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
follows by construction of the sequence	$ v_j $	. For the second part of the claim , note that the ordering of indices to form	v sub j.
is the vector whose	$ N $	components are the spins on the	N
are hybridized by	$ \Delta_ { 1 } $	. The green	Delta one.
5	$ \pm $	0.5 & 16	Plus or minus.
and	$ S_L^n $	indices to distinguish between regular and chaotic orbits in dynamical systems whose phase space dimension is higher than two . As a representative case of such a system we consider	S sub L to the power of n.
&	$ 65.1 $	& - & Theoretical &	65.1: Sixty-five point one
for each topology tree	$ \tau $	. After	Tau.
is	$ M $	-indiscernible and	M
and can only take values of	$ \pm 1 $	. We consider the following seven configurations :	Plus or minus one.
ranges from	$ 0.5 $	to	0.5: None
we have a right adjoint	$ \Res_f $	to	Residue of f.
are rigid in	$ X \oplus_\infty \mathbb { R } $	. The result now follows from Proposition	X plus infinity R.
em If	$ \|I\|=2 $	, then with probability	Absolute value of I equals 2.
em	$ \beta \in [ 0,1 ) $	: exponential decay rate for the moment squared estimates	Beta belongs to the interval from 0 to 1, excluding 1.
where each	$ g_i $	belongs to	g sub i.
and H	$ \alpha $	lines that form over broad atmospheric regions reveal a high complexity when searching for a model able to match all considered spectral lines . The temperature structure itself plays a	Alpha.
replaced by	$ p $	and	p
we have	$ c_\sB ( \xx ) \sqsubset_l\xx $	. If	c sub B of x is a proper subset of l sub x.
3	$ \pm $	1.9 & 70.1	Plus or minus.
% The coherent coupling owing to the T field depends on the length	$ L_z $	of nanorod and the background refractive index	L sub z.
By induction on	$ \alpha $	: if	Alpha.
with respect to	$ \pi^\ast M $	. Then the subvarieties	pi to the power of M.
and	$ \varepsilon_p $	, we can recast the constraints in terms of up and down quarks as shown in table	Epsilon p.
-layers are shifted by	$ \frac { 1 } { 2 } \vec { b } $	upon stacking along	one-half times vector b
Parabolic macroscopic dynamics :	$ \forall h\in\mathcal { H } $		for all h in the set H
and	$ \mu_2 $	associated with the conserved charges for	mu sub 2
transitioning to compartment	$ Y $	from	Y
	$ w ( \mathcal { P } ) \leq [ W^ { t } ] _ { ij } $	. Because all terms appearing in the	w of P is less than or equal to the quantity W raised to the power of t, subscript i j.
% When	$ n=0 $	, the reflection	n equals zero.
the original feature maps	$ f_1 , f_2 , f_3 $	and the warped feature maps	f sub 1, f sub 2, f sub 3
We extend these operation to take as input triples of tournaments	$ ( T^1 , T^2 , T^3 ) $	with different sizes of vertex sets . Suppose that	T raised to the power of 1, T raised to the power of 2, T raised to the power of 3.
	$ n=1,2 , ... $	such that	n equals one, two, ellipsis.
&	$ Y_2 $	&	Y sub 2.
By law of cosines at the vertex	$ w $	we have	w
transversely to	$ C_x $	in	C sub x.
in the fermionic theory corresponds to the combination	$ \Lambda M/u $	in the bosonic theory	Lambda M over u
constructed for three non-relativistic spin-	$ \frac { 1 } { 2 } $	particles previously described in References	One-half.
and	$ \Delta E=\omega_0 $	for the transition	Delta E equals omega sub zero.
for	$ x\in K $	and all	x belongs to K.
with	$ P_ { 0 } $	the hardware power consumed by the base station and mobile terminals , and	P sub 0.
em if	$ \circ ( \alpha_i ) > 0 $	,	Circle of alpha sub i is greater than zero.
and inaccurate ones with	$ F $	, we can derive an accuracy metric	F
	$ and the other at a top port ( TPA ) at $		and the other exponent of atop, open parenthesis, TPA, close parenthesis, at.
