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
where	$ \rho_ { A ' B } $	is the state of detectors after the entanglement harvesting , given in Eq.~	Rho sub A prime B
of	$ R $	in parallel } { % Update training set	R
general	$ n $	-point function is given exactly in the integral form , which was worked out first by Okounkov	n
we observe a transition in a more conventional observable -- the charge density-imbalance between neighboring sites	$ \Delta $	-- as	Delta.
D3-branes with	$ N_F $	probe D7-branes , i.e . , 4d	N sub F
the vertex is symmetric in each index pair	$ \tilde V_ { ijkl } = \tilde V_ { jikl } = \tilde V_ { kjil } = ... $	To solve the integral we use the series representation of the Hermite polynomials	The tilde V sub ijkl equals tilde V sub jikl equals tilde V sub kjil equals dot dot dot.
inclusively and	$ y $	exclusively . Again the precise knowledge about the effect of path halving allows us to derive useful properties so they do not have to be included in the invariant :	y
and	$ 1 $	-stable if	one
are associated with the	$ s $	largest singular values of	s
and	$ S $	is the invertible symmetric matrix relating	S
of length	$ n-1 $	.	n minus one
in the CPDAG if the Markov equivalence class contains a DAG for which	$ X_i\rightarrow X_j $	as well as a DAG for which	X sub i goes to X sub j.
such that	$ j\in \mathcal { C } _t $	. The following holds	j belongs to the set C sub t
srk	$ such that $		Such that.
when	$ r < 1 $	, the volume diminishes exponentially in	r is less than one.
represent the	$ SU ( 3 ) $	flavor multiplets : the flavor sextets are labeled by	S U three.
we propose the	$ \alpha $	-NEEP , which generalizes the NEEP . Instead of the KL divergence , the	Alpha.
in	$ V ( \bsl { k } +\bsl { G } ) n_f ( \bsl { k } +\bsl { G } ) $	as a good approximation , resulting in	center dot
We first use a numerically stable	$ \Delta t $	then adjust	Delta t.
the order parameter	$ \mathcal { B } $	in Eq.~	bee
are the rows of	$ T_ { s-1 } $	for	T sub s minus 1.
is the intersection of	$ F $	with the torsion subgroup of	F
We denote	$ I_G $	the projection of	I G.
We can take these elements since	$ B $	is atomless . Put	B
If	$ n\ge n_0 $	, then one of the components of	n greater than or equal to n sub zero.
993	$ \pm $	0.009 & 0.842	Plus or minus.
	$ + $	& 9.96e-2 ( 2.18e-2 ) & 1.03e-1 ( 2.39e-2 )	plus
	$ ^\textrm { \scriptsize 163 } $	,	One hundred sixty-three.
is separable in terms of the variables	$ { \bf w } _t $	and	bold w sub t
A natural example is to choose	$ g $	to parametrize the ( positively oriented ) unit tangent vector to	g
	$ \Sigma_ { \mathrm { hol } } $	has a non-zero pole at	sigma sub hol.
Let	$ G $	be a graph of order	G
states	$ \rho $	, and	rho.
Recall that the first	$ n-k $	entries of	n minus k
if	$ x^ * $	satisfies	x raised to the power of asterisk.
% In this appendix we report the exact expression for the	$ \yttwth $	variable	tee-wth
and	$ d_b $	is defined as follows : %	d sub b.
% Without loss of generality We illustrate results for subgroup	$ k = 1 $	. Figure	k equals one.
we conduct ablation studies to analyze the relative contribution of our policy representation module and the sensitivity of the number of candidate policies	$ M $	.	M
-- geodesic	$ \beta $	that crosses each	Beta.
&	$ \mathsf { x } $	&	x.
and	$ 2 $	, deserves	two
For	$ t \geq 1 $	, let	t is greater than or equal to 1.
in the above described unconditional way and then accepting the sample if	$ y \in Y $	. In practical applications , due to a finite runtime , we sample a batch of	y belongs to Y.
is constructed to lower bound the true	$ Q $	value ( point-wise ) . That is ,	Q
and	$ m+k-j $	are strictly between	m plus k minus j
We also compared the results for	$ \sigma_F $	to the results for the engineering stress obtained using the	sigma sub F.
node in	$ M $	( and	M
6	$ ^\circ $	N geographic latitude , 25.2	degree.
