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u _ 0 = \frac { U _ 0 } { r } \sqrt { r ^ 2 - x ^ 2 } ,
e9a5b62c-bc37-11e1-1586-001143e3f55c__mathematical-expression-and-equation_0.jpg
v \in \mathcal { N } _ \text { f . p a s t }
e9b2e049-ac0a-11e1-1154-001143e3f55c__mathematical-expression-and-equation_15.jpg
e _ \alpha e _ \beta \neq e _ \beta e _ \alpha .
e9fdcf5a-570a-11e1-1726-001143e3f55c__mathematical-expression-and-equation_1.jpg
\{ a _ i \} \subseteq \{ x _ i \} ; \{ b _ i \} \subseteq \{ y _ i \}
e9fdcff3-570a-11e1-1726-001143e3f55c__mathematical-expression-and-equation_9.jpg
\beta ( \alpha ^ 2 + \gamma ^ 2 ) = \frac { b } { 4 } ,
ea35e615-40e3-11e1-1331-001143e3f55c__mathematical-expression-and-equation_4.jpg
\{ v _ B \} = R _ A = q _ A s _ A
ea4661e0-354a-11e3-b79f-5ef3fc9bb22f__mathematical-expression-and-equation_4.jpg
A { \prime } ^ { - 1 } y = 0
ea77df49-ac0a-11e1-1431-001143e3f55c__mathematical-expression-and-equation_7.jpg
B ( r ) = - r + C _ 2 r ^ 2 + C _ 3 r ^ 3 + \dots
eaa9eee4-570a-11e1-1726-001143e3f55c__mathematical-expression-and-equation_3.jpg
q _ { j + r - 1 } q _ { j + r - 2 } \dots q _ j \le t < q _ { j + r }
eaa9ef33-570a-11e1-1726-001143e3f55c__mathematical-expression-and-equation_11.jpg
C _ t = 0 . 1 6 7
eabcc61f-a89c-11e1-1027-001143e3f55c__mathematical-expression-and-equation_1.jpg
v = \sqrt { \frac { 4 } { \pi } ( k _ 1 - k _ 2 ) }
eacd6acb-40e3-11e1-1331-001143e3f55c__mathematical-expression-and-equation_2.jpg
\mathcal { M } ( v ) = m _ { v _ 1 } m _ { v _ 2 } \dots m _ { v _ p }
eb5dd538-570a-11e1-1586-001143e3f55c__mathematical-expression-and-equation_3.jpg
[ \overline { \omega } _ 1 \overline { \omega } _ 2 ] = \{ \rho _ 1 \rho _ 2 a + ( - \rho _ 1 ^ 2 + \rho _ 2 ^ 2 ) b - \rho _ 1 \rho _ 2 c \} [ \omega _ 1 \omega _ 2 ] \neq 0
eb5dd542-570a-11e1-1586-001143e3f55c__mathematical-expression-and-equation_12.jpg
s ^ 2 = \sum _ { k = 1 } ^ { m } D W
eb5dd679-570a-11e1-1586-001143e3f55c__mathematical-expression-and-equation_8.jpg
x y = x _ m y + ( x - x _ m ) y = \Sigma x ( t _ i ) y ( t _ i ) + \Theta \epsilon | y | =
eb5dd6a1-570a-11e1-1586-001143e3f55c__mathematical-expression-and-equation_5.jpg
\overline { c _ 1 c _ 2 } = c _ 1
eb66c4fc-40e3-11e1-1418-001143e3f55c__mathematical-expression-and-equation_1.jpg
c _ L = c _ { L 0 } \times \frac { L } { L _ u } = 5 9 2 0 \times \frac { L } { L _ u }
ebc284b8-d253-44a4-88bc-eb857d90c6e1__mathematical-expression-and-equation_4.jpg
n \equiv c _ 2 e ^ { - \frac { 1 } { 2 } c _ 1 ( a - u ) ^ 2 }
ebcab92f-bc37-11e1-1586-001143e3f55c__mathematical-expression-and-equation_4.jpg
\tilde { k } _ B = \mp \frac { M _ B } { h W _ y } \dots . . ( 3 8 9 )
