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Dr
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Gr
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Stu
d
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s
itio
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T
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s
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CC B
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C
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s
p
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A
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:
Dan
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B
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alel
A.
Gar
cia
Dep
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tm
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t o
f
Ma
th
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d
Natu
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Scien
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s
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p
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Pan
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asin
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State
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C
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,
2
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Pan
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5
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ts
ch
o
s
e
to
d
r
o
p
o
u
t
o
f
th
eir
p
r
o
g
r
am
.
Aca
d
em
ic
s
tr
ess
,
in
clu
d
in
g
p
r
ess
u
r
e
f
r
o
m
th
eir
teac
h
er
s
an
d
p
ee
r
s
,
is
a
s
ig
n
if
ican
t
is
s
u
e
[
1
3
]
.
So
m
e
s
tu
d
en
ts
f
r
o
m
in
d
i
g
en
o
u
s
co
m
m
u
n
ities
s
u
ch
as
“
Aeta
s
”
,
s
tr
u
g
g
le
with
lan
g
u
ag
e
b
ar
r
ier
s
,
cu
ltu
r
al
d
if
f
er
en
ce
s
,
an
d
lo
w
s
elf
-
esteem
[
1
4
]
.
O
n
e
o
f
th
e
Ph
ilip
p
in
e
g
o
v
er
n
m
en
t’
s
in
i
tiativ
es
is
th
e
ter
tiar
y
ed
u
ca
t
io
n
s
u
b
s
id
y
(
T
E
S)
p
r
o
g
r
a
m
,
p
ar
t
o
f
R
.
A.
1
0
9
3
1
also
k
n
o
wn
as
th
e
“
u
n
iv
e
r
s
al
ac
ce
s
s
to
q
u
ality
ter
tiar
y
ed
u
c
atio
n
ac
t
”
,
aim
s
t
o
s
u
p
p
o
r
t
t
h
o
s
e
s
tu
d
en
ts
th
at
a
r
e
im
p
o
v
er
is
h
ed
y
et
d
eser
v
in
g
s
tu
d
en
ts
[
1
5
]
.
Ho
wev
er
,
c
o
n
ce
r
n
s
a
b
o
u
t
th
e
lo
n
g
-
ter
m
f
in
an
cial
v
iab
ilit
y
p
er
s
is
t,
an
d
th
e
r
e
ar
e
s
h
o
r
t
co
m
in
g
s
in
t
h
e
im
p
lem
e
n
tatio
n
p
a
r
ticu
lar
ly
th
e
d
elay
ed
f
u
n
d
allo
ca
tio
n
[
1
6
]
.
Hen
ce
,
ev
er
y
ac
ad
e
m
ic
in
s
titu
tio
n
m
u
s
t c
o
n
d
u
ct
p
r
o
ac
tiv
e
m
ea
s
u
r
es in
d
etec
tin
g
th
e
s
tu
d
en
ts
wh
o
tak
e
lo
n
g
e
r
to
f
in
is
h
th
eir
p
r
o
g
r
am
an
d
th
o
s
e
wh
o
d
r
o
p
o
u
t
to
s
u
p
p
o
r
t
s
tr
u
g
g
lin
g
s
tu
d
en
ts
an
d
im
p
r
o
v
e
r
ete
n
tio
n
r
ates
[
1
7
]
–
[
1
9
]
.
T
h
ese
id
en
tifie
d
d
il
em
m
as
em
p
h
asize
th
e
n
ee
d
f
o
r
tar
g
eted
s
u
p
p
o
r
t
an
d
in
ter
v
e
n
tio
n
s
to
h
elp
Fil
ip
in
o
s
tu
d
en
ts
s
u
cc
ee
d
in
c
o
lleg
e.
On
e
o
f
th
e
p
r
o
g
r
am
s
o
f
f
er
e
d
a
t
Pan
g
asin
an
State
Un
iv
er
s
ity
–
Ur
d
an
eta
C
ity
C
am
p
u
s
(
PS
U
Ur
d
an
eta)
is
th
e
B
ac
h
elo
r
o
f
Scien
ce
i
n
Ma
th
em
atics
(
B
S
Ma
th
)
.
T
h
e
B
S
Ma
th
p
r
o
g
r
am
h
as
p
o
s
s
ib
le
s
p
ec
ializatio
n
s
in
p
u
r
e
m
ath
em
atics,
s
tatis
tics
,
a
n
d
co
m
p
u
ter
in
f
o
r
m
atio
n
tech
n
o
lo
g
y
u
n
d
er
th
e
s
u
p
e
r
v
is
io
n
o
f
th
e
Ma
th
em
atics
an
d
Natu
r
al
Scien
ce
s
Dep
ar
tm
en
t
o
f
th
e
C
o
lleg
e
o
f
C
o
m
p
u
tin
g
.
Of
c
o
u
r
s
e,
th
e
B
S
Ma
th
p
r
o
g
r
am
is
n
o
t
ex
em
p
ted
f
r
o
m
th
e
d
ilem
m
a
o
f
co
m
p
letin
g
t
h
e
d
eg
r
ee
b
ey
o
n
d
th
e
r
eg
u
lar
n
u
m
b
e
r
o
f
y
ea
r
s
.
I
t
was o
b
s
er
v
ed
in
th
e
s
tu
d
y
o
f
Ah
m
ad
et
a
l.
[
2
0
]
wh
ich
also
alig
n
s
o
n
th
e
r
ep
o
r
t
f
r
o
m
th
e
Of
f
ice
o
f
th
e
C
am
p
u
s
R
eg
is
tr
ar
o
f
PS
U
Ur
d
an
eta,
th
at
th
er
e
h
av
e
b
ee
n
a
lar
g
e
n
u
m
b
e
r
o
f
u
n
d
er
g
r
ad
u
ate
m
ath
em
atics
o
r
B
S
Ma
th
s
tu
d
en
ts
wh
o
f
ailed
to
f
in
is
h
th
e
p
r
o
g
r
am
b
ec
au
s
e
o
f
th
e
f
o
llo
win
g
r
ea
s
o
n
s
:
i
)
th
ey
s
h
if
ted
to
an
o
th
er
p
r
o
g
r
am
o
f
f
er
e
d
with
in
th
e
u
n
iv
er
s
ity
(
B
S
M
ath
s
tu
d
en
t
wh
o
will
tr
an
s
f
er
to
a
b
o
ar
d
p
r
o
g
r
am
)
;
ii
)
tr
an
s
f
er
r
ed
to
an
o
th
e
r
ac
ad
em
ic
in
s
titu
tio
n
;
an
d
iii
)
d
r
o
p
p
ed
f
r
o
m
th
e
B
S
Ma
th
p
r
o
g
r
am
d
u
e
t
o
v
ar
i
o
u
s
r
ea
s
o
n
s
.
T
h
u
s
,
th
er
e
is
a
n
ee
d
to
s
tu
d
y
t
h
e
s
tu
d
en
t
p
r
o
g
r
ess
io
n
o
f
th
e
B
S
Ma
th
p
r
o
g
r
a
m
as
a
b
asis
f
o
r
s
tr
ateg
ic
p
lan
n
in
g
to
p
r
ev
en
t
t
h
e
s
h
if
tin
g
,
d
r
o
p
p
in
g
,
an
d
d
elay
o
f
th
e
s
tu
d
en
ts
.
Ma
r
k
o
v
c
h
ain
an
aly
s
is
h
as
b
e
en
u
tili
ze
d
b
y
m
an
y
r
esear
ch
er
s
to
an
aly
ze
th
e
s
tu
d
e
n
ts
’
p
r
o
g
r
ess
in
h
ig
h
er
e
d
u
ca
tio
n
al
in
s
titu
tio
n
s
.
Ma
r
k
o
v
m
o
d
el
was
u
s
ed
to
e
v
alu
ate
th
e
ac
a
d
em
ic
p
er
f
o
r
m
an
ce
o
f
s
tu
d
en
ts
in
a
f
o
u
r
-
y
ea
r
en
g
in
ee
r
in
g
p
r
o
g
r
am
in
an
in
s
titu
tio
n
lo
ca
te
d
in
I
n
d
ia
[
2
1
]
.