After an observation at time	$ t $	is measured , the predictive density can be updated as	t
of	$ L^2_0 ( \rho_P ) $	of associated eigenfunctions for	L squared sub 0 of rho P.
is simple and the first eigenfunction	$ u_1 $	is positive on	u sub one.
is the Dehn twist along	$ \zeta $	, see Figure	zeta.
We recall that	$ H $	is a function in	H
there must be some	$ c $	and	c
R	$ for a fraction $	0	for a fraction.
we aim to learn an effective encoder	$ f $	to improve representations against false negatives and class imbalance . Next , we will give an in-depth analysis to point out the two major issues currently troubling UCL :	f
be the first time the the chain	$ { \bf P } ^ { * } $	is in a state of the form	P star.
	$ since $	d >	Since None
we show the performance trends with the change of	$ \gamma $	in Figure~	Gamma.
we discretize this probability distribution in	$ \kappa_1 $	-	kappa sub 1
	$ s_i $	represents Pauli matrices in the spin space . An advantage of this basis is that the hole sector becomes the time-reversal partner of the electron sector :	s sub i.
\ ! \ ! \ ! &	$ 8 $		eight
	$ 5 ) $	]	five
we may start with the doublet	$ E $	state which can be expressed by (	E
& & & & & \\	$ a $	(	a
and	$ n=1 $	, the proposition has already been discussed above . For	n equals one.
we need to change the phase to obtain an	$ X $	operation from a	X
This makes the reference signal	$ x_i $	a combination of the far-end signal	x sub i.
and	$ H \in \mathcal\ { U^Z_C $	.	H belongs to the set of U raised to the power of Z sub C.
be real valued on an interval	$ I $		I
The natural transformation	$ \ov { \nu } $	is in fact an isomorphism . To observe this , we define the morphism	v nu.
generates the parameters to scale and bias each channel	$ z_ { c } $	such that the the channel values are computed as : %	z sub c
We can identify the nonlinear	$ S $	-matrix symmetry using Eqs.~	S
is nested in	$ h_ { b , c } ( 1 ) $	.	h sub b comma c open parenthesis 1 close parenthesis.
be positive measures on	$ \mathbb R^d $	with respect to which	the set of real numbers raised to the power of d.
Z_s^\prime \longrightarrow Z_s^\prime	$ & $		and
the color of each node pair	$ ( u , v ) $	can encode its distance	u comma v.
it is noticed that the effect of neutron skin in	$ r $	space on the emission of hard photons becomes larger when considering the surface term in the kinetic energy density , leading to the appearance of proton skin in	r
and	$ b =\sum_ { i=n-2t+1 } ^ { n } v_ { i } [ k ] $	. Then ,	b equals the sum from i equals n minus 2t plus 1 to n of v sub i of k.
of dimension	$ n $	with	n
then delete	$ u $	.	u
rounds no red cycle is created nor a blue copy of	$ H $	. Hence	H
of	$ \phi_x $	is %	phi sub x.
We may therefore get an estimate for	$ \gamma $	from the value of	Gamma.
be such that	$ \Xl\cap ( V_0+x ) =\emptyset $	. Then there are constants	X intersection V sub 0 plus x equals empty set.
&	$ \sigma_ { \mathrm { RV } _2 } $	& RV	Sigma sub RV 2.
and degree	$ 3 $	projection maps mapping infinite points on	three
of	$ ( X , B+A+H ) $	such that	X, B plus A plus H.
for	$ Re_c \ll 1 $	, moving to smaller	Re c much less than one.
these fronts are marginally stable if	$ d_1 \geq d_1^ * ( \delta ) $	, and strictly stable for	d sub 1 greater than or equal to d sub 1 star delta
end to	$ 0 $	( resp .	zero
	$ K_ { g } $	,	K sub g.
	$ > $	} }	greater than
is smooth and isomorphic to	$ \Ff_n $	. The surface	F sub n.
over all	$ T $	is found to correspond to a time horizon of	T
! M_2	$ of rank $		$\text{rank}$
we introduce an indicator of tradeoff relation	$ \omega $	derived from our way of quantifying a tradeoff relation . The	omega.
and	$ P_ { T } $	to denote source and target point clouds ,	P sub T.
% % The condition that	$ \phi_1 $	has no superattracting cycles turns out to be necessary . % % The reader will notice the hypothesis of no superattracting periodic points on the base - this is tight	Phi sub 1.