Furthermore we assume all schemes to be separated and of finite type over	$ k $	and rings are not necessarily commutative . For the convenience of the reader , we give the following table of notations :	k
	$ N $	S-	N
	$ \pm $	0.23	Plus or minus.
-descendants of	$ \Delta_0 $	in	Delta sub zero.
can be computed in perturbation theory as a series in	$ \alpha_s ( Q ) $	, while the matrix elements	Alpha s of Q.
where	$ R $	is a contraction ( i.e .	R
be vertices at	$ m $	-distance	m
et~	$ Q $	; en particulier , si	Q
& 26260 &	$ P $	–62	P
at a point	$ P $	, located at distance	P
Suppose there is a free action of a finite group	$ G $	on	G
For a subfield	$ F $	of	F
5 &	$ 3.2\pm0.2 $	&	3.2 plus or minus 0.2.
th tensor margin and	$ r $	th channel &	r
% &	$ P_0=2 $	,	P zero equals two.
together form an	$ r $	-scattered tuple . Indeed , the	r
and let	$ g $	be an observable function for the Koopman operator	g
Note that this operator	$ Y $	is related to the	Y
and	$ \nu $	-almost every point	nu.
versus the half-light radius	$ R_e $	.	R sub e
we are given a	$ k $	-submodular function	k
say	$ l $	, by definition of	l
66	$ \pm $	73.70 & -63.10	Plus or minus.
we require a false alarm probability of	$ 0.005 $	. Other criteria include selecting sources , where the standard deviation of all the	Zero point zero zero five.
spectrum of EXO~1846 -- 031 because we do not have these models for the	$ \delta $	-Kerr metric and their construction would be beyond the scope of the present work . In our analysis of EXO~1846 -- 031 , the electron density is supposed to be	delta.
&	$ \emptyset $	&	Empty set.
the renormalized momentum	$ k $	is	k
gives	$ c_ { 0l } =0 $	,	c sub zero l equals zero.
The functions	$ f^+ $	and	f plus
there exists	$ N\trianglelefteq F_2 $	such that	N triangle left F sub 2
\\ 3 &	$ 2 $	&	two
Index the	$ l $	the codeword in subcodebook	l
	$ \alpha $	/Fe ] is defined as the average of [ X	Alpha.
and will be made use of again to establish the SUSYQM of the associated hypergeometric-like functions of	$ H_0 ( x ) $	in Section	H sub zero of x
Generate	$ i $	th layer distribution ,	i
otherwise	$ m \le 2 $	. The strategy is the same as	m less than or equal to 2
of length	$ n+2 $	, then the Kruskal rank of	n plus 2
	$ z $	,	z
and coil sensitivities	$ C $	. Reconstruction is an ill-posed inverse problem , so a regularization term is typically employed to obtain a faithful solution	C
we can twist	$ \tau $	by the	Tau.
be the bad components of	$ H $	. We construct an auxiliary edge-weighted graph	H
the coefficients	$ c_3 $	and	c sub 3
of a function	$ u \colon \Omega \to \bR $	is defined as	u, colon, Omega, to, bR
does not depend on the variable	$ p $	. Thus , by generalizing the above equation to cover the cases of	p
is not convex for	$ \alpha < 1 $	. Nevertheless , some properties hold for all	Alpha is less than one.
Since	$ I $	is contained in the Jacobson radical ,	I
recall that we denote by	$ C_k $	the	C sub k.
for both	$ O ( \alpha ) $	and	O of alpha.
for all	$ j=s , \dots , t $	, the unit vector in the direction of	j equals s, dot dot dot, t
-GDP on the unit circle	$ S^1 $	. %	S one.
and	$ m $	. We would like to apply Corollary	m
with degree not less than	$ s $	with	s
Here	$ H ( x ) $	is the Heaviside step function . Moreover , all problem parameters are imposed as non-constant case as follows with unit domain parameters	H of x.
holds if and only if	$ y=0 $	.	y equals zero.
When classified by the level	$ i $	of the derivation tree , sentences in each class have the same number of inserted adjuncts , indicating that they have a consistent degree of variation and similar sentence structure	i
	$ k_ { \mathrm { sat , r } } = 2~\mathrm { mmh^ { -1 } } $	,	k sub s a t, r equals 2 millimeters h to the power of negative 1
Let	$ n_1 $	be the least common multiple of the set of all	n sub 1

