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This \ background sweep is assumed to be deterministic and uninfluenced by the focal \ allele. All variables are scaled. Let the advantage of the focal allele be ", Cell[BoxData[ FormBox["s", TraditionalForm]]], ", the advantage of the background allele be ", Cell[BoxData[ FormBox["S", TraditionalForm]]], ", and the recombination rate between the loci be ", Cell[BoxData[ FormBox["r", TraditionalForm]]], ", and letting ", Cell[BoxData[ FormBox["t", TraditionalForm]]], " be the number of generations from when the background sweep is at \ frequency 1/2 (so that ", Cell[BoxData[ FormBox[ RowBox[{"t", "<", "0"}], TraditionalForm]]], " corresponds to frequencies <1/2). 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of a new favorable allele \ at a locus given that there are ongoing sweeps at linked loci. These \ background sweeps are assumed to be deterministic and uninfluenced by the \ focal allele. PfixMany, Pbar, PfixManyTot, and recEqs assume that all \ background sweeps are in linkage equilibrium, while PfixManyLD and \ PfixManyTotLD allow linkage disequilibrium among the background alleles and \ assume that all the sweeping background alleles arose in the original \ ancestral background. (This is potentially problematic -- see the definition \ of Gsol in the section \[OpenCurlyDoubleQuote]Background genotype dynamics\ \[CloseCurlyDoubleQuote].) PbarLD and recEqsLD allow linkage disequilibrium \ and make no assumption about the initial state of the background sweeps.\n\n\ All variables are scaled by the advantage of the focal allele, ", Cell[BoxData[ FormBox["s", TraditionalForm]]], ". (Note that this is the standard scaling used in this manuscript, which is \ different from the scaling used above for a single background sweep.) Let ", Cell[BoxData[ FormBox[ SubscriptBox["A", "i"], TraditionalForm]]], " be the selective advantage of the ", Cell[BoxData[ FormBox[ SuperscriptBox["i", "th"], TraditionalForm]]], " background allele and ", Cell[BoxData[ FormBox[ SubscriptBox["r", "i"], TraditionalForm]]], " be its map position relative to the focal locus (with ", Cell[BoxData[ FormBox["r", TraditionalForm]]], " allowed to take both positive and negative values, so that the \ recombination rate is ", Cell[BoxData[ FormBox[ RowBox[{"|", SubscriptBox["r", "i"], "|"}], TraditionalForm]]], "), and let ", Cell[BoxData[ FormBox["t", TraditionalForm]]], " be the number of generations from some arbitrary time. The functions work \ with ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["S", "i"], "=", RowBox[{ SubscriptBox["A", "i"], "/", "s"}]}], ",", " ", RowBox[{ SubscriptBox["\[Rho]", "i"], "=", RowBox[{ SubscriptBox["r", "i"], "/", "s"}]}], ",", " ", RowBox[{"T", "=", RowBox[{"s", " ", "t"}]}]}], TraditionalForm]]], ", and return quantities related to the scaled probability of fixation, ", Cell[BoxData[ FormBox[ RowBox[{"\[CapitalPhi]", "=", RowBox[{ SubscriptBox["P", "fix"], "/", RowBox[{"(", RowBox[{"2", "s"}], ")"}]}]}], TraditionalForm]]], ". 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