(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 334652, 11738] NotebookOptionsPosition[ 205296, 8988] NotebookOutlinePosition[ 325424, 11443] CellTagsIndexPosition[ 325343, 11438] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\<\"\[FirstPage]\"\>", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\<\"\[LeftPointer]\"\>", "SR"], ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPagePrevious"]}], ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\<\"\[RightPointer]\"\>", "SR"], ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageNext"]}], ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\<\"\[LastPage]\"\>", "SR"], ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageLast"]}], ButtonNote->"Last Slide"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell["\<\ Frobenius Numbers by Toric Gr\[ODoubleDot]bner Bases\ \>", "Title"], Cell["Daniel Lichtblau", "Author"], Cell["\<\ Wolfram Research, Inc. 100 Trade Centre Dr. Champaign IL USA, 61820\ \>", "Affiliation"], Cell[TextData[ButtonBox["danl@wolfram.com", ButtonData:>{ URL["mailto:danl@wolfram.com"], None}]], "Author"], Cell["\<\ ACA 2005, Nara, Japan August 2005\ \>", "Affiliation"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\<\"\[FirstPage]\"\>", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\<\"\[LeftPointer]\"\>", "SR"], ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPagePrevious"]}], ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\<\"\[RightPointer]\"\>", "SR"], ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageNext"]}], ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\<\"\[LastPage]\"\>", "SR"], ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageLast"]}], ButtonNote->"Last Slide"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Abstract", "Section", CellOpen->True, FontSize->24], Cell[TextData[{ "Given a set ", Cell[BoxData[ FormBox[ RowBox[{"A", "=", RowBox[{"{", RowBox[{ SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "n"]}], "}"}]}], TraditionalForm]]], " of positive integers with gcd 1, it is not hard to show that all \ \"sufficiently large\" integers can be represented as a nonnegative integer \ combination of elements of ", Cell[BoxData[ FormBox["A", TraditionalForm]]], ". The Frobenius number of the set is defined as the largest integer not so \ representable. The Frobenius instance problem (also called the money changing \ or postage stamp problem) is to determine, given a positive integer M, a set \ of nonnegative integers ", Cell[BoxData[ FormBox[ RowBox[{"X", "=", RowBox[{"{", RowBox[{ SubscriptBox["x", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["x", "n"]}], "}"}]}], TraditionalForm]]], " such that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"X", ".", "A"}], "=", "n"}], TraditionalForm]]], ", or else show no such set exists. We will briefly recall how this can be \ addressed via toric Gr\[ODoubleDot]bner bases." }], "Text"], Cell[TextData[{ "It is known that the Frobenius number problem is NP\[Hyphen]hard in \ general. For dimension 2 it is trivial (Sylvester solved in two decades \ before Frobenius publicized the problem). In dimension 3 a very efficient \ method was found independently by Greenberg and Davison. For higher \ dimensions some quite effective methods are known for the case where one \ element of ", Cell[BoxData[ FormBox["A", TraditionalForm]]], " is not too large (say, less than ", Cell[BoxData[ FormBox[ SuperscriptBox["10", "7"], TraditionalForm]]], ")." }], "Text"], Cell[TextData[{ "Recent work has given rise to methods that are effective when the above \ restrictions do not hold, although the dimension must be bounded by 10 or so. \ It turns out that there is a way to recast this work using toric Gr\ \[ODoubleDot]bner bases, wherein the \"fundamental domain\" for the set ", Cell[BoxData[ FormBox["A", TraditionalForm]]], " is given by the staircase of the basis with respect to a particular \ ordering. It is reasonably efficient in dimensions 4 to 7, when the elements \ in the set are as large as ", Cell[BoxData[ FormBox[ SuperscriptBox["10", "40"], TraditionalForm]]], " or so. We will illustrate this." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\<\"\[FirstPage]\"\>", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\<\"\[LeftPointer]\"\>", "SR"], ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPagePrevious"]}], ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\<\"\[RightPointer]\"\>", "SR"], ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageNext"]}], ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\<\"\[LastPage]\"\>", "SR"], ButtonFunction:>FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageLast"]}], ButtonNote->"Last Slide"], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}]} }, GridBoxAlignment->{ "Columns" -> {{Center}}, "ColumnsIndexed" -> {}, "Rows" -> {{Center}}, "RowsIndexed" -> {}}]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Introduction: Background and brief history", "Section", CellOpen->True, FontSize->24], Cell[TextData[{ "We are given a set ", Cell[BoxData[ FormBox[ RowBox[{"A", "=", RowBox[{"{", RowBox[{ SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "n"]}], "}"}]}], TraditionalForm]]], " of positive integers with ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"gcd", "(", "A", ")"}], "=", "1"}], TraditionalForm]]], ". We assume for later purposes that the set is in ascending order." }], "Text"], Cell[TextData[{ "Problem 1 (Frobenius instance problem): Given a nonnegative integer M, find \ a set of nonnegative integers ", Cell[BoxData[ FormBox[ RowBox[{"X", "=", RowBox[{"{", RowBox[{ SubscriptBox["x", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["x", "n"]}], "}"}]}], TraditionalForm]]], " such that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"X", ".", "A"}], "=", "n"}], TraditionalForm]]], ", or else show no such set exists." }], "Text"], Cell[TextData[{ "Problem 2 (Frobenius number problem): Find the largest integer not \ representable as a nonnegative integer combination of ", Cell[BoxData[ FormBox["A", TraditionalForm]]], "." }], "Text"], Cell[TextData[{ "In the 80's and 90's Greenberg and Davison independenly found an ultrafast \ method for problem 2 when ", Cell[BoxData[ FormBox[ RowBox[{"n", "=", "3"}], TraditionalForm]]], ". Beyond this size no specialized (that is, dimension specific) methods are \ known, and we must resort to general tactics." }], "Text"], Cell[TextData[{ "Reasonably effective methods based mostly on graph theory have appeared \ also in the past 30 years or so. Some very nice new ones are presented, along \ with older approaches, in very recent work by Beihoffer, Hendry, Nijenhuis, \ and Wagon. It should be noted that the third author helped originate the \ graph theory approach. These methods are limited by the size of ", Cell[BoxData[ FormBox[ SubscriptBox["a", "1"], TraditionalForm]]], ", but not by ", Cell[BoxData[ FormBox["n", TraditionalForm]]], "." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageLast"]}]& )], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Introduction: Background and brief history", "Section"], Cell[TextData[{ "This restriction apparently rankled the fourth author, who continued to \ pursue the problem using different tactics. Forthcoming joint work by David \ Einstein, Adam Strzebonski, Stan Wagon, and myself will show how one can \ attack this problem effectively using lattice methods and integer \ programming. While we can do away with the size restriction on ", Cell[BoxData[ FormBox[ SubscriptBox["a", "0"], TraditionalForm]]], " we do get into some algorithmic complexity due to dimension, and at \ present cannot get beyond ", Cell[BoxData[ FormBox[ RowBox[{"n", "=", "11"}], TraditionalForm]]], " or so. Some of the technology we use is also applicable to problem 1, \ where we can manage higher dimension (25 or larger)." }], "Text"], Cell["\<\ It so happens that much of this can be recast in a setting of toric \ varieties. While problem 1, which boils to integer linear programming, has \ long been known to be amenable to such an approach (as per work by Conti and \ Traverso), this is apparently a new tactic for finding Frobenius numbers. We \ can exploit it to handle examples that, to the best of my knowledge, could \ not be done by methods known as of a year ago. A nice benefit we will soon \ see is that the needed code is quite short (three pages or so).\ \>", "Text"], Cell[TextData[{ "We will define a \"fundamental domain\" which is a generalization by Wagon \ that subsumes both lattice diagrams in earlier literature and a graph \ description from Beihoffer et al. The Frobenius number will be the furthest \ corner from the origin, in a suitably weighted ", Cell[BoxData[ FormBox[ SubscriptBox["l", "1"], TraditionalForm]]], " norm. As we will see, an important domain feature is what we term \"elbows\ \". The method to find those via a Gr\[ODoubleDot]bner basis \"staircase\" \ constitutes the new material in this talk." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageLast"]}]& )], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["\<\ Solving a Frobenius instance via toric Gr\[ODoubleDot]bner bases\ \>", "Section"], Cell["Say we are given the set and value", "Text"], Cell[BoxData[{ RowBox[{ RowBox[{"A", "=", RowBox[{"{", RowBox[{ "200", ",", "230", ",", "528", ",", "863", ",", "905", ",", "1355", ",", "1725", ",", "1796", ",", "1808"}], "}"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"b", "=", "7777"}], ";"}]}], "Input"], Cell[TextData[{ "We wish to know whether or how we can write ", Cell[BoxData[ FormBox["b", TraditionalForm]]], " as a nonnegative combination of elements of ", Cell[BoxData[ FormBox["A", TraditionalForm]]], ". We may do this as follows. Create a variable ", Cell[BoxData[ FormBox["t", TraditionalForm]]], " that will be raised to the powers in this set. Create a variable for each \ set element ", Cell[BoxData[ FormBox[ SubscriptBox["a", "j"], TraditionalForm]]], " and equate it to ", Cell[BoxData[ FormBox[ SuperscriptBox["t", SubscriptBox["a", "j"]], TraditionalForm]]], ". We form a Gr\[ODoubleDot]bner basis in these variables, using an order \ that makes powers of ", Cell[BoxData[ FormBox["t", TraditionalForm]]], " larger than all monomials that do not contain it. For this we use a weight \ matrix. It is structured in such a way as to be efficient for the task at \ hand. Basically it is like an elimination ordering on ", Cell[BoxData[ FormBox["t", TraditionalForm]]], " with degree\[Hyphen]reverse\[Hyphen]lexicographic on the remaining \ variables, except instead of using (homogenious) total degree we weight by \ the values in ", Cell[BoxData[ FormBox["A", TraditionalForm]]], "." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], 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We want to see that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"4", "*", "230"}], "+", RowBox[{"1", "*", "528"}], "+", RowBox[{"1", "*", "905"}], "+", RowBox[{"3", "*", "1808"}]}], "=", "7777"}], TraditionalForm]]], "." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Total", "[", RowBox[{"Apply", "[", RowBox[{"Times", ",", "exponvec", ",", "2"}], "]"}], "]"}]], "Input"], Cell[BoxData["7777"], "Output"] }, Open ]], Cell["\<\ Remark: The above illustrates more or less the original formulation for \ handling ILPs via toric Gr\[ODoubleDot]bner bases. Subsequent improvements \ have appeared and quite likely this can be done much more efficiently now. \ This concludes our brief review of how one might solve a Frobenius instance \ problem using the method of Conti and Traverso. We return to the main task at \ hand, which is computation of Frobenius numbers.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageLast"]}]& )], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["The Fundamental Domain", "Section"], Cell[TextData[{ "We define a lattice by the set of integer combinations of ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{ SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "n"]}], "}"}], TraditionalForm]]], " that are zero modulo ", Cell[BoxData[ FormBox[ SubscriptBox["a", "1"], TraditionalForm]]], ". This is a full dimensional lattice in ", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalZ]", RowBox[{"n", "-", "1"}]], TraditionalForm]]], ". The set of residues of ", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalZ]", RowBox[{"n", "-", "1"}]], TraditionalForm]]], " modulo this lattice gives rise to what we call the fundamental domain, \ which, as we see, lives in a space of dimension one less than the size of our \ set. It is not hard to see that there are ", Cell[BoxData[ FormBox[ SubscriptBox["a", "1"], TraditionalForm]]], " distinct residue classes, so we know the cardinality of this domain. We \ now define the \"weight\" of a vector ", Cell[BoxData[ FormBox[ RowBox[{"v", "\[Element]", SuperscriptBox["\[DoubleStruckCapitalZ]", RowBox[{"n", "-", "1"}]]}], TraditionalForm]]], " as ", Cell[BoxData[ FormBox[ RowBox[{"v", ".", RowBox[{"{", RowBox[{ SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "n"]}], "}"}]}], TraditionalForm]]], ". It can be shown that every residue class has at least one element with \ all nonnegative entries. Among those, we choose one of minimal weight. In \ case of tie, choose the one that is lexicographically last. This uniquely \ defines the set of residues that we take to comprise the fundamental domain." }], "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageLast"]}]& )], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["The Fundamental Domain", "Section"], Cell["\<\ This domain can be shown to have several interesting properties.