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
this resulting uncertainty is taken as the difference in the estimated	$ A\epsilon $	value when using the experimental	A epsilon
It can be found that the time-averaged Hamiltonian	$ \hat { H } _ { 0 } $	can capture the physics predicted by the method in Sec .	H naught hat.
406	$ \pm $	0.291 & 15.5	Plus or minus.
&	$ v_ { \rm wind } $	& References	v wind subscript.
and chemical potential	$ \mu_\nu $	. We assume that	mu nu.
	$ u $	is an	u
together with the local roots will be the actual input to	$ \varphi^\ell $	; those	Phi to the power of ell.
are defined and of class	$ C^0 $	, or	C raised to the power of 0.
in the first equation and using the canonical commutation relation the eigenvalue equation for	$ \psi_1 $	can be expressed as	Psi one.
-good colouring	$ \phi $	with	phi.
then there exists	$ z\in \T $	such that	z belongs to the set T.
The antipode	$ S^ { \sigma } $	is given by	S raised to the power of sigma.
The	$ 0 $	-norm of a vector	zero
in the 1950s ; they related them to the complex roots of	$ f ' $	. % Roots of modulus	f prime
-exact and	$ \bar \partial $	-exact , and therefore the K	bar partial.
017 &	$ -0.19 $	&	Negative zero point one nine.
and a theory	$ T $	, and has output a total structure	T
the derivative operator	$ \nabla d ( . , . ) $	in	Nabla d, open parenthesis, dot dot dot, comma, dot dot dot, close parenthesis.
This	$ \theta $	-motion is shown in Fig .	Theta.
The morphisms in	$ \mathcal { C } $	are taken to be morphisms of	see
where	$ t_k=k\Delta t $	is the discretized time ,	t sub k equals k delta t
	$ z $	is self-adjoint if	z
	$ f_ { j } ^ { l } ( x_i ) $	represents the feature of sample	f sub j superscript l open parenthesis x sub i close parenthesis
&	$ 0.05 $	&	zero point zero five.
For a packing in a periodic simulation box	$ \mathfrak { F } $	, which consists of	Fraktur F.
8	$ \pm $	1.0 & 66.7	Plus or minus.
If we shift all low-frequency signals of	$ \boldsymbol { \phi } $	to a center patch of size	phi.
and such that	$ y_n $	points in the	y sub n.
and letting	$ \epsilon\to 0^+ $	, we obtain	Epsilon tends to zero plus.
-th instance	$ w_i $	is computed as :	w subscript i.
	$ f $	is continuous if and only if for each sequence of points	f
contains points of	$ E $	other than	E
So the best choice is	$ \nu=1 $	.	nu equals one.
~	$ y $	, and~	y
at~	$ k $	where :	k
so that	$ \dot { \mathcal { M } } _ { \rm { p } } \ , $		dot M sub p
Two-dimensional plots of the predicted	$ \Delta c $	,	delta c
for	$ p $		p
	$ x_s $	,	x sub s.
This paper assumes that both	$ \mathcal { S } $	and	ess
After independent training on such	$ n $	microstructures , we only obtain a	n
which equals 0 outside	$ E_k $	and	E subscript k.
every lex-min coloring	$ c $	of	c
denotes the typical linear dimension of	$ A $	and and	A
we again take	$ n = 2 $	and consider two generators with	n equals 2
and batch size is set as	$ 256 $	. We pre-train compression plugins for	256: Two hundred fifty-six.
Expected immediate reward received when taking action a in state	$ s $	.	s
Using quadruple cut at four-point one-loop massive	$ \mathcal { N } =4 $	SYM amplitude , we find that the coefficient of the scalar box integral is independent of the loop momentum . Therefore , there can not be any contribution of terms	N equals four.
& Adjusted	$ R^ { 2 } $	: & 0.63 &	R squared.
and cutting along	$ \Sigma $	will not give us a closed surface component .	Sigma.
increases in the interval	$ k < k_0 $	, the QNM frequencies increases and reach their maximum values at	k is less than k sub zero.
for specific implementation details of	$ D_\theta $	. Let	D theta.
Viscosity-dominated at the end of the release	$ \mathcal { K } _ { ms } \ll1 $		Mathcal K sub m s much less than 1.
	$ util ( u ' v ) \le ( 1+\eps ) ^ { j_v+1 } $	. Thus ,	util of u prime v is less than or equal to the quantity one plus epsilon raised to the power of j sub v plus one.
