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 insignificant contribution of the	$ b $	parameter to the model as it is two orders of magnitude smaller than	b
a combination of global Hadamard	$ \mathrm { H } $	and global phase gates	H
mag	$ ] $		closing bracket
with the measurement of	$ \theta , \omega $	, ect , as the input ,	Theta, omega.
we consistently use the index	$ i $	( and its variations such as	i
letting	$ L ' $	denote the Lipschitz constant of	L prime.
and	$ h_2 $	reaches the minimum in its region . Similarly , one can conclude from the third level plot that	h sub 2
consists of the contiguous values of	$ T $	from position	T
over various	$ C_1 $	fields of interest . Some	C sub 1.
&	$ : $	&	colon
and	$ D $	chooses the round	D
and	$ y $	, where	y
Result of the	$ \Delta $	Benchmark of	Delta.
	$ \frac { 3-3\sqrt { 3 } } { \sqrt { 6 } } $	;	Fraction, 3 minus 3 times the square root of 3, divided by the square root of 6.
	$ dynamo model of \citet { OH96 } with stochastic fluctuations in the $		The dynama model of t times OH96 with stochastic fluctuations.
For every	$ t $	, both the flow map	t
Notice that the temperature	$ T $	of dilatonic black hole is ill-defined when	T
is interpreted as an active left-handed neutrino and	$ \nu_2 $	with mass	nu sub 2.
It is important to note that each flow	$ ( i , j , k ) $	is associated with a dual variable	i, j, k
By examining the results of the	$ n=1 $	test cases , we can determine whether bugs are located inside or outside the loops . During the execution of the experiments , it is worth noting that some mutations	n equals one.
and	$ \mathbf v_l $	is hyperideal . Then in the projective model , either all	v sub l
now have weight	$ ( 1+\frac { \varepsilon } { t ' } ) \cdot w'_1 $	.	Open parenthesis, one plus epsilon over t prime, close parenthesis, times w prime sub 1.
	$ w $	and	w
	$ \|\mathcal { Z } ( G ) \|=p^ { 3 } $	and	Absolute value of the set Z of G equals p raised to the power of 3.
At	$ \alpha'^3 $	, the relevant term in the action is :	Alpha prime cubed.
timeslots of a frame by	$ A_1 $	with elements given by	A sub one.
since	$ \tau $	is a single appendage node ,	Tau.
	$ i = 1,2 , ... , n $	, distribution proportion boundary :	i equals 1, 2, dot dot dot, n
	$ w $	with	w
for some sufficiently small	$ c > 0 $	we have	c greater than zero
em If	$ \mathsf { mono } _ { A ( n ) , i } = \mathsf { increasing } $	, then	mono A of n comma i equals increasing.
60578561 &	$ - $	72.15298701 & 26.59 & 25.62 & 26.68 & 25.96	minus
curves for	$ H \parallel ab $	plane of Ba	H parallel ab.
thermal potential	$ \alpha $	and 4-velocity	Alpha.
	$ & $		and
where	$ \tau_E $	is the inelastic scattering time and	Tau sub E.
shares an endpoint of	$ \alpha $	where	Alpha.
there are at most two vertices	$ v \in I $	such that	v belongs to I.
does not contain variable	$ x $	.	x
	$ \Gamma_ { D } $	, the thermal gluon-	Gamma sub D.
of degree	$ r $	, we can rewrite any	r
Let	$ \fh=\fh ( c ) $	be the Levi subalgebra corresponding to a~composition	f h equals f h of c
and move to adjacent node	$ v $	via	v
in the	$ ( \cF , \fvect ) $	-bimodule	fac, F, vect
the deviations from ideal gas behavior for both electron-electron and electron-ion structure factors vanish as expected when	$ \kappa \rightarrow 0 $	, corresponding to large electron temperature . Similarly , the ion-ion structure factor vanishes as	kappa implies zero.
if	$ k $	even } .	k
for some permutation	$ \pi $	on	pi.
-partition of	$ V ( G ) $	afforded by	V of G.
is replaced with	$ 0 $	.	zero
is a set of values and	$ k $	is the size of the pooling window .	k
