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 values of	$ Q_ { \rm rot } $	and	Q sub rot.
with coefficients in	$ F $	, the sum on the right is over all compositions	F
is an MLP with	$ 17 $	K trainable parameters .	Seventeen.
	$ \ISG_\stepG $	has	I S G, subscript step G.
342 & 1	$ \| $	1	pipe
states that there exists Darboux coordinates	$ ( x_1 , \xi_1 , x_2 , \xi_2 ) $	near	Open parenthesis, x sub 1, comma, xi sub 1, comma, x sub 2, comma, xi sub 2, close parenthesis.
detected a total of	$ \simeq $	19	approximately equal to.
9643	$ \pm $	0.0031 & 0.0083	Plus or minus.
in	$ \R^d $	is a collection of simplices such that :	R to the power of d.
with the center at	$ z_0 $	.	z sub zero.
% Define	$ \zeta : =\omega\gamma^ { -1/2 } \wedge\omega^ { 1/2 } $	. If	Zeta equals omega gamma to the power of negative one-half wedge omega to the power of one-half.
	$ \text { Cs } _ { \text { 0.05 } } $		C s sub 0.05
The quasar is at	$ \Delta $	RA ,	Delta.
~where	$ \vev { A^a } =0 $	. The VEV can then be rescaled to satisfy	vee times A to the power of a equals zero
The set-valued map	$ G $	is LB and the mapping	G
algebra representation where we found that one can write down an infinite number of consistent	$ w $	invariant OPEs where truncation happens at different values of	w
we performed	$ k $	-fold (	k
axis will be labelled by	$ 0 $	, the next by	zero
	$ u = u_0 + kq $	is a solution with	u equals u sub zero plus k times q.
The	$ z $	-soft approximation is not particularly convenient for two reasons . One is that it is not very accurate , as we shall see in section~	z
we could use the single KPO with	$ d_x > 1 $	and	d sub x is greater than one.
Differentiating	$ \zeta_n $	yields :	zeta sub n.
As	$ \bm w_t $	is the only random vector in	lambda w sub t
Here we study bond-pruning in prestressed central-force spring networks with mean coordinations of	$ z < 6 $	. For such systems , the mean coordination number at the onset of rigidity ,	z is less than 6.
each converges as	$ n\to\infty $	, see Theorem	n tends to infinity
Dia	$ 1 $	) and ( Dia	one
there exists a constant	$ C > 0 $	such that	C is greater than zero.
then a cooperator in the game is rewarded with a	$ \mu_ { R } $	amount received from a central institution when interacting with a neighbor	mu sub R.
for all	$ m\geq 1 $	,	m greater than or equal to 1
assessments in order to provide physical reasoning behind Series	$ A $	and	A
Now that we know the relation between	$ \calR_k $	and	cal R sub k.
boson decays to leptons and the	$ a $	particle decays to a photon pair are presented using the full LHC Run 2 proton -- proton (	a
&	$ \ell_2 $	&	ell sub 2.
is a homomorphism from~	$ F $	to~	F
is a set in	$ V $	of size	V
we provide below the known classification of tuples of degree	$ 3 $	admitting maps .	three
to	$ B_2 $		B sub 2.
the one with the highest	$ SNR $	. % https : //authortools.aas.org/LATEX/make-latex.html	SNR: None
	$ f\colon \rel A \to \rel B $	is defined to be a mapping	f maps from set A to set B
the overall LU direct channel	$ { \bf { H } } _ { \rm d } $	and the multi-user active beamforming	H sub d.
demonstrates the effect of selecting different sizes of	$ f_b $	during Algorithm	f sub b.
and a fixed index	$ k\in\ { 1 , \dots , n\ } $	, by the minimax principle , Lemma	k belongs to the set of integers from 1 to n (inclusive).
	$ ^\textrm { \scriptsize 100 } $	,	One hundred.
and event horizon radius	$ r_h $	is established with the aid of the corrected effective geometry which caused by the NLED term . The results shows that they have a positive correlation , which implies that	r h.
	$ K $	, and for different bending stiffness ,	K
In the case when	$ N $	is a co-algebra equipped with the Lie-Poisson structure	N
and hence so is the top horizontal map by	$ 2 $	-out-of-	two
