(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 87681, 2766] NotebookOptionsPosition[ 86867, 2734] NotebookOutlinePosition[ 87334, 2752] CellTagsIndexPosition[ 87291, 2749] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ ALGHCAGloss. Glossary for Loops, Groups, Hoops, Cliffords.\ \>", "Title", PageWidth->WindowWidth, CellChangeTimes->{{3.540386216719159*^9, 3.540386224347559*^9}, { 3.541219087733565*^9, 3.541219098247965*^9}}, FontFamily->"Times New Roman"], Cell["\<\ Roger Beresford. 10th. Mar. 2012.\ \>", "Subtitle", CellChangeTimes->{{3.509428588105162*^9, 3.509428589493562*^9}, { 3.5294196858288*^9, 3.5294196987455997`*^9}, {3.5403876608423595`*^9, 3.5403876750227594`*^9}}, FontSize->16], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"NotebookFileName", "[", "]"}], RowBox[{"(*", RowBox[{ RowBox[{ RowBox[{"10", "/", "3"}], "/", "12"}], " ", "17.01", " ", RowBox[{"version", "."}]}], "*)"}]}]], "Input", CellChangeTimes->{{3.5094285463127623`*^9, 3.509428561273162*^9}, { 3.509428594750762*^9, 3.5094286023011622`*^9}, {3.5146068273723583`*^9, 3.514606871301958*^9}, {3.5294197147044*^9, 3.5294197389624*^9}, { 3.5294230918859997`*^9, 3.5294230980948*^9}, {3.5403876930719595`*^9, 3.540387714693559*^9}, {3.541219072788765*^9, 3.541219074847965*^9}}, CellID->306117493], Cell[BoxData["\<\"C:\\\\Users\\\\Roger\\\\A2012\\\\Archive\\\\ALGHCAGloss.nb\"\ \>"], "Output", CellChangeTimes->{3.509428568168362*^9, 3.514606876886758*^9, 3.514606927259158*^9, 3.529419765888*^9, 3.541219064130765*^9, 3.5412191277007647`*^9}] }, Open ]], Cell[CellGroupData[{ Cell["Introduction,", "Subsection", CellChangeTimes->{{3.5294955562022*^9, 3.5294955654997997`*^9}}], Cell[TextData[{ "Negative, real, complex, quaternionic and octonionic numbers, subtraction, \ and the signs {+, \[ImaginaryI], -, -\[ImaginaryI]}, are a degenerate subset \ of mathematics, restricted to even symmetry. Algebraic loops (quasi-groups \ with 1) define general multiplication for Natural, Rational, and ", StyleBox["Primal", FontSlant->"Italic"], " (continuous, absolute, half line) (", StyleBox["italic names introduce new mathematical concepts", FontSlant->"Italic"], ") unsigned numbers and are created without defining division or \ subtraction. They provide Cayley tables for the multiplication of ", StyleBox["vec", FontSlant->"Italic"], "s, sets of {direction, unsigned coefficient} pairs. Signs can only be \ defined for loops having C", StyleBox["r", FontSlant->"Italic"], " (the cyclic group with ", StyleBox["r", FontSlant->"Italic"], " elements) as central subgroups. Such loops can be ", StyleBox["folded", FontSlant->"Italic"], " to smaller ", StyleBox["signed tables", FontSlant->"Italic"], " where the ", StyleBox["r", FontSlant->"Italic"], "-fold symmetry is represented by ", StyleBox["s", FontSlant->"Italic"], " (with ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"s", " "}], "r"], TraditionalForm]]], "=1) as a ", StyleBox["generalized sign.", FontSlant->"Italic"], " This introduces ", StyleBox["r", FontSlant->"Italic"], "\[CloseCurlyQuote]th roots of unity as ", StyleBox["generalised signs", FontSlant->"Italic"], ". (", StyleBox["r", FontSlant->"Italic"], "=2 and 4 give real and complex algebras and impose even symmetry). Folding \ creates algebras with generalised addition/subtraction as a second operation \ and folds vecs to vectors; Moufang loops (which have a division property \ ab/b=a=ba/b) fold to ", StyleBox["vector-division", FontSlant->"Italic"], " algebras.\nFrobenius showed that associative loops (i.e. groups) conserve \ their determinants (up to a sign) on vector multiplication. This ", StyleBox["symmetry-conserving property", FontSlant->"Italic"], " is possessed by all associative (group) and a few non-associative \ (octonionic \[DoubleStruckCapitalO]) Moufang loops, and by algebras folded \ from conservative loops. Vectors in these algebras have", StyleBox[" multiplicative inverses that split into partial fractions with \ determinant factors as numerators", FontSlant->"Italic"], ". \nI propose the name ", StyleBox["hoops", FontSlant->"Italic"], " for these symmetry-conserving partial-fraction-division algebras. I call \ the factors ", StyleBox["sizes", FontSlant->"Italic"], " and the list of conserved sizes the ", StyleBox["shape", FontSlant->"Italic"], " of a vector. Noether\[CloseCurlyQuote]s theorem, relating conserved \ symmetries to particles and forces, implies that hoops describe ", StyleBox["physical mathematics", FontSlant->"Italic"], ". Anti-symmetric hoops are the Clifford (Geometric) algebras that \ generalise many concepts to any number of dimensions. \nMost Hoops have \ multiple sizes. (Real, complex, quaternion, octonion, Planck and even-order \ Clifford algebras are degenerate, monosized, hoops.) When two vectors have ", StyleBox["disparate zero sizes", FontSlant->"Italic"], ", the product loses any sizes that are only present in one multiplicand - \ it is ", StyleBox["projected", FontSlant->"Italic"], " onto a sub-algebra of reduced symmetry. To maintain the key conservation \ property, the lost sizes are ", StyleBox["ejected", FontSlant->"Italic"], " as (left or right) ", StyleBox["remainders", FontSlant->"Italic"], ". Multiplication (and division, which is multiplication by an inverse \ vector) can create a product and two remainders. This implements ", StyleBox["vector-division-by-zero", FontSlant->"Italic"], ". Different algebras (of the same length) conserve different (but \ overlapping) ", StyleBox["sets of conserved properties", FontSlant->"Italic"], ". The analogy with particle physics, where different forces conserve \ overlapping sets of properties, has yet to be explored.\nThese concepts were \ developed empirically, but have (for groups) a theoretical basis in \ Representation Theory.\nThis material was originally developed in the ALGHC \ package [], which includes an empirical database of loops, groups, hoops and \ Clifford algebras, together with many procedures to create, identify, and \ apply them. It relied heavily on Cayley tables, and made little reference to \ standard Group theory. It was limited by the difficulty of finding symbolic \ sizes and by the lack of a mathematical basis. Work with G. Gerrard led to a \ reformulation (restricted to groups) in terms of irreducible matrix \ representations (irreps). These are based in representation theory. They give \ permutation group descriptions as well as Cayley tables and allow the \ calculation of vector sizes and inverses. The GroupHoop package [] has a \ database derived from GAP, and provides simpler access to many small groups \ and hoops." }], "Text", CellChangeTimes->{{3.529494241325*^9, 3.5294943658442*^9}, { 3.5294943983858*^9, 3.5294944462622004`*^9}, {3.5294944893962*^9, 3.5294944936394*^9}, {3.5294945549318*^9, 3.5294945998129997`*^9}, { 3.5294946309505997`*^9, 3.5294946358178*^9}, {3.5294946686401997`*^9, 3.5294947354706*^9}, {3.5294948082134*^9, 3.5294948134862003`*^9}, 3.529494990281*^9, {3.5294950243358*^9, 3.5294950637414*^9}, { 3.5294950948322*^9, 3.5294951129282*^9}, 3.5294952136418*^9, 3.5294953175222*^9, {3.5294953487846003`*^9, 3.5294953629962*^9}, { 3.5294953963646*^9, 3.529495423493*^9}, {3.529495577309*^9, 3.5294960246078*^9}, {3.5294961805766*^9, 3.5294965022018003`*^9}, { 3.529496533043*^9, 3.5294965666610003`*^9}, {3.5294965985942*^9, 3.5294966041633997`*^9}, {3.5294966513378*^9, 3.5294966589038*^9}, { 3.5294966984966*^9, 3.5294969296418*^9}, {3.5403862624427595`*^9, 3.5403862967783594`*^9}, {3.540386356838359*^9, 3.540386401657159*^9}, { 3.540386468440759*^9, 3.540386475382759*^9}, {3.540386555020759*^9, 3.540386555348359*^9}}] }, Open ]], Cell[CellGroupData[{ Cell["Glossary.", "Subsection", CellChangeTimes->{{3.5294955346274*^9, 3.529495541585*^9}}], Cell[TextData[{ StyleBox["a,b,..or", FontSlant->"Italic"], StyleBox[" ", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ","}], TraditionalForm]]], " ", StyleBox["Elements representing Primal (unsigned, non-negative) numbers, \ coefficients in a director.\n", FontWeight->"Plain"], StyleBox["\nA", FontSlant->"Italic"], "\t", StyleBox["A \"Vec\", an indexed set {", FontWeight->"Plain"], StyleBox["a,b", FontSlant->"Italic"], StyleBox["\[Ellipsis]} of ", FontWeight->"Plain"], StyleBox["m", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" primal elements; also written ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "m"], StyleBox[ RowBox[{ SubscriptBox["a", "i"], SubscriptBox["d", "i"]}], FontWeight->"Bold"]}], TraditionalForm]], "Input", FontWeight->"Plain"], StyleBox[". where ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["d", "i"], FontWeight->"Bold"], TraditionalForm]], FontWeight->"Plain"], " ", StyleBox["is a direction.\n\n", FontWeight->"Plain"], "A, B, V", StyleBox[" Vectors, indexed sets {a,b\[Ellipsis]} of ", FontWeight->"Plain"], StyleBox["m/r", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" \[OpenCurlyDoubleQuote]", FontWeight->"Plain"], StyleBox["r", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["-signed\[CloseCurlyDoubleQuote] elements; also written ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], RowBox[{"m", "/", "r"}]], RowBox[{ SubscriptBox["a", "i"], SubscriptBox["x", "i"]}]}], TraditionalForm]], "Input", FontWeight->"Plain"], StyleBox[". (They take on\n\t all the properties of vectors in some \ algebras. In Geometric Algebra ", FontWeight->"Plain"], StyleBox["Vector", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" is confusing so I use\n\t ", FontWeight->"Plain"], StyleBox["monovector", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" by analogy with bivector etc..