Datasets:
Veronika02
/
valid

Dataset card Data Studio Files
xet
 Community
1
formula
stringlengths
5
635
image
stringlengths
80
86
\alpha ( x ) - \alpha \le \alpha ( 1 - \delta ) \frac { \log \log x } { \log x } ( x \ge x _ 0 )
f50e74e3-570a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_1.jpg
r \ge r _ 0
f52a98fa-79ec-4cd9-8f01-12b1f90b6bee__mathematical-expression-and-equation_3.jpg
= \frac { 1 } { 2 b } \sqrt { 1 6 s ( s - a ) ( s - b ) ( s - c ) } ,
f56bb5ee-3336-11ec-af5b-001b63bd97ba__mathematical-expression-and-equation_2.jpg
\sin \beta = \frac { 4 } { 9 } ; d ) \cos \alpha = \frac { 1 8 } { 2 5 } \operatorname { t g } \beta = 1 \frac { 5 } { 9 } ; e ) \cos \alpha = \frac { 7 } { 2 4 } \cos \beta = \frac { 3 5 } { 3 8 }
f56fd4f0-3336-11ec-af5b-001b63bd97ba__mathematical-expression-and-equation_4.jpg
m = 5 , n = 3 , h = 3 0 , s = 1 2 , r = 2 0 .
f5726c43-3336-11ec-af5b-001b63bd97ba__mathematical-expression-and-equation_1.jpg
\tan 2 a = \frac { 2 a _ { 1 2 } } { a _ { 1 1 } - a _ { 2 2 } } .
f578d970-d3b7-11e2-b791-5ef3fc9bb22f__mathematical-expression-and-equation_0.jpg
\parallel ( I - S _ { r y } S _ { y u } ) ^ { - 1 } S _ { r y } \parallel _ M < \frac { 1 } { v _ 2 ( p ) }
f579678a-ac0a-11e1-1027-001143e3f55c__mathematical-expression-and-equation_0.jpg
B ^ 0 = \frac { h _ n } { h \prime \prime _ n } - 1 = \frac { h \prime \prime \prime _ n } { h \prime \prime _ n }
f5d6d870-73f4-11e4-9605-005056825209__mathematical-expression-and-equation_1.jpg
\int _ { \mathbf { R } ^ \infty } z _ k ^ 2 d v ( z _ 1 , z _ 2 , \dots ) < \infty
f6462e12-ac0a-11e1-1431-001143e3f55c__mathematical-expression-and-equation_3.jpg
\langle \overline { T } \delta ( u , v ) ; \delta ( u , v ) \rangle = \sum _ n \sum _ m \sum _ p \sum _ q \delta ( n - 1 , m - 1 ) \delta ( p , q ) S ( n + p , m - q ) =
f6462e57-ac0a-11e1-1431-001143e3f55c__mathematical-expression-and-equation_1.jpg
\beta : = ( \lambda , \phi _ 1 , \dots , \phi _ p ) ^ T
f6462eb2-ac0a-11e1-1431-001143e3f55c__mathematical-expression-and-equation_2.jpg
+ b _ 1 t _ 1 + b _ 2 t t _ 1 + b _ 3 t ^ 2 t _ 1 + b _ 4 t ^ 3 t _ 1 + \dots
f6a3343b-40e3-11e1-1729-001143e3f55c__mathematical-expression-and-equation_0.jpg
\sphericalangle a o p = 2 \alpha _ 3
f6a33563-40e3-11e1-1729-001143e3f55c__mathematical-expression-and-equation_7.jpg
\mathbf { f } ( s ) = \mathbf { f } _ S ( s ) + \mathbf { f } _ { N S } ( s ) = \mathbf { 0 }
f6ac1d4c-c0f3-4051-beb6-e22681639ac1__mathematical-expression-and-equation_2.jpg
v * = - \frac { 0 , 1 9 } { \mathrm { l g } \frac { z _ 0 } { H } } \frac { u _ 2 } { 1 - \frac { 0 , 2 6 } { \mathrm { l g } \frac { z _ 0 } { H } } \frac { H } { L } } ,
f6af44b2-bc37-11e1-1211-001143e3f55c__mathematical-expression-and-equation_3.jpg
\check { s } _ 1 = \frac { V _ 1 } { \check { s } _ 1 } = \frac { 1 5 } { 1 0 0 0 } = 0 , 0 1 5
f6b9b988-6bfe-11e5-aeea-001b21d0d3a4__mathematical-expression-and-equation_4.jpg
k = 1 , 2 , \dots , n
f6c2b40c-570a-11e1-5298-001143e3f55c__mathematical-expression-and-equation_10.jpg
\hat { G } _ { i k h } ^ { i } \stackrel { \mathrm { d e f } } { = } \dot { \partial } _ { h } \hat { G } _ { i k } ^ { i } \stackrel { \mathrm { d e f } } { = } \dot { \partial } _ { h } \dot { \partial } _ { k } \hat { G } _ { i } ^ { i } \stackrel { \mathrm { d e f } } { = } \dot { \partial } _ { h } \dot { \partial...