follows by construction of the sequence

$ v_j $

. For the second part of the claim , note that the ordering of indices to form

v sub j.

is the vector whose

$ N $

components are the spins on the

N

are hybridized by

$ \Delta_ { 1 } $

. The green

Delta one.

5

$ \pm $

0.5 & 16

Plus or minus.

and

$ S_L^n $

indices to distinguish between regular and chaotic orbits in dynamical systems whose phase space dimension is higher than two . As a representative case of such a system we consider

S sub L to the power of n.

&

$ 65.1 $

& - & Theoretical &

65.1: Sixty-five point one

for each topology tree

$ \tau $

. After

Tau.

is

$ M $

-indiscernible and

M

and can only take values of

$ \pm 1 $

. We consider the following seven configurations :

Plus or minus one.

ranges from

$ 0.5 $

to

0.5: None

we have a right adjoint

$ \Res_f $

to

Residue of f.

are rigid in

$ X \oplus_\infty \mathbb { R } $

. The result now follows from Proposition

X plus infinity R.

em If

$ |I|=2 $

, then with probability

Absolute value of I equals 2.

em

$ \beta \in [ 0,1 ) $

: exponential decay rate for the moment squared estimates

Beta belongs to the interval from 0 to 1, excluding 1.

where each

$ g_i $

belongs to

g sub i.

and H

$ \alpha $

lines that form over broad atmospheric regions reveal a high complexity when searching for a model able to match all considered spectral lines . The temperature structure itself plays a

Alpha.

replaced by

$ p $

and

p

we have

$ c_\sB ( \xx ) \sqsubset_l\xx $

. If

c sub B of x is a proper subset of l sub x.

3

$ \pm $

1.9 & 70.1

Plus or minus.

% The coherent coupling owing to the T field depends on the length

$ L_z $

of nanorod and the background refractive index

L sub z.

By induction on

$ \alpha $

: if

Alpha.

with respect to

$ \pi^\ast M $

. Then the subvarieties

pi to the power of M.

and

$ \varepsilon_p $

, we can recast the constraints in terms of up and down quarks as shown in table

Epsilon p.

-layers are shifted by

$ \frac { 1 } { 2 } \vec { b } $

upon stacking along

one-half times vector b

Parabolic macroscopic dynamics :

$ \forall h\in\mathcal { H } $

for all h in the set H

and

$ \mu_2 $

associated with the conserved charges for

mu sub 2

transitioning to compartment

$ Y $

from

Y

$ w ( \mathcal { P } ) \leq [ W^ { t } ] _ { ij } $

. Because all terms appearing in the

w of P is less than or equal to the quantity W raised to the power of t, subscript i j.

% When

$ n=0 $

, the reflection

n equals zero.

the original feature maps

$ f_1 , f_2 , f_3 $

and the warped feature maps

f sub 1, f sub 2, f sub 3

We extend these operation to take as input triples of tournaments

$ ( T^1 , T^2 , T^3 ) $

with different sizes of vertex sets . Suppose that

T raised to the power of 1, T raised to the power of 2, T raised to the power of 3.

$ n=1,2 , ... $

such that

n equals one, two, ellipsis.

&

$ Y_2 $

&

Y sub 2.

By law of cosines at the vertex

$ w $

we have

w

transversely to

$ C_x $

in

C sub x.

in the fermionic theory corresponds to the combination

$ \Lambda M/u $

in the bosonic theory

Lambda M over u

constructed for three non-relativistic spin-

$ \frac { 1 } { 2 } $

particles previously described in References

One-half.

and

$ \Delta E=\omega_0 $

for the transition

Delta E equals omega sub zero.

for

$ x\in K $

and all

x belongs to K.

with

$ P_ { 0 } $

the hardware power consumed by the base station and mobile terminals , and

P sub 0.

em if

$ \circ ( \alpha_i ) > 0 $

,

Circle of alpha sub i is greater than zero.

and inaccurate ones with

$ F $

, we can derive an accuracy metric

F

$ and the other at a top port ( TPA ) at $

and the other exponent of atop, open parenthesis, TPA, close parenthesis, at.

Furthermore we assume all schemes to be separated and of finite type over

$ k $

and rings are not necessarily commutative . For the convenience of the reader , we give the following table of notations :

k

$ N $

S-

N

$ \pm $

0.23

Plus or minus.