where

$ \rho_ { A ' B } $

is the state of detectors after the entanglement harvesting , given in Eq.~

Rho sub A prime B

of

$ R $

in parallel } { % Update training set

R

general

$ n $

-point function is given exactly in the integral form , which was worked out first by Okounkov

n

we observe a transition in a more conventional observable -- the charge density-imbalance between neighboring sites

$ \Delta $

-- as

Delta.

D3-branes with

$ N_F $

probe D7-branes , i.e . , 4d

N sub F

the vertex is symmetric in each index pair

$ \tilde V_ { ijkl } = \tilde V_ { jikl } = \tilde V_ { kjil } = ... $

To solve the integral we use the series representation of the Hermite polynomials

The tilde V sub ijkl equals tilde V sub jikl equals tilde V sub kjil equals dot dot dot.

inclusively and

$ y $

exclusively . Again the precise knowledge about the effect of path halving allows us to derive useful properties so they do not have to be included in the invariant :

y

and

$ 1 $

-stable if

one

are associated with the

$ s $

largest singular values of

s

and

$ S $

is the invertible symmetric matrix relating

S

of length

$ n-1 $

.

n minus one

in the CPDAG if the Markov equivalence class contains a DAG for which

$ X_i\rightarrow X_j $

as well as a DAG for which

X sub i goes to X sub j.

such that

$ j\in \mathcal { C } _t $

. The following holds

j belongs to the set C sub t

srk

$ such that $

Such that.

when

$ r < 1 $

, the volume diminishes exponentially in

r is less than one.

represent the

$ SU ( 3 ) $

flavor multiplets : the flavor sextets are labeled by

S U three.

we propose the

$ \alpha $

-NEEP , which generalizes the NEEP . Instead of the KL divergence , the

Alpha.

in

$ V ( \bsl { k } +\bsl { G } ) n_f ( \bsl { k } +\bsl { G } ) $

as a good approximation , resulting in

center dot

We first use a numerically stable

$ \Delta t $

then adjust

Delta t.

the order parameter

$ \mathcal { B } $

in Eq.~

bee

are the rows of

$ T_ { s-1 } $

for

T sub s minus 1.

is the intersection of

$ F $

with the torsion subgroup of

F

We denote

$ I_G $

the projection of

I G.

We can take these elements since

$ B $

is atomless . Put

B

If

$ n\ge n_0 $

, then one of the components of

n greater than or equal to n sub zero.

993

$ \pm $

0.009 & 0.842

Plus or minus.

$ + $

& 9.96e-2 ( 2.18e-2 ) & 1.03e-1 ( 2.39e-2 )

plus

$ ^\textrm { \scriptsize 163 } $

,

One hundred sixty-three.

is separable in terms of the variables

$ { \bf w } _t $

and

bold w sub t

A natural example is to choose

$ g $

to parametrize the ( positively oriented ) unit tangent vector to

g

$ \Sigma_ { \mathrm { hol } } $

has a non-zero pole at

sigma sub hol.

Let

$ G $

be a graph of order

G

states

$ \rho $

, and

rho.

Recall that the first

$ n-k $

entries of

n minus k

if

$ x^ * $

satisfies

x raised to the power of asterisk.

% In this appendix we report the exact expression for the

$ \yttwth $

variable

tee-wth

and

$ d_b $

is defined as follows : %

d sub b.

% Without loss of generality We illustrate results for subgroup

$ k = 1 $

. Figure

k equals one.

we conduct ablation studies to analyze the relative contribution of our policy representation module and the sensitivity of the number of candidate policies

$ M $

.

M

-- geodesic

$ \beta $

that crosses each

Beta.

&

$ \mathsf { x } $

&

x.

and

$ 2 $

, deserves

two

For

$ t \geq 1 $

, let

t is greater than or equal to 1.

in the above described unconditional way and then accepting the sample if

$ y \in Y $

. In practical applications , due to a finite runtime , we sample a batch of

y belongs to Y.

is constructed to lower bound the true

$ Q $

value ( point-wise ) . That is ,

Q

and

$ m+k-j $

are strictly between

m plus k minus j

We also compared the results for

$ \sigma_F $

to the results for the engineering stress obtained using the

sigma sub F.

node in

$ M $

( and

M

6

$ ^\circ $

N geographic latitude , 25.2

degree.

