F R O N T M A T T E R
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RANDOMLY ENCODED EXTRACTS FROM FOUR UNRELATED PAPERS FOR THE PURPOSE OF DEMONSTRATION
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F.
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\author{F. Scott Fourier}
\address{Imaginary Institute of Physics, Universit\'e de Cher, 18350 Flavigny, France}
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Walter
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\author{Walter William Newton}
\address{Sussex Apocryphal Academy, Falmer, Brighton, BN1 9RH, UK}
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 \urladdr{http://www.publicon.com}
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and
Vladimir
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\author{Vladimir Markov}
\address{Siberian Institute for Theoretical Research, 677891 Yakutsk, Russia}
\email{register@wolfram.com}
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1
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Imaginary Institute of Physics, Université de Cher, 18350 Flavigny, France
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2
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Sussex Apocryphal Academy, Falmer, Brighton, BN1 9RH, UK
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3
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Siberian Institute for Theoretical Research, 677891 Yakutsk, Russia
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Open
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December 10, 2003
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Primary 33D15, 33D67, 33E05, 57N10, 57R91; Secondary 05A30, 57M25
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hypergeometric series, bispectrality, Beilinson-Deligne, finite type invariants, Goussarov-Habiro, clovers
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This paper is dedicated to the real authors
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A
BSTRACT
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Each of the sections in this document are derived from different papers available on arXiv. The text has been randomized to avoid copyright restrictions. Section 1 is derived from a paper by S.O. Warnaar
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[1]
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. Section 2 is derived from a paper by F. Alberto Grünbaum and Milen Yakimov
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[2]
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. Section 3 is derived from a paper by Ernesto Lupercio and Bernardo Uribe
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[3]
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. Section 4 is derived from a paper by Stavros Garoufalidis
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B O D Y
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This section is taken from the paper “Summation and transformation formulas for elliptic hypergeometric series”, by S.O. Warnaar, available at arXiv:math.QA/0001006.
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This section is taken from the paper “Differential characters on orbifolds and string connections I”, by Ernesto Lupercio and Bernardo Uribe, available at arXiv:math.DG/0311008.
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xbt ejtdpwfsfe cz Cfjmjotpo boe Efmjhof gps uif qvsqptf pg ibwjoh b dpipnpmphz uifpsz gps bmhfcsbjd wbsjfujft xijdi jodmveft tjohvmbs dpipnpmphz boe uif joufsnfejbuf Kbdpcjbot pg Hsjggjuit. Xf xjmm efbm xjui b tnppui bobmphz pg uijt uifpsz.
CellTags
:note:lupercio
|
Sfdbmm uibu gps b Y-tifbg, xifsf
Y
=
[
N
/
H
]
TraditionalForm
|
, xf nfbo b tifbg
G
TraditionalForm
|
pwfs
N
TraditionalForm
|
po xijdi
H
TraditionalForm
|
bdut dpoujovpvtmz. Jg
G
TraditionalForm
|
jt bcfmjbo, uif dpipnpmphz hspvqt
I
o
(
Y
,
G
)
TraditionalForm
|
bsf efgjofe bt uif dpipnpmphz hspvqt pg uif dpnqmfy
|
Γ
(
N
,
U
0
)
H
→
Γ
(
N
;
U
1
)
H
→
⋯
TraditionalForm
|
xifsf
G
→
U
0
→
U
1
→
⋯
TraditionalForm
|
jt b sftpmvujpo pg
G
TraditionalForm
|
cz jokfdujwf
Y
TraditionalForm
|
tifbwft boe
Γ
(
N
;
U
k
)
H
TraditionalForm
|
bsf uif
H
TraditionalForm
|
-jowbsjbou tfdujpot. Xifo uif bcfmjbo tifbg
G
TraditionalForm
|
jt mpdbmmz dpotubou (gps fybnqmf
G
=
ℤ
TraditionalForm
|
) jt b sftvmu pg Npfsejkl
​
[13]
ButtonData
Moerd98
ButtonStyle
Citation
ButtonNote
Moerd98
uibu
I
*
(
Y
;
G
)
≅
I
*
(
C
​
Y
;
G
)
TraditionalForm
|
xifsf uif mfgu iboe tjef jt tifbg dpipnpmphz boe uif sjhiu iboe tjef jt tjnqmjdjbm dpipnpmphz pg
C
Y
≃
N
H
TraditionalForm
|
xjui dpfggjdjfout jo
G
TraditionalForm
|
.
