Added more annotated data

valid.csv

@@ -1974,4 +1974,146 @@ formula,image
 "\frac { u _ { 2 m - 1 } + u _ { 2 m } } { O _ { 2 m } ^ { 2 } } + \frac { u _ { 2 m } + u _ { 1 } } { O _ { 1 } ^ { 2 } } = - \frac { 1 } { 2 }" ,079fa77b-40e4-11e1-8339-001143e3f55c__mathematical-expression-and-equation_13.jpg
 "+ 2 \epsilon _ { 2 3 } ^ { 2 } [ 2 ( \epsilon _ { 2 2 } ^ { 2 } + \epsilon _ { 3 3 } ^ { 2 } ) + 3 \epsilon _ { 2 2 } \epsilon _ { 3 3 } - \epsilon _ { 1 1 } ^ { 2 } ] + 2 \epsilon _ { 3 1 } ^ { 2 } [ 2 ( \epsilon _ { 3 3 } ^ { 2 } + \epsilon _ { 1 1 } ^ { 2 } )" ,1cf70e88-3c62-11e1-7963-001143e3f55c__mathematical-expression-and-equation_10.jpg
 "M g x _ { 0 } = m _ { 1 } g x _ { 1 } + m _ { 2 } g x _ { 2 } + \dots =" ,208ff050-f0e4-11e2-9439-005056825209__mathematical-expression-and-equation_9.jpg
-"\dots \mathbf { u } _ { S 3 } = \frac { 1 } { 3 } ( u _ { V } e ^ { j 2 / 3 \pi } + u _ { C 6 } e ^ { j \pi / 3 } ) = \frac { 1 } { 3 } [ u _ { V } e ^ { j 2 / 3 \pi } + ( u _ { C 4 } + u _ { C 5 } ) e ^ { j }" ,5fb896c3-a5e9-4906-8c8d-8f3bc5faad91__mathematical-expression-and-equation_21.jpg
+"\dots \mathbf { u } _ { S 3 } = \frac { 1 } { 3 } ( u _ { V } e ^ { j 2 / 3 \pi } + u _ { C 6 } e ^ { j \pi / 3 } ) = \frac { 1 } { 3 } [ u _ { V } e ^ { j 2 / 3 \pi } + ( u _ { C 4 } + u _ { C 5 } ) e ^ { j }" ,5fb896c3-a5e9-4906-8c8d-8f3bc5faad91__mathematical-expression-and-equation_21.jpg
+"\zeta = \zeta _ { 1 } + \zeta _ { 2 } + \zeta _ { 3 } + \dots" ,959bd1a2-de39-4feb-a174-09bcacb166d7__mathematical-expression-and-equation_1.jpg
+"3 = \frac { 5 \times 6 } { 1 0 }" ,08326dca-e631-4952-9e6d-45f904520556__mathematical-expression-and-equation_5.jpg
+"\frac { P + Q } { 2 } + \frac { 1 } { 2 } \sum _ { k = 1 } \frac { F ( r _ { 2 } ) - F ( r _ { 1 } ) } { k + 4 } ( r _ { 2 } ^ { k + 4 } - r _ { 1 } ^ { k + 4 } ) \binom { \frac { 1 } { 2 } } { \frac { k + 3 } { 2 } }" ,30654e80-df3d-11e1-1287-001143e3f55c__mathematical-expression-and-equation_2.jpg
+"\frac { D w } { D t } = \frac { \partial w } { \partial t } + u \frac { \partial w } { \partial x } + v \frac { \partial w } { \partial y } + w \frac { \partial w } { \partial z } = Z - \frac { 1 } { \rho } \frac { \partial p } { \partial z }" ,1b3dee2f-3c62-11e1-7963-001143e3f55c__mathematical-expression-and-equation_4.jpg
+"Z _ { 1 } - Z _ { 2 } = \pm 2 p" ,1c1d5157-3c62-11e1-3052-001143e3f55c__mathematical-expression-and-equation_3.jpg
+"\frac { r _ { j + 1 } - r _ { j } } { 2 } = \epsilon _ { j } \prime ; \frac { r _ { j + 2 } - r _ { j + 1 } } { 2 } = \epsilon _ { j } \prime \prime ; \dots ; \frac { r _ { j + j \prime } - r _ { j + j \prime - 1 } } { 2 } = \epsilon _ { j } ^ { ( j \prime ) }" ,3f277f55-408b-11e1-8339-001143e3f55c__mathematical-expression-and-equation_3.jpg
+"Q \prime _ { \mu } = Q _ { \mu } - \frac { \partial \log F } { \partial \xi ^ { \mu } }" ,0e4660e2-40e4-11e1-1121-001143e3f55c__mathematical-expression-and-equation_3.jpg
+"Y = - 0 . 3 + 0 . 2 0 0 X" ,2a55dc88-4ce4-11e1-1726-001143e3f55c__mathematical-expression-and-equation_8.jpg
+"\sphericalangle m c b + c b m = R ," ,17a619d0-e929-11e4-a794-5ef3fc9bb22f__mathematical-expression-and-equation_2.jpg
+"\frac { \partial p _ { 0 } } { \partial t } = - p _ { 0 } \frac { \partial c } { \partial s }" ,43b6a980-d866-4b84-97b9-407007e64205__mathematical-expression-and-equation_2.jpg