Ma
r
k
o
v
ch
ai
n
was
em
p
lo
y
ed
in
E
led
u
m
a
n
d
I
d
r
is
s
[
2
2
]
,
to
ca
l
cu
late
th
e
p
r
o
b
a
b
ilit
ies
o
f
s
tu
d
en
ts
’
p
r
o
g
r
ess
io
n
b
etwe
en
s
u
cc
ess
iv
e
s
em
ester
s
o
f
th
e
s
tu
d
y
p
r
o
g
r
am
f
o
r
ea
ch
ac
ad
em
ic
y
ea
r
.
A
n
o
th
er
s
tu
d
y
em
p
l
o
y
ed
Ma
r
k
o
v
m
o
d
el
to
esti
m
ate
th
e
p
r
o
b
a
b
ilit
y
o
f
s
tu
d
en
ts
in
d
i
f
f
er
en
t
le
v
els
f
o
r
g
r
a
d
u
atin
g
with
o
u
t
d
elay
in
g
,
th
e
p
r
o
b
a
b
ilit
y
o
f
ac
ad
em
ic
d
is
m
is
s
al
an
d
d
r
o
p
p
in
g
o
u
t
o
f
th
e
s
y
s
tem
b
ef
o
r
e
attain
in
g
t
h
e
m
ax
im
u
m
q
u
alif
icatio
n
o
v
er
th
e
b
atch
es
[
2
3
]
.
T
h
e
ex
p
ec
ted
tim
e
th
at
a
s
tu
d
en
t
s
p
en
d
s
at
p
ar
ticu
lar
s
tag
e
as
well
as
th
e
ex
p
ec
te
d
d
u
r
ati
o
n
o
f
th
e
s
tu
d
y
is
d
eter
m
in
ed
u
s
in
g
Ma
r
k
o
v
c
h
ain
[
2
4
]
.
I
n
ad
d
itio
n
,
th
e
Ma
r
k
o
v
c
h
ain
m
o
d
el
was
u
s
ed
to
d
ef
i
n
e
a
n
in
d
ex
esti
m
atin
g
u
n
iv
er
s
ity
d
r
o
p
o
u
t
r
is
k
,
s
h
o
win
g
th
e
h
ig
h
est
d
r
o
p
o
u
t
p
r
o
b
a
b
ilit
y
in
f
ir
s
t
two
s
em
ester
s
(
o
r
f
ir
s
t
y
ea
r
o
f
c
o
lleg
e)
[
2
5
]
.
L
astl
y
,
a
s
tu
d
y
b
y
T
ed
esch
i
et
a
l
.
[
2
6
]
s
h
o
ws th
at
Ma
r
k
o
v
c
h
ain
ca
n
i
m
p
r
o
v
e
m
o
d
els f
o
r
esti
m
atin
g
s
tu
d
en
t
r
eten
tio
n
an
d
g
r
ad
u
atio
n
r
ates
b
y
in
cr
ea
s
in
g
co
n
f
id
e
n
ce
an
d
r
e
d
u
cin
g
th
e
esti
m
atio
n
b
iases
esp
ec
ially
f
o
r
u
n
d
er
r
e
p
r
esen
te
d
s
tu
d
en
ts
o
r
ca
s
es with
s
m
all
s
am
p
le
s
izes.
T
h
is
s
tu
d
y
aim
s
to
a
n
aly
ze
t
h
e
s
tu
d
en
t
p
r
o
g
r
ess
io
n
o
f
th
e
p
r
o
g
r
am
th
at
ca
n
h
elp
th
e
Dep
ar
tm
en
t
o
f
Ma
th
em
atics
an
d
Natu
r
al
Sci
en
ce
s
o
f
th
e
C
o
lleg
e
o
f
C
o
m
p
u
tin
g
d
ev
elo
p
a
p
lan
to
r
ed
u
ce
th
e
n
u
m
b
e
r
o
f
s
tu
d
en
ts
wh
o
ar
e
d
ela
y
ed
an
d
d
r
o
p
o
u
t.
T
h
is
s
tu
d
y
u
s
es
th
e
d
is
cr
ete
-
tim
e
Ma
r
k
o
v
c
h
ain
(
DT
MC)
m
o
d
el
to
an
aly
ze
th
e
p
er
f
o
r
m
an
ce
o
f
B
S
Ma
th
s
tu
d
en
ts
,
d
eter
m
in
e
th
e
av
er
ag
e
n
u
m
b
er
o
f
s
em
ester
s
it
tak
e
s
f
o
r
B
S
Ma
th
s
tu
d
en
ts
to
co
m
p
let
e
th
e
p
r
o
g
r
am
an
d
t
h
e
lik
elih
o
o
d
o
f
s
tu
d
en
ts
d
r
o
p
p
in
g
o
u
t
o
f
th
e
p
r
o
g
r
am
o
r
ad
v
an
cin
g
f
r
o
m
th
eir
last
s
em
ester
.
T
h
e
ap
p
licatio
n
o
f
DT
MC
in
an
aly
zin
g
s
tu
d
en
t
p
r
o
g
r
ess
io
n
in
tr
o
d
u
ce
s
a
s
to
ch
asti
c
o
r
p
r
o
b
ab
ilis
tic
ap
p
r
o
ac
h
to
u
n
d
e
r
s
tan
d
in
g
h
o
w
B
S
Ma
th
s
tu
d
en
ts
tr
av
er
s
e
f
r
o
m
o
n
e
s
tate
to
an
o
th
er
u
s
in
g
th
e
PS
U
Ur
d
an
eta
d
ata.
Un
lik
e
co
m
m
o
n
d
ete
r
m
in
is
tic
m
o
d
els,
th
e
DT
MC
m
o
d
el
ca
p
tu
r
es
th
e
laten
t
u
n
ce
r
tain
ties
an
d
v
ar
i
ab
ilit
y
in
s
tu
d
en
t
p
r
o
g
r
ess
io
n
b
y
tr
ea
tin
g
th
eir
p
r
o
g
r
ess
io
n
as
s
tates
w
ith
p
r
o
b
a
b
ilis
tic
tr
an
s
itio
n
s
.
A
s
id
e
f
r
o
m
co
n
tr
ib
u
tin
g
to
th
e
s
ca
r
ce
r
ef
er
en
ce
o
n
th
e
ap
p
licatio
n
s
o
f
DT
MC
in
th
e
Ph
ilip
p
in
e
ed
u
ca
tio
n
s
ettin
g
,
t
h
is
m
eth
o
d
o
lo
g
y
en
ab
les
a
d
ata
-
d
r
iv
en
f
o
r
ec
ast
o
f
f
u
tu
r
e
ac
ad
em
ic
o
u
tco
m
es,
h
elp
in
g
t
h
e
ad
m
in
is
tr
ato
r
s
an
d
f
ac
u
lty
m
em
b
er
s
id
en
tif
y
s
t
u
d
en
ts
wh
o
ar
e
at
-
r
is
k
an
d
o
p
tim
ize
in
ter
v
en
tio
n
s
tr
ateg
ies
f
o
r
th
e
B
S
Ma
th
p
r
o
g
r
am
.
T
h
e
n
o
v
elty
lies
in
t
h
e
i
n
co
r
p
o
r
atio
n
o
f
r
ea
l
-
wo
r
ld
s
ch
o
last
ic
r
ec
o
r
d
d
ata
with
p
r
o
b
ab
ilis
tic
m
o
d
elin
g
,
allo
win
g
a
m
o
r
e
d
y
n
am
ic
in
s
ig
h
t
c
o
m
p
ar
e
d
t
o
c
o
n
v
en
tio
n
al
s
tatis
tical
m
eth
o
d
o
l
o
g
ies.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
8
2
2
I
n
t
J
E
v
al
&
R
es E
d
u
c
,
Vo
l
.
14
,
No
.
6
,
Dec
em
b
er
20
25
:
4
4
7
8
-
4
4
8
6
4480
2.
M
E
T
H
O
D
A
r
eq
u
est
letter
n
o
te
d
b
y
th
e
ca
m
p
u
s
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u
tiv
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d
ir
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was
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to
th
e
o
f
f
ice
o
f
th
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m
p
u
s
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eg
is
tr
ar
.
T
h
e
r
eq
u
ested
d
ata
s
et
is
th
e
co
m
p
lete
en
r
o
llm
en
t
d
ata
f
r
o
m
th
e
B
S
Ma
th
p
r
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g
r
am
s
tar
tin
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f
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ac
ad
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r
(
A.
