Last-Bullet/MathBridge_Splitted · Data Studio

context_before stringlengths 0 199	equation stringlengths 5 80	context_after stringlengths 0 249	spoken_English stringlengths 0 119
for all	$ \sigma \in \cC $	. Since	Sigma belongs to the set C.
for all	$ x $	.	x
By choosing	$ T $	small enough , we can control	T
as a continuous-time Markov chain	$ ( \eta_t ) _ { t\geq 0 } $	on	Open parenthesis, eta sub t, close parenthesis, sub t greater than or equal to zero.
Introducing	$ u=u_ { h } x $	,	u equals u sub h times x.
exactly equals to	$ M_K $	. The label	M sub K.
are finite and contribute to the normalized crossed conductance for	$ E > \Delta_ { 0 } $	. For	E greater than Delta sub zero
&	$ \delta $	max &	delta.
Let	$ m\geq 0 $	and	m greater than or equal to zero.
while	$ \delta $	is determined by the relative phase among 2 orthogonal oscillations as electromagnetic fields . These 2 parameters must not be affected during the amplification process for which we used a	delta.
which is at a distance	$ x $	from the initial position , differ by unity depending on whether it is located on the right-side or the left-side of the initial position . For	x
&	$ A $		A
and	$ \epsilon $	, we start the simulation from rest , adding an infinitesimal perturbation to the otherwise linear temperature background . After a short transient , the kinetic energy grows exponentially in	epsilon.
for	$ v \in A $	and	v belongs to A
and we append two indexes to indicate the values of the parameter	$ a $	and of the potential	a
4 10	$ ^ { -4 } $	& 9.3	Negative four.
&	$ - $	& 41.0 & 5.0 & 3.2 & 15.3 &	minus
is the centralizer of	$ \pi \in S_n $	. Note that all elements	Pi belongs to the set of permutations of n.
	$ p $	? x ? o ) } is contradicted by a positive effect ( resp . precondition ) , then	p
At any time	$ t $	, one can observe the market price , i.e.~the time-	t
05	$ , $	h=0.5	comma
a vector meson	$ V\fn { Q\ol Q } $	decays into two light pseudo-scalar mesons	V f n Q ol Q
0	$ & $	1.00	and
1	$ \pm { 0.02 } $	, 0.3	Plus or minus zero point zero two.
~Zhang	$ ^ { 80 } $	, J.~J.~Zhang	to the power of 80
and the seller 's outside option value	$ x $	and market participation costs	x
the differential equation has the typical form with a non-homogeneous term	$ g ( x ) $		g of x.
2	$ & $	876	and
by	$ Y '' $	. Now on each	Y double prime.
The proof for general	$ x\in [ 0,1 ] $	follows from shifting every interval by	x belongs to the interval from 0 to 1.
He bound state under each of the	$ \eta N $	potential sets shown in Fig.~	Eta N.
and taking the expectation over	$ S , S ' $	, the second inequality follows from Jensen 's inequality , and the third inequality follows from the fact that	S comma S prime.
and	$ G_2 = ( V_2 , E_2 ) $	and suppose that	G sub 2 equals open parenthesis, V sub 2 comma E sub 2, close parenthesis.
28	$ \pm $	26.73 & 48.83	Plus or minus.
is	$ C $	-convex if and only if it is the intersection of some translates of	C
8	$ \pm $	4.9 & 96.2	Plus or minus.
In the area with	$ -1 $	pc	Minus one.
	$ We have $		We have: None
by choice of	$ A^ * $	. For every	A star.
have	$ R $	-charges 1 ,	R
After checking all the	$ n $	individual arms , it can be easily seen that by the algorithm 's construction , both sets	n
these slots have indices in the set	$ D $	, so also refer to pivot columns )	D
corresponding to a different band index	$ \nu $	. Elements of each column are ordered as	nu.
The function	$ \zeta $	plays almost no role in our analysis , as we are focused on the bulk of the system . For simplicity we keep “ the same	zeta.
independently of	$ \delta $	near which we can fix areas . %	delta.
and keep track of	$ w ( i ) $	, and plot it in Figure	w of i
26	$ \pm $	0.45	Plus or minus.
All models were highly successful in learning the distribution of the novel tokens in the fine-tuning datasets : average accuracy was	$ 99.6\ % $	for DistilBERT ,	99.6 percent.
in terms of performance on the CTW	$ 1500 $	benchmark . Fig .	One thousand five hundred.
where	$ i \in [ 2 ] $	. Let	i belongs to the set containing 2.
is unmeasured or that the effect of	$ X $	on	X
as the regularity properties of	$ \mathcal { N } $	do not suffice to control	capital N
&	$ \log\mbh/\msun $	&	log m b h divided by m s u
