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W2110540131 abstract "If<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$T$ id=E2><mml:mi>T</mml:mi></mml:math>is the parallel map associated with a<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$1$ id=E3><mml:mn>1</mml:mn></mml:math>-dimensional tessellation automaton, then we say a configuration<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$f$ id=E4><mml:mi>f</mml:mi></mml:math>is a weak Garden of Eden for<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$T$ id=E5><mml:mi>T</mml:mi></mml:math>if<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$f$ id=E6><mml:mi>f</mml:mi></mml:math>has no pre-image under<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$T$ id=E7><mml:mi>T</mml:mi></mml:math>other than a shift of itself. Let<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$WGleft( T right) = $ id=E8><mml:mi>W</mml:mi><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>T</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo></mml:math>the set of weak Gardens of Eden for<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$T$ id=E9><mml:mi>T</mml:mi></mml:math>and<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$Gleft( T right) = $ id=E10><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>T</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo></mml:math>the set of Gardens of Eden (i.e., the set of configurations not in the range of<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$T$ id=E11><mml:mi>T</mml:mi></mml:math>). Typically members of<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$WGleft( T right) - Gleft( T right)$ id=E12><mml:mi>W</mml:mi><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>T</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>T</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>satisfy an equation of the form<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$Tf = S^m f$ id=E13><mml:mi>T</mml:mi><mml:mi>f</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>S</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mi>f</mml:mi></mml:math>where<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$S^m $ id=E14><mml:msup><mml:mi>S</mml:mi><mml:mi>m</mml:mi></mml:msup></mml:math>is the shift defined by<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$left( {S^m f} right)left( j right) = fleft( {j + m} right)$ id=E15><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>S</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mi>f</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>j</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>+</mml:mo><mml:mi>m</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>. Subject to a mild restriction on<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$m$ id=E16><mml:mi>m</mml:mi></mml:math>, the equation<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$Tf = S^m f$ id=E17><mml:mi>T</mml:mi><mml:mi>f</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mi>S</mml:mi><mml:mi>m</mml:mi></mml:msup><mml:mi>f</mml:mi></mml:math>always has a solution<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$f$ id=E18><mml:mi>f</mml:mi></mml:math>, and all such solutions are periodic. We present a few other properties of weak Gardens of Eden and a characterization of<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$WGleft( T right)$ id=E19><mml:mi>W</mml:mi><mml:mi>G</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>T</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>for a class of parallel maps we call<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=$left( {0,1} right)$ id=E20><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>-characteristic transformations in the case where there are at least three cell states." @default.
W2110540131 created "2016-06-24" @default.
W2110540131 creator A5044875563 @default.
W2110540131 date "1985-01-01" @default.
W2110540131 modified "2023-10-18" @default.
W2110540131 title "Weak gardens of Eden for 1-dimensional tessellation automata" @default.
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W2110540131 doi "https://doi.org/10.1155/s0161171285000631" @default.
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