Italic"], " - ", "placed", " tuple with ", StyleBox["genes", FontSlant->"Italic"], " which are symbols from an alphabet ", StyleBox["\:0410", FontSlant->"Italic"], " = {", StyleBox["a", FontSlant->"Italic"], StyleBox["i", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], ", ", StyleBox["i ", FontSlant->"Italic"], "= 1, 2, \[Ellipsis] , ", StyleBox["d", FontSlant->"Italic"], "}. ", "The number of ", "binary", " coded chromosomes is", StyleBox[" ", FontColor->RGBColor[0, 0, 1]], StyleBox["N=", FontSlant->"Italic"], StyleBox[" ", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ \(TraditionalForm\`2\^l\)]], ".\nGenetic algorithm starts to work with initial population", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`P\_0\)]], ". Every population ", Cell[BoxData[ \(TraditionalForm\`P\_\(t + 1\)\)]], " is generated from previous population", StyleBox[" ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`P\_t\)]], " by applying operators of ", StyleBox["selection", FontSlant->"Italic"], ", ", StyleBox["mutation", FontSlant->"Italic"], " and ", StyleBox["recombination", FontSlant->"Italic"], ".\nAssociate with every population ", Cell[BoxData[ \(TraditionalForm\`P\_t\)]], " a vector ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["p", FontWeight->"Bold", FontSlant->"Plain"], \((t)\)], TraditionalForm]]], " of ", StyleBox["N", FontSlant->"Italic"], " dimension with the elements ", Cell[BoxData[ \(TraditionalForm\`p\_j\%\((t)\) = \[Nu]/N\)]], " where ", StyleBox["\[Nu]", FontFamily->"Times New Roman Greek"], " is number of elements ", StyleBox["j", FontSlant->"Italic"], " presented in population, ", Cell[BoxData[ \(TraditionalForm\`p\_j\%\((t)\)\)]], "\[GreaterEqual]0, ", Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(j = 1\)\%N p\_j\%\((t)\) = 1\)]], "\nIt is considered that population does not change its dimension ", "during generations", ". So at the end of evolution a population is filled by the best \ individuals and each of them is presented by several examples. The value of \ ", Cell[BoxData[ \(TraditionalForm\`p\_j\%\((t)\)\)]], " is interpreted as probability of the fact that element ", StyleBox["j", FontSlant->"Italic"], " is in the population on the step", StyleBox[" t", FontSlant->"Italic"], ". All the vectors ", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["p", FontWeight->"Bold", FontSlant->"Plain"], \((t)\)], TraditionalForm]]], "constitute a vector space, a ", StyleBox["population state space", FontSlant->"Italic"], ". Transition from one ", "generation to another is carried out ", "by operator \[ScriptCapitalU] so that ", Cell[BoxData[ \(TraditionalForm\`P\_\(t + 1\)\)]], " = \[ScriptCapitalU](", Cell[BoxData[ \(TraditionalForm\`P\_t\)]], "). Operator \[ScriptCapitalU] is consisted on two components: \ \[ScriptCapitalU] = \[GothicCapitalM]\[SmallCircle]\[GothicCapitalR], where \ \[GothicCapitalR] is selection operator and \[GothicCapitalM] ", StyleBox["is so called mixing scheme [10] combining recombination and \ mutation. ", FontFamily->"Times New Roman"], "Mixing scheme is a vector - function of the form" }], "Text"], Cell[TextData[{ "\[GothicCapitalM](", StyleBox["p", FontWeight->"Bold"], ") = \[LeftAngleBracket]", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["p", FontWeight->"Bold", FontSlant->"Plain"], \(\((t)\) T\)], TraditionalForm]]], StyleBox["M", FontWeight->"Bold"], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["p", FontWeight->"Bold", FontSlant->"Plain"], \((t)\)], TraditionalForm]]], ", [S(2)", Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["p", FontWeight->"Bold", FontSlant->"Plain"], \((t)\)], TraditionalForm]]], "]", Cell[BoxData[ \(TraditionalForm\`\^T\)]], Cell[BoxData[ RowBox[{ StyleBox[ RowBox[{"M", FormBox[ RowBox[{\(S(2)\), SuperscriptBox[ StyleBox["p", FontWeight->"Bold", FontSlant->"Plain"], \((t)\)]}], "TraditionalForm"]}], FontFamily->"Times New Roman"], StyleBox[",", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["...", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox[",", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], RowBox[{ RowBox[{ StyleBox["[", FontFamily->"Times New Roman"], RowBox[{ StyleBox["S", FontFamily->"Times New Roman", FontSlant->"Italic"], RowBox[{ StyleBox["(", FontFamily->"Times New Roman"], StyleBox["k", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[")", FontFamily->"Times New Roman"]}], StyleBox[ FormBox[ SuperscriptBox[ StyleBox["p", FontWeight->"Bold", FontSlant->"Plain"], \((t)\)], "TraditionalForm"], FontFamily->"Times New Roman"]}], StyleBox["]", FontFamily->"Times New Roman"]}], StyleBox[ FormBox[\(\^T\), "TraditionalForm"], FontFamily->"Times New Roman"], StyleBox["M", FontFamily->"Times New Roman", FontWeight->"Bold"], StyleBox[ FormBox[ RowBox[{\(S(k)\), SuperscriptBox[ StyleBox["p", FontWeight->"Bold", FontSlant->"Plain"], \((t)\)]}], "TraditionalForm"], FontFamily->"Times New Roman"]}], StyleBox[",", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["...", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox[",", FontFamily->"Times New Roman"], RowBox[{ RowBox[{ StyleBox["[", FontFamily->"Times New Roman"], RowBox[{ StyleBox["S", FontFamily->"Times New Roman", FontSlant->"Italic"], RowBox[{ StyleBox["(", FontFamily->"Times New Roman"], StyleBox["N", FontFamily->"Times New Roman", FontSlant->"Italic"], StyleBox[")", FontFamily->"Times New Roman"]}], StyleBox[ FormBox[ SuperscriptBox[ StyleBox["p", FontWeight->"Bold", FontSlant->"Plain"], \((t)\)], "TraditionalForm"], FontFamily->"Times New Roman"]}], StyleBox["]", FontFamily->"Times New Roman"]}], StyleBox[ FormBox[\(\^T\), "TraditionalForm"], FontFamily->"Times New Roman"], StyleBox["M", FontFamily->"Times New Roman"], StyleBox[ FormBox[ RowBox[{\(S(N)\), SuperscriptBox[ StyleBox["p", FontWeight->"Bold", FontSlant->"Plain"], \((t)\)]}], "TraditionalForm"], FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"]}]}]]], "\[RightAngleBracket]" }], "NumberedEquation"], Cell[TextData[{ "where ", StyleBox["S", FontSlant->"Italic"], "(", StyleBox["k", FontSlant->"Italic"], ") is permutation matrix, ", StyleBox["\:041c", FontWeight->"Bold"], " is mixing matrix having", StyleBox[" ", FontVariations->{"CompatibilityType"->"Subscript"}], "elements of" }], "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`m\_\(i, j\) = \ \[Sum]\+\(u = 1\)\%N p\_u\)]], Cell[BoxData[ FormBox[ SubscriptBox["\[Delta]", RowBox[{"0", StyleBox["uvm", FontSlant->"Italic"]}]], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(v = 1\)\%N\), RowBox[{\(p\_v\), SubscriptBox["\[Delta]", RowBox[{"0", StyleBox["uvm", FontSlant->"Italic"]}]]}]}], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(m = 1\)\%N p\_m\)]], Cell[BoxData[ FormBox[ SubscriptBox["\[Delta]", RowBox[{"0", StyleBox["uvm", FontSlant->"Italic"]}]], TraditionalForm]]], ". " }], "NumberedEquation"], Cell[TextData[{ "Mutation and crossover are realized by using masks, which are also strings \ from the same alphabet: ", StyleBox["m", FontSlant->"Italic"], " is mutation mask, ", StyleBox["u", FontSlant->"Italic"], ", ", StyleBox["v", FontSlant->"Italic"], ", - crossover masks taking into consideration that crossover is dyadic \ operation. In (6) the logical function on masks is defined by the following:\n\ ", StyleBox["\n", FontVariations->{"CompatibilityType"->"Subscript"}], Cell[BoxData[ FormBox[ SubscriptBox["\[Delta]", RowBox[{"0", StyleBox["uvm", FontSlant->"Italic"]}]], TraditionalForm]]], StyleBox[" = ", FontVariations->{"CompatibilityType"->"Subscript"}], Cell[BoxData[ FormBox[ TagBox[ StyleBox[ RowBox[{"{", StyleBox[GridBox[{ {\(\(1\ if\ \((i\[CirclePlus]u)\)\[CircleTimes]m\ \[CirclePlus]m\&_\ \[CircleTimes]\((j\[CirclePlus]v)\) = 0\)\(,\)\(\ \)\)}, {\(\(0\ if\ \((i\[CirclePlus]u)\)\[CircleTimes]m\ \[CirclePlus]m\&_\ \[CircleTimes]\((j\[CirclePlus]v)\) \[NotEqual] 0\)\(,\)\)} }], ShowAutoStyles->True]}], ShowAutoStyles->False], (#&)], TraditionalForm]]], "\n\nwhere \[CirclePlus] is addition modulo ", StyleBox["a", FontSlant->"Italic"], " (", StyleBox["a", FontSlant->"Italic"], " =2 for binary chromosomes). ", Cell[BoxData[ \(TraditionalForm\`p\_u\)]], ", ", Cell[BoxData[ \(TraditionalForm\`p\_v\)]], ",", Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(p\_m\)\)\)]], " are probabilities of ", StyleBox["u", FontSlant->"Italic"], ", ", StyleBox["v", FontSlant->"Italic"], ", ", StyleBox["m", FontSlant->"Italic"], " masks selection, correspondingly. Their values and the values of \ probabilities of mutation and crossover (standard genetic algorithm \ parameters) are interrelated. As it is followed from the model (5) it is \ determined only by single mixing matrix M and set of permutations specified \ by matrices ", StyleBox["S", FontSlant->"Italic"], "(", StyleBox["k", FontSlant->"Italic"], ") It is easy to prove that this set constitute a group of permutations. \n\ \tAll operations performed by genetic algorithm can be interpreted as \ permutations of codes (chromosomes) in the code space. Therefore, every \ realization of genetic algorithm is connected with certain permutation group \ acting in the code space. If in the model there is an invariant under group \ action then the model has symmetry.\n \tSymmetry is known property of \ genetic algorithms [11] interpreted as a property of the problem to solve \ with genetic algorithm or as a property of the algorithm itself. There is a \ symmetry on the alphabet, for which there exists a permutation on the \ alphabet which leaves the objective value of a string of a chromosome \ unchanged, and a symmetry on the string positions, for which there exists a \ permutation on the string positions e.g. a rotation, which leaves the \ objective value unchanged. Symmetry is also interpreted as regularity in the \ form of genetic algorithm\[CloseCurlyQuote]s ", StyleBox["fitness", FontSlant->"Italic"], " ", StyleBox["landscape", FontSlant->"Italic"], ". \n \tModel (5) describes all known kinds of symmetry of genetic \ algorithms.\n\tSymmetry is recognized in genetic algorithms as one of the \ major difficulties. Symmetry can cause a genetic algorithm to get stuck in a \ local optimum, so breaking symmetry is a way to improve algorithm\ \[CloseCurlyQuote]s performance [11].\n\tNevertheless, symmetry can also help \ an algorithm to ", "find the optima more quickly. ", " Important but not the only implementation symmetry in genetic algorithms \ is finding niches.\n\t", StyleBox["Niche", FontSlant->"Italic"], " is a region in the search space where the algorithm gets stuck. There may \ be cyclic behavior in niche or stop the search. In any case algorithm does \ not leave niche during evolution. \n\tBlock - diagonal representation of \ mixing matrix of the model (5) in the form of (4) reveals the structure of \ possible niches of the algorithm. \nThis is a variant of decomposition of \ genetic algorithm. It can be realized by applying symmetry based \ decomposition.\n\tTo find decomposition, we need to work with permutation \ groups ", Cell[BoxData[ \(TraditionalForm\`S\_N\)]], StyleBox[" ", FontSlant->"Italic"], "of high order since ", StyleBox["N", FontSlant->"Italic"], "=", Cell[BoxData[ \(TraditionalForm\`a\^l\)]], ". Unified way of representing permutation groups based of using [0, 1] \ permutation matrices is used here." