> -------------------------------------------------- ;; Loading the Kenzo program. -------------------------------------------------- > -------------------------------------------------- (LOAD-CFILES) -------------------------------------------------- > -------------------------------------------------- ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\classes.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\macros.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\various.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\combinations.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\chain-complexes.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\chcm-elementary-op.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\effective-homology.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\searching-homology.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\homology-groups.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\cones.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\tensor-products.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\koszul.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\groebner.fasl ; Fast loading C:\Docume~1\Francis\AA\Kenzo\GiftQ\smithQ.fasl --- done --- -------------------------------------------------- > -------------------------------------------------- ;; Kenzo expression of the Macaulay ideal: ;; | t5-x t3y-x2 t2y2-xz t3z-y2 t2x-y tx2-z x3-ty2 y3-x2z xy-tz | ;; = the Groebner basis of ;; ideal(x-t^5, y-t^7, z-t^11) -------------------------------------------------- > -------------------------------------------------- (SETF IDEAL (LIST (CMBN 0 1 '(5 0 0 0) -1 '(0 1 0 0)) ;; t5 - x (CMBN 0 1 '(3 0 1 0) -1 '(0 2 0 0)) ;; t3y - x2 (CMBN 0 1 '(2 0 2 0) -1 '(0 1 0 1)) ;; t2y2 - xz (CMBN 0 1 '(3 0 0 1) -1 '(0 0 2 0)) ;; t3z - y2 (CMBN 0 1 '(2 1 0 0) -1 '(0 0 1 0)) ;; t2x - y (CMBN 0 1 '(1 2 0 0) -1 '(0 0 0 1)) ;; tx2 - z (CMBN 0 1 '(0 3 0 0) -1 '(1 0 2 0)) ;; x3 - ty2 (CMBN 0 1 '(0 0 3 0) -1 '(0 2 0 1)) ;; y3 - x2z (CMBN 0 1 '(0 1 1 0) -1 '(1 0 0 1))));; xy - tz -------------------------------------------------- > -------------------------------------------------- ( ----------------------------------------------------------------------{CMBN 0} <1 * (5 0 0 0)> <-1 * (0 1 0 0)> ------------------------------------------------------------------------------ ----------------------------------------------------------------------{CMBN 0} <1 * (3 0 1 0)> <-1 * (0 2 0 0)> ------------------------------------------------------------------------------ ----------------------------------------------------------------------{CMBN 0} <1 * (2 0 2 0)> <-1 * (0 1 0 1)> ------------------------------------------------------------------------------ ----------------------------------------------------------------------{CMBN 0} <1 * (3 0 0 1)> <-1 * (0 0 2 0)> ------------------------------------------------------------------------------ ----------------------------------------------------------------------{CMBN 0} <1 * (2 1 0 0)> <-1 * (0 0 1 0)> ------------------------------------------------------------------------------ ----------------------------------------------------------------------{CMBN 0} <1 * (1 2 0 0)> <-1 * (0 0 0 1)> ------------------------------------------------------------------------------ ----------------------------------------------------------------------{CMBN 0} <1 * (0 3 0 0)> <-1 * (1 0 2 0)> ------------------------------------------------------------------------------ ----------------------------------------------------------------------{CMBN 0} <1 * (0 0 3 0)> <-1 * (0 2 0 1)> ------------------------------------------------------------------------------ ----------------------------------------------------------------------{CMBN 0} <1 * (0 1 1 0)> <-1 * (1 0 0 1)> ------------------------------------------------------------------------------ ) -------------------------------------------------- > -------------------------------------------------- ;; Computing the minimal resolution. -------------------------------------------------- > -------------------------------------------------- (DISPLAY-RESOLUTION IDEAL) -------------------------------------------------- > -------------------------------------------------- G-0-1 ----------------------------------------------------------------------{CMBN -1} ------------------------------------------------------------------------------ G-1-1 ----------------------------------------------------------------------{CMBN 0} <-1 * (G-0-1 2 1 0 0)> <1 * (G-0-1 0 0 1 0)> ------------------------------------------------------------------------------ G-1-2 ----------------------------------------------------------------------{CMBN 0} <-1 * (G-0-1 1 2 0 0)> <1 * (G-0-1 0 0 0 1)> ------------------------------------------------------------------------------ G-1-3 ----------------------------------------------------------------------{CMBN 0} <-1 * (G-0-1 5 0 0 0)> <1 * (G-0-1 0 1 0 0)> ------------------------------------------------------------------------------ G-2-1 ----------------------------------------------------------------------{CMBN 1} <1 * (G-1-2 5 0 0 0)> <-1 * (G-1-2 0 1 0 0)> <-1 * (G-1-3 1 2 0 0)> <1 * (G-1-3 0 0 0 1)> ------------------------------------------------------------------------------ G-2-2 ----------------------------------------------------------------------{CMBN 1} <1 * (G-1-1 5 0 0 0)> <-1 * (G-1-1 0 1 0 0)> <-1 * (G-1-3 2 1 0 0)> <1 * (G-1-3 0 0 1 0)> ------------------------------------------------------------------------------ G-2-3 ----------------------------------------------------------------------{CMBN 1} <1 * (G-1-1 1 2 0 0)> <-1 * (G-1-1 0 0 0 1)> <-1 * (G-1-2 2 1 0 0)> <1 * (G-1-2 0 0 1 0)> ------------------------------------------------------------------------------ G-3-1 ----------------------------------------------------------------------{CMBN 2} <-1 * (G-2-1 2 1 0 0)> <1 * (G-2-1 0 0 1 0)> <1 * (G-2-2 1 2 0 0)> <-1 * (G-2-2 0 0 0 1)> <-1 * (G-2-3 5 0 0 0)> <1 * (G-2-3 0 1 0 0)> ------------------------------------------------------------------------------ NIL -------------------------------------------------- > -------------------------------------------------- ;; How many chain-complexes in our environment? -------------------------------------------------- > -------------------------------------------------- (COUNT-IF #'(LAMBDA (ITEM) (TYPEP ITEM 'CHAIN-COMPLEX)) *K-LIST*) -------------------------------------------------- > -------------------------------------------------- 76 -------------------------------------------------- > -------------------------------------------------- ;; How many chain-complex morphisms? -------------------------------------------------- > -------------------------------------------------- (COUNT-IF #'(LAMBDA (ITEM) (TYPEP ITEM 'MORPHISM)) *K-LIST*) -------------------------------------------------- > -------------------------------------------------- 616 -------------------------------------------------- > -------------------------------------------------- ;; How