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
\neq0	$ ( $	j	opening parenthesis
	$ ^\textrm { \scriptsize 48 } $	,	48
methods by	$ 25.7\ % $	and	25.7 percent.
We need to show that some subsequence of	$ u_k $	converges strongly to a limit	u subscript k.
and	$ h $	is increasing , we can define	h
Then for every	$ i \in I $	such that	i belongs to the set I.
	$ l $	, we find the energy per unit length of the ground state %	l
&	$ < 0.0001 $	&	less than 0.0001.
the root committee is	$ C_ { 1 } $	, its left child is	C sub 1.
removing at least the	$ 2 $	edges	two
	$ d $	is the distance to the source , and	d
which underlies both the	$ \textsf { CSWAP } $	and the data-copying operations . We ignore the number of required	C W A
in	$ F $	. Then , the measure	F
In this case the warp factor	$ w $	in the metric ansatz	w
complex	$ K $	-theory , but for larger	K
coincides with the	$ n_0 $	-th eigenvalue of the limit problem	n sub zero
includes the case of stationary initial condition with parameter	$ \mu/2 $	. This is related to the fact that the corresponding characteristic direction , in this case , is the diagonal direction	mu divided by 2
in order for this to be precise we need to consider	$ F $	as a functor from	F
can be computed in	$ \Oh ( n ) $	time .	big oh of n.
-graph	$ H $	with	H
is the statistical weight of each angle	$ \theta_ { ij } $	. We account for the spatial correlation between pixels introduced by the telescope beam and set the statistical weights to	Theta sub i j.
&	$ 0.38 $	& TH & He-sdO	0.38: None
/d	$ z $	}	z
belonging to the species~	$ B $	. If the atomic motion in mode~	B
We implement suitable loss function with regard to	$ \Theta $	and train neural network using TensorFlow framework to optimize this loss function .	Theta.
is a boundary of a disk	$ D $	, and equivalent points are connected by chords	D
From the boundary conditions at	$ \kappa^ * $	, it comes	kappa star
of the Drinfeld center	$ \cZ ( \cS ) $	of	c Z, c S
21	$ \pm $	0.36 &	Plus or minus.
to be a field of characteristic coprime to	$ n $	in this definition . This means that we accept as Kummer extensions those for which this condition is not satisfied . Sometimes it will be more convenient to work with	n
for	$ \overline { v } $	,	v bar.
acts on different	$ J_z $	in one	J sub z.
	$ \beta $	, is relatively small compared with the data norm	Beta.
the queries to decode	$ \mathbf { s } _t $	in HiT/LigHT-DVAE are computed from	s sub t.
in	$ \cat_ { \infty , G } \coloneqq \func ( \orbit_G\op , \cat_ { \infty } ) $	and we will use the underline notation	The category infinity G is defined as the functor from the opposite of the orbit G to the category infinity.
the initial value	$ \hat { X } _ { 0 } ^ { i , \Delta } =X^ { i } _ { 0 } $	and	Hat X sub 0 superscript i comma Delta equals X superscript i sub 0.
	$ \mathcal { D } $	, into the training set	D
follows since	$ n_ { \mu_0 } ( \omega ) $	is decreasing . By Lemma~	n sub mu zero open parenthesis omega close parenthesis
1 \\ 1730	$ - $	3350 & 4 & 1.5 & 26 & 3.5	minus
em the prior	$ P_1 $	is such that	P sub 1.
	$ ^\dagger $	&	dagger.
and define	$ K_ { \max } = \underset { m \in [ M ] } { \max } K_ { m } $	, so we can get the second inequality .	K sub max equals the maximum of K sub m, where m belongs to the set of integers from 1 to M.
is a metric and	$ \mu $	is the minimum , we	mu.
&	$ 0.114 ( 13 ) $	&	0.114 times the quantity 13.
where	$ M $	and	M
the ACDR is able to implicitly enhance the sequence representation~	$ V $	itself by allowing	V
and	$ \gamma $	such that	Gamma.
to	$ j $	at time	j
which allows us to choose the size of the model space as	$ N=2 $	,	N equals two.
and	$ \nu_- $	N-	Nu minus.
	$ q $	-Poincar {	q
of	$ D $	.	D
Let	$ u $	be the minimizer . Then it holds that	u
instrument2	$ > $		greater than
To give a positive answer to the Modular Isomorphism Problem in this case we should prove that	$ G\cong H $	, or equivalently that	G is isomorphic to H.
	$ v ( h ) $	denotes the composition of	v of h
of	$ A $	for the	A