+"\frac { v d v } { e ^ { - \frac { v } { 2 5 } } | 1 - e ^ { \frac { v } { 2 5 } } ( 0 . 0 0 4 5 - 0 . 0 0 0 0 7 1 5 v ^ { 2 } ) | } = - 0 . 1 g d \lambda" ,3b1cc537-dbba-11e6-8be1-001b63bd97ba__mathematical-expression-and-equation_1.jpg
+"\Sigma \frac { 1 } { \cos ^ { 3 } \alpha } = 2 \times 1 9 . 5 1 4" ,2574b750-31e7-11e4-90aa-005056825209__mathematical-expression-and-equation_5.jpg
+"= \hat { 2 } < E _ { p } \frac { \tau _ { p } } { \mu _ { p } } \sum _ { k = \alpha , \beta } \kappa _ { k } [ 1 - \frac { \mu _ { k } } { \mu _ { p } } \ln ( 1 + \frac { \mu _ { p } } { \mu _ { k } } ) ]" ,9a4ea058-4334-11e1-1589-001143e3f55c__mathematical-expression-and-equation_0.jpg
+"x _ { 0 } ^ { 2 } + y _ { 0 } ^ { 2 } = ( 2 r ) ^ { 2 }" ,8e512500-7ad7-11e8-9690-005056827e51__mathematical-expression-and-equation_6.jpg
+"\frac { 1 5 } { 2 2 } : 1 7 \frac { 1 } { 4 } = x ." ,05fd0c1d-224c-11ea-bbb4-001b63bd97ba__mathematical-expression-and-equation_13.jpg
+"x = \frac { a + b } { 2 }" ,39dda871-c060-11e6-855e-001b63bd97ba__mathematical-expression-and-equation_3.jpg
+"> 1 - \lambda ," ,02b32dd6-ac0b-11e1-1589-001143e3f55c__mathematical-expression-and-equation_15.jpg
+"V = \frac { B ( v + x ) } { 3 } - \frac { b x } { 3 } = \frac { B v } { 3 } + \frac { x } { 3 } ( B - b" ,4cce0e06-2dac-11ec-b355-001b63bd97ba__mathematical-expression-and-equation_2.jpg
+"\parallel \kappa _ { 3 , 0 } + p \kappa _ { 3 , 1 } + p ^ { 2 } \kappa _ { 3 , 2 } \dots + , \kappa _ { 4 , 0 } + p \kappa _ { 4 , 1 } + p ^ { 2 } \kappa _ { 4 , 2 } + \dots" ,11f9f260-3c62-11e1-8339-001143e3f55c__mathematical-expression-and-equation_11.jpg
+"P = P ( t ) > 0" ,5841cd7d-be8f-491c-adb3-1a631f6252fd__mathematical-expression-and-equation_11.jpg
+"1 a + 1 6 b ( a + b ) ( 6 a + 1 1 b ) - 4 ( 3 a + 4 b ) ( 4 a + 5 b ) ( a + 2 b ) =" ,3790b91e-eb13-11ec-b47a-00155d01210b__mathematical-expression-and-equation_29.jpg
+"= 1 + \frac { 1 } { n } \sum _ { \lambda = 2 , 4 , \dots } ^ { \infty } \frac { ( - 1 ) ^ { \frac { \lambda } { 2 } } } { ( \lambda + 1 ) ! } ( \frac { 2 m \pi } { n } ) ^ { \lambda } \mathbf { B } _ { \lambda - 1 } ( n ) \dots" ,324aafa8-df3d-11e1-1027-001143e3f55c__mathematical-expression-and-equation_3.jpg
+"I \prime = V \prime Y \prime" ,11f9f2c4-3c62-11e1-8339-001143e3f55c__mathematical-expression-and-equation_7.jpg
+"f ( a , t ) = \sum _ { i = 1 } ^ { N } a _ { i } \phi _ { i } ( t )" ,0ecd6a07-3c62-11e1-8339-001143e3f55c__mathematical-expression-and-equation_1.jpg
+"\mathfrak { A } ) _ { v } = 8 E J _ { o } \delta \epsilon \frac { h } { l ^ { 2 } + 8 h ^ { 2 } }" ,1f382a37-9c7a-4355-af65-c71a6907cdfa__mathematical-expression-and-equation_5.jpg
+"W = 5 . 3 0 8 \times 0 . 0 1 5 = 8 0 \text { O h m } ;" ,5f9431f0-00cf-11eb-916b-5ef3fc9bb22f__mathematical-expression-and-equation_9.jpg
+"w = \frac { N p } { t g \alpha }" ,329ff308-87f8-4a23-b9cf-22021f5abd13__mathematical-expression-and-equation_7.jpg
+"R _ { 1 } = - x _ { 4 } x _ { 5 } ( 1 + \lambda _ { 1 } ) ," ,079fa6ba-40e4-11e1-8339-001143e3f55c__mathematical-expression-and-equation_5.jpg
+"\sum _ { i = 1 } ^ { n } 1 / x _ { j } = a / b" ,49b13f44-408b-11e1-1586-001143e3f55c__mathematical-expression-and-equation_7.jpg
+"\bar { b } c = \omega _ { 1 } L _ { 1 } J _ { 1 } ( 1 - \sigma ) \frac { x } { \sqrt { 1 + x ^ { 2 } } }" ,08084bbd-dbba-11e6-8be1-001b63bd97ba__mathematical-expression-and-equation_15.jpg
+"+ \Delta _ { 3 } ( e ^ { N } ) . \int _ { \{ | 2 ( \sqrt { \lambda _ { n , h } } - 1 ) | > \epsilon \} \cap B _ { n , N } }" ,0024ba86-ac0b-11e1-5298-001143e3f55c__mathematical-expression-and-equation_16.jpg