\n\t ", FontWeight->"Plain"], "\n", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{"\[DoubleStruckCapitalA]", "=", RowBox[{"0", "\[Union]", SuperscriptBox["\[DoubleStruckCapitalR]", "+"]}]}]}], TraditionalForm]]], " ", StyleBox["The Primal (unsigned continuous, Absolute) numbers, the union of \ zero and the half line.\n", FontWeight->"Plain"], "\nadditive elimination. ", StyleBox["The generalization of subtraction ", FontWeight->"Plain"], StyleBox["A", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["+(-", FontWeight->"Plain"], StyleBox["A", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[")=0 to ", FontWeight->"Plain"], StyleBox["A", FontSlant->"Italic"], StyleBox["+", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["A", "i"], TraditionalForm]]], StyleBox["+", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["A", "j"], TraditionalForm]]], StyleBox["+..=0. ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["A", "i"], TraditionalForm]]], " ", StyleBox["etc are roots of unity.\n This is the result of folding a \ loop, which equivalences the sum of a set of roots of unity to zero.\n \ ", FontWeight->"Plain"], "\nAi= ", Cell[BoxData[ FormBox[ RowBox[{ UnderscriptBox["\[Sum]", "j"], RowBox[{"(", RowBox[{ UnderscriptBox["\[Sum]", "i"], RowBox[{ SubscriptBox["\[Eta]", "i"], RowBox[{ SubscriptBox["\[PartialD]", SubscriptBox["a", RowBox[{"i", " "}]]], SubscriptBox["f", RowBox[{"j", " "}]]}]}]}], ")"}]}], TraditionalForm]]], " /", Cell[BoxData[ FormBox[ SubscriptBox["f", "j"], TraditionalForm]]], ". ", StyleBox["The multiplicative inverse of A. Calculated by ", FontWeight->"Plain"], StyleBox["hoopInverse[A], cInverse[A]", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[",\n\t", FontWeight->"Plain"], StyleBox["gapInverse[A]", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[". Ai.A={1,0,..} Also found by adding the inverses of non-zero \ irreducible matrix\n\trepresentations (irreps) after mapping with a vector.\n \ \n", FontWeight->"Plain"], "algebraic loop.", StyleBox[" (Abbreviated to loop in these notebooks.) An ", FontWeight->"Plain"], StyleBox["m\[Times]m", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" quasigroup Cayley index-table with a 1. \n This is a \ multiplication table for ", FontWeight->"Plain"], StyleBox["m", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" unsigned elements and for ", FontWeight->"Plain"], StyleBox["m", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["-element vecs. It converts to an \n ", FontWeight->"Plain"], StyleBox["r", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["-signed algebra with ", FontWeight->"Plain"], StyleBox["m/r r", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["-signed elements if it can be folded over a central cyclic \ subgroup Cr.\n \n", FontWeight->"Plain"], "annihilator.", StyleBox["\tA non-trivial vector with all sizes zero. E.g.{1,1,1,1,-2,-2} in \ S3. See also ", FontWeight->"Plain"], "selector.\n", StyleBox["\n", FontWeight->"Plain"], "anti-commutative ", StyleBox["The property (possessed by Clifford & geometric algebra elements), \ that ", FontWeight->"Plain"], StyleBox["ab=-ba", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[".\n\n", FontWeight->"Plain"], "arcTanh[a,o] ", StyleBox["A generalization of", FontWeight->"Plain"], " ", StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["'s ArcTanh[a/o] that gives a complex hyperangle depending\n\t \ on the a,o (adjacent,opposite) octant. Many mathematicians use (opposite, \ adjacent).\n\t \n", FontWeight->"Plain"], "binegation. ", StyleBox["Ordinary (unary) negation -A, with -(-A)=A, resulting from \ negation by C2 equivalencing, and\n\t arising if the hidden axiom \ \[OpenCurlyDoubleQuote]minus one exists\[CloseCurlyDoubleQuote] is valid. \ Often incorrectly conflated with\n\t subtraction (the ", FontWeight->"Plain"], StyleBox["r", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["=2 case of subtractive elimination) which is a binary operation \ B-A.\n\t \n", FontWeight->"Plain"], "bivector", StyleBox[" ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["e", "i"], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"\[Wedge]", SubscriptBox["e", "j"]}], TraditionalForm]]], ",", StyleBox[" the wedge product of two distinct ", FontWeight->"Plain"], "monovectors", StyleBox[", an element of a Geometric Algebra.\n It \ represents a directed area, but carries no shape information. Rotations occur \ in such areas", FontWeight->"Plain"], ".\n \n", StyleBox["C2, C3", FontWeight->"Plain"], " etc., ", StyleBox["The cyclic groups (associative loops) of 2,3, etc. primal \ elements. Do not confuse with Rings ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["\[DoubleStruckCapitalZ]", "2"], TraditionalForm]]], StyleBox["\n ", FontWeight->"Plain"], " \t ", StyleBox[" etc, for which negation is defined. Negation is often wrongly \ assumed for loops and groups.\n \t ", FontWeight->"Plain"], "\n", StyleBox["C8xC2", FontWeight->"Plain"], " etc. ", StyleBox["The direct composition of C8 with C2 etc. Indirect composition is \ shown as e.g. C8iC4, C8jC4.\n\t In GAP, indirect composition is \ (ambiguously) shown as C8:C2.\n\t ", FontWeight->"Plain"], "\nC3\t ", StyleBox["Ternal", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", ", FontWeight->"Plain"], StyleBox["Terplex", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["; the algebra of three elements, equivalence relationships on \ C3xC2, C3xC4", FontWeight->"Plain"], StyleBox[" ", FontSlant->"Italic"], StyleBox["etc.\n\n", FontWeight->"Plain"], "Cartesian- ", StyleBox["A vec or vector as an indexed set of coefficients with related \ directions or dimensions. Abelian\n", FontWeight->"Plain"], "Form. ", StyleBox[" algebras have dual ", FontWeight->"Plain"], "Polar Forms ", StyleBox["(q.v.) with offsets, radii, and angles. See ", FontWeight->"Plain"], "toPolar", StyleBox[", ", FontWeight->"Plain"], "toVector", StyleBox[".\n\n", FontWeight->"Plain"], "Clifford ", StyleBox["The anti-commutative sub-set of hoops, CL", FontWeight->"Plain"], StyleBox["pq", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", created by ", FontWeight->"Plain"], StyleBox["cLoop[p,q]", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", with", FontWeight->"Plain"], " ", StyleBox["2^(p+q) elements, \n", FontWeight->"Plain"], "(geometric) ", StyleBox["with p anti-commuting generators squaring to +1 and q squaring to \ -1. Other orderings are\n", FontWeight->"Plain"], "algebras. ", StyleBox[" isomorphic to one of these.", FontWeight->"Plain"], " ", StyleBox["The elements partition into a scalar (first), pseudoscalar (last), \ p+q\n\t relators (", FontWeight->"Plain"], StyleBox["monovectors", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["), and bi-, tri, etc k-vectors. A ", FontWeight->"Plain"], StyleBox["k-blade", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" is a a set of elements of one \n\t dimensionality or \ grade. A ", FontWeight->"Plain"], StyleBox["multivector", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" is a set (often sparse) of coefficients. The\n\t \ non-Abelian ", FontWeight->"Plain"], StyleBox["geometric product,", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" ", FontWeight->"Plain"], StyleBox["ab=a.b+a", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox["\[Wedge]", TraditionalForm]]], StyleBox["b,", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" extends to multiple dimensions, unlike the\n\t related \ dot and cross products.", FontWeight->"Plain"], StyleBox[" ab=a.b+a\[And]b ", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["is sometimes used to define geometric algebra in\n\t \ terms of the dot and wedge products, but corresponds to the Jordan & Lie \ products for any\n\t non-commutative hoop (not just for Cliffords). \ \n\t \n", FontWeight->"Plain"], "CL", StyleBox["pqr ", FontSlant->"Italic"], StyleBox["My extended Clifford-like algebras. CL", FontWeight->"Plain"], StyleBox["pqr", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" have 3-fold symmetry if ", FontWeight->"Plain"], StyleBox["r=", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["3, and are based on non-\n\t associative octonions or \ split-octonions if r=1 or 2. \n\t \n", FontWeight->"Plain"], "cTimes[A,B,o] ", StyleBox[" The hoopTimes multiplication procedure, modified to give AB, A.B, \ and A", FontWeight->"Plain"], Cell[BoxData["\[Wedge]"]], StyleBox["B etc. for\n Cliffords. The ", FontWeight->"Plain"], StyleBox["o", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" parameter is used to give different products (geometric, Lie, \ Jordan, etc.).\n \n", FontWeight->"Plain"], "collapse.", StyleBox[" Used in earlier versions, now replaced by ", FontWeight->"Plain"], "fold,", StyleBox[" replacement of ", FontWeight->"Plain"], StyleBox["r", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["-fold symmetry by ", FontWeight->"Plain"], StyleBox["r", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["-signs.\n\n", FontWeight->"Plain"], "conservative", StyleBox[".The property (discovered by Frobenius for group inverse tables) \ of some multiplication tables\n\t to conserve the determinant on \ vector multiplication. The defining property of hoop\n\t \ vector-division algebras - many signed tables, all groups, and some \ non-associative\n\t octonionic Moufang loops define conservative \ algebras. Generalised-conjugate determinant\n\t factors are \ conserved. Gerrard (2010) showed that these were \ \[OpenCurlyDoubleQuote]irreps\[CloseCurlyDoubleQuote].