of	$ \mathbb { I } _2 $	is nonsingular for all physically acceptable parameter values . In particular ,	I sub 2.
-bundle	$ \sG_j $	on	G sub j.
of	$ \Sch_k $	, then we write	S-k
as	$ \lambda $	goes to	Lambda.
027	$ \pm $	0.028 & ... & ... & 0.000	Plus or minus.
where	$ W_t $	is a Wiener process with	W sub t
or	$ b\neq0 $	, then	b not equal to 0
	$ \orbiflat X $	is a set since	set X
and	$ \Lambda $	is a real random diagonal matrix with a specified condition number	Lambda.
which consists in learning independently	$ Q_a $	. However , there are also CTDE methods that take the state	Q sub a.
As	$ x > y $	in	x is greater than y.
78	$ \pm $	0.13 & 6.71	Plus or minus.
we continue to expect that the levitated sensor will behave similarly to an L-shaped interferometer in the limit of small	$ x_s $	.	x sub s.
with	$ s $	active positions such that	s
have higher node degrees due to people 's strong preference for age differences around	$ 0 $	and young ages around	zero
% aim In the weak-binding limit	$ B\to 0 $	, it is shown that the compositeness	B goes to 0.
who proved that an	$ n $	-state 2DFA can be transformed to a DFA with	n
In this step we additionally normalize	$ \widehat { \delta \rho } $	by	wide hat delta rho
we focus on the cases with extended	$ SU ( 3 ) _C $	. Other cases like extended	SU ( 3 ) C: S U three C
with arbitrary functions	$ f , g $	and	f comma g
\hat \beta_1	$ & $		and
The prior constraint for	$ \epsilon $	is set to 80	epsilon.
then the interpretation of a term in	$ \mathcal { M } $	is a	M
which are subgroups of	$ \Gamma $	. Indeed , since any	Gamma.
at time	$ t $	by its	t
% \State Communicate	$ L $	coefficient vectors	L
to	$ T $	leads to the conclusion , i.e . , Eq.~	T
	$ n_0 $	represents the carrier number at transparency ,	n sub zero
and a supernova ejecta metallicity of 6	$ Z_ { \odot } $	, a standard ratio of 0.1 would yield a final cloud metallicity of 1.5	Z sub sun.
	$ \Lambda_ { 11 } $	in Eq.	Lambda sub 11.
Assume that	$ \Omega $	is a bounded , convex and open subset of	Omega.
one can find	$ m $	such that	m
as a subgraph in	$ G $	or in the complement of	G
and	$ y= ( 1,1,0,0 , \cdots ) \in \ell_p $	. Define	y equals open parenthesis one comma one comma zero comma zero comma ellipsis close parenthesis in ell p.
	$ \alpha_ { vw } $	is LeakyReLU	alpha sub v w
Here the constant	$ C $	must be determined by the initial conditions :	C
and covariance function	$ \rho $	are given respectively in~	rho.
inputs of elements of	$ \mathfrak { g } $	according to the order on the set of external vertices . For each trivalent vertex , we assign the cubic trace form	g in Fraktur font.
After obtaining	$ \pmb { W } $	by solving the convex SDP , the adversary computes its principle eigenvalue ,	W
&	$ Y $	&	Y
in	$ I $	contains a branch vertex as the middle vertex of the path . Similarly to before , this implies that	I
is uniformly distributed in the interval	$ \left [ 0,1\right ] $	, thus the infidelity from Eq.~	the closed interval from 0 to 1.
by the additivity of	$ k^ { \sigma } $	class , we can write	k raised to the power of sigma
it is clear that	$ f ( 0 ) =1 $	. This finishes the proof of	f of 0 equals 1
may freely be chosen to lie on the same slice as	$ p_i $	, as represented in bold in Figure	p sub i.
for every	$ i\in\N $	. By definition of	i belongs to the set of natural numbers.
is Lipschitz with constant	$ 1 $	and slope restricted in	one
depends only on	$ m $	( and not	m
the cases	$ \mu=\pm \frac { 1 } { 2 } $	and	mu equals plus or minus one-half.
the latter then imposes that	$ g ( x ) $	diverges as	g of x.
44	$ \pm $	17.12 & 7.42	Plus or minus.
we conclude that	$ u=0 $	, proving the injectivity of	u equals zero.
show	$ \bar { U } _\mathrm { KOAc } ( r_\mathrm { KOAc } ) $		U bar sub KOAc of r sub KOAc.

