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
with the thickness decreasing as	$ b $	increases .	b
4 GPa at	$ 1200^ { \circ } $	C . For this scenario , we trained a cGAN with the architecture described in Section~	one hundred raised to the power of degree
G &	$ 0.41\pm0.39 $	&	zero point one plus or minus zero point three nine.
-deceptive queries with respect to file index	$ k $	( required file ) for correctness ( see Tables~	k
We first show that given any block encoding of	$ H $	, we can obtain a Hamiltonian that block-encodes	H
A pair of nontrivial elements in	$ \PH_k ( X ) $	and	P H sub k of X
is related to the real momentum	$ p_m ( t ) $	as	p sub m of t
em Likelihood : diagonal Gaussian with event shape	$ 1 $	.	one
	$ \lambda_j $	's , on different categories of	Lambda sub j.
and high-	$ T $	(	T
They are prominent only for	$ \theta $	values greater than	Theta.
this proposition holds for	$ h $	-dependent symbols by taking	h
The latter is equivalent to have that	$ C \subset \LimInn C^\nu $	so by proving this we will have	C subset Lim In C nu
of finding the walker at	$ ( x , y ) $	on the lattice after ten steps of the 2DQW with~	open parenthesis x comma y close parenthesis.
has a unique non-neighbor	$ u_i $	in	u sub i.
and knowing the profile	$ \psi ( a ) $	one can find that	Psi of a.
	$ g_8^0 $	to	g sub 8 superscript 0
we see the intense AE and AO conical dispersions that form concentric rings of intensity in the	$ \Gamma $	zones . We also see streaks of intensity that extend longitudinally across the	Gamma.
and	$ \Gamma_1 $	by replacing	Gamma one.
random variables is generated by only	$ J $	independent random variables (	J
and thus is limited by	$ \chi $	. The cat size for this method is	chi.
The current state is estimated by Kalman filter as	$ \hat { x } _ { l\|l } $	. Then we reconstruct and solve the control optimization problem of the mobile agent as Problem	x hat sub l given l
the number	$ n_D $	of power stages of the demand and	n sub D
is a term with one free variable	$ { v } $	, then the formula	v
and illustrates the need for the evaluation map in the chained projection groupoid~	$ ( \G , \ve ) $	.	open parenthesis G, comma, vee, close parenthesis.
em If	$ s < 0 $	, there exists a unique	s is less than zero.
it is the set of all generalized Mycielski transformations of	$ K_2 $	obtained from	K sub 2.
the iterative method that has larger convergence zones to the roots is the	$ SW_1 $	method , so in this case it would be more recommendable to use this method .	S W sub 1.
or	$ a ' ( c ) $	. This idea can be generalised as follows .	a prime of c
is	$ T $	-periodic while , by elementary calculations , the function	T
	$ F ( R ) $	gravity [ 10 ] , quantum field theory in curved spacetime [ 7,11 ] , and Einstein-	F of R.
by setting	$ Z=X^ { 1/8 } l^ { -1/4 } $	, thus completing the proof of Theorem	Z equals X to the power of one-eighth times l to the power of negative one-fourth.
the space of standard embeddings of a sphere into	$ ( M , \xi ) $	and by	M xi.
For	$ i \in [ 7 ] $	, let	i belongs to the set of 7
A	$ \Gamma\simeq 1.4 $	power-law described the quiescent X-ray spectrum of SAX J1808.4 -- 3658	Gamma approximately equal to 1.4.
if the parameters	$ n $	and	n
Given that the total number of TSs is	$ S $	, the original information sequence is mapped to the corresponding bit sub-channel using polarization coding to state	S
em Let	$ q ( A , x ) = c ( A , x ) + f ( A , x ) \in \R^n $		q of A, x equals c of A, x plus f of A, x belongs to R to the power of n.
	$ T $	}	T
it is apparent that	$ \Lambda $	belongs to the closure of	Lambda.
whose map	$ F ' $	is given by the diagram	F prime.
is	$ \mathcal { T } ( \varphi , \vartheta ) $	.	The calligraphic T of phi and theta.
from	$ \calG $	is not saturated , compute	G: None
Since the choice of	$ \delta $	determines the temporal resolution of our classifier , the largest window duration should not exceed the order of 10 seconds . On the other hand , very short	delta.
those that are different from	$ v_ { a } $	and	v subscript a
