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
Though the interlayer ferromagnetic exchange	$ J_4 $	is crucial for introducing the 3D character , it is much smaller in comparison to the in-plane interactions and insufficient for creating long-range correlations along the	J sub 4.
&	$ 0 $	&	zero
and it consists the symbol-by-symbol detected vector	$ \check { \boldsymbol { x } } $	whose	check x
depends only on the	$ \SL_2 ( \Z ) $	-equivalence class of	S L two in parentheses Z.
such as minimal ones and certain maximal positive bases such as those formed by the coordinate vectors and their negatives in	$ \R^n $	. More OSPBs can be obtained via numerical procedures , even for intermediate sizes~	R to the power of n.
as	$ T\|_ { ( d ) } $	and	T restricted to d.
& 2 &	$ 0.23 $	&	0.23: None
by finding its	$ k $	nearest neighbor samples in	k
say	$ \underline { \alpha } _k $	and	alpha sub k.
through its image	$ \nu ( \Gamma ) \le\mrm { PGL } _2 ( \R ) $	.	Nu of Gamma is less than or equal to PGL 2 of R.
321	$ \pm $	0.041 & 0.851	Plus or minus.
is zero in an open set	$ \mathcal { S } $	, then	ess
its influence on the current probability distribution has a Total Variation distance upper bounded by	$ M_T $	.	M sub T.
is an	$ r $	-th root of unity . In particular ,	r
&	$ \ $	&	backslash
and the calibrated probability estimate	$ p $	for	p
calling	$ W $		W
	$ { \bf eg } = { \bf fh } =\beta/100 $	and the spherical angles at	e g equals f h equals beta divided by 100
em The operators	$ z_i $	and	z sub i.
The	$ \rho $	parameter determines the weight of external benchmarks in the ranking . We set	rho.
at redshift	$ z $	to have mass ratio in the range	z
and	$ \Sigma ' $	be a surface-minor of	sigma prime.
	$ \bY\coloneqq\bN^ { -1 } $	, and	b Y is defined as b N inverse
The strength of the attraction is modeled by the parameter	$ \epsilon $	, which is the depth of the square well-potential . The range of attractive interaction is given by the width of the square well , i.e . ,	epsilon.
with	$ K\leq G_ { w_0 } $	. Since the action of	K less than or equal to G sub w sub 0.
and the exponential	$ e^ { \mat { A } } $	, where	e raised to the power of A
and	$ q < \infty $	, observe that for	q is less than infinity.
This shows the negative correlation on	$ G ' $	.	the derivative of G.
Set	$ { \pi } ^ { j } _1=\pi^ { j } _1 \setminus \ { o\ } $	,	Pi to the power of j sub 1 equals pi to the power of j sub 1 set minus open brace o close brace.
Let	$ \chi $	be a fixed-point of	chi.
and standard deviation degrees up to	$ k $	-hop distance context of each node	k
An examination of the galaxy age versus star formation activity at	$ z > 8 $		z is greater than 8.
in~	$ B_x $	, as desired .	B sub x.
these frequencies only emit radiation in a certain range of values of	$ \kappa $	. In other words , the coupling between	kappa.
and	$ \phi_1 $	such that	Phi sub 1.
&	$ 2.3\pm1.1 $	&	2.3 plus or minus 1.1
the task is to find the diameter of the	$ G/K $	space , that is , the furthest distance possible pairs of points can have . Given the fact that motion in	G divided by K.
the magnetic field continuously turns the magnetization to	$ x $	direction , preserving continuously the non-zero Kondo hybridization just above	x
Due to the bounds assumed on the solutions	$ M_1 , M_2 $	, we have	M one, M two.
let	$ A_i\colon \H_i\to 2^ { \H_i } $	and	A sub i, colon, H sub i, to, 2 to the power of H sub i.
% % Variable selection update with respect to the	$ i $	th node regressing to the rest of the nodes . % % Update	i
&	$ 3.9_ { -0.3 } ^ { +0.4 } $	&	3.9 minus 0.3 raised to the power of plus 0.4.
em for all	$ x \in X $	,	x belongs to X.
and	$ d_i $	errors are rigorously less than	d sub i.
	$ i=1,2 ) $	, which implies	i equals one, two.
be the vector space of sections of	$ E $	and	E
we claim the righthand inequality holds with	$ c_1=c_0/4=1/16 $	. In terms of	c sub 1 equals c sub 0 divided by 4 equals 1 divided by 16.
is trapped in	$ ( 0 , \alpha ) $	so that not only	zero, alpha.
Layers	$ 4 $	and	four
