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
1	$ & $	30.2	and
and	$ Q $	are the largest among all forward angles of self-intersection points of	Q
009 &	$ \pm $	0.009 &	Plus or minus.
As	$ \kappa $	increases , both	kappa.
mode than in the	$ P_2 $	mode . Strong non-adiabaticity reduces the temperature variation significantly so that the radius effect in the luminosity variation	P sub 2
	$ ^\textrm { \scriptsize 139 } $	,	139
has at least two separate connected components	$ C_1 , C_2 $	. Then there must exist a closed curve in the plane separating	C sub 1, C sub 2.
identified with	$ X $	and	X
corresponding to at least one `	$ j $	' value	j
satisfies the peak throughput requirement of	$ 20 $	~Gbps as stipulated in the 5G standard~	Twenty.
is dense in	$ \mathscr { H } _ { L } $	, the condition in	H sub L.
and a sink	$ t $	; ( ii )	t
&	$ 224^2 $	& 24 & 4.5 & 299 & 82.7	The equation you provided is: 224 squared.
and the two-element group	$ ( e , \tilde e ) $	. Let us consider the following group action : for all	(e, tilde e) : e, tilde e
the entries of	$ Df_t $	depend continuously on	D f sub t
are in the same block of	$ \sigma $	. Writing	Sigma.
then go through the weight classes in decreasing order to insert the pairs into the corresponding	$ Q_v $	. We explicitly give the pseudocode below as	Q sub v.
when the context	$ \mathcal { C } ^ { 1 } $	and	C one.
% Here	$ K $	is a hyperparameter that specifies the modality of the prior . For the domain of the US politics we choose	K
Consider a subFinsler manifold	$ M $	and fix	M
against	$ 1 $	, we see that	one
admits a neighbor-locating	$ k $	-coloring . A proper	k
defined by the set of points	$ ( x,0 ) $	where	x comma zero
approaches	$ t $	,	t
which realized the smaller value of	$ J $	. % % These optimized designs are specialized for their mechanical situation under the traction force	J
with large	$ l_z $	associated to a motion farther away from the	lz: None
as	$ J ' ( t ) $	. Thus ,	J prime of t
We derive a	$ 5\sigma $	detection limit of	Five sigma.
Then	$ u_0 - v_0 $	is a pluriharmonic distribution on	u sub zero minus v sub zero
contrary to the previous terms	$ \bbU_ { z,0 } ( \bk , t ) $	and	B B U sub z comma 0 open parenthesis B k comma t close parenthesis
there is a unique flat section	$ \widetilde { A } _a $	of	A tilde sub a
	$ - $		minus
Then we can choose the	$ r $	-state in Definition~	r
un espace de Segal complet est aussi un espace simplicial où le morphisme de Segal	$ \phi_k $	est une équivalence faible pour tout	phi sub k.
denoted by	$ h $	. After	h
is an element of	$ \Sscr_ { obs } $	. Therefore , we use Corollary~	S c o s.
and	$ b_n $	,	b sub n
This can be written in terms of the induced velocity field	$ u_1 ( \cdot ) $	of one prototypical eddy .	u sub 1, open parenthesis, dot, close parenthesis.
	$ G_1 , G_2 , \cdots , G_m $	with	G sub 1, G sub 2, dot dot dot, G sub m
of	$ \Phi $	is greater than	Phi.
for	$ i=1 , \ldots , r $	subject to the requirement that	i equals 1, dot dot dot, r.
and	$ G $	is the units of a central simple algebra over	G
is the component of	$ X_ { ( 1 ) } $	and	X sub 1
is	$ 0 $	, and the latter happens if and only if	zero
since the minor semiaxis of	$ E_2 $	is	E sub 2.
at location	$ n $	, such that	n
begins with pairs of edges	$ e_i , e'_i $	,	e prime i, e prime i
which contains	$ S $	in its interior and is a deformation retract of	S
satisfying	$ f ( x ) =0 $	and	f of x equals zero.
31	$ \pm $	1.69 & 0.9773	Plus or minus.
	$ A $	, and	A
and the Lipschtiz character of	$ \partial \Omega $	, with the property that if	Partial Omega.