is the average time increment of unit vectors along directions in	$ \mathcal { A } _ { p } $	; it is a constant function of	A sub p.
: Surviving user set	$ \mathcal { U } ' $	,	U prime
Each cover tree is constructed by inserting one point of	$ Q $	at a time . Let	Q
- produces dust masses	$ 0.39 $	and	0.39: Zero point three nine
for all	$ k $	. To get a graph with quantum symmetry and trivial automorphism group , we must find a linear constraint system whose homogeneous solution group is perfect . For this we	k
	$ W $	denotes the continuous wavelet transform and	W
em a	$ \left ( \Psi , \alpha\right ) $		Left parenthesis, Psi, comma, alpha, right parenthesis.
for	$ t \geq 0 $	, almost surely ,	t is greater than or equal to zero.
a physics-based forecasting tool that calculates	$ > $	10	greater than
and	$ n_ { \textrm { vis } } ( T ) $	, we can express the RISC model in the THz band as	n sub vis of T
	$ < $	NA	less than
-instanton object	$ A $	defined by the exact triangle	A
it is evident that the sample selection algorithm can attain a	$ \mathcal { O } ( M^ { 1/2 } ) $	local regret bound . This implies that the Lagrangian function may converge to zero when	Big O of M to the power of one-half
	$ 2 $	-covariant tensor field % defined by	two
&	$ < 2 $	&	Less than two.
where	$ z $	and	z
&	$ \frac { 40 } { 255 } $	&	forty over two hundred and fifty.
there exists	$ N \in \mathbb { N } $	such that for every	N belongs to the set of natural numbers.
We consider the number of heads to be	$ 10 $	, and start the training for	Ten.
	$ i $	) Correlation- and frustration-induced nearly flat band in proximity to	i
	$ ^\textrm { \scriptsize 101 } $	,	101
4	$ \mu $	m surface roughness . The stainless steel is customized with low radioactivity and their inner surfaces are electrically polished . Another important source of Radon is the underground air ,	mu.
an	$ F $	-move corresponds to a multiply-controlled unitary operator that changes the state of one target qudit , depending on the state of four control qudits ( see Fig.~	F
measured the reddening of the star Sk-69	$ ^\circ $	203 ,	degree.
&	$ - $		minus
	$ P $	} { pressure , Pa or -- }	P
We are interested in the number	$ L_n $	of edges in the leftmost path of	L sub n.
and therefore in	$ L^p ( M , \bar g ) $	for any	L to the power of p, open parenthesis, M, comma, bar g, close parenthesis.
~	$ \mu $	m . This results in a qubit spacing of	mu.
In discrete and finite MDPs we can compute the state-transition graph	$ G= ( V , E ) $	of an MDP . In this section , we will revise how to learn the minimum action distance from the graph adjacency matrix . A state-transition graph	G equals open parenthesis V comma E close parenthesis.
or	$ g $	modes with small radial order . Therefore , we decided to apply our resonance condition to the envelope	g
where	$ \XiB_t $	is a	ksi B sub t
with	$ x\in\R $	.	x belongs to the set of real numbers.
start with individual initial value	$ v_i $	that can be a vector with finite number of elements , individually compose an	v sub i
For each	$ X $	, we fix a pair of	X
the results are not very sensitive to the amplitude of	$ \epsilon $	. The profile is rather symmetrical around zero , but shifting the zero level lower would not change the results significantly . The essential property of the	epsilon.
6242	$ \pm $	.0245 & .6424	Plus or minus.
we know that there exists a reconstruction consisting of	$ n $	cameras , not all lying on a line . It follows from the Sylvester-Gallai theorem	n
suppose	$ D_ { z_i } \in G\setminus \ { D_ { g_i } \ } $	. By definition of	D sub z sub i belongs to G set minus D sub g sub i.
and	$ 2i $	in positions	Two i.
in the cross terms is less than	$ \|k-q\| $	. Therefore by Muirhead 's inequality of Proposition~	absolute value of k minus q
We can see that the best performance of our proposed DIDA-CDR is achieved when	$ \alpha = 1 $	, which presents that	Alpha equals one.
-by-	$ n $	( possibly dense ) symmetric eigenproblem in each iteration of this algorithm , and the algorithm may converge very slowly if there is no subspace speeding up . % A	n

