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
For	$ n\geq1 $	, define	n greater than or equal to 1.
the choice of	$ \tau $	reflects the level of prudence of the policy that the PM commits to . The proposed optimal policy has another interesting interpretation that relates to the PM 's incentive .	Tau.
A change of variables may be necessary to ensure	$ \boldsymbol { X } $	is standard uniform , see	capital X.
	$ k_i=k_j+k_m $	, the bispectrum variance should be multiplied by a factor of 2	k sub i equals k sub j plus k sub m.
as well as similarity predicates and whose free variables	$ x $	and	x
vertex	$ x $	has degree 8 and	x
we usually take the exchange coupling parameters as a constant value	$ J $	( also	J
so	$ Q^1 $	is a projective cover of	Q raised to the power of 1.
and	$ K $	exceed exceed the value of the central density of	K
We find	$ c_ { 11 } $	= 20.94 eV	c sub 11.
in	$ I $	. This can all be confirmed formally , the details of which can be found in	I
for	$ \mu=0 $	, the semigroup is called	mu equals zero.
\eg either it is flat or	$ R $	and	R
The next definition is for type-	$ 3 $		three
&	$ .00 $	&	zero point zero zero
queries and determines whether	$ \Phi $	is a	Phi.
and	$ a\in I $	.	a belongs to the set I
between vertices in	$ \mathcal { V } _0 $	, which have already been determined . Equation	V zero.
such that there is a	$ j $	with	j
Q_a	$ ) $		closing parenthesis
and pharmaceutical companies are at the forefront of Industry	$ 4.0 $	adoption . It is common wisdom that in those economic sectors that are culturally resistant to change , the innovation mindset can be boosted by government-level interventions and policy-making that	Four point zero.
	$ d^ { 2 } $	in	d squared.
yields the same value for the quantity	$ \sum_ { j \in V } m_ { ij } x_j $	, hence :	Sum over j in V, m sub i j times x sub j.
in direction	$ u $	. Fix	u
which is denoted as an undirected graph	$ { G^ * } \left ( { V , { \bm { E } ^ * } } \right ) $	. Then check the connectivity of this undirected graph . The sufficient condition of a connected undirected graph is that all node pairs are connectable , which denotes as	G raised to the power of asterisk, open parenthesis, V, comma, E raised to the power of asterisk, close parenthesis.
Suppose that an	$ m $	-set	m
fails to be free over	$ \mathcal { O } _ { F } $	, one can still use this technique , applied to a free cover . However , it gets increasingly cumbersome to show that this yields a well-defined topology .	O sub F.
and	$ t-1 $	and the effects of the initial factors remain qualitatively unchanged from the baseline . In addition , the shocks at	t minus one
The Jacobian matrix	$ \nabla^2_ { xy } g ( x , y ) $	and Hessian matrix	Nabla squared sub x y g of x, y.
	$ W_n $	for	W sub n.
are stored in	$ A_u $	. The associate canonical squares of all segments are then sorted , and used to construct a compressed quadtree . % %	A sub u.
we focus on the problem of community detection with a potentially growing	$ k $	. It turns out that the problem starts to exhibit a statistical-computational gap as	k
columns of	$ \M_1 $	. Running	M one.
If	$ D $	is dense in	D
under the	$ h $	integral . We call	h
We will write	$ { \mathbb P } \colon\ul\S\to\ul\GammaS_ * ^ { \text { spc } } $	for	Mathematical double-struck capital P, colon, ul S, arrow, ul Gamma S star superscript spc.
8153	$ \pm $	0.0018	Plus or minus.
&	$ 22.78 $	&	twenty-two point seven eight.
and the fraction of collision complexes that decay into the product channel	$ f $	; that is , %	f
and the grand propulsion tensor	$ \boldsymbol { \Pi } $	,	Pi.
and its eccentricity is	$ 0.998 $	-- in this case , the minimum approach of the dust particle with the Sun is 0.5~AU , i.e . the dust particle goes inside the Earth 's orbit .	Zero point nine nine eight.
&	$ 6 $	&	six