ebe49640-e953-11e2-9439-005056825209__mathematical-expression-and-equation_10.jpg
F = [ - 1 ]
ec018b63-ac0a-11e1-1027-001143e3f55c__mathematical-expression-and-equation_4.jpg
R _ 2 = R _ 3 = 0
ec018bd4-ac0a-11e1-1027-001143e3f55c__mathematical-expression-and-equation_9.jpg
h _ 7 < \beta < H
ec20d83c-570a-11e1-1586-001143e3f55c__mathematical-expression-and-equation_11.jpg
1 2 \% + 4 \%
ec3537c0-8e76-11df-afc6-0030487be43a__mathematical-expression-and-equation_5.jpg
\int \frac { x d x } { x ^ 2 - a ^ 2 } = \frac { 1 } { 2 } \int \frac { 2 x d x } { x ^ 2 - a ^ 2 } = \frac { 1 } { 2 } l ( x ^ 2 - a ^ 2 ) + C
ec389084-20a0-4b7e-b7e5-6f8662b9b244__mathematical-expression-and-equation_5.jpg
1 - 2 F _ r ^ { - 2 / 3 } ( X _ { \beta \prime } - 1 ) - \sqrt { F _ r ^ { - 4 / 3 } ( X _ { \beta \prime } ^ 2 - 1 ) - 2 F _ r ^ { 2 / 3 } + 2 F _ { r D V } ^ { 2 / 3 } + F _ { r D V } ^ { - 4 / 3 } } = 0
ec8d6cc9-bc37-11e1-1586-001143e3f55c__mathematical-expression-and-equation_8.jpg
a = E \sqrt { \frac { C } { A } }
ec93d2a0-40e3-11e1-1418-001143e3f55c__mathematical-expression-and-equation_3.jpg
\frac { a } { r } > \frac { k } { m } \implies \Delta ( a , r , k , m ) > \Delta ( a + 1 , r , k + 1 , m )
ecce2a03-ac0a-11e1-1431-001143e3f55c__mathematical-expression-and-equation_5.jpg
\chi = \sigma _ 1 \sigma _ 2 \sigma _ 3 + s _ 1 s _ 2 s _ 3
ecce2a30-ac0a-11e1-1431-001143e3f55c__mathematical-expression-and-equation_0.jpg
x _ { i + 1 } = A x _ i + B u _ i
ecce2b19-ac0a-11e1-1431-001143e3f55c__mathematical-expression-and-equation_3.jpg
[ \begin{array} { c } z ^ { - 1 } - z ^ { - 2 } + z ^ { - 3 } ( 1 - z ^ { - 1 } ) ^ { 2 } t \\ 1 + z ^ { - 2 } - z ^ { - 3 } + z ^ { - 2 } ( 1 - z ^ { - 1 } ) ^ { 2 } t \end{array} ]
ecce2bee-ac0a-11e1-1431-001143e3f55c__mathematical-expression-and-equation_15.jpg
A _ j ^ { ( k ) } = a _ { t _ { k - j - 1 } - t _ { k - j } } ( \begin{array} { c } ( k - j ) s + t _ { j } \\ ( k - j - 1 ) s + t _ { j + 1 } \end{array} ) , t _ 0 = t - ( k - 1 )
ece734b3-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_9.jpg
N = H + H _ { 0 }
ece7351c-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_10.jpg
\int _ { x _ { 2 k } ^ { ( ) ) } } ^ { x _ { 2 k + 1 } ^ { ( 1 ) } } P ( y ) d y < - \frac { 1 } { 2 } \gamma _ 1 k ^ { \frac { r + 1 } { 2 } }
ece735ce-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_4.jpg
x _ 1 x _ 2 = \frac { r } { p } = - \frac { q ^ 2 } { p ^ 2 } \frac { \sin ^ 2 \frac { \phi } { 2 } \cos ^ 2 \frac { \phi } { 2 } } { \cos ^ 2 \phi } ,
ed2f00f3-40e3-11e1-1331-001143e3f55c__mathematical-expression-and-equation_5.jpg
v _ r = \frac { 1 } { T } \frac { \partial \phi } { \partial r } = \frac { Q } { 2 \pi T } \frac { \partial } { \partial r } \ln \sqrt { \frac { r ^ 2 } { r ^ 2 + l ^ 2 - 2 r l \cos \theta } } =
