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
The cost associated to	$ M $	is	M
the laser pulse is Gaussian with spot-size	$ w_0=4 c/\omega_p $	. The ratio between the central laser frequency ,	w sub 0 equals 4 c divided by omega sub p.
to a	$ \mathfrak { Z } $	-action . Then we define	Z in Fraktur font.
we obtain the inferred latent code	$ \hat { \mathbf { z } } $	and covariates	z-hat.
We consider an	$ SU ( 2 ) $	extension of the gauge sector of the SM . Furthermore , the scalar sector is extended with an additional scalar bi-doublet odd under a	S U two.
Although the jerk of Model 4 is inconsistent with	$ \Lambda $	CDM , since it shows different evolution at both high and low redshifts and predicts the lower value of	Lambda.
	$ v_ { 2k } \in \hat N_ { S_2 } ( b_k ) $	for all	v sub 2k belongs to the set of N sub S sub 2 open parenthesis b sub k close parenthesis
of degree	$ > N ( g ) $	, then	greater than N of g
	$ e $	) and	e
in	$ G/\Gamma $	. The problem arises when	G over Gamma.
The SWs frequencies are finally identified by the maxima of the power spectra of the	$ x $	component of magnetization	x
2	$ \pm $	0.3 & 0.66	Plus or minus.
	$ t $	}	t
&	$ 69.41 $	&	69.41: Sixty-nine point four one
and express	$ \beta ( W ) $	in terms of	beta of W
the value of	$ L $	( Denmark , capital , Copenhagen ) could be % the set containing the templates `` What is the capital of	L
the spins form an AFM double-cone structure with a modulated basal-plane moment while the moments are predominantly pointing along the	$ c $	axis~	c
	$ M ( x ) $	is nonsingular modulo	M of x.
and	$ T_D $	is the Dingle temperature defined by	T sub D
for	$ i\neq j $	. As	i is not equal to j.
89	$ \pm $	0.06	Plus or minus.
be a genus	$ g $	closed Riemann surface with	g
the collection of all	$ r $	-element subsets of	r
in the	$ k $	-th iteration is written as	k
A prior which includes negative	$ \Lambda_ { 1,2 } $	would certainly avoid the issue of vanishing prior support at	Lambda sub one comma two.
Let	$ u\in \F_ { q^2 } \backslash \fq $	. The number of	you belongs to the set of F sub q squared backslash fq
which is affected by spurious resonances if	$ \kappa_0 \in \mathfrak { S } ( \Delta , \Omega_\Sigma ) $	. On the contrary , if we assume that the	Kappa sub zero belongs to the set of S, enclosed in parentheses Delta, Omega sub Sigma.
We have shown that two wave packets in the	$ -Y $	region are annihilated ; the unbounded behavior occurs because the pulses reach the region where the potential is significant . As we can see in Fig.~	Negative Y.
Then there is a canonical one-to-one correspondence between	$ w^ { * } $	-continuous convolution semigroups of states invariant under the adjoint action of	w raised to the power of asterisk.
or for fixed	$ \tau $	and uniform	Tau.
for	$ \delta \in ( 0,1 ) $	, for any	Delta belongs to the open interval from 0 to 1.
and	$ z ' $	.	z prime
is Strongly monotone if whenever flipping a match between a team from	$ C $	and a team from	C
semigroup of contractions	$ S ( t ) $	,	S of t.
In this section we develop the vital constituent of our procedure for growing	$ 2 $	-trees by iteratively obtaining common neighbors of prescribed pairs of adjacent vertices . The main result is Theorem~	two
Here each	$ \alpha_i $	is a loop which goes around	Alpha sub i.
in	$ \bar { P } _ { n } $	.	P bar sub n
	$ r \in \ { \mathbb\ N $	, and Gaussian-weighted space	r belongs to the set of natural numbers.
	$ x $	parametrizes the dimensionless target relative speed . Small values of	x
	$ E_k $	=	E subscript k.
Changing	$ n $	into a new variable	n
is adjacent to	$ u $	. If the condition stated in	u
so the	$ \omega_0 $	term in the sum can be dropped . The phase fluctuation	omega sub zero.
for even integers	$ k $	with	k
the value of	$ \Delta $	is larger , when the components become anisotropic and this results in a larger anisotropy . We note that if we are in the part of the phase diagram where	Delta.
where	$ L $	is the length of the serpentine . Integrating , we obtain	L