+"\int _ { x \prime \prime \prime } ^ { x \prime \prime \prime \prime } y \partial x + \int _ { x \prime \prime \prime } ^ { x \prime \prime \prime \prime } u \partial x = \int _ { x \prime \prime \prime } ^ { x \prime \prime \prime \prime } y _ { 1 } \partial x + \int _ { x \prime \prime \prime } ^ { x \prime \prime \prime \prime } u _ { 1 } \partial x" ,53784c9d-30fa-4688-8585-30b9b60edc48__mathematical-expression-and-equation_4.jpg
+"+ \frac { \alpha _ { 1 } \gamma _ { 1 } } { p _ { 1 } } f _ { 1 } f _ { 3 } + \frac { \beta _ { 1 } \gamma _ { 1 } } { p _ { 1 } } f _ { 2 } f _ { 3 } + \frac { \gamma _ { 1 } \gamma _ { 1 } } { p _ { 1 } } f _ { 3 } f _ { 3 } + \dots" ,8e3eb976-f46c-11e7-ae40-001b63bd97ba__mathematical-expression-and-equation_2.jpg
+"\begin{array} { c c c c c c c } & \text { F e } & \text { A g } & \text { C u } & \text { P b } & \text { P t } & \text { N i } \\ \alpha . 1 0 ^ { 6 } & + 1 7 , 1 5 & + 2 , 1 2 & + 1 , 3 4 & 0 , 0 & - 0 , 6 0 & - 2 1 , 8 \\ \beta . 1 0 ^ { 6 } & - 0 , 0 4 8 & + 0 , 0 1 5 & + 0 , 0 0 9 & 0 , 0 & - 0 , 0 1 1 & - 0 , 0 5 \end{array}" ,45885af0-f0e4-11e2-9439-005056825209__mathematical-expression-and-equation_4.jpg
+"- \frac { r } { R a } - \frac { 2 } { a } ) + \frac { 1 6 v - 1 7 } { 4 R r } s ^ { 2 } + b ( \frac { 3 } { 2 R } + \frac { 2 r } { a } ( \frac { b } { s } ) ^ { 2 } + ( 2 v - 1 ) ( 4 - \frac { r } { R } ) ] n _ { r } +" ,9f78db5d-ae15-4865-be6f-0dc2fb2f4ea7__mathematical-expression-and-equation_6.jpg
+"\frac { Q V _ { 2 } ^ { 2 } } { 2 g } \alpha _ { 2 } = W r ( \frac { \pi - 2 } { 2 \pi } ) \dots 7 )" ,04a6f5cd-dbf5-11e6-a7df-001b63bd97ba__mathematical-expression-and-equation_6.jpg
+"w _ { i } = H _ { i } x _ { i }" ,039047c6-ac0b-11e1-7963-001143e3f55c__mathematical-expression-and-equation_3.jpg
+"B ( Z ) = \frac { \int _ { 0 } ^ { E _ { m a x } } \sigma ^ { \pi } ( E ) N ( E ) d E } { \int _ { 0 } ^ { E _ { m a x } } E N ( E ) d E }" ,9d8e639f-4334-11e1-8339-001143e3f55c__mathematical-expression-and-equation_1.jpg
+"Q _ { s i } = 1 + \frac { u _ { 2 } } { U }" ,9a4ea091-4334-11e1-1589-001143e3f55c__mathematical-expression-and-equation_5.jpg
+"- ( X d x + Y d y + Z d z ) = d U" ,51d36ba0-0c73-11e4-8413-5ef3fc9ae867__mathematical-expression-and-equation_5.jpg
+"y \prime = \frac { f _ { 1 } f _ { 2 } } { \Delta } \frac { y } { x }" ,2438f5e0-1b94-11e4-8e0d-005056827e51__mathematical-expression-and-equation_6.jpg
+"\mathbf { a } _ { k } \alpha + \mathbf { a } _ { 1 } ( x _ { 1 } - \alpha a _ { 1 k } ) + \mathbf { a } _ { 2 } ( x _ { 2 } - \alpha a _ { 2 k } ) + \dots + \mathbf { a } _ { r } ( x _ { r } - \alpha a _ { r k } ) + \dots" ,17ebdd42-3c62-11e1-8486-001143e3f55c__mathematical-expression-and-equation_4.jpg
+"\{ P _ { n ( \Lambda _ { \theta _ { 0 } } + \frac { 1 } { \sqrt { n } } h ) } \}" ,0024ba88-ac0b-11e1-5298-001143e3f55c__mathematical-expression-and-equation_9.jpg
+"\frac { 1 } { p Z ( p ) } = \int _ { 0 } ^ { \infty } A ( t ) e ^ { - p t } d t ." ,08ebfcc4-40e4-11e1-1418-001143e3f55c__mathematical-expression-and-equation_2.jpg
+"0 = a t j t + b j t + d t" ,01d92630-570b-11e1-7459-001143e3f55c__mathematical-expression-and-equation_10.jpg
+"x - x _ { 0 } = \theta" ,039b5170-9944-11de-9613-0030487be43a__mathematical-expression-and-equation_5.jpg
+"\mathrm { p } \cap \mathrm { p } \prime < \mathrm { p } ," ,4c97a13c-ab88-4e31-b611-8544a3052e6e__mathematical-expression-and-equation_4.jpg
