Last-Bullet/MathBridge_Splitted · Datasets at Hugging Face

context_before stringlengths 0 199	equation stringlengths 5 80	context_after stringlengths 0 249	spoken_English stringlengths 0 119
since empty carriers have the same length	$ \sigma_ { \myCARRIER } \equiv\sigma_ { \myCCC } $	as the corresponding CCCs , a two-species picture is sufficient if one is only interested in information contained in	Sigma sub I C R I is equivalent to Sigma sub I C C.
43	$ ^ { +0.08 } _ { -0.06 } $		-0.06
The value of	$ k $	determines the size of the three subgroups , and the size of compliers	k
and	$ \psi_2=\psi_R $	. Additionally ,	psi sub 2 equals psi sub R
at	$ p $	( in the topology Eq . ( 3.1 ) of Ref .	p
a map	$ f $	is a fibration respectively a weak equivalence if and only if	f
It focuses on learning a parameterized function for mapping image	$ Y_i $	to its pose	Y sub i.
d is that for sufficiently large	$ x $	these diagrams require resummation before the first ( tree ) diagram becomes of order one . The Feynman-diagram method we use is less powerful than the noisy FKPP equation	x
00	$ \pm $	0.00 & 0.00	Plus or minus.
if a task of	$ s_j^E $	is not associated with any CS , then	s sub j superscript E
such that	$ \Hd ( F \cap \ell ) \geq s $	for all	H-d of the intersection of F and ell is greater than or equal to s.
-clockwise rotation of its external unit normal coincide with the tangent to the boundary of	$ B_\phi $	. Then the quantities above are computed as	B sub phi.
To obtain	$ \alpha $	we calculate	Alpha.
&	$ \gamma\leq\frac { 1 } { 1+\zeta/6 } $	&	Gamma is less than or equal to one divided by the quantity one plus zeta divided by six.
-	$ 32.96 $		32.96: None
is the probability of measuring	$ n $	photon in mode 1 . One can show ( see Appendix	n
is in the same situation as it was on	$ y $	( exactly one neighbor with color	y
to process	$ P_ { q } $	.	P sub q.
at the frequency	$ \omega $	. The sum of the squares of the individual I and Q residuals as opposed to just the sum of the I and Q residuals was used to prevent the	omega.
To obtain	$ S_ { ij } $	, we need to compute	S sub i j.
but not with respect to~	$ N $	. The fastest existing algorithm~	N
for large cavity loss rate	$ \kappa $	gives	kappa.
They estimated a	$ 50\ % $	uncertainty in the positron flux by comparing the predicted flux when	Fifty percent.
The generalized angular momentum quantum numbers	$ m_J $	of the different subgap states are also shown in the	m sub J
where	$ n=2K \cdot M $	with	n equals two K dot M
x	$ is mapped to a protein fold represented by $	y	is mapped to a protein fold represented by
The entire energy spectrum is independent of	$ \mu $	under OBC . In the following , we prove that	mu.
is equal to	$ 1 $	all BSs of different types	one
of	$ \QQ_v $	let	Q v.
If the potential outcome differs depending on whether a patient takes Moderna or Pfizer booster while both fall under treatment	$ A=1 $	, then this could lead to different outcomes . Having consistency assumption enables us to	A equals one.
to be filled completely with the false vacuum at time	$ t $	,	t
and similar analyses may be possible for the additional Hubbard-Stratonovich fields	$ \chi $	,	chi.
em	$ \theta_0 $	- The	Theta zero.
here subscript for	$ F $	stands for partial differentiation )	F
the sides of	$ \Sigma_g-G $	can be labelled following the orientation in such way that the two sides coming from the same edge of	Sigma sub g minus G.
when the chosen cutoff is	$ r $	above society 's preferred cutoff .	r
If	$ V $	and	V
to construct an	$ ( n + 1 ) $	-dimensional space of asymptotically translational vacuum Killing initial data . In Section~	n plus one.
	$ y_ { k } $	is named as a	y sub k
belongs to the subgroup of order	$ d $	and	d
\Cr adds	$ D $	to the formula	D
At rate	$ R=1 $	, the SNR gain over the O-FSSQ is 0.31 dB for	R equals one.
We can then consider the evolution of	$ \Pi $	, %	Pi.
	$ \chi^2/n $		chi squared over n.
for all	$ k\in\mathbb { Z } $	. Since	k belongs to the set of integers.
Here we are faceing with two cases	$ \delta\rho_ { m } > 0 $	and	Delta rho subscript m greater than zero.