and	$ R_z $	gates correspond to single qubit rotations . The parameters of the circuit are optimised to implement the optimal POVM for a given set of parameters	R sub z.
into	$ \mathcal { Q } $		capital Q
the time-	$ t $	configuration of a MaBB which starts from a random configuration	t
	$ j=1,2. $	Here	j equals one comma two.
% We assume	$ CP $	-conserving~	CP (None)
	$ \omega_j\neq 0 $	are the oscillator frequencies , and	Omega sub j not equal to zero.
is called the Anti-Fuhrmann Circle of	$ ABC $	.	ABC
-distributive	$ \centernot\iff $		center not if and only if.
	$ > $		greater than
is	$ 2 $	-uniform . For larger values of	two
The two models are specified by selecting values for the conformal times	$ \tau_s $	and	tau sub s.
as the value	$ p^ { * } = G ' ( z^ { * } ) $	.	p star equals G prime of z star
Given a repertoire	$ D $	and an element	D
so	$ N $	is a normal subgroup . Then , since	N
&	$ 0 $		zero
and	$ \bx_2 $	, and for each initial condition , we take	x sub 2.
	$ E $	) lines detected than para (	E
It is easy to see that	$ u^C - v^C $	is uniformly bounded and converges in capacity to	u to the power of C minus v to the power of C
and	$ \eta=0 $	and terminating at a point on the imaginary axis between	eta equals zero.
We see that as	$ \epsilon_1 $	is varied , the minimum testing error is always obtained when	Epsilon sub one.
used mean-field theory to obtain a finite temperature phase diagram for the	$ d=2 $	CG model over a range of average particle density . Using their phase diagram as a reference point , we have studied ordering kinetics at	d equals two.
where the time evolutions of the conditional mean	$ \boldsymbol\mu $	and the conditional covariance	mu
for distinct pairs of	$ \mu\nu $	and be distributed as an even function in	mu nu
which we wrote in the case	$ p=2 $	as that one is easy to compute with ; we could generalise that lemma , or restrict this one . I mean , basically we are just using that	p equals 2
reduce to	$ a $	and	a
which we refer to as	$ \sigmatau $	-Hochschild cohomology . The	sigma tau
436	$ & $	3.347	and
the majority of H	$ \alpha $	flux originates at the inner edge of the NSR shell . Here , the density of hydrogen is extremely high compared to the rest of the NSR and so the	Alpha.
can be recovered from	$ v $	by using	v
~km~s	$ ^ { -1 } $	~Mpc	to the power of negative one.
is bounded below by	$ c $	. By	c
where	$ g_a $	denotes the internal degrees of freedom for species	g sub a.
further confirm the spin Hall current in the	$ x $	-direction and the spin accumulation on two sides of the 2D materials almost identical to the electronic spin Hall effect	x
Here the degree of a vertex	$ v $	is the number of vertices which share an edge with	v
	$ L_m $	, and	L sub m.
	$ m $	} . } We discuss the effect of the number of generators on the quality of the synthesized image . The results are reported in Table~	m
We also assume that each element of the unknown signal	$ \bm { w } _0 $	is i.i.d . , according to a distribution	w sub zero.
denotes the element	$ u ( p ) $	. We write the composition of a pair of morphisms	u evaluated at p
Signs	$ ( \cdot ) ^T $	,	center dot, transpose.
for any positive	$ a , b\geq 0 $	and we define	a, b greater than or equal to zero.
is equal to zero we identify this tree with the empty forest :	$ \lambda^ { 0 } =\mathbf\ { 1 $	. We now define the symbol	Lambda to the power of zero equals bold 1.
Our work reveals the connection between the toughening effectiveness and the mesoscale size interplay between	$ d $	and	d
is a set	$ S \subset X $	such that	S is a subset of X.
	$ \tau $	,	Tau.
large and	$ s $	small	s
it does not come near	$ \mu $	easily . As explained below Theorem~	mu.
Get the set of non-existing edges from the adjacency matrix	$ A $	} }	A
the potential	$ V $	is zero . The 2D Schrödinger equation for a particle confined in the quantum corral in polar coordinates is described as	V
321	$ \pm $	0.0082	Plus or minus.
and	$ \beta > 0 $	from which it follows	Beta is greater than zero.
reaches its global upper bounds when	$ \gamma_0=0 $	and	Gamma sub zero equals zero.