ed526bc9-bc37-11e1-1027-001143e3f55c__mathematical-expression-and-equation_6.jpg
[ m ^ 3 . s ^ { - 1 } ]
ed526c1e-bc37-11e1-1027-001143e3f55c__mathematical-expression-and-equation_18.jpg
\partial b = 4
ed9d6188-ac0a-11e1-1586-001143e3f55c__mathematical-expression-and-equation_15.jpg
C = C ( \mathcal { H } , s , D )
ed9d6289-ac0a-11e1-1586-001143e3f55c__mathematical-expression-and-equation_5.jpg
\bigcup H _ i = X
edb4e528-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_2.jpg
T _ 1 ; 0 < \sigma < \sigma _ 0 , s \le \tau \le \epsilon
edb4e602-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_9.jpg
2 [ 1 - \Phi ( \frac { 1 } { \lambda _ \epsilon } ) ] > 1 - \epsilon .
edb4e656-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_3.jpg
\mathbf { F } _ { \Theta R } - \mathbf { F } _ { \Theta \Theta } \mathbf { \Theta } = 0
ee62aca4-ac0a-11e1-1154-001143e3f55c__mathematical-expression-and-equation_6.jpg
G = \{ h \} + \{ \mathfrak { G } \setminus ( h ) \}
ee861730-570a-11e1-1027-001143e3f55c__mathematical-expression-and-equation_2.jpg
d a _ 2 + a _ 2 ( \omega _ { 2 2 } - 2 \omega _ { 1 1 } ) = 0
ee861799-570a-11e1-1027-001143e3f55c__mathematical-expression-and-equation_5.jpg
\int _ 0 ^ { T _ 0 ^ * } ( \frac { \partial f } { \partial \dot { x } } ) ^ * d s < 0 , \text { s i g n } \epsilon . M \prime ( \omega _ 0 ^ * ) < 0
ee861829-570a-11e1-1027-001143e3f55c__mathematical-expression-and-equation_8.jpg
\frac { d x } { d \tau } = - D A _ k ^ * ( t ) x
ee8618d8-570a-11e1-1027-001143e3f55c__mathematical-expression-and-equation_0.jpg
4 + 4 + 1 =
ee9623e1-224b-11ea-bbb4-001b63bd97ba__mathematical-expression-and-equation_43.jpg
\partial f / \partial \delta \prime _ { i n } = ( \sin \delta \prime _ { i n } \sin T \prime _ { i n } \sin \delta \prime _ { k n } + \cos \delta \prime _ { i n } \cos \delta \prime _ { k n } \sin T \prime _ { k n } ) x \prime _ { i k } +
ee9e3615-285b-481a-99ac-83e974c34521__mathematical-expression-and-equation_13.jpg
k _ 6 = 0 , 4 9 5 5 k _ 2 + 0 , 5 8 6 3 k _ z - 0 , 0 8 4 3 I _ { p n } + 0 , 0 0 0 5 W _ { k _ z } - 0 , 1 0 1 4
eee5b4ca-bc37-11e1-1027-001143e3f55c__mathematical-expression-and-equation_4.jpg
0 . 0 , 1 . 0 , 2 . 0 , 3 . 0 , 4 . 0 , 5 . 0 m
ef0324bc-40e3-11e1-1331-001143e3f55c__mathematical-expression-and-equation_2.jpg
f ( x ) = \sum _ { n = - \infty } ^ { \infty } a _ n x ^ n
ef0324e8-40e3-11e1-1331-001143e3f55c__mathematical-expression-and-equation_0.jpg
B = \{ ( ( [ \alpha _ 1 , A _ 1 ] , [ \alpha _ 2 , A _ 2 ] ) , ( [ \alpha _ 3 , A _ 2 ] , [ \alpha _ 4 , A _ 1 ] ) ) | \alpha _ 1 , \alpha _ 2 , \alpha _ 3 , \alpha _ 4 \in V _ \lambda
