Last-Bullet/MathBridge_Splitted · Data Studio

context_before stringlengths 0 199	equation stringlengths 5 80	context_after stringlengths 0 249	spoken_English stringlengths 0 119
198	$ \pm $	0.001	Plus or minus.
depends on	$ \epsilon $	and on the periodicity of the fields , see	epsilon.
if	$ s^\vee $	is Lorentzian .	s hat
on	$ G $	.	G
consider the solution for when the machine permutes	$ 3 $	people .	three
If the minimal codimension of a component of	$ C $	is greater than	C
yields a trivial basis for	$ k = 1 $	and a nontrivial one for	k equals one.
We will take the single-particle constituents to be the	$ \| \phi^1 \rangle $	scalars of the	ket phi raised to the power of one
and resampled it at	$ a_j $	, these data can be reused when the FFA reaches a fold at twice the period , as shown in Figure	a sub j.
we should choose to set	$ \epsilon \rightarrow 0 $	in the end . Computationally , this can be done by using a linear fitting of	Epsilon tends to zero.
The QD device featured a set of	$ 9 $	bottom gates with a width of	nine
denote the	$ \infty $	-operad of	Infinity.
the	$ \rho $	-PDF continues to broaden into the low-density regime . At t=0.5 , the lowest density reaches	rho.
and	$ U_ { 11 } $	are dominant comparing to the other terms . In contrast to Hermitian cases , conditions that	U subscript 11
	$ k $	and	k
helium superfluid of radius	$ R $	( Fig.4 ( a ) ) ~	R
Let	$ \mathbb { W } $	denote the set of all possible Wang prototiles , and	W double-struck.
We vary the constant radius-force range ratio	$ y $	with values of	y
and	$ W \subset V $	we denote the restriction of	W is a subset of V.
	$ \cH^ { d-1 } $	denotes the	H to the power of d minus 1.
in terms of	$ C $	.	C
ranging from 1	$ ^\circ $	to 30	degree.
uniformly/quasiuniformly/firmly uniformly/firmly quasiuniformly lower semicontinuous near a point	$ \bx\in\dom\varphi_1\cap\dom\varphi_2 $	if it is uniformly/quasiuniformly/firmly uniform	b x belongs to the set of phi sub 1 intersection phi sub 2.
of edges in	$ \mathcal M $	: for a solution	capital M.
allow also for the computation of the minimal number of	$ e $	-folds of inflation	e
The norm of an operator	$ A\in\mathcal { B } ( E ) $	is denoted by	A belongs to the set of B of E
either	$ H ( b ) $	or	H of b.
satisfies	$ \hth { 2,2 } $	by Theorem	h-th two comma two
parameters namely radial	$ EoS $	parameter and transverse	Ee oh S
&	$ 18 $	&	Eighteen.
Consistency * implies that for all	$ \omega \in \Omega $	,	Omega belongs to omega.
one can choose	$ L $	so that	L
there exists a	$ G $	and	G
For any policy	$ \pi $	, a value function	pi.
be a partial edge coloring of	$ G $	. For any vertex	G
and	$ \Gamma ' $	. For our curvature results , we need to introduce one more concept , the	Gamma prime.
the condition	$ k\geq 3 $	is still needed to ensure	k greater than or equal to 3
Gaussian unitary	$ V $	such that for	V
For	$ k \ge 3 $	, by induction hypothesis we have	k greater than or equal to 3
	$ ( \pi , V ) $	is spinorial if and only if	pi, V.
we assume that	$ Q $	is parameterized through the equation	Q
Hence the total revenue is at most	$ j\cdot1/j=1 $	, leading to an	j times 1 divided by j equals 1
which leads to a	$ \ell_\infty $	bound of the difference between the estimator	ell infinity.
	$ \tau=0 $	,	Tau equals zero.
of vertices of	$ G $	. A graph is	G
is simply equal to	$ L $	, albeit restricted to the vertices of	L
&	$ 0.94\pm 0.03 $	&	0.94 plus or minus 0.03
The predicted apparent magnitude for a passband	$ S $	depends on	S
since the distribution associated to	$ \hat K $	factorizes according to	K hat.
	$ , $	C_ { 50 }	comma
is the tangent cone at a	$ 1 $	-Lipschitz	one
fix an arbitrary	$ \sigma\in S $	.	Sigma belongs to S.
In Sheet~B	$ _1 $	the maximum count is 620 , and the counts in the adjacent cells are 300 and 442 . It is notable that the galaxy counts at the two maxima are	One.
