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
we do not include a weight function applied to each	$ \bx_i $	in the latter . A weight function can be used to decrease the weight of high-leverage points . Such a function can be included in our approach when viewed as	x sub i.
including	$ 92\ % $	of targets at	92 percent.
and	$ j^\delta $	for	j raised to the power of delta
	$ d_ { t , x , y } $	. The time step is defined as	dee sub tee, eks, why
in	$ \mathcal { R } $	. As	capital R
of	$ v $	with respect to	v
We varied these design parameters approximately around the values estimated in HPTN 037 to assess the sensitivity of	$ K $	to these changes . In particular , we set the the effect size of both the AIE and the ASpE to 0.5 times , 1 times , and 1.5 times	K
as we are assuming that	$ G $	is torsion-free ,	G
In the following proposition we aim to show that if a quantum state	$ \rho $	satisfies the quantum Poincar	rho.
in	$ \mathcal { G } $	and	G
for	$ 0 < \alpha\leq 1 $	, we have	0 is less than alpha and alpha is less than or equal to 1.
and	$ \Gamma_2 $	with	Gamma 2.
We define the energy functional	$ M ( t ) $	as follows	M of t.
and	$ \mu ( g,0 ) \propto g $	. The weak-field expansion in Eq.~	Mu of g comma zero is proportional to g.
That is	$ \mathbb { E } [ \frac { 1 } { m } A^TA ] =I_d $	. Now we consider the variance of each	The expectation of one over m times A transpose equals the identity matrix.
the percentage of trajectories that exceed the cost limit	$ K $	decreases ( last column ) , and the expected negated reward increases . The statistical properties of the total cost incurred by our method under different risk level	K
where	$ M $	is some ( unknown ) constant .	M
	$ N $	) between an inner region where	N
and the image scale of	$ 0.15 $	arcsec pixel	0.15: None
illustrates the energy values obtained using the AV8	$ ' $	+central 3N potential~	apostrophe
and operators	$ \phi ( f ) $	also aligns with our intuition that we can measure the amplitude of a given mode at different points in time and/or space .	phi of f.
we obtain	$ CZ_\tau ( \gamma_1 ) = CZ_\tau ( \overline { \gamma } _1 ) \geq 7 $	. Thus ,	CZ sub tau of gamma 1 equals CZ sub tau of the conjugate of gamma 1, greater than or equal to 7.
the	$ s_d $	and	s subscript d.
with simple gadgetry to get the desired graph	$ G ' $	where we can prove the correspondence between independent sets in	the derivative of G.
and	$ I^A\equiv \gamma^ { AB } I_B $	. The structure constants are defined by	I to the power of A is equivalent to gamma to the power of A B times I sub B.
diseases explain	$ 4,000 $	findings . The probability of a finding given diseases is expressed by Equation	Four thousand.
	$ \pi_t $	satisfies	Pi sub t.
The ensemble size is	$ N = 10^3 $	. We fix	N equals ten to the power of three.
with support contained in	$ S $	that satisfy	S
and	$ c_ { 2 } $	with their corresponding entanglement quantified by means of the Gaussian R	c sub 2.
designing high-dimensional sparse codebooks for simultaneously supporting devices for a large	$ K $	is quite challenging , since the sparse teletraffic of massive communication can not be readily exploited . Considering the sparse codebooks designed in~	K
we mainly discuss the zero	$ E $	. The triangles , circles , + and x marks represent the cases	E
	$ , $	{	comma
	$ P_2 $	,	P sub 2
and	$ Q_l $	equal to	Q sub l.
of an arbitrary integrable function	$ G_ { d } $	over a d-dimensional polyhedron	G sub d.
As	$ A $	is a fibration , there exists , for each	A
and let	$ S_ { X } $	be a vector bundle over	S sub X.
the monotone submodular function	$ g_ { d } $	associated with	g subscript d.
has the same dp-rank as	$ R $	.	R
k	$ -plane trees and $	k	Negative plane and
The 2H stacking is obtained by a	$ 180^ { \circ } $	rotation about the	180 degrees
	$ l_ { 1,2 } =r_ { 1,2 } -r_ { ps } $	. In the Schwarzschild case , this number equals	L sub one comma two equals r sub one comma two minus r sub p s.