many Koszul complexes? -------------------------------------------------- > -------------------------------------------------- (COUNT-IF #'(LAMBDA (ITEM) (MEMBER (FIRST (DFNT ITEM)) '(K-COMPLEX/GI K-COMPLEX/I K-COMPLEX))) *K-LIST*) -------------------------------------------------- > -------------------------------------------------- 19 -------------------------------------------------- > -------------------------------------------------- ;; How many cones? -------------------------------------------------- > -------------------------------------------------- (COUNT 'CONE *K-LIST* :KEY #'(LAMBDA (ITEM) (FIRST (DFNT ITEM)))) -------------------------------------------------- > -------------------------------------------------- 47 -------------------------------------------------- > -------------------------------------------------- ;; Examining the morphisms which have been called ;; more than 1,000 times. -------------------------------------------------- > -------------------------------------------------- (DOLIST (ITEM *K-LIST*) (WHEN (AND (TYPEP ITEM 'MORPHISM) (< 1000 (???-CLNM ITEM))) (FORMAT T " ~7D = ~A~%" (???-CLNM ITEM) ITEM))) -------------------------------------------------- > -------------------------------------------------- 1109 = [K732 Morphism (degree 0): K5 -> K745] 1109 = [K718 Morphism (degree 0): K18 -> K713] 1139 = [K703 Morphism (degree 0): K11 -> K681] 1139 = [K687 Morphism (degree 0): K30 -> K681] 2248 = [K670 Morphism (degree 0): K9 -> K648] 1994 = [K669 Morphism (degree 0): K648 -> K9] 2248 = [K654 Morphism (degree 0): K557 -> K648] 1994 = [K653 Morphism (degree 0): K648 -> K557] 4514 = [K637 Morphism (degree 0): K550 -> K615] 4006 = [K636 Morphism (degree 0): K615 -> K550] 4514 = [K621 Morphism (degree 0): K570 -> K615] 4006 = [K620 Morphism (degree 0): K615 -> K570] 9044 = [K604 Morphism (degree 0): K563 -> K580] 8028 = [K603 Morphism (degree 0): K580 -> K563] 17072 = [K590 Morphism (degree 0): K580 -> K580] 9044 = [K583 Morphism (degree 0): K563 -> K580] 8028 = [K582 Morphism (degree 0): K580 -> K563] 9044 = [K579 Morphism (degree 0): K563 -> K1] 8028 = [K578 Morphism (degree 0): K1 -> K563] 9044 = [K577 Morphism (degree 0): K1 -> K1] 4514 = [K573 Morphism (degree 0): K550 -> K570] 4006 = [K572 Morphism (degree 0): K570 -> K550] 4514 = [K568 Morphism (degree 0): K550 -> K563] 4006 = [K567 Morphism (degree 0): K563 -> K550] 4514 = [K566 Morphism (degree 0): K563 -> K563] 4514 = [K564 Morphism (degree -1): K563 -> K563] 2248 = [K560 Morphism (degree 0): K9 -> K557] 1994 = [K559 Morphism (degree 0): K557 -> K9] 2248 = [K555 Morphism (degree 0): K9 -> K550] 1994 = [K554 Morphism (degree 0): K550 -> K9] 2248 = [K553 Morphism (degree 0): K550 -> K550] 2248 = [K551 Morphism (degree -1): K550 -> K550] 1209 = [K544 Morphism (degree 0): K24 -> K522] 1209 = [K528 Morphism (degree 0): K45 -> K522] 1209 = [K511 Morphism (degree 0): K36 -> K492] 1005 = [K510 Morphism (degree 0): K492 -> K36] 1209 = [K497 Morphism (degree 0): K482 -> K492] 1005 = [K496 Morphism (degree 0): K492 -> K482] 1209 = [K485 Morphism (degree 0): K36 -> K482] 1005 = [K484 Morphism (degree 0): K482 -> K36] 1209 = [K481 Morphism (degree 0): K36 -> K66] 1005 = [K480 Morphism (degree 0): K66 -> K36] 1209 = [K479 Morphism (degree 0): K66 -> K66] 1267 = [K472 Morphism (degree 0): K38 -> K450] 1267 = [K456 Morphism (degree 0): K60 -> K450] 1267 = [K439 Morphism (degree 0): K51 -> K420] 1025 = [K438 Morphism (degree 