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
with respect to	$ n_j $	and	n sub j
If	$ X= [ 0,1 ] $	,	X equals the interval from 0 to 1.
with	$ 0\leq u ' < u $	. %	Zero is less than or equal to u prime, which is less than u.
% to be the number of edges of the	$ \mathcal A_ { \mathcal Y } $	containing	A sub Y.
on a variable	$ x $	but	x
fluxes from uniform flux phase at	$ h=0 $	is estimated by Exact Diagonalization . The ED calculations are done with matter Majorana fermions by fixing the gauge in accordance with uniform flux configuration . The	h equals zero.
in the boundary decomposition of	$ \Sigma $	. Let	Sigma.
The idea is to require that the putative analytic continuation	$ f ( z ) $	obeys Carlson 's conditions . Thanks to Carlson 's theorem this analytic continuation would then be unique . For the analytic continuation of	f of z.
	$ & $		and
represents the memory bank containing	$ K^ { \prime } $	embeddings after eliminating false negative samples through pseudo-labels , which will be described later . Finally , the instance-level contrastive loss is defined as follows :	K prime.
&	$ 23.85 $	&	Twenty-three point eight five.
&	$ \checkmark $	&	Checkmark.
such that for every	$ z $	,	z
The domain	$ \Omega $	can be represented by	Omega.
be a finite group acting on smooth algebraic variety	$ X $	. If	X
the evaluation network is trained every	$ M $	steps by a batch of data samples randomly selected from the replay memory . After every	M
while for	$ E_2 $	, there is a rank jump and	E sub 2.
of radius	$ \delta_1 $	such that for any	delta one.
Our method of integrating the field equations is equivalent to using the dimensionality reduction directly to the	$ D $	-dimensional action to arrive at an effective 5-dimensional theory . % The curvature of the extra	D
-collection	$ M $	the	M
as it exhibits the maximum population of	$ 0.4 $	on state	0.4: None
when	$ l < n $	, we ignore the input	l is less than n.
	$ \forall t\in \ { 0 , \Delta t , \dots , ( n-1 ) \Delta t\ } $	and where	For all t in the set of 0, Delta t, dot dot dot, n minus 1 Delta t.
initially demonstrated that the stationary version of AA without damping is locally	$ r $	-linearly convergent if the fixed-point map is a contraction . Subsequently , Evans et al .	r
in each of the iterations	$ k\in\N $	,	k belongs to the set of natural numbers.
be the set of paths	$ p \in S $	that use	p belongs to S
The error originates from numerical uncertainty	$ \lesssim 0.5\ % $	and the ambiguity in the sign of	less than or approximately equal to 0.5 percent.
-parent of	$ w $	}	w
	$ p $	is also odd . We split	p
We may therefore choose a representative	$ ( Z ' _\eta , \cX ' _\eta ) $	in the same equivalence class as	Open parenthesis, Z, prime, subscript eta, comma, c, X, prime, subscript eta, close parenthesis.
-valued	$ 1 $	-cochain . Per	one
\\	$ 4 $	&	four
the latter can be extended to be order invariant by a slight modification : avoid the use of the lower triangular parameterization of	$ \bB_0 $	in	B sub zero
Let	$ n $	be the current time step of the MD . Given previous	n
&	$ 0 $	&	zero
A model	$ ( \mathcal { Z } , \mathcal { L } ) $	of	Z, L
is equivalent to computing the vertex cover for	$ \GrA $	. Since	Gr A.
Write	$ D_Y=rE_Y $	where	D sub Y equals r times E sub Y.
For	$ \epsilon $	-accurate computation of any	epsilon.
mag in the	$ i $	band . Sources	i
we multiply them by a sufficiently large scalar value -- -	$ 10^7 $	.	Ten to the power of seven.
27	$ \pm $	1.85 & 0.83 & 75.85	Plus or minus.
obtained via averaging the daily temperature curves and the daily precipitation curves yearly over the period	$ 1960 $	to	1960
where the matrix	$ M $	is defined by~	M
is the computing capability of TD	$ n $	. The corresponding energy consumption is	n
with	$ m_e $	the electron mass ,	m sub e.
We use a smaller value of	$ w=30 $	for the disentangled deformation fields and a larger value of	w equals thirty.
and	$ v $	segment defining a bag . Each bag contains a set of points	v
is the	$ B $	meson DA , and the two-hadron DA	B