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)
2
0
0
0
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2
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0
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to
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2
2
-
2
0
2
3
wh
ich
i
n
clu
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th
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ts
’
en
tr
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en
r
o
llm
en
t)
e
v
er
y
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ester
in
th
e
B
S
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th
p
r
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am
an
d
th
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e
x
it
(
eith
er
g
r
a
d
u
ated
,
s
h
if
ted
o
r
d
r
o
p
p
ed
)
in
th
e
p
r
o
g
r
am
.
Af
ter
d
ata
clea
n
in
g
an
d
v
e
r
if
icatio
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,
th
e
s
tu
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en
ts
wh
o
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en
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lle
d
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m
A.
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2
0
0
0
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2
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to
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0
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wer
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m
o
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u
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co
m
p
lete
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o
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atio
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.
A
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tal
o
f
2
1
1
s
tu
d
en
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lled
f
r
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m
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2011
-
2
0
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2
to
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0
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2
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2
0
2
3
we
r
e
k
ep
t f
o
r
th
is
s
tu
d
y
.
2
.
1
.
So
f
t
wa
re
s
pecif
ica
t
io
n
R
Stu
d
io
was
u
s
ed
to
ap
p
ly
Ma
r
k
o
v
ch
ain
a
n
aly
s
is
in
th
e
f
in
al
d
ata
s
et.
I
t
is
an
in
teg
r
ated
d
ev
elo
p
m
e
n
t
en
v
ir
o
n
m
en
t
(
I
DE
)
f
o
r
R
th
at
in
clu
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es
a
co
n
s
o
le,
an
d
s
y
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h
i
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h
tin
g
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ito
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th
at
s
u
p
p
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ts
d
ir
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t
co
d
e
ex
ec
u
tio
n
,
an
d
to
o
ls
f
o
r
p
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,
h
is
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y
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s
tatis
tical
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aly
s
es,
d
eb
u
g
g
in
g
,
an
d
wo
r
k
s
p
ac
e
m
an
ag
em
en
t.
Mo
r
eo
v
er
,
th
e
Py
th
o
n
ed
ito
r
in
No
tep
ad
++
was
u
s
ed
to
g
en
er
ate
a
m
o
r
e
d
ec
en
t
tr
an
s
itio
n
m
atr
ix
p
lo
t.
2
.
2
.
Dis
cr
et
e
-
t
i
m
e
M
a
rko
v
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in
a
na
ly
s
is
T
h
e
ex
p
lo
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atio
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o
f
th
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e
o
r
y
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d
ap
p
licatio
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s
o
f
th
e
DT
MC
is
ev
id
en
t
in
v
ar
io
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s
f
ield
s
o
f
s
tu
d
y
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DT
MC
is
u
s
ed
in
an
aly
zin
g
an
d
p
r
e
d
ictin
g
tim
e
s
er
ies
[
2
7
]
–
[
2
9
]
,
d
em
o
n
s
tr
atin
g
th
e
ir
ef
f
ec
tiv
en
ess
in
th
e
co
n
tex
t
o
f
t
h
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s
tu
d
y
.
Asi
d
e
f
r
o
m
th
e
th
eo
r
y
o
f
DT
MC,
its
ap
p
licatio
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s
ca
n
b
e
ex
ten
d
ed
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is
cr
ete
-
tim
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two
-
tim
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-
s
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Ma
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ain
s
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g
th
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s
e
in
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ilter
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atio
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,
c
o
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o
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o
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tim
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n
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an
d
f
in
a
n
cial
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ee
r
in
g
[
3
0
]
.
Ov
er
all,
DT
MC
ca
n
b
e
u
s
ed
to
d
escr
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th
e
b
e
h
av
io
r
o
f
an
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s
y
s
tem
th
at
tr
av
er
s
es f
r
o
m
o
n
e
s
tate
to
an
o
th
er
s
tate
with
a
d
eter
m
in
ed
p
r
o
b
ab
ilit
y
,
an
d
th
is
p
r
o
b
ab
ilit
y
o
f
tr
an
s
itio
n
to
th
e
n
ex
t
s
tate
is
d
ep
en
d
en
t
o
n
ly
o
n
th
e
cu
r
r
en
t
s
tate
o
f
th
e
s
y
s
tem
.
T
h
is
also
m
ea
n
s
th
at
th
e
p
r
o
b
ab
ilit
y
o
f
tr
an
s
itio
n
is
n
o
t d
ep
e
n
d
en
t
o
n
wh
ich
s
tate
th
e
s
y
s
tem
was in
b
ef
o
r
e
t
h
e
cu
r
r
en
t state.
I
n
th
is
s
tu
d
y
,
s
em
ester
s
(
1
s
t
an
d
2
n
d
)
p
er
y
ea
r
le
v
el
(
1
s
t
y
ea
r
to
4
t
h
y
ea
r
)
,
g
r
ad
u
ate
d
,
s
h
if
ted
,
an
d
d
r
o
p
p
ed
ar
e
t
h
e
s
ta
tes.
th
e
m
o
v
em
en
t
o
f
th
e
s
y
s
tem
s
tar
ts
in
th
e
f
ir
s
t
-
y
ea
r
,
f
ir
s
t
s
em
este
r
an
d
will
m
o
v
e
to
an
o
th
er
s
tate
u
n
til
it
r
ea
ch
es
eith
er
th
e
g
r
ad
u
ated
s
tate,
s
h
if
ted
s
tate,
o
r
d
r
o
p
p
ed
s
tate
.
T
h
u
s
,
th
e
s
em
ester
ev
er
y
y
ea
r
lev
el
is
th
e
t
r
an
s
itio
n
s
tate,
th
e
g
r
ad
u
ated
,
s
h
if
ted
,
an
d
d
r
o
p
p
ed
ar
e
th
e
ab
s
o
r
b
i
n
g
s
tates
.
T
h
e
s
y
s
tem
allo
ws
th
e
m
o
v
em
en
t
f
r
o
m
th
e
cu
r
r
en
t
s
tate
to
th
e
p
r
ev
io
u
s
s
tate
in
d
icatin
g
th
e
s
tatu
s
o
f
a
s
tu
d
en
t
m
ay
r
etu
r
n
to
th
e
p
r
e
v
io
u
s
y
e
ar
lev
el
if
s
h
e/h
e
f
ails
to
p
ass
h
er
/h
is
co
r
e
m
ath
em
atics
s
u
b
jects
in
th
e
cu
r
r
en
t
s
tate.
Fu
r
th
er
m
o
r
e,
th
e
f
o
llo
win
g
co
n
d
itio
n
s
ar
e
ass
u
m
ed
:
i
)
th
e
s
y
s
tem
ap
p
lies
to
all
B
S
Ma
th
s
tu
d
en
ts
;
ii
)
th
e
p
r
o
b
ab
ilit
ies
o
f
m
o
v
i
n
g
f
r
o
m
o
n
e
s
tate
to
all
o
th
er
s
ta
tes
ar
e
eq
u
al
to
o
n
e;
an
d
iii
)
t
h
e
p
r
o
b
ab
ilit
ies
ar
e
co
n
s
tan
t
in
th
e
lo
n
g
r
u
n
.
T
h
is
s
y
s
tem
ca
n
b
e
ex
p
lain
ed
f
u
r
th
er
th
r
o
u
g
h
a
m
ath
em
a
tical
m
o
d
el
in
th
e
s
u
cc
ee
d
in
g
d
is
cu
s
s
io
n
s
.
C
o
n
s
id
er
a
f
in
ite
d
is
cr
ete
tim
e
h
o
m
o
g
en
eo
u
s
s
eq
u
en
ce
{
:
∈
ℤ
}
o
f
r
an
d
o
m
v
ar
iab
les.
I
n
th
is
ca
s
e,
in
d
icate
s
th
e
c
u
r
r
en
t
s
tate
o
f
th
e
p
r
o
ce
s
s
at
a
t
im
e
.
A
s
et
co
n
tain
in
g
all
s
tates
is
ca
lled
th
e
s
tate
s
p
ac
e
,
th
u
s
,
∈
.
Mo
r
eo
v
e
r
,
th
e
s
to
ch
asti
c
p
r
o
ce
s
s
h
er
e
s
atis
f
ies
th
e
m
em
o
r
y
less
o
f
th
e
s
to
ch
asti
c
p
r
o
ce
s
s
o
r
th
e
Ma
r
k
o
v
p
r
o
p
e
r
ty
,
th
at
is
,
g
iv
en
th
e
p
r
esen
t sem
ester
an
d
y
ea
r
lev
el
o
f
a
s
tu
d
en
t,
th
e
n
ex
t step
is
in
d
ep
en
d
en
t o
f
th
e
p
r
io
r
s
em
e
s
ter
s
an
d
y
ea
r
lev
els o
f
th
e
cu
r
r
en
t state.