converges as	$ \delta \to 0 $	to that of	delta tends to zero.
let	$ Y_k^\ast $	denote a missing value inputed using the proposed methodology . We shall obtain	Y sub k star
with index	$ b $	to be added to	b
5	$ & $	195.4	and
	$ d $	} & 1 & 4 & 16 & 64 & 256 & 1024 & 4096 & 16384 & 65536 & 262144 & 1048576 & 4194304 & 16777216 & 67108864 &	d
are connected in	$ S $	by the chordality of	S
Suppose	$ \sigma ( u_ { \ell } ) = D $	for all	Sigma of u sub ell equals D.
Qualitative results do not change for different values of	$ \delta_k $	and	Delta k.
of the	$ Y_j $	variables corresponding to the at most	Y sub j.
-geodesic joining	$ x $	and	x
holds or we can replace	$ P_1 $	by	P sub 1.
after the	$ k $	th jump . Our initial condition suggests	k
The above scheme allows one to construe this disambiguation internally for	$ F $	and for	F
we find that the surroundings of the 2000 massive halos were ionized before	$ z=8 $	in most cases when less than 20	z equals 8.
for all	$ x\in E $	, contradicting Equation	x belongs to E.
or the minor-axis sample	$ b $	) and the fiducial sample (	b
where the third Mackey functor can be computed from	$ \upi_ { \star } H\M^ { \sharp C_2 } $	and	up i star H M sharp C 2
We denote by	$ A_ { \mathcal B } $	the family of	A sub B.
and since agent	$ 3 $	has an empty bundle , agent	three
that equals the number of minimal generators of degree	$ j $	in the	j
and	$ H_4 $	are such that the remaining elements of the basis are eigenvectors of	H sub 4.
is consistent among	$ BVri $	bands , but gives larger values in	bee-eye
is only defined in the cube	$ ( i , j , k ) $	and is identically zero in all other cubes . It is a tensor product of three one-dimensional functions	i, j, k
and node	$ j $	. CAF also uses orthogonal constraint to ensure that	j
The binary field is denoted as	$ \mathbb { F } _2 $	. We use	F two.
where in the first inequality we have used	$ T_\sigma\geq T_\alpha > r-\tau_N $	. Hence the first expectation in	T sub sigma greater than or equal to T sub alpha greater than r minus tau sub N.
with the order type	$ \alpha $	of	Alpha.
face weightings for perfect matchings of the Aztec diamond of order	$ n $	, defined by	n
~Spataro	$ ^ { 74A,74C } $	, F.~Stieler	to the power of 74A, 74C.
% Any algebra	$ \frA \defeq\langle A , f , g\rangle $	such that	angle bracket A, f, g angle bracket
where	$ \gamma = 5/3 $	is the adiabatic index of a monoatomic gas , and	Gamma equals five-thirds.
the optimal upper bounds for these parameters when	$ g $	is small are still unknown . For small values of	g
the plaquette	$ p $	is said to be threaded by one flux . In the Kitaev model the spin operator	p
	$ k $	vertices are marked in the local neighborhood and showed that it is more expressive than deletion . It identifies	k
	$ y $	is any ( and hence the momentum	y
and	$ n $	. Nevertheless , for any point	n
at each decision step	$ t $	, the real-time decision variables are determined according to the features of observation from the traffic environments , as the key factors to modulate the cost functions of MPC .	t
the function	$ L_\alpha^t $	is differentiable~	L sub alpha superscript t.
is a normal subgroup of	$ G $	, i.e . , for all	G
&	$ +1 $	&	Plus one.
46	$ \pm $	0.03 . For the stationary shock , the integrated spectral index ,	Plus or minus.
where	$ \mathbf { s } $	is defined in	s
Visualization of the real/fake attention map	$ M_ { df } $	and the domain attention map	M sub d f
sites and	$ q=3 $	. We see that the structure of a central band surrounded by deviant lines corresponding to corner sites and their immediate neighbours becomes even more apparent . Again we see	q equals three
for a fixed time	$ t $	. For a vector	t
all	$ \pi\in \mcS_3 $	give rise to a linked diagram	Pi belongs to the set S sub 3.

