Newer
Older
{
"cells": [
{
"cell_type": "markdown",
"id": "8ebd0216",
"metadata": {},
"source": [
"# Utilities"
]
},
{
"cell_type": "markdown",
"id": "b22eb2a9",
"metadata": {},
"source": [
"Define \\chern command in latex $\\newcommand{\\chern}{\\operatorname{ch}}$"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "b87a49bc",
"metadata": {},
"outputs": [],
"source": [
"# Requires extra package:\n",
"#! sage -pip install \"pseudowalls==0.0.3\" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple\n",
"%display latex\n",
"\n",
"from pseudowalls import *\n",
"\n",
"Δ = lambda v: v.Q_tilt()\n",
"alpha = stability.Tilt().alpha\n",
"beta = stability.Tilt().beta\n",
"\n",
"def beta_minus(v):\n",
" solutions = solve(\n",
" stability.Tilt(alpha=0).degree(v)==0,\n",
" beta)\n",
" return min(map(lambda s: s.rhs(), solutions))\n",
"\n",
"class Object(object):\n",
" pass"
]
},
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
{
"cell_type": "markdown",
"id": "6d374a2a",
"metadata": {},
"source": [
"Fix a Chern character $v$ with positive rank and $\\originalDelta(v) \\geq 0$"
]
},
{
"cell_type": "code",
"execution_count": 42,
"id": "ab162897",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = R \\\\ \\mathrm{ch}_{1} = C \\ell^{1} \\\\ \\mathrm{ch}_{2} = D \\ell^{2} \\end{array}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = R \\\\ \\mathrm{ch}_{1} = C \\ell^{1} \\\\ \\mathrm{ch}_{2} = D \\ell^{2} \\end{array}$"
],
"text/plain": [
"<pseudowalls.chern_character.Chern_Char object at 0x7f554bd7dc90>"
]
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v = Chern_Char(*var(\"R C D\", domain=\"real\"))\n",
"v"
]
},
{
"cell_type": "markdown",
"id": "06d8357b",
"metadata": {},
"source": [
"Let $u$ be a semistabilizer fitting problem 1 or 2 (destabilizing $v$ going down $\\Theta_v^{-}$)"
]
},
{
"cell_type": "code",
"execution_count": 43,
"id": "0d33d7e1",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = r \\\\ \\mathrm{ch}_{1} = c \\ell^{1} \\\\ \\mathrm{ch}_{2} = d \\ell^{2} \\end{array}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = r \\\\ \\mathrm{ch}_{1} = c \\ell^{1} \\\\ \\mathrm{ch}_{2} = d \\ell^{2} \\end{array}$"
],
"text/plain": [
"<pseudowalls.chern_character.Chern_Char object at 0x7f554bd7fd00>"
]
},
"execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"u = Chern_Char(*var(\"r c d\", domain=\"real\"))\n",
"u"
]
},
{
"cell_type": "markdown",
"id": "cb9c11e7",
"metadata": {},
"source": [
"# Bounds on $\\operatorname{ch}_2(u)=d$"
]
},
{
"cell_type": "code",
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
"execution_count": 44,
"id": "23d48b0b",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle q\\)</html>"
],
"text/latex": [
"$\\displaystyle q$"
],
"text/plain": [
"q"
]
},
"execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"var(\"q\", domain=\"real\") # Symbol for q=\\chern_1^{\\beta}(u)"
]
},
{
"cell_type": "markdown",
"id": "377c2843",
"metadata": {},
"source": [
"Express $c$ in terms of $q:=\\chern_1^{\\beta}(u)$"
]
},
{
"cell_type": "code",
"execution_count": 49,
"id": "49be9bb7",
"metadata": {},
"outputs": [],
"source": [
"c_in_terms_of_q = solve(q == u.twist(beta).ch[1], c)[0]\n",
"assert c_in_terms_of_q.lhs() == c, \"Meant to be an expression for c\""
]
},
{
"cell_type": "code",
"execution_count": 51,