while atoms with low ionization potential are excited at temperatures	$ \sim 10^4 $	K , as MgII and SiII	Approximately ten to the power of four.
of lists in which every vertex of	$ G $	which is not of odd degree is majority colored and each vertex of odd degree in	G
The author defines three real numbers	$ \eta=\pm 1 $	,	eta equals plus or minus one.
-stationary and stable	$ C^ { 1,1 } $		C one comma one.
concerns the restriction functor associated to the operad morphism	$ f\colon \mathbf { Ass } \to \mathbf { Com } $	, while Question	f, colon, bold ass, to, bold comp.
as a subset of	$ \C $	, one finds that , for	C: None
of the proposition about	$ 0 $	-- and moreover with each certificate/proof	zero
we set	$ a_ { \frac { 1 } { 2 } } =a_ { { K } -\frac { 1 } { 2 } } =0 $	, where a term of	a sub one-half equals a sub K minus one-half equals zero.
and	$ b_ { j } =a_ { j } $	for all	b sub j equals a sub j.
which we will call	$ HU $	and	HU: None
at	$ 2\sigma $	CL for	Two sigma.
Note that	$ \rho $	is jointly continuous . By Theorem	rho.
At	$ T=130 $	K , we deduce two	T equals 130.
The polarization	$ \phi $	, defined as	phi.
	$ \pm $	2.79 &	Plus or minus.
and	$ I $	a small index set together with a functor	I
&	$ 1.7_ { -1.0 } ^ { +0.8 } $	&	1.7 sub negative 1.0 superscript positive 0.8.
and stabilizes around a value that is smaller than 1 but does not depend on	$ N $	. Indeed , such local maxima of	N
63	$ \pm $	2.60 & 29.04	Plus or minus.
which applies to the	$ B $	meson . However , we note that also for the charm-quark , with	B
be a form of degree	$ d $	on a finite-dimensional vector space	d
&	$ 0.05 $	&	zero point zero five.
and	$ X_t $	. Note that we have already obtained the following three realisations for the Lie algebra	X sub t.
&	$ 1 $	&	one
	$ w_t $	is located in	w at time t.
for each	$ \varepsilon^ * > 0 $	there is a	Epsilon star is greater than zero.
as well as all	$ \mu $	and	mu.
because if we have bubbles	$ A $	and	A
at the nodes	$ \node { w } _i \in W $	. For classification purposes , the kernel predictor	w sub i belongs to capital W
We stress that	$ G_1 ( b_\sigma ) $		G sub 1, open parenthesis, b sub sigma, close parenthesis.
such that the Voronoi regions of the	$ k $	elements in	k
\checkmark	$ & $		and
number of developmental cycles	$ \mathcal { C } $	, pruning threshold	see
at heights	$ 2 $	and	two
with	$ n $	designated vertices of degree	n
then	$ f $	defines a splitting map on	f
and	$ \Gamma $	are Lebesgue-almost everywhere	Gamma.
	$ ^\textrm { \scriptsize 105 } $	,	One zero five.
that the proposed algorithm can improved travel distance compared with the case without RIS by	$ 0.45\ % $	,	0.45 percent.
and it can be approximated with the second order function of	$ u $	. Thus , the lowest order approximation of such function is	u
a disjunction of	$ a $	distinct variables is wrong on	a
	$ x_3 $	, or	x sub 3.
identity elements	$ e $	:	e
how closely the target direction	$ \theta_0 $	is adhered to . The motion becomes increasingly unidirectional the larger	Theta zero.
is in	$ \mathrm { M_\odot yr^ { -1 } kpc^ { -2 } } $	,	M sub solar to the power of negative one, kpc to the power of negative two.
and any minimal labelling	$ \lambda $	such that	Lambda.
	$ L_ { k } $	is the undirected binary phylogenetic network on	L sub k.
which describes the exchange energy as a function of the orbital overlap	$ S^2 $	,	S squared.
; the	$ H^1 $	-bounds for the later three variables would suffice , as in Definition~	H raised to the power of 1.
under which	$ \mathbb { P } _3 $	is solved again with the new	P sub 3