this resulting uncertainty is taken as the difference in the estimated

$ A\epsilon $

value when using the experimental

A epsilon

It can be found that the time-averaged Hamiltonian

$ \hat { H } _ { 0 } $

can capture the physics predicted by the method in Sec .

H naught hat.

406

$ \pm $

0.291 & 15.5

Plus or minus.

&

$ v_ { \rm wind } $

& References

v wind subscript.

and chemical potential

$ \mu_\nu $

. We assume that

mu nu.

$ u $

is an

u

together with the local roots will be the actual input to

$ \varphi^\ell $

; those

Phi to the power of ell.

are defined and of class

$ C^0 $

, or

C raised to the power of 0.

in the first equation and using the canonical commutation relation the eigenvalue equation for

$ \psi_1 $

can be expressed as

Psi one.

-good colouring

$ \phi $

with

phi.

then there exists

$ z\in \T $

such that

z belongs to the set T.

The antipode

$ S^ { \sigma } $

is given by

S raised to the power of sigma.

The

$ 0 $

-norm of a vector

zero

in the 1950s ; they related them to the complex roots of

$ f ' $

. % Roots of modulus

f prime

-exact and

$ \bar \partial $

-exact , and therefore the K

bar partial.

017 &

$ -0.19 $

&

Negative zero point one nine.

and a theory

$ T $

, and has output a total structure

T

the derivative operator

$ \nabla d ( . , . ) $

in

Nabla d, open parenthesis, dot dot dot, comma, dot dot dot, close parenthesis.

This

$ \theta $

-motion is shown in Fig .

Theta.

The morphisms in

$ \mathcal { C } $

are taken to be morphisms of

see

where

$ t_k=k\Delta t $

is the discretized time ,

t sub k equals k delta t

$ z $

is self-adjoint if

z

$ f_ { j } ^ { l } ( x_i ) $

represents the feature of sample

f sub j superscript l open parenthesis x sub i close parenthesis

&

$ 0.05 $

&

zero point zero five.

For a packing in a periodic simulation box

$ \mathfrak { F } $

, which consists of

Fraktur F.

8

$ \pm $

1.0 & 66.7

Plus or minus.

If we shift all low-frequency signals of

$ \boldsymbol { \phi } $

to a center patch of size

phi.

and such that

$ y_n $

points in the

y sub n.

and letting

$ \epsilon\to 0^+ $

, we obtain

Epsilon tends to zero plus.

-th instance

$ w_i $

is computed as :

w subscript i.

$ f $

is continuous if and only if for each sequence of points

f

contains points of

$ E $

other than

E

So the best choice is

$ \nu=1 $

.

nu equals one.

~

$ y $

, and~

y

at~

$ k $

where :

k

so that

$ \dot { \mathcal { M } } _ { \rm { p } } \ , $

dot M sub p

Two-dimensional plots of the predicted

$ \Delta c $

,

delta c

for

$ p $

p

$ x_s $

,

x sub s.

This paper assumes that both

$ \mathcal { S } $

and

ess

After independent training on such

$ n $

microstructures , we only obtain a

n

which equals 0 outside

$ E_k $

and

E subscript k.

every lex-min coloring

$ c $

of

c

denotes the typical linear dimension of

$ A $

and and

A

we again take

$ n = 2 $

and consider two generators with

n equals 2

and batch size is set as

$ 256 $

. We pre-train compression plugins for

256: Two hundred fifty-six.

Expected immediate reward received when taking action a in state

$ s $

.

s

Using quadruple cut at four-point one-loop massive

$ \mathcal { N } =4 $

SYM amplitude , we find that the coefficient of the scalar box integral is independent of the loop momentum . Therefore , there can not be any contribution of terms

N equals four.

& Adjusted

$ R^ { 2 } $

: & 0.63 &

R squared.

and cutting along

$ \Sigma $

will not give us a closed surface component .

Sigma.

increases in the interval

$ k < k_0 $

, the QNM frequencies increases and reach their maximum values at

k is less than k sub zero.

for specific implementation details of

$ D_\theta $

. Let

D theta.

Viscosity-dominated at the end of the release

$ \mathcal { K } _ { ms } \ll1 $

Mathcal K sub m s much less than 1.