No
w,
f
o
r
,
,
0
,
1
,
⋯
,
−
1
∈
.
(
+
1
=
|
0
=
0
,
1
=
1
,
⋯
,
−
1
=
−
1
,
=
)
=
(
+
1
=
|
=
)
=
W
h
er
e
,
is
th
e
p
r
o
b
a
b
ilit
y
o
f
m
o
v
in
g
f
r
o
m
s
tate
to
an
o
th
er
s
tate
in
ju
s
t
o
n
e
tr
an
s
itio
n
an
d
all
th
ese
q
u
an
titi
es d
ef
in
e
th
e
m
atr
ix
o
f
o
n
e
-
s
tep
tr
an
s
itio
n
p
r
o
b
ab
ilit
ies
,
th
at
is
,
=
[
11
12
⋯
1
21
22
⋯
2
⋮
⋮
⋱
⋮
1
2
⋯
]
W
h
er
e
,
th
e
f
in
ite
s
et
{
1
,
2
,
⋯
,
}
is
th
e
s
tate
s
p
ac
e
o
f
th
e
Ma
r
k
o
v
ch
ain
.
E
ac
h
en
tr
y
P
m
u
s
t
s
atis
f
y
th
e
co
n
d
itio
n
s
.
T
h
is
m
atr
ix
is
ca
ll
ed
th
e
tr
an
s
itio
n
m
atr
ix
P.
T
h
e
r
ec
u
r
r
en
t
s
tate
i
is
ca
lled
th
e
ab
s
o
r
b
in
g
s
tate
if
=
1
.
T
h
e
tr
an
s
itio
n
m
atr
i
x
ca
n
b
e
wr
itten
in
its
ca
n
o
n
ical
f
o
r
m
.
T
h
e
ca
n
o
n
ical
f
o
r
m
is
ju
s
t
a
p
ar
titi
o
n
ed
tr
an
s
itio
n
m
atr
ix
in
to
f
o
u
r
m
a
tr
ices.
T
h
u
s
,
th
e
ca
n
o
n
ical
f
o
r
m
o
f
th
e
tr
an
s
itio
n
m
at
r
ix
P
o
f
th
e
s
y
s
tem
o
f
th
is
s
tu
d
y
is
:
=
(
0
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t
J
E
v
al
&
R
es E
d
u
c
I
SS
N:
2252
-
8
8
2
2
Dete
r
min
in
g
s
tu
d
en
t p
r
o
g
r
ess
i
o
n
r
a
tes u
s
in
g
d
is
crete
-
time
Ma
r
ko
v
ch
a
in
m
o
d
el
(
Ma
r
k
Jo
h
n
T.
Ma
n
g
s
a
t
)
4481
W
h
er
e
,
I
is
an
3
×
3
id
en
tity
m
atr
ix
th
at
co
r
r
esp
o
n
d
s
to
th
r
ee
ab
s
o
r
b
in
g
s
tates,
0
is
a
3
×
8
ze
r
o
m
atr
ix
,
Q
is
an
8
×
8
m
atr
ix
o
f
tr
an
s
itio
n
p
r
o
b
ab
ilit
ies
f
o
r
th
e
tr
a
n
s
ien
t
s
tates,
an
d
R
is
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8
×
3
m
atr
ix
o
f
tr
an
s
itio
n
p
r
o
b
a
b
ilit
ies
f
o
r
th
e
ab
s
o
r
b
in
g
s
tates.
I
n
g
en
er
al,
t
h
e
tr
an
s
i
tio
n
m
atr
ix
is
a
s
q
u
ar
e
m
atr
i
x
th
at
d
escr
ib
es
th
e
p
r
o
b
a
b
ilit
y
o
f
m
o
v
in
g
f
r
o
m
o
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e
to
an
o
t
h
er
with
in
th
e
s
y
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tem
an
d
ca
n
b
e
u
s
ed
to
ex
ten
d
th
is
co
n
ce
p
t
in
to
lo
n
g
er
tim
e
in
te
r
v
als.
An
o
th
er
im
p
o
r
tan
t
co
n
ce
p
t
r
el
ated
to
DT
MC
i
s
th
e
f
u
n
d
am
en
tal
m
atr
ix
.
T
h
e
f
u
n
d
a
m
en
ta
l
m
atr
ix
N
o
f
a
Ma
r
k
o
v
c
h
ain
is
a
m
atr
ix
th
at
g
iv
es
in
f
o
r
m
atio
n
r
e
g
ar
d
in
g
th
e
ex
p
ec
ted
n
u
m
b
e
r
o
f
ti
m
es
a
p
r
o
ce
s
s
is
in
ea
ch
s
tate,
g
iv
en
th
at
th
e
s
y
s
tem
s
tar
ts
in
a
p
ar
ticu
lar
s
tate.
Ma
th
em
atica
lly
,
ea
ch
en
tr
y
o
f
th
e
m
atr
i
x
N
,
s
ay
,
ca
n
b
e
th
o
u
g
h
t
o
f
as
t
h
e
ex
p
ec
ted
n
u
m
b
er
o
f
tim
es
th
e
ch
ain
is
i
n
j
if
it
s
tar
ted
in
i
.
Mo
r
eo
v
er
,
th
e
f
u
n
d
am
e
n
tal
m
atr
ix
N
ca
n
b
e
d
ef
in
ed
as
(
×
−
×
)
−
1
,
wh
er
e
it
is
k
n
o
wn
th
at
×
−
×
is
an
in
v
e
r
tib
le
m
atr
ix
.
An
o
th
e
r
im
p
o
r
tan
t
q
u
an
tity
is
th
e
p
r
o
b
ab
ilit
y
.
T
h
is
v
alu
e
r
ep
r
esen
ts
th
e
p
r
o
b
a
b
ilit
y
o
f
m
o
v
i
n
g
f
r
o
m
a
tr
a
n
s
ien
t state
to
an
ab
s
o
r
p
tio
n
s
tate
.
I
f
t
h
e
m
atr
ix
c
o
n
tain
s
ea
ch
,
th
en
=
[
3
1
]
–
[
3
3
]
.
3.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
T
ab
le
1
s
h
o
ws
th
e
v
alu
es
f
o
r
th
e
tr
an
s
itio
n
p
r
o
b
ab
ilit
y
m
atr
ix
wh
ich
ar
e
in
ter
p
r
et
ed
as
th
e
p
r
o
b
a
b
ilit
ies
o
f
a
s
tu
d
en
t’
s
p
r
o
g
r
ess
io
n
f
r
o
m
o
n
e
s
em
ester
to
th
e
n
ex
t
s
em
ester
.
As
s
h
o
w
n
i
n
th
e
tab
le,
th
e
t
o
p
th
r
ee
h
ig
h
est
p
r
o
b
ab
ilit
y
o
f
p
r
o
g
r
ess
io
n
ar
e
f
r
o
m
s
ec
o
n
d
-
y
e
ar
f
ir
s
t
s
em
ester
to
s
ec
o
n
d
-
y
e
ar
,
s
ec
o
n
d
s
em
ester
with
9
5
.
9
1
%,
f
r
o
m
th
ir
d
-
y
ea
r
f
ir
s
t
s
em
ester
to
th
ir
d
-
y
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r
s
ec
o
n
d
s
em
ester
with
9
5
.
1
1
%,
an
d
f
r
o
m
f
i
r
s
t
-
y
ea
r
s
ec
o
n
d
s
em
ester
to
s
ec
o
n
d
-
y
e
ar
f
ir
s
t
s
em
ester
with
9
3
.
3
7
%.
On
t
h
e
o
th
er
h
an
d
,
s
tu
d
en
ts
h
av
e
a
lo
wer
ch
an
c
e
o
f
m
o
v
i
n
g
s
u
cc
ess
f
u
lly
to
th
e
n
ex
t
s
em
ester
wh
en
th
ey
ar
e
in
th
e
f
ir
s
t
-
y
ea
r
,
f
ir
s
t
s
em
ester
with
7
8
.
3
0
%
an
d
wh
en
th
ey
ar
e
in
th
e
t
h
ir
d
-
y
ea
r
,
s
ec
o
n
d
s
em
ester
with
7
3
.