\ \>", "Text"], Cell[TextData[{ "\[Bullet] It is a staircase. If it contains a lattice element then it \ contains all nonnegative vectors with any coordinate strictly smaller.\n\ \[Bullet] It tiles ", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckCapitalZ]", RowBox[{"n", "-", "1"}]], TraditionalForm]]], ".\n\[Bullet] It is a cyclic group ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[DoubleStruckCapitalZ]", "/", SubscriptBox["a", "1"]}], " ", "\[DoubleStruckCapitalZ]"}], TraditionalForm]]], ".\n", "\[Bullet]", " It can be given a circulant graph structure. It is this structure that was \ utilized in various shortest\[Hyphen]path graph methods. Old and new methods \ for this are discussed at length in the recent work by Beihoffer et al." }], "Text"], Cell["\<\ For our purposes the property of most interest is the first one. We can \ recover this staircase by computing a toric Gr\[ODoubleDot]bner basis.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageLast"]}]& )], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Definitions related to the Fundamental Domain", "Section"], Cell[TextData[{ "From the staircase property the fundamental domain has turning points we \ refer to as elbows. It has extremal points called corners. Specifically, a \ corner is a point ", Cell[BoxData[ FormBox["c", TraditionalForm]]], " in the domain, such that ", Cell[BoxData[ FormBox[ RowBox[{"c", "+", SubscriptBox["e", "j"]}], TraditionalForm]]], " is not in the domain, where ", Cell[BoxData[ FormBox[ SubscriptBox["e", "j"], TraditionalForm]]], " is the ", Cell[BoxData[ FormBox[ SubscriptBox["j", "th"], TraditionalForm]]], " coordinate vector. An elbow is a point ", Cell[BoxData[ FormBox["x", TraditionalForm]]], " that is not in the domain, but is such that, for each ", Cell[BoxData[ FormBox["j", TraditionalForm]]], ", either ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "j"], "=", "0"}], TraditionalForm]]], " or ", Cell[BoxData[ FormBox[ RowBox[{"x", "-", SubscriptBox["e", "j"]}], TraditionalForm]]], " is in the domain." }], "Text"], Cell["\<\ There are two other definitions that play a role in the algorithm. We will \ not descibe them too carefully but, roughly, there are as follows. (i) Protoelbows. These have both positive and negative coordinates and \ correspond to certain \"minimal\" equivalences (that is, reducing relations) \ in the lattice. In Gr\[ODoubleDot]bner basis terms, these are given as \ exponent vectors of binomial pairs in the basis. (ii) Preelbows. These are the \"positive parts\" of the protoelbows. Elbows \ are minimal elements in the partially ordered (ascending by inclusion) set of \ preelbows.\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageLast"]}]& )], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["Fundamental Domain, illustrated", "Section"], Cell[TextData[{ "Since this region is one dimension smaller than the input set (as verything \ is done modulo ", Cell[BoxData[ FormBox[ SubscriptBox["a", "1"], TraditionalForm]]], "), it can be illustrated for the cases ", Cell[BoxData[ FormBox[ RowBox[{"n", "=", "3"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"n", "=", "4"}], TraditionalForm]]], ". The illustrations are due to Stan Wagon." }], "Text"], Cell[TextData[{ "With respect to these domains, the Frobenius number corresponds to the \ farthest corner from the origin, with distance an ", Cell[BoxData[ FormBox[ SubscriptBox["l", "1"], TraditionalForm]]], " metric weighted by element sizes. In the planar diagram the \"elbows\" are \ the lattice points on the axes that bound the diagram, and the lattice point \ in the interior just outside the \"ell\". The corners are the two extremal \ points reached by intersecting vertical and horizontal lines through the \ elbows. This picture tells the entire story as regards the ", Cell[BoxData[ FormBox[ RowBox[{"n", "=", "3"}], TraditionalForm]]], " case, because it can be shown that there is at most one interior elbow and \ two such corners, and finding them is easy." }], "Text"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1.78947 %%ImageSize: 100 178.947 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations -0.00125313 0.125313 0.0175439 0.125313 [ [ 0 0 0 0 ] [ 1 1.78947 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 1 0 0 r .901 1.29574 m .901 1.37093 L .97619 1.37093 L .97619 1.29574 L F .39975 1.67168 m .39975 1.74687 L .47494 1.74687 L .47494 1.67168 L F .77569 1.29574 m .77569 1.37093 L .85088 1.37093 L .85088 1.29574 L F .901 1.17043 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closepath p F P 0 g s 1 1 0 r .56937 .65739 m .58671 .61814 L .5744 .63959 L closepath p F P 0 g s 1 .1 .15 r .49748 .48229 m .52265 .49721 L .49806 .51174 L p F P 0 g s 1 .1 .15 r .49806 .51174 m .4729 .49721 L .49748 .48229 L p F P 0 g s 1 .1 .15 r .47228 .52594 m .4729 .49721 L .49806 .51174 L p F P 0 g s 1 .1 .15 r .49806 .51174 m .49777 .54029 L .47228 .52594 L p F P 0 g s 1 .1 .15 r .52268 .52594 m .49777 .54029 L .49806 .51174 L p F P 0 g s 1 .1 .15 r .49806 .51174 m .52265 .49721 L .52268 .52594 L p F P 0 g s 1 .1 .15 r .46148 .54465 m .48711 .55875 L .46298 .57249 L p F P 0 g s 1 .1 .15 r .46298 .57249 m .43737 .55875 L .46148 .54465 L p F P 0 g s 1 .1 .15 r .43628 .5883 m .43737 .55875 L .46298 .57249 L p F P 0 g s 1 .1 .15 r .46298 .57249 m .46224 .60183 L .43628 .5883 L p F P 0 g s 1 .1 .15 r .48668 .5883 m .46224 .60183 L .46298 .57249 L p F P 0 g s 1 .1 .15 r .46298 .57249 m .48711 .55875 L .48668 .5883 L p F P 0 g s 1 .1 .15 r .56949 .48229 m .59371 .49721 L .56821 .51174 L p F P 0 g s 1 .1 .15 r .56821 .51174 m .54397 .49721 L .56949 .48229 L p F P 0 g s 1 .1 .15 r .54429 .52594 m .54397 .49721 L .56821 .51174 L p F P 0 g s 1 .1 .15 r .56821 .51174 m .56884 .54029 L .54429 .52594 L p F P 0 g s 1 .1 .15 r .59469 .52594 m .56884 .54029 L .56821 .51174 L p F P 0 g s 1 .1 .15 r .56821 .51174 m .59371 .49721 L .59469 .52594 L p F P 0 g s 1 .1 .15 r .53349 .54465 m .55818 .55875 L .53313 .57249 L p F P 0 g s 1 .1 .15 r .53313 .57249 m .50843 .55875 L .53349 .54465 L p F P 0 g s 1 .1 .15 r .50828 .5883 m .50843 .55875 L .53313 .57249 L p F P 0 g s 1 .1 .15 r .53313 .57249 m .53331 .60183 L .50828 .5883 L p F P 0 g s 1 .1 .15 r .55869 .5883 m .53331 .60183 L .53313 .57249 L p F P 0 g s 1 .1 .15 r .53313 .57249 m .55818 .55875 L .55869 .5883 L p F P 0 g s 1 .1 .15 r .49748 .60701 m .52265 .6203 L .49806 .63324 L p F P 0 g s 1 .1 .15 r .49806 .63324 m .4729 .6203 L .49748 .60701 L p F P 0 g s 1 .1 .15 r .47228 .65066 m .4729 .6203 L .49806 .63324 L p F P 0 g s 1 .1 .15 r .49806 .63324 m .49777 .66338 L .47228 .65066 L p F P 0 g s 1 .1 .15 r .52268 .65066 m .49777 .66338 L .49806 .63324 L p F P 0 g s 1 .1 .15 r .49806 .63324 m .52265 .6203 L .52268 .65066 L p F P 0 g s 1 .1 .15 r .6415 .48229 m .66478 .49721 L .63836 .51174 L p F P 0 g s 1 .1 .15 r .63836 .51174 m .61503 .49721 L .6415 .48229 L p F P 0 g s 1 .1 .15 r .61629 .52594 m .61503 .49721 L .63836 .51174 L p F P 0 g s 1 .1 .15 r .63836 .51174 m .63991 .54029 L .61629 .52594 L p F P 0 g s 1 .1 .15 r .6667 .52594 m .63991 .54029 L .63836 .51174 L p F P 0 g s 1 .1 .15 r .63836 .51174 m .66478 .49721 L .6667 .52594 L p F P 0 g s 1 .1 .15 r .53349 .66937 m .55818 .68184 L .53313 .694 L p F P 0 g s 1 .1 .15 r .53313 .694 m .50843 .68184 L .53349 .66937 L p F P 0 g s 1 .1 .15 r .50828 .71302 m .50843 .68184 L .53313 .694 L p F P 0 g s 1 .1 .15 r .53313 .694 m .53331 .72493 L .50828 .71302 L p F P 0 g s 1 .1 .15 r .55869 .71302 m .53331 .72493 L .53313 .694 L p F P 0 g s 1 .1 .15 r .53313 .694 m .55818 .68184 L .55869 .71302 L p F P 0 g s 1 .1 .15 r .60549 .66937 m .62925 .68184 L .60328 .694 L p F P 0 g s 1 .1 .15 r .60328 .694 m .5795 .68184 L .60549 .66937 L p F P 0 g s 1 .1 .15 r .58029 .71302 m .5795 .68184 L .60328 .694 L p F P 0 g s 1 .1 .15 r .60328 .694 m .60437 .72493 L .58029 .71302 L p F P 0 g s 1 .1 .15 r .6307 .71302 m .60437 .72493 L .60328 .694 L p F P 0 g s 1 .1 .15 r .60328 .694 m .62925 .68184 L .6307 .71302 L p F P 0 g s 1 .1 .15 r .42547 .48229 m .45158 .49721 L .42791 .51174 L p F P 0 g s 1 .1 .15 r .42791 .51174 m .40183 .49721 L .42547 .48229 L p F P 0 g s 1 .1 .15 r .40027 .52594 m .40183 .49721 L .42791 .51174 L p F P 0 g s 1 .1 .15 r .42791 .51174 m .42671 .54029 L .40027 .52594 L p F P 0 g s 1 .1 .15 r .45068 .52594 m .42671 .54029 L .42791 .51174 L p F P 0 g s 1 .1 .15 r .42791 .51174 m .45158 .49721 L .45068 .52594 L p F P 0 g s 1 .1 .15 r .7135 .48229 m .73585 .49721 L .70851 .51174 L p F P 0 g s 1 .1 .15 r .70851 .51174 m .6861 .49721 L .7135 .48229 L p F P 0 g s 1 .1 .15 r .6883 .52594 m .6861 .49721 L .70851 .51174 L p F P 0 g s 1 .1 .15 r .70851 .51174 m .71097 .54029 L .6883 .52594 L p F P 0 g s 1 .1 .15 r .73871 .52594 m .71097 .54029 L .70851 .51174 L p F P 0 g s 1 .1 .15 r .70851 .51174 m .73585 .49721 L .73871 .52594 L p F P 0 g s 1 .1 .15 r .56949 .73173 m .59371 .74339 L .56821 .75475 L p F P 0 g s 1 .1 .15 r .56821 .75475 m .54397 .74339 L .56949 .73173 L p F P 0 g s 1 .1 .15 r .54429 .77538 m .54397 .74339 L .56821 .75475 L p F P 0 g s 1 .1 .15 r .56821 .75475 m .56884 .78647 L .54429 .77538 L p F P 0 g s 1 .1 .15 r .59469 .77538 m .56884 .78647 L .56821 .75475 L p F P 0 g s 1 .1 .15 r .56821 .75475 m .59371 .74339 L .59469 .77538 L p F P 0 g s 1 .1 .15 r .49718 .51121 m .52268 .52594 L .49777 .54029 L p F P 0 g s 1 .1 .15 r .49777 .54029 m .47228 .52594 L .49718 .51121 L p F P 0 g s 1 .1 .15 r .49718 .51121 m .47228 .52594 L .4729 .49721 L p F P 0 g s 1 .1 .15 r .4729 .49721 m .49748 .48229 L .49718 .51121 L p F P 0 g s 1 .1 .15 r .4607 .57441 m .46148 .54465 L .48711 .55875 L p F P 0 g s 1 .1 .15 r .48711 .55875 m .48668 .5883 L .4607 .57441 L p F P 0 g s 1 .1 .15 r .4607 .57441 m .43628 .5883 L .43737 .55875 L p F P 0 g s 1 .1 .15 r .43737 .55875 m .46148 .54465 L .4607 .57441 L p F P 0 g s 1 .1 .15 r .57016 .51121 m .59469 .52594 L .56884 .54029 L p F P 0 g s 1 .1 .15 r .56884 .54029 m .54429 .52594 L .57016 .51121 L p F P 0 g s 1 .1 .15 r .53367 .57441 m .55869 .5883 L .53331 .60183 L p F P 0 g s 1 .1 .15 r .53331 .60183 m .50828 .5883 L .53367 .57441 L p F P 0 g s 1 .1 .15 r .53367 .57441 m .53349 .54465 L .55818 .55875 L p F P 0 g s 1 .1 .15 r .55818 .55875 m .55869 .5883 L .53367 .57441 L p F P 0 g s 1 .1 .15 r .53367 .57441 m .50828 .5883 L .50843 .55875 L p F P 0 g s 1 .1 .15 r .50843 .55875 m .53349 .54465 L .53367 .57441 L p F P 0 g s 1 .1 .15 r .49718 .6376 m .49748 .60701 L .52265 .6203 L p F P 0 g s 1 .1 .15 r .52265 .6203 m .52268 .65066 L .49718 .6376 L p F P 0 g s 1 .1 .15 r .64313 .51121 m .6667 .52594 L .63991 .54029 L p F P 0 g s 1 .1 .15 r .63991 .54029 m .61629 .52594 L .64313 .51121 L p F P 0 g s 1 .1 .15 r .53367 .7008 m .53349 .66937 L .55818 .68184 L p F P 0 g s 1 .1 .15 r .55818 .68184 m .55869 .71302 L .53367 .7008 L p F P 0 g s 1 .1 .15 r .53367 .7008 m .50828 .71302 L .50843 .68184 L p F P 0 g s 1 .1 .15 r .50843 .68184 m .53349 .66937 L .53367 .7008 L p F P 0 g s 1 .1 .15 r .7161 .51121 m .7135 .48229 L .73585 .49721 L p F P 0 g s 1 .1 .15 r .73585 .49721 m .73871 .52594 L .7161 .51121 L p F P 0 g s 1 .1 .15 r .60664 .7008 m .60549 .66937 L .62925 .68184 L p F P 0 g s 1 .1 .15 r .62925 .68184 m .6307 .71302 L .60664 .7008 L p F P 0 g s 1 .1 .15 r .60664 .7008 m .58029 .71302 L .5795 .68184 L p F P 0 g s 1 .1 .15 r .5795 .68184 m .60549 .66937 L .60664 .7008 L p F P 0 g s 1 .1 .15 r .57016 .764 m .59469 .77538 L .56884 .78647 L p F P 0 g s 1 .1 .15 r .56884 .78647 m .54429 .77538 L .57016 .764 L p F P 0 g s 1 .1 .15 r .42421 .51121 m .45068 .52594 L .42671 .54029 L p F P 0 g s 1 .1 .15 r .42671 .54029 m .40027 .52594 L .42421 .51121 L p F P 0 g s 1 .1 .15 r .42421 .51121 m .42547 .48229 L .45158 .49721 L p F P 0 g s 1 .1 .15 r .45158 .49721 m .45068 .52594 L .42421 .51121 L p F P 0 g s 1 .1 .15 r .42421 .51121 m .40027 .52594 L .40183 .49721 L p F P 0 g s 1 .1 .15 r .40183 .49721 m .42547 .48229 L .42421 .51121 L p F P 0 g s 1 .1 .15 r .49718 .51121 m .49748 .48229 L .52265 .49721 L p F P 0 g s 1 .1 .15 r .52265 .49721 m .52268 .52594 L .49718 .51121 L p F P 0 g s 1 .1 .15 r .4607 .57441 m .48668 .5883 L .46224 .60183 L p F P 0 g s 1 .1 .15 r .46224 .60183 m .43628 .5883 L .4607 .57441 L p F P 0 g s 1 .1 .15 r .57016 .51121 m .56949 .48229 L .59371 .49721 L p F P 0 g s 1 .1 .15 r .59371 .49721 m .59469 .52594 L .57016 .51121 L p F P 0 g s 1 .1 .15 r .57016 .51121 m .54429 .52594 L .54397 .49721 L p F P 0 g s 1 .1 .15 r .54397 .49721 m .56949 .48229 L .57016 .51121 L p F P 0 g s 1 .1 .15 r .49718 .6376 m .52268 .65066 L .49777 .66338 L p F P 0 g s 1 .1 .15 r .49777 .66338 m .47228 .65066 L .49718 .6376 L p F P 0 g s 1 .1 .15 r .49718 .6376 m .47228 .65066 L .4729 .6203 L p F P 0 g s 1 .1 .15 r .4729 .6203 m .49748 .60701 L .49718 .6376 L p F P 0 g s 1 .1 .15 r .64313 .51121 m .6415 .48229 L .66478 .49721 L p F P 0 g s 1 .1 .15 r .66478 .49721 m .6667 .52594 L .64313 .51121 L p F P 0 g s 1 .1 .15 r .64313 .51121 m .61629 .52594 L .61503 .49721 L p F P 0 g s 1 .1 .15 r .61503 .49721 m .6415 .48229 L .64313 .51121 L p F P 0 g s 1 .1 .15 r .53367 .7008 m .55869 .71302 L .53331 .72493 L p F P 0 g s 1 .1 .15 r .53331 .72493 m .50828 .71302 L .53367 .7008 L p F P 0 g s 1 .1 .15 r .7161 .51121 m .73871 .52594 L .71097 .54029 L p F P 0 g s 1 .1 .15 r .71097 .54029 m .6883 .52594 L .7161 .51121 L p F P 0 g s 1 .1 .15 r .7161 .51121 m .6883 .52594 L .6861 .49721 L p F P 0 g s 1 .1 .15 r .6861 .49721 m .7135 .48229 L .7161 .51121 L p F P 0 g s 1 .1 .15 r .60664 .7008 m .6307 .71302 L .60437 .72493 L p F P 0 g s 1 .1 .15 r .60437 .72493 m .58029 .71302 L .60664 .7008 L p F P 0 g s 1 .1 .15 r .57016 .764 m .56949 .73173 L .59371 .74339 L p F P 0 g s 1 .1 .15 r .59371 .74339 m .59469 .77538 L .57016 .764 L p F P 0 g s 1 .1 .15 r .57016 .764 m .54429 .77538 L .54397 .74339 L p F P 0 g s 1 .1 .15 r .54397 .74339 m .56949 .73173 L .57016 .764 L p F P 0 g s 1 1 0 r .56937 .65739 m .54509 .62398 L .58671 .61814 L closepath p F P 0 g s 1 .1 .15 r .46035 .46028 m .48649 .47578 L .46192 .49086 L p F P 0 g s 1 .1 .15 r .46192 .49086 m .4358 .47578 L .46035 .46028 L p F P 0 g s 1 .1 .15 r .43467 .50478 m .4358 .47578 L .46192 .49086 L p F P 0 g s 1 .1 .15 r .46192 .49086 m .46115 .51968 L .43467 .50478 L p F P 0 g s 1 .1 .15 r .48604 .50478 m .46115 .51968 L .46192 .49086 L p F P 0 g s 1 .1 .15 r .46192 .49086 m .48649 .47578 L .48604 .50478 L p F P 0 g s 1 .1 .15 r .42366 .52384 m .45028 .53849 L .42618 .55275 L p F P 0 g s 1 .1 .15 r .42618 .55275 m .39959 .53849 L .42366 .52384 L p F P 0 g s 1 .1 .15 r .39797 .56834 m .39959 .53849 L .42618 .55275 L p F P 0 g s 1 .1 .15 r .42618 .55275 m .42494 .58239 L .39797 .56834 L p F P 0 g s 1 .1 .15 r .44935 .56834 m .42494 .58239 L .42618 .55275 L p F P 0 g s 1 .1 .15 r .42618 .55275 m .45028 .53849 L .44935 .56834 L p F P 0 g s 1 .1 .15 r .53375 .46028 m .55891 .47578 L .53338 .49086 L p F P 0 g s 1 .1 .15 r .53338 .49086 m .50822 .47578 L .53375 .46028 L p F P 0 g s 1 .1 .15 r .50806 .50478 m .50822 .47578 L .53338 .49086 L p F P 0 g s 1 .1 .15 r .53338 .49086 m .53356 .51968 L .50806 .50478 L p F P 0 g s 1 .1 .15 r .55944 .50478 m .53356 .51968 L .53338 .49086 L p F P 0 g s 1 .1 .15 r .53338 .49086 m .55891 .47578 L .55944 .50478 L p F P 0 g s 1 .1 .15 r .49705 .52384 m .5227 .53849 L .49765 .55275 L p F P 0 g s 1 .1 .15 r .49765 .55275 m .47201 .53849 L .49705 .52384 L p F P 0 g s 1 .1 .15 r .47136 .56834 m .47201 .53849 L .49765 .55275 L p F P 0 g s 1 .1 .15 r .49765 .55275 m .49735 .58239 L .47136 .56834 L p F P 0 g s 1 .1 .15 r .52274 .56834 m .49735 .58239 L .49765 .55275 L p F P 0 g s 1 .1 .15 r .49765 .55275 m .5227 .53849 L .52274 .56834 L p F P 0 g s 1 .1 .15 r .46035 .58741 m .48649 .60121 L .46192 .61465 L p F P 0 g s 1 .1 .15 r .46192 .61465 m .4358 .60121 L .46035 .58741 L p F P 0 g s 1 .1 .15 r .43467 .6319 m .4358 .60121 L .46192 .61465 L p F P 0 g s 1 .1 .15 r .46192 .61465 m .46115 .64511 L .43467 .6319 L p F P 0 g s 1 .1 .15 r .48604 .6319 m .46115 .64511 L .46192 .61465 L p F P 0 g s 1 .1 .15 r .46192 .61465 m .48649 .60121 L .48604 .6319 L p F P 0 g s 1 .1 .15 r .60714 .46028 m .63133 .47578 L .60485 .49086 L p F P 0 g s 1 .1 .15 r .60485 .49086 m .58064 .47578 L .60714 .46028 L p F P 0 g s 1 .1 .15 r .58146 .50478 m .58064 .47578 L .60485 .49086 L p F P 0 g s 1 .1 .15 r .60485 .49086 m .60598 .51968 L .58146 .50478 L p F P 0 g s 1 .1 .15 r .63283 .50478 m .60598 .51968 L .60485 .49086 L p F P 0 g s 1 .1 .15 r .60485 .49086 m .63133 .47578 L .63283 .50478 L p F P 0 g s 1 .1 .15 r .49705 .65097 m .5227 .66392 L .49765 .67654 L p F P 0 g s 1 .1 .15 r .49765 .67654 m .47201 .66392 L .49705 .65097 L p F P 0 g s 1 .1 .15 r .47136 .69546 m .47201 .66392 L .49765 .67654 L p F P 0 g s 1 .1 .15 r .49765 .67654 m .49735 .70782 L .47136 .69546 L p F P 0 g s 1 .1 .15 r .52274 .69546 m .49735 .70782 L .49765 .67654 L p F P 0 g s 1 .1 .15 r .49765 .67654 m .5227 .66392 L .52274 .69546 L p F P 0 g s 1 .1 .15 r .53375 .71453 m .55891 .72664 L .53338 .73843 L p F P 0 g s 1 .1 .15 r .53338 .73843 m .50822 .72664 L .53375 .71453 L p F P 0 g s 1 .1 .15 r .50806 .75902 m .50822 .72664 L .53338 .73843 L p F P 0 g s 1 .1 .15 r .53338 .73843 m .53356 .77054 L .50806 .75902 L p F P 0 g s 1 .1 .15 r .55944 .75902 m .53356 .77054 L .53338 .73843 L p F P 0 g s 1 .1 .15 r .53338 .73843 m .55891 .72664 L .55944 .75902 L p F P 0 g s 1 .1 .15 r .60714 .71453 m .63133 .72664 L .60485 .73843 L p F P 0 g s 1 .1 .15 r .60485 .73843 m .58064 .72664 L .60714 .71453 L p F P 0 g s 1 .1 .15 r .58146 .75902 m .58064 .72664 L .60485 .73843 L p F P 0 g s 1 .1 .15 r .60485 .73843 m .60598 .77054 L .58146 .75902 L p F P 0 g s 1 .1 .15 r .63283 .75902 m .60598 .77054 L .60485 .73843 L p F P 0 g s 1 .1 .15 r .60485 .73843 m .63133 .72664 L .63283 .75902 L p F P 0 g s 1 .1 .15 r .38696 .46028 m .41407 .47578 L .39045 .49086 L p F P 0 g s 1 .1 .15 r .39045 .49086 m .36338 .47578 L .38696 .46028 L p F P 0 g s 1 .1 .15 r .36127 .50478 m .36338 .47578 L .39045 .49086 L p F P 0 g s 1 .1 .15 r .39045 .49086 m .38873 .51968 L .36127 .50478 L p F P 0 g s 1 .1 .15 r .41265 .50478 m .38873 .51968 L .39045 .49086 L p F P 0 g s 1 .1 .15 r .39045 .49086 m .41407 .47578 L .41265 .50478 L p F P 0 g s 1 .1 .15 r .68054 .46028 m .70375 .47578 L .67632 .49086 L p F P 0 g s 1 .1 .15 r .67632 .49086 m .65305 .47578 L .68054 .46028 L p F P 0 g s 1 .1 .15 r .65485 .50478 m .65305 .47578 L .67632 .49086 L p F P 0 g s 1 .1 .15 r .67632 .49086 m .6784 .51968 L .65485 .50478 L p F P 0 g s 1 .1 .15 r .70623 .50478 m .6784 .51968 L .67632 .49086 L p F P 0 g s 1 .1 .15 r .67632 .49086 m .70375 .47578 L .70623 .50478 L p F P 0 g s 1 .1 .15 r .75393 .46028 m .77616 .47578 L .74778 .49086 L p F P 0 g s 1 .1 .15 r .74778 .49086 m .72547 .47578 L .75393 .46028 L p F P 0 g s 1 .1 .15 r .72825 .50478 m .72547 .47578 L .74778 .49086 L p F P 0 g s 1 .1 .15 r .74778 .49086 m .75082 .51968 L .72825 .50478 L p F P 0 g s 1 .1 .15 r .77962 .50478 m .75082 .51968 L .74778 .49086 L p F P 0 g s 1 .1 .15 r .74778 .49086 m .77616 .47578 L .77962 .50478 L p F P 0 g s 1 .1 .15 r .57045 .77809 m .59512 .78935 L .56912 .80032 L p F P 0 g s 1 .1 .15 r .56912 .80032 m .54443 .78935 L .57045 .77809 L p F P 0 g s 1 .1 .15 r .54476 .82258 m .54443 .78935 L .56912 .80032 L p F P 0 g s 1 .1 .15 r .56912 .80032 m .56977 .83326 L .54476 .82258 L p F P 0 g s 1 .1 .15 r .59613 .82258 m .56977 .83326 L .56912 .80032 L p F P 0 g s 1 .1 .15 r .56912 .80032 m .59512 .78935 L .59613 .82258 L p F P 0 g s 1 .1 .15 r .38514 .48947 m 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.49827 m .54525 .48278 L .57216 .46686 L p F P 0 g s 1 .1 .15 r .57216 .46686 m .57144 .43741 L .59658 .45351 L p F P 0 g s 1 .1 .15 r .59658 .45351 m .59764 .48278 L .57216 .46686 L p F P 0 g s 1 .1 .15 r .57216 .46686 m .54525 .48278 L .5449 .45351 L p F P 0 g s 1 .1 .15 r .5449 .45351 m .57144 .43741 L .57216 .46686 L p F P 0 g s 1 .1 .15 r .45834 .664 m .48538 .67721 L .46001 .69006 L p F P 0 g s 1 .1 .15 r .46001 .69006 m .43299 .67721 L .45834 .664 L p F P 0 g s 1 .1 .15 r .45834 .664 m .45919 .63184 L .48585 .64531 L p F P 0 g s 1 .1 .15 r .48585 .64531 m .48538 .67721 L .45834 .664 L p F P 0 g s 1 .1 .15 r .45834 .664 m .43299 .67721 L .43417 .64531 L p F P 0 g s 1 .1 .15 r .43417 .64531 m .45919 .63184 L .45834 .664 L p F P 0 g s 1 .1 .15 r .64804 .46686 m .67247 .48278 L .64456 .49827 L p F P 0 g s 1 .1 .15 r .64456 .49827 m .62009 .48278 L .64804 .46686 L p F P 0 g s 1 .1 .15 r .64804 .46686 m .64628 .43741 L .6704 .45351 L p F P 0 g s 1 .1 .15 r .6704 .45351 m .67247 .48278 L .64804 .46686 L p F P 0 g s 1 .1 .15 r .49628 .72971 m .5228 .74202 L .49692 .75399 L p F P 0 g s 1 .1 .15 r .49692 .75399 m .47041 .74202 L .49628 .72971 L p F P 0 g s 1 .1 .15 r .49628 .72971 m .49661 .69665 L .52276 .70924 L p F P 0 g s 1 .1 .15 r .52276 .70924 m .5228 .74202 L .49628 .72971 L p F P 0 g s 1 .1 .15 r .72392 .46686 m .74731 .48278 L .71838 .49827 L p F P 0 g s 1 .1 .15 r .71838 .49827 m .69492 .48278 L .72392 .46686 L p F P 0 g s 1 .1 .15 r .72392 .46686 m .72112 .43741 L .74422 .45351 L p F P 0 g s 1 .1 .15 r .74422 .45351 m .74731 .48278 L .72392 .46686 L p F P 0 g s 1 .1 .15 r .72392 .46686 m .69492 .48278 L .69255 .45351 L p F P 0 g s 1 .1 .15 r .69255 .45351 m .72112 .43741 L .72392 .46686 L p F P 0 g s 1 .1 .15 r .53422 .79543 m .56022 .80683 L .53383 .81792 L p F P 0 g s 1 .1 .15 r .53383 .81792 m .50783 .80683 L .53422 .79543 L p F P 0 g s 1 .1 .15 r .53422 .79543 m .53402 .76146 L .55967 .77317 L p F P 0 g s 1 .1 .15 r .55967 .77317 m .56022 .80683 L .53422 .79543 L p F P 0 g s 1 .1 .15 r .53422 .79543 m .50783 .80683 L .50799 .77317 L p F P 0 g s 1 .1 .15 r .50799 .77317 m .53402 .76146 L .53422 .79543 L p F P 0 g s 1 .1 .15 r .79981 .46686 m .79595 .43741 L .81804 .45351 L p F P 0 g s 1 .1 .15 r .81804 .45351 m .82215 .48278 L .79981 .46686 L p F P 0 g s 1 .1 .15 r .34452 .46686 m .37312 .48278 L .34928 .49827 L p F P 0 g s 1 .1 .15 r .34928 .49827 m .32074 .48278 L .34452 .46686 L p F P 0 g s 1 .1 .15 r .34452 .46686 m .34693 .43741 L .37511 .45351 L p F P 0 g s 1 .1 .15 r .37511 .45351 m .37312 .48278 L .34452 .46686 L p F P 0 g s 1 .1 .15 r .38246 .53257 m .41054 .54759 L .38619 .5622 L p F P 0 g s 1 .1 .15 r .38619 .5622 m .35816 .54759 L .38246 .53257 L p F P 0 g s 1 .1 .15 r .4204 .59829 m .44796 .6124 L .4231 .62613 L p F P 0 g s 1 .1 .15 r .4231 .62613 m .39558 .6124 L .4204 .59829 L p F P 0 g s 1 .1 .15 r .64804 .46686 m .62009 .48278 L .61873 .45351 L p F P 0 g s 1 .1 .15 r .61873 .45351 m .64628 .43741 L .64804 .46686 L p F P 0 g s 1 .1 .15 r .49628 .72971 m .47041 .74202 L .47108 .70924 L p F P 0 g s 1 .1 .15 r .47108 .70924 m .49661 .69665 L .49628 .72971 L p F P 0 g s 1 .1 .15 r .79981 .46686 m .82215 .48278 L .79221 .49827 L p F P 0 g s 1 .1 .15 r .79221 .49827 m .76976 .48278 L .79981 .46686 L p F P 0 g s 1 .1 .15 r .79981 .46686 m .76976 .48278 L .76637 .45351 L p F P 0 g s 1 .1 .15 r .76637 .45351 m .79595 .43741 L .79981 .46686 L p F P 0 g s 1 1 0 r .57131 .87807 m .54607 .84116 L .58932 .83641 L closepath p F P 0 g s 1 1 0 r .36321 .49237 m .3205 .49932 L .35523 .5179 L closepath p F P 0 g s 1 1 0 r .34393 .53346 m .35523 .5179 L .3205 .49932 L closepath p F P 0 g s 1 1 0 r .34393 .53346 m .36321 .49237 L .35523 .5179 L closepath p F P 0 g s .3 .8 .3 r .4198 .47964 m .44761 .49552 L .42254 .51097 L p F P 0 g s .3 .8 .3 r .42254 .51097 m .39477 .49552 L .4198 .47964 L p F P 0 g s .3 .8 .3 r .39301 .52605 m .39477 .49552 L .42254 .51097 L p F P 0 g s .3 .8 .3 r .42254 .51097 m .42119 .54129 L .39301 .52605 L p F P 0 g s .3 .8 .3 r .4466 .52605 m .42119 .54129 L .42254 .51097 L p F P 0 g s .3 .8 .3 r .42254 .51097 m .44761 .49552 L .4466 .52605 L p F P 0 g s 1 .1 .15 r .4198 .47973 m .44754 .49557 L .42253 .51097 L p F P 0 g s 1 .1 .15 r .42253 .51097 m .39484 .49557 L .4198 .47973 L p F P 0 g s 1 .1 .15 r .39309 .52601 m .39484 .49557 L .42253 .51097 L p F P 0 g s 1 .1 .15 r .42253 .51097 m .42119 .5412 L .39309 .52601 L p F P 0 g s 1 .1 .15 r .44652 .52601 m .42119 .5412 L .42253 .51097 L p F P 0 g s 1 .1 .15 r .42253 .51097 m .44754 .49557 L .44652 .52601 L p F P 0 g s 1 .1 .15 r .3053 .41362 m .33461 .43037 L .31115 .44667 L p F P 0 g s 1 .1 .15 r .31115 .44667 m .28192 .43037 L .3053 .41362 L p F P 0 g s 1 .1 .15 r .27858 .4599 m .28192 .43037 L .31115 .44667 L p F P 0 g s 1 .1 .15 r .31115 .44667 m .30827 .47601 L .27858 .4599 L p F P 0 g s 1 .1 .15 r .33202 .4599 m .30827 .47601 L .31115 .44667 L p F P 0 g s 1 .1 .15 r .31115 .44667 m .33461 .43037 L .33202 .4599 L p F P 0 g s 1 .1 .15 r .38163 .41362 m .40989 .43037 L .38541 .44667 L p F P 0 g s 1 .1 .15 r .38541 .44667 m .3572 .43037 L .38163 .41362 L p F P 0 g s 1 .1 .15 r .35492 .4599 m .3572 .43037 L .38541 .44667 L p F P 0 g s 1 .1 .15 r .38541 .44667 m .38355 .47601 L .35492 .4599 L p F P 0 g s 1 .1 .15 r .40835 .4599 m .38355 .47601 L .38541 .44667 L p F P 0 g s 