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8/20/2019 Research Plan Beasiswa
http://slidepdf.com/reader/full/research-plan-beasiswa 1/4
( rt*ft)
4t9/n'W}-(-fffiXi+@
Field of Srudy
and
Srudy Program
Fuil name in
your
native language BALADMM
MOHAMMAD
SAMY
(r$4
(Etr;E))
(Farnily
namelSumame)
(First
nme)
(Middle
name)
INDONESIA
ationality
(E
#)
Proposed
study
program
in
Japan
(State
the outline
ofyour
major field ofstudy
on
this
side and
the concrete
details
ofyour study
program
on
the
back side
of this sheet.
This
section will be used as one of the most importart
references for selection. The statement must be
typewritten
or witten in block letters.
Additional sheets of
paper
may be
attached,
if necessary.
)
/E^roEftfratE,;
:onfrf;il@t*,
iEe&LNt+6tEr@4tr84Lt6ar, *frt
F:4..hwaffig.?,
F.ffit
Efi4;+E,)E+#EaEf6r:;ts\
\-n-g-u.
L.
ie^&r4inaffi=r:tr
aLaL.t-,
,L'g./r*e&ErrffEaEtuL<til'."
)
If
you
have
Japanese
ianguage ability, write in
Japanese.
1 Present field of
study
GAtraq-*lSW)
My
present
field
of
study is Mathematics.
Here,
I
focused
myself
on
Algebra and Combinatorics.
For my flnal
project,
I
took
one of Combinatorics
topic that is the construction
of interval t-design
and spherical t-design.
2 Your research
theme
after arrivaL
in
Japan:
CIearIy explain
the research
you
vish to
carry out
in
Japan.
(ffiA'&AEfrX7*'z
:
E
Ai.*ir.(
D
)
r,tcfzlfift,f\
Lf:l,
\i'[email protected]=d,^-t6:
l
)
Attached
in
separate
paper
(study
program
on
the
back
side of this
sheet)
8/20/2019 Research Plan Beasiswa
http://slidepdf.com/reader/full/research-plan-beasiswa 2/4
Detail
of
The
Proposed
Research
1 Research
Theme
Extended
construction
of spherical l-design
aud
Euclidean
l-design
2 Classification
Based
on
Mathematics Subject
Classification
2010,
this
research will be
under
these
subjects
:
Primary:
05899
None of above
[Subjects
in
Designs
and Configuration
Section],
but
il
this
section
Secondary:
05E30
Associatiou
schetnes,
strongly
regular
graphs
65D32
Quadrature
and cubature
formulas
05E05
Symmetricfunctions
and
generalizations
41A55
Approxirnate
quadrature
05830
Other
design,
configuratiotl
12D10
Polynomials
:
location of
zeroes
3 Introduction
Cornbirratorics
is a brauch
of rnathematics
concernitg
the study
of finite
or countable
discrete
structures.
It
has
many applications
in
optimization,
computer
science,
ergodic
theory
and
statistical
physics.
One
main
aspects
of combinatorics
is
about
deciding
when certain
criteria
can be
met,
including
how
to
construct
and analize
objects
meeting the
criteria.
In this
aspect,
combinatoriai
design
play
a big
roie,
furthermore
it is
one
of
important
objects
in combinatorics'
Combinatorial
design,
especially
block
design,
might be viewed,
in a sense,
as
au
approximation
of
the
discrete
sphcre
56
of all
k-subsets
by
the sub-coilection
X
of
St,
,
where
Sp
::
{r
e
IR'
:
r]
+
r}+...+
?
:
k,*.i€
{0,1}}.
Later,
Delsarte, Goethals
and Seidel
17]
introduced
an analogue
concept
ofdesigns
for
(continuous)
sphere
by deflning
what
they called
spheri,cal
t-de,si'gn.
A spherical
l-desigu
is
a
finite
subset
X iu the
unit
sphere
5,-1 6
lR which
replaces
the
value of
the
integral on
the
sphere
of
any polynomial
of
degree
at
most
f
by the
average
of the
values
of
the
polynomial on the
finite
subset X,
formally
the
fcxmula
t
f
'x)do(r)=*f
ft )
i1sa1
J
't|
4t
7x
is exact
for ali
poiynomials
/(r)
:
l@o,nt,rz,....ra)
of
degree at
most
I
(where
o
denotes
the
surface
rleasule
orr
S'l).
Generalizing
the
concept
of
spherical
designs,
Neumaier and
Seidel
l11l
defined
the
concept
of Eu-
ciiclean
i-design
in lR.
as a
finite
set
X
in IR'
for
which
P r,,
'
,
)-','j, I
f@)a o@)
:Iw(r)f(r)
= ,
15,
I
J
s,'
r€l
holds
for
any
polynomial
/(c)
of
deg(/)
(
l,
where
{,9,,
i
<
i
< p}
is
the
set
of
ail
the concentric
spheres
centerecl
at the origin
and
intersect
with
X,
X;.:
X n S,. and
u:
X
-)
IR;'s
is
a
weight
function
of X'
We
inight viewed
the spherical
f-clesign
as
the
Euclidean
t-design
with
p
:
1 aud
a
coustant
weight'
4 Motivation
The rnai,
problem i1 the
stutly
of spherical
design
and
Euclideau
desigu
is to
provide
aD
expliclt
constluc-
tion
of
it.