\n\t \nc", FontWeight->"Plain"], "onformal ", StyleBox["Spaces with signature p,q are described as null vectors in \ cl[p+1,q+1]. Points, lines,\n ", FontWeight->"Plain"], "geometry. ", StyleBox["circles and spheres are geometrical primitives in conformal \ geometry.\n \n", FontWeight->"Plain"], "det.", StyleBox["\tGeneralised Determinant. (Handles rectangular and degenerate \ tables.) The primal parent of\n Determinant, Permanent, and \ Moore-Penrose inverse. det[a_?MatrixQ, r_:2, t_:Times] calculates \n \ a generalized determinant, as a list containing m components, for a (not \ necessarily square)\n matrix. Putting \[OpenCurlyQuote]r\ \[CloseCurlyQuote] ={1,2,3,4,5} gives the {permanent, ditermanent, \ tritermanent, teterminant,\n penterminant}; the traditional \ Determinant is the signed sum of the two ditermanent terms. \n In \ effect, each element of \[OpenCurlyQuote]a\[CloseCurlyQuote] is multiplied by \ a power of the sign \[OpenCurlyQuote]\[DoubleStruckS]\[CloseCurlyQuote], with \ \[DoubleStruckS]^r=1; the final\n result comprises the signed sums \ of all the determinant-like terms with a given power of \[OpenCurlyQuote]\ \[DoubleStruckS]\[CloseCurlyQuote].\n The calculation is recursive \ via a stripped down version, \[OpenCurlyQuote]det1\[CloseCurlyQuote], a \ signed row or column\n matrix being the basis. Short matrices are \ transposed to become \[OpenCurlyQuote]long\[CloseCurlyQuote], i.e. with more \ columns\n than rows. \[OpenCurlyQuote]det\[CloseCurlyQuote] sums \ all the square det\[CloseCurlyQuote]s, as in the Moore-Penrose inverse.\n \ Replacing \[OpenCurlyQuote]t_:Times\[CloseCurlyQuote] by another \ binary operation creates new functions.\n \n", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["d", "i"], TraditionalForm]]], " ", StyleBox["A basis direction for a loop. ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["x", "i"], TraditionalForm]], FontWeight->"Plain"], StyleBox[" is a basis dimension, folded from 2 or more directions.\n\t \ Directions can only be associated with operators or primal coefficients.\n\t \ \n", FontWeight->"Plain"], "dozal, doz. ", StyleBox["The algebra of 12-element directors and vecs, and the unital \ algebra derived from it.\n\n", FontWeight->"Plain"], "dual.\t ", StyleBox["Polar Duals exist in Abelian hoops as a set of offsets, squared \ radii or ulnae, and angles.\n\t Clifford Duals result from exchanging ", FontWeight->"Plain"], StyleBox["k", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" grades with ", FontWeight->"Plain"], StyleBox["p+q-k", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" grades, and are not polar.\n\t ", FontWeight->"Plain"], "\n", StyleBox["D6,D8,", FontWeight->"Plain"], " etc. ", StyleBox["Dihedral groups, symmetries of the triangle, square, etc. Written ", FontWeight->"Plain"], "S3, D8", StyleBox[" when used as hoops.\n \t WikiMaths says that algebraists \ prefer D2n (as here & in GAP) but geometers prefer Dn.\n \t \n", FontWeight->"Plain"], "element.", StyleBox[" ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["a", "1"], FontWeight->"Bold"], TraditionalForm]]], Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["d", "1"], FontWeight->"Bold"], TraditionalForm]]], StyleBox[",", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["b", "2"], TraditionalForm]], FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["x", "2"], TraditionalForm]], FontWeight->"Plain"], StyleBox[", ", FontWeight->"Plain"], StyleBox["c", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" etc. A constituent part of a vec or vector, consisting of a \ coefficient (", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ SubscriptBox["a", "1"], FontWeight->"Bold"], ","}], TraditionalForm]], FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["b", "2"], TraditionalForm]], FontWeight->"Plain"], StyleBox[")\n\t and a direction or dimension (", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["d", "1"], TraditionalForm]]], StyleBox[", ", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox[ SubscriptBox["x", "2"], TraditionalForm]], FontWeight->"Plain"], StyleBox[") (often implied by the index or the position in the set).\n\t \ ", FontWeight->"Plain"], "\nequivalencing. ", StyleBox["The creation of a signed hoop table or vector by folding, an \ equivalence relationship\n\t between sets (pairs, triples, quads etc) of \ primal elements ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["r", "i"], "~", RowBox[{"(", RowBox[{ SubscriptBox["a", "i"], "-", SubscriptBox["a", RowBox[{"i", "+", RowBox[{"m", "/", "r"}]}]]}], ")"}]}], SubscriptBox["x", "i"]}], TraditionalForm]], FontWeight->"Plain"], StyleBox[". Impossible for\n \t some loops, such as C3 & S3, which can \ only be folded after first composing with", FontWeight->"Plain"], " ", StyleBox["C2, etc.\n\t Complex hoops and vectors are folded from quads \ of primal elements, i.e. ", FontWeight->"Plain"], StyleBox["r", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["=4. Generalized or \n\t extended signs are distinct powers of \ roots of unity, created by folding. (See ", FontWeight->"Plain"], "fold", StyleBox[".)\n\t ", FontWeight->"Plain"], "\nextended signs. ", StyleBox["Generalizations of +,-,+\[ImaginaryI],-\[ImaginaryI];", FontWeight->"Plain"], " ", StyleBox["e.g. ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckS]", "\[DoubleStruckR]"], TraditionalForm]], FontWeight->"Plain"], StyleBox[", ", FontWeight->"Plain"], Cell[BoxData[ SuperscriptBox[ StyleBox["\[DoubleStruckN]", FontWeight->"Plain"], "9"]], PageWidth->PaperWidth, CellMargins->{{Inherited, 0.5625}, {Inherited, Inherited}}, InitializationCell->True, FontWeight->"Plain"], StyleBox[", ", FontWeight->"Plain"], Cell[BoxData[ SuperscriptBox["\[DoubleStruckO]", "8"]], PageWidth->PaperWidth, CellMargins->{{Inherited, 0.5625}, {Inherited, Inherited}}, InitializationCell->True, FontWeight->"Plain"], StyleBox[", ", FontWeight->"Plain"], Cell[BoxData[ SuperscriptBox["\[DoubleStruckV]", "7"]], PageWidth->PaperWidth, CellMargins->{{Inherited, 0.5625}, {Inherited, Inherited}}, InitializationCell->True, FontWeight->"Plain"], StyleBox[", ", FontWeight->"Plain"], Cell[BoxData[ SuperscriptBox["\[DoubleStruckH]", "6"]], PageWidth->PaperWidth, CellMargins->{{Inherited, 0.5625}, {Inherited, Inherited}}, InitializationCell->True, FontWeight->"Plain"], StyleBox[", ", FontWeight->"Plain"], Cell[BoxData[ SuperscriptBox["\[DoubleStruckP]", "5"]], PageWidth->PaperWidth, CellMargins->{{Inherited, 0.5625}, {Inherited, Inherited}}, InitializationCell->True, FontWeight->"Plain"], StyleBox[",", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ StyleBox[" ", FontWeight->"Plain"], StyleBox[ SuperscriptBox["\[DoubleStruckCapitalX]", "4"], FontFamily->"Courier New"]}], TraditionalForm]], FontWeight->"Plain"], ".", StyleBox[" ", FontWeight->"Plain"], Cell[BoxData[ SuperscriptBox["\[DoubleStruckCapitalY]", "3"]], PageWidth->PaperWidth, CellMargins->{{Inherited, 0.5625}, {Inherited, Inherited}}, InitializationCell->True, FontWeight->"Plain"], StyleBox[", ", FontWeight->"Plain"], Cell[BoxData[ SuperscriptBox[ StyleBox["\[DoubleStruckCapitalJ]", FontWeight->"Plain"], "3"]], PageWidth->PaperWidth, CellMargins->{{Inherited, 0.5625}, {Inherited, Inherited}}, InitializationCell->True, FontWeight->"Plain"], StyleBox[", ", FontWeight->"Plain"], Cell[BoxData[ SuperscriptBox["\[DoubleStruckK]", "2"]], PageWidth->PaperWidth, CellMargins->{{Inherited, 0.5625}, {Inherited, Inherited}}, InitializationCell->True, FontWeight->"Plain"], StyleBox[", ", FontWeight->"Plain"], Cell[BoxData[ SuperscriptBox["\[DoubleStruckM]", "2"]], PageWidth->PaperWidth, CellMargins->{{Inherited, 0.5625}, {Inherited, Inherited}}, InitializationCell->True, FontWeight->"Plain"], " ", StyleBox["all = +1\n", FontWeight->"Plain"], "\n", Cell[BoxData[ FormBox[ SubscriptBox["f", "j"], TraditionalForm]]], "\t ", StyleBox["The j'th distinct factor of the determinant of ", FontWeight->"Plain"], "G", StyleBox[". It is conserved, if ", FontWeight->"Plain"], "G", StyleBox[" is a hoop, on multiplication;\n\t if it is linear (L1) it is \ also conserved on addition. Distinct factors are ", FontWeight->"Plain"], StyleBox["sizes", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" for the hoop algebra.\n\t The list of ", FontWeight->"Plain"], StyleBox["sizes", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" is the ", FontWeight->"Plain"], StyleBox["shape", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" of a vector. If GAPdb is available, it can be found from smat.\n\t\ \n", FontWeight->"Plain"], "faithful representation. ", StyleBox["Sets of matrices that act as elements of a specified group. \ Created by ", FontWeight->"Plain"], StyleBox["ma", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" as 2x2,\n 3x3, or 4x4 unitary monomial matrices", FontWeight->"Plain"], " ", StyleBox["from basic sets chosen from matrix lists ", FontWeight->"Plain"], StyleBox["u2, u3,", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" or ", FontWeight->"Plain"], StyleBox["u4", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[".\n \n", FontWeight->"Plain"], "flat vector.", StyleBox[" The usual representation of a vector as a flat list. cf ", FontWeight->"Plain"], StyleBox["sparse, graded.