of order	$ 2k $	, from which the standard CS theory ensures very likely a robust signal recovery from	Two k.
	$ L $	-2 }	L
for	$ \mathcal { H } $	such that	H: None
whose cardinality	$ \|\Xi\| $	is much larger than	the absolute value of Xi.
For each	$ i\in C $	there is some	i belongs to C
where	$ V $	is the mode volume , this occurs when the size of the device exceeds the characteristic localization length of the considered mode . Under these circumstances , the localized modes	V
and	$ \mathcal\ P ( p+1-s , q ) $	be the set of all partitions of	P of p plus 1 minus s, q
an accurate solution of metric variables is determined when the infinity norm	$ L_ { \infty } $	of the residual of the metric equation , namely , the maximum absolute value of the residual of the CFC equations , falls below a chosen tolerance . However ,	L infinity.
is an	$ L^q $	averaged version of the coefficients in	L to the power of q.
is the step-wise optimized controller to be executed at each	$ t $	that renders the occlusion-free set	t
with an expected value of	$ 0.4 $	at	0.4: None
The commutativity of~	$ d_H $	and~	d sub H.
and	$ j = 1 $	. Since	j equals one.
We observe that	$ M^ { * } _ { \rm op } $	is weakly dependent on	M raised to the power of op star.
may have many monomial preimages under	$ \varphi $	, a monomial in	phi.
is required for computing	$ \mathsf { G } $	and	G
changes little so we do not re-optimize this in larger	$ N $	but simply take the average from smaller sizes . On the other hand , we find	N
% where we set the sign so that	$ x < 0 $	is subsonic and	x is less than zero.
and	$ S $	of the same shape is defined by	S
69	$ \pm $	6.52	Plus or minus.
admits an	$ S $	-left adjoint ( that is , the underlying	S
_1	$ , $		comma
that contains all of the vertices of~	$ X $	) , and	X
denote the restriction of	$ \mu $	on	mu.
we consider	$ 100 $	such porphyrin films stacked one over the other along the	One hundred.
This is consistent with the simulations predicting a stronger response to	$ \Omega_ { \rm m } $	than the theory in Fig .	Omega sub m.
We choose a minimal parabolic	$ \wh { \cO } _i $	-subgroup	omega hat O sub i
	$ . For larger levels $		for large numbers.
is a projection and	$ a=1 $	. In	a equals one
Referring all these calculations back to the equation	$ ( \ref { evolution } ) $	, after cancellation we shall get :	ref delta.
561	$ \pm $	3e-3 & 0.685	Plus or minus.
=2 %	$ \forall $	i , j	for every
For each	$ r=1 , \ldots , \bar { N } $	,	r equals 1, dot dot dot, bar N.
is independent of	$ \kappa $	. That is , there exists a	kappa.
&	$ -1.04742 $		negative one point zero four seven four.
-differentiable in	$ t\in I $	, that is ,	t belongs to I.
of	$ G $	. For each	G
	$ E ( \lambda_ { i } ) $	and	E of lambda sub i
&	$ \alpha $	& PSNR	Alpha.
of	$ \varphi $	at	phi.
we can only prove that the Ginzburg Landau energy for	$ \bar { u } $	is controlled by	u bar.
as the H	$ \alpha $	reservoirs in BCGs are mainly filamentary . % The sizes of the H	Alpha.
where	$ J $	is given by	J
On the background Milne geometry the tensors	$ \Kc_ { ij } , \tau \Xi^i_j $	and	K sub i j, tau Xi superscript i subscript j.
and	$ f_a $	. In general , we treat	f sub a.
	$ k_\text { B } $	the Boltzmann constant . Noise is non-negligible when the typical frequency of the system , say	k sub B
computes both	$ \ { \rm\ CH^b ( X , -p-q ) $	and	Phi superscript b open parenthesis X comma negative p minus q close parenthesis
	$ j $		j
&	$ \pm1 $	,	Plus or minus one.
4	$ \pm $	1.9 & 50.5	Plus or minus.
	$ \pi_\phi $		pi sub phi.
For larger values of	$ \epsilon $	, the results improve until	epsilon.
and	$ \eta $	to indicate both the objects on	eta.
by	$ [ \a_j ] ( U ) \bar { N } _j $	. Here is a formal definition of	Open square bracket, a subscript j, close square bracket, open parenthesis, U, close parenthesis, bar, N subscript j.

since empty carriers have the same length

$ \sigma_ { \myCARRIER } \equiv\sigma_ { \myCCC } $

as the corresponding CCCs , a two-species picture is sufficient if one is only interested in information contained in

Sigma sub I C R I is equivalent to Sigma sub I C C.