ef286ef0-ac0a-11e1-1586-001143e3f55c__mathematical-expression-and-equation_7.jpg
\beta _ v < \beta _ v ^ * \text { f o r } v > 0
ef286f20-ac0a-11e1-1586-001143e3f55c__mathematical-expression-and-equation_12.jpg
s \in [ 0 , t ]
ef286f84-ac0a-11e1-1586-001143e3f55c__mathematical-expression-and-equation_15.jpg
\omega _ p = 2 , 6 1 6 1 / s
ef4fc9f8-98e0-4460-9684-bedb6cbf4118__mathematical-expression-and-equation_8.jpg
d = \frac { \frac { 1 } { 2 } \sqrt { 2 } } { \sqrt { 2 } - 1 } = \frac { 1 } { 2 } \sqrt { 2 } ( \sqrt { 2 } + 1 ) = 1 + \frac { 1 } { 2 } \sqrt { 2 } = 1 . 7 0 7 1 1 \dots
ef509743-3a61-488d-8d12-571901ed0946__mathematical-expression-and-equation_0.jpg
H _ { p _ 0 } \cap \{ H _ p \} _ { p \in \Pi _ 0 } = G
ef5610bc-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_0.jpg
| \frac { U ( x ) \phi \prime ( x ) } { U \prime ( x ) } | \le \frac { | U ^ { 2 } ( b _ 3 ) \phi \prime ( b _ 3 ) | } { U ( x ) U \prime ( x ) } + \epsilon
ef5612c9-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_3.jpg
\frac { S a P } { S e \Pi }
efcbbca2-e055-11ea-b978-00155d012102__mathematical-expression-and-equation_2.jpg
a ( r ) = \text { c o n s t a n t } = a
efed1cf3-ac0a-11e1-1121-001143e3f55c__mathematical-expression-and-equation_1.jpg
= ( A _ { \alpha , \beta } - A _ { \gamma , \delta } ) ^ { - 1 } \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { m } ( p ^ { \alpha } ( x _ i , y _ j ) q ^ { \beta - \alpha } ( x _ i , y _ j ) - p ^ { \gamma } ( x _ i , y _ j ) q ^ { \delta - \gamma } ( x _ i , y _ j ) )
efed1db0-ac0a-11e1-1121-001143e3f55c__mathematical-expression-and-equation_7.jpg
| \phi | = A ^ m , m > 0
efed1de9-ac0a-11e1-1121-001143e3f55c__mathematical-expression-and-equation_3.jpg
W ^ * ( s , \zeta ) = \bigcap _ { n = 1 } ^ \infty W ^ * ( n , s , \zeta )
f02b1421-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_3.jpg
\mathbf { w } _ { \mathcal { S } } ( \hat { \xi } _ a , \hat { \xi } _ b , \epsilon ) \equiv \mathbf { u } _ { \mathcal { S } } ( \hat { \eta } ( a ) , \hat { \eta } ( b _ 0 ) ) + \tau \int _ { 0 } ^ { 1 } [ \frac { \mathbf { D u } _ { \mathcal { S } } } { \mathbf { D c } _ { \bar { \nu } } } \frac { \partial \sigma } ...
f02b155d-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_3.jpg
( A ) _ 0 = A _ 0
f02b1581-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_15.jpg
\theta _ n [ \sqrt { m _ 1 m _ 2 } , \sqrt { p _ 1 p _ 2 } ] \le \sqrt { \theta _ n ( m _ 1 , p _ 1 ) \theta _ n ( m _ 2 , p _ 2 ) } ,
f02b15d2-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_6.jpg
+ \frac { i } { e c } \lim . \sum _ { m = 0 } ^ { n - 1 } \sum _ { \mu = 0 } ^ { n - 1 } ( - 1 ) ^ { m + \mu } \psi _ 1 ( m , \mu ) .