1
4
%.
Ad
d
itio
n
ally
,
it
ca
n
b
e
o
b
s
er
v
ed
th
at
th
e
s
em
ester
th
at
a
s
tu
d
en
t
h
as
th
e
h
ig
h
est
ch
an
ce
o
f
s
h
if
tin
g
to
an
o
th
er
d
e
g
r
ee
p
r
o
g
r
am
is
wh
en
th
e
s
tu
d
en
t
is
in
its
f
ir
s
t
-
y
ea
r
,
f
ir
s
t
s
em
ester
w
ith
1
6
.
0
4
%
ch
a
n
ce
o
f
s
h
if
tin
g
.
I
t
is
also
n
o
ted
th
at
d
u
r
in
g
th
e
f
ir
s
t
-
y
ea
r
,
f
ir
s
t
s
em
ester
,
th
e
s
tu
d
en
t
h
as
5
.
1
9
%
ch
an
ce
o
f
d
r
o
p
p
in
g
f
r
o
m
t
h
e
p
r
o
g
r
am
wh
ich
is
also
th
e
h
ig
h
est
am
o
n
g
t
h
e
s
em
ester
s
.
T
h
is
r
esu
lt
is
alig
n
ed
with
th
e
s
tu
d
y
o
f
G
o
n
zá
lez
-
C
am
p
o
s
et
a
l
.
[
2
5
]
.
Mo
r
e
o
v
er
,
s
tu
d
en
ts
wh
o
p
r
o
g
r
ess
ed
s
u
cc
ess
f
u
lly
f
r
o
m
th
e
f
ir
s
t
s
em
ester
u
p
to
th
e
last
s
em
ester
h
av
e
8
8
.
5
5
%
ch
an
ce
o
f
g
r
ad
u
atin
g
f
r
o
m
th
e
B
S
Ma
th
p
r
o
g
r
am
.
L
astl
y
,
it
was
d
u
r
in
g
th
e
t
r
an
s
itio
n
f
r
o
m
th
e
s
ec
o
n
d
-
y
ea
r
,
s
ec
o
n
d
s
em
ester
to
th
ir
d
-
y
ea
r
,
f
ir
s
t
s
em
ester
an
d
th
ir
d
-
y
ea
r
,
s
ec
o
n
d
s
em
ester
to
f
o
u
r
th
-
y
ea
r
,
f
ir
s
t
s
em
ester
wh
er
e
th
er
e
ar
e
s
tu
d
en
ts
wh
o
h
a
v
e
r
etu
r
n
ed
to
p
r
ev
io
u
s
y
ea
r
le
v
el,
p
a
r
ticu
lar
ly
o
n
s
ec
o
n
d
-
y
ea
r
,
f
ir
s
t
s
em
ester
an
d
th
ir
d
-
y
ea
r
,
f
ir
s
t
s
em
ester
,
r
esp
ec
tiv
ely
.
T
h
ese
ar
e
th
e
o
b
s
er
v
ed
b
o
ttlen
e
ck
s
wh
er
e
B
S
Ma
th
s
tu
d
en
ts
s
tar
t
to
b
e
class
if
ied
as
r
eg
u
lar
a
n
d
i
r
r
eg
u
la
r
s
tu
d
en
ts
.
On
e
o
f
t
h
e
p
o
s
s
ib
le
f
ac
to
r
s
th
at
ca
n
b
e
attr
ib
u
te
d
to
t
h
is
ev
en
t
is
th
e
lev
el
o
f
th
e
d
if
f
icu
lty
o
f
th
e
co
r
e
c
o
u
r
s
es
d
u
r
in
g
th
ese
s
em
ester
s
s
u
ch
as
th
e
Ad
v
a
n
ce
d
C
alcu
lu
s
I
,
Dif
f
er
e
n
tial
E
q
u
atio
n
s
,
an
d
Ab
s
tr
ac
t
Alg
e
b
r
a
I
f
o
r
s
ec
o
n
d
-
y
ea
r
an
d
Mo
d
er
n
Geo
m
etr
y
,
Nu
m
b
er
T
h
e
o
r
y
,
Ma
th
/
Stat
/C
I
T
E
lectiv
es
an
d
C
o
g
n
ates
f
o
r
th
ir
d
-
y
ea
r
,
m
o
s
t
o
f
wh
ich
a
r
e
p
r
e
-
r
eq
u
is
ites
f
o
r
th
e
n
ex
t
co
r
e
co
u
r
s
es.
Failu
r
e
to
p
ass
th
ese
co
u
r
s
es will m
ea
n
th
at
th
e
s
tu
d
en
ts
will n
o
t p
r
o
g
r
ess
to
th
e
n
ex
t y
ea
r
’
s
lev
el.
T
h
e
en
tr
ies
in
th
e
f
u
n
d
am
en
t
al
m
atr
ix
N
ar
e
s
h
o
wn
in
T
a
b
le
2
.
T
h
is
m
atr
ix
co
n
tain
s
th
e
ex
p
ec
ted
n
u
m
b
er
o
f
tim
es
o
r
s
tep
s
th
at
th
e
p
r
o
ce
s
s
is
in
th
e
n
ex
t
tr
a
n
s
ien
t
s
tate
if
i
t
s
tar
ted
in
th
e
tr
an
s
ien
t
s
tate
.
Fo
r
in
s
tan
ce
,
th
e
en
tr
y
in
s
ev
en
th
r
o
w,
eig
h
th
c
o
lu
m
n
m
ea
n
s
th
at
if
th
e
s
tu
d
en
t
is
cu
r
r
en
tly
a
f
o
u
r
th
y
ea
r
en
r
o
lled
in
f
ir
s
t
s
em
ester
,
th
en
th
e
ex
p
ec
te
d
n
u
m
b
er
o
f
s
tep
s
in
th
e
n
ex
t
tr
a
n
s
ien
t
s
tate
wh
ich
is
f
o
u
r
th
-
y
ea
r
,
s
ec
o
n
d
s
em
ester
is
ap
p
r
o
x
i
m
ately
1
(
1
.
0
2
3
4
)
b
ef
o
r
e
b
ein
g
ab
s
o
r
b
ed
.
T
ab
le
1
.
State
tr
an
s
itio
n
p
r
o
b
a
b
ili
ty
m
atr
ix
P
S
t
a
t
e
1
Y
S
1
1
Y
S
2
2
Y
S
1
2
Y
S
2
3
Y
S
1
3
Y
S
2
4
Y
S
1
4
Y
S
2
G
S
D
1
Y
S
1
0
.
0
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4
7
0
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7
8
3
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1
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0
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0
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0
5
1
9
1
Y
S
2
0
.
0
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0
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0
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9
3
3
7
0
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0
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3
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2
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.
0
0
0
0
0
.
0
0
0
0
0
.
0
0
0
0
1
.
0
0
0
0
N
o
t
e
:
1
Y
S
1
=
f
i
r
st
y
e
a
r
,
f
i
r
s
t
se
mes
t
e
r
;
1
Y
S
2
=
f
i
r
s
t
y
e
a
r
,
sec
o
n
d
sem
e
st
e
r
;
2
Y
S
1
=
se
c
o
n
d
y
e
a
r
,
f
i
r
st
s
e
mes
t
e
r
;
2
Y
S
2
=
sec
o
n
d
y
e
a
r
,
sec
o
n
d
sem
e
st
e
r
;
3
Y
S
1
=
t
h
i
r
d
y
e
a
r
,
f
i
r
st
seme
st
e
r
;
3
Y
S
2
=
t
h
i
r
d
y
e
a
r
,
se
c
o
n
d
seme
s
t
e
r
;
4
Y
S
1
=
f
o
u
r
t
h
y
e
a
r
,
f
i
r
s
t
sem
e
st
e
r
;
4
Y
S
2
=
f
o
u
r
t
h
y
e
a
r
,
se
c
o
n
d
s
e
mes
t
e
r
;
G
=
g
r
a
d
u
a
t
e
d
;
S
=
sh
i
f
t
e
r
;
a
n
d
D
=
d
r
o
p
p
e
d
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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N
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2
2
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2
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8
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t
J
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14
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No
.
6
,
Dec
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b
er
20
25
:
4
4
7
8
-
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4482
T
ab
le
2
.
