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
	$ \lambda_ { max } $	, and	Lambda sub max.
99	$ \pm $	0.15 & 1.00	Plus or minus.
and	$ \Gamma $	on spheres , deforming the boundary action of	Gamma.
settle and the similarity between features can be uniformly measured by	$ \phi ( \cdot , \cdot ) $	. BN and ReLU are the batch normalization and ReLU activation layers .	phi of dot comma dot.
and	$ c $	becomes slightly larger . Therefore , the inter-layer coupling is more pronounced in the bulk.	c
To understand	$ \sigma $	intuitively , we first note that 1-minus each token 's unigram probability :	Sigma.
and	$ j \in [ n ] $	. Without loss of generality , we assume	j belongs to the set of integers from 1 to n.
% for % some constant	$ c > 0 $	depending only on	c greater than zero
there exists	$ K \in \N $	such that	K belongs to the set of natural numbers.
The dominant instability at given	$ \mu $	and	mu.
_2	$ ) $		closing parenthesis
we see that the appearance of the peak at	$ k = 1 $	precedes that of the flattening of the structure factor near the origin , and that there exists a range of temperatures for which the peak exists but the structure factor	k equals one.
-norm of	$ x $	, see	x
where	$ \gamma_ { ry } =0 $	indicates full discount and	gamma sub r y equals zero.
surface states present AM because L	$ _1 $	does not merge with L	One.
as	$ x ' $	. Based on how we have defined the partitions in	the derivative of x.
if	$ V $	satisfies the following conditions :	V
For a given scenario	$ s\in \mathcal { S } $	, the inequalities	s belongs to the set S.
and	$ j $	is determined by the shortest path between them . The efficiency is then defined as the inverse of the shortest path length :	j
The top left panel shows the effect of changing	$ \Omega_m $	. For low	Omega m.
&	$ R_ { 8 } $	&	R sub 8.
since each of two cycles of	$ G $	is in	G
06 & 374	$ \pm $	31 & -0.58	Plus or minus.
write	$ f = [ \phi ] _ { \wh M } $	and	f equals the phi sub Q M.
we design a process with a global autoregressive effect of order	$ p = 1 $	. Also , the stage-1 neighbourhood is considered together with two exogenous regressors and corresponding time lags	p equals one.
	$ 1 $	} &	one
fix a generator	$ h_i $	of the cyclic subgroup	h sub i.
plots the	$ u-r $	color versus the stellar mass for NGC 7392 and for the latest sample of TDE candidates from the ZTF survey	u minus r.
-	$ y $	cross-section area , which leads to	y
in	$ G $	.	G
	$ F_2 $	=	F sub 2.
em	$ f_\theta $	transforms each element of the partition to	f sub theta
belongs to	$ \bb_\eta $	.	b sub eta
we say that the odd girth of	$ G $	is the length of the smallest odd cycle . If there are no odd cycles , we say that the odd girth is infinite .	G
Let	$ R $	be a monomial subring . Fix	R
denote the total length of the hospitals ' preference lists in	$ J $	. We claim that	J
-constructible sheaves on	$ X $	is denoted by	X
fermions coupling to	$ H_1 $	and	H sub 1.
&	$ \frac32 \mathbf { k } \cdot \mathbf { k } ^\prime $	&	Fraction 32 k dot k prime.
The result shows a remarkably small dependence on the renormalization-scale parameter	$ X $	( see Fig.~	X
or	$ \pm $	~20	Plus or minus.
Observe that	$ e \in { \prod } _ { k \in l } { { 2 } ^ { \Pset ( k ) } } $	and that for all	e belongs to the Cartesian product over k in l of 2 raised to the power of the P set of k.
of	$ ( r_ { t } ) _ { t\in [ 0 , T ] } $	is independent of	r sub t, sub t in the interval from 0 to T.
& 2 & 2 & N &	$ ( 0,0 ) $	&	zero comma zero.
on	$ \hat { X } $	, and	X hat.
probability of	$ \fn $	to obtain the	eff en
For	$ N=1 $	we have a	N equals one.
to	$ \hat { P } ( i ) $	, for	P hat of i
~Peng	$ ^ { 71,58 } $	, K.~Peters	to the power of seventy-one, fifty-eight.
The MV model result is applied at the initial scale	$ Q_0 $	=3 GeV . The gluon	Q zero.
strong decays and branching ratios of	$ b\overline { b } $	meson upto higher states with	b bar b