\n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["\n", FontWeight->"Plain"], "fold. ", StyleBox["See Equivalencing, signed table, multi-fold. The conversion of a \ loop with ", FontWeight->"Plain"], StyleBox["m", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" unsigned elements to\n a hoop with ", FontWeight->"Plain"], StyleBox["m/r", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" signed elements, generating a signed multiplication table. All \ groups can operate a\n algebras over fields because any group X can \ be considered to be folded from XF, where F is a field\n group such \ as C2 (real) or C4 (complex). Vecs (unsigned sets) fold to generalized signed \ vectors.\n If GAPdb is available, fromdb[m,n,o] creates the hoop \ folded over o-1 central subgroups.\n ", FontWeight->"Plain"], "\nfragment. ", StyleBox["A linear term in a compact size expression e.g. ", FontWeight->"Plain"], StyleBox["(a-b)", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" in (", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"a", "-", "b"}], ")"}], "2"], TraditionalForm]], FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{"+", FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"b", "-", "c"}], ")"}], "2"], TraditionalForm]}], TraditionalForm]], FontWeight->"Plain"], StyleBox["+", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"c", "-", "a"}], ")"}], "2"], TraditionalForm]], FontWeight->"Plain"], StyleBox[")/2; also the\n\t quadratic terms in compact expressions for \ quartic sizes. As they always occur as squares, they\n\t are \"symmetric \ differences\" that do not involve subtraction (despite the minus sign!). The \ entries\n\t in irreducible matrix representations are fragments.\n\t \ \n", FontWeight->"Plain"], "Frobenius conservation. ", StyleBox["The property that Det[A] Det[B]=\[PlusMinus]Det[AB] when vectors \ are multiplied using\n\t some Cayley Tables. Det is the conjugate of that \ of the Cayley table mapped with the vector\n\t elements. Frobenius proved \ that all associative tables (groups) are conservative. I found that\n\t \ Octonions and a few other non-associative Moufang loops are conservative, and \ the property\n\t survives on folding to a signed table. As with fragments, \ the \[PlusMinus] sign indicates a \"symmetric\n\t difference\". For real \ tables, conjugation has no effect, so the sizes are factors of the \ determinant.\n\t \n", FontWeight->"Plain"], "GAP ", StyleBox["The immense Groups, Algorithms, Programmes package, which does not \ handle real or complex\n\t numbers, but uses roots of unity extensively. \ The source of most of the ALGHC and GAPdb data.\n\t \n", FontWeight->"Plain"], "Gaussian Pulse.", StyleBox[" Multi-dimensional", FontWeight->"Plain"], " ", StyleBox["Exp[go + \[ImaginaryI] \[Omega] T+ ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["\[Sum]", RowBox[{"x", ",", "y", ",", ".."}]], TraditionalForm]], FontWeight->"Plain"], StyleBox["[\[ImaginaryI] kx X - jx\.b2(X+xi-vx T)\.b2].\n", FontWeight->"Plain"], "\nG", Cell[BoxData[ FormBox[ RowBox[{":", RowBox[{ RowBox[{ SubscriptBox["d", "i"], "\[Times]", SubscriptBox["d", "j"]}], "\[RightArrow]", " ", SubscriptBox["d", StyleBox["ij", FontWeight->"Plain", FontSlant->"Italic"]]}]}], TraditionalForm]]], " ", StyleBox["A finite loop, group, hoop or groupoid giving the direction or \ basis dimension ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["d", StyleBox["ij", FontWeight->"Plain", FontSlant->"Italic"]], TraditionalForm]]], StyleBox[" resulting\n\t from the multiplication of ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["d", "i"], "&"}], SubscriptBox["d", "j"]}], TraditionalForm]]], ".", StyleBox[" Det[G] is that of the symbolic Cayley Table.\n\t ", FontWeight->"Plain"], "\ngeneralized conjugate.", StyleBox[" An involute set that may give the conserved determinant factors. \ Real sets are their\n\town conjugate. The terplex conjugate of {a,b,c} is \ {a,c,b}. The complex conjugate of {r,i} is {r,-i}.\n\t", FontWeight->"Plain"], "\ngeneralized negation. ", StyleBox["Equivalence relations on sets of ", FontWeight->"Plain"], StyleBox["r", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" primals ", FontWeight->"Plain"], Cell[BoxData[ SubscriptBox["n", "i"]], FontWeight->"Plain"], "~", StyleBox["{", FontWeight->"Plain"], Cell[BoxData[ SubscriptBox["a", "i"]], FontWeight->"Plain"], StyleBox[", ", FontWeight->"Plain"], Cell[BoxData[ SubscriptBox["a", RowBox[{"i", "+", RowBox[{"m", "/", "r"}]}]]], FontWeight->"Plain"], StyleBox[",...", FontWeight->"Plain"], Cell[BoxData[ SubscriptBox["a", RowBox[{"i", "+", RowBox[{"m", RowBox[{ RowBox[{"(", RowBox[{"r", "-", "1"}], ")"}], "/", "r"}]}]}]]], FontWeight->"Plain"], StyleBox["}.\n\t Subtraction is 2-fold ", FontWeight->"Plain"], "additive elimination.\n\t ", StyleBox["\n", FontWeight->"Plain"], "generalized signs. ", StyleBox["Generalizations of +, -, +\[ImaginaryI], -\[ImaginaryI];", FontWeight->"Plain"], " ", StyleBox["with", FontWeight->"Plain"], " ", Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckS]", "\[DoubleStruckR]"], TraditionalForm]], FontWeight->"Plain"], StyleBox["=+1. See \"Extended Signs\" for a proposed standard.\n\n", FontWeight->"Plain"], StyleBox["gi, gp ", FontSlant->"Italic"], " ", StyleBox["Lists created by ", FontWeight->"Plain"], StyleBox["sd", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[", used by ", FontWeight->"Plain"], StyleBox["hoopInverse, cInverse & toPolar", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[". \n\n", FontWeight->"Plain"], "g", StyleBox["mmnn ", FontSlant->"Italic"], StyleBox["A mnemonic for groups given by GAP>SmallGroup(mm,nn), or loops in \ the extended GAP Atlas.\n\n", FontWeight->"Plain"], "grade.", StyleBox[" In Clifford (geometric) algebras, the number of monovectors in \ the dimension of an element. Scalars\n\tare grade 0; bivector elements are \ grade 2 and may represent an oriented area; trivector elements\n\tare grade 3 \ and may represent an oriented volume. ", FontWeight->"Plain"], StyleBox["tocGr[A]", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" converts a flat vector into a list of\n\telements grouped by \ grade; ", FontWeight->"Plain"], StyleBox["tocGr[A,k]", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" implements ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["", "k"], FontWeight->"Plain"], TraditionalForm]]], ",", StyleBox[" selecting every element of grade k.\n\t\n", FontWeight->"Plain"], "graded vector", StyleBox[". A vector stored as a list of lists of increasing grade. cf ", FontWeight->"Plain"], StyleBox["flat, grade, sparse.\n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["\n", FontWeight->"Plain"], "hoopInverse[A,hoop] ", StyleBox["The hoop left-multiplicative inverse Ai, permitting division in \ the specified hoop", FontWeight->"Plain"], ";\n ", StyleBox["AB/A= Ai\[CenterDot]AB=B. Calculated as sums of derivatives of \ shapes wrt \"specific variables\" that are fixed\n by the location \ of the 1's in the index table. In signed tables, each term has to be \ multiplied by the sign\n of each 1. The location and multiplier are \ supplied as ", FontWeight->"Plain"], StyleBox["gi", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" in the ", FontWeight->"Plain"], StyleBox["shape", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" of the hoop. This is a Cramer's\n rule calculation, so \ the inverse has the determinant as a divisor. When this factorises, the \ inverse\n splits into partial-fractions with each size becoming a \ denominator. Division is left-multiplication by\n an inverse. \ Multiplication and division both eliminate infinite partial-fractions \ (created by zero sizes)\n by ", FontWeight->"Plain"], StyleBox["ejecting", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" remainders, giving a renormalised result matching a \ pseudo-inverse. The result is\n\t", FontWeight->"Plain"], StyleBox["projected", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" onto a constrained sub-algebra. ", FontWeight->"Plain"], StyleBox["cInverse ", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["is the same, but A can be sparse or graded.\n\t", FontWeight->"Plain"], StyleBox["gapInverse", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" uses ", FontWeight->"Plain"], StyleBox["smat", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" and avoids the inversion of the full Cayley table.\n\t\n", FontWeight->"Plain"], "hoopPower[A,power] ", StyleBox[" A procedure to convert a vector to a power or root. It raises \ offsets and radii to the\n\t required power, and multiplies angles by the \ power. The target hoop polar form must be known.\n\t If angles exceed 2\[Pi] \ they \"wrap round\", and then ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", FormBox[ SuperscriptBox["A", "p"], TraditionalForm], ")"}], RowBox[{"1", "/", "p"}]], TraditionalForm]]], StyleBox[" will be a rotated version of A. Not\n\t defined for non-abelian \ hoops, which may introduce uncertainty.\n\t \n", FontWeight->"Plain"], "hoopTimes[A,B,hoop] ", StyleBox["The hoop multiplication rule, multiplying vectors A & B according \ to the specified\n\t hoop", FontWeight->"Plain"], ".", StyleBox[" If the multiplicands have disparate zeroes, sizes that become \ zero in the product are\n\t ", FontWeight->"Plain"], StyleBox["ejected", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" into ", FontWeight->"Plain"], StyleBox["remainders", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" to maintain size conservation. The product is then ", FontWeight->"Plain"], StyleBox["projected", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" into a\n\t sub-algebra. ", FontWeight->"Plain"], StyleBox["cTimes, gapTimes", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" are multiplications that ignore remainders.