is the union of	$ \Delta_a $	and the triangle that is formed by the rays defining	Delta a.
high resolution and detection ability of the	$ e^ { + } e^ { - } $	collider . The future Super-	e raised to the power of positive e raised to the power of negative.
of	$ T $	and	T
and	$ \lvert E_ { ( 5,5 ) } \rvert=6l $	,	The absolute value of E sub (5,5) equals 6l.
~Biesuz	$ ^ { 21 } $		Twenty-one.
of	$ \pi $	. Minimal recurrent configurations are in natural bijection with certain acyclic orientations of the graph . The connection between CNATs and the ASM allows us to solve a number of	pi.
while the factor of	$ 1/\delta $	leads to the canonically normalized K	One divided by delta.
steps of the shuffle using the Heads-Tails differentials of	$ i $	from the times it is in position	i
is odd and squeezed-squeezed states when	$ n $	is even .	n
The right picture is a decomposition of~	$ \Delta_2 $	, where each face~	Delta 2.
the wind-to-ISM transition time	$ T_r $	at transition radius	T sub r.
or	$ z $	does not occur in	z
the source or destination 's	$ 3 $	D orientations can be estimated jointly with either the source or destination 's	three
Then public knots	$ \ { u_i\ } $	and public normalized knots	u sub i.
We assume that	$ G $	is finite . The following assertions are equivalent .	G
and a formula	$ \phi $	,	phi.
-refinable function	$ \phi $	is supported inside	phi.
which we denote by	$ \mathcal { M } _n $	for	M sub n.
We train policies across the distribution	$ \Omega $	with a method that learns a	Omega.
at	$ 95\ % $	CL . Since	95 percent.
is closer to	$ s $	than to	s
~Tramontano	$ ^ { a } $		a raised to the power of a.
&	$ 0.77168 $	&	zero point seven seven one six.
is the antisymmetric tensor with	$ \mu , \nu=1,2 $	, and the summation convention for repeated indices is adopted . The notation	mu, nu equals one, two.
and the null hypothesis that the	$ K $	independent samples are equally distributed , we have	K
For any	$ d'\geq 2 $	, with the input as the projected data	d prime greater than or equal to 2.
has	$ d_Y $	components , and the size of the matrix on the right-hand side is	d sub Y
graph	$ K_n $	. If	K sub n.
and	$ a\in\mathbb { C } $	. Assume	a belongs to the set of complex numbers
a small angle	$ \theta $	can be simulated with a larger one	Theta.
rather than	$ \alpha_ { i_r } $	to increase readability .	Alpha sub i sub r.
09E+21 & 1 \\ 1013 & 03116	$ + $	0451	plus
which can be computed at a given time	$ t $	for all	t
There is a maximal extriangulated structure	$ \mathbb { E } ^T $	on	E raised to the power of T.
if	$ F $	does not contain	F
	$ t $	in the loss function is sampled uniformly within the range of diffusion steps during the training process . As for the covariance	t
The number of groups per integration	$ N_ { group } $	was determined to avoid saturation while maximizing counts . Then , we chose the number of integrations to reach a	N group.
the gradient of	$ f $	at point	f
is the square root of the estimated error variance in the	$ k $	th regression model after rescaling . By symmetry , analogous confidence intervals can be obtained for	k
is not maximized through the case	$ ( v=1 ) $	, as it only contributes two additional flips compared to	v equals 1.
be a random variable uniformly distributed on	$ \cur { -1,1 } $	and , for	curly brace negative one, one.
We can choose	$ \gamma $	so that	Gamma.
& coordinate on the	$ \hat { z } $	-- axis of the rotating reference frame	z-hat symbol.
	$ \alpha $	is the charge transfer coefficient , and	Alpha.
where	$ x_ { i } ^ { t } $	represents the spatial coordinates of x for the	x sub i superscript t.
It simply selects each time the edges with local maximum weights and yields an approximation ratio of	$ 1/2 $	as compared to the optimum . A parallel algorithm is further proposed in	One-half.
and	$ \tilde { u } _k $		u tilde sub k.
which are computable by width	$ s $	depth	s
and	$ f $	. % For example , given	f
together with the smallest parabolic	$ P \subset G $	such that	P is a subset of G.
it is evident that the parameters	$ v_S $	,	v sub s.