+"\frac { e ^ { 2 } } { 4 } = D ^ { 1 } _ { 2 }" ,5c30c26f-6bff-11e5-aeea-001b21d0d3a4__mathematical-expression-and-equation_22.jpg
+"i _ { 1 } = 2 I _ { 1 } + 3" ,71d128dd-4abf-42d2-adf2-36f7c75c3efe__mathematical-expression-and-equation_7.jpg
+"= \int _ { B } \int _ { B } \frac { 1 } { A } f ( z ) d \kappa _ { m - 1 } ( z ) d v ( x ) = 0" ,3bb9a7d3-408b-11e1-1586-001143e3f55c__mathematical-expression-and-equation_1.jpg
+"m _ { T } \cdot C ^ { 2 } | v _ { T } ( z , t ) | e ^ { i \omega _ { c } t } - G I _ { T } \cdot D ^ { 2 } | v _ { T } ( z , t ) | e ^ { i \omega _ { c } t } ( 1 + i \delta _ { T } / \pi ) =" ,1f7999cd-3c62-11e1-7459-001143e3f55c__mathematical-expression-and-equation_7.jpg
+"L P ( 0 ) = L P _ { m i n } - \frac { F P _ { m i n } } { k ^ { * } }" ,227c99f0-ed5f-11ec-95f3-005056827e51__mathematical-expression-and-equation_1.jpg
+"\rcases t _ { 0 } = \alpha _ { 1 } x _ { 0 } + x _ { 1 } \\ t _ { 1 } = x _ { 0 } + \alpha _ { 1 } x _ { 1 } \rcases" ,6e081ab0-148e-11de-b5d5-0030487be43a__mathematical-expression-and-equation_0.jpg
+"[ A _ { 1 1 } + n ( n + 1 ) p ^ { 2 } ] u ^ { 2 } + 2 n ( n + 1 ) p q u v + [ A _ { 2 2 } + n ( n + 1 ) q ^ { 2 } ] v ^ { 2 } +" ,3f3ca384-df28-11e1-1154-001143e3f55c__mathematical-expression-and-equation_5.jpg
+"w = a _ { 0 } + a _ { 1 } z + a _ { 2 } z ^ { 2 } + \dots + a _ { n } z _ { n } + \dots ," ,9acc1850-7d97-11e7-921c-5ef3fc9ae867__mathematical-expression-and-equation_1.jpg
+"[ p v ] = 0 ," ,0561c666-a56b-f851-ec31-04d636e3359c__mathematical-expression-and-equation_9.jpg
+"T = | \begin{array} { c c c c } A _ { r , \alpha - r } & A _ { r , \beta - r } & A _ { r , \gamma - r } & . \\ A _ { r + \alpha \prime - \alpha , \alpha - r } & A _ { r + \alpha \prime - \alpha , \beta - r } & \dots & . \\ A _ { r + \alpha \prime \prime - \alpha , \alpha - r } & A _ { r + \alpha \prime \prime - \alpha , \beta - r } & \dots & . \\ \dots & \dots & \dots & . \end{array} |" ,0551bf31-40e4-11e1-1726-001143e3f55c__mathematical-expression-and-equation_5.jpg
+"= W ( x _ { 1 } , x _ { 2 } , \dots , x _ { k } , x _ { k + 1 } ) ( a _ { j } + \epsilon )" ,46fe54a9-408b-11e1-1586-001143e3f55c__mathematical-expression-and-equation_9.jpg
+"= e ^ { 2 \pi ( f _ { * } ( x ) + n + j + 1 ) i } = e ^ { 2 \pi f _ { * } ( x ) i } = f ( e ^ { 2 \pi x i } )" ,07df0b68-570b-11e1-5298-001143e3f55c__mathematical-expression-and-equation_1.jpg
+"Q = 2 4 . 3 6 0 0 \frac { 1 0 2 N } { 1 0 0 0 H \eta }" ,2bbdfa50-fc9a-11e2-9439-005056825209__mathematical-expression-and-equation_0.jpg
+"\le c ^ { * } \cdot \parallel ( u _ { \delta } ) ^ { 0 } ( \cdot , t ) \parallel _ { W _ { p } ^ { 2 } ( \mathbb { R } ^ { n - 1 } \times ( - \delta / 2 , \infty ) ) } \le c ^ { * } \cdot \parallel u ( \cdot , t ) \parallel _ { W _ { p } ^ { 2 } ( Q _ { + } ^ { n } ( \alpha , \beta }" ,0c39d915-570b-11e1-5298-001143e3f55c__mathematical-expression-and-equation_8.jpg
+"1 1 0 \times 1 . 1 8 6 = 1 3 0 . 5" ,94db40e5-2b65-4b8a-9d30-2943ed982e0d__mathematical-expression-and-equation_1.jpg
+"[ p ] x - [ p b ] y - [ p c ] z - [ p l ] = 0" ,9b195470-c4a3-a7cc-1cc9-3fa25cf1d5e6__mathematical-expression-and-equation_6.jpg
+"A _ { n _ { 1 } \dots n _ { p } } = \frac { 1 } { \Gamma ( s ) } \sum _ { \alpha = 1 } ^ { p } C _ { \alpha } \frac { \pi } { \sin s \pi } [ \frac { 2 \pi i } { c _ { \alpha } } ( n _ { \alpha } - v _ { \alpha } ) ] ^ { s - 1 } ," ,89e0ffb6-0634-4e2a-9db7-cc211142cc98__mathematical-expression-and-equation_1.jpg