$ util ( u ' v ) \le ( 1+\eps ) ^ { j_v+1 } $

. Thus ,

util of u prime v is less than or equal to the quantity one plus epsilon raised to the power of j sub v plus one.

converges as

$ \delta \to 0 $

to that of

delta tends to zero.

let

$ Y_k^\ast $

denote a missing value inputed using the proposed methodology . We shall obtain

Y sub k star

with index

$ b $

to be added to

b

5

$ & $

195.4

and

$ d $

} & 1 & 4 & 16 & 64 & 256 & 1024 & 4096 & 16384 & 65536 & 262144 & 1048576 & 4194304 & 16777216 & 67108864 &

d

are connected in

$ S $

by the chordality of

S

Suppose

$ \sigma ( u_ { \ell } ) = D $

for all

Sigma of u sub ell equals D.

Qualitative results do not change for different values of

$ \delta_k $

and

Delta k.

of the

$ Y_j $

variables corresponding to the at most

Y sub j.

-geodesic joining

$ x $

and

x

holds or we can replace

$ P_1 $

by

P sub 1.

after the

$ k $

th jump . Our initial condition suggests

k

The above scheme allows one to construe this disambiguation internally for

$ F $

and for

F

we find that the surroundings of the 2000 massive halos were ionized before

$ z=8 $

in most cases when less than 20

z equals 8.

for all

$ x\in E $

, contradicting Equation

x belongs to E.

or the minor-axis sample

$ b $

) and the fiducial sample (

b

where the third Mackey functor can be computed from

$ \upi_ { \star } H\M^ { \sharp C_2 } $

and

up i star H M sharp C 2

We denote by

$ A_ { \mathcal B } $

the family of

A sub B.

and since agent

$ 3 $

has an empty bundle , agent

three

that equals the number of minimal generators of degree

$ j $

in the

j

and

$ H_4 $

are such that the remaining elements of the basis are eigenvectors of

H sub 4.

is consistent among

$ BVri $

bands , but gives larger values in

bee-eye

is only defined in the cube

$ ( i , j , k ) $

and is identically zero in all other cubes . It is a tensor product of three one-dimensional functions

i, j, k

and node

$ j $

. CAF also uses orthogonal constraint to ensure that

j

The binary field is denoted as

$ \mathbb { F } _2 $

. We use

F two.

where in the first inequality we have used

$ T_\sigma\geq T_\alpha > r-\tau_N $

. Hence the first expectation in

T sub sigma greater than or equal to T sub alpha greater than r minus tau sub N.

with the order type

$ \alpha $

of

Alpha.

face weightings for perfect matchings of the Aztec diamond of order

$ n $

, defined by

n

~Spataro

$ ^ { 74A,74C } $

, F.~Stieler

to the power of 74A, 74C.

% Any algebra

$ \frA \defeq\langle A , f , g\rangle $

such that

angle bracket A, f, g angle bracket

where

$ \gamma = 5/3 $

is the adiabatic index of a monoatomic gas , and

Gamma equals five-thirds.

the optimal upper bounds for these parameters when

$ g $

is small are still unknown . For small values of

g

the plaquette

$ p $

is said to be threaded by one flux . In the Kitaev model the spin operator

p

$ k $

vertices are marked in the local neighborhood and showed that it is more expressive than deletion . It identifies

k

$ y $

is any ( and hence the momentum

y

and

$ n $

. Nevertheless , for any point

n

at each decision step

$ t $

, the real-time decision variables are determined according to the features of observation from the traffic environments , as the key factors to modulate the cost functions of MPC .

t

the function

$ L_\alpha^t $

is differentiable~

L sub alpha superscript t.

is a normal subgroup of

$ G $

, i.e . , for all

G

&

$ +1 $

&

Plus one.

46

$ \pm $

0.03 . For the stationary shock , the integrated spectral index ,

Plus or minus.

where

$ \mathbf { s } $

is defined in

s

Visualization of the real/fake attention map

$ M_ { df } $

and the domain attention map

M sub d f

sites and

$ q=3 $

. We see that the structure of a central band surrounded by deviant lines corresponding to corner sites and their immediate neighbours becomes even more apparent . Again we see

q equals three

for a fixed time

$ t $

. For a vector

t

all

$ \pi\in \mcS_3 $

give rise to a linked diagram

Pi belongs to the set S sub 3.