1 .1 .15 r .38541 .44667 m .40989 .43037 L .40835 .4599 L p F P 0 g s 1 .1 .15 r .45797 .41362 m .48518 .43037 L .45966 .44667 L p F P 0 g s 1 .1 .15 r .45966 .44667 m .43248 .43037 L .45797 .41362 L p F P 0 g s 1 .1 .15 r .43125 .4599 m .43248 .43037 L .45966 .44667 L p F P 0 g s 1 .1 .15 r .45966 .44667 m .45883 .47601 L .43125 .4599 L p F P 0 g s 1 .1 .15 r .48469 .4599 m .45883 .47601 L .45966 .44667 L p F P 0 g s 1 .1 .15 r .45966 .44667 m .48518 .43037 L .48469 .4599 L p F P 0 g s 1 .1 .15 r .38163 .54584 m .40989 .56076 L .38541 .57528 L p F P 0 g s 1 .1 .15 r .38541 .57528 m .3572 .56076 L .38163 .54584 L p F P 0 g s 1 .1 .15 r .35492 .59212 m .3572 .56076 L .38541 .57528 L p F P 0 g s 1 .1 .15 r .38541 .57528 m .38355 .6064 L .35492 .59212 L p F P 0 g s 1 .1 .15 r .40835 .59212 m .38355 .6064 L .38541 .57528 L p F P 0 g s 1 .1 .15 r .38541 .57528 m .40989 .56076 L .40835 .59212 L p F P 0 g s 1 .1 .15 r .53431 .41362 m .56046 .43037 L .53391 .44667 L p F P 0 g s 1 .1 .15 r .53391 .44667 m .50776 .43037 L .53431 .41362 L p F P 0 g s 1 .1 .15 r .50759 .4599 m .50776 .43037 L .53391 .44667 L p F P 0 g s 1 .1 .15 r .53391 .44667 m .53411 .47601 L .50759 .4599 L p F P 0 g s 1 .1 .15 r .56103 .4599 m .53411 .47601 L .53391 .44667 L p F P 0 g s 1 .1 .15 r .53391 .44667 m .56046 .43037 L .56103 .4599 L p F P 0 g s 1 .1 .15 r .4198 .61195 m .44754 .62596 L .42253 .63958 L p F P 0 g s 1 .1 .15 r .42253 .63958 m .39484 .62596 L .4198 .61195 L p F P 0 g s 1 .1 .15 r .39309 .65823 m .39484 .62596 L .42253 .63958 L p F P 0 g s 1 .1 .15 r .42253 .63958 m .42119 .67159 L .39309 .65823 L p F P 0 g s 1 .1 .15 r .44652 .65823 m .42119 .67159 L .42253 .63958 L p F P 0 g s 1 .1 .15 r .42253 .63958 m .44754 .62596 L .44652 .65823 L p F P 0 g s 1 .1 .15 r .61065 .41362 m .63574 .43037 L .60817 .44667 L p F P 0 g s 1 .1 .15 r .60817 .44667 m .58304 .43037 L .61065 .41362 L p F P 0 g s 1 .1 .15 r .58393 .4599 m .58304 .43037 L .60817 .44667 L p F P 0 g s 1 .1 .15 r .60817 .44667 m .60939 .47601 L .58393 .4599 L p F P 0 g s 1 .1 .15 r .63736 .4599 m .60939 .47601 L .60817 .44667 L p F P 0 g s 1 .1 .15 r .60817 .44667 m .63574 .43037 L .63736 .4599 L p F P 0 g s 1 .1 .15 r .45797 .67806 m .48518 .69115 L .45966 .70389 L p F P 0 g s 1 .1 .15 r .45966 .70389 m .43248 .69115 L .45797 .67806 L p F P 0 g s 1 .1 .15 r .43125 .72434 m .43248 .69115 L .45966 .70389 L p F P 0 g s 1 .1 .15 r .45966 .70389 m .45883 .73679 L .43125 .72434 L p F P 0 g s 1 .1 .15 r .48469 .72434 m .45883 .73679 L .45966 .70389 L p F P 0 g s 1 .1 .15 r .45966 .70389 m .48518 .69115 L .48469 .72434 L p F P 0 g s 1 .1 .15 r .68698 .41362 m .71102 .43037 L .68242 .44667 L p F P 0 g s 1 .1 .15 r .68242 .44667 m .65832 .43037 L .68698 .41362 L p F P 0 g s 1 .1 .15 r .66027 .4599 m .65832 .43037 L .68242 .44667 L p F P 0 g s 1 .1 .15 r .68242 .44667 m .68467 .47601 L .66027 .4599 L p F P 0 g s 1 .1 .15 r .7137 .4599 m .68467 .47601 L .68242 .44667 L p F P 0 g s 1 .1 .15 r .68242 .44667 m .71102 .43037 L .7137 .4599 L p F P 0 g s 1 .1 .15 r .49614 .74417 m .52282 .75635 L .49679 .76819 L p F P 0 g s 1 .1 .15 r .49679 .76819 m .47012 .75635 L .49614 .74417 L p F P 0 g s 1 .1 .15 r .46942 .79045 m .47012 .75635 L .49679 .76819 L p F P 0 g s 1 .1 .15 r .49679 .76819 m .49647 .80198 L .46942 .79045 L p F P 0 g s 1 .1 .15 r .52286 .79045 m .49647 .80198 L .49679 .76819 L p F P 0 g s 1 .1 .15 r .49679 .76819 m .52282 .75635 L .52286 .79045 L p F P 0 g s 1 .1 .15 r .76332 .41362 m .7863 .43037 L .75667 .44667 L p F P 0 g s 1 .1 .15 r .75667 .44667 m .7336 .43037 L .76332 .41362 L p F P 0 g s 1 .1 .15 r .7366 .4599 m .7336 .43037 L .75667 .44667 L p F P 0 g s 1 .1 .15 r .75667 .44667 m .75995 .47601 L .7366 .4599 L p F P 0 g s 1 .1 .15 r .79004 .4599 m .75995 .47601 L .75667 .44667 L p F P 0 g s 1 .1 .15 r .75667 .44667 m .7863 .43037 L .79004 .4599 L p F P 0 g s 1 .1 .15 r .83966 .41362 m .86158 .43037 L .83093 .44667 L p F P 0 g s 1 .1 .15 r .83093 .44667 m .80888 .43037 L .83966 .41362 L p F P 0 g s 1 .1 .15 r .81294 .4599 m .80888 .43037 L .83093 .44667 L p F P 0 g s 1 .1 .15 r .83093 .44667 m .83523 .47601 L .81294 .4599 L p F P 0 g s 1 .1 .15 r .86638 .4599 m .83523 .47601 L .83093 .44667 L p F P 0 g s 1 .1 .15 r .83093 .44667 m .86158 .43037 L .86638 .4599 L p F P 0 g s 1 .1 .15 r .41838 .51038 m .39309 .52601 L .39484 .49557 L p F P 0 g s 1 .1 .15 r .41838 .51038 m .44652 .52601 L .42119 .5412 L p F P 0 g s 1 .1 .15 r .42119 .5412 m .39309 .52601 L .41838 .51038 L p F P 0 g s .3 .8 .3 r .41837 .51038 m .4466 .52605 L .42119 .54129 L p F P 0 g s .3 .8 .3 r .42119 .54129 m .39301 .52605 L .41837 .51038 L p F P 0 g s 1 .1 .15 r .39484 .49557 m .4198 .47973 L .41838 .51038 L p F P 0 g s 1 .1 .15 r .41838 .51038 m .4198 .47973 L .44754 .49557 L p F P 0 g s .3 .8 .3 r .41837 .51038 m .4198 .47964 L .44761 .49552 L p F P 0 g s 1 .1 .15 r .44754 .49557 m .44652 .52601 L .41838 .51038 L p F P 0 g s .3 .8 .3 r .44761 .49552 m .4466 .52605 L .41837 .51038 L p F P 0 g s .3 .8 .3 r .41837 .51038 m .39301 .52605 L .39477 .49552 L p F P 0 g s .3 .8 .3 r .39477 .49552 m .4198 .47964 L .41837 .51038 L p F P 0 g s 1 .1 .15 r .30225 .44333 m .33202 .4599 L .30827 .47601 L p F P 0 g s 1 .1 .15 r .30827 .47601 m .27858 .4599 L .30225 .44333 L p F P 0 g s 1 .1 .15 r .30225 .44333 m .3053 .41362 L .33461 .43037 L p F P 0 g s 1 .1 .15 r .33461 .43037 m .33202 .4599 L .30225 .44333 L p F P 0 g s 1 .1 .15 r .30225 .44333 m .27858 .4599 L .28192 .43037 L p F P 0 g s 1 .1 .15 r .28192 .43037 m .3053 .41362 L .30225 .44333 L p F P 0 g s 1 .1 .15 r .37967 .44333 m .40835 .4599 L .38355 .47601 L p F P 0 g s 1 .1 .15 r .38355 .47601 m .35492 .4599 L .37967 .44333 L p F P 0 g s 1 .1 .15 r .37967 .44333 m .38163 .41362 L .40989 .43037 L p F P 0 g s 1 .1 .15 r .40989 .43037 m .40835 .4599 L .37967 .44333 L p F P 0 g s 1 .1 .15 r .37967 .44333 m .35492 .4599 L .3572 .43037 L p F P 0 g s 1 .1 .15 r .3572 .43037 m .38163 .41362 L .37967 .44333 L p F P 0 g s 1 .1 .15 r .45709 .44333 m .48469 .4599 L .45883 .47601 L p F P 0 g s 1 .1 .15 r .45883 .47601 m .43125 .4599 L .45709 .44333 L p F P 0 g s 1 .1 .15 r .45709 .44333 m .45797 .41362 L .48518 .43037 L p F P 0 g s 1 .1 .15 r .48518 .43037 m .48469 .4599 L .45709 .44333 L p F P 0 g s 1 .1 .15 r .45709 .44333 m .43125 .4599 L .43248 .43037 L p F P 0 g s 1 .1 .15 r .43248 .43037 m .45797 .41362 L .45709 .44333 L p F P 0 g s 1 .1 .15 r .37967 .57743 m .40835 .59212 L .38355 .6064 L p F P 0 g s 1 .1 .15 r .38355 .6064 m .35492 .59212 L .37967 .57743 L p F P 0 g s 1 .1 .15 r .37967 .57743 m .38163 .54584 L .40989 .56076 L p F P 0 g s 1 .1 .15 r .40989 .56076 m .40835 .59212 L .37967 .57743 L p F P 0 g s 1 .1 .15 r .37967 .57743 m .35492 .59212 L .3572 .56076 L p F P 0 g s 1 .1 .15 r .3572 .56076 m .38163 .54584 L .37967 .57743 L p F P 0 g s 1 .1 .15 r .53452 .44333 m .56103 .4599 L .53411 .47601 L p F P 0 g s 1 .1 .15 r .53411 .47601 m .50759 .4599 L .53452 .44333 L p F P 0 g s 1 .1 .15 r .53452 .44333 m .53431 .41362 L .56046 .43037 L p F P 0 g s 1 .1 .15 r .56046 .43037 m .56103 .4599 L .53452 .44333 L p F P 0 g s 1 .1 .15 r .53452 .44333 m .50759 .4599 L .50776 .43037 L p F P 0 g s 1 .1 .15 r .50776 .43037 m .53431 .41362 L .53452 .44333 L p F P 0 g s 1 .1 .15 r .41838 .64448 m .44652 .65823 L .42119 .67159 L p F P 0 g s 1 .1 .15 r .42119 .67159 m .39309 .65823 L .41838 .64448 L p F P 0 g s 1 .1 .15 r .41838 .64448 m .4198 .61195 L .44754 .62596 L p F P 0 g s 1 .1 .15 r .44754 .62596 m .44652 .65823 L .41838 .64448 L p F P 0 g s 1 .1 .15 r .41838 .64448 m .39309 .65823 L .39484 .62596 L p F P 0 g s 1 .1 .15 r .39484 .62596 m .4198 .61195 L .41838 .64448 L p F P 0 g s 1 .1 .15 r .61194 .44333 m .63736 .4599 L .60939 .47601 L p F P 0 g s 1 .1 .15 r .60939 .47601 m .58393 .4599 L .61194 .44333 L p F P 0 g s 1 .1 .15 r .61194 .44333 m .61065 .41362 L .63574 .43037 L p F P 0 g s 1 .1 .15 r .63574 .43037 m .63736 .4599 L .61194 .44333 L p F P 0 g s 1 .1 .15 r .61194 .44333 m .58393 .4599 L .58304 .43037 L p F P 0 g s 1 .1 .15 r .58304 .43037 m .61065 .41362 L .61194 .44333 L p F P 0 g s 1 .1 .15 r .45709 .71153 m .48469 .72434 L .45883 .73679 L p F P 0 g s 1 .1 .15 r .45883 .73679 m .43125 .72434 L .45709 .71153 L p F P 0 g s 1 .1 .15 r .45709 .71153 m .45797 .67806 L .48518 .69115 L p F P 0 g s 1 .1 .15 r .48518 .69115 m .48469 .72434 L .45709 .71153 L p F P 0 g s 1 .1 .15 r .45709 .71153 m .43125 .72434 L .43248 .69115 L p F P 0 g s 1 .1 .15 r .43248 .69115 m .45797 .67806 L .45709 .71153 L p F P 0 g s 1 .1 .15 r .68936 .44333 m .7137 .4599 L .68467 .47601 L p F P 0 g s 1 .1 .15 r .68467 .47601 m .66027 .4599 L .68936 .44333 L p F P 0 g s 1 .1 .15 r .68936 .44333 m .68698 .41362 L .71102 .43037 L p F P 0 g s 1 .1 .15 r .71102 .43037 m .7137 .4599 L .68936 .44333 L p F P 0 g s 1 .1 .15 r .68936 .44333 m .66027 .4599 L .65832 .43037 L p F P 0 g s 1 .1 .15 r .65832 .43037 m .68698 .41362 L .68936 .44333 L p F P 0 g s 1 .1 .15 r .4958 .77858 m .52286 .79045 L .49647 .80198 L p F P 0 g s 1 .1 .15 r .49647 .80198 m .46942 .79045 L .4958 .77858 L p F P 0 g s 1 .1 .15 r .4958 .77858 m .49614 .74417 L .52282 .75635 L p F P 0 g s 1 .1 .15 r .52282 .75635 m .52286 .79045 L .4958 .77858 L p F P 0 g s 1 .1 .15 r .4958 .77858 m .46942 .79045 L .47012 .75635 L p F P 0 g s 1 .1 .15 r .47012 .75635 m .49614 .74417 L .4958 .77858 L p F P 0 g s 1 .1 .15 r .76679 .44333 m .79004 .4599 L .75995 .47601 L p F P 0 g s 1 .1 .15 r .75995 .47601 m .7366 .4599 L .76679 .44333 L p F P 0 g s 1 .1 .15 r .76679 .44333 m .76332 .41362 L .7863 .43037 L p F P 0 g s 1 .1 .15 r .7863 .43037 m .79004 .4599 L .76679 .44333 L p F P 0 g s 1 .1 .15 r .76679 .44333 m .7366 .4599 L .7336 .43037 L p F P 0 g s 1 .1 .15 r .7336 .43037 m .76332 .41362 L .76679 .44333 L p F P 0 g s 1 .1 .15 r .84421 .44333 m .86638 .4599 L .83523 .47601 L p F P 0 g s 1 .1 .15 r .83523 .47601 m .81294 .4599 L .84421 .44333 L p F P 0 g s 1 .1 .15 r .84421 .44333 m .83966 .41362 L .86158 .43037 L p F P 0 g s 1 .1 .15 r .86158 .43037 m .86638 .4599 L .84421 .44333 L p F P 0 g s 1 .1 .15 r .84421 .44333 m .81294 .4599 L .80888 .43037 L p F P 0 g s 1 .1 .15 r .80888 .43037 m .83966 .41362 L .84421 .44333 L p F P 0 g s 1 1 0 r .34393 .53346 m .3205 .49932 L .36321 .49237 L closepath p F P 0 g s 1 1 0 r .74689 .40237 m .7009 .41003 L .72992 .43048 L closepath p F P 0 g s 1 1 0 r .72879 .44392 m .72992 .43048 L .7009 .41003 L closepath p F P 0 g s 1 1 0 r .72879 .44392 m .74689 .40237 L .72992 .43048 L closepath p F P 0 g s 1 1 0 r .90163 .40237 m .85467 .41003 L .8811 .43048 L closepath p F P 0 g s 1 1 0 r .88418 .44392 m .8811 .43048 L .85467 .41003 L closepath p F P 0 g s 1 1 0 r .88418 .44392 m .90163 .40237 L .8811 .43048 L closepath p F P 0 g s .3 .8 .3 r .45671 .72608 m .48456 .7388 L .45847 .75116 L p F P 0 g s .3 .8 .3 r .45847 .75116 m .43064 .7388 L .45671 .72608 L p F P 0 g s .3 .8 .3 r .42936 .77344 m .43064 .7388 L .45847 .75116 L p F P 0 g s .3 .8 .3 r .45847 .75116 m .4576 .78548 L .42936 .77344 L p F P 0 g s .3 .8 .3 r .48405 .77344 m .4576 .78548 L .45847 .75116 L p F P 0 g s .3 .8 .3 r .45847 .75116 m .48456 .7388 L .48405 .77344 L p F P 0 g s .3 .8 .3 r .80726 .38876 m .83014 .40625 L .79923 .42325 L p F P 0 g s .3 .8 .3 r .79923 .42325 m .77623 .40625 L .80726 .38876 L p F P 0 g s .3 .8 .3 r .77991 .43612 m .77623 .40625 L .79923 .42325 L p F P 0 g s .3 .8 .3 r .79923 .42325 m .80319 .45294 L .77991 .43612 L p F P 0 g s .3 .8 .3 r .8346 .43612 m .80319 .45294 L .79923 .42325 L p F P 0 g s .3 .8 .3 r .79923 .42325 m .83014 .40625 L .8346 .43612 L p F P 0 g s 1 .1 .15 r .26196 .38886 m .29248 .40629 L .26914 .42325 L p F P 0 g s 1 .1 .15 r .26914 .42325 m .23872 .40629 L .26196 .38886 L p F P 0 g s 1 .1 .15 r .2347 .43608 m .23872 .40629 L .26914 .42325 L p F P 0 g s 1 .1 .15 r .26914 .42325 m .2656 .45285 L .2347 .43608 L p F P 0 g s 1 .1 .15 r .28923 .43608 m .2656 .45285 L .26914 .42325 L p F P 0 g s 1 .1 .15 r .26914 .42325 m .29248 .40629 L .28923 .43608 L p F P 0 g s 1 .1 .15 r .33986 .38886 m .36928 .40629 L .34487 .42325 L p F P 0 g s 1 .1 .15 r .34487 .42325 m .31552 .40629 L .33986 .38886 L p F P 0 g s 1 .1 .15 r .3126 .43608 m .31552 .40629 L .34487 .42325 L p F P 0 g s 1 .1 .15 r .34487 .42325 m .3424 .45285 L .3126 .43608 L p F P 0 g s 1 .1 .15 r .36712 .43608 m .3424 .45285 L .34487 .42325 L p F P 0 g s 1 .1 .15 r .34487 .42325 m .36928 .40629 L .36712 .43608 L p F P 0 g s 1 .1 .15 r .41776 .38886 m .44608 .40629 L .4206 .42325 L p F P 0 g s 1 .1 .15 r .4206 .42325 m .39232 .40629 L .41776 .38886 L p F P 0 g s 1 .1 .15 r .39049 .43608 m .39232 .40629 L .4206 .42325 L p F P 0 g s 1 .1 .15 r .4206 .42325 m .4192 .45285 L .39049 .43608 L p F P 0 g s 1 .1 .15 r .44502 .43608 m .4192 .45285 L .4206 .42325 L p F P 0 g s 1 .1 .15 r .4206 .42325 m .44608 .40629 L .44502 .43608 L p F P 0 g s 1 .1 .15 r .49566 .38886 m .52288 .40629 L .49633 .42325 L