with respect to

$ n_j $

and

n sub j

If

$ X= [ 0,1 ] $

,

X equals the interval from 0 to 1.

with

$ 0\leq u ' < u $

. %

Zero is less than or equal to u prime, which is less than u.

% to be the number of edges of the

$ \mathcal A_ { \mathcal Y } $

containing

A sub Y.

on a variable

$ x $

but

x

fluxes from uniform flux phase at

$ h=0 $

is estimated by Exact Diagonalization . The ED calculations are done with matter Majorana fermions by fixing the gauge in accordance with uniform flux configuration . The

h equals zero.

in the boundary decomposition of

$ \Sigma $

. Let

Sigma.

The idea is to require that the putative analytic continuation

$ f ( z ) $

obeys Carlson 's conditions . Thanks to Carlson 's theorem this analytic continuation would then be unique . For the analytic continuation of

f of z.

$ & $

and

represents the memory bank containing

$ K^ { \prime } $

embeddings after eliminating false negative samples through pseudo-labels , which will be described later . Finally , the instance-level contrastive loss is defined as follows :

K prime.

&

$ 23.85 $

&

Twenty-three point eight five.

&

$ \checkmark $

&

Checkmark.

such that for every

$ z $

,

z

The domain

$ \Omega $

can be represented by

Omega.

be a finite group acting on smooth algebraic variety

$ X $

. If

X

the evaluation network is trained every

$ M $

steps by a batch of data samples randomly selected from the replay memory . After every

M

while for

$ E_2 $

, there is a rank jump and

E sub 2.

of radius

$ \delta_1 $

such that for any

delta one.

Our method of integrating the field equations is equivalent to using the dimensionality reduction directly to the

$ D $

-dimensional action to arrive at an effective 5-dimensional theory . % The curvature of the extra

D

-collection

$ M $

the

M

as it exhibits the maximum population of

$ 0.4 $

on state

0.4: None

when

$ l < n $

, we ignore the input

l is less than n.

$ \forall t\in \ { 0 , \Delta t , \dots , ( n-1 ) \Delta t\ } $

and where

For all t in the set of 0, Delta t, dot dot dot, n minus 1 Delta t.

initially demonstrated that the stationary version of AA without damping is locally

$ r $

-linearly convergent if the fixed-point map is a contraction . Subsequently , Evans et al .

r

in each of the iterations

$ k\in\N $

,

k belongs to the set of natural numbers.

be the set of paths

$ p \in S $

that use

p belongs to S

The error originates from numerical uncertainty

$ \lesssim 0.5\ % $

and the ambiguity in the sign of

less than or approximately equal to 0.5 percent.

-parent of

$ w $

}

w

$ p $

is also odd . We split

p

We may therefore choose a representative

$ ( Z ' _\eta , \cX ' _\eta ) $

in the same equivalence class as

Open parenthesis, Z, prime, subscript eta, comma, c, X, prime, subscript eta, close parenthesis.

-valued

$ 1 $

-cochain . Per

one

\\

$ 4 $

&

four

the latter can be extended to be order invariant by a slight modification : avoid the use of the lower triangular parameterization of

$ \bB_0 $

in

B sub zero

Let

$ n $

be the current time step of the MD . Given previous

n

&

$ 0 $

&

zero

A model

$ ( \mathcal { Z } , \mathcal { L } ) $

of

Z, L

is equivalent to computing the vertex cover for

$ \GrA $

. Since

Gr A.

Write

$ D_Y=rE_Y $

where

D sub Y equals r times E sub Y.

For

$ \epsilon $

-accurate computation of any

epsilon.

mag in the

$ i $

band . Sources

i

we multiply them by a sufficiently large scalar value -- -

$ 10^7 $

.

Ten to the power of seven.

27

$ \pm $

1.85 & 0.83 & 75.85

Plus or minus.

obtained via averaging the daily temperature curves and the daily precipitation curves yearly over the period

$ 1960 $

to

1960

where the matrix

$ M $

is defined by~

M

is the computing capability of TD

$ n $

. The corresponding energy consumption is

n

with

$ m_e $

the electron mass ,

m sub e.