f6c2b493-570a-11e1-5298-001143e3f55c__mathematical-expression-and-equation_5.jpg
p _ k ( X ) = \sum _ { r = 1 } ^ k \frac { \partial ^ r a ^ n } { \partial y ^ { \alpha _ 1 } \dots \partial y ^ { \alpha _ r } } H _ { i _ 1 \dots i _ k } ^ { \alpha _ 1 \dots \alpha _ r } \frac { \partial } { \partial y _ { i _ 1 \dots i _ k } ^ \eta } \text { f o r } k \ge 1 ,
f6c2b50b-570a-11e1-5298-001143e3f55c__mathematical-expression-and-equation_10.jpg
k = 0 , 1 ; \phi ^ { ( 4 ) } ( 0 , 0 ) = \phi ^ { ( 4 ) } ( \pi , 0 ) = 0 , \epsilon \ge 0
f6c2b52d-570a-11e1-5298-001143e3f55c__mathematical-expression-and-equation_7.jpg
\delta \frac { \tau } { h } ( \alpha + \beta x + \delta \beta h ) \ge 0
f6c2b599-570a-11e1-5298-001143e3f55c__mathematical-expression-and-equation_1.jpg
t > s
f7123076-ac0a-11e1-1431-001143e3f55c__mathematical-expression-and-equation_14.jpg
R ( s , t ) = \lim _ { \substack { a , a \prime \rightarrow - \infty \\ b , b \prime \rightarrow + \infty } } \int _ { a } ^ { b } \int _ { a \prime } ^ { b \prime } f ( s , \lambda ) \overline { f ( t , \mu ) } \text { d d } F ( \lambda , \mu )
f71230d1-ac0a-11e1-1431-001143e3f55c__mathematical-expression-and-equation_9.jpg
u _ i = ( K _ { 1 i } , K _ { 2 i } ) \tilde { y } _ i
f7123147-ac0a-11e1-1431-001143e3f55c__mathematical-expression-and-equation_2.jpg
u = C e ^ { - b t } \sin ( \omega _ 1 t + \gamma )
f739f2b0-1b93-11e4-8e0d-005056827e51__mathematical-expression-and-equation_6.jpg
u = 0 . 5
f7608d90-7a06-11e4-964c-5ef3fc9bb22f__mathematical-expression-and-equation_2.jpg
( P _ { s m } ) _ { i j } = - p \delta _ { i j } + \mu _ { s m } ( \Delta v _ { s m } ) _ { i j } + \mu _ { s m 2 } \Theta v _ { s m } \delta _ { i j }
f78ab0ef-bc37-11e1-1211-001143e3f55c__mathematical-expression-and-equation_2.jpg
\beta _ 2 \prime = \sigma ^ { - 2 } \beta _ 2 + \sigma ^ { - 1 } \sum _ { t = 2 n + 1 } ^ { i } \gamma _ 1 ^ t c _ t ^ { 2 n }
f7a461fa-570a-11e1-1090-001143e3f55c__mathematical-expression-and-equation_20.jpg
\rho ^ w = ( \rho _ { A A } \cup \rho _ { B A } \cup \rho _ { B B } ) ^ w
f7a4634d-570a-11e1-1090-001143e3f55c__mathematical-expression-and-equation_0.jpg
d e g \{ \tilde { p } _ { i i } ( z ) \} > d e g \{ \tilde { p } _ { j i } ( z ) \}
f7dd9661-ac0a-11e1-1431-001143e3f55c__mathematical-expression-and-equation_9.jpg
\mathbf { a b } = ( x _ 1 \mathbf { i } + y _ 1 \mathbf { j } + z _ 1 \mathbf { k } ) ( x _ 2 \mathbf { i } + y _ 2 \mathbf { j } + z _ 2 \mathbf { k } )
f815b01e-40e3-11e1-1586-001143e3f55c__mathematical-expression-and-equation_9.jpg
\overline { \mathbf { K } } \subset \overline { \mathbf { R } } \subset \overline { \mathbf { L } } .
