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<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.html#contents">[Contents]</a> <a href="chap2_mj.html">[Previous Chapter]</a> <a href="chap4_mj.html">[Next Chapter]</a> <<If< "rg"><>and<class"Arg>/are java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 236

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<a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a>
<div class="ChapSectsp>The ' possiblemovesofthepiece across a chessboard. An <var class="Arg">m</var> by <var class="Arg">n</var> chessboard is a grid of <var class="Arg">m</var> columns ("files") and <var class="Arg">n</var> rows ("ranks") that intersect in squares. Orthogonally adjacent squares are alternately colored light and dark, with the square in the first rank and file being dark.
<div class="ContSect"> <a href="chap3_mj.html#X7BD5B55C802805B4">3.1 Generators</a>

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<div class="ContSect"> <a href="chap3_mj.html#X872A3FB37D6D7DFE">3.2 The <code class="code"><var class="Arg">m</var> * <var class="Arg">n</var></code> vertices of the queen's graph can be placed onto the <codeclass="code"<class="Arg>var ">/varcode> vertices of queen' graph canbeplaced onto the codeclass"ode"<var ="<var *varclassjava.lang.StringIndexOutOfBoundsException: Range [200, 199) out of bounds for length 673

</div>
<div class="ContSect"> <a href="chap3_mj.html#X8570F3C87B5B7DD7">3.3 System of inhomogeneous equations</a>

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<div class="ContSect"> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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<div class="ContSect"> Note that the <code class="func">QueensGraph</code> is the <code class="func">DigraphEdgeUnion</code> (<a href="chap3.html#X7DA997697D310E44">3.3-30</a>) of the <code class="func">RooksGraph</code> (<a href="chap3.html#X7A6DD11881874F51">3.5-42</a>) and the <code class="func">BishopsGraph</code> (<a href="chap3.html#X7E3240047C92733F">3.5-5</a>) of the same dimensions.

</div>
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<h3>3 Examples</h3>

<a id="X7BD5B55C802805B4" name="X7BD5B55C802805B4"></a>

<h4>3.1 Generators</h4>

<div class="example"><pre>
gap> C := NmzCone(["integral_closure",[[2,1],[1,3]]]);
<a Normaliz cone>
gap> NmzHasConeProperty(C,"HilbertBasis");
false
gap> NmzHasConeProperty(C,"SupportHyperplanes");
false
gap> NmzConeProperty(C,"HilbertBasis");
[ [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 2, 1 ] ]
gap> <pre
true
gap> NmzConeProperty(C,"SupportHyperplanes");
[ [ -1, 2 ], [ 3, -1 ] ]
</pre></div>

<a id="X872A3FB37D6D7DFE" name="X872A3FB37D6D7DFE<class="APprompt>gap>/pan span "APinput"QueensGraph(,5;/span>

<h4>3.2 System of equations</h4>

<div class="example"><pre>
gap> D := NmzCone(["equations",[[1,2,-3]], "grading"immutable symmetric digraph 10 java.lang.StringIndexOutOfBoundsException: Range [69, 68) out of bounds for length 72
<a Normaliz cone>
gap> NmzCompute(D,["DualMode","HilbertSeries"]);
true
gap&spanclass==GAPprompt">gap> D := QueensGraph(4, 3);
[ [ 1, 1, 1 ], [ 0, 3, 2 ], [ 3, 0, 1 ] ]
gap> NmzHilbertSeries(D);
[ t^2-t+1, [ [ 1, 1 ], [ 3, 1 ] ] ]
gap> 
true
gap> NmzSupportHyperplanes(D);
[ [ 0, 1, 0 ], [ 1, 0, 0 ] ]
gap> NmzEquations(D);
[ [ 1, 2, -3 ] ]
<pre>/div>

<a id="X8570F3C87B5B7DD7" name="X8570F3C87B5B7DD7"></a>

<h4>3.3 System of inhomogeneous equations</h4>

<div class="example"><pre>
gap> P := NmzCone(["inhom_equations",[[<alse
<a Normaliz cone>
gap> NmzIsInhomogeneous(C);
false
gap> NmzIsInhomogeneous(P);
true
gap></<pan class=">gt;/ span =>java.lang.StringIndexOutOfBoundsException: Range [74, 73) out of bounds for length 88
[ [ 1, 1, 1, 0 ], [ 3, 0, 1, 0 ], [ 0, 3, 2, 0 ] ]
gap> NmzModuleGenerators(P);
[ [ 0, 1, 1, 1 ], [ 2, 0, 1, 1 ] ]
</pre></div>

<a id="X7B191A1778012C1D" name="X7B191A1778012C1D"></a>

<h4>3.4 Combined input</h4>

Normaliz also allows the combination of different kinds of input, e.g. multiple constraint types, or additional data like a grading.