The
curreutly
known construction
methods
are varied,
mostiy
involving
algebraic
combinatorics
urethods.
Thus,
i1 19g1. Rabau
a1d
Bajlok
[12]
stated
t]rat
the constructioil
of spherical
design
in
1R might
be
viewed
as
the
constructlon
of
interval
t-design
with Gegenbauer
weight
function.
By
having
the
interval
8/20/2019 Research Plan Beasiswa
http://slidepdf.com/reader/full/research-plan-beasiswa 3/4
\
\
f-design,
any
spherical
d"esign
in ihe
unit
sphere
,5d
1
C
lRd
can
be
lifted
to the
tight
or
non-tight
i-desi*n
]
in
the
unit
sphere
5d
q
Pd+r'
I
trntcrval
t-6esign
itself
can
be
studied
frorn
the
perspective
of
poiynomial
theory-in
this
context
zeto2/
of
poiy,o*riats
ptay
the
roie
in
the
construction.
The
possible
evaluation
rnethod
is
bv
usi[g
Sturnl's
theorem
(see
[13]).
On
the
other
hand,
spherical
clesign
can
be
said
as
the
geometrical
interpretaiion
of
combinatorial
desig,.
This
rnakes tire
corrstructiou
reler,'aut
for
sorne
combinatorial
techniques,
larnely association
scheme,
coherent
configuration,
and
distance-regular
graph
(see
[1]
and
[a])
-Since
spherical
desig,
can
be
generalized
into
Euclidea,n
design,
these
techniques
can
also
be
applied
in
the
construction
(sec
l2l
and
16l)
The
goal
of
this
research
is
to
investigate
soure
construction
methods
of
spherical
t-desig[
aild
Eu-
clidean
t-design
from
these
points of
view'
5 Related
Result
Inspired.
by
the
Kuperberg's
method
(see
[8]),
I
have
proposed
these
new theorems
iu
[3]
as
part
of
the
result
in
my
previous
research.
Theorerrr
L Let
s
>
0
be
an,inte,ger
and,a:
(or,rr....,4,)
€
lR".
iet
alsoY
be
the
set oJ2u
points
of
the
forrns
*oi
*
az
1...
+ a".
Tlrc
set
lorrns
a'n
'interual
3-tlesi,grt
on
l-L,L)
with
'res'pect to tlt'e
Gege'nbaue'r
weiglfi
Jurtctzon
Tr'(r)
:
(t
*
r)Lz-
i'J a'nrt
o'nlg
if
llall
:+
"
t/d+t
Theorern
2
The
setY
oJ22
points
o/
the
Jorms
*a1
la2,
wi'th
a;>
0,
satisiyi'ng
the
t'nteruaiS-desi:gn
onlr-7.1)with,r'especttoth,eGe4enbatteru;ezgh'tfunctzonut(r):
(1
-n2)(d-2)12
ea'istsonlyford:1'
-t
rt
^^,1-')
i
-
L. ut u
-
o
Theorem
3
The
23
poi,nts
fJA
t/rr*
JA
Jorm
an
i,nterual
S-d'es'ign
on
l-l,I)
wi'th
th'e
Gegenbauer
wei,qhi.functionto(r):(l-rz;ta-zl/'i'fon'donlyi'ft'hezi'sarerootsoJpolynornial
a
-
r."
Q@):t"-,-,;*
dr
-p,
2(d+t)2(d+3)
u;here-
p
i,s
i,n,
th,e
interual
18d2-27d+5d3+ri2-108
5(13
Ol
or
:
54116
-t
648r)b
+
3078d4 +
rc44{P +
9234d2
+
5832d *
t458,
a'nd
)
+02+108
7d-
t8d2
*
O1
,r:
uE@+
3)3/2
Moreouer,
i,t'ts
only
erists
for
L
<
d
<
6'
_d3+L8d2_108d+216.
Theorem
4 ThesetY
of
23
poi,ntsof
theform .a11,21.a3.uti,t,ha;>0,sati'sfyi'ngt'lt'ei'nterua'l
7-clest'gn
,n
l-l,l)
witlt
.r.es,pect
to tlte
Gege,nbiuer
we'igltt
fu:nction
u:(r)
:
(1
*'*2)td-2)12
eaists
ortly
Jor
d:
L or
d:2.
Theorern
5
TheKuperbergsetY
oJ2a
poi'nts'L\/4+J"+'/4+:@ :T*taninterual7-desi'gnon
l-l.ll
w1tlt
respect
to
the
Cegenbatter
we'ight
J"'n':tionu(r):
(1
-
n2)(d'-2)l'
il
th"
zi's
are'
th'e'
roots
of
poLynorni,al
^,
\ o
,3
r2d
*({-7)-.
=-=.