f87a0099-570a-11e1-7963-001143e3f55c__mathematical-expression-and-equation_6.jpg
d \alpha = 2 \omega _ { 2 1 \wedge } \omega _ { 2 3 }
f87a01fc-570a-11e1-7963-001143e3f55c__mathematical-expression-and-equation_2.jpg
S = [ \bigcup _ { \alpha \in \Lambda } P _ \alpha ] \cup M ^ *
f87a0204-570a-11e1-7963-001143e3f55c__mathematical-expression-and-equation_1.jpg
d \sigma \prime = \sin \theta \prime d \theta \prime d \phi \prime
f87a024e-570a-11e1-7963-001143e3f55c__mathematical-expression-and-equation_7.jpg
\Sigma \frac { \psi ( x _ k ) } { f \prime ( x _ k ) } = \alpha _ 1
f8cf62f6-40e3-11e1-1586-001143e3f55c__mathematical-expression-and-equation_1.jpg
+ \frac { x } { r ^ 5 } ( 3 x y \cos \alpha \prime + ( - x ^ 2 + 2 y ^ 2 - z ^ 2 ) \cos \beta \prime + 3 y z \cos
f8cf63e5-40e3-11e1-1586-001143e3f55c__mathematical-expression-and-equation_9.jpg
\cos 4 \alpha = \cos ^ 4 \alpha - 6 \cos ^ 2 \alpha \sin ^ 2 \alpha + \sin ^ 4 \alpha
f8fbdad0-853a-11e4-a354-005056825209__mathematical-expression-and-equation_7.jpg
= \overline { \lim } _ { h \rightarrow 0 ^ + } \frac { 1 } { h } \{ W ( t + h , x ^ * ( t + h ; t , x ) ) - W ( t + h , x ( t + h ; t , x ) ) \} +
f9562fec-570a-11e1-2028-001143e3f55c__mathematical-expression-and-equation_4.jpg
P _ 2 - P a a r
f9563082-570a-11e1-2028-001143e3f55c__mathematical-expression-and-equation_2.jpg
\langle 0 , 1 \rangle \le x \le \langle 1 , 1 \rangle
f9563112-570a-11e1-2028-001143e3f55c__mathematical-expression-and-equation_0.jpg
X = [ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & - 1 & 1 \end{array} ]
f96ce7ef-ac0a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_8.jpg
Q _ { h k } = [ q _ { 0 0 } ^ T , q _ { 1 0 } ^ T , \dots , q _ { h 0 } ^ T , q _ { 0 1 } ^ T , \dots , q _ { 0 k } ^ T , \dots , q _ { 1 1 } ^ T , \dots , q _ { h k } ^ T ] T
f96ce8f2-ac0a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_4.jpg
\mathbf { V } ^ { ( 2 ) } = \operatorname { I m } [ \begin{array} { c c c c } 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \end{array} ]
f96ce905-ac0a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_6.jpg
z ^ * ( z _ 1 - l ( x _ 1 ) ) \ge \alpha \ge z ^ * ( z _ 2 - l ( x _ 2 ) )
f96cea10-ac0a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_8.jpg
\mathbf { W } . \mathbf { c _ 1 } = \iint \nabla ( \mathbf { v } . d \mathbf { p } ) . \mathbf { c _ 1 } - \iint ( d i v \mathbf { v } \mathbf { c _ 1 } ) . d \mathbf { p }
f98cbec7-40e3-11e1-1121-001143e3f55c__mathematical-expression-and-equation_3.jpg
y _ v = b ^ 2 - \frac { b ^ 2 \xi ^ 2 } { a ^ 2 } ,
f98cbeec-40e3-11e1-1121-001143e3f55c__mathematical-expression-and-equation_0.jpg
x _ 1 = y _ 0 \cdot \cos \alpha
f98cc0d7-40e3-11e1-1121-001143e3f55c__mathematical-expression-and-equation_6.jpg
\frac { 1 } { 2 } m ( v \prime ^ { 2 } - v ^ { 2 } ) = h \nu _ { r a d }