where for a fixed	$ u $	we used H	u
factors as	$ q : W\to H_3\to H_2 $	. Since	q: W to H 3 to H 2
In	$ \ { \modd\text\ { \- $	, let	mod-
8 \\	$ B $	( IV2 : 2	B
of	$ w $	.	w
a computer stores information in	$ m $	-tuples called _words_	m
be the boundary of	$ \Omega $	within	Omega.
	$ be the irreducible representation of $		bee-inducible representation.
to third-order of	$ \zeta $	yields the interaction action	zeta.
are	$ \gamma $	-related if , and only if ,	Gamma.
	$ r_1 $	,	r sub 1
32	$ \pm $	0.00 & 7.02	Plus or minus.
that is allowed to traverse the boundary of	$ [ 0,1 ] ^d $	at zero cost . For a proof of the following lemma , we refer to the monograph by Yukich~	Zero to one raised to the power of d.
	$ l_1=l_2 $	. From this ,	l sub 1 equals l sub 2
may be independent of	$ i $	} )	i
As the fiber	$ F_z $	is a conic with	F sub z.
When	$ h\sim 0 $	,	h is approximately zero.
is	$ \|T\|_ { \Sigma^\bfalpha_1 } $	. Now working inside	Absolute value of T, sub Sigma to the power of alphapha 1.
	$ b_ { s^2 } $	, can be easily introduced following	b sub s squared.
is the dust diffusivity and	$ W_t $	denotes a Wiener process . We define the dust diffusivity as	W sub t
	$ \gamma $	and	Gamma.
	$ ( w_ { k22 } ^ { `` } ) ^2 $		w sub k 22 superscript tilde squared
	$ \gamma $	,	Gamma.
the diameter is exactly	$ 4 $	, in all	four
% Let	$ L $	be a component of a flat FAL and suppose that distinct reflection surfaces	L
is shown to be the generating function for permutations of	$ [ n ] $	with exactly	n in brackets.
central large surface brightness contour within the produced pseudo-narrowband Ly	$ \alpha $	images . Each target 's total Ly	Alpha.
	$ T_s \in D $	and	T sub s belongs to D.
As expected we can observe that the DG-norm of the error decreases as	$ h^p $	, while the	h raised to the power of p.
as well as the absolute value of	$ \bar { \lambda } _\ell $	, are plotted in fig.~	bar lambda sub ell
Then	$ Y $	contains	Y
Let during the round	$ t+T_f $	,	t plus T sub f
	$ \ { \la_\ell , \rho_\ell\ } _ { \ell\in\mathbb\ K } $	can be uniquely determined by	Curly braces l sub l, comma, rho sub l, sub l in the set of K.
Since the statement of the theorem gets weaker as	$ \alpha_i $	increases in the interval	Alpha sub i.
This leads to	$ 2N $	FLOP due to the dot product of two vectors .	Two N.
with a probability of at least	$ 1-\delta $	over our calibration datasets	One minus delta.
All the numerical data reported here is determined with precision up to at least the	$ 10^ { \rm th } $	decimal place . To accurately determine the precise value of a given	ten to the power of th
let us assume that	$ \mathbb { M } $	allocates	M double-struck.
and spin	$ \sigma ' $	. Hence , the averages in the Green 's function are taken with respect to the state	Sigma prime.
which is an equal weight superposition of all	$ n $	qubit states with Hamming Weight	n
then it vanishes at at most	$ n_j $	points on	n sub j
let us discuss how to estimate the internal energy	$ \mathcal { U } $	of our system . First , note that a classical system can be described by a density matrix	U
This is true for any partition	$ \lambda $	, which will be proved in Section	Lambda.
we observe that	$ G_L $	is contained in the kernel of every coboundary . So if	G sub L.
^T	$ , $		comma
at timestep	$ t $	in the transient value function update rule is	t
and	$ \mathbf { u } $	the eigenvalues and the eigenvector matrix of	u
the local parameters at	$ \tilde p $	are	p tilde.
was first proved in the case of	$ \ { \mathcal\ B^\infty_\kappa $	. The extension to	B infinity sub kappa.
	$ C^ { e\gamma } $	,	C raised to the power of e gamma.
and locally Lipschitz in	$ y $	and	y
where	$ u_\pm=1\pm \frac { \mathcal { J } } { 2\pi } $	and	u plus or minus equals one plus or minus the fraction J divided by two pi.
with	$ n $	conical nanopores is the sum of the conductance of each individual nanopore	n
	$ M $	is	M
The main observation in this proof is a decomposition of h-paths in	$ X $	with respect to a subgraph	X