0): K420 -> K51] 1267 = [K425 Morphism (degree 0): K410 -> K420] 1025 = [K424 Morphism (degree 0): K420 -> K410] 1267 = [K413 Morphism (degree 0): K51 -> K410] 1025 = [K412 Morphism (degree 0): K410 -> K51] 1267 = [K409 Morphism (degree 0): K51 -> K108] 1025 = [K408 Morphism (degree 0): K108 -> K51] 1267 = [K407 Morphism (degree 0): K108 -> K108] 1345 = [K400 Morphism (degree 0): K53 -> K381] 1345 = [K386 Morphism (degree 0): K75 -> K381] 1421 = [K371 Morphism (degree 0): K68 -> K349] 1421 = [K355 Morphism (degree 0): K87 -> K349] 5198 = [K338 Morphism (degree 0): K66 -> K319] 4027 = [K337 Morphism (degree 0): K319 -> K66] 5198 = [K324 Morphism (degree 0): K309 -> K319] 4027 = [K323 Morphism (degree 0): K319 -> K309] 5198 = [K312 Morphism (degree 0): K66 -> K309] 4027 = [K311 Morphism (degree 0): K309 -> K66] 5198 = [K308 Morphism (degree 0): K66 -> K93] 4027 = [K307 Morphism (degree 0): K93 -> K66] 5198 = [K306 Morphism (degree 0): K93 -> K93] 1454 = [K299 Morphism (degree 0): K81 -> K277] 1454 = [K283 Morphism (degree 0): K102 -> K277] 11872 = [K266 Morphism (degree 0): K93 -> K242] 9045 = [K265 Morphism (degree 0): K242 -> K93] 20917 = [K252 Morphism (degree 0): K242 -> K242] 11872 = [K245 Morphism (degree 0): K93 -> K242] 9045 = [K244 Morphism (degree 0): K242 -> K93] 11872 = [K241 Morphism (degree 0): K93 -> K1] 9045 = [K240 Morphism (degree 0): K1 -> K93] 11872 = [K239 Morphism (degree 0): K1 -> K1] 1476 = [K232 Morphism (degree 0): K95 -> K210] 1476 = [K216 Morphism (degree 0): K117 -> K210] 4020 = [K199 Morphism (degree 0): K108 -> K175] 3045 = [K198 Morphism (degree 0): K175 -> K108] 7065 = [K185 Morphism (degree 0): K175 -> K175] 4020 = [K178 Morphism (degree 0): K108 -> K175] 3045 = [K177 Morphism (degree 0): K175 -> K108] 4020 = [K174 Morphism (degree 0): K108 -> K1] 3045 = [K173 Morphism (degree 0): K1 -> K108] 4020 = [K172 Morphism (degree 0): K1 -> K1] 1514 = [K165 Morphism (degree 0): K110 -> K128] 2461 = [K149 Morphism (degree 0): K128 -> K128] 95030 = [K134 Morphism (degree 0): K1 -> K1] 1514 = [K131 Morphism (degree 0): K110 -> K128] 1514 = [K126 Morphism (degree 0): K110 -> K1] 1514 = [K124 Morphism (degree 0): K1 -> K1] 1476 = [K120 Morphism (degree 0): K95 -> K117] 1476 = [K115 Morphism (degree 0): K95 -> K110] 1476 = [K113 Morphism (degree 0): K110 -> K108] 1476 = [K111 Morphism (degree -1): K110 -> K110] 1267 = [K109 Morphism (degree -1): K108 -> K108] 1454 = [K105 Morphism (degree 0): K81 -> K102] 1454 = [K100 Morphism (degree 0): K81 -> K95] 1454 = [K98 Morphism (degree 0): K95 -> K93] 1454 = [K96 Morphism (degree -1): K95 -> K95] 5198 = [K94 Morphism (degree -1): K93 -> K93] 1421 = [K90 Morphism (degree 0): K68 -> K87] 1421 = [K86 Morphism (degree 0): K68 -> K81] 1421 = [K84 Morphism (degree 0): K81 -> K66] 1421 = [K82 Morphism (degree -1): K81 -> K81] 1345 = [K78 Morphism (degree 0): K53 -> K75] 1345 = [K73 Morphism (degree 0): K53 -> K68] 1345 = [K71 Morphism (degree 0): K68 -> K66] 1345 = [K69 Morphism (degree -1): K68 -> K68] 1209 = [K67 Morphism (degree -1): K66 -> K66] 1267 = [K63 Morphism (degree 0): K38 -> K60] 1267 = [K58 Morphism (degree 0): K38 -> K53] 1267 = [K56 Morphism (degree 0): K53 -> K51] 1267 = [K54 Morphism (degree -1): K53 -> K53] 1209 = [K48 Morphism (degree 0): K24 -> K45] 1209 = [K43 Morphism (degree 0): K24 -> K38] 1209 = [K41 Morphism (degree 0): K38 -> K36] 1209 = [K39 Morphism (degree -1): K38 -> K38] 1139 = [K33 Morphism (degree 0): K11 -> K30] 1139 = [K29 Morphism (degree 0): K11 -> K24] 1139 = [K27 Morphism (degree 0): K24 -> K9] 1139 = [K25 Morphism (degree -1): K24 -> K24] 1109 = [K21 Morphism (degree 0): K7 -> K18] 1109 = [K16 Morphism (degree 0): K7 -> K11] 1109 = [K14 Morphism (degree 0): K11 -> K9] 1109 = [K12 Morphism (degree -1): K11 -> K11] 52828 = [K2 Morphism (degree -1): K1 -> K1] NIL -------------------------------------------------- > -------------------------------------------------- ;; How many homotopy operators ?? -------------------------------------------------- > -------------------------------------------------- (COUNT-IF #'(LAMBDA (ITEM) (AND (TYPEP ITEM 'MORPHISM) (= 1 (DEGR ITEM)))) *K-LIST*) -------------------------------------------------- > -------------------------------------------------- 199 -------------------------------------------------- > -------------------------------------------------- ;; Sorting these operators according to work amount. -------------------------------------------------- > -------------------------------------------------- (LET ((RSLT +EMPTY-LIST+)) (DOLIST (ITEM *K-LIST*) (WHEN (AND (TYPEP ITEM 'MORPHISM) (= 1 (DEGR ITEM))) (LET ((CLNM (+ (?-CLNM ITEM) (???-CLNM ITEM)))) (WHEN (PLUSP CLNM) (PUSH (LIST CLNM ITEM) RSLT))))) (SETF RSLT (SORT RSLT #'<= :KEY #'FIRST)) (MAP NIL #'PRINT RSLT)) -------------------------------------------------- > -------------------------------------------------- (1 [K596 Morphism (degree 1): K137 -> K1]) (1 [K595 Morphism (degree 1): K137 -> K1]) (1 [K258 Morphism (degree 1): K137 -> K1]) (1 [K257 Morphism (degree 1): K137 -> K1]) (1 [K191 Morphism (degree 1): K137 -> K1]) (1 [K190 Morphism (degree 1): K137 -> K1]) (1 [K156 Morphism (degree 1): K137 -> K1]) (1 [K155 Morphism (degree 1): K137 -> K1]) (2 [K628 Morphism (degree 1): K588 -> K580]) (2 [K627 Morphism (degree 1): K588 -> K580]) (2 [K431 Morphism (degree 1): K183 -> K175]) (2 [K430 Morphism (degree 1): K183 -> K175]) (2 [K330 Morphism (degree 1): K250 -> K242]) (2 [K329 Morphism (degree 1): K250 -> K242]) (2 [K290 Morphism (degree 1): K250 -> K210]) (2 [K289 Morphism (degree 1): K250 -> K210]) (2 [K223 Morphism (degree 1): K183 -> K128]) (2 [K222 Morphism (degree 1): K183 -> K128]) (3 [K500 Morphism (degree 1): K319 -> K321]) (3 [K428 Morphism (degree 1): K175 -> K183]) (4 [K661 Morphism (degree 1): K618 -> K615]) (4 [K660 Morphism (degree 1): K618 -> K615]) (4 [K503 Morphism (degree 1): K321 -> K319]) (4 [K502 Morphism (degree 1): K321 -> K319]) (4 [K463 Morphism (degree 1): K422 -> K381]) (4 [K462 Morphism (degree 1): K422 -> K381]) (4 [K392 Morphism (degree 1): K321 -> K349]) (4 [K391 Morphism (degree 1): K321 -> K349]) (4 [K362 Morphism (degree 1): K321 -> K277]) (4 [K361 Morphism (degree 1): K321 -> K277]) (5 [K663 Morphism (degree 1): K615 -> K615]) (5 [K658 Morphism (degree 1): K615 -> K618]) (6 [K630 Morphism (degree 1): K580 -> K580]) (6 [K625 Morphism (degree 1): K580 -> K588]) (6 [K499 Morphism (degree 1): K319 -> K319]) (6 [K427 Morphism (degree 1): K175 -> K175]) (7 [K721 Morphism (degree 1): K648 -> K684]) (8 [K724 Morphism (degree 1): K651 -> K681]) (8 [K723 Morphism (degree 1): K651 -> K681]) (8 [K694 Morphism (degree 1): K651 -> K522]) (8 [K693 Morphism (degree 1): K651 -> K522]) (8 [K535 Morphism (degree 1): K494 -> K450]) (8 [K534 Morphism (degree 1): K494 -> K450]) (9 [K327 Morphism (degree 1): K242 -> K250]) (9 [K287 Morphism (degree 1): K242 -> K213]) (10 [K657 Morphism (degree 1): K615 -> K615]) (10 [K593 Cohomology-Class on K1 of degree -1]) (11 [K255 Cohomology-Class on K1 