+"x \cdot \frac { \partial F } { \partial x } + y \cdot \frac { \partial F } { \partial y } + z \cdot \frac { \partial F } { \partial z } = 0" ,2fd77d12-dbba-11e6-8be1-001b63bd97ba__mathematical-expression-and-equation_0.jpg
+"Q _ { 1 } = Q _ { 2 } = Q _ { 3 }" ,2168a3b5-c5fa-4012-a710-43918d3ab685__mathematical-expression-and-equation_2.jpg
+"3 r \prime + l \prime = 3 ( r \prime + 1 ) - 2 l \prime = 3 ( r + 1 ) - 2 l \prime" ,5d3bd029-972d-4471-b711-26afadf395c2__mathematical-expression-and-equation_14.jpg
+"\Delta \phi = \frac { v \Delta t } { r } , \Delta v _ { n } = v \prime \Delta \phi = \frac { v \prime v } { r } \Delta t ," ,1da46920-f0e4-11e2-9439-005056825209__mathematical-expression-and-equation_2.jpg
+"M S f g" ,5ef7f9d8-c417-4682-9905-8c0a5beef123__mathematical-expression-and-equation_0.jpg
+"\xi = \alpha _ { 1 } t" ,53d28b90-d7ae-11ea-b03f-5ef3fc9bb22f__mathematical-expression-and-equation_4.jpg
+"- 2 x + 4 T ( x , y ) - T ( x , x ) + T ( y , y ) \le 0" ,0469dd78-ac0b-11e1-1090-001143e3f55c__mathematical-expression-and-equation_2.jpg
+"| \tau _ { 2 } - \tau _ { 1 } | \le \sigma" ,4094a2c7-408b-11e1-8339-001143e3f55c__mathematical-expression-and-equation_5.jpg
+"T = \frac { Q } { 2 }" ,4e80e19c-c073-11e6-ae7e-001b63bd97ba__mathematical-expression-and-equation_8.jpg
+"\prime \delta u ^ { h } = d u ^ { h } + \Lambda ^ { k } _ { J i } u ^ { i } ( d \xi ) ^ { J }" ,0efce001-40e4-11e1-2755-001143e3f55c__mathematical-expression-and-equation_3.jpg
+"Z ^ { ( k - 1 ) } ( t _ { 0 } ^ { k } ) \neq 0" ,0c39d9e9-570b-11e1-5298-001143e3f55c__mathematical-expression-and-equation_1.jpg
+"\frac { d ^ { 2 } u } { d x ^ { 2 } } + ( q _ { 2 } - \frac { 1 } { 4 } q _ { 1 } ^ { 2 } - \frac { 1 } { 2 } q _ { 1 } \prime ) u = 0 ." ,6d40e55e-39b9-4212-85ed-b190817cc1dc__mathematical-expression-and-equation_0.jpg
+"\mathcal { L } ^ { i } = \frac { 1 } { 2 } \int _ { V } \sigma _ { i j } e _ { i j } d V" ,5fbe9849-e504-4a7d-a78b-28e44a4421c0__mathematical-expression-and-equation_0.jpg
+"\align* 1 7 & - 7 & & = & 1 0 \\ 1 7 & - 8 & ( - 7 - 1 ) & = & 9 \\ 1 7 & - 9 & ( - 7 - 2 ) & = & 8 \\ 1 7 & - 1 0 & ( - 7 - 3 ) & = & 7 \align*" ,3378af50-1030-11e5-ae7e-001018b5eb5c__mathematical-expression-and-equation_11.jpg
+"\sqrt { x } ;" ,084a3f02-40e4-11e1-1418-001143e3f55c__mathematical-expression-and-equation_5.jpg
+"R _ { x \prime } ( \phi , t ) = ( R _ { x \prime _ { 1 } } ( \phi , t ) , \dots , R _ { x \prime _ { n } } ( \phi , t ) ) ^ { T }" ,47babbfc-f33d-11e1-1154-001143e3f55c__mathematical-expression-and-equation_8.jpg
+"t > 0" ,08bd5ccd-570b-11e1-7459-001143e3f55c__mathematical-expression-and-equation_8.jpg
+"v = 1 . 2 0 m" ,28baa56d-6761-11e9-bca3-001999480be2__mathematical-expression-and-equation_12.jpg
+"n Z _ { 0 } z ^ { n - 1 } + ( n - 1 ) Z _ { 1 } z ^ { n - 2 } + \dots + Z _ { n - 1 } = 0" ,9c7b00d0-7d97-11e7-921c-5ef3fc9ae867__mathematical-expression-and-equation_2.jpg
+"( \frac { F _ { o } } { F _ { \iota 0 } } ) ^ { 2 } = 0 , 3 6 8 1 6" ,0011ea20-bc38-11e1-1211-001143e3f55c__mathematical-expression-and-equation_4.jpg
+"T = \epsilon \log \Delta ^ { \circ } + 1 ^ { \circ }" ,3d1f72ce-ec08-49a7-a9ec-45180104d497__mathematical-expression-and-equation_0.jpg
+"N = 0 ; \gamma ^ { * } = 1 / 3 0 ; \theta ^ { * } = 1 . 4 ; \mu = 0 . 3 ; \kappa = 1 0 ^ { - 2 } ; \lambda = 0 . 0 3 ; \rho = 1 / 6 0" ,153e6ec1-3c62-11e1-1589-001143e3f55c__mathematical-expression-and-equation_4.jpg