with respect to the basis	$ v_ { \lambda 1 } , \ldots , v_ { \lambda \|\lambda\| } $	. Let	v sub lambda 1, ellipsis, v sub lambda absolute value of lambda
letting	$ \dfrac { \partderiv { \mathcal { Q } } } { \partderiv { \wght { } } } = 0 $	and rearranging , %	The derivative of the Q with respect to the derivative of delta wgtht equals zero.
-valued simple functions in	$ L^ { p ' } ( \Omega , \omega^ { \frac { 1 } { p-1 } } ; Y ) $	is also norming for	L prime, open parenthesis, Omega, comma, omega, to the power of one over p minus one, semicolon, Y, close parenthesis.
and any value of	$ \bar { p } $	, whereas the second group satisfies the following formulas :	p bar.
there exists a constant	$ \mathfrak { C } > 0 $	does n't depending neither on	C is greater than zero.
because it is not consistent with the median 's optimal choice of	$ r^ * $	. In this example , both potential local optima are global optima . Moreover , there is also a null equilibrium , because at	r star.
and the label	$ y $	is in	y
and	$ X $	. % , which is similar to Algorithm	X
	$ b $	} { edges } & 30 & 35 & 40 & 45 & 50	b
which approximately coincides with the duration of the ultra-slow-roll phase	$ \Delta N_\text { USR } =\Or ( 1 ) $	. Using Eqs.~	Delta N sub U R equals big O r of 1.
and	$ \psi $	can be removed by use of their EOMs . After this manipulation , we have	Psi.
let	$ P ( x ) $	be the proposition that	P of x
	$ b=b\sigma^\dagger\sigma+b\sigma\sigma^\dagger $	, which gives two dynamical equations corresponding to the operator	b equals b sigma dagger sigma plus b sigma sigma dagger.
Since we assume	$ \hat\Gamma $	to be Lipschitz , there exists a decomposition	gamma-hat.
for all	$ u\in U $	such that the mapping	u belongs to U.
The ‘	$ \alpha $	’ and	Alpha.
where the matrix elements of	$ V_ { n } $	are	V sub n.
and	$ \\ { \tilde\ { Z $	.	curly brace, tilde, Z
where	$ \|\cdot\| $	is the absolute value norm . Consequently ,	Absolute value of center dot.
is the polynomial in	$ q $	defined by	q
is the reduction threshold used in the definition of a reduced in	$ G_\al $	word . Informally , a reduced in	G sub alpha.
Taking into account the previous constraint in	$ \beta ' $	, and that thus	Beta prime.
if	$ i \in V_1 $	and	i belongs to V sub 1.
contains all	$ 2 $	-faces of	two
we compare the values of	$ N_ * $	for NFD and NSPS schemes by simulating PEPQCs possessing noisy CNOT gates with the channel action in~	N star.
as well as to obtain a general equation relating the quantities	$ E $	,	E
for some finite	$ S'\supseteq S $	. Then one can imitate the proof of	S prime is a superset of S.
002 &	$ - $	2.551 & & & &	minus
In the case	$ d $	even , the vertices are even , but the internal edges are odd so that an ordering of this set is chosen ( up to permutation ) . This rule	d
the cross section	$ \sigma ( E ) $	is substantially larger for the	sigma of E.
in thermoelectric materials with ultralow	$ \kappa $	at 300 K. The comparison clearly illustrates that the oscillation of Ag atoms in AgTlI	kappa.
we conclude that	$ \beta $	swaps colors	Beta.
060	$ \pm $	0.002	Plus or minus.
the masses of the chimney and edge clouds are 3800 and 2300	$ M_ { \odot } $	, respectively . Also , the masses of these clouds are 1100 and 700	M sub-sun.
is satisfied in case	$ \sigma $	is as in Example	Sigma.
is an	$ n $	-sequence of disjoint clopens on a Stone-space	n
and the action of	$ H $	on	H
for any convex body	$ C $	. See also	C
of dimension	$ 0 $	and consider the two arrows , say	zero
where	$ \alpha $	and	Alpha.
	$ M $	, is presented in Figure	M
that are exactly at distance	$ \eps $	from some point in	epsilon.
our choice of the symmetric	$ T^ { \mu\nu } $	is more preferable . % % % % % % % % % % % % % % % % % % % %	T raised to the mu nu.
for	$ Q_1 > 0 $	:	Q one is greater than zero.
-dimension subspace of	$ \mathcal { K } ^ * _b $	.	K star sub b.
we have	$ x > y $	in	x is greater than y.
let	$ ( i , i ' ) $	be a minimal element of	open parenthesis i, comma, i prime, close parenthesis.

for all

$ \sigma \in \cC $

. Since

Sigma belongs to the set C.

for all

$ x $

.

x

By choosing

$ T $

small enough , we can control

T

as a continuous-time Markov chain

$ ( \eta_t ) _ { t\geq 0 } $

on

Open parenthesis, eta sub t, close parenthesis, sub t greater than or equal to zero.