f0c405f8-3342-4ac9-b368-b44c8c269544__mathematical-expression-and-equation_6.jpg
f = a _ 1 x _ 1 ^ 2 + a _ 2 x _ 2 ^ 2 + a _ 3 x _ 3 ^ 2 ,
f0da2f00-40e3-11e1-1418-001143e3f55c__mathematical-expression-and-equation_7.jpg
\tilde { G } _ 2 ^ { ( p ) } = \{ C , g _ i ^ { ( \alpha ) } ( i = m + 1 , \dots , r ; \alpha = 1 , 2 , \dots ) \} / C
f0fd3117-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_4.jpg
\omega _ { 0 0 } + \omega _ { 3 3 } = - \omega _ { 1 1 } - \omega _ { 2 2 } = 0
f0fd31d1-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_9.jpg
\frac { t _ 2 + c t _ 1 } { t \prime _ 2 + c \prime t \prime _ 1 } = \frac { t _ 1 + c t _ 2 } { t \prime _ 1 + c \prime t \prime _ 2 } ( = j )
f0fd3212-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_1.jpg
\mathrm { c l } t _ 1 \ge r + 1 ,
f0fd327f-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_10.jpg
u ( x , 0 ) = \phi ( x )
f0fd338c-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_7.jpg
\sin \frac { \alpha } { 2 } \cos ^ 2 \frac { \alpha } { 2 } = \sin 3 0 ^ \circ \cos ^ 2 3 0 ^ \circ
f178b9c2-40e3-11e1-1331-001143e3f55c__mathematical-expression-and-equation_2.jpg
\Phi ( x y ) = \Phi ( x ) + \Phi ( y )
f17fcc65-ac0a-11e1-1431-001143e3f55c__mathematical-expression-and-equation_4.jpg
U _ T = \{ a _ 1 , a _ 2 , a _ 3 , a _ 4 \}
f17fcc98-ac0a-11e1-1431-001143e3f55c__mathematical-expression-and-equation_3.jpg
= \frac { 1 } { g } + ( b - 0 . 0 0 3 6 6 a + c t \prime ) \frac { 1 } { K _ 1 }
f18fcbc0-c2a2-4ee4-9e77-d34a8e950173__mathematical-expression-and-equation_7.jpg
\frac { \Delta ^ 2 u _ { i k } } { h ^ 2 }
f1cd530a-570a-11e1-1027-001143e3f55c__mathematical-expression-and-equation_4.jpg
\frac { \Delta q _ { i k } } { h } = \frac { \Delta p _ { i k } } { l }
f1cd530c-570a-11e1-1027-001143e3f55c__mathematical-expression-and-equation_0.jpg
x ^ 3 \equiv z
f1cd536f-570a-11e1-1027-001143e3f55c__mathematical-expression-and-equation_8.jpg
P ( \alpha _ 1 \frac { d y } { d \tau } - \alpha _ 2 \frac { d x } { d \tau } - \alpha _ 0 \frac { d t } { d \tau } ) = 0
f1cd5385-570a-11e1-1027-001143e3f55c__mathematical-expression-and-equation_1.jpg
\omega y \prime \prime - \omega \prime y \prime + ( \omega \prime \prime + 2 A \omega ) y = 0
f1cd5482-570a-11e1-1027-001143e3f55c__mathematical-expression-and-equation_0.jpg
\tilde { \mathcal { D } } : v ^ * \mapsto \frac { \partial v ^ * } { \partial t } + A v ^ *
f247d5c7-ac0a-11e1-1431-001143e3f55c__mathematical-expression-and-equation_0.jpg
| \bar { \Phi } ( \lambda , \mu _ { i + 1 } , k , C ) - \bar { \Phi } ( \lambda , \mu _ i , k , C ) | =
f247d6c7-ac0a-11e1-1431-001143e3f55c__mathematical-expression-and-equation_4.jpg
m = \frac { S _ 0 } { S }