\n\t ", FontWeight->"Plain"], "\nhoop. ", StyleBox["An algebra (H,+,\[CenterDot],\\,/,1) with addition +, Moufang \ \[CenterDot], left \\ & right / division, 1, generalized signs,\n\t and the ", FontWeight->"Plain"], "Frobenius determinant conservation", StyleBox[" property. All groups, a few non-associative\n\t Moufang Loops, \ and signed tables folded from them, are Hoops. They are in the database as\n\t\ ", FontWeight->"Plain"], StyleBox["Protoloop", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" preferred Cayley table isomorphs. Loops with names ending in ", FontWeight->"Plain"], StyleBox["n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" are not hoops.\n\t \n", FontWeight->"Plain"], "hoop algebra", StyleBox[". Multiplication and division of vectors using a hoop \ multiplication table, with addition and\n\t generalized subtraction. \ Powers & roots exist in Abelian hoop algebras.\n\t \n", FontWeight->"Plain"], "helix identity.", StyleBox[" ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubsuperscriptBox["\[Sum]", RowBox[{"n", "=", "1"}], RowBox[{"n", "=", "m"}]], TraditionalForm]], FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"Sin", "[", RowBox[{"t", "+", RowBox[{"2", "n", " ", RowBox[{"\[Pi]", "/", "m"}]}]}], "]"}], RowBox[{"2", "p"}]], TraditionalForm]], FontWeight->"Plain", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox["=m Binomial[2p,p]/", FontWeight->"Plain", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SuperscriptBox["4", "p"], TraditionalForm]], FontWeight->"Plain", FontSlant->"Italic", FontVariations->{"CompatibilityType"->0}], StyleBox[" (p"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["\n\t The generalisation of ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["Sin", "2"], "[", "x", "]"}], "+", RowBox[{ SuperscriptBox["Cos", "2"], "[", "x", "]"}]}], "=", "1."}], TraditionalForm]], FontWeight->"Plain"], " ", StyleBox["Derived by J.J.Thwaites (personal communication).\n\t \n", FontWeight->"Plain"], "hexal, hex. ", StyleBox["The algebra of 6-element directors and vectors, and the unitary \ algebra derived from it.\n\n", FontWeight->"Plain"], "Hub. \t", StyleBox["(provisional term). non-Abelian (non-commutative) hoop vectors \ have inverses that consist of a\n\thub or central vector, and a set of \ null-vector spokes. These introduce uncertainty into division.\n\t", FontWeight->"Plain"], "\nhyperbolic, ", StyleBox["The analog of polar, where a size includes a ", FontWeight->"Plain"], "difference", StyleBox[" of 2 squared fragments. The Klein\n ", FontWeight->"Plain"], "Hypolar. ", StyleBox[" group is", FontWeight->"Plain"], " ", StyleBox["the archetype, with squared \"ulnae\" (a\[PlusMinus]c)^2-(b\ \[PlusMinus]d)^2 and angles arcTanh[a\[PlusMinus]c,b\[PlusMinus]d].\n \n", FontWeight->"Plain"], "\[DoubleStruckH],\[DoubleStruckCapitalJ],\[DoubleStruckK],\[DoubleStruckM] \ etc.", StyleBox[" Generalized signs, different roots of unity. Proposed usage in \ Extended Signs. 'h etc. in ASCII.\n", FontWeight->"Plain"], "\n\[ImaginaryI]\t ", Cell[BoxData[ FormBox[ RadicalBox[ RowBox[{"+", "1"}], "4"], TraditionalForm]], FontWeight->"Plain"], StyleBox[", usually written ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RadicalBox[ RowBox[{"-", "1"}], "2"], TraditionalForm]], FontWeight->"Plain"], StyleBox[", a complex unit element. Ambiguous in Dav, CL", FontWeight->"Plain"], StyleBox["pq", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" & C4C4. etc.\n\n", FontWeight->"Plain"], "incantation. ", StyleBox["My name for the expressions used to create a Cayley table using \ any of the procedures ", FontWeight->"Plain"], StyleBox["ca, cd,\n\t cl, co, fromdb, ge, ma, md, ms, pe,sgroup,", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" or", FontWeight->"Plain"], StyleBox[" ts.\n\t \n", FontWeight->"Plain", FontSlant->"Italic"], "irreducible matrix representations", StyleBox["\t", FontWeight->"Plain"], " (irreps).", StyleBox["\n ", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["\n", FontWeight->"Plain"], "isomorphs", StyleBox[", i", FontWeight->"Plain"], "sotopes", StyleBox[" I define ", FontWeight->"Plain"], StyleBox["isomorphs", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" as loops with renamed (shuffled) elements whilst ", FontWeight->"Plain"], StyleBox["isotopes", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" are\n signed loops with some changed signs that conserve \ the structure, but not the details, of the shape.\n ", FontWeight->"Plain"], "\ninvolutions.", StyleBox[" See Plex-conjugates. Vectors with changed signs of some elements. \ [Lounesto p29]\n\n", FontWeight->"Plain"], "indexed set", StyleBox[". A set in which an index identifies each element, so that \ permuted sets are set isomorphs.\n", FontWeight->"Plain"], "\nJordan product.", StyleBox[" The Abelian component of a non-commutative product, ", FontWeight->"Plain"], StyleBox["a.b", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" in ", FontWeight->"Plain"], StyleBox["ab=a.b+a", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox["\[Wedge]", TraditionalForm]]], StyleBox["b,", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" the inner\n or dot product. Non-associative. See \ Clifford Algebras.\n ", FontWeight->"Plain"], "\nK\t ", StyleBox["The Klein four-group, C2xC2, and the corresponding algebra, which \ has three square roots of\n\t +1, and equivalences to a \"double algebra\" \ with two. See Hyperbolic, Study numbers.\n \n", FontWeight->"Plain"], "KdVB", StyleBox["\t Korteweg-deVries broadened equation.", FontWeight->"Plain"], " ", Cell[BoxData[ FormBox[ RowBox[{"[", " ", RowBox[{ RowBox[{"n", "(", RowBox[{"1", "+", RowBox[{"n", " ", "r"}]}], ")"}], SuperscriptBox["A", RowBox[{"1", "+", RowBox[{"2", "/", "n"}]}]], SubscriptBox["A", "x"]}]}], TraditionalForm]], FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{"+", RowBox[{"(", RowBox[{"r", "-", "1"}], ")"}]}], RowBox[{ SuperscriptBox[ SubscriptBox["A", "x"], "2"], "/", "A"}]}], FontWeight->"Plain"], TraditionalForm]]], Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ RowBox[{" ", RowBox[{"+", SubscriptBox["A", RowBox[{"x", " ", "x"}]]}], "]"}], "x"], FontWeight->"Plain"], TraditionalForm]]], " +\n\t ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ SubscriptBox[ RowBox[{"[", RowBox[{ SuperscriptBox["A", RowBox[{"1", "+", "m"}]], RowBox[{ SuperscriptBox["n", "2"], "(", RowBox[{"r", "+", "m"}], ")"}]}], "]"}], RowBox[{" ", "t"}]], FontWeight->"Plain"], " "}], TraditionalForm]]], StyleBox["=0", FontWeight->"Plain"], ".", StyleBox[" ", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["Some multi-dimensional cases have travelling pulse solutions.\n\t \ \n", FontWeight->"Plain"], "large number ", StyleBox[" The observation that several ratios in physics are \ (approximately) low powers of ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ SuperscriptBox["10", "40"], FontWeight->"Plain"], TraditionalForm]]], ";", StyleBox[" ", FontWeight->"Plain"], "\n hypothesis. ", StyleBox["expressed", FontWeight->"Plain"], " ", StyleBox["by Dirac as", FontWeight->"Plain"], " ", Cell[BoxData[ FormBox[ SuperscriptBox["e", "2"], TraditionalForm]], FontWeight->"Plain"], StyleBox["\[TildeEqual]", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox["10", "40"], TraditionalForm]], FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["G", "m"], TraditionalForm]], FontWeight->"Plain"], StyleBox["t. It may be related to multi-radius polar-orbits. \n ", FontWeight->"Plain"], "\nL1, L2, etc. ", StyleBox["Linear (L1), quadratic (L2) etc. sizes, conserved in an algebra, \ and given by ", FontWeight->"Plain"], StyleBox["shape", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[". L26\[Zeta]\[Lambda]\[Phi] is\n\t the quadratic (L2) size \ which splits into six squared fragments and contains \[Zeta], \[Lambda], & \ \[Phi] in its polar\n\t form. L1 sizes are conserved on addition, all \ sizes are conserved on multiplication for hoops.\n\t Originally \ misnamed \"lengths\"; the n'th root of Ln may be a length, and may provide a \ metric.\n\t \n", FontWeight->"Plain"], "length. ", StyleBox[" A constituent of a Polar, corresponding to either a scalar or, \ if the ", FontWeight->"Plain"], StyleBox["n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["'th root of Ln, to a radius or\n\t ulna in polar or hypolar \ coordinates. (The other constituents are angles and offsets.)\n\t \n", FontWeight->"Plain"], "Lie product.", StyleBox[" The non-Abelian component of a non-commutative product, ", FontWeight->"Plain"], StyleBox["a", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox["\[Wedge]", TraditionalForm]]], StyleBox["b", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" in ", FontWeight->"Plain"], StyleBox["ab=a.b+a", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox["\[Wedge]", TraditionalForm]]], StyleBox["b,", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" the outer\n or dot product. Non-conservative. \ Possesses the Jacobi property, so it is a Lie algebra.