the catalogue contains all of the brightest galaxies giving 90\ % of the total	$ B $	-band and	B
increases a bit to	$ 0.55\pi $	. Now , when	0.55 times pi.
and	$ g_i $	denotes an	g sub i.
small than	$ \min\ { \epsilon_D,1\ } /10 $	, where	Minimum of epsilon sub D, 1 divided by 10.
Let	$ Z_0=\cL_y^ { max } ( \F^h ) \subset X $	be the fiber at the central point	Z sub zero equals c times L sub y to the power of max times F to the power of h, subset X.
by less than	$ 20\ % $	. This is a trivial consequence of the fact that for large values of	Twenty percent.
	$ ^\textrm { \scriptsize 62b } $	,	62b
This type of modified theories are usually known as	$ f ( R ) $	theories , and there exists a vast literature on these theories , for which a review on them is beyond our scope . Nevertheless , a construction of such theories	f of R
These traveling waves are `` running '' in the positive or negative	$ x $	-direction depending on the sign of	x
V	$ } , which is obtained from $	D	where there exists a variable from.
and	$ \Omega_ { \Lambda } $	is the dark energy density parameter , i.e . , the parameters of the standard	Omega sub Lambda.
that	$ e_i $	is an edge of	e sub i.
for	$ n = 1 , 2 , \dots $	, then	n equals one, two, ellipsis.
color temperature	$ T_ { \rm C } $	through the Wouthuysenâ€ “ Field coupling	T sub C.
has a definite value or magnitude at some initial time	$ 0 $	. Then the subject system must be in a specific configuration with probability	zero
on	$ \Sigma $	is a finite collection	Sigma.
It must be noted that the CPR might change if	$ P $	is too high due to non-equilibrium effects~	P
&	$ \checkmark $	&	Checkmark.
is	$ \Pi_1 $	-conservative over	Pi sub 1.
updates	$ G $	whenever the rule in ( 1 ) is satisfied by the temporal pattern of events , i.e . ,	G
top anion layer with the same concentration	$ x $	as in the underlying	x
-covered by the set	$ \calB_1 $	, i.e . ,	B sub 1.
&	$ 0 $	&	zero
The following lemma shows we can solve for	$ F $	in the above equation , with high probability .	F
of	$ q $	nodes are leaders , and the remaining	q
be the tangent vector at	$ x_0 $	to the unit speed geodesic	x sub zero.
and	$ p^ { n+1 } $	as follows :	p to the power of n plus 1.
since it is impossible for more than	$ s $	candidates to have more than this number of votes ) .	s
If	$ \vecF_0 $	is an	vector F sub zero.
	$ F $	is a finite extension of	F
	$ \zeta_l $	is the 1d FI parameter associated with	zeta sub l.
and	$ b $	are both dimensionless constants . In particular ,	b
we can observe this by measuring the	$ L^ { \infty } $	and	L to the power of infinity.
where each %	$ \beta_i $	is an SCS of	Beta sub i.
Thus it suffices to show that if	$ C $	is any connected component of	C
be the vertex of	$ C_j $	which is adjacent to both	C sub j.
&	$ 0 $	&	zero
The goal of the agent is then to find an optimal policy	$ \pi^ * $	satisfying	pi star.
Let the trial data	$ Y $	with the fill distance	Y
	$ \bm { \beta } ^ * $	denoting	beta star.
where	$ e_ { ij } $	are the	e sub i j.
-invariant	$ 1 $	-form ( or even a contact form ) , but this is clearly in error . Lutz also discusses the classification up to equivariant diffeomorphism .	one
:326	$ - $	31 . %	minus
-class estimators that takes	$ ( \tilde { y } , \tilde { X } , \tilde { Z } ) $	is proportional to	Open parenthesis, tilde y, comma, tilde X, comma, tilde Z, close parenthesis.
The degrees of	$ G $	are then	G
the evaluation of significance of LSP peaks	$ - $	commonly called False Alarm Probabilities	minus
We have assumed each orbital is in the Fermi liquid state with a bandwidth governed by	$ \rho_r $	. Therefore , doing the integral and replacing	rho sub r.
We crawl over	$ 20 $	k timelines and collect over	Twenty.
Of another interesting point is that we found the range of critical polarization values for the appearance of vortices in the momentum spectra for different cycles with	$ \Delta\phi=0 $	and	Delta phi equals zero.