The constant-order term	$ \order { 1 } $	is given by	order one.
private good neighbors of	$ x $	and	x
	$ \lambda_ { A } $	is uniformly distributed in the interval	Lambda sub A.
and	$ \ { A , B\ } = ( AB+BA ) /2 $	. When the coordinate axes	A, B equals the quantity A times B plus B times A, divided by 2.
observe that~	$ p $	ties~	p
503	$ \pm $	0.2 & 32.31	Plus or minus.
- and	$ p $	-wave components , increases as the	p
namely	$ F $	-crystals	F
	$ ( H ) $		H
instead of a value of	$ \widehat \rho $	.	rho hat.
within the	$ w $	CDM model , using the	w
We sample simulated terrains with a variety of terrain properties	$ e_t $	and command a locomotion policy from prior work~	e sub t
and for any	$ j\in S $		j belongs to S.
Let	$ \psi_M ( m ) $	be a real , non-negative coefficient sequence which is smooth at scale	psi sub M of m
and	$ \mu^ { -1 } $	, and	mu to the power of negative one.
or	$ 1 $	. Consider the norm	one
since any	$ \Xi $	of	Xi
For each vertex	$ v $	of	v
then	$ M_S $	contains a positive proportion of irreducibles by Corollary	M sub S.
	$ \lambda $	of a persistence diagram	Lambda.
Note that	$ Z $	and	Z
	$ p_s $	walkers are spawned . Otherwise if the condition	p sub s.
after the	$ \hat { \sigma } _z $	-measurement on electron	sigma hat sub z.
be some log resolution of	$ Y $	and let	Y
we set	$ \epsilon = 0.01 $	in	Epsilon equals zero point zero one.
to	$ 90^ { \circ } $	and the detection distance range is up to	Ninety degrees.
% We develop two loss functions upon transforming the standard form of variational lower bound	$ \mathcal { L } $	in	ell
	$ a $	and	a
&	$ w_0 $	&	w sub zero.
and the covariance matrix is	$ \C $	. Then by definition ,	C: None
and therefore again	$ 1 $	solution with	one
the density of the second model at	$ x $	is	x
and inner iteration	$ j $	, we generate a copy of	j
with	$ \vert \beta_i\vert +2 $	nodes :	Absolute value of beta sub i plus 2.
for all these	$ k $	cases , the total expected cost of all home-moves are	k
By removing the components of	$ \gamma $	inside the sphere	Gamma.
with length no less than	$ l $	has	l
of	$ ( \Theta , p ) $	:	Theta comma p
em For	$ K_\alpha $		K sub alpha.
to	$ c_i+\delta $	.	c sub i plus delta
and the element of	$ \bar Y=\ { \bar y_ { k_0 } , \dots , \bar y_ { k_ { m-1 } } \ } $	are not same as the previously computed but they can be computed by the aggregation scheme . Then , Let us take a newly computed curvature pair	Y bar
Substituting	$ r $	for	r
Since	$ \Psi $	is continuous , by Lemma~	Psi.
with all	$ N_Q $	qubits set to	N sub Q
and	$ d \ge 0 $	, we consider the task of testing whether a given function	d is greater than or equal to zero.
until	$ t = 0 $	where	t equals zero.
of	$ B $	and	B
the following equations hold for small	$ \epsilon $	with probability	epsilon.
at	$ z\sim 3.5 $	;	z is approximately distributed according to a normal distribution with a mean of 3.5.
In this section we calculate perturbations of the eigenenergies of	$ H $	in the limit	H
at real places	$ v $	. It seems reasonable to guess that whenever	v
for all	$ s\in S $	, such that the resulting incidence graph	s belongs to S.
and	$ \ket { 1 } $	states in a single readout . However , a readout is subject to uncertainties in the discrimination leading to non-ideal measurements and overlap between the probability distributions	ket one.
The	$ h = 0 $	lattice action is	h equals zero.
We examine the improvement in constraining power on these parameters when jointly-fitting mock data of the high-	$ z $	galaxy rest-frame ultraviolet ( UV ) luminosity function ( LF ) in addition to the global 21-cm signal . We present the first nested sampling constraints on {	z
with a step-size	$ h = T/N $	leads to the following layer equation %	h equals T divided by N.
	$ T=20 $	~ks and	T equals twenty.
is served by vehicle	$ k $	through the link	k