T
h
e
f
u
n
d
am
en
tal
m
atr
ix
N
f
o
r
P
S
t
a
t
e
1
Y
S
1
1
Y
S
2
2
Y
S
1
2
Y
S
2
3
Y
S
1
3
Y
S
2
4
Y
S
1
4
Y
S
2
1
Y
S
1
1
.
0
0
4
7
0
.
7
8
6
7
0
.
8
1
0
4
0
.
7
7
7
3
0
.
8
7
2
0
0
.
8
2
9
4
0
.
6
6
8
2
0
.
6
2
0
9
1
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S
2
0
.
0
0
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0
1
.
0
0
0
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.
0
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0
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9
8
8
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1
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0
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4
2
0
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8
4
9
4
0
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8
9
2
2
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0
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1
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7
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9
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0
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4
5
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0
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0
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1
0
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0
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0
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7
0
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3
0
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8
7
6
1
3
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S
1
0
.
0
0
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0
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0
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0
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3
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9
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8
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0
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5
9
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4
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S
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0
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7
4
T
ab
le
3
s
h
o
ws
th
e
v
alu
es
o
f
p
r
o
b
a
b
ilit
ies
o
f
g
r
ad
u
atin
g
,
s
h
if
tin
g
,
o
r
d
r
o
p
p
in
g
o
u
t
f
r
o
m
a
g
iv
en
s
em
ester
.
T
h
e
p
r
o
b
ab
ilit
ies
o
f
g
r
ad
u
atio
n
f
o
r
f
ir
s
t
-
y
ea
r
,
f
ir
s
t
s
em
ester
,
u
p
to
f
o
u
r
th
-
y
ea
r
,
s
ec
o
n
d
s
em
ester
ar
e
5
6
.
8
7
%,
7
2
.
2
9
%,
7
7
.
4
2
%,
8
0
.
2
5
%,
8
4
.
5
1
%,
8
7
.
8
9
%,
9
3
.
7
5
%,
an
d
9
6
.
4
8
%,
r
esp
ec
tiv
ely
.
I
t
s
h
o
ws
th
at
th
e
p
r
o
b
a
b
ilit
y
o
f
p
r
o
g
r
ess
in
g
to
g
r
ad
u
atio
n
in
c
r
ea
s
es
as
a
s
tu
d
e
n
t
p
r
o
g
r
ess
f
r
o
m
o
n
e
s
em
ester
to
an
o
th
er
.
On
t
h
e
o
th
er
h
a
n
d
,
t
h
e
p
r
o
b
ab
ilit
y
v
a
lu
es
o
f
s
h
if
tin
g
o
r
d
r
o
p
p
in
g
o
u
t
o
f
th
e
p
r
o
g
r
am
d
ec
r
ea
s
es
a
s
a
s
tu
d
en
t
m
o
v
es
f
r
o
m
o
n
e
s
em
ester
to
an
o
th
er
s
em
ester
.
I
t
ca
n
also
b
e
n
o
ted
th
at
s
h
if
tin
g
ten
d
s
to
b
e
r
am
p
a
n
t
in
th
e
f
ir
s
t
th
r
ee
s
em
ester
s
,
wh
ile
d
r
o
p
-
o
u
ts
ar
e
ess
en
tially
ex
is
ten
t
ev
en
u
n
til
th
e
f
ir
s
t
s
em
ester
o
f
th
e
f
o
u
r
th
y
ea
r
.
A
s
tu
d
y
b
y
Hala
s
z
an
d
B
lo
o
m
[
3
4
]
i
n
d
ica
tes
th
at
a
s
ig
n
if
ica
n
t
n
u
m
b
er
o
f
co
lle
g
e
s
tu
d
en
ts
c
h
an
g
e
d
t
h
eir
d
eg
r
ee
p
r
o
g
r
am
d
u
r
in
g
th
e
f
ir
s
t
two
s
em
ester
s
o
r
d
u
r
in
g
th
ei
r
f
ir
s
t
y
ea
r
in
c
o
lleg
e.
T
h
e
s
h
if
tin
g
o
f
s
tu
d
e
n
ts
d
u
r
in
g
th
eir
f
ir
s
t
y
ea
r
in
co
lleg
e
m
i
g
h
t
b
e
in
f
l
u
en
ce
d
b
y
v
ar
io
u
s
f
ac
to
r
s
s
u
ch
as
p
r
o
x
im
ity
a
n
d
m
o
b
ilit
y
[
6
]
,
p
r
e
v
io
u
s
an
d
cu
r
r
en
t
ac
ad
em
ic
p
er
f
o
r
m
an
c
e
[
7
]
–
[
9
]
,
p
ar
en
tal
d
esire
s
,
lack
o
f
in
ter
est
in
th
e
d
eg
r
ee
p
r
o
g
r
a
m
,
p
er
ce
iv
ed
av
ailab
ilit
y
o
f
jo
b
o
p
p
o
r
t
u
n
it
ies
[
3
5
]
,
[
3
6
]
o
r
ev
en
th
e
C
OVI
D
-
1
9
p
a
n
d
em
ic
[
3
7
]
w
h
ich
ca
n
b
e
ex
p
lo
r
ed
f
u
r
th
er
f
o
r
t
h
e
b
etter
m
en
t
o
f
t
ea
ch
in
g
-
lear
n
in
g
s
er
v
ice
d
eliv
er
y
o
f
PS
U
-
Ur
d
a
n
eta
C
ity
C
am
p
u
s
p
ar
ticu
lar
ly
in
th
e
B
S M
ath
p
r
o
g
r
am
an
d
i
m
p
r
o
v
e
in
s
titu
tio
n
al
p
e
r
f
o
r
m
an
ce
.
T
h
e
ex
p
ec
ted
n
u
m
b
e
r
o
f
s
tep
s
to
r
ea
c
h
an
ab
s
o
r
p
tio
n
s
tate
(
g
r
ad
u
ated
,
s
h
if
ted
,
o
r
d
r
o
p
p
e
d
)
s
tar
tin
g
at
a
tr
an
s
ien
t
s
tate
ca
n
b
e
g
le
an
ed
in
T
ab
le
4
.
T
h
e
r
esu
lts
wer
e
o
b
tain
ed
af
ter
m
u
ltip
ly
i
n
g
th
e
f
u
n
d
a
m
en
tal
m
atr
ix
N
with
a
co
lu
m
n
v
ec
t
o
r
wh
o
s
e
en
t
r
ies
ar
e
all
1
.
T
o
r
ea
ch
an
ab
s
o
r
p
tio
n
s
tate,
a
f
ir
s
t
-
y
ea
r
B
S
Ma
th
s
tu
d
en
t
tak
es
ap
p
r
o
x
im
ately
s
ix
to
s
ev
en
s
tep
s
,
in
d
icatin
g
th
at
af
ter
th
r
ee
s
em
ester
s
,
a
f
r
esh
m
an
B
S
Ma
th
s
tu
d
en
t
is
ex
p
ec
ted
to
h
av
e
eit
h
er
d
r
o
p
p
ed
o
r
s
h
if
ted
o
u
t o
f
t
h
e
p
r
o
g
r
am
.
Me
an
wh
ile,
a
s
ec
o
n
d
-
y
ea
r
s
tu
d
en
t
is
ex
p
ec
ted
to
b
e
‘
ab
s
o
r
b
ed
’
af
ter
s
ix
s
em
ester
s
,
a
s
ig
n
th
at
m
o
s
t
o
f
th
e
s
ec
o
n
d
-
y
ea
r
s
tu
d
en
ts
wo
u
ld
h
av
e
g
r
ad
u
ated
,
s
h
if
ted
,
o
r
d
r
o
p
p
e
d
o
u
t
o
f
th
e
p
r
o
g
r
am
.
A
f
o
u
r
th
-
y
ea
r
s
tu
d
en
t
e
n
r
o
lled
in
th
e
s
ec
o
n
d
s
em
ester
tak
es a
p
p
r
o
x
im
ately
o
n
e
s
tep
to
b
e
ab
s
o
r
b
ed
,
m
o
s
t lik
ely
g
r
a
d
u
atin
g
f
r
o
m
t
h
e
p
r
o
g
r
am
af
te
r
th
e
s
aid
s
em
ester
.
Fu
r
th
er
m
o
r
e
,
it
ca
n
b
e
o
b
s
er
v
ed
th
at
th
e
e
x
p
ec
ted
tim
es
o
f
ab
s
o
r
p
ti
o
n
f
o
r
f
ir
s
t
y
ea
r
s
t
u
d
en
ts
ar
e
b
elo
w
th
e
m
in
im
u
m
n
u
m
b
e
r
o
f
s
em
ester
s
r
eq
u
ir
ed
to
f
in
is
h
th
e
p
r
o
g
r
am
(
8
s
tep
s
)
.