"id": "4cacebc7",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle c = \\beta r + q\\)</html>"
"$\\displaystyle c = \\beta r + q$"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c_in_terms_of_q"
]
},
{
"cell_type": "markdown",
"id": "900f332b",
"metadata": {},
"source": [
"## $\\Delta(u) \\geq 0$"
]
},
{
"cell_type": "markdown",
"id": "62529298",
"metadata": {},
"source": [
"Express this inequality in terms of $q$"
]
},
{
"cell_type": "code",
"execution_count": 76,
"id": "5991acb3",
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle 0 \\leq {\\left(\\beta r + q\\right)}^{2} - 2 \\, d r\\)</html>"
],
"text/latex": [
"$\\displaystyle 0 \\leq {\\left(\\beta r + q\\right)}^{2} - 2 \\, d r$"
],
"text/plain": [
"0 <= (beta*r + q)^2 - 2*d*r"
]
},
"execution_count": 76,
"metadata": {},
"output_type": "execute_result"
}
],
"bgmlv2_with_q = ((0 <= Δ(u))\n",
" .subs(c_in_terms_of_q))\n",
"bgmlv2_with_q"
]
},
{
"cell_type": "markdown",
"id": "723511fa",
"metadata": {},
"source": [
"Rearrange expression for $d$"
]
},
{
"cell_type": "code",
"execution_count": 77,
"id": "1f66d604",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle d \\leq \\frac{{\\left(\\beta r + q\\right)}^{2}}{2 \\, r}\\)</html>"
"$\\displaystyle d \\leq \\frac{{\\left(\\beta r + q\\right)}^{2}}{2 \\, r}$"
"d <= 1/2*(beta*r + q)^2/r"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_ineq = (bgmlv2_with_q\n",
" + 2*d*r # move d to rhs\n",
") / (2*r) # scale-out d coefficient (r>0)\n",
"\n",
"assert bgmlv2_d_ineq.lhs() == d, \"Should be ineq for d\"\n",
"\n",
"bgmlv2_d_ineq"
]
},
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
{
"cell_type": "markdown",
"id": "425bcb7c",
"metadata": {},
"source": [
"Keep hold of the upper bound for $d$:"
]
},
{
"cell_type": "code",
"execution_count": 80,
"id": "6ae4f2e7",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\frac{q^{2}}{2 \\, r}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\frac{q^{2}}{2 \\, r}$"
],
"text/plain": [
"1/2*beta^2*r + beta*q + 1/2*q^2/r"
]
},
"execution_count": 80,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_upperbound = bgmlv2_d_ineq.rhs().expand()\n",
"bgmlv2_d_upperbound"
]
},
{
"cell_type": "markdown",
"id": "988aaf5b",
"metadata": {},
"source": [
"Separate out the terms of this lower bound for d"
]
},
{
"cell_type": "code",
"id": "653a3340",
"metadata": {},
"outputs": [],
"source": [
"bgmlv2_d_upperbound_terms = Object()\n",
"\n",
"bgmlv2_d_upperbound_without_hyp = (\n",
" bgmlv2_d_upperbound\n",
" .subs(1/r == 0)\n",
")\n",
"\n",
"bgmlv2_d_upperbound_terms.const = (\n",
" bgmlv2_d_upperbound_without_hyp\n",
" .subs(r==0)\n",
")\n",
"\n",
"bgmlv2_d_upperbound_terms.linear = (\n",
" bgmlv2_d_upperbound_without_hyp\n",
" - bgmlv2_d_upperbound_terms.const\n",
").expand()\n",
"\n",
"bgmlv2_d_upperbound_terms.hyperbolic = (\n",
" bgmlv2_d_upperbound\n",
" - bgmlv2_d_upperbound_without_hyp\n",
").expand()"
]
},
{
"cell_type": "code",
"id": "326bb656",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\beta q\\)</html>"
],
"text/latex": [
"$\\displaystyle \\beta q$"
],
"text/plain": [
"beta*q"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_upperbound_terms.const"
]
},
{
"cell_type": "code",
"id": "fcd9fa2a",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r$"
],
"text/plain": [
"1/2*beta^2*r"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_upperbound_terms.linear"
