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
& 1 &	$ - $	1.764 & 0.282 & 2 &	minus
for large	$ N $	,	N
of	$ P $	,	P
running from 1 to	$ 2D $	. If we define it as	Two D.
which are similar to the Chmutov -- Varchenko relations for	$ w_ { \mathfrak\ { sl ( 2 ) } $	are deduced . In~	w sub s l of 2
The kinetic matrix	$ K_ { \alpha\beta } $	is usually quadratic in the momentum . In the direct Fourier expansion of the SFT action	K sub alpha beta.
If	$ \hat { \varphi } $	is satisfiable , both	phi hat.
As	$ G-E ( F ) $	is of girth at least 9 , we obtain that	G minus E of F.
it is customary to neglect all	$ C_n $	's while only keeping the two leading-order squirming modes . Figure	C sub n.
where	$ \rho $	is the matter density . Whenever	rho.
by	$ f_n^ { j , i } $	. We shall illustrate the influence of the parameter	f sub n superscript j comma i
in the angular correlation function	$ C_\ell $	. Concretely , the parity asymmetry estimator is taken to be	C sub L.
Under the deviation mechanism	$ m '' $	, because the post-trial rate	m double prime.
In a single-server queue with feedback with probability	$ p $	the probability that an arriving job overtakes any job it finds in the queue is	p
	$ k $	-chain } . If	k
	$ n $	is the batch size . Specifically , we employ the non-saturating GAN loss function	n
since	$ W $	admits no essential lamination (	W
and	$ * $	represent the polarization component .	asterisk
in the convex hull of	$ C $	, then it is finite .	C
since	$ D_ { k+1 } = r $	, the next side around	D sub k plus one equals r.
are the atomic concentration and radius of	$ i^ { th } $	element , and	i to the power of th.
of positive real-valued increasing functions with domain	$ \mathbb { Z } _ { + } $	satisfying	Z positive.
	$ X ( t ) $	. The difference in distance between two adjacent clone boundaries ,	X of t.
For any	$ \epsilon > 0 $	, when	Epsilon is greater than zero.
	$ n $	} homogeneous linear forms .	n
we obtain the widths	$ dx $	and	d x.
The learning rate starts at	$ 0.01 $	and is decayed twice during training : after 50	zero point zero one.
bins and central-	$ q^2 $	bins in neutral B decay	q squared
expert locations	$ E $	, and their rankings	E
each	$ Q_l $	, where	Q sub l.
	$ 4 $	} ;	four
and	$ D $	should be interpreted as mathematical models for events . To clarify what we mean by this , we will treat the simplest possible case of an experiment involving two SG	D
in episode	$ i $	. DTS is the distance of the agent from the success threshold boundary when the episode ends . The boundary is set to	i
Let	$ A $	be a complete	A
and obtained from	$ F ' $	of	F prime.
&	$ 6.37 $	&	six point three seven.
but substituting the initial differential operator	$ d $	with the algebraic part	d
where	$ b ( r ) $	represents the shape function of the WH , which must obey the following conditions.	b of r
where	$ \varphi_\rho $	is the	Phi sub rho.
	$ < $	3	less than
in	$ ( X , d ) $	for some	X comma d.
	$ . The second uses [ O II ] $		Two times the quantity O sub I.
	$ A^- $	; finite subsets of	A minus.
for	$ t\in [ T ] $	. The informativeness of	t belongs to the set of T.
-	$ Z $	type , with which we achieve the interchanging of the logical operators . Indeed , both approaches can be thought of as exploiting domain walls if we think of transversal	Z
unconnected line	$ k_1 $	, it seems to be imposed in the form of	k sub 1
As we employ the coreset	$ \mathcal { S } _c $	and introduce the weights	S sub c.
	$ ^\textrm { \scriptsize 128 } $	,	128
The vertical dashed lines in the left hand plot denote the discontinuities of the double layer solutions at	$ z=z_c $	.	z equals z sub c.
	$ k $	in top-	k
for	$ \gamma < 1 $	, the observe that is far from the singularity measures a finite time for a freely falling particle to hit the singularity .	Gamma is less than one.