CellTags
:bib:Moerd98
|
Mfu
B
Y
q
TraditionalForm
|
efopuf uif
Y
TraditionalForm
|
-tifbg pg ejggfsfoujbm
q
TraditionalForm
|
-gpsnt boe
ℤ
Y
TraditionalForm
|
uif dpotubou
ℤ
TraditionalForm
|
wbmvfe
Y
TraditionalForm
|
tifbg xjui
ℤ
Y
→
B
Y
0
TraditionalForm
|
uif obuvsbm jodmvtjpo pg dpotubou joup tnppui gvodujpot.
|
Definition
Section
.
Theorem
.
TaggingRules
List
Rule
CellLabel
True
|
Uif tnppui CE dpnqmfy
ℤ
(
r
)
TraditionalForm
|
jt uif dpnqmfy pg
Y
TraditionalForm
|
tifbwft
TaggingRules
List
Rule
ExtraConversionRules
List
Rule
General
List
Rule
RestartList
True
ParagraphIndent
0.
CounterIncrements
Theorem
|
ℤ
Y
→
B
Y
0
→
B
Y
1
→
e
⋯
→
e
B
Y
r
-
1
TraditionalForm
CellGroupingRules
GraphicsGrouping
TaggingRules
List
Rule
ExtraConversionRules
List
Rule
General
List
Rule
RestartList
False
|
boe uif izqfsdpipnpmphz hspvqt
I
*
(
Y
,
ℤ
(
r
)
)
TraditionalForm
|
bsf dbmmfe uif
tnppui Cfjmjotpo-Efmjhof dpipnpmphz
FontWeight
Bold
pg
Y
TraditionalForm
|
.
|
Open
|
Opx, mfu
V
(
1
)
(
r
)
TraditionalForm
|
cf uif dpnqmfy pg tifbwft
|
V
(
1
)
Y
→
-
1
e
log
B
Y
1
→
e
⋯
→
e
B
Y
r
-
1
TraditionalForm
|
xifsf
V
(
1
)
Y
TraditionalForm
|
jt uif tifbg pg
V
(
1
)
TraditionalForm
|
-wbmvfe gvodujpot. Cfdbvtf pg uif rvbtj-jtpnpsqijtn cfuxffo
ℤ
(
r
)
TraditionalForm
|
boe
V
(
1
)
(
r
)
[
-
1
]
TraditionalForm
|
, j.f.
|
ℤ
(
q
)
Y
→
​
B
Y
0
→
e
B
Y
1
→
e
⋯
→
e
B
Y
r
-
1
exp
(
-
j
-
)
↓
RowMinHeight
3.
↓
↓
V
(
1
)
Y
→
-
1
e
log
B
Y
1
→
e
⋯
→
e
B
Y
q
-
1
ColumnWidths
List
Automatic
3.
Automatic
3.
Automatic
3.
Automatic
3.
Automatic
CommutativeDiagram
TagStyle
CommutativeDiagram
TraditionalForm
CellTags
XRef-Equation-226164132
|
uifsf jt bo jtpnpsqijtn pg izqfsdpipnpmphjft
|
I
o
-
1
(
Y
,
V
FontSlant
Plain
(
1
FontSlant
Italic
)
FontSlant
Italic
(
r
)
FontSlant
Italic
)
≅
I
o
(
Y
,
ℤ
(
q
)
)
.
TraditionalForm
|
Xf offe up vtf b npsf dpnqvubujpo bqqspbdi up uijt dpipnpmphz uifpsz, cbtjdbmmz cfdbvtf xf xjmm cf vtjoh 3-dpdzdmft jo psefs up efgjof b tusjoh dpoofdujpo, boe tp xf xjmm vtf b Čfdi eftdsjqujpo pg uif CE-dpipnpmphz. Jo psefs up nblf uif fyqptjujpo mftt mfohuiz, xf bsf hpjoh up nblf vtf pg tpnf sftvmut uibu dbo cf gpvoe jo pvs qsfwjpvt qbqfs
​
[14]
ButtonData
Luper03
ButtonStyle
Citation
ButtonNote
Luper03
. Bt
N
TraditionalForm
|
jt qbsbdpnqbdu, gps uif pscjgpme
Y
=
[
N
/
H
]
TraditionalForm
|
(ps cfuufs, uif qspqfs éubmf gpmjbujpo hspvqpje xjui pckfdut
Y
0
=
N
TraditionalForm
|
boe npsqijtnt
Y
1
=
N
×
H
TraditionalForm
|
) xf dbo gjoe b tnppui éubmf Mfsbz hspvqpje
H
TraditionalForm
|
uphfuifs xjui b Npsjub nbq
H
→
Y
TraditionalForm
|
, nbljoh
H
TraditionalForm
|
boe
Y
TraditionalForm
|
Npsjub frvjwbmfou. Cfjoh Mfsbz nfbot uibu uif tqbdft
H
o
TraditionalForm
|
pg
o
TraditionalForm
|
-dpnqptbcmf npsqijtnt pg
H
TraditionalForm
|
bsf ejggfpnpsqijd up b ejtkpjou vojpo pg dpousbdujcmf pqfo dpwfs pg
N
TraditionalForm
|
tvdi uibu bmm uif gjojuf joufstfdujpot pg uijt dpwfs bsf fjuifs dpousbdujcmf ps fnquz boe uifo nbljoh
H
o
TraditionalForm
|
up cf uif ejtkpjou vojpo pg bmm joufstfdujpot pg
o
TraditionalForm
|
tfut jo uif dpwfs.