p F P 0 g s 1 .1 .15 r .49633 .42325 m .46912 .40629 L .49566 .38886 L p F P 0 g s 1 .1 .15 r .46839 .43608 m .46912 .40629 L .49633 .42325 L p F P 0 g s 1 .1 .15 r .49633 .42325 m .496 .45285 L .46839 .43608 L p F P 0 g s 1 .1 .15 r .52292 .43608 m .496 .45285 L .49633 .42325 L p F P 0 g s 1 .1 .15 r .49633 .42325 m .52288 .40629 L .52292 .43608 L p F P 0 g s 1 .1 .15 r .37881 .59124 m .40768 .60582 L .38273 .62 L p F P 0 g s 1 .1 .15 r .38273 .62 m .35392 .60582 L .37881 .59124 L p F P 0 g s 1 .1 .15 r .35154 .63847 m .35392 .60582 L .38273 .62 L p F P 0 g s 1 .1 .15 r .38273 .62 m .3808 .65238 L .35154 .63847 L p F P 0 g s 1 .1 .15 r .40607 .63847 m .3808 .65238 L .38273 .62 L p F P 0 g s 1 .1 .15 r .38273 .62 m .40768 .60582 L .40607 .63847 L p F P 0 g s 1 .1 .15 r .57356 .38886 m .59968 .40629 L .57206 .42325 L p F P 0 g s 1 .1 .15 r .57206 .42325 m .54592 .40629 L .57356 .38886 L p F P 0 g s 1 .1 .15 r .54629 .43608 m .54592 .40629 L .57206 .42325 L p F P 0 g s 1 .1 .15 r .57206 .42325 m .5728 .45285 L .54629 .43608 L p F P 0 g s 1 .1 .15 r .60082 .43608 m .5728 .45285 L .57206 .42325 L p F P 0 g s 1 .1 .15 r .57206 .42325 m .59968 .40629 L .60082 .43608 L p F P 0 g s 1 .1 .15 r .41776 .6587 m .44608 .67233 L .4206 .68558 L p F P 0 g s 1 .1 .15 r .4206 .68558 m .39232 .67233 L .41776 .6587 L p F P 0 g s 1 .1 .15 r .39049 .70593 m .39232 .67233 L .4206 .68558 L p F P 0 g s 1 .1 .15 r .4206 .68558 m .4192 .71889 L .39049 .70593 L p F P 0 g s 1 .1 .15 r .44502 .70593 m .4192 .71889 L .4206 .68558 L p F P 0 g s 1 .1 .15 r .4206 .68558 m .44608 .67233 L .44502 .70593 L p F P 0 g s 1 .1 .15 r .65145 .38886 m .67647 .40629 L .64779 .42325 L p F P 0 g s 1 .1 .15 r .64779 .42325 m .62272 .40629 L .65145 .38886 L p F P 0 g s 1 .1 .15 r .62419 .43608 m .62272 .40629 L .64779 .42325 L p F P 0 g s 1 .1 .15 r .64779 .42325 m .6496 .45285 L .62419 .43608 L p F P 0 g s 1 .1 .15 r .67872 .43608 m .6496 .45285 L .64779 .42325 L p F P 0 g s 1 .1 .15 r .64779 .42325 m .67647 .40629 L .67872 .43608 L p F P 0 g s 1 .1 .15 r .45671 .72617 m .48448 .73884 L .45846 .75117 L p F P 0 g s 1 .1 .15 r .45846 .75117 m .43072 .73884 L .45671 .72617 L p F P 0 g s 1 .1 .15 r .42944 .77339 m .43072 .73884 L .45846 .75117 L p F P 0 g s 1 .1 .15 r .45846 .75117 m .4576 .7854 L .42944 .77339 L p F P 0 g s 1 .1 .15 r .48397 .77339 m .4576 .7854 L .45846 .75117 L p F P 0 g s 1 .1 .15 r .45846 .75117 m .48448 .73884 L .48397 .77339 L p F P 0 g s 1 .1 .15 r .80725 .38886 m .83007 .40629 L .79925 .42325 L p F P 0 g s 1 .1 .15 r .79925 .42325 m .77631 .40629 L .80725 .38886 L p F P 0 g s 1 .1 .15 r .77999 .43608 m .77631 .40629 L .79925 .42325 L p F P 0 g s 1 .1 .15 r .79925 .42325 m .80319 .45285 L .77999 .43608 L p F P 0 g s 1 .1 .15 r .83452 .43608 m .80319 .45285 L .79925 .42325 L p F P 0 g s 1 .1 .15 r .79925 .42325 m .83007 .40629 L .83452 .43608 L p F P 0 g s 1 .1 .15 r .81143 .41882 m .83452 .43608 L .80319 .45285 L p F P 0 g s 1 .1 .15 r .83007 .40629 m .83452 .43608 L .81143 .41882 L p F P 0 g s .3 .8 .3 r .81144 .41881 m .8346 .43612 L .80319 .45294 L p F P 0 g s 1 .1 .15 r .80319 .45285 m .77999 .43608 L .81143 .41882 L p F P 0 g s .3 .8 .3 r .80319 .45294 m .77991 .43612 L .81144 .41881 L p F P 0 g s 1 .1 .15 r .45579 .76103 m .48397 .77339 L .4576 .7854 L p F P 0 g s 1 .1 .15 r .4576 .7854 m .42944 .77339 L .45579 .76103 L p F P 0 g s .3 .8 .3 r .45579 .76104 m .48405 .77344 L .4576 .78548 L p F P 0 g s .3 .8 .3 r .4576 .78548 m .42936 .77344 L .45579 .76104 L p F P 0 g s 1 .1 .15 r .45579 .76103 m .45671 .72617 L .48448 .73884 L p F P 0 g s 1 .1 .15 r .48448 .73884 m .48397 .77339 L .45579 .76103 L p F P 0 g s 1 .1 .15 r .45579 .76103 m .42944 .77339 L .43072 .73884 L p F P 0 g s 1 .1 .15 r .43072 .73884 m .45671 .72617 L .45579 .76103 L p F P 0 g s .3 .8 .3 r .45579 .76104 m .45671 .72608 L .48456 .7388 L p F P 0 g s .3 .8 .3 r .48456 .7388 m .48405 .77344 L .45579 .76104 L p F P 0 g s .3 .8 .3 r .45579 .76104 m .42936 .77344 L .43064 .7388 L p F P 0 g s .3 .8 .3 r .43064 .7388 m .45671 .72608 L .45579 .76104 L p F P 0 g s 1 .1 .15 r .81143 .41882 m .80725 .38886 L .83007 .40629 L p F P 0 g s .3 .8 .3 r .81144 .41881 m .80726 .38876 L .83014 .40625 L p F P 0 g s .3 .8 .3 r .83014 .40625 m .8346 .43612 L .81144 .41881 L p F P 0 g s 1 .1 .15 r .81143 .41882 m .77999 .43608 L .77631 .40629 L p F P 0 g s 1 .1 .15 r .77631 .40629 m .80725 .38886 L .81143 .41882 L p F P 0 g s .3 .8 .3 r .81144 .41881 m .77991 .43612 L .77623 .40625 L p F P 0 g s .3 .8 .3 r .77623 .40625 m .80726 .38876 L .81144 .41881 L p F P 0 g s 1 .1 .15 r .25822 .41882 m .28923 .43608 L .2656 .45285 L p F P 0 g s 1 .1 .15 r .2656 .45285 m .2347 .43608 L .25822 .41882 L p F P 0 g s 1 .1 .15 r .25822 .41882 m .26196 .38886 L .29248 .40629 L p F P 0 g s 1 .1 .15 r .29248 .40629 m .28923 .43608 L .25822 .41882 L p F P 0 g s 1 .1 .15 r .25822 .41882 m .2347 .43608 L .23872 .40629 L p F P 0 g s 1 .1 .15 r .23872 .40629 m .26196 .38886 L .25822 .41882 L p F P 0 g s 1 .1 .15 r .33725 .41882 m .36712 .43608 L .3424 .45285 L p F P 0 g s 1 .1 .15 r .3424 .45285 m .3126 .43608 L .33725 .41882 L p F P 0 g s 1 .1 .15 r .33725 .41882 m .33986 .38886 L .36928 .40629 L p F P 0 g s 1 .1 .15 r .36928 .40629 m .36712 .43608 L .33725 .41882 L p F P 0 g s 1 .1 .15 r .33725 .41882 m .3126 .43608 L .31552 .40629 L p F P 0 g s 1 .1 .15 r .31552 .40629 m .33986 .38886 L .33725 .41882 L p F P 0 g s 1 .1 .15 r .41628 .41882 m .44502 .43608 L .4192 .45285 L p F P 0 g s 1 .1 .15 r .4192 .45285 m .39049 .43608 L .41628 .41882 L p F P 0 g s 1 .1 .15 r .41628 .41882 m .39049 .43608 L .39232 .40629 L p F P 0 g s 1 .1 .15 r .39232 .40629 m .41776 .38886 L .41628 .41882 L p F P 0 g s 1 .1 .15 r .49531 .41882 m .52292 .43608 L .496 .45285 L p F P 0 g s 1 .1 .15 r .496 .45285 m .46839 .43608 L .49531 .41882 L p F P 0 g s 1 .1 .15 r .49531 .41882 m .46839 .43608 L .46912 .40629 L p F P 0 g s 1 .1 .15 r .46912 .40629 m .49566 .38886 L .49531 .41882 L p F P 0 g s 1 .1 .15 r .37676 .62415 m .37881 .59124 L .40768 .60582 L p F P 0 g s 1 .1 .15 r .40768 .60582 m .40607 .63847 L .37676 .62415 L p F P 0 g s 1 .1 .15 r .37676 .62415 m .35154 .63847 L .35392 .60582 L p F P 0 g s 1 .1 .15 r .35392 .60582 m .37881 .59124 L .37676 .62415 L p F P 0 g s 1 .1 .15 r .57434 .41882 m .60082 .43608 L .5728 .45285 L p F P 0 g s 1 .1 .15 r .5728 .45285 m .54629 .43608 L .57434 .41882 L p F P 0 g s 1 .1 .15 r .57434 .41882 m .57356 .38886 L .59968 .40629 L p F P 0 g s 1 .1 .15 r .59968 .40629 m .60082 .43608 L .57434 .41882 L p F P 0 g s 1 .1 .15 r .41628 .69259 m .44502 .70593 L .4192 .71889 L p F P 0 g s 1 .1 .15 r .4192 .71889 m .39049 .70593 L .41628 .69259 L p F P 0 g s 1 .1 .15 r .65337 .41882 m .67872 .43608 L .6496 .45285 L p F P 0 g s 1 .1 .15 r .6496 .45285 m .62419 .43608 L .65337 .41882 L p F P 0 g s 1 .1 .15 r .65337 .41882 m .65145 .38886 L .67647 .40629 L p F P 0 g s 1 .1 .15 r .67647 .40629 m .67872 .43608 L .65337 .41882 L p F P 0 g s 1 .1 .15 r .65337 .41882 m .62419 .43608 L .62272 .40629 L p F P 0 g s 1 .1 .15 r .62272 .40629 m .65145 .38886 L .65337 .41882 L p F P 0 g s 1 .1 .15 r .41628 .41882 m .41776 .38886 L .44608 .40629 L p F P 0 g s 1 .1 .15 r .44608 .40629 m .44502 .43608 L .41628 .41882 L p F P 0 g s 1 .1 .15 r .49531 .41882 m .49566 .38886 L .52288 .40629 L p F P 0 g s 1 .1 .15 r .52288 .40629 m .52292 .43608 L .49531 .41882 L p F P 0 g s 1 .1 .15 r .37676 .62415 m .40607 .63847 L .3808 .65238 L p F P 0 g s 1 .1 .15 r .3808 .65238 m .35154 .63847 L .37676 .62415 L p F P 0 g s 1 .1 .15 r .57434 .41882 m .54629 .43608 L .54592 .40629 L p F P 0 g s 1 .1 .15 r .54592 .40629 m .57356 .38886 L .57434 .41882 L p F P 0 g s 1 .1 .15 r .41628 .69259 m .41776 .6587 L .44608 .67233 L p F P 0 g s 1 .1 .15 r .44608 .67233 m .44502 .70593 L .41628 .69259 L p F P 0 g s 1 .1 .15 r .41628 .69259 m .39049 .70593 L .39232 .67233 L p F P 0 g s 1 .1 .15 r .39232 .67233 m .41776 .6587 L .41628 .69259 L p F P 0 g s 1 1 0 r .72879 .44392 m .7009 .41003 L .74689 .40237 L closepath p F P 0 g s 1 1 0 r .88418 .44392 m .85467 .41003 L .90163 .40237 L closepath p F P 0 g s 1 1 0 r .43569 .719 m .39075 .72479 L .42569 .74025 L closepath p F P 0 g s 1 1 0 r .41591 .76276 m .42569 .74025 L .39075 .72479 L closepath p F P 0 g s 1 1 0 r .41591 .76276 m .43569 .719 L .42569 .74025 L closepath p F P 0 g s .3 .8 .3 r .37586 .63845 m .40545 .65269 L .37996 .66653 L p F P 0 g s .3 .8 .3 r .37996 .66653 m .35043 .65269 L .37586 .63845 L p F P 0 g s .3 .8 .3 r .34795 .6868 m .35043 .65269 L .37996 .66653 L p F P 0 g s .3 .8 .3 r .37996 .66653 m .37794 .70034 L .34795 .6868 L p F P 0 g s .3 .8 .3 r .40378 .6868 m .37794 .70034 L .37996 .66653 L p F P 0 g s .3 .8 .3 r .37996 .66653 m .40545 .65269 L .40378 .6868 L p F P 0 g s 1 .1 .15 r .21682 .36306 m .24861 .38123 L .22542 .39888 L p F P 0 g s 1 .1 .15 r .22542 .39888 m .19375 .38123 L .21682 .36306 L p F P 0 g s 1 .1 .15 r .18898 .41127 m .19375 .38123 L .22542 .39888 L p F P 0 g s 1 .1 .15 r .22542 .39888 m .22118 .42874 L .18898 .41127 L p F P 0 g s 1 .1 .15 r .24465 .41127 m .22118 .42874 L .22542 .39888 L p F P 0 g s 1 .1 .15 r .22542 .39888 m .24861 .38123 L .24465 .41127 L p F P 0 g s 1 .1 .15 r .29634 .36306 m .32699 .38123 L .30269 .39888 L p F P 0 g s 1 .1 .15 r .30269 .39888 m .27213 .38123 L .29634 .36306 L p F P 0 g s 1 .1 .15 r .26851 .41127 m .27213 .38123 L .30269 .39888 L p F P 0 g s 1 .1 .15 r .30269 .39888 m .29956 .42874 L .26851 .41127 L p F P 0 g s 1 .1 .15 r .32417 .41127 m .29956 .42874 L .30269 .39888 L p F P 0 g s 1 .1 .15 r .30269 .39888 m .32699 .38123 L .32417 .41127 L p F P 0 g s 1 .1 .15 r .37587 .36306 m .40537 .38123 L .37995 .39888 L p F P 0 g s 1 .1 .15 r .37995 .39888 m .35051 .38123 L .37587 .36306 L p F P 0 g s 1 .1 .15 r .34803 .41127 m .35051 .38123 L .37995 .39888 L p F P 0 g s 1 .1 .15 r .37995 .39888 m .37794 .42874 L .34803 .41127 L p F P 0 g s 1 .1 .15 r .4037 .41127 m .37794 .42874 L .37995 .39888 L p F P 0 g s 1 .1 .15 r .37995 .39888 m .40537 .38123 L .4037 .41127 L p F P 0 g s 1 .1 .15 r .45539 .36306 m .48375 .38123 L .45722 .39888 L p F P 0 g s 1 .1 .15 r .45722 .39888 m .42889 .38123 L .45539 .36306 L p F P 0 g s 1 .1 .15 r .42756 .41127 m .42889 .38123 L .45722 .39888 L p F P 0 g s 1 .1 .15 r .45722 .39888 m .45632 .42874 L .42756 .41127 L p F P 0 g s 1 .1 .15 r .48322 .41127 m .45632 .42874 L .45722 .39888 L p F P 0 g s 1 .1 .15 r .45722 .39888 m .48375 .38123 L .48322 .41127 L p F P 0 g s 1 .1 .15 r .61444 .36306 m .64051 .38123 L .61175 .39888 L p F P 0 g s 1 .1 .15 r .61175 .39888 m .58564 .38123 L .61444 .36306 L p F P 0 g s 1 .1 .15 r .58661 .41127 m .58564 .38123 L .61175 .39888 L p F P 0 g s 1 .1 .15 r .61175 .39888 m .61308 .42874 L .58661 .41127 L p F P 0 g s 1 .1 .15 r .64227 .41127 m .61308 .42874 L .61175 .39888 L p F P 0 g s 1 .1 .15 r .61175 .39888 m .64051 .38123 L .64227 .41127 L p F P 0 g s 1 .1 .15 r .53492 .36306 m .56213 .38123 L .53449 .39888 L p F P 0 g s 1 .1 .15 r .53449 .39888 m .50726 .38123 L .53492 .36306 L p F P 0 g s 1 .1 .15 r .50708 .41127 m .50726 .38123 L .53449 .39888 L p F P 0 g s 1 .1 .15 r .53449 .39888 m .5347 .42874 L .50708 .41127 L p F P 0 g s 1 .1 .15 r .56275 .41127 m .5347 .42874 L .53449 .39888 L p F P 0 g s 1 .1 .15 r .53449 .39888 m .56213 .38123 L .56275 .41127 L p F P 0 g s 1 .1 .15 r .37587 .63854 m .40537 .65274 L .37995 .66654 L p F P 0 g s 1 .1 .15 r .37995 .66654 m .35051 .65274 L .37587 .63854 L p F P 0 g s 1 .1 .15 r .34803 .68675 m .35051 .65274 L .37995 .66654 L p F P 0 g s 1 .1 .15 r .37995 .66654 m .37794 .70026 L .34803 .68675 L p F P 0 g s 1 .1 .15 r .4037 .68675 m .37794 .70026 L .37995 .66654 L p F P 0 g s 1 .1 .15 r .37995 .66654 m .40537 .65274 L .4037 .68675 L p F P 0 g s 1 .1 .15 r .37373 .67284 m .4037 .68675 L .37794 .70026 L p F P 0 g s 1 .1 .15 r .37794 .70026 m .34803 .68675 L .37373 .67284 L p F P 0 g s .3 .8 .3 r .37372 .67285 m .40378 .6868 L .37794 .70034 L p F P 0 g s .3 .8 .3 r .37794 .70034 m .34795 .6868 L .37372 .67285 L p F P 0 g s 1 .1 .15 r .37373 .67284 m .37587 .63854 L .40537 .65274 L p F P 0 g s 1 .1 .15 r .40537 .65274 m .4037 .68675 L .37373 .67284 L p F P 0 g s 1 .1 .15 r .37373 .67284 m .34803 .68675 L .35051 .65274 L p F P 0 g s 1 .1 .15 r .35051 .65274 m .37587 .63854 L .37373 .67284 L p F P 0 g s .3 .8 .3 r .37372 .67285 m .37586 .63845 L .40545 .65269 L p F P 0 g s .3 .8 .3 r .40545 .65269 m .40378 .6868 L .37372 .67285 L p F P 0 g s .3 .8 .3 r .37372 .67285 m .34795 .6868 L .35043 .65269 L p F P 0 g s .3 .8 .3 r .35043 .65269 m .37586 .63845 L .37372 .67285 L p F P 0 g s 1 .1 .15 r .21232 .39327 m .24465 .41127 L .22118 .42874 L p F P 0 g s 1 .1 .15 r .22118 .42874 m .18898 .41127 L .21232 .39327 L p F P 0 g s 1 .1 .15 r .21232 .39327 m .21682 .36306 L .24861 .38123 L p F P 0 g s 1 .1 .15 r .24861 .38123 m .24465 .41127 L .21232 .39327 L p F P 0 g s 1 .1 .15 r .29303 .39327 m .32417 .41127 L .29956 .42874 L p F P 0 g s 1 .1 .15 r .29956 .42874 m .26851 .41127 L .29303 .39327 L p F P 0 g s 1 .1 .15 r .29303 .39327 m .26851 .41127 L .27213 .38123 L p F P 0 g s 1 .1 .15 r .27213 .38123 m .29634 .36306 L .29303 .39327 L p F P 0 g s 1 .1 .15 r .37373 .39327 m .4037 .41127 L .37794 .42874 L p F P 0 g s 1 .1 .15 r .37794 .42874 m .34803 .41127 L .37373 .39327 L p F P 0 g s 1 .1 .15 r .45444 .39327 m .48322 .41127 L .45632 .42874 L p F P 0 g s 1 .1 .15 r .45632 .42874 m .42756 .41127 L .45444 .39327 L p F P 0 g s 1 .1 .15 r .45444 .39327 m .42756 .41127 L .42889 .38123 L p F P 0 g s 1 .1 .15 r .42889 .38123 m .45539 .36306 L .45444 .39327 L p F P 0 g s 1 .1 .15 r .61584 .39327 m .64227 .41127 L .61308 .42874 L p F P 0 g s 1 .1 .15 r .61308 .42874 m .58661 .41127 L .61584 .39327 L p F P 0 g s 1 .1 .15 r .21232 .39327 m .18898 .41127 L .19375 .38123 L p F P 0 g s 1 .1 .15 r .19375 .38123 m .21682 .36306 L .21232 .39327 L p F P 0 g s 1 .1 .15 r .29303 .39327 m .29634 .36306 L .32699 .38123 L p F P 0 g s 1 .1 .15 r .32699 .38123 m .32417 .41127 L .29303 .39327 L p F P 0 g s 1 .1 .15 r .37373 .39327 m .37587 .36306 L .40537 .38123 L p F P 0 g s 1 .1 .15 r .40537 .38123 m .4037 .41127 L .37373 .39327 L p F P 0 g s 1 .1 .15 r .37373 .39327 m .34803 .41127 L .35051 .38123 L p F P 0 g s 1 .1 .15 r .35051 .38123 m .37587 .36306 L .37373 .39327 L p F P 0 g s 1 .1 .15 r .45444 .39327 m .45539 .36306 L .48375 .38123 L p F P 0 g s 1 .1 .15 r .48375 .38123 m .48322 .41127 L .45444 .39327 L p F P 0 g s 1 .1 .15 r .53514 .39327 m .56275 .41127 L .5347 .42874 L p F P 0 g s 1 .1 .15 r .5347 .42874 m .50708 .41127 L .53514 .39327 L p F P 0 g s 1 .1 .15 r .53514 .39327 m .53492 .36306 L .56213 .38123 L p F P 0 g s 1 .1 .15 r .56213 .38123 m .56275 .41127 L .53514 .39327 L p F P 0 g s 1 .1 .15 r .53514 .39327 m .50708 .41127 L .50726 .38123 L p F P 0 g s 1 .1 .15 r .50726 .38123 m .53492 .36306 L .53514 .39327 L p F P 0 g s 1 .1 .15 r .61584 .39327 m .61444 .36306 L .64051 .38123 L p F P 0 g s 1 .1 .15 r .64051 .38123 m .64227 .41127 L .61584 .39327 L p F P 0 g s 1 .1 .15 r .61584 .39327 m .58661 .41127 L .58564 .38123 L p F P 0 g s 1 .1 .15 r .58564 .38123 m .61444 .36306 L .61584 .39327 L p F P 0 g s 1 1 0 r .41591 .76276 m .39075 .72479 L .43569 .719 L closepath p F P 0 g s 1 .1 .15 r .25097 .33616 m .28293 .35511 L .25876 .37351 L p F P 0 g s 1 .1 .15 r .25876 .37351 m .22691 .35511 L .25097 .33616 L p F P 0 g s 1 .1 .15 r .22254 .3854 m .22691 .35511 L .25876 .37351 L p F P 0 g s 1 .1 .15 r .25876 .37351 m .25492 .40362 L .22254 .3854 L p F P 0 g s 1 .1 .15 r .27939 .3854 m .25492 .40362 L .25876 .37351 L p F P 0 g s 1 .1 .15 r .25876 .37351 m .28293 .35511 L .27939 .3854 L p F P 0 g s 1 .1 .15 r .41341 .33616 m .44298 .35511 L .41649 .37351 L p F P 0 g s 1 .1 .15 r .41649 .37351 m .38696 .35511 L .41341 .33616 L p F P 0 g s 1 .1 .15 r .38498 .3854 m .38696 .35511 L .41649 .37351 L p F P 0 g s 1 .1 .15 r .41649 .37351 m .41497 .40362 L .38498 .3854 L p F P 0 g s 1 .1 .15 r .44183 .3854 m .41497 .40362 L .41649 .37351 L p F P 0 g s 1 .1 .15 r .41649 .37351 m .44298 .35511 L .44183 .3854 L p F P 0 g s 1 .1 .15 r .16974 .33616 m .2029 .35511 L .17989 .37351 L p F P 0 g s 1 .1 .15 r .17989 .37351 m .14688 .35511 L .16974 .33616 L p F P 0 g s 1 .1 .15 r .14132 .3854 m .14688 .35511 L .17989 .37351 L p F P 0 g s 1 .1 .15 r .17989 .37351 m .17489 .40362 L .14132 .3854 L p F P 0 g s 1 .1 .15 r .19817 .3854 m .17489 .40362 L .17989 .37351 L p F P 0 g s 1 .1 .15 r .17989 .37351 m .2029 .35511 L .19817 .3854 L p F P 0 g s 1 .1 .15 r .33219 .33616 m .36295 .35511 L .33762 .37351 L p F P 0 g s 1 .1 .15 r .33762 .37351 m .30694 .35511 L .33219 .33616 L p F P 0 g s 1 .1 .15 r .30376 .3854 m .30694 .35511 L .33762 .37351 L p F P 0 g s 1 .1 .15 r .33762 .37351 m .33495 .40362 L .30376 .3854 L p F P 0 g s 1 .1 .15 r .36061 .3854 m .33495 .40362 L .33762 .37351 L p F P 0 g s 1 .1 .15 r .33762 .37351 m .36295 .35511 L .36061 .3854 L p F P 0 g s 1 .1 .15 r .49463 .33616 m .52301 .35511 L .49536 .37351 L p F P 0 g s 1 .1 .15 r .49536 .37351 m .46699 .35511 L .49463 .33616 L p F P 0 g s 1 .1 .15 r .4662 .3854 m .46699 .35511 L .49536 .37351 L p F P 0 g s 1 .1 .15 r .49536 .37351 m .495 .40362 L .4662 .3854 L p F P 0 g s 1 .1 .15 r .52306 .3854 m .495 .40362 L .49536 .37351 L p F P 0 g s 1 .1 .15 r .49536 .37351 m .52301 .35511 L .52306 .3854 L p F P 0 g s 1 .1 .15 r .57585 .33616 m .60303 .35511 L .57422 .37351 L p F P 0 g s 1 .1 .15 r .57422 .37351 m .54702 .35511 L .57585 .33616 L p F P 0 g s 1 .1 .15 r .54742 .3854 m .54702 .35511 L .57422 .37351 L p F P 0 g s 1 .1 .15 r .57422 .37351 m .57502 .40362 L .54742 .3854 L p F P 0 g s 1 .1 .15 r .60428 .3854 m .57502 .40362 L .57422 .37351 L p F P 0 g s 1 .1 .15 r .57422 .37351 m .60303 .35511 L .60428 .3854 L p F P 0 g s 1 .1 .15 r .24689 .36662 m .27939 .3854 L .25492 .40362 L p F P 0 g s 1 .1 .15 r .25492 .40362 m .22254 .3854 L .24689 .36662 L p F P 0 g s 1 .1 .15 r .41179 .36662 m .44183 .3854 L .41497 .40362 L p F P 0 g s 1 .1 .15 r .41497 .40362 m .38498 .3854 L .41179 .36662 L p F P 0 g s 1 .1 .15 r .16444 .36662 m .19817 .3854 L .17489 .40362 L p F P 0 g s 1 .1 .15 r .17489 .40362 m .14132 .3854 L .16444 .36662 L p F P 0 g s 1 .1 .15 r .16444 .36662 m .16974 .33616 L .2029 .35511 L p F P 0 g s 1 .1 .15 r .2029 .35511 m .19817 .3854 L .16444 .36662 L p F P 0 g s 1 .1 .15 r .16444 .36662 m .14132 .3854 L .14688 .35511 L p F P 0 g s 1 .1 .15 r .14688 .35511 m .16974 .33616 L .16444 .36662 L p F P 0 g s 1 .1 .15 r .24689 .36662 m .25097 .33616 L .28293 .35511 L p F P 0 g s 1 .1 .15 r .28293 .35511 m .27939 .3854 L .24689 .36662 L p F P 0 g s 1 .1 .15 r .24689 .36662 m .22254 .3854 L .22691 .35511 L p F P 0 g s 1 .1 .15 r .22691 .35511 m .25097 .33616 L .24689 .36662 L p F P 0 g s 1 .1 .15 r .32934 .36662 m .36061 .3854 L .33495 .40362 L p F P 0 g s 1 .1 .15 r .33495 .40362 m .30376 .3854 L .32934 .36662 L p F P 0 g s 1 .1 .15 r .32934 .36662 m .33219 .33616 L .36295 .35511 L p F P 0 g s 1 .1 .15 r .36295 .35511 m .36061 .3854 L .32934 .36662 L p F P 0 g s 1 .1 .15 r .32934 .36662 m .30376 .3854 L .30694 .35511 L p F P 0 g s 1 .1 .15 r .30694 .35511 m .33219 .33616 L .32934 .36662 L p F P 0 g s 1 .1 .15 r .41179 .36662 m .41341 .33616 L .44298 .35511 L p F P 0 g s 1 .1 .15 r .44298 .35511 m .44183 .3854 L .41179 .36662 L p F P 0 g s 1 .1 .15 r .41179 .36662 m .38498 .3854 L .38696 .35511 L p F P 0 g s 1 .1 .15 r .38696 .35511 m .41341 .33616 L .41179 .36662 L p F P 0 g s 1 .1 .15 r .49425 .36662 m .52306 .3854 L .495 .40362 L p F P 0 g s 1 .1 .15 r .495 .40362 m .4662 .3854 L .49425 .36662 L p F P 0 g s 1 .1 .15 r .49425 .36662 m .49463 .33616 L .52301 .35511 L p F P 0 g s 1 .1 .15 r .52301 .35511 m .52306 .3854 L .49425 .36662 L p F P 0 g s 1 .1 .15 r .49425 .36662 m .4662 .3854 L .46699 .35511 L p F P 0 g s 1 .1 .15 r .46699 .35511 m .49463 .33616 L .49425 .36662 L p F P 0 g s 1 .1 .15 r .5767 .36662 m .60428 .3854 L .57502 .40362 L p F P 0 g s 1 .1 .15 r .57502 .40362 m .54742 .3854 L .5767 .36662 L p F P 0 g s 1 .1 .15 r .5767 .36662 m .57585 .33616 L .60303 .35511 L p F P 0 g s 1 .1 .15 r .60303 .35511 m .60428 .3854 L .5767 .36662 L p F P 0 g s 1 .1 .15 r .5767 .36662 m .54742 .3854 L .54702 .35511 L p F P 0 g s 1 .1 .15 r .54702 .35511 m .57585 .33616 L .5767 .36662 L p F P 0 g s 1 1 0 r .47325 .32306 m .42613 .33175 L .46191 .35489 L closepath p F P 0 g s 1 1 0 r .45277 .36698 m .46191 .35489 L .42613 .33175 L closepath p F P 0 g s 1 1 0 r .45277 .36698 m .47325 .32306 L .46191 .35489 L closepath p F P 0 g s 1 1 0 r .14368 .32306 m .09876 .33175 L .1404 .35489 L closepath p F P 0 g s 1 1 0 r .12175 .36698 m .1404 .35489 L .09876 .33175 L closepath p F P 0 g s 1 1 0 r .12175 .36698 m .14368 .32306 L .1404 .35489 L closepath p F P 0 g s .3 .8 .3 r .53558 .30799 m .56403 .32782 L .53511 .34707 L p F P 0 g s .3 .8 .3 r .53511 .34707 m .50665 .32782 L .53558 .30799 L p F P 0 g s .3 .8 .3 r .50645 .35845 m .50665 .32782 L .53511 .34707 L p F P 0 g s .3 .8 .3 r .53511 .34707 m .53534 .37752 L .50645 .35845 L p F P 0 g s .3 .8 .3 r .56471 .35845 m .53534 .37752 L .53511 .34707 L p F P 0 g s .3 .8 .3 r .53511 .34707 m .56403 .32782 L .56471 .35845 L p F P 0 g s 1 .1 .15 r .20361 .30809 m .23697 .32787 L .21297 .34706 L p F P 0 g s 1 .1 .15 r .21297 .34706 m .17975 .32787 L .20361 .30809 L p F P 0 g s 1 .1 .15 r .17457 .3584 m .17975 .32787 L .21297 .34706 L p F P 0 g s 1 .1 .15 r .21297 .34706 m .20836 .37742 L .17457 .3584 L p F P 0 g s 1 .1 .15 r .23266 .3584 m .20836 .37742 L .21297 .34706 L p F P 0 g s 1 .1 .15 r .21297 .34706 m .23697 .32787 L .23266 .3584 L p F P 0 g s 1 .1 .15 r .2866 .30809 m .31872 .32787 L .29351 .34706 L p F P 0 g s 1 .1 .15 r .29351 .34706 m .2615 .32787 L .2866 .30809 L p F P 0 g s 1 .1 .15 r .25756 .3584 m .2615 .32787 L .29351 .34706 L p F P 0 g s 1 .1 .15 r .29351 .34706 m .29011 .37742 L .25756 .3584 L p F P 0 g s 1 .1 .15 r .31565 .3584 m .29011 .37742 L .29351 .34706 L p F P 0 g s 1 .1 .15 r .29351 .34706 m .31872 .32787 L .31565 .3584 L p F P 0 g s 1 .1 .15 r .36959 .30809 m .40046 .32787 L .37404 .34706 L p F P 0 g s 1 .1 .15 r .37404 .34706 m .34324 .32787 L .36959 .30809 L p F P 0 g s 1 .1 .15 r .34055 .3584 m .34324 .32787 L .37404 .34706 L p F P 0 g s 1 .1 .15 r .37404 .34706 m .37185 .37742 L .34055 .3584 L p F P 0 g s 1 .1 .15 r .39864 .3584 m .37185 .37742 L .37404 .34706 L p F P 0 g s 1 .1 .15 r .37404 .34706 m .40046 .32787 L .39864 .3584 L p F P 0 g s 1 .1 .15 r .53558 .30809 m .56395 .32787 L .53511 .34706 L p F P 0 g s 1 .1 .15 r .53511 .34706 m .50673 .32787 L .53558 .30809 L p F P 0 g s 1 .1 .15 r .50653 .3584 m .50673 .32787 L .53511 .34706 L p F P 0 g s 1 .1 .15 r .53511 .34706 m .53534 .37742 L .50653 .3584 L p F P 0 g s 1 .1 .15 r .56462 .3584 m .53534 .37742 L .53511 .34706 L p F P 0 g s 1 .1 .15 r .53511 .34706 m .56395 .32787 L .56462 .3584 L p F P 0 g s 1 .1 .15 r .53582 .33879 m .56462 .3584 L .53534 .37742 L p F P 0 g s 1 .1 .15 r .53582 .33879 m .53558 .30809 L .56395 .32787 L p F P 0 g s 1 .1 .15 r .56395 .32787 m .56462 .3584 L .53582 .33879 L p F P 0 g s .3 .8 .3 r .53582 .33878 m .56471 .35845 L .53534 .37752 L p F P 0 g s 1 .1 .15 r .53534 .37742 m .50653 .3584 L .53582 .33879 L p F P 0 g s 1 .1 .15 r .53582 .33879 m .50653 .3584 L .50673 .32787 L p F P 0 g s 1 .1 .15 r .50673 .32787 m .53558 .30809 L .53582 .33879 L p F P 0 g s .3 .8 .3 r .53534 .37752 m .50645 .35845 L .53582 .33878 L p F P 0 g s .3 .8 .3 r .53582 .33878 m .53558 .30799 L .56403 .32782 L p F P 0 g s .3 .8 .3 r .56403 .32782 m .56471 .35845 L .53582 .33878 L p F P 0 g s .3 .8 .3 r .53582 .33878 m .50645 .35845 L .50665 .32782 L p F P 0 g s .3 .8 .3 r .50665 .32782 m .53558 .30799 L .53582 .33878 L p F P 0 g s 1 .1 .15 r .19871 .33879 m .23266 .3584 L .20836 .37742 L p F P 0 g s 1 .1 .15 r .20836 .37742 m .17457 .3584 L .19871 .33879 L p F P 0 g s 1 .1 .15 r .19871 .33879 m .17457 .3584 L .17975 .32787 L p F P 0 g s 1 .1 .15 r .17975 .32787 m .20361 .30809 L .19871 .33879 L p F P 0 g s 1 .1 .15 r .28299 .33879 m .31565 .3584 L .29011 .37742 L p F P 0 g s 1 .1 .15 r .29011 .37742 m .25756 .3584 L .28299 .33879 L p F P 0 g s 1 .1 .15 r .28299 .33879 m .2866 .30809 L .31872 .32787 L p F P 0 g s 1 .1 .15 r .31872 .32787 m .31565 .3584 L .28299 .33879 L p F P 0 g s 1 .1 .15 r .36727 .33879 m .39864 .3584 L .37185 .37742 L p F P 0 g s 1 .1 .15 r .37185 .37742 m .34055 .3584 L .36727 .33879 L p F P 0 g s 1 .1 .15 r .36727 .33879 m .34055 .3584 L .34324 .32787 L p F P 0 g s 1 .1 .15 r .34324 .32787 m .36959 .30809 L .36727 .33879 L p F P 0 g s 1 .1 .15 r .19871 .33879 m .20361 .30809 L .23697 .32787 L p F P 0 g s 1 .1 .15 r .23697 .32787 m .23266 .3584 L .19871 .33879 L p F P 0 g s 1 .1 .15 r .28299 .33879 m .25756 .3584 L .2615 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containing elbows, then use a method David and Stan devised \ to go from elbows to corners. We finish when we have the furthest corner. \ While pathological cases (too many elbows) arise even at ", Cell[BoxData[ FormBox[ RowBox[{"n", "=", "4"}], TraditionalForm]]], ", the average case performance (e.g. random examples) is quite nice. Some \ of the ILP ideas appear in work by Aardal, Hurkins and Lenstra with \ subsequent refinement by Aardal and Lenstra. I also had techniques similar to \ those in AHL which I used to find large examples of what are known Keith \ numbers." }], "Text"], Cell["\<\ Getting back to Frobenius numbers, it turns out that one can use toric Gr\ \[ODoubleDot]bner bases to find what are called \"protoelbows\". These are \ lattice points where positive combinations of one subset equal positive \ combinations of another. From these we get a superset of the elbows, and we \ use a domination algorithm of Bentley, Clarkson, and Levine to get the \ \"kernel\" set which comprise the actual elbows. The hardest step, \ algorithmically, is in coming up with the protoelbows. This can be done with \ a toric Gr\[ODoubleDot]bner basis.