for any complex coefficients	$ a_n $	,	a sub n.
For any graph	$ G $	, we have	G
Let	$ \mathcal { C } $	be an abelian category . An inverse system consists of a pair	see
the computing cost of the full gradient of the function	$ H ( x , y ) $	is often prohibitively expensive . In order to overcome this difficulty , stochastic gradient approximations were applied , see , e.g .	H of x, y
whenever	$ e_ { ij } $	and	e sub i j.
	$ 0 $	--	zero
we expect it will have the	$ \alpha $	-PbO	Alpha.
	$ n $	-manifold without conjugate points , with visibility universal covering and with continuous Green bundles . Suppose the geodesic flow has a periodic hyperbolic point then quotient space	n
	$ , $	Y	comma
	$ n=50 $	; the squares show problems with	n equals 50
gives the upper-bound on the received information to the	$ k^\text { th } $	user . On the other hand , in the second case , consider that	k to the power of th
of order 1 and stop it when	$ X $	reaches a fixed ( small ) level	X
G &	$ 3.26 $	M &	three point two six.
	$ ^\textrm { \scriptsize 48 } $	,	48
Now we can take	$ \widetilde { \mu } $	to be	mu tilde.
with	$ x_0 $	=	x sub zero.
such that for all	$ \aA > 0 $	, any sufficiently large	a times A greater than zero.
	$ of the input variable $		of the input variable.
and defined the moduli space of pseudoholomorphic curves on the	$ \mathfrak { lcs } $	-fication	l-c-s
we focus on the numerical analysis of the time-dependence of	$ A_ { \alpha , a } $	for various	A sub alpha, a
& MN	$ E_6 $	& MN	E six.
Recall that for any	$ p \in [ 1 , \infty ) $	, the	p belongs to the interval from 1 to infinity, excluding 1.
and	$ y $	and	y
This is related to deformations of the hypercomplex structure on the fibers of	$ f $	, cf.~	f
and	$ \mathcal { T } $	symmetries are still broken in the macroscopic sense . As shown in Fig.~	tee
&	$ \mathcal { O } ( N_tN_x ) $	&	Big O of N sub t N sub x
and	$ q=p $	, which do not require the full power of	q equals p
	$ \mathcal { L } _ { \rm X-ray } $	, which is similar to Xspec	L sub X-ray.
to lighten the notation and assuming	$ d \neq 1 $	with no loss of generality , we posed	d is not equal to 1
We thus fix system designs	$ D_j $	, each obtained by a capacity expansion based on a weather year	D sub j.
We use the deflection angle	$ \alpha $	to measure the deflection of light . In the metric	Alpha.
required to reach an accuracy	$ \epsilon_1 $	with probability	Epsilon sub one.
which contradicts our assumption that	$ \tau_ { \Pi^+ ( u , C ( u ) ) } $	is non-deterministic , thereby implying	tau sub pi superscript plus open parenthesis u comma C open parenthesis u close parenthesis close parenthesis
have the same orders	$ 8 $	and	eight
write	$ [ \ifi ] $	for the cylinder set	square brackets, i
	$ 2n $	-	Two n.
is the inverse of the bijection	$ \mu $	. Finally , an inverse of a bijective minion homomorphism	mu.
is the delta function which evaluates to	$ 1 $	at	one
is the product of	$ n-1 $	oriented frame bundles over	n minus one
are computed with angular numbers	$ l , m $	and	l, m: None
There is generally a lot of similarity between the two methods for estimating	$ \dot { \nu } ( t ) $	. Some differences appear around glitches , e.g . in PSR B0740	dot nu of t
since	$ Z $	and the collection of eigenvalues of	Z
We note that given a	$ G $	-labelled diagram from Definition	G
While our analysis found	$ \xi=-0.00036^ { +0.00070 } _ { -0.00014 } $	and	Xi equals negative zero point zero zero three six raised to the power of plus zero point zero zero zero one four.
or	$ n=5 $	. In both cases	n equals 5
when an online vertex of type	$ i $	arrives at time	i
where	$ \lambda $	is the camera focal length , and	Lambda.
be a shape whose boundary contains the segment	$ S $	. Suppose that	S
Then	$ \Delta $	is also not stably rational .	Delta.