for	$ i=1,2,3,4 $	. We recapitulate the above relations with the diagram ,	i equals one, two, three, four.
By the fact that	$ X $	is a vertex set with binding number	X
and	$ \Gamma^ * $	are the conjugates of	Gamma star.
The overall inter-rater agreement in terms of Fleiss	$ \kappa $	was 0.22 which corresponds to a fair agreement . The	kappa.
\\ \hline	$ 3 $	&	three
A lower value of	$ K $	or	K
for	$ 1 \leq n \leq 4 $	are given in Eqn.	One is less than or equal to n, which is less than or equal to four.
with	$ i=\ { A , B\ } $	, are chosen according to the Akaike Information Criterion	i equals the set containing A and B.
in the asymptotic regime as	$ n\to\infty $	.	n tends to infinity
as functions of the radius	$ r $	. The EATS used here reads	r
we define its volume as the sum of degrees of nodes in	$ A $	, that is ,	A
exists for all	$ t > 0 $	. If	t greater than zero
	$ 1 $	&	one
each denoted by	$ L_i [ f ] : \mathcal { F } \mapsto \mathbb { R } $	and such that	L sub i of f, colon F maps to R
or	$ E_8 $	, let	E sub 8.
Consider the deformation of a body initially occupying a domain	$ \Omega $	through a displacement field	Omega.
	$ U $	and mark embeddings	U
-sub-Gaussian for some	$ K > 0 $	whenever	K is greater than zero.
at times	$ r_1 , r_2 $	. Therefore by Lemma	r sub 1, r sub 2
can provide it with probability	$ p $	. When cooperation is substitutable ,	p
016	$ \pm $		Plus or minus.
maps the problem of the reconstruction of the higher-order interactions into that of solving a system of algebraic equations in the unknown variables given by the	$ H $	entries of	H
em	$ \partial \Omega^ { 3 , - } \cap B_r ( x_s ) $	is the unique	Partial Omega to the power of three, minus, intersection B sub r of x sub s.
be an ample line bundle on	$ T $	with	T
	$ 0.25 $	,	0.25: None
to any number	$ > 4 $	. Let	Greater than four.
we use the parameters as follows :	$ L= $	8 ,	L equals None.
	$ E $	the total number of epochs , and	E
We set	$ \psi_3 ( x ) : =\bot $	.	Psi sub 3 open parenthesis x close parenthesis equals bottom.
commutes with	$ A $	for any continuous function	A
this formally leads to an infinite number of discontinuities for any mode	$ k = 2\pi n/N $	, which are labeled by the integer	k equals two pie n over N.
&	$ 4.0 $	&	Four point zero.
be a composition and let	$ \P_c $	be the set of all vector partitions that are cylindric partitions with profile	P sub c.
375	$ \pm $	0.2 . Related to Eq .	Plus or minus.
for some	$ 1d $	-local	One d.
is such that	$ \alpha-\zeta_3 $	is not	alpha minus zeta sub 3
10	$ ^ { -24 } $	g being the mass of hydrogen ,	Negative 24.
This is important because the actual message delay	$ \delta $	may be much smaller than	delta.
the qualification	$ \nu $	of the Tikhonov regularization scheme is equal to	nu.
unknown parameter	$ \theta $	in parameter space	Theta.
and	$ S_2 $	is of degree	S sub 2.
em The point	$ \xi $	is	xi.
this means	$ y $	must have a neighbor inside	y
	$ M ' \ba f $	, and	M prime b f
are only observed at times	$ t $	when	t
is a modular flat of	$ M\|H $	. Therefore	M H
We fix	$ i \in \ { 1,2\ } $	and thus drop the superscript	i belongs to the set containing 1 and 2.
homogeneous part of	$ f $	. Using ~	f
and its spatially associated decoder feature point set	$ P_l $	. To simplify our analysis , we assume	P sub l.
that for the first drop corresponds to the drop radius	$ R $	. To confirm this scaling , we measured grounded drops of different volumes	R
& &	$ \bm { 0.005 } $	&	0.005: zero point zero zero five
SPITZER reduced with the High-	$ z $	sample with SNR	z
with	$ i \neq j $	. Then , the relative order of	i is not equal to j.
exhibits the intensity at	$ \bm { q } _1+\bm { q } _2 $	~	q one plus q two
138	$ \pm $	0.012 & 0.163	Plus or minus.
and	$ P ( \mathcal { L } ) $	become the	P of L
& & &	$ \checkmark $	&	Checkmark.