-descendants of

$ \Delta_0 $

in

Delta sub zero.

can be computed in perturbation theory as a series in

$ \alpha_s ( Q ) $

, while the matrix elements

Alpha s of Q.

where

$ R $

is a contraction ( i.e .

R

be vertices at

$ m $

-distance

m

et~

$ Q $

; en particulier , si

Q

& 26260 &

$ P $

–62

P

at a point

$ P $

, located at distance

P

Suppose there is a free action of a finite group

$ G $

on

G

For a subfield

$ F $

of

F

5 &

$ 3.2\pm0.2 $

&

3.2 plus or minus 0.2.

th tensor margin and

$ r $

th channel &

r

% &

$ P_0=2 $

,

P zero equals two.

together form an

$ r $

-scattered tuple . Indeed , the

r

and let

$ g $

be an observable function for the Koopman operator

g

Note that this operator

$ Y $

is related to the

Y

and

$ \nu $

-almost every point

nu.

versus the half-light radius

$ R_e $

.

R sub e

we are given a

$ k $

-submodular function

k

say

$ l $

, by definition of

l

66

$ \pm $

73.70 & -63.10

Plus or minus.

we require a false alarm probability of

$ 0.005 $

. Other criteria include selecting sources , where the standard deviation of all the

Zero point zero zero five.

spectrum of EXO~1846 -- 031 because we do not have these models for the

$ \delta $

-Kerr metric and their construction would be beyond the scope of the present work . In our analysis of EXO~1846 -- 031 , the electron density is supposed to be

delta.

&

$ \emptyset $

&

Empty set.

the renormalized momentum

$ k $

is

k

gives

$ c_ { 0l } =0 $

,

c sub zero l equals zero.

The functions

$ f^+ $

and

f plus

there exists

$ N\trianglelefteq F_2 $

such that

N triangle left F sub 2

\\ 3 &

$ 2 $

&

two

Index the

$ l $

the codeword in subcodebook

l

$ \alpha $

/Fe ] is defined as the average of [ X

Alpha.

and will be made use of again to establish the SUSYQM of the associated hypergeometric-like functions of

$ H_0 ( x ) $

in Section

H sub zero of x

Generate

$ i $

th layer distribution ,

i

otherwise

$ m \le 2 $

. The strategy is the same as

m less than or equal to 2

of length

$ n+2 $

, then the Kruskal rank of

n plus 2

$ z $

,

z

and coil sensitivities

$ C $

. Reconstruction is an ill-posed inverse problem , so a regularization term is typically employed to obtain a faithful solution

C

we can twist

$ \tau $

by the

Tau.

be the bad components of

$ H $

. We construct an auxiliary edge-weighted graph

H

the coefficients

$ c_3 $

and

c sub 3

of a function

$ u \colon \Omega \to \bR $

is defined as

u, colon, Omega, to, bR

does not depend on the variable

$ p $

. Thus , by generalizing the above equation to cover the cases of

p

is not convex for

$ \alpha < 1 $

. Nevertheless , some properties hold for all

Alpha is less than one.

Since

$ I $

is contained in the Jacobson radical ,

I

recall that we denote by

$ C_k $

the

C sub k.

for both

$ O ( \alpha ) $

and

O of alpha.

for all

$ j=s , \dots , t $

, the unit vector in the direction of

j equals s, dot dot dot, t

-GDP on the unit circle

$ S^1 $

. %

S one.

and

$ m $

. We would like to apply Corollary

m

with degree not less than

$ s $

with

s

Here

$ H ( x ) $

is the Heaviside step function . Moreover , all problem parameters are imposed as non-constant case as follows with unit domain parameters

H of x.

holds if and only if

$ y=0 $

.

y equals zero.

When classified by the level

$ i $

of the derivation tree , sentences in each class have the same number of inserted adjuncts , indicating that they have a consistent degree of variation and similar sentence structure

i

$ k_ { \mathrm { sat , r } } = 2~\mathrm { mmh^ { -1 } } $

,

k sub s a t, r equals 2 millimeters h to the power of negative 1

Let

$ n_1 $

be the least common multiple of the set of all

n sub 1