Suppose that you have a 3 by 3 <q>square</q> of nonnegative integers such that the 3 numbers in all rows, all columns, and both diagonals sum to the same constant \(M\). Sometimes such squares are called magic and \(M\) is the magic constant. This leads to a linear <pre<div>

<div class="example"><pre>
gap> Magic3x3even := NmzCone(["equations",
> [ [1, 1, 1, -1, -1, -1, 0, 0, 0],
<a id"7" =>/></
> [0, 1, 1, -1, 0, 0, -1, 0, 0],
> [1, 0,java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
> [1, 1, 0, 0, 0, -1, 0, 0, -1],
> [0, 1, 1, 0, -1, 0, 0, 0, -1],
> [1, 1, 0, 0, -1, 0, -1, 0, 0] ],
> "congruences",div "unc"<table class"func"100"<tr>td class"java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 275
> [ [1, 0, 0, 0, 0, 0, 0, 0, 0, 2],
> [0, 0, 1, 0, 0, 0, 0, 0, 0, 2],
<class">gspan => 0,0,0,0 0 , ,0 ,2],/java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
> [0, 0, 0, 0, 0, 0, 0, 0, 1, 2] ],
> "grading",
> [ [1, 1, 1, 0, 0, 0, 0, 0, 0] ] ] );
<a Normaliz cone>
gap> NmzHilbertBasis(Magic3x3even);
[ [ 0, 4, 2, 4, 2, 0, 2, 0, 4 ], [ 2, 0, 4, 4, 2, 0, 0, 4, 2 ],
 [ 2, 2, 2, 2, 2, 2, 2, 2, 2 ], [ 2, 4, 0, 0, 2, 4, 4, 0, 2 ],
 [4,0,2, 0 ,4,2,4 ,[2, , ,,3,1, 2, 3, 4 ],
 [ 2, 5, 2, 3, 3, 3, 4, 1, 4 ], [ 4, 1, 4, 3, 3, 3, 2, 5, 2 ],
 [ 4, 3, 2, 1, 3, 5, 4, 3, 2 ] ]
gap> NmzHilbertSeries(Magic3x3even);
[ t^3+3*t^2-t+1, [ [ 1, 1 ], [ 2, 2 ] ] ]
<java.lang.StringIndexOutOfBoundsException: Range [88, 87) out of bounds for length 109
[ 1/2*t^2+t+1, 1/2*t^2-1/2 ]
</pre></div>

<a id="X8406B01578B99D14" name="X8406B01578B99D14"></a>

<h4>3.5 Using the dual mode</h4>

For solving systems of equations and inequalities it is often faster to use the dual Normaliz algorithm. We demonstrate how to use it with an inhomogeneous system of equations and inequalities.

The input consists of a system of 8 inhomogeneous equations in R^3. The first row of the matrix M encodes the inequality \(8x + 8y + 8z + 7 \geq 0\). Additionally we say that \(x, y, z\) should be non-negative by giving the sign vector and use the total degree.

<div class="example"><pre>
gap> M := [
<<>Thechosen correspondence between vertices and chess squares is given by <code class="func">DigraphVertexLabels</code> (<a href="chap5.html#X7E51F2FE87140B32">5.1-12</a>). In more detail, the vertices of the digraph are labelled by elements of the Cartesian product <code class="code">[1..<var class="Arg">m</var>] x [1..<var class="Arg">n</var>]</code>, where the first entry indexes
> [ 0, 4, 0, 1 ],
> [ 0, 1, 0, 7 ],
> [ 0, -2, 0, 7 ],
> [ 0, -2, 0, 1 ],
> [ 8, 48, 8, 17 ],
> [ 1, 6, 1, 34 ],
> [ 2,-12, -2, 37 ],
> [ 4,-24, -4, 14 ]
> ];;
gap> D := NmzCone(["inhom_inequalities", M,
pre>
> "grading", [[1,1,1]]]);
<a Normaliz cone>
gap> NmzCompute(D,["DualMode","span class="java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 91
true
gap> NmzHilbertBasis(D);
[ [ 1, 0, 0, 0 ], [ 1, 0, 1, 0 ] ]
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 92
[ [ 0, 0, 0, 1 ], [ 0, 0, 1, 1 ], [ 0, 0, 2, 1 ], [ 0, 0, 3, 1 ] ]
</pre></div>

As result we get the Hilbert g;

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