the insignificant contribution of the

$ b $

parameter to the model as it is two orders of magnitude smaller than

b

a combination of global Hadamard

$ \mathrm { H } $

and global phase gates

H

mag

$ ] $

closing bracket

with the measurement of

$ \theta , \omega $

, ect , as the input ,

Theta, omega.

we consistently use the index

$ i $

( and its variations such as

i

letting

$ L ' $

denote the Lipschitz constant of

L prime.

and

$ h_2 $

reaches the minimum in its region . Similarly , one can conclude from the third level plot that

h sub 2

consists of the contiguous values of

$ T $

from position

T

over various

$ C_1 $

fields of interest . Some

C sub 1.

&

$ : $

&

colon

and

$ D $

chooses the round

D

and

$ y $

, where

y

Result of the

$ \Delta $

Benchmark of

Delta.

$ \frac { 3-3\sqrt { 3 } } { \sqrt { 6 } } $

;

Fraction, 3 minus 3 times the square root of 3, divided by the square root of 6.

$ dynamo model of \citet { OH96 } with stochastic fluctuations in the $

The dynama model of t times OH96 with stochastic fluctuations.

For every

$ t $

, both the flow map

t

Notice that the temperature

$ T $

of dilatonic black hole is ill-defined when

T

is interpreted as an active left-handed neutrino and

$ \nu_2 $

with mass

nu sub 2.

It is important to note that each flow

$ ( i , j , k ) $

is associated with a dual variable

i, j, k

By examining the results of the

$ n=1 $

test cases , we can determine whether bugs are located inside or outside the loops . During the execution of the experiments , it is worth noting that some mutations

n equals one.

and

$ \mathbf v_l $

is hyperideal . Then in the projective model , either all

v sub l

now have weight

$ ( 1+\frac { \varepsilon } { t ' } ) \cdot w'_1 $

.

Open parenthesis, one plus epsilon over t prime, close parenthesis, times w prime sub 1.

$ w $

and

w

$ |\mathcal { Z } ( G ) |=p^ { 3 } $

and

Absolute value of the set Z of G equals p raised to the power of 3.

At

$ \alpha'^3 $

, the relevant term in the action is :

Alpha prime cubed.

timeslots of a frame by

$ A_1 $

with elements given by

A sub one.

since

$ \tau $

is a single appendage node ,

Tau.

$ i = 1,2 , ... , n $

, distribution proportion boundary :

i equals 1, 2, dot dot dot, n

$ w $

with

w

for some sufficiently small

$ c > 0 $

we have

c greater than zero

em If

$ \mathsf { mono } _ { A ( n ) , i } = \mathsf { increasing } $

, then

mono A of n comma i equals increasing.

60578561 &

$ - $

72.15298701 & 26.59 & 25.62 & 26.68 & 25.96

minus

curves for

$ H \parallel ab $

plane of Ba

H parallel ab.

thermal potential

$ \alpha $

and 4-velocity

Alpha.

$ & $

and

where

$ \tau_E $

is the inelastic scattering time and

Tau sub E.

shares an endpoint of

$ \alpha $

where

Alpha.

there are at most two vertices

$ v \in I $

such that

v belongs to I.

does not contain variable

$ x $

.

x

$ \Gamma_ { D } $

, the thermal gluon-

Gamma sub D.

of degree

$ r $

, we can rewrite any

r

Let

$ \fh=\fh ( c ) $

be the Levi subalgebra corresponding to a~composition

f h equals f h of c

and move to adjacent node

$ v $

via

v

in the

$ ( \cF , \fvect ) $

-bimodule

fac, F, vect

the deviations from ideal gas behavior for both electron-electron and electron-ion structure factors vanish as expected when

$ \kappa \rightarrow 0 $

, corresponding to large electron temperature . Similarly , the ion-ion structure factor vanishes as

kappa implies zero.

if

$ k $

even } .

k

for some permutation

$ \pi $

on

pi.

