research plan beasiswa

8/20/2019 Research Plan Beasiswa

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( 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)



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



\

\

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).



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]

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