This principle leads to universal scaling for variance on angular scale	$ \Theta $	,	Theta.
that homogeneous local Poisson structures of differential degree	$ d=2 $	,	d equals two.
-figure-eight	$ \gamma_ { \lambda } $	as in Lemma	Gamma sub lambda.
	$ E $	( of dimension	E
shows that the second term tends to	$ 0 $	as	zero
&	$ 1^ { st } $	&	One to the power of st.
% % Assume now	$ g\geq 1 $	and	g greater than or equal to 1
where nodes must adopt colors from lists of size	$ \Delta^2+1 $	or	Delta squared plus one.
is infinite for these	$ X $	, and an infinite index subgroup of a finitely generated group is not necessarily finitely generated .	X
65 mJy is due to the non-uniform noise distribution and the resulting variation in the	$ 7\ , \sigma $	detection limit over the survey region . It is to be noted that there is a significant population of unresolved sources with	Seven comma sigma.
another equivalent equation reformulation of the	$ \VIXF $	is as follows	vee eye X eff.
the upper bounds above illustrate the need for norms such as	$ \left\Vert F_ { j } \right\Vert _ { \infty } $	,	The norm of F sub j, sub infinity.
for O	$ ( N ) $	model and Eq.~	N
we define a slender filament of uniform radius	$ \epsilon $	as	epsilon.
varying from 10 to 14 for	$ m $	= 1 for Vietnamese , whereas for English and Russian the sequences are shorter -- -	m
26	$ \pm $	0.1 nm , the corresponding distribution is shown in Fig.~S6 %	Plus or minus.
implies that for some	$ \delta > 0 $	,	Delta is greater than zero.
and	$ k ' $	th atom type	k prime
A node	$ n $	without an outgoing red arrow is its own root . A set of nodes connected to the same node with red edges forms an equivalence class . In this example	n
where	$ \gamma=1 $	or	Gamma equals one.
and the dominated convergence theorem to interchange the limit	$ t\to0 $	with the integral over	t goes to zero.
180	$ ^\circ $	. Their order in Table~	degree.
where we used the continuity of	$ \F_\epsilon $	in the second equality . This , however , means that	F sub epsilon.
at	$ N\to\infty $	. Consequently , in the case	N tends to infinity.
across all feature maps indexed by	$ k $	.	k
&	$ -0.03 $	&	Negative zero point zero three.
and	$ b \in \N $	, we use the following notation for integer intervals	b belongs to the set of natural numbers.
and	$ \tilde { \theta } _n $	. Since the final estimator is obtained by measuring at angle	theta tilde sub n.
878	$ \pm $	0.003 & & 0.652	Plus or minus.
and	$ \widetilde { M } = \widetilde { M } _d = M_u^1 = 10^2 M_u^3 $	, respectively , with	The tilde of M equals the tilde of M sub d equals M sub u to the power of 1 equals 10 squared M sub u cubed.
enabling electron injection with energy	$ E $	from the left and calculate the steady-state current by~	E
	$ mb $	should be	pee bee
the argument of the exponential in	$ \Theta_ { 2 \lambda } ( g , \beta , \bm { \mu_ { 2 \lambda } } ) $	is	Theta sub 2 lambda open parenthesis g, beta, bm mu sub 2 lambda close parenthesis.
instead of	$ K $	) , hence the gain .	K
	$ \beta $	,	Beta.
and	$ g=1 $	, Eq.	g equals one
Because of the prefactor	$ \Delta ( \ { \bf\ z ) $	the states	Delta of bold z
em for all	$ j \in \mathcal { J } $	,	j belongs to the set J.
of	$ P_2 $	. Our goal is to describe the face lattice of	P sub 2
only in columns	$ \le i $	and	Less than or equal to i.
rather than	$ p_0 $	, where	p sub 0
allowing for enhanced ITG-driven transport as	$ \beta $	is gradually increased . This work examines the effect of high	Beta.
denotes the fused features which share the same channels with	$ v^k $	. Although there are other fusion manners ,	v to the power of k
be a closed connected smooth 4-manifold and let	$ X_S $	be the 4-manifold that is obtained by performing a Gluck twist along a smoothly embedded 2-sphere	X sub S
and~	$ w $	are contained in the clique~	w
The value in front of	$ i $	is known when we anticipate the	i
&	$ h=5 $	&	h equals five.