\n \n", FontWeight->"Plain"], "limiting ", StyleBox["The non-linear partial differential equation that has any \ fixed-shape fixed speed wave, travelling", FontWeight->"Plain"], "\n velocity ", StyleBox["in any direction", FontWeight->"Plain"], " ", StyleBox["in d spatial and 1 time direction, as a solution. Massive and \ light-like particles arise", FontWeight->"Plain"], "\n eq'n. ", StyleBox["if pairs or directions combine to give dimensions, or if some \ directions are \"rolled-up\" (Kaluza\n\t -Klein). Gtt-\[Sum][Gxx]=0 (with \ Gxx=", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox["d", "2"], TraditionalForm]], FontWeight->"Plain"], StyleBox["G/d", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox["X", "2"], TraditionalForm]], FontWeight->"Plain"], StyleBox[" etc. and summation is over x1...xd).\n\t \n", FontWeight->"Plain"], "loop.\t ", StyleBox["An algebraic loop. An ", FontWeight->"Plain"], StyleBox["m\[Times]m", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" quasigroup Cayley table with a 1, for a set with ", FontWeight->"Plain"], StyleBox["m unsigned elements", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[".\n See Moufang Loop. Protoloops include signed \ tables as well as strict (unsigned) loops.\n ", FontWeight->"Plain"], "\n", StyleBox["mm", FontSlant->"Italic"], "\t ", StyleBox["The number of primal elements in a loop, group or director. Hoops \ have ", FontWeight->"Plain"], StyleBox["mm/r", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" signed elements.\n\n", FontWeight->"Plain"], "monovector.", StyleBox[" My name for the generators (aka relators, gammas) of groups, \ Cayley-Dickson & Clifford\n\t algebras. The unit is scalar, products are \ bivectors, trivectors, etc. Commonly and confusingly\n\t called vectors in \ the Geometric Algebra literature.\n \n", FontWeight->"Plain"], "Moufang Loop. ", StyleBox["An ", FontWeight->"Plain"], StyleBox["m\[Times]m ", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["multiplication table for a set with ", FontWeight->"Plain"], StyleBox["m", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" elements with L & R inverses ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["L", "i"], TraditionalForm]], FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["R", "i"], TraditionalForm]], FontWeight->"Plain"], StyleBox["=", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["R", "i"], TraditionalForm]], FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["L", "i"], TraditionalForm]], FontWeight->"Plain"], StyleBox[" and\n\t the Moufang properties, including ", FontWeight->"Plain"], StyleBox["zx.yz=(z.xy)z,", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" relaxed to \"up to a sign\" for signed tables. All\n\t \ groups are Moufang Loops. A few non-associative Moufang Loops (e.g. \ octonions) are hoops. \n\n", FontWeight->"Plain"], "Multifolding. ", StyleBox["A loop with one or more central cyclic subgroups can be folded on \ each. Abelian hoops can\n\t\tbe folded all the way down to C1 (in a single \ step for prime groups). Folding different\n\t\tisomorphs may produce an \ unsigned algebra, but will usually require the intoduction of\n\t\tpowers of \ the r\[CloseCurlyQuote]th root(s) of unity as generalised signs.\n\t\t\n", FontWeight->"Plain"], "multi-phase ", StyleBox[" ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"n", "=", "1"}], "m"], RowBox[{"c", " ", RowBox[{"Cos", "[", RowBox[{"\[Omega]t", "+", RowBox[{"2", "n", " ", "k", " ", RowBox[{"\[Pi]", "/", "m"}]}], "+", "\[Phi]"}], "]"}], SubscriptBox["\[Zeta]", "n"]}]}], StyleBox["+", FontWeight->"Plain"], StyleBox[ RowBox[{"f", "[", "t", "]"}], FontWeight->"Plain"]}], TraditionalForm]], FontWeight->"Plain"], StyleBox[", a solution to ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["y", "n"], TraditionalForm]], FontWeight->"Plain"], StyleBox["\" = ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox["a", "2"], TraditionalForm]], FontWeight->"Plain"], StyleBox["(", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["y", RowBox[{"n", "-", "1"}]], TraditionalForm]], FontWeight->"Plain"], StyleBox["-2", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["y", "n"], TraditionalForm]], FontWeight->"Plain"], StyleBox["+", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["y", RowBox[{"n", "+", "1"}]], TraditionalForm]], FontWeight->"Plain"], StyleBox[")", FontWeight->"Plain"], "+", StyleBox["f\"[t]", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" or \n ", FontWeight->"Plain"], "sinusoid.", StyleBox["\t", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["y", "n"], TraditionalForm]], FontWeight->"Plain"], StyleBox["' = ", FontWeight->"Plain"], Cell[BoxData[ FormBox["a", TraditionalForm]], FontWeight->"Plain"], StyleBox["(", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["y", RowBox[{"n", "-", "1"}]], TraditionalForm]], FontWeight->"Plain"], StyleBox["-", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["y", RowBox[{"n", "+", "1"}]], TraditionalForm]], FontWeight->"Plain"], StyleBox[")", FontWeight->"Plain"], "+", StyleBox["f'[t]. ", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["e", "r"], RowBox[{"\[DoubleStruckS]", " ", "x"}]], TraditionalForm]]], StyleBox["generalizes ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox["e", RowBox[{"\[ImaginaryI]", " ", "x"}]], TraditionalForm]]], StyleBox["to define them.\n \n", FontWeight->"Plain"], "multivector. ", StyleBox["The Clifford (geometric) algebra name for a general indexed set of \ elements or blades.\n\n", FontWeight->"Plain"], "non-commutative. ", StyleBox["The non-Abelian property that ab", FontWeight->"Plain"], "\[NotEqual]", StyleBox["ba for at least one pair of elements.\n\t Non-commutative \ hoops have repeated determinant factors and so cannot revert polar\n\t \ duals to the Cartesian form.\n\t \n", FontWeight->"Plain"], StyleBox["n", FontSlant->"Italic"], "-potent. ", StyleBox["A vector which is its own ", FontWeight->"Plain"], StyleBox["n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["'th power in a particular algebra, ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox["A", "n"], TraditionalForm]], FontWeight->"Plain"], StyleBox["=", FontWeight->"Plain"], StyleBox["A", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[". An Idempotent if ", FontWeight->"Plain"], StyleBox["n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["=2.\n", FontWeight->"Plain"], "\nOC311. ", StyleBox["A typical continuous polar-orbit, being the ", FontWeight->"Plain"], "C3", StyleBox[" orbit with both sizes equal to 1.\n\n", FontWeight->"Plain"], "octr, octi.", StyleBox[" The non-associative 8-element Octonion & split-Octonion algebras. \ Created by ", FontWeight->"Plain"], StyleBox["cd[-1,-1,", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["\[PlusMinus]", FontSlant->"Italic"], StyleBox["1]\n ", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["&", FontWeight->"Plain"], StyleBox[", cLoop[p,q,1 or 2]", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[". 2-folded from Oct and Q8M.\n \n", FontWeight->"Plain"], "offset.", StyleBox[" The L1 sizes after conversion to polar form. See Radius, Polar \ Form. Symbols \[Alpha], \[Beta], \[Gamma], \[Delta].\n", FontWeight->"Plain"], "\npartition.", StyleBox[" One of the monosized vectors", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{" ", SubscriptBox["B", "n"]}], FontWeight->"Plain"], TraditionalForm]]], StyleBox[" that sum to a vector ", FontWeight->"Plain"], StyleBox["B", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[". Created by multiplication by a\n\t projector. The inverse is \ obtained by summing 1/", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox["B", "n"], FontWeight->"Plain"], TraditionalForm]]], ".\n\t ", StyleBox["\n", FontWeight->"Plain"], "Pauli-\[Sigma] algebra. ", StyleBox["The algebra defined by the Pauli-\[Sigma] 2\[Times]2 matrix \ multiplication tables.\n\tConserves ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ SuperscriptBox["t", "2"], FontWeight->"Plain"], TraditionalForm]]], "-", Cell[BoxData[ FormBox[ StyleBox[ SuperscriptBox["x", "2"], FontWeight->"Plain"], TraditionalForm]]], "-", Cell[BoxData[ FormBox[ StyleBox[ SuperscriptBox["y", "2"], FontWeight->"Plain"], TraditionalForm]]], "-", Cell[BoxData[ FormBox[ StyleBox[ SuperscriptBox["z", "2"], FontWeight->"Plain"], TraditionalForm]]], ".\n", StyleBox["\n", FontWeight->"Plain"], "phase.", StyleBox["\t One or more cyclically varying element in an orbit. A phase has \ a phase angle and a radius or ulna.\n\n", FontWeight->"Plain"], "plex-conjugate. ", StyleBox["The generalization of complex-conjugate to some hoops, a vector \ with some elements\n\t negated or (in ternary algebras) with \ \[DoubleStruckCapitalJ]\[LeftRightArrow]", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["\[DoubleStruckCapitalJ]", FontWeight->"Plain"], "2"], TraditionalForm]]], StyleBox[" ", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["exchanges. Elements of the product of the vector and the\n\t \ conjugates (which are involutions) are related to the sizes of the vec, often \ as signed sums of squared\n\t elements. The generalised complex-conjugate ", FontWeight->"Plain"], StyleBox["plex", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" is supplied (if known) in ", FontWeight->"Plain"], StyleBox["sd", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" for a hoop.\n\t \n", FontWeight->"Plain"], "polar form. ", StyleBox["A Dual representation of a vector as scalars (displacements), \ radii and angles, generalised\n\tpolar coordinates.