We use a smaller value of

$ w=30 $

for the disentangled deformation fields and a larger value of

w equals thirty.

and

$ v $

segment defining a bag . Each bag contains a set of points

v

is the

$ B $

meson DA , and the two-hadron DA

B

is the union of

$ \Delta_a $

and the triangle that is formed by the rays defining

Delta a.

high resolution and detection ability of the

$ e^ { + } e^ { - } $

collider . The future Super-

e raised to the power of positive e raised to the power of negative.

of

$ T $

and

T

and

$ \lvert E_ { ( 5,5 ) } \rvert=6l $

,

The absolute value of E sub (5,5) equals 6l.

~Biesuz

$ ^ { 21 } $

Twenty-one.

of

$ \pi $

. Minimal recurrent configurations are in natural bijection with certain acyclic orientations of the graph . The connection between CNATs and the ASM allows us to solve a number of

pi.

while the factor of

$ 1/\delta $

leads to the canonically normalized K

One divided by delta.

steps of the shuffle using the Heads-Tails differentials of

$ i $

from the times it is in position

i

is odd and squeezed-squeezed states when

$ n $

is even .

n

The right picture is a decomposition of~

$ \Delta_2 $

, where each face~

Delta 2.

the wind-to-ISM transition time

$ T_r $

at transition radius

T sub r.

or

$ z $

does not occur in

z

the source or destination 's

$ 3 $

D orientations can be estimated jointly with either the source or destination 's

three

Then public knots

$ \ { u_i\ } $

and public normalized knots

u sub i.

We assume that

$ G $

is finite . The following assertions are equivalent .

G

and a formula

$ \phi $

,

phi.

-refinable function

$ \phi $

is supported inside

phi.

which we denote by

$ \mathcal { M } _n $

for

M sub n.

We train policies across the distribution

$ \Omega $

with a method that learns a

Omega.

at

$ 95\ % $

CL . Since

95 percent.

is closer to

$ s $

than to

s

~Tramontano

$ ^ { a } $

a raised to the power of a.

&

$ 0.77168 $

&

zero point seven seven one six.

is the antisymmetric tensor with

$ \mu , \nu=1,2 $

, and the summation convention for repeated indices is adopted . The notation

mu, nu equals one, two.

and the null hypothesis that the

$ K $

independent samples are equally distributed , we have

K

For any

$ d'\geq 2 $

, with the input as the projected data

d prime greater than or equal to 2.

has

$ d_Y $

components , and the size of the matrix on the right-hand side is

d sub Y

graph

$ K_n $

. If

K sub n.

and

$ a\in\mathbb { C } $

. Assume

a belongs to the set of complex numbers

a small angle

$ \theta $

can be simulated with a larger one

Theta.

rather than

$ \alpha_ { i_r } $

to increase readability .

Alpha sub i sub r.

09E+21 & 1 \\ 1013 & 03116

$ + $

0451

plus

which can be computed at a given time

$ t $

for all

t

There is a maximal extriangulated structure

$ \mathbb { E } ^T $

on

E raised to the power of T.

if

$ F $

does not contain

F

$ t $

in the loss function is sampled uniformly within the range of diffusion steps during the training process . As for the covariance

t

The number of groups per integration

$ N_ { group } $

was determined to avoid saturation while maximizing counts . Then , we chose the number of integrations to reach a

N group.

the gradient of

$ f $

at point

f

is the square root of the estimated error variance in the

$ k $

th regression model after rescaling . By symmetry , analogous confidence intervals can be obtained for

k

is not maximized through the case

$ ( v=1 ) $

, as it only contributes two additional flips compared to

v equals 1.

be a random variable uniformly distributed on

$ \cur { -1,1 } $

and , for

curly brace negative one, one.

We can choose

$ \gamma $

so that

Gamma.

& coordinate on the

$ \hat { z } $

-- axis of the rotating reference frame

z-hat symbol.

$ \alpha $

is the charge transfer coefficient , and

Alpha.

where

$ x_ { i } ^ { t } $

represents the spatial coordinates of x for the

x sub i superscript t.

It simply selects each time the edges with local maximum weights and yields an approximation ratio of

$ 1/2 $

as compared to the optimum . A parallel algorithm is further proposed in

One-half.

and

$ \tilde { u } _k $

u tilde sub k.

which are computable by width

$ s $

depth

s

and

$ f $

. % For example , given

f

together with the smallest parabolic

$ P \subset G $

such that

P is a subset of G.

it is evident that the parameters

$ v_S $

,

v sub s.