43

$ ^ { +0.08 } _ { -0.06 } $

-0.06

The value of

$ k $

determines the size of the three subgroups , and the size of compliers

k

and

$ \psi_2=\psi_R $

. Additionally ,

psi sub 2 equals psi sub R

at

$ p $

( in the topology Eq . ( 3.1 ) of Ref .

p

a map

$ f $

is a fibration respectively a weak equivalence if and only if

f

It focuses on learning a parameterized function for mapping image

$ Y_i $

to its pose

Y sub i.

d is that for sufficiently large

$ x $

these diagrams require resummation before the first ( tree ) diagram becomes of order one . The Feynman-diagram method we use is less powerful than the noisy FKPP equation

x

00

$ \pm $

0.00 & 0.00

Plus or minus.

if a task of

$ s_j^E $

is not associated with any CS , then

s sub j superscript E

such that

$ \Hd ( F \cap \ell ) \geq s $

for all

H-d of the intersection of F and ell is greater than or equal to s.

-clockwise rotation of its external unit normal coincide with the tangent to the boundary of

$ B_\phi $

. Then the quantities above are computed as

B sub phi.

To obtain

$ \alpha $

we calculate

Alpha.

&

$ \gamma\leq\frac { 1 } { 1+\zeta/6 } $

&

Gamma is less than or equal to one divided by the quantity one plus zeta divided by six.

-

$ 32.96 $

32.96: None

is the probability of measuring

$ n $

photon in mode 1 . One can show ( see Appendix

n

is in the same situation as it was on

$ y $

( exactly one neighbor with color

y

to process

$ P_ { q } $

.

P sub q.

at the frequency

$ \omega $

. The sum of the squares of the individual I and Q residuals as opposed to just the sum of the I and Q residuals was used to prevent the

omega.

To obtain

$ S_ { ij } $

, we need to compute

S sub i j.

but not with respect to~

$ N $

. The fastest existing algorithm~

N

for large cavity loss rate

$ \kappa $

gives

kappa.

They estimated a

$ 50\ % $

uncertainty in the positron flux by comparing the predicted flux when

Fifty percent.

The generalized angular momentum quantum numbers

$ m_J $

of the different subgap states are also shown in the

m sub J

where

$ n=2K \cdot M $

with

n equals two K dot M

x

$ is mapped to a protein fold represented by $

y

is mapped to a protein fold represented by

The entire energy spectrum is independent of

$ \mu $

under OBC . In the following , we prove that

mu.

is equal to

$ 1 $

all BSs of different types

one

of

$ \QQ_v $

let

Q v.

If the potential outcome differs depending on whether a patient takes Moderna or Pfizer booster while both fall under treatment

$ A=1 $

, then this could lead to different outcomes . Having consistency assumption enables us to

A equals one.

to be filled completely with the false vacuum at time

$ t $

,

t

and similar analyses may be possible for the additional Hubbard-Stratonovich fields

$ \chi $

,

chi.

em

$ \theta_0 $

- The

Theta zero.

here subscript for

$ F $

stands for partial differentiation )

F

the sides of

$ \Sigma_g-G $

can be labelled following the orientation in such way that the two sides coming from the same edge of

Sigma sub g minus G.

when the chosen cutoff is

$ r $

above society 's preferred cutoff .

r

If

$ V $

and

V

to construct an

$ ( n + 1 ) $

-dimensional space of asymptotically translational vacuum Killing initial data . In Section~

n plus one.

$ y_ { k } $

is named as a

y sub k

belongs to the subgroup of order

$ d $

and

d

\Cr adds

$ D $

to the formula

D

At rate

$ R=1 $

, the SNR gain over the O-FSSQ is 0.31 dB for

R equals one.

We can then consider the evolution of

$ \Pi $

, %

Pi.

$ \chi^2/n $

chi squared over n.

for all

$ k\in\mathbb { Z } $

. Since

k belongs to the set of integers.