we do not include a weight function applied to each

$ \bx_i $

in the latter . A weight function can be used to decrease the weight of high-leverage points . Such a function can be included in our approach when viewed as

x sub i.

including

$ 92\ % $

of targets at

92 percent.

and

$ j^\delta $

for

j raised to the power of delta

$ d_ { t , x , y } $

. The time step is defined as

dee sub tee, eks, why

in

$ \mathcal { R } $

. As

capital R

of

$ v $

with respect to

v

We varied these design parameters approximately around the values estimated in HPTN 037 to assess the sensitivity of

$ K $

to these changes . In particular , we set the the effect size of both the AIE and the ASpE to 0.5 times , 1 times , and 1.5 times

K

as we are assuming that

$ G $

is torsion-free ,

G

In the following proposition we aim to show that if a quantum state

$ \rho $

satisfies the quantum Poincar

rho.

in

$ \mathcal { G } $

and

G

for

$ 0 < \alpha\leq 1 $

, we have

0 is less than alpha and alpha is less than or equal to 1.

and

$ \Gamma_2 $

with

Gamma 2.

We define the energy functional

$ M ( t ) $

as follows

M of t.

and

$ \mu ( g,0 ) \propto g $

. The weak-field expansion in Eq.~

Mu of g comma zero is proportional to g.

That is

$ \mathbb { E } [ \frac { 1 } { m } A^TA ] =I_d $

. Now we consider the variance of each

The expectation of one over m times A transpose equals the identity matrix.

the percentage of trajectories that exceed the cost limit

$ K $

decreases ( last column ) , and the expected negated reward increases . The statistical properties of the total cost incurred by our method under different risk level

K

where

$ M $

is some ( unknown ) constant .

M

$ N $

) between an inner region where

N

and the image scale of

$ 0.15 $

arcsec pixel

0.15: None

illustrates the energy values obtained using the AV8

$ ' $

+central 3N potential~

apostrophe

and operators

$ \phi ( f ) $

also aligns with our intuition that we can measure the amplitude of a given mode at different points in time and/or space .

phi of f.

we obtain

$ CZ_\tau ( \gamma_1 ) = CZ_\tau ( \overline { \gamma } _1 ) \geq 7 $

. Thus ,

CZ sub tau of gamma 1 equals CZ sub tau of the conjugate of gamma 1, greater than or equal to 7.

the

$ s_d $

and

s subscript d.

with simple gadgetry to get the desired graph

$ G ' $

where we can prove the correspondence between independent sets in

the derivative of G.

and

$ I^A\equiv \gamma^ { AB } I_B $

. The structure constants are defined by

I to the power of A is equivalent to gamma to the power of A B times I sub B.

diseases explain

$ 4,000 $

findings . The probability of a finding given diseases is expressed by Equation

Four thousand.

$ \pi_t $

satisfies

Pi sub t.

The ensemble size is

$ N = 10^3 $

. We fix

N equals ten to the power of three.

with support contained in

$ S $

that satisfy

S

and

$ c_ { 2 } $

with their corresponding entanglement quantified by means of the Gaussian R

c sub 2.

designing high-dimensional sparse codebooks for simultaneously supporting devices for a large

$ K $

is quite challenging , since the sparse teletraffic of massive communication can not be readily exploited . Considering the sparse codebooks designed in~

K

we mainly discuss the zero

$ E $

. The triangles , circles , + and x marks represent the cases

E

$ , $

{

comma

$ P_2 $

,

P sub 2

and

$ Q_l $

equal to

Q sub l.

of an arbitrary integrable function

$ G_ { d } $

over a d-dimensional polyhedron

G sub d.

As

$ A $

is a fibration , there exists , for each

A

and let

$ S_ { X } $

be a vector bundle over

S sub X.

the monotone submodular function

$ g_ { d } $

associated with

g subscript d.

has the same dp-rank as

$ R $

.

R

k

$ -plane trees and $

k

Negative plane and

The 2H stacking is obtained by a

$ 180^ { \circ } $

rotation about the

180 degrees

$ l_ { 1,2 } =r_ { 1,2 } -r_ { ps } $

. In the Schwarzschild case , this number equals

L sub one comma two equals r sub one comma two minus r sub p s.

for

$ i=1,2,3,4 $

. We recapitulate the above relations with the diagram ,

i equals one, two, three, four.