-partition of

$ V ( G ) $

afforded by

V of G.

is replaced with

$ 0 $

.

zero

is a set of values and

$ k $

is the size of the pooling window .

k

while atoms with low ionization potential are excited at temperatures

$ \sim 10^4 $

K , as MgII and SiII

Approximately ten to the power of four.

of lists in which every vertex of

$ G $

which is not of odd degree is majority colored and each vertex of odd degree in

G

The author defines three real numbers

$ \eta=\pm 1 $

,

eta equals plus or minus one.

-stationary and stable

$ C^ { 1,1 } $

C one comma one.

concerns the restriction functor associated to the operad morphism

$ f\colon \mathbf { Ass } \to \mathbf { Com } $

, while Question

f, colon, bold ass, to, bold comp.

as a subset of

$ \C $

, one finds that , for

C: None

of the proposition about

$ 0 $

-- and moreover with each certificate/proof

zero

we set

$ a_ { \frac { 1 } { 2 } } =a_ { { K } -\frac { 1 } { 2 } } =0 $

, where a term of

a sub one-half equals a sub K minus one-half equals zero.

and

$ b_ { j } =a_ { j } $

for all

b sub j equals a sub j.

which we will call

$ HU $

and

HU: None

at

$ 2\sigma $

CL for

Two sigma.

Note that

$ \rho $

is jointly continuous . By Theorem

rho.

At

$ T=130 $

K , we deduce two

T equals 130.

The polarization

$ \phi $

, defined as

phi.

$ \pm $

2.79 &

Plus or minus.

and

$ I $

a small index set together with a functor

I

&

$ 1.7_ { -1.0 } ^ { +0.8 } $

&

1.7 sub negative 1.0 superscript positive 0.8.

and stabilizes around a value that is smaller than 1 but does not depend on

$ N $

. Indeed , such local maxima of

N

63

$ \pm $

2.60 & 29.04

Plus or minus.

which applies to the

$ B $

meson . However , we note that also for the charm-quark , with

B

be a form of degree

$ d $

on a finite-dimensional vector space

d

&

$ 0.05 $

&

zero point zero five.

and

$ X_t $

. Note that we have already obtained the following three realisations for the Lie algebra

X sub t.

&

$ 1 $

&

one

$ w_t $

is located in

w at time t.

for each

$ \varepsilon^ * > 0 $

there is a

Epsilon star is greater than zero.

as well as all

$ \mu $

and

mu.

because if we have bubbles

$ A $

and

A

at the nodes

$ \node { w } _i \in W $

. For classification purposes , the kernel predictor

w sub i belongs to capital W

We stress that

$ G_1 ( b_\sigma ) $

G sub 1, open parenthesis, b sub sigma, close parenthesis.

such that the Voronoi regions of the

$ k $

elements in

k

\checkmark

$ & $

and

number of developmental cycles

$ \mathcal { C } $

, pruning threshold

see

at heights

$ 2 $

and

two

with

$ n $

designated vertices of degree

n

then

$ f $

defines a splitting map on

f

and

$ \Gamma $

are Lebesgue-almost everywhere

Gamma.

$ ^\textrm { \scriptsize 105 } $

,

One zero five.

that the proposed algorithm can improved travel distance compared with the case without RIS by

$ 0.45\ % $

,

0.45 percent.

and it can be approximated with the second order function of

$ u $

. Thus , the lowest order approximation of such function is

u

a disjunction of

$ a $

distinct variables is wrong on

a

$ x_3 $

, or

x sub 3.

identity elements

$ e $

:

e

how closely the target direction

$ \theta_0 $

is adhered to . The motion becomes increasingly unidirectional the larger

Theta zero.

is in

$ \mathrm { M_\odot yr^ { -1 } kpc^ { -2 } } $

,

M sub solar to the power of negative one, kpc to the power of negative two.

and any minimal labelling

$ \lambda $

such that

Lambda.

$ L_ { k } $

is the undirected binary phylogenetic network on

L sub k.

which describes the exchange energy as a function of the orbital overlap

$ S^2 $

,

S squared.

; the

$ H^1 $

-bounds for the later three variables would suffice , as in Definition~

H raised to the power of 1.

under which

$ \mathbb { P } _3 $

is solved again with the new

P sub 3