$ \lambda_ { max } $

, and

Lambda sub max.

99

$ \pm $

0.15 & 1.00

Plus or minus.

and

$ \Gamma $

on spheres , deforming the boundary action of

Gamma.

settle and the similarity between features can be uniformly measured by

$ \phi ( \cdot , \cdot ) $

. BN and ReLU are the batch normalization and ReLU activation layers .

phi of dot comma dot.

and

$ c $

becomes slightly larger . Therefore , the inter-layer coupling is more pronounced in the bulk.

c

To understand

$ \sigma $

intuitively , we first note that 1-minus each token 's unigram probability :

Sigma.

and

$ j \in [ n ] $

. Without loss of generality , we assume

j belongs to the set of integers from 1 to n.

% for % some constant

$ c > 0 $

depending only on

c greater than zero

there exists

$ K \in \N $

such that

K belongs to the set of natural numbers.

The dominant instability at given

$ \mu $

and

mu.

_2

$ ) $

closing parenthesis

we see that the appearance of the peak at

$ k = 1 $

precedes that of the flattening of the structure factor near the origin , and that there exists a range of temperatures for which the peak exists but the structure factor

k equals one.

-norm of

$ x $

, see

x

where

$ \gamma_ { ry } =0 $

indicates full discount and

gamma sub r y equals zero.

surface states present AM because L

$ _1 $

does not merge with L

One.

as

$ x ' $

. Based on how we have defined the partitions in

the derivative of x.

if

$ V $

satisfies the following conditions :

V

For a given scenario

$ s\in \mathcal { S } $

, the inequalities

s belongs to the set S.

and

$ j $

is determined by the shortest path between them . The efficiency is then defined as the inverse of the shortest path length :

j

The top left panel shows the effect of changing

$ \Omega_m $

. For low

Omega m.

&

$ R_ { 8 } $

&

R sub 8.

since each of two cycles of

$ G $

is in

G

06 & 374

$ \pm $

31 & -0.58

Plus or minus.

write

$ f = [ \phi ] _ { \wh M } $

and

f equals the phi sub Q M.

we design a process with a global autoregressive effect of order

$ p = 1 $

. Also , the stage-1 neighbourhood is considered together with two exogenous regressors and corresponding time lags

p equals one.

$ 1 $

} &

one

fix a generator

$ h_i $

of the cyclic subgroup

h sub i.

plots the

$ u-r $

color versus the stellar mass for NGC 7392 and for the latest sample of TDE candidates from the ZTF survey

u minus r.

-

$ y $

cross-section area , which leads to

y

in

$ G $

.

G

$ F_2 $

=

F sub 2.

em

$ f_\theta $

transforms each element of the partition to

f sub theta

belongs to

$ \bb_\eta $

.

b sub eta

we say that the odd girth of

$ G $

is the length of the smallest odd cycle . If there are no odd cycles , we say that the odd girth is infinite .

G

Let

$ R $

be a monomial subring . Fix

R

denote the total length of the hospitals ' preference lists in

$ J $

. We claim that

J

-constructible sheaves on

$ X $

is denoted by

X

fermions coupling to

$ H_1 $

and

H sub 1.

&

$ \frac32 \mathbf { k } \cdot \mathbf { k } ^\prime $

&

Fraction 32 k dot k prime.

The result shows a remarkably small dependence on the renormalization-scale parameter

$ X $

( see Fig.~

X

or

$ \pm $

~20

Plus or minus.

Observe that

$ e \in { \prod } _ { k \in l } { { 2 } ^ { \Pset ( k ) } } $

and that for all

e belongs to the Cartesian product over k in l of 2 raised to the power of the P set of k.

of

$ ( r_ { t } ) _ { t\in [ 0 , T ] } $

is independent of

r sub t, sub t in the interval from 0 to T.

& 2 & 2 & N &

$ ( 0,0 ) $

&

zero comma zero.

on

$ \hat { X } $

, and

X hat.

probability of

$ \fn $

to obtain the

eff en

For

$ N=1 $

we have a

N equals one.

to

$ \hat { P } ( i ) $

, for

P hat of i

~Peng

$ ^ { 71,58 } $

, K.~Peters

to the power of seventy-one, fifty-eight.