fa0366e8-bfe4-495d-830d-c4112f8583ef__mathematical-expression-and-equation_5.jpg
y = \sqrt { \frac { 1 } { n } } = \pm b
fa09ac30-d3b7-11e2-b791-5ef3fc9bb22f__mathematical-expression-and-equation_2.jpg
\frac { d p } { d \rho } = a ^ 2 ,
fa17eb95-bc37-11e1-7963-001143e3f55c__mathematical-expression-and-equation_1.jpg
\frac { \sin ( z a + \alpha \prime ) } { \sin z a } = 0 . 6 8 8 2 = p ;
fa38fbed-2b9d-4c24-9b1b-deec60d9cce8__mathematical-expression-and-equation_6.jpg
V ^ * ( x ) : = \inf _ { \pi } V ( \pi , x ) , x \in X ,
fa42fd63-ac0a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_6.jpg
h a t { S } _ n = \prod _ { i = 1 } ^ { n } \hat { \underline { b } } _ i ( \underline { X } _ 1 , \underline { X } _ 2 , \dots , \underline { X } _ { i - 1 } ) \underline { X } _ i
fa42fe94-ac0a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_3.jpg
x \cdot 1 = 1 \star ( 1 \star x )
fa42fef7-ac0a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_13.jpg
W _ { I I } = \frac { 1 } { 3 2 } 3 , 1 4 . 0 , 0 7 ^ { 3 } ( 2 0 - 1 4 ) = 2 , 0 2 . 1 0 ^ { - 4 } m ^ { 3 }
fab828f0-d6b1-11ea-903c-5ef3fc9ae867__mathematical-expression-and-equation_8.jpg
\tilde { J } _ x ^ { r + 1 } E \xrightarrow { [ \kappa _ { r + 1 } ( X , Y ) ] ^ { - 1 } } J _ x ^ 1 ( B , \tilde { J } _ x ^ r E )
fb0eb612-570a-11e1-1278-001143e3f55c__mathematical-expression-and-equation_6.jpg
i _ 1 = 1 , \dots , k
fb0eb73a-570a-11e1-1278-001143e3f55c__mathematical-expression-and-equation_8.jpg
\int _ 0 ^ T E ( t ) d t \le C ( | ( w _ t , w ) _ \Omega | _ 0 ^ T ] | + \gamma ^ 2 | ( \nabla w _ t , \nabla w ) _ \Omega | _ 0 ^ T | + | ( b w , w ) _ \Omega | _ 0 ^ T |
fb0f9c98-ac0a-11e1-1027-001143e3f55c__mathematical-expression-and-equation_10.jpg
u ( s ) = - F ( s ) x ( s ) + G v ( s )
fb0f9d13-ac0a-11e1-1027-001143e3f55c__mathematical-expression-and-equation_3.jpg
B _ 1 \equiv \frac { u _ 2 } { A _ { 1 3 } } - \frac { u _ 3 } { A _ { 1 2 } } = 0
fb1affd8-40e3-11e1-1121-001143e3f55c__mathematical-expression-and-equation_0.jpg
\delta _ 2 ^ 5 \equiv c _ 2 ( \bmod \lambda ^ 5 ) ,
fb1b00ef-40e3-11e1-1121-001143e3f55c__mathematical-expression-and-equation_13.jpg
\frac { 2 7 . 3 \text { k r . } } { 8 9 } = 0 . 3 0 \text { k r }
fb438693-e256-11e6-95ce-001999480be2__mathematical-expression-and-equation_1.jpg
m ^ 2 ( \Delta N _ g ) = m _ { \Sigma } ^ 2 + m _ E ^ 2
fb9393a0-2883-4226-943a-0603aa66ba16__mathematical-expression-and-equation_1.jpg
\frac { 1 } { \delta _ 1 } = \frac { a \prime } { \delta a \prime _ 1 }
fbb2d6b0-287f-11e9-844c-005056827e51__mathematical-expression-and-equation_5.jpg
+ r _ 1 r _ 2 \cdot \tan \frac { \alpha } { 2 } \tan \frac { \beta } { 2 } ]
fbdd1787-40e3-11e1-1027-001143e3f55c__mathematical-expression-and-equation_1.jpg
I I \equiv \frac { x } { \alpha } + \frac { y } { \beta } - 1 = 0