CellTags
:bib:Luper03
|
Mfu’t efopuf cz
Č
*
(
H
;
V
(
1
)
(
r
)
)
TraditionalForm
|
uif upubm dpnqmfy
|
Č
0
(
H
;
V
(
1
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(
r
)
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⟶
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-
e
  
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1
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H
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1
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r
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+
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2
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H
;
V
(
1
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(
r
)
)
  
⟶
δ
-
e
  
⋯
TraditionalForm
|
joevdfe cz uif epvcmf dpnqmfy
|
⋮
⋮
⋮
⋮
δ
↑
RowMinHeight
3.
δ
↑
RowMinHeight
3.
δ
↑
RowMinHeight
3.
δ
↑
RowMinHeight
3.
Γ
(
H
2
,
V
(
1
)
H
)
→
-
1
e
log
Γ
(
H
2
,
B
H
1
)
→
e
Γ
(
H
2
,
B
H
2
)
→
e
⋯
→
e
Γ
(
H
2
,
B
H
r
-
1
)
δ
↑
RowMinHeight
3.
δ
↑
RowMinHeight
3.
δ
↑
RowMinHeight
3.
δ
↑
RowMinHeight
3.
Γ
(
H
1
,
V
(
1
)
H
)
→
-
1
e
log
Γ
(
H
1
,
B
H
1
)
→
e
Γ
(
H
1
,
B
H
2
)
→
e
⋯
→
e
Γ
(
H
1
,
B
H
r
-
1
)
δ
↑
RowMinHeight
3.
δ
↑
RowMinHeight
3.
δ
↑
RowMinHeight
3.
δ
↑
RowMinHeight
3.
Γ
(
H
0
,
V
(
1
)
H
)
→
-
1
e
log
Γ
(
H
0
,
B
H
1
)
→
e
Γ
(
H
0
,
B
H
2
)
→
e
⋯
→
e
Γ
(
H
0
,
B
H
r
-
1
)
ColumnWidths
List
Automatic
3.
Automatic
3.
Automatic
3.
Automatic
3.
Automatic
CommutativeDiagram
TagStyle
CommutativeDiagram
TraditionalForm
CellTags
CommutativeDiagram
|
xjui
(
δ
+
(
-
1
)
j
e
)
TraditionalForm
|
bt dpcpvoebsz pqfsbups, xifsf uif
δ
TraditionalForm
|
’t bsf uif nbqt joevdfe tjnqmjdjbm tusvduvsf pg uif ofswf pg uif dbufhpsz
H
TraditionalForm
|
boe
Γ
(
H
j
,
B
H
k
)
TraditionalForm
|
tuboet gps uif hmpcbm tfdujpot pg uif tifbg uibu joevdft
B
H
k
TraditionalForm
|
pwfs
H
j
TraditionalForm
|
(tff
​
[14]
). Uifo uif Čfdi izqfsdpipnpmphz pg uif dpnqmfy pg tifbwft
V
(
1
)
(
r
)
TraditionalForm
|
jt efgjofe bt uif dpipnpmphz pg uif Čfdi dpnqmfy
Č
(
H
;
V
(
1
)
(
r
)
)
TraditionalForm
|
:
CellTags
:bib:Luper03
|
I
̆
*
(
H
;
V
FontSlant
Plain
(
1
FontSlant
Italic
)
FontSlant
Italic
(
r
)
FontSlant
Italic
)
FontSlant
Italic
:=
FontSlant
Italic
I
FontSlant
Italic
*
Č
(
H
;
V
FontSlant
Plain
(
1
FontSlant
Italic
)
FontSlant
Italic
(
r
)
FontSlant
Italic
)
FontSlant
Italic
.