Images are collected every	$ 2 $	m from the trajectory with no preprocessing . We construct a point cloud submap with the global coordinates by accumulating 3D LiDAR data and sampling the points visible in the	two
80	$ \pm $	1.90 km s	Plus or minus.
Notice that	$ \mathcal { L } _ { \pm } $	have the form of a generator of a quantum dynamical semigroup~	L sub plus minus.
We assume that	$ \bm { Y } $	can be inferred from	Y
and	$ V_S $		V sub S
So	$ Y\rightarrow Y/H $	factors through	Y goes to Y over H
&	$ 0.38 $	&	0.38: None
Their calculation only requires a knowledge of the absorptive parts of CFFs and the	$ D $	-term form factor in the physical domain of the DVCS reaction . Moreover , it does not necessarily imply the analytic continuation in	D
The idea is now to find a linear combination of these	$ p ( x+k ) $	that is equal to	pee of eks plus kay
region	$ M_s $	is bounded , with	M sub s
em Defining	$ T $	as the index set of the non-bound constraints in	T
in	$ \GrVect_ { \mathbb\ { F } ^ { \mathsf\ { G } $	.	Gr-Vect sub F to the power of G
We write	$ \alpha\colon i \to j $	to indicate that	Alpha, colon, i, to, j.
Define a vector	$ u_X $	by	u sub X.
is the string operator that moves anyon	$ v $	downward along the	v
Denote the	$ k $	-th square in	k
Consider a simple error model where each two-qubit Clifford	$ V $	is independently affected by a unitary noise	V
em The scale dependent factor	$ f $	, which is calculated from the lowest lying charmonium state , introduces a correction of	f
The discriminator	$ D $	, classifies	D
or	$ h=2 , p > 3 $		h equals 2, p greater than 3.
619e	$ - $	9 & 2091 & 7.033e	minus
counts the number of free neighbors of	$ w $	given the configuration	w
in	$ K $	;	K
can be computed exactly only in the cases	$ p=2 $	using Mehta 's integral formula and	p equals 2
is	$ \H^n $	-contained in	H to the power of n
-complex does not interact with the new components that contain the points	$ p_i ( 0 ) $	. We will see shortly that the contributions from such vertices vanish . In order to force some interaction with the new points , consider the same	p sub i of 0
near the horizon	$ x=x_a $	: %	x equals x sub a.
or there exists an isoperimetric set	$ A $	such that	A
and Ag	$ _ { 1-x } $	Mn	One minus x.
We can write	$ g_j = \sigma_1^2 $	after applying a global conjugation move by a power of	g sub j equals sigma sub 1 squared.
changes sign whenever	$ u^2 $	takes the value	u squared
as	$ \mc G_s $	. Any fixed	m c G sub s
Let	$ \tilde B $	denote the image of	tilde B.
-modules over with	$ k $	an arbitrary unital commutative ring . Let us denote	k
and for	$ i=2,3 $	,	i equals 2, 3.
This shows that	$ \mu $	satisfies the associative axiom . One can similarly show that	mu.
8	$ \pm $	0.1 &	Plus or minus.
are functions of	$ f $	and do not involve parameters	f
shows that when	$ \xi_0 $	is super-heavy tailed , then the limiting behavior of	Xi sub zero.
associated with topics	$ T $	. Let	T
-secant variety to the Veronese embedding	$ v_ { d } ( \mathbb { P } ^ { m } ) $	is computed for	v sub d, capital P to the power of m
-th infinitesimal neighborhood of	$ 0 $	in	zero
is due to the definition of	$ d_\mu $	, which is a result of the definition of	d sub mu.
each VR headset	$ u $	}	u
and	$ H_t $	given , distribution of	H sub t.
the induced actions of	$ t_iq_ { jk } $	and	tee sub i q sub j k
for 1-Well and	$ 32^4 = \text { 1,048,576 } $	for 2-Wells . Note that for a fixed grid size , the state grows exponentially with the number of Wells -- -i.e.	32 to the power of 4 equals 1,048,576.
connects	$ P_0 $	with	P sub zero.
&	$ 0.28 $	&	0.28: None
Latent space image	$ \Vec { z } _0 $	and source embedding	vector z sub zero.
in the training set	$ \mathcal { P } $	. Step 6 employs the SELECTION process to navigate from the root node	P in calligraphic font.
for task	$ \alpha $	is defined as :	Alpha.
and	$ \pm\ { \rm\ Im\omega_\ { \boldsymbol\ q $	in the CM frame	Plus or minus, the imaginary part of omega sub q.
with	$ X $	not dependent on	X