fbdd17cb-40e3-11e1-1027-001143e3f55c__mathematical-expression-and-equation_0.jpg
x _ 1 = x \prime _ 1 + \lambda _ { 1 2 } x \prime _ 2 + \dots + \lambda \prime _ { 1 n } x \prime _ n
fbdd191a-40e3-11e1-1027-001143e3f55c__mathematical-expression-and-equation_7.jpg
i = 1 , \dots , r
fbe38f59-ac0a-11e1-1589-001143e3f55c__mathematical-expression-and-equation_7.jpg
y ( t ) = C _ C ( \beta ) x _ C ( t ) + N _ C ( \beta ) d ( t )
fbe39027-ac0a-11e1-1589-001143e3f55c__mathematical-expression-and-equation_3.jpg
a _ 1 \parallel y \parallel ^ 2 + a _ 2 \parallel y \prime \parallel ^ 2 \le \bar { c } + \int _ { t _ 0 } ^ t ( \phi _ 1 \parallel y \parallel ^ 2 + \phi _ 2 \parallel y \prime \parallel ^ 2 ) d s ,
fbe6ed1c-570a-11e1-7459-001143e3f55c__mathematical-expression-and-equation_0.jpg
\frac { 2 f \prime _ 1 ( t ) } { f \prime ( t ) } = \frac { 2 f _ 2 ( t ) } { f _ 1 ( t ) } + \frac { f \prime \prime ( t ) f _ 1 ( t ) } { f \prime ^ 2 ( t ) }
fcab1629-40e3-11e1-3052-001143e3f55c__mathematical-expression-and-equation_0.jpg
h ( \omega | x ) = c ( x ) [ 1 - \frac { 2 } { N } \sum _ { i = 1 } ^ { k } \frac { P ( \omega _ i ) } { Q } ] ^ { - \frac { N } { 2 } + k }
fcbef91a-ac0a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_2.jpg
f ( \nabla ) = \bar { f } ( \nabla ) S ^ { - 1 } ( \nabla )
fcbefb6d-ac0a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_0.jpg
\lim _ { t \rightarrow \infty } \inf f ( t ) = \inf _ { t \ge 0 } f ( t )
fcbefb78-ac0a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_4.jpg
3 4 \frac { 1 } { 2 } m - 1 6 \frac { 3 } { 4 } m = ?
fcbf773b-0943-453a-82d9-038b145ad193__mathematical-expression-and-equation_4.jpg
u \prime ( t ) = ( A + B ) u ( t ) + f ( t )
fcc4062e-570a-11e1-7963-001143e3f55c__mathematical-expression-and-equation_4.jpg
\sum _ \nu \cos a x _ \nu \sin b x _ \nu = \frac { 1 } { 2 } \sum _ \nu \sin ( a + b ) x _ \nu - \frac { 1 } { 2 } \sum _ \nu \sin ( a - b ) x _ \nu ,
fd4bf0aa-7127-635e-b733-ce2826e93884__mathematical-expression-and-equation_2.jpg
p = \pm \frac { b ^ 2 } { a }
fd4ee380-5c0f-11e7-85f6-005056825209__mathematical-expression-and-equation_6.jpg
\phi ( v ) \psi \prime \prime ( v ) - \psi ( v ) \phi \prime \prime ( v ) = 0
fd79b035-40e3-11e1-1431-001143e3f55c__mathematical-expression-and-equation_6.jpg
\int _ 0 ^ r \phi ( r ) d r = \frac { r } { \pi } \int _ 0 ^ { \frac { \pi } { 2 } } \psi ( r \sin \phi ) \sin \theta d \theta
fd79b18c-40e3-11e1-1431-001143e3f55c__mathematical-expression-and-equation_3.jpg
d s ^ 2 = \frac { k ^ 4 } { 4 } \frac { 1 + f _ 1 ^ 2 } { f _ 1 ^ 4 } ( d \alpha ^ 2 + f _ 1 ^ 2 d \beta ^ 2 )
fd79b1c0-40e3-11e1-1431-001143e3f55c__mathematical-expression-and-equation_5.jpg
\frac { l _ v } { d } = 7 2 0 0 P e ^ { - 0 . 6 }