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Application in Evolutionary Modelling System\ \>", "Subsection"], Cell[TextData[{ "Symmetry based decomposition can be applied to solve problems which have \ ", StyleBox["niched search space", FontSlant->"Italic"], ". Sometimes not global but all local extremes are of interest in such \ problems. The similar problem arises in the research of internal-combustion \ engine regimes.\n\tWe use symmetry analysis and decomposition of genetic \ algorithms in the project of creating evolutionary modelling system for \ optimizing parameters of internal-combustion engines of various number of \ cylinders. \n\tThere are some symmetries in the engine including evident \ geometrical symmetry due to regularity of placing cylinders. Other kinds of \ symmetries are allowed by equations of the engine model we apply. \n\tThe \ model is presented as the system of 4 ", StyleBox["m", FontSlant->"Italic"], " partial differential equations added by the system of algebraic equations \ representing additional conditions, where ", StyleBox["m", FontSlant->"Italic"], " is number of cylinders. There are Lie symmetries also in the model.\n\t\ The problem is to find optimal values of certain parameters of engine. These \ parameters are coefficients of model\[CloseCurlyQuote]s equations. It is \ impossible to transform engine model so that problem\[CloseCurlyQuote]s \ solution will be analytical. Using evolutionary modeling we get solutions \ from computational experiments with genetic algorithms. We apply software \ which can realize various genetic algorithms with several variants of \ selection and special variants of crossovers. \n\tGenetic algorithm must be \ tuned up by establishing its parameters. \nAmong them the length of \ chromosomes and population size are of great importance. Chromosome length \ determines accuracy of calculation values of fitness function and also \ determines dimension of search space. \nAs dimension of search space grows \ the time needed to find solution increases very much. So chromosome length \ is crucial parameter of the algorithm\nSince every permutation group of ", Cell[BoxData[ \(TraditionalForm\`S\_N\)]], StyleBox[" ", FontVariations->{"CompatibilityType"->"Subscript"}], "has a set of nested subgroups ", Cell[BoxData[ \(TraditionalForm\`S\_\(N - 1\)\)]], ", ", StyleBox[" ", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], Cell[BoxData[ \(TraditionalForm\`S\_\(N - 2\)\)]], StyleBox[" ", FontSlant->"Italic", FontVariations->{"CompatibilityType"->"Subscript"}], StyleBox["\[Ellipsis]", FontSlant->"Italic"], " there is unified object (a group of permutations) to analyze for every \ search space dimension. \n\tWhen population size increases then more probably \ to find global extreme. In the problem of finding local extremes it is \ possible to work on smaller populations if they contain decisions from the \ niches.\n \tWe make experiments with various probability distributions for \ mutation and crossovers. These distributions determine the structure of \ mixing matrix ", StyleBox["M", FontWeight->"Bold"], ". Correspondingly, various probability distributions of mutation and \ crossovers set various symmetries in the mixing matrix ", StyleBox["M", FontWeight->"Bold"], ".\n\tThere are other factors which restrict maximal symmetry in the model \ of genetic algorithm. They are from the problem conditions. As result, the \ number of niches is determined from the physically grounded ranges of \ changing engine\[CloseCurlyQuote]s parameters during evolution.\n\tThe \ evolutionary modelling system has open architecture. There are three \ independed interacting subsystems in the system. Manager subsystem is for \ organizing interactions between user and subsystems. The subsystem which \ realizes genetic algorithms is created on Java platform and works with \ Windows DLL - libraries which represent the model to optimize, particularly \ engine model. The next, symmetry based decomposition subsystem, interacts \ with ", StyleBox["Mathematica ", FontSlant->"Italic"], "kernel by evaluating notebooks and packages. \n\tThe whole system is on \ developing stage. First experiments made with it show its efficiency in \ tuning genetic algorithms by using symmetry based decomposition." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["References.", FontWeight->"Bold"]], "Subsection"], Cell[TextData[{ StyleBox["\n", FontWeight->"Bold"], StyleBox["1.", "Reference", FontFamily->"Times New Roman"], StyleBox[" ", FontFamily->"Times New Roman"], StyleBox["L.V. Ovsyannikov Group Analysis of Differential Equations. \ Moscow, Nauka. English translation, Academic Press, 1982.\n2. P. Olver \ Applications of Lie Groups To Differential Equations. Springer - Verlag, New \ York, 1986. \n3. B. J. Cantwell Introduction to Symmetry Analysis. Cambridge \ University Press, 2002.\n4. V. F. Zaitsev, et al. Handbook of First-Order \ Partial Differential Equations (Differential and Integral Equations and Their \ Applications). Taylor & Francis, 2002\n5. Hamermesh, M., Group Theory and Its \ Application to Physical Problems. New York, Dover, 1989. \n6. M. Bogatyrev \ Modelling Systems With Symmetry In: Proc. IMACS Symposium on Math. \ Modelling, Vienna, February 5-7, 2003. ARGESIM-Verlag, Vienna. - pp. \ 270-275.\n7. R. A. Wilson, R.A.Parker, J.N.Bray ATLAS of Finite Group \ Representation. [www http://for.mat.bham.ac.uk/atlas/]\n8. N.J. Radcliffe, \ The algebra of genetic algorithms. - Annals of", FontFamily->"Times New Roman"], " Mathematics and Artificial Intelligence", StyleBox[". - V. 10 p. 339 - 384.", FontFamily->"Times New Roman"], StyleBox["\n", FontSize->14], StyleBox["9. M. D. Vose, The Simple Genetic Algorithm: Foundations and \ Theory. MIT press, 1999.\n10. M. D. Vose, A. Wright, J. Rowe ", FontFamily->"Times New Roman"], "Group Properties of Crossover and Mutation ", "Evolutionary Computation 1", "0(2), 2002, pages 151-184. \n", StyleBox["11. M. Pelikan, D. Goldberg. Genetic Algorithms, Clustering, and \ the Breaking of Symmetry. IlliGAL Report No. 2000013 March 2000.", FontFamily->"Times New Roman"], " " }], "Reference"] }, Open ]], Cell[CellGroupData[{ Cell["Biographies of authors", "Subsection"], Cell[TextData[{ StyleBox["Michael Bogatyrev", "Author", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold"], StyleBox[" is professor at the department of Automatic and Remote Control \ in Tula State University and the head of Laboratory of Information Systems. \ His research interests are in the field of applying symmetry in engineering \ and information systems.\n\n", "Author", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Plain"], StyleBox["Vladimir Latov", "Author", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold"], StyleBox[" is PhD student at the department of Automatic and Remote Control \ in Tula State University. His thesis is devoted to genetic algorithms \ imlementaions in engineering problems of multicriteria optimization.\n\n", "Author", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Plain"], StyleBox["Sergey Behterev", "Author", FontFamily->"Times New Roman", FontSize->12, FontWeight->"Bold"], StyleBox[" is research officer in the ", "Author", FontFamily->"Times New Roman", FontSize->12, FontSlant->"Plain"], StyleBox["Institute of Earth Phisics of Russian Acadeny of Sciences. He \ works in the field of creating instruments for measuring gravitation using \ symmetry. ", FontSize->12] }], "TextAboutAuthor"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.2 for X", ScreenRectangle->{{0, 1920}, {0, 1200}}, WindowToolbars->"EditBar", WindowSize->{1016, 1115}, WindowMargins->{{58, Automatic}, {Automatic, 25}}, PrintingPageRange->{Automatic, Automatic}, PageHeaders->{{ Inherited, Inherited, Cell[ "Michael Y. 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