]
},
{
"cell_type": "code",
"id": "ade0f7b2",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{q^{2}}{2 \\, r}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{q^{2}}{2 \\, r}$"
],
"text/plain": [
"1/2*q^2/r"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_upperbound_terms.hyperbolic"
]
},
{
"cell_type": "markdown",
"id": "5cc08c62",
"metadata": {},
"source": [
"Sanity check:"
]
},
{
"cell_type": "code",
"execution_count": 96,
"id": "7ff937ff",
"metadata": {},
"outputs": [],
"source": [
"assert ( bgmlv2_d_upperbound\n",
"- bgmlv2_d_upperbound_terms.const\n",
"- bgmlv2_d_upperbound_terms.linear\n",
"- bgmlv2_d_upperbound_terms.hyperbolic) == 0, \"Error in terms separation\""
]
},
{
"cell_type": "markdown",
"id": "024e8c41",
"metadata": {},
"source": [
"## $\\Delta(v-u) \\geq 0$"
]
},
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
{
"cell_type": "markdown",
"id": "a2647f43",
"metadata": {},
"source": [
"Express this inequality in terms of $q$"
]
},
{
"cell_type": "code",
"execution_count": 101,
"id": "87544e6e",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle 0 \\leq {\\left(\\beta r - C + q\\right)}^{2} - 2 \\, {\\left(D - d\\right)} {\\left(R - r\\right)}\\)</html>"
],
"text/latex": [
"$\\displaystyle 0 \\leq {\\left(\\beta r - C + q\\right)}^{2} - 2 \\, {\\left(D - d\\right)} {\\left(R - r\\right)}$"
],
"text/plain": [
"0 <= (beta*r - C + q)^2 - 2*(D - d)*(R - r)"
]
},
"execution_count": 101,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_with_q = ((0 <= Δ(v-u))\n",
" .subs(c_in_terms_of_q)\n",
")\n",
"\n",
"bgmlv3_with_q"
]
},
{
"cell_type": "markdown",
"id": "d36504bb",
"metadata": {},
"source": [
"Rearrange in terms of $d$ assuming $r>R$"
]
},
{
"cell_type": "code",
"execution_count": 108,
"id": "8986af54",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle d \\leq D - \\frac{{\\left(\\beta r - C + q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
],
"text/latex": [
"$\\displaystyle d \\leq D - \\frac{{\\left(\\beta r - C + q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
],
"text/plain": [
"d <= D - 1/2*(beta*r - C + q)^2/(R - r)"
]
},
"execution_count": 108,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_ineq = (\n",
" (\n",
" bgmlv3_with_q\n",
" + 2*(D-d)*(R-r) # move d term to lhs\n",
" )/2/(r-R) # assume r>R\n",
") + D\n",
"\n",
"assert bgmlv3_d_ineq.lhs() == d, \"Should be bound for d\"\n",
"assert not bgmlv3_d_ineq.rhs().has(d), \"Should be bound for d\"\n",
"\n",
"bgmlv3_d_ineq"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "b7799156",
"metadata": {},
"outputs": [],
"source": [
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
"var(\"r_alt\",domain=\"real\") # r_alt = r - R temporary substitution\n",
"\n",
"bgmlv3_with_q_reparam = (\n",
" bgmlv3_with_q\n",
" .subs(r == r_alt + R)\n",
" /r_alt # This operation assumes r_alt > 0\n",
").expand()\n",
"\n",
"bgmlv3_d_ineq = (\n",
" ((0 <= bgmlv3_with_q_reparam)/2 + d) # Rearrange for d\n",
" .subs(r_alt == r - R) # Resubstitute r back in\n",
" .expand()\n",
")\n",
"\n",
"# Check that this equation represents a bound for d\n",
"assert bgmlv3_d_ineq.lhs() == d\n",
"\n",
"bgmlv3_d_upperbound = bgmlv3_d_ineq.rhs() # Keep hold of lower bound for d\n",
"\n",
"# Seperate out the terms of the lower bound for d\n",
"\n",
"bgmlv3_d_upperbound_without_hyp = (\n",