fd7f0b3d-bc37-11e1-1027-001143e3f55c__mathematical-expression-and-equation_0.jpg
k b = k \prime b \prime , k a = k \prime a \prime
fd92eb3c-ac0a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_7.jpg
\parallel T _ { \Delta t } ^ { \gamma } \mu _ p \parallel _ { \mu p } \le ( 1 + h C )
fd92eb82-ac0a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_0.jpg
| Q _ n - P | = O ( \epsilon _ n )
fd92ec83-ac0a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_10.jpg
u _ 1 ^ N ( t , x ) = \sum _ { j = 1 } ^ { N - 1 } w _ j ( t ) L _ j ( x ) + ( \Delta + d ( t ) ) L _ N ( x )
fd92ed16-ac0a-11e1-1211-001143e3f55c__mathematical-expression-and-equation_2.jpg
\le \mu \phi ( \mu ^ 2 . \mathbf { 1 } , \delta ^ 2 . 1 , \dots , \delta ^ 2 . \mathbf { 1 } ) \int _ { t } ^ { t + 1 } f ( s ; \mu ^ 2 . \mathbf { 1 } ) d s + \delta \le \infty ,
fd9efd6b-570a-11e1-1278-001143e3f55c__mathematical-expression-and-equation_6.jpg
j = 0 , 1 , \dots , n + 3
fd9eff4a-570a-11e1-1278-001143e3f55c__mathematical-expression-and-equation_7.jpg
\mathbf { D } \prime \prime \prime = \frac { 0 \prime 5 2 } { V \sqrt { \mathbf { B } \prime \prime \prime } }
fdd34d68-6182-b600-6a06-c0af8786d669__mathematical-expression-and-equation_3.jpg
I _ 2 + 6 O H _ 2 + 5 C l _ 2 = 1 0 C l H , 2 I O _ 3 H
fde92b10-0af5-11e5-b0b8-5ef3fc9ae867__mathematical-expression-and-equation_1.jpg
u _ z = \frac { u \prime _ z } { \beta ( 1 + \frac { v } { c ^ 2 } u \prime _ x ) }
fe48260a-40e3-11e1-1278-001143e3f55c__mathematical-expression-and-equation_0.jpg
P _ 1 \equiv p x - y _ 0 y + p x _ 0 = 0 .
fe482659-40e3-11e1-1278-001143e3f55c__mathematical-expression-and-equation_0.jpg
\sum _ { j = 1 } ^ { m } x _ { i j } = 1 ,
fe69c5ad-ac0a-11e1-7963-001143e3f55c__mathematical-expression-and-equation_0.jpg
G _ m ^ k ( d ) \le k m [ ( d - 2 ) ( k - 1 ) + 1 ] i f d \ge 3
fe781f05-570a-11e1-2028-001143e3f55c__mathematical-expression-and-equation_2.jpg
e ^ { ( n + 1 ) \sqrt { ( - 1 ) } \phi } \eta _ { 2 1 } \dots \eta _ { n + 1 , n } = \det | X , X \prime , \dots , X ^ { ( n ) } | .
fe781fe4-570a-11e1-2028-001143e3f55c__mathematical-expression-and-equation_2.jpg
y _ j = a _ j + \lambda _ 0 ( u _ 1 - p a _ 1 ) + \sum _ { i = 1 } ^ { l } \lambda _ i ( v _ i - p a _ { i + 1 } + a _ i ) + W
fe782000-570a-11e1-2028-001143e3f55c__mathematical-expression-and-equation_8.jpg
\Phi ( z ) = \sum _ { j = 1 } ^ { m } \int _ { 0 } ^ { 1 } \frac { f _ j ( t ) } { \psi _ j ( t ) - z } d \psi _ j ( t )
fe782064-570a-11e1-2028-001143e3f55c__mathematical-expression-and-equation_2.jpg
x = S _ { k \in K } \{ x _ k ^ * \}
fe782086-570a-11e1-2028-001143e3f55c__mathematical-expression-and-equation_8.jpg
| | | u _ L ( t ) | | | \le C \eta < C _ 1 \eta < R _ 1 \le R \text { f o r } t \ge t _ 0 .
fe7820e0-570a-11e1-2028-001143e3f55c__mathematical-expression-and-equation_10.jpg