Though the interlayer ferromagnetic exchange

$ J_4 $

is crucial for introducing the 3D character , it is much smaller in comparison to the in-plane interactions and insufficient for creating long-range correlations along the

J sub 4.

&

$ 0 $

&

zero

and it consists the symbol-by-symbol detected vector

$ \check { \boldsymbol { x } } $

whose

check x

depends only on the

$ \SL_2 ( \Z ) $

-equivalence class of

S L two in parentheses Z.

such as minimal ones and certain maximal positive bases such as those formed by the coordinate vectors and their negatives in

$ \R^n $

. More OSPBs can be obtained via numerical procedures , even for intermediate sizes~

R to the power of n.

as

$ T|_ { ( d ) } $

and

T restricted to d.

& 2 &

$ 0.23 $

&

0.23: None

by finding its

$ k $

nearest neighbor samples in

k

say

$ \underline { \alpha } _k $

and

alpha sub k.

through its image

$ \nu ( \Gamma ) \le\mrm { PGL } _2 ( \R ) $

.

Nu of Gamma is less than or equal to PGL 2 of R.

321

$ \pm $

0.041 & 0.851

Plus or minus.

is zero in an open set

$ \mathcal { S } $

, then

ess

its influence on the current probability distribution has a Total Variation distance upper bounded by

$ M_T $

.

M sub T.

is an

$ r $

-th root of unity . In particular ,

r

&

$ \ $

&

backslash

and the calibrated probability estimate

$ p $

for

p

calling

$ W $

W

$ { \bf eg } = { \bf fh } =\beta/100 $

and the spherical angles at

e g equals f h equals beta divided by 100

em The operators

$ z_i $

and

z sub i.

The

$ \rho $

parameter determines the weight of external benchmarks in the ranking . We set

rho.

at redshift

$ z $

to have mass ratio in the range

z

and

$ \Sigma ' $

be a surface-minor of

sigma prime.

$ \bY\coloneqq\bN^ { -1 } $

, and

b Y is defined as b N inverse

The strength of the attraction is modeled by the parameter

$ \epsilon $

, which is the depth of the square well-potential . The range of attractive interaction is given by the width of the square well , i.e . ,

epsilon.

with

$ K\leq G_ { w_0 } $

. Since the action of

K less than or equal to G sub w sub 0.

and the exponential

$ e^ { \mat { A } } $

, where

e raised to the power of A

and

$ q < \infty $

, observe that for

q is less than infinity.

This shows the negative correlation on

$ G ' $

.

the derivative of G.

Set

$ { \pi } ^ { j } _1=\pi^ { j } _1 \setminus \ { o\ } $

,

Pi to the power of j sub 1 equals pi to the power of j sub 1 set minus open brace o close brace.

Let

$ \chi $

be a fixed-point of

chi.

and standard deviation degrees up to

$ k $

-hop distance context of each node

k

An examination of the galaxy age versus star formation activity at

$ z > 8 $

z is greater than 8.

in~

$ B_x $

, as desired .

B sub x.

these frequencies only emit radiation in a certain range of values of

$ \kappa $

. In other words , the coupling between

kappa.

and

$ \phi_1 $

such that

Phi sub 1.

&

$ 2.3\pm1.1 $

&

2.3 plus or minus 1.1

the task is to find the diameter of the

$ G/K $

space , that is , the furthest distance possible pairs of points can have . Given the fact that motion in

G divided by K.

the magnetic field continuously turns the magnetization to

$ x $

direction , preserving continuously the non-zero Kondo hybridization just above

x

Due to the bounds assumed on the solutions

$ M_1 , M_2 $

, we have

M one, M two.

let

$ A_i\colon \H_i\to 2^ { \H_i } $

and

A sub i, colon, H sub i, to, 2 to the power of H sub i.

% % Variable selection update with respect to the

$ i $

th node regressing to the rest of the nodes . % % Update

i

&

$ 3.9_ { -0.3 } ^ { +0.4 } $

&

3.9 minus 0.3 raised to the power of plus 0.4.

em for all

$ x \in X $

,

x belongs to X.

and

$ d_i $

errors are rigorously less than

d sub i.

$ i=1,2 ) $

, which implies

i equals one, two.

be the vector space of sections of

$ E $

and

E

we claim the righthand inequality holds with

$ c_1=c_0/4=1/16 $

. In terms of

c sub 1 equals c sub 0 divided by 4 equals 1 divided by 16.

is trapped in

$ ( 0 , \alpha ) $

so that not only

zero, alpha.