+"r x _ { 0 } + - p _ { 1 } \sigma = 0 ." ,27e827a1-df3d-11e1-1027-001143e3f55c__mathematical-expression-and-equation_5.jpg
+"A = \frac { P . x b } { a b }" ,7b144b10-04d9-11e5-91f2-005056825209__mathematical-expression-and-equation_2.jpg
+"6 m - 2 m = 4 m" ,1c278840-14e4-11e5-9192-001018b5eb5c__mathematical-expression-and-equation_3.jpg
+"\sin ( a x + \alpha ) \sin ( b x + \beta )" ,14fefca0-0a0b-11e3-9439-005056825209__mathematical-expression-and-equation_10.jpg
+"\frac { c } { b } \sqrt { b b - x x } ^ { 9 } )" ,3dd2f884-e41f-4c6a-aacf-062a1f31be33__mathematical-expression-and-equation_3.jpg
+"\sum _ { i = 1 } ^ { \infty } \psi _ { a } ( \alpha _ { i } )" ,0024baab-ac0b-11e1-5298-001143e3f55c__mathematical-expression-and-equation_1.jpg
+"\int _ { a } ^ { b } w ( x ) ( \int _ { a } ^ { x } p ^ { - 1 } ( t ) d t ) ^ { \eta } d x = \infty" ,01d92655-570b-11e1-7459-001143e3f55c__mathematical-expression-and-equation_0.jpg
+"\mu = \mu _ { 0 } = - \frac { 1 } { 2 } ( 2 + \gamma ) + \frac { 1 } { 2 } \sqrt { ( 4 + \gamma ^ { 2 } ) }" ,3b0c5023-408b-11e1-1586-001143e3f55c__mathematical-expression-and-equation_3.jpg
+"+ \frac { 9 } { 4 } ( 1 - \frac { | \Delta l | } { l } ) ^ { 2 } ] ^ { 1 / 2 }" ,19c168f7-7c23-4cd3-99dd-07ef5510ffe8__mathematical-expression-and-equation_2.jpg
+"\sin ^ { 2 } \phi = \frac { z ^ { 2 } } { a ^ { 2 } }" ,5fac6741-9c00-44ba-aebf-f1bc7045e74b__mathematical-expression-and-equation_8.jpg
+"2 . \genfrac { ( } { ) } { 0 p t } { } { 4 } { 3 } = 1 2 0 ." ,05f7ea40-40e4-11e1-1418-001143e3f55c__mathematical-expression-and-equation_8.jpg
+"+ | | V _ { 1 } | - | V _ { 2 } | | + | | V _ { 2 } | - | V _ { 3 } | | - | | V _ { 1 } | - | V _ { 3 } | | ," ,3b0c51c3-408b-11e1-1586-001143e3f55c__mathematical-expression-and-equation_4.jpg
+"t = \frac { t _ { 1 } - t _ { 2 } } { \ln r _ { 1 } - \ln r _ { 2 } } \cdot \ln r + \frac { t _ { 2 } \ln r _ { 1 } - t _ { 1 } \ln r _ { 2 } } { \ln r _ { 1 } - \ln r _ { 2 } }" ,0f6d5b4a-5308-11ea-8ddc-00155d012102__mathematical-expression-and-equation_6.jpg
+"A _ { 2 3 } = A _ { 2 1 }" ,249e4a60-3c62-11e1-1211-001143e3f55c__mathematical-expression-and-equation_7.jpg
+"\Delta Q = \frac { F _ { m } + F _ { m + 1 } } { 2 } \Delta u" ,79bf0e00-e3eb-11e2-b28b-001018b5eb5c__mathematical-expression-and-equation_0.jpg
+"a _ { 2 0 } + a _ { 2 1 } x _ { 0 } + a _ { 2 2 } y _ { 0 } + a _ { 2 3 } z _ { 0 } = 0" ,0a6a7c00-3e68-4146-a8c5-66b7dfef6b35__mathematical-expression-and-equation_9.jpg
+"A _ { 1 } B _ { 2 } - A _ { 2 } B _ { 1 } = 0" ,11f9f18c-3c62-11e1-8339-001143e3f55c__mathematical-expression-and-equation_3.jpg
+"[ p c c . 2 ] = [ p c c . 1 ] - \frac { [ p b c . 1 ] } { [ p b b . 1 ] } . [ p b c . 1 ] ," ,4a88cd72-c406-4784-a346-12a713d0c33c__mathematical-expression-and-equation_3.jpg
+"\vec { p } ( n ) = ( p _ { 1 } ( n ) , p _ { 2 } ( n ) , \dots , p _ { 5 } ( n ) )" ,53c77add-420f-11e1-8339-001143e3f55c__mathematical-expression-and-equation_4.jpg
+"t \in \mathbb { G } _ { 3 } ( x ) \cup \mathbb { G } _ { 4 } ( x )" ,0c39d9ee-570b-11e1-5298-001143e3f55c__mathematical-expression-and-equation_4.jpg
+"+ K e ^ { j \alpha } e ^ { \mathbf { p } _ { 2 } t } [ \frac { \mathbf { C } _ { R 2 } } { \mathbf { p } _ { 2 } } e ^ { j \beta } ( 1 - e ^ { - \mathbf { p } _ { 2 } t } ) - ( \frac { \mathbf { C } _ { S 1 } } { \mathbf { p } _ { 1 } } + \frac { \mathbf { C } _ { S 2 } } { \mathbf { p } _ { 2 } } )" ,1f799c43-3c62-11e1-7459-001143e3f55c__mathematical-expression-and-equation_2.jpg