\neq0

$ ( $

j

opening parenthesis

$ ^\textrm { \scriptsize 48 } $

,

48

methods by

$ 25.7\ % $

and

25.7 percent.

We need to show that some subsequence of

$ u_k $

converges strongly to a limit

u subscript k.

and

$ h $

is increasing , we can define

h

Then for every

$ i \in I $

such that

i belongs to the set I.

$ l $

, we find the energy per unit length of the ground state %

l

&

$ < 0.0001 $

&

less than 0.0001.

the root committee is

$ C_ { 1 } $

, its left child is

C sub 1.

removing at least the

$ 2 $

edges

two

$ d $

is the distance to the source , and

d

which underlies both the

$ \textsf { CSWAP } $

and the data-copying operations . We ignore the number of required

C W A

in

$ F $

. Then , the measure

F

In this case the warp factor

$ w $

in the metric ansatz

w

complex

$ K $

-theory , but for larger

K

coincides with the

$ n_0 $

-th eigenvalue of the limit problem

n sub zero

includes the case of stationary initial condition with parameter

$ \mu/2 $

. This is related to the fact that the corresponding characteristic direction , in this case , is the diagonal direction

mu divided by 2

in order for this to be precise we need to consider

$ F $

as a functor from

F

can be computed in

$ \Oh ( n ) $

time .

big oh of n.

-graph

$ H $

with

H

is the statistical weight of each angle

$ \theta_ { ij } $

. We account for the spatial correlation between pixels introduced by the telescope beam and set the statistical weights to

Theta sub i j.

&

$ 0.38 $

& TH & He-sdO

0.38: None

/d

$ z $

}

z

belonging to the species~

$ B $

. If the atomic motion in mode~

B

We implement suitable loss function with regard to

$ \Theta $

and train neural network using TensorFlow framework to optimize this loss function .

Theta.

is a boundary of a disk

$ D $

, and equivalent points are connected by chords

D

From the boundary conditions at

$ \kappa^ * $

, it comes

kappa star

of the Drinfeld center

$ \cZ ( \cS ) $

of

c Z, c S

21

$ \pm $

0.36 &

Plus or minus.

to be a field of characteristic coprime to

$ n $

in this definition . This means that we accept as Kummer extensions those for which this condition is not satisfied . Sometimes it will be more convenient to work with

n

for

$ \overline { v } $

,

v bar.

acts on different

$ J_z $

in one

J sub z.

$ \beta $

, is relatively small compared with the data norm

Beta.

the queries to decode

$ \mathbf { s } _t $

in HiT/LigHT-DVAE are computed from

s sub t.

in

$ \cat_ { \infty , G } \coloneqq \func ( \orbit_G\op , \cat_ { \infty } ) $

and we will use the underline notation

The category infinity G is defined as the functor from the opposite of the orbit G to the category infinity.

the initial value

$ \hat { X } _ { 0 } ^ { i , \Delta } =X^ { i } _ { 0 } $

and

Hat X sub 0 superscript i comma Delta equals X superscript i sub 0.

$ \mathcal { D } $

, into the training set

D

follows since

$ n_ { \mu_0 } ( \omega ) $

is decreasing . By Lemma~

n sub mu zero open parenthesis omega close parenthesis

1 \\ 1730

$ - $

3350 & 4 & 1.5 & 26 & 3.5

minus

em the prior

$ P_1 $

is such that

P sub 1.

$ ^\dagger $

&

dagger.

and define

$ K_ { \max } = \underset { m \in [ M ] } { \max } K_ { m } $

, so we can get the second inequality .

K sub max equals the maximum of K sub m, where m belongs to the set of integers from 1 to M.

is a metric and

$ \mu $

is the minimum , we

mu.

&

$ 0.114 ( 13 ) $

&

0.114 times the quantity 13.

where

$ M $

and

M

the ACDR is able to implicitly enhance the sequence representation~

$ V $

itself by allowing

V

and

$ \gamma $

such that

Gamma.

to

$ j $

at time

j

which allows us to choose the size of the model space as

$ N=2 $

,

N equals two.

and

$ \nu_- $

N-

Nu minus.

$ q $

-Poincar {

q

of

$ D $

.

D

Let

$ u $

be the minimizer . Then it holds that

u

instrument2

$ > $

greater than

To give a positive answer to the Modular Isomorphism Problem in this case we should prove that

$ G\cong H $

, or equivalently that

G is isomorphic to H.