" bgmlv3_d_upperbound\n",
" .subs(1/(R-r) == 0)\n",
")\n",
"\n",
"bgmlv3_d_upperbound_const_term = (\n",
" bgmlv3_d_upperbound_without_hyp\n",
" .subs(r==0)\n",
")\n",
"\n",
"bgmlv3_d_upperbound_linear_term = (\n",
" bgmlv3_d_upperbound_without_hyp\n",
" - bgmlv3_d_upperbound_const_term\n",
").expand()\n",
"\n",
"bgmlv3_d_upperbound_exp_term = (\n",
" bgmlv3_d_upperbound\n",
" - bgmlv3_d_upperbound_without_hyp\n",
").expand()\n",
"\n",
"# Verify the simplified forms of the terms that will be mentioned in text\n",
"\n",
"var(\"psi phi\", domain=\"real\") # symbol to represent ch_1^\\beta(v) and\n",
"# ch_2^\\beta(v)\n",
"\n",
"assert bgmlv3_d_upperbound_const_term == ( \n",
" (\n",
" # keep hold of this alternative expression:\n",
" bgmlv3_d_upperbound_const_term_alt := (\n",
" phi\n",
" + beta*q\n",
" )\n",
" )\n",
" .subs(phi == v.twist(beta).ch[2]) # subs real val of ch_1^\\beta(v)\n",
" .expand()\n",
")\n",
"\n",
"assert bgmlv3_d_upperbound_exp_term == (\n",
" (\n",
" # Keep hold of this alternative expression:\n",
" bgmlv3_d_upperbound_exp_term_alt :=\n",
" (\n",
" R*phi\n",
" + (C - q)^2/2\n",
" + R*beta*q\n",
" - D*R\n",
" )/(r-R)\n",
" )\n",
" .subs(phi == v.twist(beta).ch[2]) # subs real val of ch_1^\\beta(v)\n",
" .expand()\n",
")\n",
"\n",
"assert bgmlv3_d_upperbound_exp_term == (\n",
" (\n",
" # Keep hold of this alternative expression:\n",
" bgmlv3_d_upperbound_exp_term_alt2 :=\n",
" (\n",
" (psi - q)^2/2/(r-R)\n",
" )\n",
" )\n",
" .subs(psi == v.twist(beta).ch[1]) # subs real val of ch_1^\\beta(v)\n",
" .expand()\n",
")"
]
},
{
"cell_type": "markdown",
"id": "9809bd85",
"metadata": {},
"source": [
"$\\renewcommand{\\psi}{\\chern_1^{\\beta}(v)}$\n",
"$\\renewcommand{\\phi}{\\chern_2^{\\beta}(v)}$\n",
"Redefine psi and phi in latex to be $\\psi$ and $\\phi$"
]
},
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
{
"cell_type": "code",
"execution_count": 11,
"id": "38bbbcd2",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r$"
],
"text/plain": [
"1/2*beta^2*r"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_upperbound_linear_term"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "6701d8a7",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\beta q + \\phi\\)</html>"
],
"text/latex": [
"$\\displaystyle \\beta q + \\phi$"
],
"text/plain": [
"beta*q + phi"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_upperbound_const_term_alt"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "7a82d747",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
],
"text/latex": [
"$\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
],
"text/plain": [
"-1/2*(psi - q)^2/(R - r)"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_upperbound_exp_term_alt2"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "927e6929",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r$"
],
"text/plain": [
"1/2*beta^2*r"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_upperbound_linear_term"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "ff25eea1",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\beta q\\)</html>"
],
"text/latex": [
"$\\displaystyle \\beta q$"
],
"text/plain": [
"beta*q"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_upperbound_const_term"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "e8359c0f",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{q^{2}}{2 \\, r}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{q^{2}}{2 \\, r}$"