+"d _ { 0 } = \sqrt { \frac { 1 6 P l _ { 2 } } { 3 \pi k _ { 3 } l } }" ,4e80e1af-c073-11e6-ae7e-001b63bd97ba__mathematical-expression-and-equation_1.jpg
+"q = \frac { 2 \cdot 1 1 0 \cdot 6 \cdot 7 } { 6 0 \cdot 4 } = 2 4 \cdot 4 m m ^ { 2 }" ,3a238330-d411-4242-89cd-bc02279b37a2__mathematical-expression-and-equation_0.jpg
+"t \ge t _ { 1 }" ,0c39d9ab-570b-11e1-5298-001143e3f55c__mathematical-expression-and-equation_11.jpg
+"T _ { 0 } = 4 ^ { s } , V _ { 0 } = 6 0 , \epsilon : 1 = 4 , r / T _ { 0 } ^ { 2 } = 0 . 0 4 \text { m m } / \text { s } ^ { 2 } ." ,64bae653-18ca-4083-aa51-7c473bab2de7__mathematical-expression-and-equation_0.jpg
+"\int _ { 0 } ^ { \infty } | \mu | ( Q _ { \tau } ) d \tau \le 2 ^ { N } \int _ { 0 } ^ { ( s r ) ^ { - p _ { j } } } v ^ { m } ( \tau ^ { - 1 / \beta _ { j } } , \mu ) d \tau = 2 ^ { N } \beta _ { j } \int _ { s r } ^ { \infty } v ^ { m } ( t , \mu ) t ^ { - \beta _ { j } - 1 } d t" ,39b0dff2-408b-11e1-1586-001143e3f55c__mathematical-expression-and-equation_1.jpg
+"- \frac { \pi \cdot d ^ { 3 } } { 6 } \cdot \rho _ { p } \cdot f \cdot \frac { ( C \cdot t - D ) } { [ 1 + \frac { B ^ { 2 } } { v _ { s } ^ { 2 } } ] ^ { \frac { 1 } { 2 } } } = 0" ,01b191b0-bc38-11e1-1586-001143e3f55c__mathematical-expression-and-equation_4.jpg
+"L _ { n } ^ { m } = q _ { 3 } P _ { n } ^ { m } ( \eta _ { 0 } ) Q _ { n } ^ { m } ( E _ { 2 } )" ,6f47237f-2895-479b-8f15-a7dec7c22569__mathematical-expression-and-equation_2.jpg
+"\le c [ \frac { 1 } { m } + \int _ { T } ^ { t } \frac { d s } { s \omega ( g ( s ) ) } + \int _ { T } ^ { t } \frac { | [ \omega ( g ( s ) ) ] \prime | } { \omega ^ { 2 } ( g ( s ) ) } d s ] < \infty" ,03921ff6-570b-11e1-1211-001143e3f55c__mathematical-expression-and-equation_1.jpg
+"\lim _ { x \rightarrow \infty } \epsilon f ( x ) = \infty" ,47babc60-f33d-11e1-1154-001143e3f55c__mathematical-expression-and-equation_6.jpg
+"\sqrt { v _ { 2 } } = \phi ^ { 2 } ( \epsilon ) + \theta ^ { 2 } \phi ^ { 2 } ( \alpha \epsilon ) + \dots + \theta ^ { 2 n - 2 } \phi ^ { 2 } ( \alpha ^ { n - 1 } \epsilon ) ," ,4791abf5-1bc8-42b4-821a-f98f5c6e8465__mathematical-expression-and-equation_7.jpg
+"d _ { 6 X 5 } = 1" ,3f277f27-408b-11e1-8339-001143e3f55c__mathematical-expression-and-equation_13.jpg
+"\langle ( \lambda _ { 1 } I - A ) z _ { n } , z _ { n } \rangle \ge ( \lambda _ { 1 } - M ) \parallel z _ { n } \parallel ^ { 2 }" ,37a928d0-408b-11e1-1586-001143e3f55c__mathematical-expression-and-equation_8.jpg
+"[ r , \frac { a ^ { 2 } } { b } , \frac { m a } { b } ]" ,68c27795-2993-442a-95f7-bd48a2dad6e8__mathematical-expression-and-equation_4.jpg
+"V ^ { ( d ) N } _ { ( t ) } = \sum _ { 1 } ^ { t } K P ^ { ( d ) } ( 1 + i ) ^ { ( t + 1 - K ) } + \sum _ { 1 } ^ { t } K q _ { ( K - 1 ) } V ^ { ( d ) N } _ { ( K ) } ( 1 + i ) ^ { ( t - K ) }" ,9e9bd45c-0e5a-11eb-b87e-005056a54372__mathematical-expression-and-equation_1.jpg
+"\align* 1 & = \frac { 6 0 } { 6 0 } \\ \frac { 1 } { 3 } & = \frac { 2 0 } { 6 0 } , \\ \frac { 1 } { } & - \frac { 1 0 } { } \align*" ,7b478b67-3637-11ec-8869-001b63bd97ba__mathematical-expression-and-equation_6.jpg
+"= 1 4 + 1 \frac { 1 } { 4 } + 1 = 1 6 \frac { 1 } { 4 }" ,2bf5673d-435f-11dd-b505-00145e5790ea__mathematical-expression-and-equation_40.jpg
+"c = c ^ { * } , e = e ^ { * } ," ,9686c344-4334-11e1-1331-001143e3f55c__mathematical-expression-and-equation_22.jpg