and the fact that	$ \nabla $	is torsion-free . 4 follows because for all unit timelike vector fields	nabla.
be uncountable cardinals with	$ \kappa $	regular and assume that	kappa.
bosonic degrees of freedom must couple to in order to be able to extract the most entanglement with	$ \bar A $	. As we will see in Section~	A bar.
-th largest element of	$ \diag ( X ) $	, and	diag of X
&	$ P_y $	&	P sub y.
	$ \alpha^1 $	and	Alpha raised to the power of 1.
can be obtained by using the round-trip light phases from satellites	$ B $	and	B
sum over	$ i=1 , \ldots , n $	and	i equals 1, dot dot dot, n.
solutions in the presence of an asymptotically small activator diffusivity	$ \ep^2\ll 1 $	. We find that two types of BLS solutions , one of which is unconditionally linearly stable and the other unstable , can be constructed provided that the activator flux	e p squared is much less than one
that this yields the correct PN results for	$ U_L $	and	U sub L
six coupling modifiers corresponding to the tree-level Higgs boson couplings are defined :	$ \kappa_W $	,	kappa sub W
if	$ \vec { A } $	is a complex matrix and/or the constellation symbols are complex , the code rate is	vector A.
with elements	$ H_ { ij } = ( M_iM_j ) ^ { -1/2 } K_ { ij } $	where	H sub i j equals the inverse square root of the product of M sub i and M sub j, multiplied by K sub i j.
then the desired move	$ j $	is the one at the current node	j
A relation algebra is a Q-relation algebra iff it contains elements	$ p , q $	such that	p, q
described by coordinate	$ x_ { i } $	, is captured by the equipartition theorem ,	x subscript i
em For all	$ i , j $	the points	i, j
\boldmath	$ a $	} }	a
for all	$ x\in G $	, where	x belongs to G.
fluctuations of	$ z $	are suppressed and we recover N	z
&	$ 26.2\pm { 1.8 } $	&	Two point six plus or minus one point eight.
&	$ - $	& 57.7 & 71.1 & 69.6 & 69.9	minus
is infinite cyclic except the case where	$ M $	is covered by	M
as the hyper-parameter	$ \eta $	and iteration steps increase , the performance decreases , which suggests that excessive calibration is detrimental .	eta.
We write the dot product on	$ \mathbb { R } ^d $	as	R to the power of d.
% in a way that every cell in	$ \mathcal { P } $	is contained in a	P in calligraphic font.
does not affect the definition of the oracle parameter	$ \Psi_0 $	. Condition	Psi sub zero.
We conclude this section on a more optimistic note in regards to	$ Q $	queries that occur in randomly chosen indices . Recall ( from Section~	Q
&	$ -0.19688 $	&	Negative zero point one nine six eight eight.
The posterior-	$ L_2 $	pseudometric between	L sub 2.
the dissipation rate	$ \Gamma_ { \rm ED } $	from this effect is	Gamma sub E D
and	$ \pi $	being the CPDP algorithm , we have that %	pi.
The reason for the first equality is that since the first term of	$ u $	is	u
for a suitable constant	$ C > 0 $	. Taking	C is greater than zero.
and	$ V_2 $	, the individual with	V sub 2
for all	$ i\in\ { 1 , \cdots , n\ } $	. We estimate each vector	i belongs to the set of numbers from 1 to n.
Given distinct points	$ x , y \in X $	, set	x, y belong to X.
from	$ x_j $	. Take a power	x sub j.
	$ a_ { j } $		a sub j.
partition of	$ b $	} is a subset	b
and	$ m $	, nor on the momentum~	m
&	$ L_ { \sun } $		L sub sun.
It	$ \hat { o } $	diffusion process ]	o-hat symbol.
is Turing reducible to the degree	$ c $	of	c
has the same area as	$ K $	but a smaller perimeter . Note the following result , known as the	K
then we may abbreviate	$ W_I $	as	W subscript I.
is again an	$ N $	by	N
must contain a short rank-one element for any symmetric finite generating set for	$ G $	. It 's not hard to see that if	G