Introducing

$ u=u_ { h } x $

,

u equals u sub h times x.

exactly equals to

$ M_K $

. The label

M sub K.

are finite and contribute to the normalized crossed conductance for

$ E > \Delta_ { 0 } $

. For

E greater than Delta sub zero

&

$ \delta $

max &

delta.

Let

$ m\geq 0 $

and

m greater than or equal to zero.

while

$ \delta $

is determined by the relative phase among 2 orthogonal oscillations as electromagnetic fields . These 2 parameters must not be affected during the amplification process for which we used a

delta.

which is at a distance

$ x $

from the initial position , differ by unity depending on whether it is located on the right-side or the left-side of the initial position . For

x

&

$ A $

A

and

$ \epsilon $

, we start the simulation from rest , adding an infinitesimal perturbation to the otherwise linear temperature background . After a short transient , the kinetic energy grows exponentially in

epsilon.

for

$ v \in A $

and

v belongs to A

and we append two indexes to indicate the values of the parameter

$ a $

and of the potential

a

4 10

$ ^ { -4 } $

& 9.3

Negative four.

&

$ - $

& 41.0 & 5.0 & 3.2 & 15.3 &

minus

is the centralizer of

$ \pi \in S_n $

. Note that all elements

Pi belongs to the set of permutations of n.

$ p $

? x ? o ) } is contradicted by a positive effect ( resp . precondition ) , then

p

At any time

$ t $

, one can observe the market price , i.e.~the time-

t

05

$ , $

h=0.5

comma

a vector meson

$ V\fn { Q\ol Q } $

decays into two light pseudo-scalar mesons

V f n Q ol Q

0

$ & $

1.00

and

1

$ \pm { 0.02 } $

, 0.3

Plus or minus zero point zero two.

~Zhang

$ ^ { 80 } $

, J.~J.~Zhang

to the power of 80

and the seller 's outside option value

$ x $

and market participation costs

x

the differential equation has the typical form with a non-homogeneous term

$ g ( x ) $

g of x.

2

$ & $

876

and

by

$ Y '' $

. Now on each

Y double prime.

The proof for general

$ x\in [ 0,1 ] $

follows from shifting every interval by

x belongs to the interval from 0 to 1.

He bound state under each of the

$ \eta N $

potential sets shown in Fig.~

Eta N.

and taking the expectation over

$ S , S ' $

, the second inequality follows from Jensen 's inequality , and the third inequality follows from the fact that

S comma S prime.

and

$ G_2 = ( V_2 , E_2 ) $

and suppose that

G sub 2 equals open parenthesis, V sub 2 comma E sub 2, close parenthesis.

28

$ \pm $

26.73 & 48.83

Plus or minus.

is

$ C $

-convex if and only if it is the intersection of some translates of

C

8

$ \pm $

4.9 & 96.2

Plus or minus.

In the area with

$ -1 $

pc

Minus one.

$ We have $

We have: None

by choice of

$ A^ * $

. For every

A star.

have

$ R $

-charges 1 ,

R

After checking all the

$ n $

individual arms , it can be easily seen that by the algorithm 's construction , both sets

n

these slots have indices in the set

$ D $

, so also refer to pivot columns )

D

corresponding to a different band index

$ \nu $

. Elements of each column are ordered as

nu.

The function

$ \zeta $

plays almost no role in our analysis , as we are focused on the bulk of the system . For simplicity we keep “ the same

zeta.

independently of

$ \delta $

near which we can fix areas . %

delta.

and keep track of

$ w ( i ) $

, and plot it in Figure

w of i

26

$ \pm $

0.45

Plus or minus.

All models were highly successful in learning the distribution of the novel tokens in the fine-tuning datasets : average accuracy was

$ 99.6\ % $

for DistilBERT ,

99.6 percent.

in terms of performance on the CTW

$ 1500 $

benchmark . Fig .