with the thickness decreasing as

$ b $

increases .

b

4 GPa at

$ 1200^ { \circ } $

C . For this scenario , we trained a cGAN with the architecture described in Section~

one hundred raised to the power of degree

G &

$ 0.41\pm0.39 $

&

zero point one plus or minus zero point three nine.

-deceptive queries with respect to file index

$ k $

( required file ) for correctness ( see Tables~

k

We first show that given any block encoding of

$ H $

, we can obtain a Hamiltonian that block-encodes

H

A pair of nontrivial elements in

$ \PH_k ( X ) $

and

P H sub k of X

is related to the real momentum

$ p_m ( t ) $

as

p sub m of t

em Likelihood : diagonal Gaussian with event shape

$ 1 $

.

one

$ \lambda_j $

's , on different categories of

Lambda sub j.

and high-

$ T $

(

T

They are prominent only for

$ \theta $

values greater than

Theta.

this proposition holds for

$ h $

-dependent symbols by taking

h

The latter is equivalent to have that

$ C \subset \LimInn C^\nu $

so by proving this we will have

C subset Lim In C nu

of finding the walker at

$ ( x , y ) $

on the lattice after ten steps of the 2DQW with~

open parenthesis x comma y close parenthesis.

has a unique non-neighbor

$ u_i $

in

u sub i.

and knowing the profile

$ \psi ( a ) $

one can find that

Psi of a.

$ g_8^0 $

to

g sub 8 superscript 0

we see the intense AE and AO conical dispersions that form concentric rings of intensity in the

$ \Gamma $

zones . We also see streaks of intensity that extend longitudinally across the

Gamma.

and

$ \Gamma_1 $

by replacing

Gamma one.

random variables is generated by only

$ J $

independent random variables (

J

and thus is limited by

$ \chi $

. The cat size for this method is

chi.

The current state is estimated by Kalman filter as

$ \hat { x } _ { l|l } $

. Then we reconstruct and solve the control optimization problem of the mobile agent as Problem

x hat sub l given l

the number

$ n_D $

of power stages of the demand and

n sub D

is a term with one free variable

$ { v } $

, then the formula

v

and illustrates the need for the evaluation map in the chained projection groupoid~

$ ( \G , \ve ) $

.

open parenthesis G, comma, vee, close parenthesis.

em If

$ s < 0 $

, there exists a unique

s is less than zero.

it is the set of all generalized Mycielski transformations of

$ K_2 $

obtained from

K sub 2.

the iterative method that has larger convergence zones to the roots is the

$ SW_1 $

method , so in this case it would be more recommendable to use this method .

S W sub 1.

or

$ a ' ( c ) $

. This idea can be generalised as follows .

a prime of c

is

$ T $

-periodic while , by elementary calculations , the function

T

$ F ( R ) $

gravity [ 10 ] , quantum field theory in curved spacetime [ 7,11 ] , and Einstein-

F of R.

by setting

$ Z=X^ { 1/8 } l^ { -1/4 } $

, thus completing the proof of Theorem

Z equals X to the power of one-eighth times l to the power of negative one-fourth.

the space of standard embeddings of a sphere into

$ ( M , \xi ) $

and by

M xi.

For

$ i \in [ 7 ] $

, let

i belongs to the set of 7

A

$ \Gamma\simeq 1.4 $

power-law described the quiescent X-ray spectrum of SAX J1808.4 -- 3658

Gamma approximately equal to 1.4.

if the parameters

$ n $

and

n

Given that the total number of TSs is

$ S $

, the original information sequence is mapped to the corresponding bit sub-channel using polarization coding to state

S

em Let

$ q ( A , x ) = c ( A , x ) + f ( A , x ) \in \R^n $

q of A, x equals c of A, x plus f of A, x belongs to R to the power of n.

$ T $

}

T

it is apparent that

$ \Lambda $

belongs to the closure of

Lambda.

whose map

$ F ' $

is given by the diagram

F prime.

is

$ \mathcal { T } ( \varphi , \vartheta ) $

.

The calligraphic T of phi and theta.

from

$ \calG $

is not saturated , compute

G: None

Since the choice of

$ \delta $

determines the temporal resolution of our classifier , the largest window duration should not exceed the order of 10 seconds . On the other hand , very short

delta.

those that are different from

$ v_ { a } $

and

v subscript a

where for a fixed

$ u $

we used H

u

factors as

$ q : W\to H_3\to H_2 $

. Since

q: W to H 3 to H 2

In

$ \ { \modd\text\ { \- $

, let

mod-

8 \\

$ B $

( IV2 : 2

B

of

$ w $

.

w

a computer stores information in

$ m $

-tuples called _words_

m

be the boundary of

$ \Omega $

within

Omega.