Here we are faceing with two cases

$ \delta\rho_ { m } > 0 $

and

Delta rho subscript m greater than zero.

of order

$ 2k $

, from which the standard CS theory ensures very likely a robust signal recovery from

Two k.

$ L $

-2 }

L

for

$ \mathcal { H } $

such that

H: None

whose cardinality

$ |\Xi| $

is much larger than

the absolute value of Xi.

For each

$ i\in C $

there is some

i belongs to C

where

$ V $

is the mode volume , this occurs when the size of the device exceeds the characteristic localization length of the considered mode . Under these circumstances , the localized modes

V

and

$ \mathcal\ P ( p+1-s , q ) $

be the set of all partitions of

P of p plus 1 minus s, q

an accurate solution of metric variables is determined when the infinity norm

$ L_ { \infty } $

of the residual of the metric equation , namely , the maximum absolute value of the residual of the CFC equations , falls below a chosen tolerance . However ,

L infinity.

is an

$ L^q $

averaged version of the coefficients in

L to the power of q.

is the step-wise optimized controller to be executed at each

$ t $

that renders the occlusion-free set

t

with an expected value of

$ 0.4 $

at

0.4: None

The commutativity of~

$ d_H $

and~

d sub H.

and

$ j = 1 $

. Since

j equals one.

We observe that

$ M^ { * } _ { \rm op } $

is weakly dependent on

M raised to the power of op star.

may have many monomial preimages under

$ \varphi $

, a monomial in

phi.

is required for computing

$ \mathsf { G } $

and

G

changes little so we do not re-optimize this in larger

$ N $

but simply take the average from smaller sizes . On the other hand , we find

N

% where we set the sign so that

$ x < 0 $

is subsonic and

x is less than zero.

and

$ S $

of the same shape is defined by

S

69

$ \pm $

6.52

Plus or minus.

admits an

$ S $

-left adjoint ( that is , the underlying

S

_1

$ , $

comma

that contains all of the vertices of~

$ X $

) , and

X

denote the restriction of

$ \mu $

on

mu.

we consider

$ 100 $

such porphyrin films stacked one over the other along the

One hundred.

This is consistent with the simulations predicting a stronger response to

$ \Omega_ { \rm m } $

than the theory in Fig .

Omega sub m.

We choose a minimal parabolic

$ \wh { \cO } _i $

-subgroup

omega hat O sub i

$ . For larger levels $

for large numbers.

is a projection and

$ a=1 $

. In

a equals one

Referring all these calculations back to the equation

$ ( \ref { evolution } ) $

, after cancellation we shall get :

ref delta.

561

$ \pm $

3e-3 & 0.685

Plus or minus.

=2 %

$ \forall $

i , j

for every

For each

$ r=1 , \ldots , \bar { N } $

,

r equals 1, dot dot dot, bar N.

is independent of

$ \kappa $

. That is , there exists a

kappa.

&

$ -1.04742 $

negative one point zero four seven four.

-differentiable in

$ t\in I $

, that is ,

t belongs to I.

of

$ G $

. For each

G

$ E ( \lambda_ { i } ) $

and

E of lambda sub i

&

$ \alpha $

& PSNR

Alpha.

of

$ \varphi $

at

phi.

we can only prove that the Ginzburg Landau energy for

$ \bar { u } $

is controlled by

u bar.

as the H

$ \alpha $

reservoirs in BCGs are mainly filamentary . % The sizes of the H

Alpha.

where

$ J $

is given by

J

On the background Milne geometry the tensors

$ \Kc_ { ij } , \tau \Xi^i_j $

and

K sub i j, tau Xi superscript i subscript j.

and

$ f_a $

. In general , we treat

f sub a.

$ k_\text { B } $

the Boltzmann constant . Noise is non-negligible when the typical frequency of the system , say

k sub B

computes both

$ \ { \rm\ CH^b ( X , -p-q ) $

and

Phi superscript b open parenthesis X comma negative p minus q close parenthesis

$ j $

j

&

$ \pm1 $

,

Plus or minus one.

4

$ \pm $

1.9 & 50.5

Plus or minus.

$ \pi_\phi $

pi sub phi.

For larger values of

$ \epsilon $

, the results improve until

epsilon.

and

$ \eta $

to indicate both the objects on

eta.

by

$ [ \a_j ] ( U ) \bar { N } _j $

. Here is a formal definition of

Open square bracket, a subscript j, close square bracket, open parenthesis, U, close parenthesis, bar, N subscript j.