One thousand five hundred.

where

$ i \in [ 2 ] $

. Let

i belongs to the set containing 2.

is unmeasured or that the effect of

$ X $

on

X

as the regularity properties of

$ \mathcal { N } $

do not suffice to control

capital N

&

$ \log\mbh/\msun $

&

log m b h divided by m s u

with respect to the basis

$ v_ { \lambda 1 } , \ldots , v_ { \lambda |\lambda| } $

. Let

v sub lambda 1, ellipsis, v sub lambda absolute value of lambda

letting

$ \dfrac { \partderiv { \mathcal { Q } } } { \partderiv { \wght { } } } = 0 $

and rearranging , %

The derivative of the Q with respect to the derivative of delta wgtht equals zero.

-valued simple functions in

$ L^ { p ' } ( \Omega , \omega^ { \frac { 1 } { p-1 } } ; Y ) $

is also norming for

L prime, open parenthesis, Omega, comma, omega, to the power of one over p minus one, semicolon, Y, close parenthesis.

and any value of

$ \bar { p } $

, whereas the second group satisfies the following formulas :

p bar.

there exists a constant

$ \mathfrak { C } > 0 $

does n't depending neither on

C is greater than zero.

because it is not consistent with the median 's optimal choice of

$ r^ * $

. In this example , both potential local optima are global optima . Moreover , there is also a null equilibrium , because at

r star.

and the label

$ y $

is in

y

and

$ X $

. % , which is similar to Algorithm

X

$ b $

} { edges } & 30 & 35 & 40 & 45 & 50

b

which approximately coincides with the duration of the ultra-slow-roll phase

$ \Delta N_\text { USR } =\Or ( 1 ) $

. Using Eqs.~

Delta N sub U R equals big O r of 1.

and

$ \psi $

can be removed by use of their EOMs . After this manipulation , we have

Psi.

let

$ P ( x ) $

be the proposition that

P of x

$ b=b\sigma^\dagger\sigma+b\sigma\sigma^\dagger $

, which gives two dynamical equations corresponding to the operator

b equals b sigma dagger sigma plus b sigma sigma dagger.

Since we assume

$ \hat\Gamma $

to be Lipschitz , there exists a decomposition

gamma-hat.

for all

$ u\in U $

such that the mapping

u belongs to U.

The ‘

$ \alpha $

’ and

Alpha.

where the matrix elements of

$ V_ { n } $

are

V sub n.

and

$ \\ { \tilde\ { Z $

.

curly brace, tilde, Z

where

$ |\cdot| $

is the absolute value norm . Consequently ,

Absolute value of center dot.

is the polynomial in

$ q $

defined by

q

is the reduction threshold used in the definition of a reduced in

$ G_\al $

word . Informally , a reduced in

G sub alpha.

Taking into account the previous constraint in

$ \beta ' $

, and that thus

Beta prime.

if

$ i \in V_1 $

and

i belongs to V sub 1.

contains all

$ 2 $

-faces of

two

we compare the values of

$ N_ * $

for NFD and NSPS schemes by simulating PEPQCs possessing noisy CNOT gates with the channel action in~

N star.

as well as to obtain a general equation relating the quantities

$ E $

,

E

for some finite

$ S'\supseteq S $

. Then one can imitate the proof of

S prime is a superset of S.

002 &

$ - $

2.551 & & & &

minus

In the case

$ d $

even , the vertices are even , but the internal edges are odd so that an ordering of this set is chosen ( up to permutation ) . This rule

d

the cross section

$ \sigma ( E ) $

is substantially larger for the

sigma of E.

in thermoelectric materials with ultralow

$ \kappa $

at 300 K. The comparison clearly illustrates that the oscillation of Ag atoms in AgTlI

kappa.

we conclude that

$ \beta $

swaps colors

Beta.

060

$ \pm $

0.002

Plus or minus.

the masses of the chimney and edge clouds are 3800 and 2300

$ M_ { \odot } $

, respectively . Also , the masses of these clouds are 1100 and 700

M sub-sun.

is satisfied in case

$ \sigma $

is as in Example

Sigma.

is an

$ n $

-sequence of disjoint clopens on a Stone-space

n

and the action of

$ H $

on

H

for any convex body

$ C $

. See also

C

of dimension

$ 0 $

and consider the two arrows , say

zero

where

$ \alpha $

and

Alpha.

$ M $

, is presented in Figure

M

that are exactly at distance

$ \eps $

from some point in

epsilon.

our choice of the symmetric

$ T^ { \mu\nu } $

is more preferable . % % % % % % % % % % % % % % % % % % % %

T raised to the mu nu.

for

$ Q_1 > 0 $

:

Q one is greater than zero.

-dimension subspace of

$ \mathcal { K } ^ * _b $

.

K star sub b.

we have

$ x > y $

in

x is greater than y.

let

$ ( i , i ' ) $

be a minimal element of

open parenthesis i, comma, i prime, close parenthesis.