\n\n", FontWeight->"Plain"], "polar angle.", StyleBox[" An angle parameter in a polar form or orbit. Symbols \[Sigma], \ \[Tau], \[Phi], \[CurlyPhi], \[Chi], \[Psi].\n\n", FontWeight->"Plain"], "polar-orbit. ", StyleBox["A unital parametric representation of a director or vector, with \ multi-phase sinusoidal elements\n\t depending on at least one free angle \ parameter {\[Sigma],\[Tau]..}. Greek letters \[Alpha]\[Ellipsis]\[Kappa] are \ used as the\n\t size parameters. An example is given under ", FontWeight->"Plain"], "Size", StyleBox[". Putting \[Tau] = -(", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SubscriptBox["E", "p"], " ", "t"}], "-"}], TraditionalForm]], FontWeight->"Plain"], StyleBox["P.x)/\[HBar] gives the orbit the\n\t properties of a multi-phase \ deBroglie wave. (Related to the \[OpenCurlyDoubleQuote]discrete orbits\ \[CloseCurlyDoubleQuote] in a group.)\t \n\n", FontWeight->"Plain"], "polyhelix identity.", StyleBox[" ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubsuperscriptBox["\[Sum]", RowBox[{"i", "=", "0"}], RowBox[{"i", "=", RowBox[{"m", "-", "1"}]}]], TraditionalForm]], FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"Sin", "[", RowBox[{"t", "+", RowBox[{"2", RowBox[{"i\[Pi]", "/", "m"}]}]}], "]"}], ")"}], RowBox[{"2", "p"}]], TraditionalForm]], FontWeight->"Plain"], StyleBox["= ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox["4", RowBox[{"-", "p"}]], TraditionalForm]], FontWeight->"Plain"], StyleBox["m ", FontWeight->"Plain", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ { RowBox[{"2", "p"}]}, {"p"} }], "\[NegativeThinSpace]", ")"}], TraditionalForm]], FontWeight->"Plain"], StyleBox[", {p"Plain"], "powerSinusoid. ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox["v", FontWeight->"Plain"], StyleBox["=", FontWeight->"Plain"], FormBox[ StyleBox[ SuperscriptBox[ RowBox[{"Sin", "[", RowBox[{"\[Omega]t", "+", "\[Phi]"}], "]"}], "p"], FontWeight->"Plain"], TraditionalForm]}], TraditionalForm]]], StyleBox[", which is a solution to ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ OverscriptBox["v", "\[DoubleDot]"], " ", "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"1", "/", "p"}], "-", "1"}], ")"}], RowBox[{ FormBox[ RowBox[{" ", SuperscriptBox[ RowBox[{ FormBox[ OverscriptBox["v", "."], TraditionalForm], " "}], "2"]}], TraditionalForm], "/", "v"}]}], "+", RowBox[{"p", " ", FormBox[ SuperscriptBox["\[Omega]", "2"], TraditionalForm], "v"}]}], "=", "0"}], TraditionalForm]], "Input", Evaluatable->False, FontWeight->"Plain"], ". ", StyleBox["The travelling\n\t wave V=(a Sin[", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["\[Sum]", RowBox[{"x", ",", "y", ",", ".."}]], TraditionalForm]], FontWeight->"Plain"], StyleBox["kx*(x+xi-jx*t)]", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox[")", "p"], TraditionalForm]], FontWeight->"Plain"], StyleBox[" is a solution to Vtt+(1/p-1) ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox["Vt", "2"], TraditionalForm]], FontWeight->"Plain"], StyleBox["/V-pV", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["\[Sum]", RowBox[{"x", ",", "y", ",", ".."}]], TraditionalForm]], FontWeight->"Plain"], " \n\t ", StyleBox["\n", FontWeight->"Plain"], "primal, \[DoubleStruckCapitalA]. ", StyleBox["Non-negative continuous, Absolute, half-line numbers. Real and \ Complex numbers are \n\t equivalence-relationships on pairs or quads of \ primal numbers. Other equivalence relations\n\t on ", FontWeight->"Plain"], StyleBox["r", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" primals give Terplex numbers (", FontWeight->"Plain"], StyleBox["r", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["=3) etc.\n\t \n", FontWeight->"Plain"], "protoloop. ", StyleBox["A prefered isomorph of a table, showing any structure; ", FontWeight->"Plain"], StyleBox["shapes", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" are defined for the protoloop. The \n\t ", FontWeight->"Plain"], StyleBox["makeProtoloop (mp)", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" procedure generates the Cayley table (with options to create some \ isomorphs).\n\t The canonical form is the \[OpenCurlyDoubleQuote]smallest\ \[CloseCurlyDoubleQuote] isomorph when flattened to an ordered set.\n\t \n", FontWeight->"Plain"], "projector. ", StyleBox["A vector ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["A", FontWeight->"Plain"], "n"], FontWeight->"Plain"], TraditionalForm]]], StyleBox[" that projects any vector onto a monosized partition ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["B", FontWeight->"Plain"], "n"], FontWeight->"Plain"], TraditionalForm]]], StyleBox[": ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ SubscriptBox["A", "n"], "*", "B"}], FontWeight->"Plain"], TraditionalForm]]], " ", StyleBox["= ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ SubscriptBox["B", "n"], "+", RowBox[{ SubscriptBox["R", "n"], "."}]}], FontWeight->"Plain"], TraditionalForm]]], " ", StyleBox["(", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["R", "n"], TraditionalForm]], FontWeight->"Plain"], " ", StyleBox["is a\n\tremainder.) It is a 2-potent vector (", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox["A", "2"], TraditionalForm]], FontWeight->"Plain"], StyleBox["=", FontWeight->"Plain"], StyleBox["A", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[") that commutes with everything. Aka sub-space identity.\n\t\n", FontWeight->"Plain"], "projective geometry. ", StyleBox["Spaces with signature p,q are described as null vectors in \ cl[p+1,q]. This has\n\t points and line as geometrical primitives.\n\t\ \t\n", FontWeight->"Plain"], "pseudo-root. ", StyleBox[" Non-abelian polar duals involve non-conserved size parameters \ and angle that do not add on\n\t multiplication, but which have valid \ reversions to the cartesian form. When used to calculate roots\n\t and \ powers, arbitrary angles and uncertainty are introduced, giving rotated ", FontWeight->"Plain"], StyleBox["pseudo-roots", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[".\n\t \n", FontWeight->"Plain"], "pseudoscalar. ", StyleBox["\[GothicCapitalI], the highest grade in a Clifford algebra, ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["e", "1"], TraditionalForm]]], Cell[BoxData[ FormBox["\[Wedge]", TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox["e", "1"], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"\[Wedge]", "..."}], TraditionalForm]]], Cell[BoxData[ FormBox[ SubscriptBox["e", "n"], TraditionalForm]]], ", ", StyleBox["which replaces", FontWeight->"Plain"], " \[ImaginaryI]", StyleBox[" in many\n algebras, but which anti-commutes with \ other elements if ", FontWeight->"Plain"], StyleBox["n", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" is even.\n \n", FontWeight->"Plain"], "radius. ", StyleBox["The length associated with an angle in a polar representation. \ Symbols ", FontWeight->"Plain"], "\[Epsilon], \[Zeta], \[Eta], \[Kappa], \[Lambda]", StyleBox[". See Ulna.\n\n", FontWeight->"Plain"], "rational pulse.", StyleBox[" ", FontWeight->"Plain"], StyleBox["v", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["=", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"a", "/", RowBox[{"(", RowBox[{"1", "+", RowBox[{ SubscriptBox["\[Sum]", RowBox[{"x", ",", "y", ",", ".."}]], RowBox[{ RowBox[{"jx", "(", RowBox[{"xi", "+", "x", "-", RowBox[{"kx", " ", "t"}]}], ")"}], "^", "2"}]}]}], ")"}]}], ")"}], "n"], TraditionalForm]], FontWeight->"Plain"], StyleBox[" \n\n", FontWeight->"Plain"], "relator. ", StyleBox["The GAP term for a generator, gamma, or monovector, a primitive \ root of unity used to define\n a table in terms of powers and \ relationships.\n \n", FontWeight->"Plain"], "remainder. ", StyleBox["Multiplication (or division) by vectors with zero-sizes creates \ left and right \"remainders\" Rr, Rl\n\twith the sizes that are present in \ the multiplicands but lost from the product, ensuring that hoops are\n\t\ conservative and \"division by zero\" is eliminated. \ A*B\[RightArrow]AB+Rr+Rl, AB/B+Rr=A, AB/A+Rl=B.\n\t\n", FontWeight->"Plain"], "\[DoubleStruckS] ", StyleBox["The generalised sign or ", FontWeight->"Plain"], StyleBox["\[DoubleStruckR]", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["'th root of unity, ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox["\[DoubleStruckS]", StyleBox["\[DoubleStruckR]", FontWeight->"Plain", FontSlant->"Italic"]], TraditionalForm]], FontWeight->"Plain"], StyleBox["=1, often incorrectly treated as a point on the complex\n\tunit \ circle, but best considered as a ", FontWeight->"Plain"], StyleBox["direction", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[". I write it as 's in printable ASCII and in irreps or smats.\n\t\ \n", FontWeight->"Plain"], "selector. ", StyleBox["A vector with one size=1and the rest=0 in a particular hoop, that \ commutes with all elements.\n\tMultiplying a vector by a full set of \ selectors creates a partition.\n\t", FontWeight->"Plain"], "\nsh[mnemonic] ", StyleBox["The symbolic determinant factors of the named loop, for use with ", FontWeight->"Plain"], StyleBox["genInverse, gi", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" and ", FontWeight->"Plain"], StyleBox["gp\n\t ", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["for use with ", FontWeight->"Plain"], StyleBox["hoopInverse,", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" ", FontWeight->"Plain"], StyleBox["toPolar, toVector", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[". It includes a ", FontWeight->"Plain"], StyleBox["plex", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" for many small hoops.