1

$ & $

30.2

and

$ Q $

are the largest among all forward angles of self-intersection points of

Q

009 &

$ \pm $

0.009 &

Plus or minus.

As

$ \kappa $

increases , both

kappa.

mode than in the

$ P_2 $

mode . Strong non-adiabaticity reduces the temperature variation significantly so that the radius effect in the luminosity variation

P sub 2

$ ^\textrm { \scriptsize 139 } $

,

139

has at least two separate connected components

$ C_1 , C_2 $

. Then there must exist a closed curve in the plane separating

C sub 1, C sub 2.

identified with

$ X $

and

X

corresponding to at least one `

$ j $

' value

j

satisfies the peak throughput requirement of

$ 20 $

~Gbps as stipulated in the 5G standard~

Twenty.

is dense in

$ \mathscr { H } _ { L } $

, the condition in

H sub L.

and a sink

$ t $

; ( ii )

t

&

$ 224^2 $

& 24 & 4.5 & 299 & 82.7

The equation you provided is: 224 squared.

and the two-element group

$ ( e , \tilde e ) $

. Let us consider the following group action : for all

(e, tilde e) : e, tilde e

the entries of

$ Df_t $

depend continuously on

D f sub t

are in the same block of

$ \sigma $

. Writing

Sigma.

then go through the weight classes in decreasing order to insert the pairs into the corresponding

$ Q_v $

. We explicitly give the pseudocode below as

Q sub v.

when the context

$ \mathcal { C } ^ { 1 } $

and

C one.

% Here

$ K $

is a hyperparameter that specifies the modality of the prior . For the domain of the US politics we choose

K

Consider a subFinsler manifold

$ M $

and fix

M

against

$ 1 $

, we see that

one

admits a neighbor-locating

$ k $

-coloring . A proper

k

defined by the set of points

$ ( x,0 ) $

where

x comma zero

approaches

$ t $

,

t

which realized the smaller value of

$ J $

. % % These optimized designs are specialized for their mechanical situation under the traction force

J

with large

$ l_z $

associated to a motion farther away from the

lz: None

as

$ J ' ( t ) $

. Thus ,

J prime of t

We derive a

$ 5\sigma $

detection limit of

Five sigma.

Then

$ u_0 - v_0 $

is a pluriharmonic distribution on

u sub zero minus v sub zero

contrary to the previous terms

$ \bbU_ { z,0 } ( \bk , t ) $

and

B B U sub z comma 0 open parenthesis B k comma t close parenthesis

there is a unique flat section

$ \widetilde { A } _a $

of

A tilde sub a

$ - $

minus

Then we can choose the

$ r $

-state in Definition~

r

un espace de Segal complet est aussi un espace simplicial où le morphisme de Segal

$ \phi_k $

est une équivalence faible pour tout

phi sub k.

denoted by

$ h $

. After

h

is an element of

$ \Sscr_ { obs } $

. Therefore , we use Corollary~

S c o s.

and

$ b_n $

,

b sub n

This can be written in terms of the induced velocity field

$ u_1 ( \cdot ) $

of one prototypical eddy .

u sub 1, open parenthesis, dot, close parenthesis.

$ G_1 , G_2 , \cdots , G_m $

with

G sub 1, G sub 2, dot dot dot, G sub m

of

$ \Phi $

is greater than

Phi.

for

$ i=1 , \ldots , r $

subject to the requirement that

i equals 1, dot dot dot, r.

and

$ G $

is the units of a central simple algebra over

G

is the component of

$ X_ { ( 1 ) } $

and

X sub 1

is

$ 0 $

, and the latter happens if and only if

zero

since the minor semiaxis of

$ E_2 $

is

E sub 2.

at location

$ n $

, such that

n

begins with pairs of edges

$ e_i , e'_i $

,

e prime i, e prime i

which contains

$ S $

in its interior and is a deformation retract of

S

satisfying

$ f ( x ) =0 $

and

f of x equals zero.

31

$ \pm $

1.69 & 0.9773

Plus or minus.