+"L \prime _ { 1 } = \frac { X _ { 1 2 } ( \frac { N _ { 3 } } { N _ { 1 } } ) ^ { 2 } + X _ { 1 3 } ( \frac { N _ { 3 } } { N _ { 1 } } ) ^ { 2 } - X _ { 2 3 } ( \frac { N _ { 3 } } { N _ { 2 } } ) ^ { 2 } } { 2 \omega }" ,0d342c41-3c62-11e1-1586-001143e3f55c__mathematical-expression-and-equation_4.jpg
+"| \Psi [ v _ { a } ] - \Psi [ v _ { b } ] | \le C | v _ { a } - v _ { b } | ," ,173d34b5-ea58-4b92-b017-33a2ad44288b__mathematical-expression-and-equation_3.jpg
+"F ( x , y , z , a ) = 0" ,5a75688e-9ea7-4484-aa62-7d33263db1f0__mathematical-expression-and-equation_7.jpg
+"\xi = r . t g . \alpha _ { o }" ,8ee789e0-de9d-11e7-8cdd-5ef3fc9bb22f__mathematical-expression-and-equation_2.jpg
+"s ^ { * } ( q ) = \xi ( q ) p ^ { * } + [ 1 - \xi ( q ) ] p ^ { 0 }" ,31eb76bd-4d7c-4478-a5fa-1a1a04fccc19__mathematical-expression-and-equation_1.jpg
+"( 2 \prime ) \{ \begin{array} { c } N _ { i k } = b _ { k i } + \sum _ { \rho = 1 } ^ { 3 } b _ { k , \rho } + s \tau _ { \rho ^ { i } } \\ \sum _ { \sigma = 1 } ^ { s } N _ { i \sigma } \tau _ { k \sigma } = b _ { k + s , i } + \sum _ { \rho = 1 } ^ { s } b _ { k + s , \rho } + s \tau _ { \rho ^ { i } } \end{array}" ,019971d0-40e4-11e1-1331-001143e3f55c__mathematical-expression-and-equation_1.jpg
+"\sum _ { k = 1 } ^ { t } \frac { 1 } { 4 \mu _ { k } S _ { k } } \{ A _ { 0 k } [ ( y _ { 1 k } - y _ { 2 k } ) ^ { 2 } + ( x _ { 1 k } - x _ { 2 k } ) ^ { 2 } ] +" ,23c01e57-3c62-11e1-7963-001143e3f55c__mathematical-expression-and-equation_13.jpg
+"a \le t \le b" ,47babc57-f33d-11e1-1154-001143e3f55c__mathematical-expression-and-equation_10.jpg
+"7 : 5 = \frac { 1 } { 1 6 5 } : \frac { 1 } { 2 4 5 }" ,60cc5dc9-d663-459f-9c4a-932061679bb7__mathematical-expression-and-equation_0.jpg
+"\vec { D } = [ \epsilon ] \epsilon _ { 0 } \vec { E }" ,1c1d502d-3c62-11e1-3052-001143e3f55c__mathematical-expression-and-equation_6.jpg
+"d = \sqrt { \frac { V } { 2 m p _ { s } } }" ,509a37e4-c99e-413f-adbf-c2c7d7c03000__mathematical-expression-and-equation_0.jpg
+"| ( ( \dot { x } , h ) ) _ { 0 } | \le M _ { 3 } \cdot \parallel h \parallel _ { L ^ { 2 } ( X ) }" ,0ee4af5d-570b-11e1-1589-001143e3f55c__mathematical-expression-and-equation_2.jpg
+"s = c \prime + c t" ,186f799f-6c55-450c-b493-d0e38aeced8e__mathematical-expression-and-equation_7.jpg
+"E _ { a } ^ { * } ( \omega ) = \mathcal { G } _ { 3 3 } ( \omega ) - \frac { 2 \mathcal { G } _ { 1 2 } ^ { 2 } ( \omega ) } { \mathcal { G } _ { 1 1 } ( \omega ) + \mathcal { G } _ { 1 2 } ( \omega ) }" ,1f7999f7-3c62-11e1-7459-001143e3f55c__mathematical-expression-and-equation_1.jpg
+"\frac { d x } { d \tau } = D [ A ( t ) x + C ( t ) x ( a ) + D ( t ) x ( b ) + \int _ { a } ^ { b } [ d _ { s } G ( t , s ) ] x ( s ) + f ( t ) ]" ,49b13f30-408b-11e1-1586-001143e3f55c__mathematical-expression-and-equation_1.jpg
+"X = f - \frac { 1 } { 2 ! } f \prime \prime - \frac { 1 } { 4 ! } f \prime \prime \prime \prime + \dots + \frac { ( - 1 ) ^ { n } } { ( 2 n ) ! } f ^ { ( 2 n ) } \pm \dots" ,62262c48-886a-435c-ae8a-f3cbe87f0fe1__mathematical-expression-and-equation_3.jpg
+"0 . 4 8 3 5 C O _ { 2 } = 6 2 . 3 2 \% C" ,157266ec-6edd-4370-b076-4c1645105ddf__mathematical-expression-and-equation_0.jpg
+"y _ { k } \prime = H ^ { v } e ^ { b H } \{ f \prime [ 1 + o ( 1 ) ] + f H \prime ( v H ^ { - 1 } + b ) [ 1 + o ( 1 ) ]" ,47ac1ff7-408b-11e1-8339-001143e3f55c__mathematical-expression-and-equation_7.jpg

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