$ be the irreducible representation of $

bee-inducible representation.

to third-order of

$ \zeta $

yields the interaction action

zeta.

are

$ \gamma $

-related if , and only if ,

Gamma.

$ r_1 $

,

r sub 1

32

$ \pm $

0.00 & 7.02

Plus or minus.

that is allowed to traverse the boundary of

$ [ 0,1 ] ^d $

at zero cost . For a proof of the following lemma , we refer to the monograph by Yukich~

Zero to one raised to the power of d.

$ l_1=l_2 $

. From this ,

l sub 1 equals l sub 2

may be independent of

$ i $

} )

i

As the fiber

$ F_z $

is a conic with

F sub z.

When

$ h\sim 0 $

,

h is approximately zero.

is

$ |T|_ { \Sigma^\bfalpha_1 } $

. Now working inside

Absolute value of T, sub Sigma to the power of alphapha 1.

$ b_ { s^2 } $

, can be easily introduced following

b sub s squared.

is the dust diffusivity and

$ W_t $

denotes a Wiener process . We define the dust diffusivity as

W sub t

$ \gamma $

and

Gamma.

$ ( w_ { k22 } ^ { `` } ) ^2 $

w sub k 22 superscript tilde squared

$ \gamma $

,

Gamma.

the diameter is exactly

$ 4 $

, in all

four

% Let

$ L $

be a component of a flat FAL and suppose that distinct reflection surfaces

L

is shown to be the generating function for permutations of

$ [ n ] $

with exactly

n in brackets.

central large surface brightness contour within the produced pseudo-narrowband Ly

$ \alpha $

images . Each target 's total Ly

Alpha.

$ T_s \in D $

and

T sub s belongs to D.

As expected we can observe that the DG-norm of the error decreases as

$ h^p $

, while the

h raised to the power of p.

as well as the absolute value of

$ \bar { \lambda } _\ell $

, are plotted in fig.~

bar lambda sub ell

Then

$ Y $

contains

Y

Let during the round

$ t+T_f $

,

t plus T sub f

$ \ { \la_\ell , \rho_\ell\ } _ { \ell\in\mathbb\ K } $

can be uniquely determined by

Curly braces l sub l, comma, rho sub l, sub l in the set of K.

Since the statement of the theorem gets weaker as

$ \alpha_i $

increases in the interval

Alpha sub i.

This leads to

$ 2N $

FLOP due to the dot product of two vectors .

Two N.

with a probability of at least

$ 1-\delta $

over our calibration datasets

One minus delta.

All the numerical data reported here is determined with precision up to at least the

$ 10^ { \rm th } $

decimal place . To accurately determine the precise value of a given

ten to the power of th

let us assume that

$ \mathbb { M } $

allocates

M double-struck.

and spin

$ \sigma ' $

. Hence , the averages in the Green 's function are taken with respect to the state

Sigma prime.

which is an equal weight superposition of all

$ n $

qubit states with Hamming Weight

n

then it vanishes at at most

$ n_j $

points on

n sub j

let us discuss how to estimate the internal energy

$ \mathcal { U } $

of our system . First , note that a classical system can be described by a density matrix

U

This is true for any partition

$ \lambda $

, which will be proved in Section

Lambda.

we observe that

$ G_L $

is contained in the kernel of every coboundary . So if

G sub L.

^T

$ , $

comma

at timestep

$ t $

in the transient value function update rule is

t

and

$ \mathbf { u } $

the eigenvalues and the eigenvector matrix of

u

the local parameters at

$ \tilde p $

are

p tilde.

was first proved in the case of

$ \ { \mathcal\ B^\infty_\kappa $

. The extension to

B infinity sub kappa.

$ C^ { e\gamma } $

,

C raised to the power of e gamma.

and locally Lipschitz in

$ y $

and

y

where

$ u_\pm=1\pm \frac { \mathcal { J } } { 2\pi } $

and

u plus or minus equals one plus or minus the fraction J divided by two pi.

with

$ n $

conical nanopores is the sum of the conductance of each individual nanopore

n

$ M $

is

M

The main observation in this proof is a decomposition of h-paths in

$ X $

with respect to a subgraph

X