The cost associated to

$ M $

is

M

the laser pulse is Gaussian with spot-size

$ w_0=4 c/\omega_p $

. The ratio between the central laser frequency ,

w sub 0 equals 4 c divided by omega sub p.

to a

$ \mathfrak { Z } $

-action . Then we define

Z in Fraktur font.

we obtain the inferred latent code

$ \hat { \mathbf { z } } $

and covariates

z-hat.

We consider an

$ SU ( 2 ) $

extension of the gauge sector of the SM . Furthermore , the scalar sector is extended with an additional scalar bi-doublet odd under a

S U two.

Although the jerk of Model 4 is inconsistent with

$ \Lambda $

CDM , since it shows different evolution at both high and low redshifts and predicts the lower value of

Lambda.

$ v_ { 2k } \in \hat N_ { S_2 } ( b_k ) $

for all

v sub 2k belongs to the set of N sub S sub 2 open parenthesis b sub k close parenthesis

of degree

$ > N ( g ) $

, then

greater than N of g

$ e $

) and

e

in

$ G/\Gamma $

. The problem arises when

G over Gamma.

The SWs frequencies are finally identified by the maxima of the power spectra of the

$ x $

component of magnetization

x

2

$ \pm $

0.3 & 0.66

Plus or minus.

$ t $

}

t

&

$ 69.41 $

&

69.41: Sixty-nine point four one

and express

$ \beta ( W ) $

in terms of

beta of W

the value of

$ L $

( Denmark , capital , Copenhagen ) could be % the set containing the templates `` What is the capital of

L

the spins form an AFM double-cone structure with a modulated basal-plane moment while the moments are predominantly pointing along the

$ c $

axis~

c

$ M ( x ) $

is nonsingular modulo

M of x.

and

$ T_D $

is the Dingle temperature defined by

T sub D

for

$ i\neq j $

. As

i is not equal to j.

89

$ \pm $

0.06

Plus or minus.

be a genus

$ g $

closed Riemann surface with

g

the collection of all

$ r $

-element subsets of

r

in the

$ k $

-th iteration is written as

k

A prior which includes negative

$ \Lambda_ { 1,2 } $

would certainly avoid the issue of vanishing prior support at

Lambda sub one comma two.

Let

$ u\in \F_ { q^2 } \backslash \fq $

. The number of

you belongs to the set of F sub q squared backslash fq

which is affected by spurious resonances if

$ \kappa_0 \in \mathfrak { S } ( \Delta , \Omega_\Sigma ) $

. On the contrary , if we assume that the

Kappa sub zero belongs to the set of S, enclosed in parentheses Delta, Omega sub Sigma.

We have shown that two wave packets in the

$ -Y $

region are annihilated ; the unbounded behavior occurs because the pulses reach the region where the potential is significant . As we can see in Fig.~

Negative Y.

Then there is a canonical one-to-one correspondence between

$ w^ { * } $

-continuous convolution semigroups of states invariant under the adjoint action of

w raised to the power of asterisk.

or for fixed

$ \tau $

and uniform

Tau.

for

$ \delta \in ( 0,1 ) $

, for any

Delta belongs to the open interval from 0 to 1.

and

$ z ' $

.

z prime

is Strongly monotone if whenever flipping a match between a team from

$ C $

and a team from

C

semigroup of contractions

$ S ( t ) $

,

S of t.

In this section we develop the vital constituent of our procedure for growing

$ 2 $

-trees by iteratively obtaining common neighbors of prescribed pairs of adjacent vertices . The main result is Theorem~

two

Here each

$ \alpha_i $

is a loop which goes around

Alpha sub i.

in

$ \bar { P } _ { n } $

.

P bar sub n

$ r \in \ { \mathbb\ N $

, and Gaussian-weighted space

r belongs to the set of natural numbers.

$ x $

parametrizes the dimensionless target relative speed . Small values of

x

$ E_k $

=

E subscript k.

Changing

$ n $

into a new variable

n

is adjacent to

$ u $

. If the condition stated in

u

so the

$ \omega_0 $

term in the sum can be dropped . The phase fluctuation

omega sub zero.

for even integers

$ k $

with

k

the value of

$ \Delta $

is larger , when the components become anisotropic and this results in a larger anisotropy . We note that if we are in the part of the phase diagram where

Delta.

where

$ L $

is the length of the serpentine . Integrating , we obtain

L

Images are collected every

$ 2 $

m from the trajectory with no preprocessing . We construct a point cloud submap with the global coordinates by accumulating 3D LiDAR data and sampling the points visible in the

two

80

$ \pm $

1.90 km s

Plus or minus.