],
"text/plain": [
"1/2*q^2/r"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_upperbound_exp_term"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "6823c5b8",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r$"
],
"text/plain": [
"1/2*beta^2*r"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_upperbound_linear_term"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "160871f2",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\beta q\\)</html>"
],
"text/latex": [
"$\\displaystyle \\beta q$"
],
"text/plain": [
"beta*q"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_upperbound_const_term_alt.subs(phi == 0)"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "1da23b29",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
],
"text/latex": [
"$\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
],
"text/plain": [
"-1/2*(psi - q)^2/(R - r)"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_upperbound_exp_term_alt2"
]
},
{
"cell_type": "markdown",
"id": "2997ec1a",
"metadata": {},
"source": [
"## Plots for all Bounds on $d$"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "d7235dc3",
"metadata": {},
"outputs": [],
"source": [
"positive_radius_condition = (\n",
" (\n",
" (0 > - u.twist(beta).ch[2])\n",
" + d # rearrange for d\n",
" )\n",
" .subs(solve(q == u.twist(beta).ch[1], c)[0]) # express c in term of q\n",
" .expand()\n",
")\n",
"\n",
"\n",
"v_example = Chern_Char(3,2,-2)\n",
"q_example = 7/3\n",
"\n",
"def plot_d_bound(\n",
" v_example,\n",
" q_example,\n",
" ymax=5,\n",
" ymin=-2,\n",
" xmax=20,\n",
" aspect_ratio=None):\n",
"\n",
" # Equations to plot imminently representing the bounds on d:\n",
" eq2 = (bgmlv2_d_upperbound\n",
" .subs(R == v_example.ch[0])\n",
" .subs(C == v_example.ch[1])\n",
" .subs(D == v_example.ch[2])\n",
" .subs(beta = beta_minus(v_example))\n",
" .subs(q == q_example)\n",
" )\n",
"\n",
" eq3 = (bgmlv3_d_upperbound\n",
" .subs(R == v_example.ch[0])\n",
" .subs(C == v_example.ch[1])\n",
" .subs(D == v_example.ch[2])\n",
" .subs(beta = beta_minus(v_example))\n",
" .subs(q == q_example)\n",
" )\n",
"\n",
" eq4 = (positive_radius_condition.rhs()\n",
" .subs(q == q_example)\n",
" .subs(beta = beta_minus(v_example))\n",
" )\n",
"\n",
" example_bounds_on_d_plot = (\n",
" plot(\n",
" eq3,\n",
" (r,v_example.ch[0],xmax),\n",
" color='green',\n",
" linestyle = \"dashed\",\n",
" legend_label=r\"upper bound: $\\Delta(v-u) \\geq 0$\",\n",
" )\n",
" + plot(\n",
" eq2,\n",
" (r,0,xmax),\n",
" color='blue',\n",
" linestyle = \"dashed\",\n",
" legend_label=r\"upper bound: $\\Delta(u) \\geq 0$\"\n",
" )\n",
" + plot(\n",
" eq4,\n",
" (r,0,xmax),\n",
" color='orange',\n",
" linestyle = \"dotted\",\n",
" legend_label=r\"lower bound: $\\mathrm{ch}_2^{\\beta_{-}}(u)>0$\"\n",
" )\n",
" )\n",
" example_bounds_on_d_plot.ymin(ymin)\n",
" example_bounds_on_d_plot.ymax(ymax)\n",
" example_bounds_on_d_plot.axes_labels(['$r$', '$d$'])\n",
" if aspect_ratio:\n",
" example_bounds_on_d_plot.set_aspect_ratio(aspect_ratio)\n",
" return example_bounds_on_d_plot"
]
},
{
"cell_type": "markdown",
"id": "683ac3f7",
"metadata": {},
"source": [
"### Bounds on $d$ with Minimal $q=\\operatorname{ch}^{\\beta}_1(u)$"
]
},
{
"cell_type": "code",
"execution_count": 21,