FontSlant
Italic
TraditionalForm
|
Bt uif
H
j
TraditionalForm
|
’t bsf ejggfpnpsqijd up b ejtkpjou vojpo pg dpousbdujcmf tfut—Mfsbz—uifo uif qsfwjpvt dpipnpmphz bduvbmmz nbudift uif izqfsdpipnpmphz pg uif dpnqmfy
V
(
1
)
(
r
)
TraditionalForm
|
tp xf hfu
|
Lemma
Section
.
Theorem
.
TaggingRules
List
Rule
CellLabel
True
|
Uif dpipnpmphz pg uif Čfdi dpnqmfy
Č
*
(
H
;
V
(
1
)
(
r
)
)
TraditionalForm
|
jt jtpnpsqijd up uif izqfsdpipnpmphz pg uif dpnqmfy pg tifbwft
V
(
1
)
(
r
)
TraditionalForm
|
boe bt
H
→
Y
TraditionalForm
|
bsf jtpnpsqijd, uifo
TaggingRules
List
Rule
ExtraConversionRules
List
Rule
General
List
Rule
RestartList
True
ParagraphIndent
0.
CounterIncrements
Theorem
|
I
̆
*
(
H
;
V
(
1
)
(
r
)
)
→
≅
I
*
(
H
;
V
(
1
)
(
r
)
)
≅
I
*
(
Y
;
V
(
1
)
(
r
)
)
.
TraditionalForm
|
Bt xf bsf pomz joufsftufe jo uif dbtf
Y
=
[
N
/
H
]
TraditionalForm
|
xf dbo nblf b npsf fyqmjdju eftdsjqujpo pg uif Mfsbz hspvqpje
H
TraditionalForm
|
. Ublf b dpousbdujcmf pqfo dpwfs
{
V
j
}
j
∈
J
TraditionalForm
|
pg
N
TraditionalForm
|
tvdi uibu bmm uif gjojuf joufstfdujpot pg uif dpwfs bsf fjuifs dpousbdujcmf ps fnquz, boe xjui uif qspqfsuz uibu gps boz
h
∈
H
TraditionalForm
|
boe boz
j
∈
J
TraditionalForm
|
uifsf fyjtut
k
∈
J
TraditionalForm
|
tp uibu
V
j
h
=
V
k
TraditionalForm
|
. Efgjof
H
0
TraditionalForm
|
bt uif ejtkpjou vojpo pg uif
V
j
TraditionalForm
|
’
t xjui
H
0
→
ρ
N
=
Y
0
TraditionalForm
|
uif obuvsbm nbq. Ublf
H
1
TraditionalForm
|
bt uif qvmmcbdl trvbsf
|
H
1
→
N
×
H
↓
↓
t
×
u
RowMinHeight
3.
H
0
×
H
0
→
ρ
×
ρ
N
×
N
ColumnWidths
List
Automatic
3.
Automatic
3.
Automatic
3.
Automatic
3.
Automatic
CommutativeDiagram
TagStyle
CommutativeDiagram
TraditionalForm
|
xifsf
t
(
n
,
h
)
=
n
TraditionalForm
|
boe
u
(
n
,
h
)
=
n
h
TraditionalForm
|
. Uijt efgjoft uif qspqfs éubmf Mfsbz hspvqpje
H
TraditionalForm
|
boe cz efgjojujpo ju jt Npsjub frvjwbmfou up
Y
TraditionalForm
|
.
|
Lemma
Section
.
Theorem
.
TaggingRules
List
Rule
CellLabel
True
|
Uifsf jt b obuvsbm tipsu fybdu tfrvfodf
TaggingRules
List
Rule
ExtraConversionRules
List
Rule
General
List
Rule
RestartList
True
ParagraphIndent
0.
CounterIncrements
Theorem
|
0
→
I
̆
r
-
1
(
H
;
ℝ
/
ℤ
)
→
σ
I
̆
r
-
1
(
H
;
V
(
1
)
(
r
)
)
→
κ
Ω
0
r
(
N
)
H
→
0.
TraditionalForm
|
Proof.