$ v ( h ) $

denotes the composition of

v of h

of

$ A $

for the

A

is the average time increment of unit vectors along directions in

$ \mathcal { A } _ { p } $

; it is a constant function of

A sub p.

: Surviving user set

$ \mathcal { U } ' $

,

U prime

Each cover tree is constructed by inserting one point of

$ Q $

at a time . Let

Q

- produces dust masses

$ 0.39 $

and

0.39: Zero point three nine

for all

$ k $

. To get a graph with quantum symmetry and trivial automorphism group , we must find a linear constraint system whose homogeneous solution group is perfect . For this we

k

$ W $

denotes the continuous wavelet transform and

W

em a

$ \left ( \Psi , \alpha\right ) $

Left parenthesis, Psi, comma, alpha, right parenthesis.

for

$ t \geq 0 $

, almost surely ,

t is greater than or equal to zero.

a physics-based forecasting tool that calculates

$ > $

10

greater than

and

$ n_ { \textrm { vis } } ( T ) $

, we can express the RISC model in the THz band as

n sub vis of T

$ < $

NA

less than

-instanton object

$ A $

defined by the exact triangle

A

it is evident that the sample selection algorithm can attain a

$ \mathcal { O } ( M^ { 1/2 } ) $

local regret bound . This implies that the Lagrangian function may converge to zero when

Big O of M to the power of one-half

$ 2 $

-covariant tensor field % defined by

two

&

$ < 2 $

&

Less than two.

where

$ z $

and

z

&

$ \frac { 40 } { 255 } $

&

forty over two hundred and fifty.

there exists

$ N \in \mathbb { N } $

such that for every

N belongs to the set of natural numbers.

We consider the number of heads to be

$ 10 $

, and start the training for

Ten.

$ i $

) Correlation- and frustration-induced nearly flat band in proximity to

i

$ ^\textrm { \scriptsize 101 } $

,

101

4

$ \mu $

m surface roughness . The stainless steel is customized with low radioactivity and their inner surfaces are electrically polished . Another important source of Radon is the underground air ,

mu.

an

$ F $

-move corresponds to a multiply-controlled unitary operator that changes the state of one target qudit , depending on the state of four control qudits ( see Fig.~

F

measured the reddening of the star Sk-69

$ ^\circ $

203 ,

degree.

&

$ - $

minus

$ P $

} { pressure , Pa or -- }

P

We are interested in the number

$ L_n $

of edges in the leftmost path of

L sub n.

and therefore in

$ L^p ( M , \bar g ) $

for any

L to the power of p, open parenthesis, M, comma, bar g, close parenthesis.

~

$ \mu $

m . This results in a qubit spacing of

mu.

In discrete and finite MDPs we can compute the state-transition graph

$ G= ( V , E ) $

of an MDP . In this section , we will revise how to learn the minimum action distance from the graph adjacency matrix . A state-transition graph

G equals open parenthesis V comma E close parenthesis.

or

$ g $

modes with small radial order . Therefore , we decided to apply our resonance condition to the envelope

g

where

$ \XiB_t $

is a

ksi B sub t

with

$ x\in\R $

.

x belongs to the set of real numbers.

start with individual initial value

$ v_i $

that can be a vector with finite number of elements , individually compose an

v sub i

For each

$ X $

, we fix a pair of

X

the results are not very sensitive to the amplitude of

$ \epsilon $

. The profile is rather symmetrical around zero , but shifting the zero level lower would not change the results significantly . The essential property of the

epsilon.

6242

$ \pm $

.0245 & .6424

Plus or minus.

we know that there exists a reconstruction consisting of

$ n $

cameras , not all lying on a line . It follows from the Sylvester-Gallai theorem

n

suppose

$ D_ { z_i } \in G\setminus \ { D_ { g_i } \ } $

. By definition of

D sub z sub i belongs to G set minus D sub g sub i.

and

$ 2i $

in positions

Two i.

in the cross terms is less than

$ |k-q| $

. Therefore by Muirhead 's inequality of Proposition~

absolute value of k minus q

We can see that the best performance of our proposed DIDA-CDR is achieved when

$ \alpha = 1 $

, which presents that

Alpha equals one.

-by-

$ n $

( possibly dense ) symmetric eigenproblem in each iteration of this algorithm , and the algorithm may converge very slowly if there is no subspace speeding up . % A

n