\n\t ", FontWeight->"Plain"], "\n", StyleBox["shape. ", FontFamily->"Times New Roman"], StyleBox["The sizes ", FontFamily->"Times New Roman", FontWeight->"Plain"], StyleBox["of a director or orbit, found by ", FontWeight->"Plain"], StyleBox["shape[vec,mnemonic]", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["; needed by ", FontWeight->"Plain"], StyleBox["genInverse, toPol", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" etc.\n", FontWeight->"Plain"], StyleBox["\n", FontWeight->"Plain", FontSlant->"Italic"], "shape[{\[OSlash]_,a_,b_...}] ", StyleBox["Factor (size) data for the target hoop, supplying the shape of {\ \[OSlash],a,b,..} as a list of individual\n\t sizes {", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["f", "1"], TraditionalForm]], FontWeight->"Plain"], StyleBox[",\[Ellipsis]", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["f", "k"], TraditionalForm]], FontWeight->"Plain"], StyleBox["} & setting ", FontWeight->"Plain"], StyleBox["gi, gp", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" for ", FontWeight->"Plain"], StyleBox["genInverse, toPolar", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[". It also sets ", FontWeight->"Plain"], StyleBox["plex", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" for many small hoops.\n\t ", FontWeight->"Plain"], "\nsigned table. ", StyleBox["A multiplication table for a set which forms a group or loop on \ appending its signed elements.\n\tThe multiplication table is the upper left \ quarter (ninth, sixteenth, etc.) of the full Cayley table, with\n\t (possibly \ signed) elements folded, by equivalence relationships, from the full table \ elements.\n\t \n", FontWeight->"Plain"], "size.", StyleBox["\tThe value of a single conserved property of a director or orbit, \ for a given hoop algebra. Thus the\n polar-orbit ", FontWeight->"Plain"], StyleBox["OC3\[Alpha]\[Gamma]={\[Alpha]+2\[Gamma] Cos[\[Tau]],\[Alpha]+2\ \[Gamma] Cos[\[Tau]+2\[Pi]/3 ],\[Alpha]+2\[Gamma] Cos[\[Tau]+4 \[Pi]/3]}/3 \ has individual sizes\n L1\[Alpha]=\[Alpha], L23\[Zeta]2=", FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox[ SuperscriptBox["\[Zeta]", "2"], FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Outline"->False, "Shadow"->False, "Underline"->False}], TraditionalForm]]], ", ", StyleBox["and so has a shape {", FontWeight->"Plain"], StyleBox["\[Alpha],", FontFamily->"Times New Roman", FontWeight->"Plain"], Cell[BoxData[ FormBox[ StyleBox["\[Zeta]", FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Outline"->False, "Shadow"->False, "Underline"->False}], TraditionalForm]]], StyleBox["}, in the algebra ", FontFamily->"Times New Roman", FontWeight->"Plain"], StyleBox["C3", FontFamily->"Times New Roman"], StyleBox["; (hyper)complex-conjugate\n Eigenvector pairs ", FontFamily->"Times New Roman", FontWeight->"Plain"], StyleBox["are sizes factorized into complex pairs (etc) but are not always \ conserved. Sizes\n are always symmetrical ideals, and may ", FontWeight->"Plain"], StyleBox["fragment", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" to compact expressions. Irreducible matrix\n \ representations become sizes when mapped with vectors\n\t\n", FontWeight->"Plain"], "spin. \t", StyleBox["A property of an orbit, which is reversed by reversing the \ sequence of the phases. Clockwise spin \n\t (Cos, -Sin, -Cos, Sin) is taken \ to be positive or Up.\n\t \n", FontWeight->"Plain"], "splitting. ", StyleBox["The primal vector equivalent of subtraction, Split[A,B]=C if \ B+C=A. See SymDiff.\n\n", FontWeight->"Plain"], "Spoke. ", StyleBox["(provisional term) See Hub.\n\n", FontWeight->"Plain"], "Study numbers. ", StyleBox[" a+\[DoubleStruckK] b, ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SuperscriptBox[ StyleBox["\[DoubleStruckK]", FontWeight->"Plain"], "2"], TraditionalForm]], FontWeight->"Plain"], StyleBox["=1,", FontWeight->"Plain"], " ", StyleBox["folded C2kC2, the algebra of the hyperbolic plane. Louenesto 2001 \ p24.\n\n", FontWeight->"Plain"], "symDiff.", StyleBox[" Unsigned difference between numbers s=|a,b| : \ s+Min[a,b]=Max[a,b]. Primal \"additive inverse\".\n\n", FontWeight->"Plain"], "tocGr. ", StyleBox["Converts a flat vector to a graded vector. ", FontWeight->"Plain"], StyleBox["tocGr[A,k]", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" selects the ", FontWeight->"Plain"], StyleBox["k", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["'th grade of A, i.e. ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["", "k"], TraditionalForm]]], "\n", StyleBox["\n", FontWeight->"Plain"], "toPolar[v,h]. ", StyleBox["The procedures to convert cartesian representations ", FontWeight->"Plain"], StyleBox["v", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" into polar forms ", FontWeight->"Plain"], StyleBox["p", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" using hoop ", FontWeight->"Plain"], StyleBox["h", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[".\n\n", FontWeight->"Plain"], "toVector[p,h].\t", StyleBox["The procedures to convert polar representations ", FontWeight->"Plain"], StyleBox["p", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" into cartesian forms ", FontWeight->"Plain"], StyleBox["v", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" using hoop ", FontWeight->"Plain"], StyleBox["h", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[".\n\n", FontWeight->"Plain"], "ternegation.\t", StyleBox["-(-(-A))=A in systems with ternary symmetry, instead of -(-A)=A in \ simple (bi)negation.\n\n", FontWeight->"Plain"], "terplex. ", StyleBox["The algebra of the C3 table over the real or complex field.\n\n", FontWeight->"Plain"], "tetral. ", StyleBox["The algebra of the C4 and K tables over the real or complex field.\ \n\n", FontWeight->"Plain"], "triality", StyleBox[". (Louenesto p309) 'sends a simple rotation to a positive \ isoclinic rotation and a positive isoclinic\n rotation to a \ negative isoclinic rotation'. The Moufang identity results in a special case \ of Cartan's\n triality. See 'Loops' in the GAP package.\n \ \n", FontWeight->"Plain"], "trivector ", StyleBox[" ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["e", "i"], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{"\[Wedge]", RowBox[{ SubscriptBox["e", "j"], "\[Wedge]", SubscriptBox["e", "k"]}]}], TraditionalForm]]], ",", StyleBox[" the wedge product of three distinct monovectors, an element of a \ Geometric\n Algebra. It represents a directed volume, but \ carries no shape information. \n \n", FontWeight->"Plain"], "ulna.\t", StyleBox["The hyperbolic analogue of radius.", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SuperscriptBox["u", "2"], TraditionalForm]], FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["=", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SuperscriptBox["x", "2"], TraditionalForm]], FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], StyleBox["-", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], Cell[BoxData[ FormBox[ SuperscriptBox["y", "2"], TraditionalForm]], FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], ".\n\nunity. ", StyleBox["The object {1,0,\[Ellipsis],0} in unitary algebras, which leaves \ any object unchanged after multiplication.\n\t Not to be confused with the \ identity matrix. The product of a vector and its inverse is unity.\n\t ", FontWeight->"Plain"], "\nunit. ", StyleBox["An object (e.g. {2/3,-1/3,-1/3} in ", FontWeight->"Plain"], "C3", StyleBox[") having all sizes 0 or 1 in a particular algebra.\n\n", FontWeight->"Plain"], "unital.\t", StyleBox["A vec or vector with sizes that multiply to 1 in a specific \ algebra, ignoring zeroes, is unital. Orbits\n\t are unital. (Clashes with \ \"unital algebras\", which are monoids and include all hoops.)\n\t \n", FontWeight->"Plain"], "vec. ", StyleBox["An ordered list of m primal elements. Folds to an m/2 real, m/3 \ ternary or m/4 complex vector,\n\t in algebras with 2, 3, or 4-fold symmetry.\ \n\t \n", FontWeight->"Plain"], "vector-division algebra.\t", StyleBox["A vector space with an invertible bilinear vector-multiplication \ operation\n\tand an\taddition operation with additive elimination. The \ multiplication table will have the\n\tMoufang division property, so every \ vector has a multiplicative inverse; if it is folded over a\n\tcentral C2 or \ C4 subgroup the addition will have subtraction as its inverse.\n\t \n", FontWeight->"Plain"], "\[Alpha],\[Beta],\[Ellipsis]\[Lambda] ", StyleBox["Size parameters in an orbit, also the corresponding length \ parameters in a Polar Dual.\n\n", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["x", "i"], TraditionalForm]]], StyleBox["\t The i'th basis dimension. Two or more directions may fold to a \ real dimension ", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["x", "i"], TraditionalForm]], FontWeight->"Plain"], StyleBox["~{", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["d", "i"], TraditionalForm]], FontWeight->"Plain"], StyleBox[",", FontWeight->"Plain"], Cell[BoxData[ FormBox[ SubscriptBox["d", RowBox[{"i", "+", RowBox[{"m", "/", "2"}]}]], TraditionalForm]], FontWeight->"Plain"], StyleBox["}. \n\n", FontWeight->"Plain"], "\[Sigma],\[Tau],\[Ellipsis],\[Psi]", StyleBox[" A frequency parameter in an orbit. 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