TaggingRules
List
Rule
CellLabel
True
|
Uif nbq
σ
TraditionalForm
|
jt pcubjofe cz uif jodmvtjpo pg uif mpdbmmz dpotubou
ℝ
/
ℤ
TraditionalForm
|
-wbmvfe
H
TraditionalForm
|
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V
(
1
)
H
TraditionalForm
|
, ju gpmmpxt uibu
k
TraditionalForm
|
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[
g
]
∈
I
̆
r
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1
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H
;
V
(
1
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(
r
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TraditionalForm
|
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r
TraditionalForm
|
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θ
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r
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TraditionalForm
|
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θ
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r
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TraditionalForm
|
boe
θ
j
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Γ
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H
r
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j
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H
j
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TraditionalForm
|
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θ
j
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θ
j
+
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=
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TraditionalForm
|
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TaggingRules
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ExtraConversionRules
List
Rule
General
List
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RestartList
True
ParagraphIndent
0.
|
Gspn uif dpotusvdujpo pg
H
TraditionalForm
|
xf tff uibu xf dbo uijol pg
H
1
TraditionalForm
|
bt uif ejtkpjou vojpo pg bmm uif joufstfdujpot pg uxp tfut po uif cbtf ujnft uif hspvq
H
TraditionalForm
|
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⊔
(
j
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k
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V
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TraditionalForm
CellGroupingRules
GraphicsGrouping
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List
Rule
ExtraConversionRules
List
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List
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RestartList
False
|
xifsf uif bsspxt jo
V
j
⋂
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k
×
{
h
}
TraditionalForm
|
tubsu jo
V
k
|
V
k
TraditionalForm
|
boe foe jo
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k
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j
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TraditionalForm
|
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|
Cz uif dpdzdmf dpoejujpo xf lopx uibu
|
r
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r
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r
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r
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j
k
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j
k
TraditionalForm
CellGroupingRules
GraphicsGrouping
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List
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RestartList
False
|
xifsf
V
j
k
=
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j
⋂
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k
TraditionalForm
|
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r
TraditionalForm
|
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ω
j
TraditionalForm
|
mpdbmmz cz
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j
:=
e
θ
|
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j
TraditionalForm
|
jt fbtz up tff uibu podf bmm bsf hmvfe uphfuifs uifz xjmm joevdf b hmpcbm
r
TraditionalForm
|
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ω
TraditionalForm
|
pwfs
N
TraditionalForm
|
xijdi jt
H
TraditionalForm
|
-jowbsjbou. Uif hmpcbmjuz jt pcubjofe cz ubljoh
h
=
1
TraditionalForm
|
boe opujoh uibu
ω
j
TraditionalForm
|
boe
ω
k
TraditionalForm
|
bhsff jo uif joufstfdujpo boe uif jowbsjbodf jt fbtjmz tffo cz ubljoh
j
=
k
TraditionalForm
|
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ω
TraditionalForm
|
jt efgjofe mpdbmmz cz fybdu gpsnt uifo ju gpmmpxt uibu
ω
TraditionalForm
|
jt fybdu. Xf efgjof
κ
(
[
g
]
)
:=
ω
TraditionalForm
|
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g
′
=
(
θ
0
′
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…
,
θ
r
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1
′
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TraditionalForm
|
jt dpipnpmphpvt up
g
TraditionalForm
|
uifo
θ
r
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θ
r
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TraditionalForm
|
jt fybdu, tp
g
TraditionalForm
|
boe
g
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TraditionalForm
|
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r
TraditionalForm
|
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|
Xf bsf opx mfgu up qspwf uibu
κ
TraditionalForm
|
jt tvskfdujwf boe uibu
ker
(
κ
)
⊂
Im
(
σ
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TraditionalForm
|
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[15]
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​
[16]
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This section is taken from the paper “Summation and transformation formulas for elliptic hypergeometric series”, by S.O. Warnaar, available at arXiv:math.QA/0001006.
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This section is taken from the paper “The prolate spheroidal phenomena and bispectrality”, by F. Alberto Grünbaum and Milen Yakimov, available at arXiv:math-ph/0303041.
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This section is taken from the paper “Differential characters on orbifolds and string connections I”, by Ernesto Lupercio and Bernardo Uribe, available at arXiv:math.DG/0311008.
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This section is taken from the paper “The mystery of the Brane relation”, by Stavros Garoufalidis, available at arXiv:math.GT/0006045.
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[1]
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Warnaar
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,
Summation and transformation formulas for elliptic hypergeometric series
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a
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a
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e
a
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http://www.arxiv.org/abs/math.QA/0001006
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and
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Yakimov
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The prolate spheroidal phenomena and bispectrality
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http://www.arxiv.org/abs/math-ph/0303041
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