& 1 &

$ - $

1.764 & 0.282 & 2 &

minus

for large

$ N $

,

N

of

$ P $

,

P

running from 1 to

$ 2D $

. If we define it as

Two D.

which are similar to the Chmutov -- Varchenko relations for

$ w_ { \mathfrak\ { sl ( 2 ) } $

are deduced . In~

w sub s l of 2

The kinetic matrix

$ K_ { \alpha\beta } $

is usually quadratic in the momentum . In the direct Fourier expansion of the SFT action

K sub alpha beta.

If

$ \hat { \varphi } $

is satisfiable , both

phi hat.

As

$ G-E ( F ) $

is of girth at least 9 , we obtain that

G minus E of F.

it is customary to neglect all

$ C_n $

's while only keeping the two leading-order squirming modes . Figure

C sub n.

where

$ \rho $

is the matter density . Whenever

rho.

by

$ f_n^ { j , i } $

. We shall illustrate the influence of the parameter

f sub n superscript j comma i

in the angular correlation function

$ C_\ell $

. Concretely , the parity asymmetry estimator is taken to be

C sub L.

Under the deviation mechanism

$ m '' $

, because the post-trial rate

m double prime.

In a single-server queue with feedback with probability

$ p $

the probability that an arriving job overtakes any job it finds in the queue is

p

$ k $

-chain } . If

k

$ n $

is the batch size . Specifically , we employ the non-saturating GAN loss function

n

since

$ W $

admits no essential lamination (

W

and

$ * $

represent the polarization component .

asterisk

in the convex hull of

$ C $

, then it is finite .

C

since

$ D_ { k+1 } = r $

, the next side around

D sub k plus one equals r.

are the atomic concentration and radius of

$ i^ { th } $

element , and

i to the power of th.

of positive real-valued increasing functions with domain

$ \mathbb { Z } _ { + } $

satisfying

Z positive.

$ X ( t ) $

. The difference in distance between two adjacent clone boundaries ,

X of t.

For any

$ \epsilon > 0 $

, when

Epsilon is greater than zero.

$ n $

} homogeneous linear forms .

n

we obtain the widths

$ dx $

and

d x.

The learning rate starts at

$ 0.01 $

and is decayed twice during training : after 50

zero point zero one.

bins and central-

$ q^2 $

bins in neutral B decay

q squared

expert locations

$ E $

, and their rankings

E

each

$ Q_l $

, where

Q sub l.

$ 4 $

} ;

four

and

$ D $

should be interpreted as mathematical models for events . To clarify what we mean by this , we will treat the simplest possible case of an experiment involving two SG

D

in episode

$ i $

. DTS is the distance of the agent from the success threshold boundary when the episode ends . The boundary is set to

i

Let

$ A $

be a complete

A

and obtained from

$ F ' $

of

F prime.

&

$ 6.37 $

&

six point three seven.

but substituting the initial differential operator

$ d $

with the algebraic part

d

where

$ b ( r ) $

represents the shape function of the WH , which must obey the following conditions.

b of r

where

$ \varphi_\rho $

is the

Phi sub rho.

$ < $

3

less than

in

$ ( X , d ) $

for some

X comma d.

$ . The second uses [ O II ] $

Two times the quantity O sub I.

$ A^- $

; finite subsets of

A minus.

for

$ t\in [ T ] $

. The informativeness of

t belongs to the set of T.

-

$ Z $

type , with which we achieve the interchanging of the logical operators . Indeed , both approaches can be thought of as exploiting domain walls if we think of transversal

Z

unconnected line

$ k_1 $

, it seems to be imposed in the form of

k sub 1

As we employ the coreset

$ \mathcal { S } _c $

and introduce the weights

S sub c.

$ ^\textrm { \scriptsize 128 } $

,

128

The vertical dashed lines in the left hand plot denote the discontinuities of the double layer solutions at

$ z=z_c $

.

z equals z sub c.

$ k $

in top-

k

for

$ \gamma < 1 $

, the observe that is far from the singularity measures a finite time for a freely falling particle to hit the singularity .

Gamma is less than one.

for any complex coefficients

$ a_n $

,

a sub n.