The values of

$ Q_ { \rm rot } $

and

Q sub rot.

with coefficients in

$ F $

, the sum on the right is over all compositions

F

is an MLP with

$ 17 $

K trainable parameters .

Seventeen.

$ \ISG_\stepG $

has

I S G, subscript step G.

342 & 1

$ | $

1

pipe

states that there exists Darboux coordinates

$ ( x_1 , \xi_1 , x_2 , \xi_2 ) $

near

Open parenthesis, x sub 1, comma, xi sub 1, comma, x sub 2, comma, xi sub 2, close parenthesis.

detected a total of

$ \simeq $

19

approximately equal to.

9643

$ \pm $

0.0031 & 0.0083

Plus or minus.

in

$ \R^d $

is a collection of simplices such that :

R to the power of d.

with the center at

$ z_0 $

.

z sub zero.

% Define

$ \zeta : =\omega\gamma^ { -1/2 } \wedge\omega^ { 1/2 } $

. If

Zeta equals omega gamma to the power of negative one-half wedge omega to the power of one-half.

$ \text { Cs } _ { \text { 0.05 } } $

C s sub 0.05

The quasar is at

$ \Delta $

RA ,

Delta.

~where

$ \vev { A^a } =0 $

. The VEV can then be rescaled to satisfy

vee times A to the power of a equals zero

The set-valued map

$ G $

is LB and the mapping

G

algebra representation where we found that one can write down an infinite number of consistent

$ w $

invariant OPEs where truncation happens at different values of

w

we performed

$ k $

-fold (

k

axis will be labelled by

$ 0 $

, the next by

zero

$ u = u_0 + kq $

is a solution with

u equals u sub zero plus k times q.

The

$ z $

-soft approximation is not particularly convenient for two reasons . One is that it is not very accurate , as we shall see in section~

z

we could use the single KPO with

$ d_x > 1 $

and

d sub x is greater than one.

Differentiating

$ \zeta_n $

yields :

zeta sub n.

As

$ \bm w_t $

is the only random vector in

lambda w sub t

Here we study bond-pruning in prestressed central-force spring networks with mean coordinations of

$ z < 6 $

. For such systems , the mean coordination number at the onset of rigidity ,

z is less than 6.

each converges as

$ n\to\infty $

, see Theorem

n tends to infinity

Dia

$ 1 $

) and ( Dia

one

there exists a constant

$ C > 0 $

such that

C is greater than zero.

then a cooperator in the game is rewarded with a

$ \mu_ { R } $

amount received from a central institution when interacting with a neighbor

mu sub R.

for all

$ m\geq 1 $

,

m greater than or equal to 1

assessments in order to provide physical reasoning behind Series

$ A $

and

A

Now that we know the relation between

$ \calR_k $

and

cal R sub k.

boson decays to leptons and the

$ a $

particle decays to a photon pair are presented using the full LHC Run 2 proton -- proton (

a

&

$ \ell_2 $

&

ell sub 2.

is a homomorphism from~

$ F $

to~

F

is a set in

$ V $

of size

V

we provide below the known classification of tuples of degree

$ 3 $

admitting maps .

three

to

$ B_2 $

B sub 2.

the one with the highest

$ SNR $

. % https : //authortools.aas.org/LATEX/make-latex.html

SNR: None

$ f\colon \rel A \to \rel B $

is defined to be a mapping

f maps from set A to set B

the overall LU direct channel

$ { \bf { H } } _ { \rm d } $

and the multi-user active beamforming

H sub d.

demonstrates the effect of selecting different sizes of

$ f_b $

during Algorithm

f sub b.

and a fixed index

$ k\in\ { 1 , \dots , n\ } $

, by the minimax principle , Lemma

k belongs to the set of integers from 1 to n (inclusive).

$ ^\textrm { \scriptsize 100 } $

,

One hundred.

and event horizon radius

$ r_h $

is established with the aid of the corrected effective geometry which caused by the NLED term . The results shows that they have a positive correlation , which implies that

r h.

$ K $

, and for different bending stiffness ,

K

In the case when

$ N $

is a co-algebra equipped with the Lie-Poisson structure

N

and hence so is the top horizontal map by

$ 2 $

-out-of-

two

of

$ \mathbb { I } _2 $

is nonsingular for all physically acceptable parameter values . In particular ,

I sub 2.