Layers

$ 4 $

and

four

and

$ R_z $

gates correspond to single qubit rotations . The parameters of the circuit are optimised to implement the optimal POVM for a given set of parameters

R sub z.

into

$ \mathcal { Q } $

capital Q

the time-

$ t $

configuration of a MaBB which starts from a random configuration

t

$ j=1,2. $

Here

j equals one comma two.

% We assume

$ CP $

-conserving~

CP (None)

$ \omega_j\neq 0 $

are the oscillator frequencies , and

Omega sub j not equal to zero.

is called the Anti-Fuhrmann Circle of

$ ABC $

.

ABC

-distributive

$ \centernot\iff $

center not if and only if.

$ > $

greater than

is

$ 2 $

-uniform . For larger values of

two

The two models are specified by selecting values for the conformal times

$ \tau_s $

and

tau sub s.

as the value

$ p^ { * } = G ' ( z^ { * } ) $

.

p star equals G prime of z star

Given a repertoire

$ D $

and an element

D

so

$ N $

is a normal subgroup . Then , since

N

&

$ 0 $

zero

and

$ \bx_2 $

, and for each initial condition , we take

x sub 2.

$ E $

) lines detected than para (

E

It is easy to see that

$ u^C - v^C $

is uniformly bounded and converges in capacity to

u to the power of C minus v to the power of C

and

$ \eta=0 $

and terminating at a point on the imaginary axis between

eta equals zero.

We see that as

$ \epsilon_1 $

is varied , the minimum testing error is always obtained when

Epsilon sub one.

used mean-field theory to obtain a finite temperature phase diagram for the

$ d=2 $

CG model over a range of average particle density . Using their phase diagram as a reference point , we have studied ordering kinetics at

d equals two.

where the time evolutions of the conditional mean

$ \boldsymbol\mu $

and the conditional covariance

mu

for distinct pairs of

$ \mu\nu $

and be distributed as an even function in

mu nu

which we wrote in the case

$ p=2 $

as that one is easy to compute with ; we could generalise that lemma , or restrict this one . I mean , basically we are just using that

p equals 2

reduce to

$ a $

and

a

which we refer to as

$ \sigmatau $

-Hochschild cohomology . The

sigma tau

436

$ & $

3.347

and

the majority of H

$ \alpha $

flux originates at the inner edge of the NSR shell . Here , the density of hydrogen is extremely high compared to the rest of the NSR and so the

Alpha.

can be recovered from

$ v $

by using

v

~km~s

$ ^ { -1 } $

~Mpc

to the power of negative one.

is bounded below by

$ c $

. By

c

where

$ g_a $

denotes the internal degrees of freedom for species

g sub a.

further confirm the spin Hall current in the

$ x $

-direction and the spin accumulation on two sides of the 2D materials almost identical to the electronic spin Hall effect

x

Here the degree of a vertex

$ v $

is the number of vertices which share an edge with

v

$ L_m $

, and

L sub m.

$ m $

} . } We discuss the effect of the number of generators on the quality of the synthesized image . The results are reported in Table~

m

We also assume that each element of the unknown signal

$ \bm { w } _0 $

is i.i.d . , according to a distribution

w sub zero.

denotes the element

$ u ( p ) $

. We write the composition of a pair of morphisms

u evaluated at p

Signs

$ ( \cdot ) ^T $

,

center dot, transpose.

for any positive

$ a , b\geq 0 $

and we define

a, b greater than or equal to zero.

is equal to zero we identify this tree with the empty forest :

$ \lambda^ { 0 } =\mathbf\ { 1 $

. We now define the symbol

Lambda to the power of zero equals bold 1.

Our work reveals the connection between the toughening effectiveness and the mesoscale size interplay between

$ d $

and

d

is a set

$ S \subset X $

such that

S is a subset of X.

$ \tau $

,

Tau.

large and

$ s $

small

s

it does not come near

$ \mu $

easily . As explained below Theorem~

mu.

Get the set of non-existing edges from the adjacency matrix

$ A $

} }

A

the potential

$ V $

is zero . The 2D Schrödinger equation for a particle confined in the quantum corral in polar coordinates is described as

V

321

$ \pm $

0.0082

Plus or minus.

and

$ \beta > 0 $

from which it follows

Beta is greater than zero.

reaches its global upper bounds when

$ \gamma_0=0 $

and

Gamma sub zero equals zero.