==+p,
(''):
rq
-
dt
1+
2d3+t0dr-Lltd+6
-
6d5;66d4+2nat"'t20d2+318dt90
wi'rere
p
zs
r,n
the
open
interual
as
prouzded
i'n
the
table
below:
d\
p
,
J
4
----(o-o
o
oooo
1
62880800
1
)
(0.00000i440015579,
0.000002166860188)
io.oooootoza
2t9404,0.000001951463832)
Tlcse
theorerns
also
wiil
be
further
generalized
ald
observed
iri
future
research
(as
ilescribed
i[ t]re
following
section).
8/20/2019 Research Plan Beasiswa
http://slidepdf.com/reader/full/research-plan-beasiswa 4/4
6
Objectiver
\
l
Objective
1 :
Concerning
the application
of Sturm's
theorem,
I
propose
to use this
method for
I
tire
further
construction of spherical
design
fbr
d
>
6.
In my
previous result,
the
spherical i-design
I
is constmcted
for
d
< 6.
Objective 2
:
I
plan to
investigate how
association
scheme,
coherent
corfiguration,
and
distance-
regular
graph relate with
the
construction
of spherical design.
This
may lead
to other
characteri-
zalioirs of
Euclidearr
desigrr.
Objective 3
: By using the
result of Objective
2, I aim
to
construct
Euclidean design usitg
those
algebraic
combinatorics
methods.
7
People
and
Places
Tiris researcir
is
expected
to be conducted
at
Graduate
School
of
hrformation
Science,
Tohoku University,
with
the
supervision
of
Professor
Akihiro
Munemasa
in his iab
Mathematics
Structure
I .
Earlier
u,-ork
done by the supervisor
which
is of
relevance
fbr
this research
includes
[9],
[10],
and
l5].
The
iab
environment
is
also expected
to
support
this
research development.
8
Research
Planning
Objective 1
and
2 are
going
to
be
ca,rried
out
first.
Technicaily, they
are
easier. Conceptually,
Objective
1
attempts to compiete
my
previous
research
;
Objective
2
relates the study
of
previously known approaches
to coustruct sphericai
and
Euclideau design.
This
understairding
will be
a
good
start
to
tackle
the
last
obiectives
which
are
the
most ambitious
ones.
An approximative
schedule
may
be
the
following
:
Objective 1
and
2
seems
to
be within
rear:h
in
the
first
year
of the
research
if accepted.
If
possible,
the
preparation
for
Objective
3
may
conducted
in
this
first
year.
Since
I
predict that
the construction
of Euclidean
design
would take
a
greater effort, Objective
3
may be
held
in the
second
vear,
hopefully,
in my
future
graduate
school.
References
t1l
l2l
t3l
B. Br;xon,
Construction
of spherical
l-designs ,
Geom
Dedicala
43
(1990),
rro. 2,767-179.
B. B.+..ruoN,
On
Euciidean
designs ,
Adt.
Geom.6
(2006), 431-446.
[,I.S-
B.cr-4DnAM
AND D. SupRuenro,
Anoteinnewconstructionof
spherical
(2t+l)-<lesigns ,
(2012)
In,
prog'ress
Er.
B-4NN.tr
aNo
Er.
BANNAT,
A survey
on
splielical
designs
and
algebraic
combitatorics
on
spheres ,
Europeu'n J.
Cort,bi,n,,30
(2009), no.
6,
1392-1425.
Ei.
B.tNNa.r,
A.
Mulrer.a.tsA)
AND
B.
VENKOV
The
nonexistence
of
certain tiglrt
spherical
designs ,
Algebru,
z
An,uliz.16
(2004),
1-23.
Er. BaNN-q.r
New
exarnples
of Euciitlean
tight
4-desigrts,
Europeu'n
J. Cornbin.
S0
(2009),
655-667
P.
DerSasrB,
J.-M.
Gr.ie:rnalS,
aNo
J.
J.
SaIoel, Spherical
codes and
designs ,
Geom.
Ded'i,cata6
(1977),
no.
3,
363-388.
G. KupeReBRG,
Special
ntoments ,
Adu.
in
Appl'
Math.34
(2005), no.
4,
853-870'
A.
MuNsN4ase,
Sphericai
5-designs
obtained
lrorn
finite
unita.ry
gtoups , European
J.
Combin.25
(2004),261-267
A.
Mu5BuLs.r, sphericai
design ,
Hand,book
of
Comb'inatorial
Desi,gns, Second
Edi,ti,on
{2006),61'7-622
A. Nnulr-qtra4No
J.
J.
Sstoor,,
Discrete
neasures
for spherical
designs,
eutactic
stars,
and
lattices ,
Ned,erl.
Akad,. Wetensch.
Proc Ser.
A 91:Indag'
Math.,50
(1988),
32L-334'
p.
R.{B-A.g
AND B.
Be;uox,
Bounds
for the
number
of
nodes
in
Chebyshev
type
quadrature
formulas ,
J.
Approrimati,on
Th.eory
67
(1991),
199-214.
c.
Y,qp, Fund,amental
problems
i.n
algorzthrni.c
algebra,
oxford
University
Press,
2000.
t4l
l5l
t6l
l7l
t8l
tel
[10]
11
1l
L12l
[13]