For

$ n\geq1 $

, define

n greater than or equal to 1.

the choice of

$ \tau $

reflects the level of prudence of the policy that the PM commits to . The proposed optimal policy has another interesting interpretation that relates to the PM 's incentive .

Tau.

A change of variables may be necessary to ensure

$ \boldsymbol { X } $

is standard uniform , see

capital X.

$ k_i=k_j+k_m $

, the bispectrum variance should be multiplied by a factor of 2

k sub i equals k sub j plus k sub m.

as well as similarity predicates and whose free variables

$ x $

and

x

vertex

$ x $

has degree 8 and

x

we usually take the exchange coupling parameters as a constant value

$ J $

( also

J

so

$ Q^1 $

is a projective cover of

Q raised to the power of 1.

and

$ K $

exceed exceed the value of the central density of

K

We find

$ c_ { 11 } $

= 20.94 eV

c sub 11.

in

$ I $

. This can all be confirmed formally , the details of which can be found in

I

for

$ \mu=0 $

, the semigroup is called

mu equals zero.

\eg either it is flat or

$ R $

and

R

The next definition is for type-

$ 3 $

three

&

$ .00 $

&

zero point zero zero

queries and determines whether

$ \Phi $

is a

Phi.

and

$ a\in I $

.

a belongs to the set I

between vertices in

$ \mathcal { V } _0 $

, which have already been determined . Equation

V zero.

such that there is a

$ j $

with

j

Q_a

$ ) $

closing parenthesis

and pharmaceutical companies are at the forefront of Industry

$ 4.0 $

adoption . It is common wisdom that in those economic sectors that are culturally resistant to change , the innovation mindset can be boosted by government-level interventions and policy-making that

Four point zero.

$ d^ { 2 } $

in

d squared.

yields the same value for the quantity

$ \sum_ { j \in V } m_ { ij } x_j $

, hence :

Sum over j in V, m sub i j times x sub j.

in direction

$ u $

. Fix

u

which is denoted as an undirected graph

$ { G^ * } \left ( { V , { \bm { E } ^ * } } \right ) $

. Then check the connectivity of this undirected graph . The sufficient condition of a connected undirected graph is that all node pairs are connectable , which denotes as

G raised to the power of asterisk, open parenthesis, V, comma, E raised to the power of asterisk, close parenthesis.

Suppose that an

$ m $

-set

m

fails to be free over

$ \mathcal { O } _ { F } $

, one can still use this technique , applied to a free cover . However , it gets increasingly cumbersome to show that this yields a well-defined topology .

O sub F.

and

$ t-1 $

and the effects of the initial factors remain qualitatively unchanged from the baseline . In addition , the shocks at

t minus one

The Jacobian matrix

$ \nabla^2_ { xy } g ( x , y ) $

and Hessian matrix

Nabla squared sub x y g of x, y.

$ W_n $

for

W sub n.

are stored in

$ A_u $

. The associate canonical squares of all segments are then sorted , and used to construct a compressed quadtree . % %

A sub u.

we focus on the problem of community detection with a potentially growing

$ k $

. It turns out that the problem starts to exhibit a statistical-computational gap as

k

columns of

$ \M_1 $

. Running

M one.

If

$ D $

is dense in

D

under the

$ h $

integral . We call

h

We will write

$ { \mathbb P } \colon\ul\S\to\ul\GammaS_ * ^ { \text { spc } } $

for

Mathematical double-struck capital P, colon, ul S, arrow, ul Gamma S star superscript spc.

8153

$ \pm $

0.0018

Plus or minus.

&

$ 22.78 $

&

twenty-two point seven eight.

and the fraction of collision complexes that decay into the product channel

$ f $

; that is , %

f

and the grand propulsion tensor

$ \boldsymbol { \Pi } $

,

Pi.

and its eccentricity is

$ 0.998 $

-- in this case , the minimum approach of the dust particle with the Sun is 0.5~AU , i.e . the dust particle goes inside the Earth 's orbit .

Zero point nine nine eight.