After an observation at time

$ t $

is measured , the predictive density can be updated as

t

of

$ L^2_0 ( \rho_P ) $

of associated eigenfunctions for

L squared sub 0 of rho P.

is simple and the first eigenfunction

$ u_1 $

is positive on

u sub one.

is the Dehn twist along

$ \zeta $

, see Figure

zeta.

We recall that

$ H $

is a function in

H

there must be some

$ c $

and

c

R

$ for a fraction $

0

for a fraction.

we aim to learn an effective encoder

$ f $

to improve representations against false negatives and class imbalance . Next , we will give an in-depth analysis to point out the two major issues currently troubling UCL :

f

be the first time the the chain

$ { \bf P } ^ { * } $

is in a state of the form

P star.

$ since $

d >

Since None

we show the performance trends with the change of

$ \gamma $

in Figure~

Gamma.

we discretize this probability distribution in

$ \kappa_1 $

-

kappa sub 1

$ s_i $

represents Pauli matrices in the spin space . An advantage of this basis is that the hole sector becomes the time-reversal partner of the electron sector :

s sub i.

\ ! \ ! \ ! &

$ 8 $

eight

$ 5 ) $

]

five

we may start with the doublet

$ E $

state which can be expressed by (

E

& & & & & \\

$ a $

(

a

and

$ n=1 $

, the proposition has already been discussed above . For

n equals one.

we need to change the phase to obtain an

$ X $

operation from a

X

This makes the reference signal

$ x_i $

a combination of the far-end signal

x sub i.

and

$ H \in \mathcal\ { U^Z_C $

.

H belongs to the set of U raised to the power of Z sub C.

be real valued on an interval

$ I $

I

The natural transformation

$ \ov { \nu } $

is in fact an isomorphism . To observe this , we define the morphism

v nu.

generates the parameters to scale and bias each channel

$ z_ { c } $

such that the the channel values are computed as : %

z sub c

We can identify the nonlinear

$ S $

-matrix symmetry using Eqs.~

S

is nested in

$ h_ { b , c } ( 1 ) $

.

h sub b comma c open parenthesis 1 close parenthesis.

be positive measures on

$ \mathbb R^d $

with respect to which

the set of real numbers raised to the power of d.

Z_s^\prime \longrightarrow Z_s^\prime

$ & $

and

the color of each node pair

$ ( u , v ) $

can encode its distance

u comma v.

it is noticed that the effect of neutron skin in

$ r $

space on the emission of hard photons becomes larger when considering the surface term in the kinetic energy density , leading to the appearance of proton skin in

r

and

$ b =\sum_ { i=n-2t+1 } ^ { n } v_ { i } [ k ] $

. Then ,

b equals the sum from i equals n minus 2t plus 1 to n of v sub i of k.

of dimension

$ n $

with

n

then delete

$ u $

.

u

rounds no red cycle is created nor a blue copy of

$ H $

. Hence

H

of

$ \phi_x $

is %

phi sub x.

We may therefore get an estimate for

$ \gamma $

from the value of

Gamma.

be such that

$ \Xl\cap ( V_0+x ) =\emptyset $

. Then there are constants

X intersection V sub 0 plus x equals empty set.

&

$ \sigma_ { \mathrm { RV } _2 } $

& RV

Sigma sub RV 2.

and degree

$ 3 $

projection maps mapping infinite points on

three

of

$ ( X , B+A+H ) $

such that

X, B plus A plus H.

for

$ Re_c \ll 1 $

, moving to smaller

Re c much less than one.

these fronts are marginally stable if

$ d_1 \geq d_1^ * ( \delta ) $

, and strictly stable for

d sub 1 greater than or equal to d sub 1 star delta

end to

$ 0 $

( resp .

zero

$ K_ { g } $

,

K sub g.

$ > $

} }

greater than

is smooth and isomorphic to

$ \Ff_n $

. The surface

F sub n.

over all

$ T $

is found to correspond to a time horizon of

T

! M_2

$ of rank $

$\text{rank}$

we introduce an indicator of tradeoff relation

$ \omega $

derived from our way of quantifying a tradeoff relation . The

omega.

and

$ P_ { T } $

to denote source and target point clouds ,

P sub T.

% % The condition that

$ \phi_1 $

has no superattracting cycles turns out to be necessary . % % The reader will notice the hypothesis of no superattracting periodic points on the base - this is tight

Phi sub 1.