By the fact that

$ X $

is a vertex set with binding number

X

and

$ \Gamma^ * $

are the conjugates of

Gamma star.

The overall inter-rater agreement in terms of Fleiss

$ \kappa $

was 0.22 which corresponds to a fair agreement . The

kappa.

\\ \hline

$ 3 $

&

three

A lower value of

$ K $

or

K

for

$ 1 \leq n \leq 4 $

are given in Eqn.

One is less than or equal to n, which is less than or equal to four.

with

$ i=\ { A , B\ } $

, are chosen according to the Akaike Information Criterion

i equals the set containing A and B.

in the asymptotic regime as

$ n\to\infty $

.

n tends to infinity

as functions of the radius

$ r $

. The EATS used here reads

r

we define its volume as the sum of degrees of nodes in

$ A $

, that is ,

A

exists for all

$ t > 0 $

. If

t greater than zero

$ 1 $

&

one

each denoted by

$ L_i [ f ] : \mathcal { F } \mapsto \mathbb { R } $

and such that

L sub i of f, colon F maps to R

or

$ E_8 $

, let

E sub 8.

Consider the deformation of a body initially occupying a domain

$ \Omega $

through a displacement field

Omega.

$ U $

and mark embeddings

U

-sub-Gaussian for some

$ K > 0 $

whenever

K is greater than zero.

at times

$ r_1 , r_2 $

. Therefore by Lemma

r sub 1, r sub 2

can provide it with probability

$ p $

. When cooperation is substitutable ,

p

016

$ \pm $

Plus or minus.

maps the problem of the reconstruction of the higher-order interactions into that of solving a system of algebraic equations in the unknown variables given by the

$ H $

entries of

H

em

$ \partial \Omega^ { 3 , - } \cap B_r ( x_s ) $

is the unique

Partial Omega to the power of three, minus, intersection B sub r of x sub s.

be an ample line bundle on

$ T $

with

T

$ 0.25 $

,

0.25: None

to any number

$ > 4 $

. Let

Greater than four.

we use the parameters as follows :

$ L= $

8 ,

L equals None.

$ E $

the total number of epochs , and

E

We set

$ \psi_3 ( x ) : =\bot $

.

Psi sub 3 open parenthesis x close parenthesis equals bottom.

commutes with

$ A $

for any continuous function

A

this formally leads to an infinite number of discontinuities for any mode

$ k = 2\pi n/N $

, which are labeled by the integer

k equals two pie n over N.

&

$ 4.0 $

&

Four point zero.

be a composition and let

$ \P_c $

be the set of all vector partitions that are cylindric partitions with profile

P sub c.

375

$ \pm $

0.2 . Related to Eq .

Plus or minus.

for some

$ 1d $

-local

One d.

is such that

$ \alpha-\zeta_3 $

is not

alpha minus zeta sub 3

10

$ ^ { -24 } $

g being the mass of hydrogen ,

Negative 24.

This is important because the actual message delay

$ \delta $

may be much smaller than

delta.

the qualification

$ \nu $

of the Tikhonov regularization scheme is equal to

nu.

unknown parameter

$ \theta $

in parameter space

Theta.

and

$ S_2 $

is of degree

S sub 2.

em The point

$ \xi $

is

xi.

this means

$ y $

must have a neighbor inside

y

$ M ' \ba f $

, and

M prime b f

are only observed at times

$ t $

when

t

is a modular flat of

$ M|H $

. Therefore

M H

We fix

$ i \in \ { 1,2\ } $

and thus drop the superscript

i belongs to the set containing 1 and 2.

homogeneous part of

$ f $

. Using ~

f

and its spatially associated decoder feature point set

$ P_l $

. To simplify our analysis , we assume

P sub l.

that for the first drop corresponds to the drop radius

$ R $

. To confirm this scaling , we measured grounded drops of different volumes

R

& &

$ \bm { 0.005 } $

&

0.005: zero point zero zero five

SPITZER reduced with the High-

$ z $

sample with SNR

z

with

$ i \neq j $

. Then , the relative order of

i is not equal to j.

exhibits the intensity at

$ \bm { q } _1+\bm { q } _2 $

~

q one plus q two

138

$ \pm $

0.012 & 0.163

Plus or minus.

and

$ P ( \mathcal { L } ) $

become the

P of L

& & &

$ \checkmark $

&

Checkmark.