&

$ 6 $

&

six

The constant-order term

$ \order { 1 } $

is given by

order one.

private good neighbors of

$ x $

and

x

$ \lambda_ { A } $

is uniformly distributed in the interval

Lambda sub A.

and

$ \ { A , B\ } = ( AB+BA ) /2 $

. When the coordinate axes

A, B equals the quantity A times B plus B times A, divided by 2.

observe that~

$ p $

ties~

p

503

$ \pm $

0.2 & 32.31

Plus or minus.

- and

$ p $

-wave components , increases as the

p

namely

$ F $

-crystals

F

$ ( H ) $

H

instead of a value of

$ \widehat \rho $

.

rho hat.

within the

$ w $

CDM model , using the

w

We sample simulated terrains with a variety of terrain properties

$ e_t $

and command a locomotion policy from prior work~

e sub t

and for any

$ j\in S $

j belongs to S.

Let

$ \psi_M ( m ) $

be a real , non-negative coefficient sequence which is smooth at scale

psi sub M of m

and

$ \mu^ { -1 } $

, and

mu to the power of negative one.

or

$ 1 $

. Consider the norm

one

since any

$ \Xi $

of

Xi

For each vertex

$ v $

of

v

then

$ M_S $

contains a positive proportion of irreducibles by Corollary

M sub S.

$ \lambda $

of a persistence diagram

Lambda.

Note that

$ Z $

and

Z

$ p_s $

walkers are spawned . Otherwise if the condition

p sub s.

after the

$ \hat { \sigma } _z $

-measurement on electron

sigma hat sub z.

be some log resolution of

$ Y $

and let

Y

we set

$ \epsilon = 0.01 $

in

Epsilon equals zero point zero one.

to

$ 90^ { \circ } $

and the detection distance range is up to

Ninety degrees.

% We develop two loss functions upon transforming the standard form of variational lower bound

$ \mathcal { L } $

in

ell

$ a $

and

a

&

$ w_0 $

&

w sub zero.

and the covariance matrix is

$ \C $

. Then by definition ,

C: None

and therefore again

$ 1 $

solution with

one

the density of the second model at

$ x $

is

x

and inner iteration

$ j $

, we generate a copy of

j

with

$ \vert \beta_i\vert +2 $

nodes :

Absolute value of beta sub i plus 2.

for all these

$ k $

cases , the total expected cost of all home-moves are

k

By removing the components of

$ \gamma $

inside the sphere

Gamma.

with length no less than

$ l $

has

l

of

$ ( \Theta , p ) $

:

Theta comma p

em For

$ K_\alpha $

K sub alpha.

to

$ c_i+\delta $

.

c sub i plus delta

and the element of

$ \bar Y=\ { \bar y_ { k_0 } , \dots , \bar y_ { k_ { m-1 } } \ } $

are not same as the previously computed but they can be computed by the aggregation scheme . Then , Let us take a newly computed curvature pair

Y bar

Substituting

$ r $

for

r

Since

$ \Psi $

is continuous , by Lemma~

Psi.

with all

$ N_Q $

qubits set to

N sub Q

and

$ d \ge 0 $

, we consider the task of testing whether a given function

d is greater than or equal to zero.

until

$ t = 0 $

where

t equals zero.

of

$ B $

and

B

the following equations hold for small

$ \epsilon $

with probability

epsilon.

at

$ z\sim 3.5 $

;

z is approximately distributed according to a normal distribution with a mean of 3.5.

In this section we calculate perturbations of the eigenenergies of

$ H $

in the limit

H

at real places

$ v $

. It seems reasonable to guess that whenever

v

for all

$ s\in S $

, such that the resulting incidence graph

s belongs to S.

and

$ \ket { 1 } $

states in a single readout . However , a readout is subject to uncertainties in the discrimination leading to non-ideal measurements and overlap between the probability distributions

ket one.

The

$ h = 0 $

lattice action is

h equals zero.

We examine the improvement in constraining power on these parameters when jointly-fitting mock data of the high-

$ z $

galaxy rest-frame ultraviolet ( UV ) luminosity function ( LF ) in addition to the global 21-cm signal . We present the first nested sampling constraints on {

z

with a step-size

$ h = T/N $

leads to the following layer equation %

h equals T divided by N.

$ T=20 $

~ks and

T equals twenty.

is served by vehicle

$ k $

through the link

k