-bundle

$ \sG_j $

on

G sub j.

of

$ \Sch_k $

, then we write

S-k

as

$ \lambda $

goes to

Lambda.

027

$ \pm $

0.028 & ... & ... & 0.000

Plus or minus.

where

$ W_t $

is a Wiener process with

W sub t

or

$ b\neq0 $

, then

b not equal to 0

$ \orbiflat X $

is a set since

set X

and

$ \Lambda $

is a real random diagonal matrix with a specified condition number

Lambda.

which consists in learning independently

$ Q_a $

. However , there are also CTDE methods that take the state

Q sub a.

As

$ x > y $

in

x is greater than y.

78

$ \pm $

0.13 & 6.71

Plus or minus.

we continue to expect that the levitated sensor will behave similarly to an L-shaped interferometer in the limit of small

$ x_s $

.

x sub s.

with

$ s $

active positions such that

s

have higher node degrees due to people 's strong preference for age differences around

$ 0 $

and young ages around

zero

% aim In the weak-binding limit

$ B\to 0 $

, it is shown that the compositeness

B goes to 0.

who proved that an

$ n $

-state 2DFA can be transformed to a DFA with

n

In this step we additionally normalize

$ \widehat { \delta \rho } $

by

wide hat delta rho

we focus on the cases with extended

$ SU ( 3 ) _C $

. Other cases like extended

SU ( 3 ) C: S U three C

with arbitrary functions

$ f , g $

and

f comma g

\hat \beta_1

$ & $

and

The prior constraint for

$ \epsilon $

is set to 80

epsilon.

then the interpretation of a term in

$ \mathcal { M } $

is a

M

which are subgroups of

$ \Gamma $

. Indeed , since any

Gamma.

at time

$ t $

by its

t

% \State Communicate

$ L $

coefficient vectors

L

to

$ T $

leads to the conclusion , i.e . , Eq.~

T

$ n_0 $

represents the carrier number at transparency ,

n sub zero

and a supernova ejecta metallicity of 6

$ Z_ { \odot } $

, a standard ratio of 0.1 would yield a final cloud metallicity of 1.5

Z sub sun.

$ \Lambda_ { 11 } $

in Eq.

Lambda sub 11.

Assume that

$ \Omega $

is a bounded , convex and open subset of

Omega.

one can find

$ m $

such that

m

as a subgraph in

$ G $

or in the complement of

G

and

$ y= ( 1,1,0,0 , \cdots ) \in \ell_p $

. Define

y equals open parenthesis one comma one comma zero comma zero comma ellipsis close parenthesis in ell p.

$ \alpha_ { vw } $

is LeakyReLU

alpha sub v w

Here the constant

$ C $

must be determined by the initial conditions :

C

and covariance function

$ \rho $

are given respectively in~

rho.

inputs of elements of

$ \mathfrak { g } $

according to the order on the set of external vertices . For each trivalent vertex , we assign the cubic trace form

g in Fraktur font.

After obtaining

$ \pmb { W } $

by solving the convex SDP , the adversary computes its principle eigenvalue ,

W

&

$ Y $

&

Y

in

$ I $

contains a branch vertex as the middle vertex of the path . Similarly to before , this implies that

I

is uniformly distributed in the interval

$ \left [ 0,1\right ] $

, thus the infidelity from Eq.~

the closed interval from 0 to 1.

by the additivity of

$ k^ { \sigma } $

class , we can write

k raised to the power of sigma

it is clear that

$ f ( 0 ) =1 $

. This finishes the proof of

f of 0 equals 1

may freely be chosen to lie on the same slice as

$ p_i $

, as represented in bold in Figure

p sub i.

for every

$ i\in\N $

. By definition of

i belongs to the set of natural numbers.

is Lipschitz with constant

$ 1 $

and slope restricted in

one

depends only on

$ m $

( and not

m

the cases

$ \mu=\pm \frac { 1 } { 2 } $

and

mu equals plus or minus one-half.

the latter then imposes that

$ g ( x ) $

diverges as

g of x.

44

$ \pm $

17.12 & 7.42

Plus or minus.

we conclude that

$ u=0 $

, proving the injectivity of

u equals zero.

show

$ \bar { U } _\mathrm { KOAc } ( r_\mathrm { KOAc } ) $

U bar sub KOAc of r sub KOAc.