The MV model result is applied at the initial scale

$ Q_0 $

=3 GeV . The gluon

Q zero.

strong decays and branching ratios of

$ b\overline { b } $

meson upto higher states with

b bar b

the catalogue contains all of the brightest galaxies giving 90\ % of the total

$ B $

-band and

B

increases a bit to

$ 0.55\pi $

. Now , when

0.55 times pi.

and

$ g_i $

denotes an

g sub i.

small than

$ \min\ { \epsilon_D,1\ } /10 $

, where

Minimum of epsilon sub D, 1 divided by 10.

Let

$ Z_0=\cL_y^ { max } ( \F^h ) \subset X $

be the fiber at the central point

Z sub zero equals c times L sub y to the power of max times F to the power of h, subset X.

by less than

$ 20\ % $

. This is a trivial consequence of the fact that for large values of

Twenty percent.

$ ^\textrm { \scriptsize 62b } $

,

62b

This type of modified theories are usually known as

$ f ( R ) $

theories , and there exists a vast literature on these theories , for which a review on them is beyond our scope . Nevertheless , a construction of such theories

f of R

These traveling waves are `` running '' in the positive or negative

$ x $

-direction depending on the sign of

x

V

$ } , which is obtained from $

D

where there exists a variable from.

and

$ \Omega_ { \Lambda } $

is the dark energy density parameter , i.e . , the parameters of the standard

Omega sub Lambda.

that

$ e_i $

is an edge of

e sub i.

for

$ n = 1 , 2 , \dots $

, then

n equals one, two, ellipsis.

color temperature

$ T_ { \rm C } $

through the Wouthuysenâ€ “ Field coupling

T sub C.

has a definite value or magnitude at some initial time

$ 0 $

. Then the subject system must be in a specific configuration with probability

zero

on

$ \Sigma $

is a finite collection

Sigma.

It must be noted that the CPR might change if

$ P $

is too high due to non-equilibrium effects~

P

&

$ \checkmark $

&

Checkmark.

is

$ \Pi_1 $

-conservative over

Pi sub 1.

updates

$ G $

whenever the rule in ( 1 ) is satisfied by the temporal pattern of events , i.e . ,

G

top anion layer with the same concentration

$ x $

as in the underlying

x

-covered by the set

$ \calB_1 $

, i.e . ,

B sub 1.

&

$ 0 $

&

zero

The following lemma shows we can solve for

$ F $

in the above equation , with high probability .

F

of

$ q $

nodes are leaders , and the remaining

q

be the tangent vector at

$ x_0 $

to the unit speed geodesic

x sub zero.

and

$ p^ { n+1 } $

as follows :

p to the power of n plus 1.

since it is impossible for more than

$ s $

candidates to have more than this number of votes ) .

s

If

$ \vecF_0 $

is an

vector F sub zero.

$ F $

is a finite extension of

F

$ \zeta_l $

is the 1d FI parameter associated with

zeta sub l.

and

$ b $

are both dimensionless constants . In particular ,

b

we can observe this by measuring the

$ L^ { \infty } $

and

L to the power of infinity.

where each %

$ \beta_i $

is an SCS of

Beta sub i.

Thus it suffices to show that if

$ C $

is any connected component of

C

be the vertex of

$ C_j $

which is adjacent to both

C sub j.

&

$ 0 $

&

zero

The goal of the agent is then to find an optimal policy

$ \pi^ * $

satisfying

pi star.

Let the trial data

$ Y $

with the fill distance

Y

$ \bm { \beta } ^ * $

denoting

beta star.

where

$ e_ { ij } $

are the

e sub i j.

-invariant

$ 1 $

-form ( or even a contact form ) , but this is clearly in error . Lutz also discusses the classification up to equivariant diffeomorphism .

one

:326

$ - $

31 . %

minus

-class estimators that takes

$ ( \tilde { y } , \tilde { X } , \tilde { Z } ) $

is proportional to

Open parenthesis, tilde y, comma, tilde X, comma, tilde Z, close parenthesis.

The degrees of

$ G $

are then

G

the evaluation of significance of LSP peaks

$ - $

commonly called False Alarm Probabilities

minus

We have assumed each orbital is in the Fermi liquid state with a bandwidth governed by

$ \rho_r $

. Therefore , doing the integral and replacing

rho sub r.

We crawl over

$ 20 $

k timelines and collect over

Twenty.

Of another interesting point is that we found the range of critical polarization values for the appearance of vortices in the momentum spectra for different cycles with

$ \Delta\phi=0 $

and

Delta phi equals zero.