T
h
is
is
an
in
d
icatio
n
th
at
f
r
esh
m
en
ar
e
m
o
s
t
s
u
s
ce
p
tib
le
to
s
h
if
tin
g
an
d
d
r
o
p
p
in
g
o
u
t
o
f
th
e
p
r
o
g
r
am
.
On
th
e
o
th
er
h
an
d
,
f
r
o
m
t
h
e
Seco
n
d
-
y
ea
r
o
n
war
d
s
,
th
e
ex
p
ec
ted
n
u
m
b
er
o
f
s
tep
s
b
ef
o
r
e
ab
s
o
r
p
tio
n
is
ar
o
u
n
d
th
e
n
u
m
b
er
o
f
r
eq
u
ir
ed
s
em
ester
s
to
f
in
is
h
th
e
p
r
o
g
r
am
(
6
d
o
wn
to
1
)
,
h
en
ce
,
th
e
c
h
an
ce
o
f
a
s
tu
d
en
t
g
r
a
d
u
atin
g
in
cr
ea
s
es
as
h
i
s
/h
er
s
tay
in
th
e
p
r
o
g
r
am
also
i
n
cr
e
ases
.
T
h
e
f
o
r
ec
ast
f
u
tu
r
e
r
ates
o
f
B
S
Ma
th
s
tu
d
en
ts
f
o
r
th
e
n
ex
t
1
2
s
em
ester
s
ar
e
ex
h
ib
ited
in
T
ab
le
5
.
Af
ter
eig
h
t
s
em
ester
s
eq
u
iv
ale
n
t
to
f
o
u
r
y
ea
r
s
,
th
e
f
o
r
ec
ast
g
r
ad
u
atio
n
r
ate
is
3
5
.
2
2
%
wh
ic
h
im
p
lies
th
at
th
er
e
ar
e
s
tu
d
en
ts
wh
o
r
etain
e
d
th
eir
cu
r
r
e
n
t
y
ea
r
lev
el
(
2
3
.
5
0
%),
an
d
s
tu
d
en
ts
wh
o
s
h
if
te
d
to
an
o
th
e
r
d
eg
r
ee
p
r
o
g
r
a
m
(
2
2
.
6
8
%)
o
r
d
r
o
p
p
ed
f
r
o
m
th
e
B
S
Ma
th
p
r
o
g
r
am
(
1
8
.
6
0
%).
T
h
is
m
ea
n
s
th
at
a
s
u
b
s
tan
tial
p
o
r
tio
n
o
f
s
tu
d
en
ts
will
n
o
t
g
r
ad
u
ate
o
n
tim
e
o
r
f
in
is
h
t
h
e
B
S
Ma
th
p
r
o
g
r
a
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7
On
e
o
f
th
e
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ai
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r
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lts
o
f
th
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s
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y
is
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at
at
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%
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f
th
e
B
S
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th
s
tu
d
en
ts
is
ex
p
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in
is
h
ed
th
e
p
r
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am
a
f
ter
1
2
s
em
ester
s
.
T
h
is
is
with
in
th
e
ex
p
ec
tatio
n
s
in
th
e
s
tu
d
y
o
f
Aik
en
et
a
l
.
[
3
8
]
s
tatin
g
th
at
s
tu
d
en
ts
ten
d
t
o
f
i
n
is
h
an
u
n
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er
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ate
p
r
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in
8
to
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2
s
em
ester
s
.
T
h
ey
f
u
r
th
er
attr
ib
u
ted
th
is
to
th
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s
tu
d
en
ts
’
g
r
ad
e
p
o
in
t a
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ag
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(
GPA)
wh
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we
c
o
u
ld
i
n
v
esti
g
ate
f
u
r
th
e
r
in
th
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ca
s
e
o
f
PS
U
–
Ur
d
an
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Fig
u
r
e
1
s
h
o
ws th
e
s
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en
t
p
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a
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b
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A
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,
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→
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→
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→
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1
→
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→
A
m
o
v
em
en
t
f
r
o
m
th
e
cu
r
r
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n
t
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an
s
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t
s
tate
o
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em
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its
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r
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d
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ailed
to
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co
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r
s
es n
ee
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ed
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e
to
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e
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t sem
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.
Fig
u
r
e
1
.
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r
k
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v
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atr
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p
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4.
CO
NCLU
SI
O
N
T
h
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ap
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f
DT
MC
m
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m
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h
e
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ter
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ath
p
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t
h
e
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s
h
o
w
th
at
th
e
p
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b
ab
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y
o
f
g
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u
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g
in
cr
ea
s
es a
s
th
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tu
d
en
t
s
u
cc
ess
f
u
lly
co
n
q
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er
s
th
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s
u
cc
ee
d
in
g
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em
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ile
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e
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ig
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t
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Af
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zin
g
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e
d
ata
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ati
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
8
2
2
I
n
t
J
E
v
al
&
R
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d
u
c
,
Vo
l
.
14
,
No
.
6
,
Dec
em
b
er
20
25
:
4
4
7
8
-
4
4
8
6
4484
th
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it
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ir
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t
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th
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m
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atin
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o
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ar
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Pre
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.
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x
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,
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.
R
az
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ale
G.
R
esu
ltay
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an
d
th
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C
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ter
Hea
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f
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Scien
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,
T
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n
g
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,
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n
v
ir
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n
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tal
R
esear
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Dr
.
R
h
o
wel
M.
Dello
s
a,
f
o
r
ex
te
n
d
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t
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ig
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ts
lead
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to
th
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c
o
m
p
letio
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o
f
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ap
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.
F
UNDING
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NF
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h
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s
tu
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y
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p
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ty
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B
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4
3
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s
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f
2
0
2
3
.
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R
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UT
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u
r
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s
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C
o
n
t
r
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to
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o
les
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y
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ed
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ize
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iv
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ip
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Aut
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k
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n
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.
Ma
n
g
s
at
✓
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Dan
iel
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ez
alel
A.
Gar
cia
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d
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ac
o
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e
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Ma
r
icel
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o
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p
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ar
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s
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x
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last
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d
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ata
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led
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tr
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n
f
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en
tiality
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d
h
i
g
h
eth
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co
n
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id
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s
f
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th
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Data
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f
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0
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R
.
A.
1
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f
th
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ep
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f
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ilip
p
in
es.
DATA AV
AI
L
AB
I
L
I
T
Y
T
h
e
f
in
al
d
ata
s
et
th
at
s
u
p
p
o
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ts
th
e
r
esu
lts
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f
in
d
i
n
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s
ar
e
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o
m
th
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co
r
r
esp
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n
d
in
g
au
th
o
r
[
DB
A
G
]
,
u
p
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n
r
ea
s
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ab
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e
q
u
est.
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r
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e
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ata,
w
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co
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tain
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n
f
o
r
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th
at
co
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ld
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t p
u
b
licly
a
v
ailab
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d
u
e
to
R
.
A.
1
0
1
7
3
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t
J
E
v
al
&
R
es E
d
u
c
I
SS
N:
2252
-
8
8
2
2
Dete
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min
in
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s
tu
d
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ess
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r
a
tes u
s
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is
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time
Ma
r
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(
Ma
r
k
Jo
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n
T.
Ma
n
g
s
a
t
)
4485
RE
F
E
R
E
NC
E
S
[
1
]
L.
Z
h
a
n
g
,
X
.
L
i
u
,
a
n
d
Y
.
H
u
,
“
D
e
g
r
e
e
s
o
f
r
e
t
u
r
n
:
e
st
i
ma
t
i
n
g
i
n
t
e
r
n
a
l
r
a
t
e
s
o
f
r
e
t
u
r
n
f
o
r
c
o
l
l
e
g
e
ma
j
o
r
s
u
s
i
n
g
q
u
a
n
t
i
l
e
r
e
g
r
e
s
s
i
o
n
,
”
Am
e
ri
c
a
n
E
d
u
c
a
t
i
o
n
a
l
Re
s
e
a
r
c
h
J
o
u
r
n
a
l
,
v
o
l
.
6
1
,
n
o
.
3
,
p
p
.
5
7
7
–
6
0
9
,
Ju
n
.