For any graph

$ G $

, we have

G

Let

$ \mathcal { C } $

be an abelian category . An inverse system consists of a pair

see

the computing cost of the full gradient of the function

$ H ( x , y ) $

is often prohibitively expensive . In order to overcome this difficulty , stochastic gradient approximations were applied , see , e.g .

H of x, y

whenever

$ e_ { ij } $

and

e sub i j.

$ 0 $

--

zero

we expect it will have the

$ \alpha $

-PbO

Alpha.

$ n $

-manifold without conjugate points , with visibility universal covering and with continuous Green bundles . Suppose the geodesic flow has a periodic hyperbolic point then quotient space

n

$ , $

Y

comma

$ n=50 $

; the squares show problems with

n equals 50

gives the upper-bound on the received information to the

$ k^\text { th } $

user . On the other hand , in the second case , consider that

k to the power of th

of order 1 and stop it when

$ X $

reaches a fixed ( small ) level

X

G &

$ 3.26 $

M &

three point two six.

$ ^\textrm { \scriptsize 48 } $

,

48

Now we can take

$ \widetilde { \mu } $

to be

mu tilde.

with

$ x_0 $

=

x sub zero.

such that for all

$ \aA > 0 $

, any sufficiently large

a times A greater than zero.

$ of the input variable $

of the input variable.

and defined the moduli space of pseudoholomorphic curves on the

$ \mathfrak { lcs } $

-fication

l-c-s

we focus on the numerical analysis of the time-dependence of

$ A_ { \alpha , a } $

for various

A sub alpha, a

& MN

$ E_6 $

& MN

E six.

Recall that for any

$ p \in [ 1 , \infty ) $

, the

p belongs to the interval from 1 to infinity, excluding 1.

and

$ y $

and

y

This is related to deformations of the hypercomplex structure on the fibers of

$ f $

, cf.~

f

and

$ \mathcal { T } $

symmetries are still broken in the macroscopic sense . As shown in Fig.~

tee

&

$ \mathcal { O } ( N_tN_x ) $

&

Big O of N sub t N sub x

and

$ q=p $

, which do not require the full power of

q equals p

$ \mathcal { L } _ { \rm X-ray } $

, which is similar to Xspec

L sub X-ray.

to lighten the notation and assuming

$ d \neq 1 $

with no loss of generality , we posed

d is not equal to 1

We thus fix system designs

$ D_j $

, each obtained by a capacity expansion based on a weather year

D sub j.

We use the deflection angle

$ \alpha $

to measure the deflection of light . In the metric

Alpha.

required to reach an accuracy

$ \epsilon_1 $

with probability

Epsilon sub one.

which contradicts our assumption that

$ \tau_ { \Pi^+ ( u , C ( u ) ) } $

is non-deterministic , thereby implying

tau sub pi superscript plus open parenthesis u comma C open parenthesis u close parenthesis close parenthesis

have the same orders

$ 8 $

and

eight

write

$ [ \ifi ] $

for the cylinder set

square brackets, i

$ 2n $

-

Two n.

is the inverse of the bijection

$ \mu $

. Finally , an inverse of a bijective minion homomorphism

mu.

is the delta function which evaluates to

$ 1 $

at

one

is the product of

$ n-1 $

oriented frame bundles over

n minus one

are computed with angular numbers

$ l , m $

and

l, m: None

There is generally a lot of similarity between the two methods for estimating

$ \dot { \nu } ( t ) $

. Some differences appear around glitches , e.g . in PSR B0740

dot nu of t

since

$ Z $

and the collection of eigenvalues of

Z

We note that given a

$ G $

-labelled diagram from Definition

G

While our analysis found

$ \xi=-0.00036^ { +0.00070 } _ { -0.00014 } $

and

Xi equals negative zero point zero zero three six raised to the power of plus zero point zero zero zero one four.

or

$ n=5 $

. In both cases

n equals 5

when an online vertex of type

$ i $

arrives at time

i

where

$ \lambda $

is the camera focal length , and

Lambda.

be a shape whose boundary contains the segment

$ S $

. Suppose that

S

Then

$ \Delta $

is also not stably rational .

Delta.