Notice that

$ \mathcal { L } _ { \pm } $

have the form of a generator of a quantum dynamical semigroup~

L sub plus minus.

We assume that

$ \bm { Y } $

can be inferred from

Y

and

$ V_S $

V sub S

So

$ Y\rightarrow Y/H $

factors through

Y goes to Y over H

&

$ 0.38 $

&

0.38: None

Their calculation only requires a knowledge of the absorptive parts of CFFs and the

$ D $

-term form factor in the physical domain of the DVCS reaction . Moreover , it does not necessarily imply the analytic continuation in

D

The idea is now to find a linear combination of these

$ p ( x+k ) $

that is equal to

pee of eks plus kay

region

$ M_s $

is bounded , with

M sub s

em Defining

$ T $

as the index set of the non-bound constraints in

T

in

$ \GrVect_ { \mathbb\ { F } ^ { \mathsf\ { G } $

.

Gr-Vect sub F to the power of G

We write

$ \alpha\colon i \to j $

to indicate that

Alpha, colon, i, to, j.

Define a vector

$ u_X $

by

u sub X.

is the string operator that moves anyon

$ v $

downward along the

v

Denote the

$ k $

-th square in

k

Consider a simple error model where each two-qubit Clifford

$ V $

is independently affected by a unitary noise

V

em The scale dependent factor

$ f $

, which is calculated from the lowest lying charmonium state , introduces a correction of

f

The discriminator

$ D $

, classifies

D

or

$ h=2 , p > 3 $

h equals 2, p greater than 3.

619e

$ - $

9 & 2091 & 7.033e

minus

counts the number of free neighbors of

$ w $

given the configuration

w

in

$ K $

;

K

can be computed exactly only in the cases

$ p=2 $

using Mehta 's integral formula and

p equals 2

is

$ \H^n $

-contained in

H to the power of n

-complex does not interact with the new components that contain the points

$ p_i ( 0 ) $

. We will see shortly that the contributions from such vertices vanish . In order to force some interaction with the new points , consider the same

p sub i of 0

near the horizon

$ x=x_a $

: %

x equals x sub a.

or there exists an isoperimetric set

$ A $

such that

A

and Ag

$ _ { 1-x } $

Mn

One minus x.

We can write

$ g_j = \sigma_1^2 $

after applying a global conjugation move by a power of

g sub j equals sigma sub 1 squared.

changes sign whenever

$ u^2 $

takes the value

u squared

as

$ \mc G_s $

. Any fixed

m c G sub s

Let

$ \tilde B $

denote the image of

tilde B.

-modules over with

$ k $

an arbitrary unital commutative ring . Let us denote

k

and for

$ i=2,3 $

,

i equals 2, 3.

This shows that

$ \mu $

satisfies the associative axiom . One can similarly show that

mu.

8

$ \pm $

0.1 &

Plus or minus.

are functions of

$ f $

and do not involve parameters

f

shows that when

$ \xi_0 $

is super-heavy tailed , then the limiting behavior of

Xi sub zero.

associated with topics

$ T $

. Let

T

-secant variety to the Veronese embedding

$ v_ { d } ( \mathbb { P } ^ { m } ) $

is computed for

v sub d, capital P to the power of m

-th infinitesimal neighborhood of

$ 0 $

in

zero

is due to the definition of

$ d_\mu $

, which is a result of the definition of

d sub mu.

each VR headset

$ u $

}

u

and

$ H_t $

given , distribution of

H sub t.

the induced actions of

$ t_iq_ { jk } $

and

tee sub i q sub j k

for 1-Well and

$ 32^4 = \text { 1,048,576 } $

for 2-Wells . Note that for a fixed grid size , the state grows exponentially with the number of Wells -- -i.e.

32 to the power of 4 equals 1,048,576.

connects

$ P_0 $

with

P sub zero.

&

$ 0.28 $

&

0.28: None

Latent space image

$ \Vec { z } _0 $

and source embedding

vector z sub zero.

in the training set

$ \mathcal { P } $

. Step 6 employs the SELECTION process to navigate from the root node

P in calligraphic font.

for task

$ \alpha $

is defined as :

Alpha.

and

$ \pm\ { \rm\ Im\omega_\ { \boldsymbol\ q $

in the CM frame

Plus or minus, the imaginary part of omega sub q.

with

$ X $

not dependent on

X