2
0
2
4
,
d
o
i
:
1
0
.
3
1
0
2
/
0
0
0
2
8
3
1
2
2
4
1
2
3
1
5
1
2
.
[
2
]
A
.
D
e
l
b
a
n
c
o
,
“
W
h
a
t
i
s
c
o
l
l
e
g
e
f
o
r
?
”
i
n
C
o
l
l
e
g
e
:
W
h
a
t
I
t
W
a
s,
I
s
,
a
n
d
S
h
o
u
l
d
Be
,
A
.
D
e
l
b
a
n
c
o
,
E
d
s.
,
P
r
i
n
c
e
t
o
n
,
N
J:
P
r
i
n
c
e
t
o
n
U
n
i
v
e
r
si
t
y
P
r
e
ss,
2
0
1
8
,
p
p
.
9
–
35
,
d
o
i
:
1
0
.
2
3
0
7
/
j
.
c
t
t
1
2
8
7
k
h
f
.
6
.
[
3
]
K.
S
.
C
h
o
i
a
n
d
C
.
P
a
r
k
,
“
T
h
e
v
a
l
u
e
o
f
c
o
l
l
e
g
e
e
d
u
c
a
t
i
o
n
a
n
d
g
e
n
e
r
a
l
e
d
u
c
a
t
i
o
n
-
e
c
o
n
o
m
i
c
p
e
r
s
p
e
c
t
i
v
e
s,”
T
h
e
K
o
rea
n
Ass
o
c
i
a
t
i
o
n
o
f
G
e
n
e
r
a
l
Ed
u
c
a
t
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o
n
,
v
o
l
.
1
5
,
n
o
.
4
,
p
p
.
1
1
–
2
1
,
A
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g
.
2
0
2
1
,
d
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:
1
0
.
4
6
3
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2
/
k
j
g
e
.
2
0
2
1
.
1
5
.
4
.
1
1
.
[
4
]
S
.
H
e
a
r
n
a
n
d
M
.
Lo
n
g
,
“
A
d
u
l
t
l
e
a
r
n
e
r
s’
c
o
l
l
e
g
e
a
c
c
e
ss
&
c
h
o
i
c
e
p
r
o
c
e
ss
:
a
l
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t
e
r
a
t
u
r
e
r
e
v
i
e
w
,
”
S
t
u
d
e
n
t
A
f
f
a
i
rs
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B
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rtme
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tate
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y
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s.
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2
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a
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Un
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h
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g
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o
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re
se
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c
a
n
b
e
c
o
n
tac
ted
a
t
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m
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il
:
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m
a
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g
sa
t@p
su
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e
d
u
.
p
h
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n
iel
Be
z
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lel
A.
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a
r
c
ia
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n
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ss
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t
p
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c
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ra
l
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c
ien
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De
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m
p
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ti
n
g
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t
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a
n
g
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sin
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n
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tate
Un
i
v
e
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,
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a
n
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ta
Cit
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m
p
u
s
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n
d
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d
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c
to
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p
h
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h
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d
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ta
sc
ien
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d
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n
t
a
t
th
e
S
c
h
o
o
l
o
f
S
tatisti
c
s,
Un
iv
e
rsity
o
f
th
e
P
h
il
i
p
p
in
e
s
,
D
il
ima
n
.
He
fin
ish
e
d
h
is
b
a
c
h
e
l
o
r
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f
sc
ien
c
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in
m
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ti
c
s
m
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jo
r
in
sta
ti
stics
a
t
th
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P
a
n
g
a
sin
a
n
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tate
Un
iv
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rsity
-
Ur
d
a
n
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ta Ci
ty
Ca
m
p
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s (2
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)
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m
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th
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Un
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rsity
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f
t
h
e
P
h
il
i
p
p
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n
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s
-
Ba
g
u
io
(2
0
2
2
).
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re
se
a
rc
h
in
tere
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a
re
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c
u
se
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Ba
y
e
si
a
n
sta
ti
stics
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m
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m
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ti
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s
ti
c
s,
a
p
p
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d
a
ta
m
in
in
g
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d
m
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th
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ti
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s
e
d
u
c
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ti
o
n
.
He
c
a
n
b
e
c
o
n
tac
ted
a
t
e
m
a
il
:
d
a
g
a
rc
ia@
u
p
.
e
d
u
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p
h
;
d
a
g
a
rc
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a
lu
m
.
u
p
.
e
d
u
.
p
h
;
d
g
a
rc
ia@
p
su
.
e
d
u
.
p
h
.
And
h
e
e
M.
J
a
c
o
b
e
is
a
fo
rm
e
r
fa
c
u
lt
y
m
e
m
b
e
r
o
f
th
e
M
a
th
e
m
a
ti
c
s
a
n
d
Na
tu
ra
l
S
c
ien
c
e
s
De
p
a
rtme
n
t,
Co
ll
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g
e
o
f
Co
m
p
u
ti
n
g
,
P
a
n
g
a
si
n
a
n
S
tat
e
Un
iv
e
rsity
-
Ur
d
a
n
e
ta
Cit
y
Ca
m
p
u
s
a
n
d
c
u
rre
n
tl
y
a
S
tatisti
c
ian
II
u
n
d
e
r
t
h
e
Tax
Re
se
a
rc
h
G
ro
u
p
o
f
th
e
Na
ti
o
n
a
l
Ta
x
Re
se
a
rc
h
Ce
n
ter.
He
fin
ish
e
d
h
i
s
b
a
c
h
e
lo
r
o
f
sc
ien
c
e
i
n
m
a
th
e
m
a
ti
c
s
d
e
g
re
e
in
P
a
n
g
a
si
n
a
n
S
tate
Un
iv
e
rsit
y
-
Urd
a
n
e
ta
Cit
y
Ca
m
p
u
s
(2
0
1
4
)
,
sp
e
c
ializin
g
in
sta
ti
stics
.
He
is
tak
in
g
u
p
m
a
ste
r
o
f
sc
ien
c
e
in
m
a
th
e
m
a
ti
c
s
a
t
th
e
Un
iv
e
rsity
o
f
t
h
e
P
h
i
li
p
p
in
e
s
,
Ba
g
u
io
.
His
re
se
a
rc
h
in
tere
sts
in
c
lu
d
e
sto
c
h
a
stic
p
r
o
c
e
ss
e
s,
d
iffere
n
ti
a
l
e
q
u
a
ti
o
n
s
,
sta
ti
stica
l
m
o
d
e
li
n
g
,
a
n
d
m
a
th
e
m
a
ti
c
a
l
m
o
d
e
li
n
g
a
n
d
sim
u
l
a
ti
o
n
s.
He
c
a
n
b
e
c
o
n
tac
ted
a
t
e
m
a
il
:
a
m
jac
o
b
e
1
@u
p
.
e
d
u
.
p
h
,
a
n
d
h
e
e
.
jac
o
b
e
@
n
trc.g
o
v
.
p
h
.
Ma
r
ice
l
A.
Bo
n
g
o
l
a
n
is
th
e
c
a
m
p
u
s
re
g
istrar
o
f
P
a
n
g
a
si
n
a
n
S
tate
Un
iv
e
rsity
,
Urd
a
n
e
ta
Cit
y
Ca
m
p
u
s.
S
h
e
fin
is
h
e
d
h
e
r
b
a
c
h
e
l
o
r
o
f
sc
ien
c
e
in
i
n
fo
rm
a
ti
o
n
,
c
o
m
m
u
n
ica
ti
o
n
s
,
a
n
d
tec
h
n
o
lo
g
y
a
t
P
a
n
g
a
sin
a
n
S
tate
U
n
iv
e
rsit
y
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Urd
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n
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ta
Ci
t
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Ca
m
p
u
s
a
n
d
m
a
ste
r
in
in
fo
rm
a
ti
o
n
tec
h
n
o
lo
g
y
a
t
th
e
Un
iv
e
rsid
a
d
d
e
Da
g
u
p
a
n
(f
o
rm
e
rly
C
o
leg
i
o
d
e
Da
g
u
p
a
n
,
Co
m
p
u
tr
o
n
i
x
C
o
ll
e
g
e
).
S
h
e
c
a
n
b
e
c
o
n
tac
ted
a
t
e
m
a
il
:
re
g
istraru
r
d
a
n
e
ta@
p
su
.
e
d
u
.
p
h
.
Evaluation Warning : The document was created with Spire.PDF for Python.