of degree -1]) (12 [K624 Morphism (degree 1): K580 -> K580]) (12 [K359 Morphism (degree 1): K319 -> K280]) (13 [K188 Cohomology-Class on K1 of degree -1]) (14 [K720 Morphism (degree 1): K648 -> K681]) (17 [K220 Morphism (degree 1): K175 -> K147]) (18 [K326 Morphism (degree 1): K242 -> K242]) (18 [K292 Morphism (degree 1): K242 -> K242]) (18 [K286 Morphism (degree 1): K242 -> K210]) (20 [K592 Morphism (degree 1): K1 -> K1]) (20 [K225 Morphism (degree 1): K175 -> K175]) (21 [K537 Morphism (degree 1): K492 -> K492]) (21 [K532 Morphism (degree 1): K492 -> K453]) (22 [K254 Morphism (degree 1): K1 -> K1]) (25 [K358 Morphism (degree 1): K319 -> K277]) (26 [K187 Morphism (degree 1): K1 -> K1]) (27 [K691 Morphism (degree 1): K648 -> K525]) (34 [K696 Morphism (degree 1): K648 -> K648]) (34 [K389 Morphism (degree 1): K319 -> K352]) (34 [K219 Morphism (degree 1): K175 -> K128]) (35 [K465 Morphism (degree 1): K420 -> K420]) (35 [K460 Morphism (degree 1): K420 -> K383]) (42 [K531 Morphism (degree 1): K492 -> K450]) (46 [K632 Morphism (degree 1): K615 -> K615]) (47 [K153 Cohomology-Class on K1 of degree -1]) (50 [K364 Morphism (degree 1): K319 -> K319]) (54 [K690 Morphism (degree 1): K648 -> K522]) (55 [K599 Morphism (degree 1): K580 -> K580]) (63 [K506 Morphism (degree 1): K492 -> K492]) (68 [K388 Morphism (degree 1): K319 -> K349]) (70 [K459 Morphism (degree 1): K420 -> K381]) (81 [K158 Morphism (degree 1): K1 -> K1]) (88 [K785 Morphism (degree 1): K779 -> K779]) (94 [K152 Morphism (degree 1): K1 -> K1]) (97 [K194 Morphism (degree 1): K175 -> K175]) (102 [K665 Morphism (degree 1): K648 -> K648]) (105 [K434 Morphism (degree 1): K420 -> K420]) (105 [K261 Morphism (degree 1): K242 -> K242]) (174 [K333 Morphism (degree 1): K319 -> K319]) (178 [K777 Morphism (degree 1): K5 -> K5]) (178 [K776 Morphism (degree 1): K5 -> K745]) (178 [K771 Morphism (degree 1): K745 -> K745]) (178 [K770 Morphism (degree 1): K745 -> K745]) (178 [K769 Morphism (degree 1): K745 -> K759]) (178 [K767 Morphism (degree 1): K759 -> K759]) (287 [K698 Morphism (degree 1): K681 -> K681]) (306 [K160 Morphism (degree 1): K128 -> K128]) (308 [K227 Morphism (degree 1): K210 -> K210]) (309 [K539 Morphism (degree 1): K522 -> K522]) (309 [K294 Morphism (degree 1): K277 -> K277]) (319 [K366 Morphism (degree 1): K349 -> K349]) (321 [K467 Morphism (degree 1): K450 -> K450]) (327 [K395 Morphism (degree 1): K381 -> K381]) (452 [K727 Morphism (degree 1): K713 -> K713]) (526 [K751 Morphism (degree 1): K745 -> K745]) (670 [K141 Morphism (degree 1): K1 -> K1]) -------------------------------------------------- > -------------------------------------------------- ;; What is the nature of the most active ?? -------------------------------------------------- > -------------------------------------------------- (DFNT (K 141)) -------------------------------------------------- > -------------------------------------------------- (K-COMPLEX-H 4) -------------------------------------------------- > -------------------------------------------------- ;; +-----------------+ ;; + | ;; + The END | ;; + | ;; +-----------------+ -------------------------------------------------- > -------------------------------------------------- ;; It is the "ROOT" Homotopy-Operator ;; ;; of the initial Koszul complex. -------------------------------------------------- > -------------------------------------------------- ;; +-----------------+ ;; + | ;; + The END | ;; + | ;; +-----------------+ -------------------------------------------------- >