\ \>", "Text"], Cell[TextData[{ "As with instance solving, the idea is again to set up relations (that is, \ generating polynomials for a toric ideal) of the form ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["x", "j"], "-", SuperscriptBox["t", SubscriptBox["a", "j"]]}], TraditionalForm]]], ", and eliminate t. 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Unlike \ most applications of Gr\[ODoubleDot]bner bases involving finite varieties, \ toric and otherwise, what is paramount in this case is the actual staircase \ for the \"normal set\" with respect to a certain monomial order. The fact \ that Gr\[ODoubleDot]bner bases can be used at all to find Frobenius numbers, \ and the fact that we use them to gain staircase information, both seem to be \ of interest.\ \>", "Text"], Cell["\<\ It is also of interest that this approach can handle Frobenius number \ problems that were not amenable to ANY method know as recently as a year or \ two ago.\ \>", "Text"], Cell[TextData[{ "The more efficient methods in our work in preparation heavily involve \ integer linear programming. Much of this has been incorporated into the ", StyleBox["Mathematica", FontSlant->"Italic"], " kernel, in the functions ", StyleBox["Reduce", "Program"], ", ", StyleBox["FindInstance", "Program"], ", and ", StyleBox["Minimize", "Program"], "/", StyleBox["Maximize", "Program"], "." }], "Text"], Cell["\<\ Open questions: (1) Would dedicated toric basis code do better? Probably so. How much better? (2) Are there better ways to do the term ordering so that we can still get \ adequate information about the fundamental domain, but faster?\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[Cell[BoxData[GridBox[{ { ButtonBox[ StyleBox["\[FirstPage]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageFirst"]}]& ), ButtonNote->"First Slide"], ButtonBox[ StyleBox["\[LeftPointer]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPagePrevious"]}]& ), ButtonNote->"Previous Slide"], ButtonBox[ StyleBox["\[RightPointer]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageNext"]}]& ), ButtonNote->"Next Slide"], ButtonBox[ StyleBox["\[LastPage]", "SR"], ButtonFunction:>(FrontEndExecute[{ FrontEndToken[ FrontEnd`SelectedNotebook[], "ScrollPageLast"]}]& )], " ", ButtonBox[ StyleBox[ RowBox[{ CounterBox["SlideShowNavigationBar"], " ", "of", " ", CounterBox["SlideShowNavigationBar", {None, "SlideShowHeader", -1}]}], "SR"], Appearance->{Automatic, None}]} }]]]], "SlideShowNavigationBar", CellTags->"SlideShowHeader"], Cell[CellGroupData[{ Cell["References", "Section"], Cell[TextData[{ "K. Aardal, C. A. J. Hurkens, and A. K. Lenstra. Solving a system of linear \ diophantine equations with lower and upper bounds on the variables. \ Mathematics of Operations Research ", StyleBox["25", FontWeight->"Bold"], ":427\[Hyphen]442, 2000." }], "SmallText", PageBreakAbove->Automatic, FontSize->18], Cell["\<\ K. Aardal and A. K. Lenstra. Hard equality constrained knapsacks. Proceedings \ of the 9th Conference on Integer Programming and Combinatorial Optimization \ (IPCO 2002), W. J. Cook and A. S. Schulz, eds. Lecture Notes in Computer \ Science 233, 350\[Hyphen]366. Springer\[Hyphen]Verlag, 2002.\ \>", "SmallText", PageBreakAbove->Automatic, FontSize->18], Cell["\<\ D. Beihoffer, J. Hendry, A. Nijenhuis, and S. Wagon. Faster algorithms for \ Frobenius numbers. The Electronic Journal of Combinatorics 12. 2005.\ \>", "SmallText", FontSize->18], Cell["\<\ J. L. Bentley, K. L. Clarkson, and D. B. Levine. Fast linear .expected\ \[Hyphen]time algorithms for computing maxima and convex hulls. Algorithmica \ 9(2): 168-183. 1993.\ \>", "SmallText", FontSize->18], Cell["\<\ P. Conti and C. Traverso. Gr\[ODoubleDot]bner bases and integer programming. \ Proceedings of AAECC-9. Springer-Verlag LNCS 539 130-139. 1991.\ \>", "SmallText", FontSize->18], Cell["\<\ J. L. Davison. On the linear Diophantine problem of Frobenius. J. Number \ Theory 48 353\[Hyphen]363. 1994.\ \>", "SmallText", FontSize->18], Cell["\<\ D. Einstein, D. Lichtblau, A. Strzebonski, and S. Wagon. Frobenius numbers by \ lattice enumeration. In preparation. 2005.\ \>", "SmallText", FontSize->18], Cell["\<\ H. Greenberg. Solution to a linear Diophantine equation for nonnegative \ integers. J. Algorithms 9 353\[Hyphen]363. 1988.\ \>", "SmallText", FontSize->18], Cell["\<\ M. Keith. (Web page discussing Keith numbers) http://users.aol.com/s6sj7gt/mikekeit.htm\ \>", "SmallText", FontSize->18], Cell["\<\ D. Lichtblau. Solving knapsack and related problems. Manuscript.\ \>", "SmallText", FontSize->18], Cell["\<\ L. Pottier. Gr\[ODoubleDot]bner bases of toric ideals. INRIA Rapport de \ recherche 2224. 1994.\ \>", "SmallText", FontSize->18] }, Open ]] }, Open ]] }, ScreenStyleEnvironment->"SlideShow", PrintingStyleEnvironment->"Presentation", WindowSize->{873, 657}, WindowMargins->{{2, Automatic}, {Automatic, 2}}, DockedCells->(FrontEndExecute[{ FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, Placeholder]}]& ), ScrollingOptions->{"PagewiseDisplay"->True}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"Magnification"->1, "PaperOrientation"->"Portrait", "PaperSize"->{594.25, 840.5}, "PostScriptOutputFile":>FrontEnd`FileName[{$RootDirectory, "home", "usr2", "danl", "Notebooks"}, "ACA2005_Frobenius_slides.ps", CharacterEncoding -> "iso8859-1"], "PrintCellBrackets"->False, "PrintMultipleHorizontalPages"->False, "PrintRegistrationMarks"->False, "PrintingMargins"->{{43.1875, 43.1875}, {57.5625, 64.75}}}, ShowSelection->True, FrontEndVersion->"7.0 for Mac OS X x86 (32-bit) (November 11, 2008)", StyleDefinitions->Notebook[{ Cell[ CellGroupData[{ Cell["Generic Conference StyleSheet", "Title"], Cell[ "Modify the definitions below to change the default appearance of all \ cells in a given style. 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Inherited}}, ImageMargins -> {{30, Inherited}, {Inherited, 0}}, Magnification -> 0.8]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["CellLabel"], LanguageCategory -> None, StyleMenuListing -> None, FontFamily -> "Helvetica", FontSize -> 9, FontColor -> RGBColor[0.392157, 0.396078, 0.717647]], Cell[ StyleData["CellLabel", "Presentation"], FontSize -> 12], Cell[ StyleData["CellLabel", "SlideShow"], FontSize -> 12], Cell[ StyleData["CellLabel", "Printout"], FontFamily -> "Courier", FontSize -> 8, FontSlant -> "Italic", FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["FrameLabel"], LanguageCategory -> None, StyleMenuListing -> None, FontFamily -> "Helvetica", FontSize -> 9], Cell[ StyleData["FrameLabel", "Presentation"], FontSize -> 12], Cell[ StyleData["FrameLabel", "SlideShow"], FontSize -> 12], Cell[ StyleData["FrameLabel", "Printout"], FontFamily -> "Courier", FontSize -> 8, FontSlant -> "Italic", FontColor -> GrayLevel[0]]}, Closed]]}, Open]], Cell[ CellGroupData[{ Cell["Presentation Specific", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["Author"], CellMargins -> {{100, 27}, {2, 20}}, FontFamily -> "Times", FontSize -> 24, FontSlant -> "Italic"], Cell[ StyleData["Author", "Presentation"], CellMargins -> {{200, 27}, {2, 50}}, FontSize -> 28], Cell[ StyleData["Author", "SlideShow"], CellMargins -> {{162, 27}, {2, 50}}, FontSize -> 28], Cell[ StyleData["Author", "Printout"], CellMargins -> {{100, 27}, {2, 20}}, FontSize -> 24]}, Open]], Cell[ CellGroupData[{ Cell[ StyleData["Affiliation"], CellMargins -> {{100, 27}, {30, 12}}, FontFamily -> "Times", FontSize -> 24, FontSlant -> "Italic"], Cell[ StyleData["Affiliation", "Presentation"], CellMargins -> {{200, 27}, {2, 10}}, FontSize -> 28], Cell[ StyleData["Affiliation", "SlideShow"], CellMargins -> {{162, 27}, {2, 10}}, FontSize -> 28], Cell[ StyleData["Affiliation", "Printout"], CellMargins -> {{100, 27}, {2, 12}}, FontSize -> 24]}, Open]]}, Closed]], Cell[ CellGroupData[{ Cell["Header Graphic", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["ConferenceGraphicCell"], ShowCellBracket -> False, CellMargins -> {{0, 0}, {0, 0}}, Evaluatable -> False, PageBreakBelow -> False, ImageMargins -> {{0, 0}, {0, 0}}, ImageRegion -> {{0, 1}, {0, 1}}, Magnification -> 1, Background -> GrayLevel[1]], Cell[ StyleData["ConferenceGraphicCell", "Presentation"]], Cell[ StyleData["ConferenceGraphicCell", "SlideShow"]], Cell[ StyleData["ConferenceGraphicCell", "Printout"], FontSize -> 8]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["Inline Formatting", "Section"], Cell[ "These styles are for modifying individual words or letters in a \ cell exclusive of the cell tag.", "Text"], Cell[ StyleData["RM"], StyleMenuListing -> None, FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["BF"], StyleMenuListing -> None, FontWeight -> "Bold"], Cell[ StyleData["IT"], StyleMenuListing -> None, FontSlant -> "Italic"], Cell[ StyleData["TR"], StyleMenuListing -> None, FontFamily -> "Times", FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["TI"], StyleMenuListing -> None, FontFamily -> "Times", FontWeight -> "Plain", FontSlant -> "Italic"], Cell[ StyleData["TB"], StyleMenuListing -> None, FontFamily -> "Times", FontWeight -> "Bold", FontSlant -> "Plain"], Cell[ StyleData["TBI"], StyleMenuListing -> None, FontFamily -> "Times", FontWeight -> "Bold", FontSlant -> "Italic"], Cell[ StyleData["MR"], "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> {"HyphenationCharacter" -> "\[Continuation]"}, StyleMenuListing -> None, FontFamily -> "Courier", FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["MO"], "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> {"HyphenationCharacter" -> "\[Continuation]"}, StyleMenuListing -> None, FontFamily -> "Courier", FontWeight -> "Plain", FontSlant -> "Italic"], Cell[ StyleData["MB"], "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> {"HyphenationCharacter" -> "\[Continuation]"}, StyleMenuListing -> None, FontFamily -> "Courier", FontWeight -> "Bold", FontSlant -> "Plain"], Cell[ StyleData["MBO"], "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> {"HyphenationCharacter" -> "\[Continuation]"}, StyleMenuListing -> None, FontFamily -> "Courier", FontWeight -> "Bold", FontSlant -> "Italic"], Cell[ StyleData["SR"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontWeight -> "Plain", FontSlant -> "Plain"], Cell[ StyleData["SO"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontWeight -> "Plain", FontSlant -> "Italic"], Cell[ StyleData["SB"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontWeight -> "Bold", FontSlant -> "Plain"], Cell[ StyleData["SBO"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontWeight -> "Bold", FontSlant -> "Italic"], Cell[ CellGroupData[{ Cell[ StyleData["SO10"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontSize -> 10, FontWeight -> "Plain", FontSlant -> "Italic"], Cell[ StyleData["SO10", "Printout"], StyleMenuListing -> None, FontFamily -> "Helvetica", FontSize -> 7, FontWeight -> "Plain", FontSlant -> "Italic"]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["Formulas and Programming", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["InlineFormula"], CellMargins -> {{10, 4}, {0, 8}}, CellHorizontalScrolling -> True, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, LanguageCategory -> "Formula", ScriptLevel -> 1, SingleLetterItalics -> True, FontSize -> 16], Cell[ StyleData["InlineFormula", "Presentation"], LineSpacing -> {1, 5}, FontSize -> 16], Cell[ StyleData["InlineFormula", "SlideShow"], LineSpacing -> {1, 5}, FontSize -> 16], Cell[ StyleData["InlineFormula", "Printout"], CellMargins -> {{2, 0}, {6, 6}}, FontSize -> 16]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["DisplayFormula"], CellMargins -> {{60, Inherited}, {Inherited, Inherited}}, CellHorizontalScrolling -> True, DefaultFormatType -> DefaultInputFormatType, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, LanguageCategory -> "Formula", ScriptLevel -> 0, SingleLetterItalics -> True, FontSize -> 16, UnderoverscriptBoxOptions -> {LimitsPositioning -> True}], Cell[ StyleData["DisplayFormula", "Presentation"], LineSpacing -> {1, 5}, FontSize -> 16], Cell[ StyleData["DisplayFormula", "SlideShow"], CellMargins -> {{100, 50}, {Inherited, Inherited}}, LineSpacing -> {1, 5}, FontSize -> 16], Cell[ StyleData["DisplayFormula", "Printout"], FontSize -> 16]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Program"], CellFrame -> {{0, 0}, {0.5, 0.5}}, CellMargins -> {{60, 4}, {0, 8}}, CellHorizontalScrolling -> True, Hyphenation -> False, LanguageCategory -> "Formula", ScriptLevel -> 1, FontFamily -> "Courier", FontSize -> 22], Cell[ StyleData["Program", "Presentation"], CellMargins -> {{24, 50}, {10, 10}}, LineSpacing -> {1, 5}, FontSize -> 22], Cell[ StyleData["Program", "SlideShow"], CellMargins -> {{100, 50}, {10, 10}}, LineSpacing -> {1, 5}, FontSize -> 22], Cell[ StyleData["Program", "Printout"], CellMargins -> {{2, 0}, {6, 6}}, FontSize -> 20]}, Open]]}, Open]], Cell[ CellGroupData[{ Cell["Outline Styles", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["Outline1"], CellDingbat -> "\[EmptyDiamond]", CellMargins -> {{60, 10}, {7, 7}}, CellGroupingRules -> {"SectionGrouping", 50}, ParagraphIndent -> -38, CounterIncrements -> "Outline1", CounterAssignments -> {{"Outline2", 0}, {"Outline3", 0}, { "Outline4", 0}}, FontSize -> 18, FontWeight -> "Plain", CounterBoxOptions -> {CounterFunction :> CapitalRomanNumeral}], Cell[ StyleData["Outline1", "SlideShow"]], Cell[ StyleData["Outline1", "Printout"], CounterBoxOptions -> {CounterFunction :> CapitalRomanNumeral}]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Outline2"], CellDingbat -> "\[EmptyDiamond]", CellMargins -> {{90, 10}, {7, 7}}, CellGroupingRules -> {"SectionGrouping", 60}, ParagraphIndent -> -27, CounterIncrements -> "Outline2", CounterAssignments -> {{"Outline3", 0}, {"Outline4", 0}}, FontSize -> 15, FontWeight -> "Bold", CounterBoxOptions -> {CounterFunction :> (Part[ CharacterRange["A", "Z"], #]& )}], Cell[ StyleData["Outline2", "SlideShow"]], Cell[ StyleData["Outline2", "Printout"], CounterBoxOptions -> {CounterFunction :> (Part[ CharacterRange["A", "Z"], #]& )}]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Outline3"], CellMargins -> {{120, 10}, {7, 7}}, CellGroupingRules -> {"SectionGrouping", 70}, ParagraphIndent -> -21, CounterIncrements -> "Outline3", CounterAssignments -> {{"Outline4", 0}}, FontSize -> 12, CounterBoxOptions -> {CounterFunction :> Identity}], Cell[ StyleData["Outline3", "SlideShow"]], Cell[ StyleData["Outline3", "Printout"], CounterBoxOptions -> {CounterFunction :> Identity}]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["Outline4"], CellMargins -> {{150, 10}, {7, 7}}, CellGroupingRules -> {"SectionGrouping", 80}, ParagraphIndent -> -18, CounterIncrements -> "Outline4", FontSize -> 10, CounterBoxOptions -> {CounterFunction :> (Part[ CharacterRange["a", "z"], #]& )}], Cell[ StyleData["Outline4", "SlideShow"]], Cell[ StyleData["Outline4", "Printout"]]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["Hyperlink Styles", "Section"], Cell[ "The cells below define styles useful for making hypertext \ ButtonBoxes. The \"Hyperlink\" style is for links within the same Notebook, \ or between Notebooks.", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["Hyperlink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookLocate[#2]}]& ), ButtonNote -> ButtonData}], Cell[ StyleData["Hyperlink", "Presentation"], FontSize -> 16], Cell[ StyleData["Hyperlink", "SlideShow"]], Cell[ StyleData["Hyperlink", "Printout"], FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ "The following styles are for linking automatically to the on-line \ help system.", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["MainBookLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["MainBook", #]}]& )}], Cell[ StyleData["MainBookLink", "Presentation"], FontSize -> 16], Cell[ StyleData["MainBookLink", "SlideShow"]], Cell[ StyleData["MainBookLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["AddOnsLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontFamily -> "Courier", FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["AddOns", #]}]& )}], Cell[ StyleData["AddOnsLink", "Presentation"], FontSize -> 16], Cell[ StyleData["AddOnsLink", "SlideShow"]], Cell[ StyleData["AddOnsLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["RefGuideLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontFamily -> "Courier", FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["RefGuide", #]}]& )}], Cell[ StyleData["RefGuideLink", "Presentation"], FontSize -> 16], Cell[ StyleData["RefGuideLink", "SlideShow"]], Cell[ StyleData["RefGuideLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["RefGuideLinkText"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["RefGuide", #]}]& )}], Cell[ StyleData["RefGuideLinkText", "Presentation"], FontSize -> 16], Cell[ StyleData["RefGuideLinkText", "SlideShow"]], Cell[ StyleData["RefGuideLinkText", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["GettingStartedLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["GettingStarted", #]}]& )}], Cell[ StyleData["GettingStartedLink", "Presentation"], FontSize -> 16], Cell[ StyleData["GettingStartedLink", "SlideShow"]], Cell[ StyleData["GettingStartedLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["DemosLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["Demos", #]}]& )}], Cell[ StyleData["DemosLink", "SlideShow"]], Cell[ StyleData["DemosLink", "Printout"], FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["TourLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["Tour", #]}]& )}], Cell[ StyleData["TourLink", "SlideShow"]], Cell[ StyleData["TourLink", "Printout"], FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["OtherInformationLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["OtherInformation", #]}]& )}], Cell[ StyleData["OtherInformationLink", "Presentation"], FontSize -> 16], Cell[ StyleData["OtherInformationLink", "SlideShow"]], Cell[ StyleData["OtherInformationLink", "Printout"], FontSize -> 10, FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["MasterIndexLink"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, FontVariations -> {"Underline" -> True}, FontColor -> RGBColor[0.269993, 0.308507, 0.6], ButtonBoxOptions -> { Active -> True, Appearance -> {Automatic, None}, ButtonFunction :> (FrontEndExecute[{ FrontEnd`HelpBrowserLookup["MasterIndex", #]}]& )}], Cell[ StyleData["MasterIndexLink", "SlideShow"]], Cell[ StyleData["MasterIndexLink", "Printout"], FontVariations -> {"Underline" -> False}, FontColor -> GrayLevel[0]]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["Styles for Headers and Footers", "Section"], Cell[ StyleData["Header"], CellMargins -> {{0, 0}, {4, 1}}, DefaultNewInlineCellStyle -> "None", LanguageCategory -> "NaturalLanguage", StyleMenuListing -> None, FontSize -> 10, FontSlant -> "Italic"], Cell[ StyleData["Footer"], CellMargins -> {{0, 0}, {0, 4}}, DefaultNewInlineCellStyle -> "None", LanguageCategory -> "NaturalLanguage", StyleMenuListing -> None, FontSize -> 9, FontSlant -> "Italic"], Cell[ StyleData["PageNumber"], CellMargins -> {{0, 0}, {4, 1}}, StyleMenuListing -> None, FontFamily -> "Times", FontSize -> 10]}, Closed]], Cell[ CellGroupData[{ Cell["Palette Styles", "Section"], Cell[ "The cells below define styles that define standard ButtonFunctions, \ for use in palette buttons.", "Text"], Cell[ StyleData["Paste"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, Placeholder]}]& )}], Cell[ StyleData["Evaluate"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[], All]}]& )}], Cell[ StyleData["EvaluateCell"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionMove[ FrontEnd`InputNotebook[], All, Cell, 1], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[], All]}]& )}], Cell[ StyleData["CopyEvaluate"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionEvaluate[ FrontEnd`InputNotebook[], All]}]& )}], Cell[ StyleData["CopyEvaluateCell"], StyleMenuListing -> None, ButtonStyleMenuListing -> Automatic, ButtonBoxOptions -> {ButtonFunction :> (FrontEndExecute[{ FrontEnd`SelectionCreateCell[ FrontEnd`InputNotebook[], All], FrontEnd`NotebookApply[ FrontEnd`InputNotebook[], #, All], FrontEnd`SelectionEvaluateCreateCell[ FrontEnd`InputNotebook[], All]}]& )}]}, Closed]], Cell[ CellGroupData[{ Cell["Placeholder Styles", "Section"], Cell[ "The cells below define styles useful for making placeholder objects \ in palette templates.", "Text"], Cell[ CellGroupData[{ Cell[ StyleData["Placeholder"], Placeholder -> True, StyleMenuListing -> None, FontSlant -> "Italic", FontColor -> RGBColor[0.890623, 0.864698, 0.384756], TagBoxOptions -> { Editable -> False, Selectable -> False, StripWrapperBoxes -> False}], Cell[ StyleData["Placeholder", "Presentation"]], Cell[ StyleData["Placeholder", "SlideShow"]], Cell[ StyleData["Placeholder", "Printout"]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["PrimaryPlaceholder"], StyleMenuListing -> None, DrawHighlighted -> True, FontSlant -> "Italic", Background -> RGBColor[0.912505, 0.891798, 0.507774], TagBoxOptions -> { Editable -> False, Selectable -> False, StripWrapperBoxes -> False}], Cell[ StyleData["PrimaryPlaceholder", "Presentation"]], Cell[ StyleData["PrimaryPlaceholder", "SlideShow"]], Cell[ StyleData["PrimaryPlaceholder", "Printout"]]}, Closed]]}, Closed]], Cell[ CellGroupData[{ Cell["FormatType Styles", "Section"], Cell[ "The cells below define styles that are mixed in with the styles of \ most cells. If a cell's FormatType matches the name of one of the styles \ defined below, then that style is applied between the cell's style and its \ own options. This is particularly true of Input and Output.", "Text"], Cell[ StyleData["CellExpression"], PageWidth -> Infinity, CellMargins -> {{6, Inherited}, {Inherited, Inherited}}, ShowCellLabel -> False, ShowSpecialCharacters -> False, AllowInlineCells -> False, Hyphenation -> False, AutoItalicWords -> {}, StyleMenuListing -> None, FontFamily -> "Courier", FontSize -> 12, Background -> GrayLevel[1]], Cell[ StyleData["InputForm"], InputAutoReplacements -> {}, AllowInlineCells -> False, Hyphenation -> False, StyleMenuListing -> None, FontFamily -> "Courier"], Cell[ StyleData["OutputForm"], PageWidth -> Infinity, TextAlignment -> Left, LineSpacing -> {0.6, 1}, StyleMenuListing -> None, FontFamily -> "Courier"], Cell[ StyleData["StandardForm"], InputAutoReplacements -> { "->" -> "\[Rule]", ":>" -> "\[RuleDelayed]", "<=" -> "\[LessEqual]", ">=" -> "\[GreaterEqual]", "!=" -> "\[NotEqual]", "==" -> "\[Equal]", Inherited}, "TwoByteSyntaxCharacterAutoReplacement" -> True, LineSpacing -> {1.25, 0}, StyleMenuListing -> None, FontFamily -> "Courier"], Cell[ StyleData["TraditionalForm"], InputAutoReplacements -> { "->" -> "\[Rule]", ":>" -> "\[RuleDelayed]", "<=" -> "\[LessEqual]", ">=" -> "\[GreaterEqual]", "!=" -> "\[NotEqual]", "==" -> "\[Equal]", Inherited}, "TwoByteSyntaxCharacterAutoReplacement" -> True, LineSpacing -> {1.25, 0}, SingleLetterItalics -> True, TraditionalFunctionNotation -> True, DelimiterMatching -> None, StyleMenuListing -> None], Cell[ "The style defined below is mixed in to any cell that is in an \ inline cell within another.", "Text"], Cell[ StyleData["InlineCell"], LanguageCategory -> "Formula", ScriptLevel -> 1, StyleMenuListing -> None], Cell[ StyleData["InlineCellEditing"], StyleMenuListing -> None, Background -> RGBColor[0.964706, 0.929412, 0.839216]]}, Closed]], Cell[ CellGroupData[{ Cell["Automatic Styles", "Section"], Cell[ "The cells below define styles that are used to affect the display \ of certain types of objects in typeset expressions. For example, \ \"UnmatchedBracket\" style defines how unmatched bracket, curly bracket, and \ parenthesis characters are displayed (typically by coloring them to make them \ stand out).", "Text"], Cell[ StyleData["UnmatchedBracket"], StyleMenuListing -> None, FontColor -> RGBColor[0.760006, 0.330007, 0.8]], Cell[ StyleData["Completions"], StyleMenuListing -> None, FontFamily -> "Courier"]}, Closed]], Cell[ CellGroupData[{ Cell["Styles from HelpBrowser", "Section"], Cell[ CellGroupData[{ Cell[ StyleData["MathCaption"], CellFrame -> {{0, 0}, {0, 0.5}}, CellMargins -> {{66, 12}, {2, 24}}, PageBreakBelow -> False, CellFrameMargins -> {{8, 8}, {8, 2}}, CellFrameColor -> GrayLevel[0.700008], CellFrameLabelMargins -> 4, LineSpacing -> {1, 1}, ParagraphSpacing -> {0, 8}, StyleMenuListing -> None, FontColor -> GrayLevel[0.2]], Cell[ StyleData["MathCaption", "Presentation"], FontSize -> 18], Cell[ StyleData["MathCaption", "SlideShow"], CellMargins -> {{100, 50}, {Inherited, Inherited}}], Cell[ StyleData["MathCaption", "Printout"], CellMargins -> {{39, 0}, {0, 14}}, Hyphenation -> True, FontSize -> 9, FontColor -> GrayLevel[0]]}, Closed]], Cell[ CellGroupData[{ Cell[ StyleData["ObjectName"], ShowCellBracket -> True, CellMargins -> {{66, 4}, {8, 8}}, Evaluatable -> True, CellGroupingRules -> "InputGrouping", PageBreakWithin -> False, GroupPageBreakWithin -> False, CellLabelAutoDelete -> False, CellLabelMargins -> {{14, Inherited}, {Inherited, Inherited}}, DefaultFormatType -> DefaultInputFormatType, ShowSpecialCharacters -> Automatic, "TwoByteSyntaxCharacterAutoReplacement" -> True, HyphenationOptions -> { "HyphenationCharacter" -> "\[Continuation]"}, LanguageCategory -> "Mathematica", FormatType -> StandardForm, ShowStringCharacters -> True, NumberMarks -> True, StyleMenuListing -> None, FontWeight -> "Bold"], Cell[ StyleData["ObjectName", "Presentation"], FontSize -> 18], Cell[ StyleData["ObjectName", "SlideShow"], CellMargins -> {{100, 50}, {Inherited, Inherited}}], Cell[ 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