$ A $

, and

A

and the Lipschtiz character of

$ \partial \Omega $

, with the property that if

Partial Omega.

and the fact that

$ \nabla $

is torsion-free . 4 follows because for all unit timelike vector fields

nabla.

be uncountable cardinals with

$ \kappa $

regular and assume that

kappa.

bosonic degrees of freedom must couple to in order to be able to extract the most entanglement with

$ \bar A $

. As we will see in Section~

A bar.

-th largest element of

$ \diag ( X ) $

, and

diag of X

&

$ P_y $

&

P sub y.

$ \alpha^1 $

and

Alpha raised to the power of 1.

can be obtained by using the round-trip light phases from satellites

$ B $

and

B

sum over

$ i=1 , \ldots , n $

and

i equals 1, dot dot dot, n.

solutions in the presence of an asymptotically small activator diffusivity

$ \ep^2\ll 1 $

. We find that two types of BLS solutions , one of which is unconditionally linearly stable and the other unstable , can be constructed provided that the activator flux

e p squared is much less than one

that this yields the correct PN results for

$ U_L $

and

U sub L

six coupling modifiers corresponding to the tree-level Higgs boson couplings are defined :

$ \kappa_W $

,

kappa sub W

if

$ \vec { A } $

is a complex matrix and/or the constellation symbols are complex , the code rate is

vector A.

with elements

$ H_ { ij } = ( M_iM_j ) ^ { -1/2 } K_ { ij } $

where

H sub i j equals the inverse square root of the product of M sub i and M sub j, multiplied by K sub i j.

then the desired move

$ j $

is the one at the current node

j

A relation algebra is a Q-relation algebra iff it contains elements

$ p , q $

such that

p, q

described by coordinate

$ x_ { i } $

, is captured by the equipartition theorem ,

x subscript i

em For all

$ i , j $

the points

i, j

\boldmath

$ a $

} }

a

for all

$ x\in G $

, where

x belongs to G.

fluctuations of

$ z $

are suppressed and we recover N

z

&

$ 26.2\pm { 1.8 } $

&

Two point six plus or minus one point eight.

&

$ - $

& 57.7 & 71.1 & 69.6 & 69.9

minus

is infinite cyclic except the case where

$ M $

is covered by

M

as the hyper-parameter

$ \eta $

and iteration steps increase , the performance decreases , which suggests that excessive calibration is detrimental .

eta.

We write the dot product on

$ \mathbb { R } ^d $

as

R to the power of d.

% in a way that every cell in

$ \mathcal { P } $

is contained in a

P in calligraphic font.

does not affect the definition of the oracle parameter

$ \Psi_0 $

. Condition

Psi sub zero.

We conclude this section on a more optimistic note in regards to

$ Q $

queries that occur in randomly chosen indices . Recall ( from Section~

Q

&

$ -0.19688 $

&

Negative zero point one nine six eight eight.

The posterior-

$ L_2 $

pseudometric between

L sub 2.

the dissipation rate

$ \Gamma_ { \rm ED } $

from this effect is

Gamma sub E D

and

$ \pi $

being the CPDP algorithm , we have that %

pi.

The reason for the first equality is that since the first term of

$ u $

is

u

for a suitable constant

$ C > 0 $

. Taking

C is greater than zero.

and

$ V_2 $

, the individual with

V sub 2

for all

$ i\in\ { 1 , \cdots , n\ } $

. We estimate each vector

i belongs to the set of numbers from 1 to n.

Given distinct points

$ x , y \in X $

, set

x, y belong to X.

from

$ x_j $

. Take a power

x sub j.

$ a_ { j } $

a sub j.

partition of

$ b $

} is a subset

b

and

$ m $

, nor on the momentum~

m

&

$ L_ { \sun } $

L sub sun.

It

$ \hat { o } $

diffusion process ]

o-hat symbol.

is Turing reducible to the degree

$ c $

of

c

has the same area as

$ K $

but a smaller perimeter . Note the following result , known as the

K

then we may abbreviate

$ W_I $

as

W subscript I.

is again an

$ N $

by

N

must contain a short rank-one element for any symmetric finite generating set for

$ G $

. It 's not hard to see that if

G