198

$ \pm $

0.001

Plus or minus.

depends on

$ \epsilon $

and on the periodicity of the fields , see

epsilon.

if

$ s^\vee $

is Lorentzian .

s hat

on

$ G $

.

G

consider the solution for when the machine permutes

$ 3 $

people .

three

If the minimal codimension of a component of

$ C $

is greater than

C

yields a trivial basis for

$ k = 1 $

and a nontrivial one for

k equals one.

We will take the single-particle constituents to be the

$ | \phi^1 \rangle $

scalars of the

ket phi raised to the power of one

and resampled it at

$ a_j $

, these data can be reused when the FFA reaches a fold at twice the period , as shown in Figure

a sub j.

we should choose to set

$ \epsilon \rightarrow 0 $

in the end . Computationally , this can be done by using a linear fitting of

Epsilon tends to zero.

The QD device featured a set of

$ 9 $

bottom gates with a width of

nine

denote the

$ \infty $

-operad of

Infinity.

the

$ \rho $

-PDF continues to broaden into the low-density regime . At t=0.5 , the lowest density reaches

rho.

and

$ U_ { 11 } $

are dominant comparing to the other terms . In contrast to Hermitian cases , conditions that

U subscript 11

$ k $

and

k

helium superfluid of radius

$ R $

( Fig.4 ( a ) ) ~

R

Let

$ \mathbb { W } $

denote the set of all possible Wang prototiles , and

W double-struck.

We vary the constant radius-force range ratio

$ y $

with values of

y

and

$ W \subset V $

we denote the restriction of

W is a subset of V.

$ \cH^ { d-1 } $

denotes the

H to the power of d minus 1.

in terms of

$ C $

.

C

ranging from 1

$ ^\circ $

to 30

degree.

uniformly/quasiuniformly/firmly uniformly/firmly quasiuniformly lower semicontinuous near a point

$ \bx\in\dom\varphi_1\cap\dom\varphi_2 $

if it is uniformly/quasiuniformly/firmly uniform

b x belongs to the set of phi sub 1 intersection phi sub 2.

of edges in

$ \mathcal M $

: for a solution

capital M.

allow also for the computation of the minimal number of

$ e $

-folds of inflation

e

The norm of an operator

$ A\in\mathcal { B } ( E ) $

is denoted by

A belongs to the set of B of E

either

$ H ( b ) $

or

H of b.

satisfies

$ \hth { 2,2 } $

by Theorem

h-th two comma two

parameters namely radial

$ EoS $

parameter and transverse

Ee oh S

&

$ 18 $

&

Eighteen.

Consistency * implies that for all

$ \omega \in \Omega $

,

Omega belongs to omega.

one can choose

$ L $

so that

L

there exists a

$ G $

and

G

For any policy

$ \pi $

, a value function

pi.

be a partial edge coloring of

$ G $

. For any vertex

G

and

$ \Gamma ' $

. For our curvature results , we need to introduce one more concept , the

Gamma prime.

the condition

$ k\geq 3 $

is still needed to ensure

k greater than or equal to 3

Gaussian unitary

$ V $

such that for

V

For

$ k \ge 3 $

, by induction hypothesis we have

k greater than or equal to 3

$ ( \pi , V ) $

is spinorial if and only if

pi, V.

we assume that

$ Q $

is parameterized through the equation

Q

Hence the total revenue is at most

$ j\cdot1/j=1 $

, leading to an

j times 1 divided by j equals 1

which leads to a

$ \ell_\infty $

bound of the difference between the estimator

ell infinity.

$ \tau=0 $

,

Tau equals zero.

of vertices of

$ G $

. A graph is

G

is simply equal to

$ L $

, albeit restricted to the vertices of

L

&

$ 0.94\pm 0.03 $

&

0.94 plus or minus 0.03

The predicted apparent magnitude for a passband

$ S $

depends on

S

since the distribution associated to

$ \hat K $

factorizes according to

K hat.

$ , $

C_ { 50 }

comma

is the tangent cone at a

$ 1 $

-Lipschitz

one

fix an arbitrary

$ \sigma\in S $

.

Sigma belongs to S.

In Sheet~B

$ _1 $

the maximum count is 620 , and the counts in the adjacent cells are 300 and 442 . It is notable that the galaxy counts at the two maxima are

One.