f257d1b2-bc37-11e1-1360-001143e3f55c__mathematical-expression-and-equation_3.jpg
y = x \cdot \frac { q } { Q }
f26a8c00-dade-11e2-9439-005056825209__mathematical-expression-and-equation_2.jpg
[ A _ 1 + ( 4 + h ) ^ { - 1 } ( a _ 0 ^ 0 + a _ 2 ^ 2 - 2 a _ 1 ^ 1 + \frac { \partial \log \beta } { \partial u } ) A _ 0 , A _ 2 +
f2a16b7a-570a-11e1-1027-001143e3f55c__mathematical-expression-and-equation_4.jpg
x = \sqrt { - 1 }
f2ac0936-40e3-11e1-1726-001143e3f55c__mathematical-expression-and-equation_8.jpg
( a _ k - u ) ( a _ k - K u ) = a _ k ^ 2 - ( u + K u ) a _ k + u . K u ,
f2ac0952-40e3-11e1-1726-001143e3f55c__mathematical-expression-and-equation_10.jpg
\frac { \partial } { \partial x } ( \Delta H _ 2 ) = - \frac { W } { \mu \sqrt { \pi a t } }
f3381f1a-bc37-11e1-1119-001143e3f55c__mathematical-expression-and-equation_3.jpg
= | \int ^ t \{ [ f ( x ( \sigma , \lambda ) , \dot { x } ( \sigma , \lambda ) , \sigma , \lambda ) - \dot { x } ( \sigma , \lambda ) \Phi _ x ( x ( \sigma , \lambda ) , \sigma , \lambda ) ] .
f36e0a8b-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_18.jpg
i = 1 , 2 , \dots , n - 1 ; k = 0 , 1 , \dots , i
f36e0bf2-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_18.jpg
\times \exp \{ - \frac { 1 } { 2 } \parallel \mathbf { P } ^ { \theta } ( \eta ( \theta ) - \eta ) \parallel ^ { 2 } \}
f3dea332-ac0a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_4.jpg
\frac { d \tilde { x } } { d t } + \tilde { \theta } A \tilde { x } = \tilde { \theta } \tilde { f } , \tilde { x } = x _ 0
f3dea403-ac0a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_12.jpg
f ( x ) = [ f ( \frac { x } { 2 } ) ] ^ 2 ,
f3f06fe1-40e3-11e1-1726-001143e3f55c__mathematical-expression-and-equation_1.jpg
L _ 1 = L _ 2
f3f07089-40e3-11e1-1726-001143e3f55c__mathematical-expression-and-equation_4.jpg
a _ y = \frac { a } { 1 } + e
f415394f-bc37-11e1-1119-001143e3f55c__mathematical-expression-and-equation_9.jpg
\frac { f ( x + t _ n h + t _ n g ) - f ( x + t _ n h ) } { t _ n } \in \overline { c o } \{ \frac { \partial ^ * } { \partial g } f ( x + t _ n h + \Theta t _ n g ) : 0 \le \Theta \le 1 \} \rightarrow \frac { \partial ^ * } { \partial g } f ( x )
f43f1706-570a-11e1-1027-001143e3f55c__mathematical-expression-and-equation_1.jpg
d \sigma _ 3 = \frac { 1 } { 2 } \sqrt { \frac { ( \lambda _ 3 - \lambda _ 1 ) ( \lambda _ 3 - \lambda _ 2 ) } { ( a + \lambda _ 3 ) ( b + \lambda _ 3 ) ( c + \lambda _ 3 ) } } d \lambda _ 3 ,
f466f5f0-9319-11e2-9142-5ef3fc9bb22f__mathematical-expression-and-equation_0.jpg
T ^ 2 : T \prime { ^ 2 } = \frac { a ^ 3 } { 1 + \frac { m } { M } } : \frac { a \prime { ^ 3 } } { 1 + \frac { m \prime } { M } }
f497abc5-40e3-11e1-1418-001143e3f55c__mathematical-expression-and-equation_0.jpg