This principle leads to universal scaling for variance on angular scale

$ \Theta $

,

Theta.

that homogeneous local Poisson structures of differential degree

$ d=2 $

,

d equals two.

-figure-eight

$ \gamma_ { \lambda } $

as in Lemma

Gamma sub lambda.

$ E $

( of dimension

E

shows that the second term tends to

$ 0 $

as

zero

&

$ 1^ { st } $

&

One to the power of st.

% % Assume now

$ g\geq 1 $

and

g greater than or equal to 1

where nodes must adopt colors from lists of size

$ \Delta^2+1 $

or

Delta squared plus one.

is infinite for these

$ X $

, and an infinite index subgroup of a finitely generated group is not necessarily finitely generated .

X

65 mJy is due to the non-uniform noise distribution and the resulting variation in the

$ 7\ , \sigma $

detection limit over the survey region . It is to be noted that there is a significant population of unresolved sources with

Seven comma sigma.

another equivalent equation reformulation of the

$ \VIXF $

is as follows

vee eye X eff.

the upper bounds above illustrate the need for norms such as

$ \left\Vert F_ { j } \right\Vert _ { \infty } $

,

The norm of F sub j, sub infinity.

for O

$ ( N ) $

model and Eq.~

N

we define a slender filament of uniform radius

$ \epsilon $

as

epsilon.

varying from 10 to 14 for

$ m $

= 1 for Vietnamese , whereas for English and Russian the sequences are shorter -- -

m

26

$ \pm $

0.1 nm , the corresponding distribution is shown in Fig.~S6 %

Plus or minus.

implies that for some

$ \delta > 0 $

,

Delta is greater than zero.

and

$ k ' $

th atom type

k prime

A node

$ n $

without an outgoing red arrow is its own root . A set of nodes connected to the same node with red edges forms an equivalence class . In this example

n

where

$ \gamma=1 $

or

Gamma equals one.

and the dominated convergence theorem to interchange the limit

$ t\to0 $

with the integral over

t goes to zero.

180

$ ^\circ $

. Their order in Table~

degree.

where we used the continuity of

$ \F_\epsilon $

in the second equality . This , however , means that

F sub epsilon.

at

$ N\to\infty $

. Consequently , in the case

N tends to infinity.

across all feature maps indexed by

$ k $

.

k

&

$ -0.03 $

&

Negative zero point zero three.

and

$ b \in \N $

, we use the following notation for integer intervals

b belongs to the set of natural numbers.

and

$ \tilde { \theta } _n $

. Since the final estimator is obtained by measuring at angle

theta tilde sub n.

878

$ \pm $

0.003 & & 0.652

Plus or minus.

and

$ \widetilde { M } = \widetilde { M } _d = M_u^1 = 10^2 M_u^3 $

, respectively , with

The tilde of M equals the tilde of M sub d equals M sub u to the power of 1 equals 10 squared M sub u cubed.

enabling electron injection with energy

$ E $

from the left and calculate the steady-state current by~

E

$ mb $

should be

pee bee

the argument of the exponential in

$ \Theta_ { 2 \lambda } ( g , \beta , \bm { \mu_ { 2 \lambda } } ) $

is

Theta sub 2 lambda open parenthesis g, beta, bm mu sub 2 lambda close parenthesis.

instead of

$ K $

) , hence the gain .

K

$ \beta $

,

Beta.

and

$ g=1 $

, Eq.

g equals one

Because of the prefactor

$ \Delta ( \ { \bf\ z ) $

the states

Delta of bold z

em for all

$ j \in \mathcal { J } $

,

j belongs to the set J.

of

$ P_2 $

. Our goal is to describe the face lattice of

P sub 2

only in columns

$ \le i $

and

Less than or equal to i.

rather than

$ p_0 $

, where

p sub 0

allowing for enhanced ITG-driven transport as

$ \beta $

is gradually increased . This work examines the effect of high

Beta.

denotes the fused features which share the same channels with

$ v^k $

. Although there are other fusion manners ,

v to the power of k

be a closed connected smooth 4-manifold and let

$ X_S $

be the 4-manifold that is obtained by performing a Gluck twist along a smoothly embedded 2-sphere

X sub S

and~

$ w $

are contained in the clique~

w

The value in front of

$ i $

is known when we anticipate the

i

&

$ h=5 $

&

h equals five.