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
containing	$ C $	classes .	C
is a short Gorenstein	$ k $	-algebra . Moreover , prescribing	k
Note that if	$ \lambda = 0 $	i.e . for Sutherland potential the stable liquid branch of the binodal does not exist at	Lambda equals zero.
Each of the determinants admits an integral representation in terms of a special real solution to the	$ n $	-th member of the Painlev	n
C_D	$ & $	C_L	and
no edge of	$ \Gamma_e $	has positive vertical slope . By balancing , it follows that no edge of	Gamma sub e.
if	$ M/\mathfrak { m } _RM $	is ramified , then the	M divided by the R-sub-M times M.
We consider a photonic system with	$ m $	modes that is transformed by a Haar-random linear optical unitary	m
and	$ E_k > 0 $	,	E sub k is greater than zero.
65	$ ^\circ $	and +90	degree.
&	$ 0.07_ { -0.03 } ^ { +0.04 } $	& & & 12821.9/11368	0.07 with a lower uncertainty of 0.03 and an upper uncertainty of 0.04.
is the	$ c $	-th channel of	c
The proposed dark fermion can be produced through the process of pair annihilation of two quarks	$ q\bar { q } $	mediated by the heavy vector boson Z	q bar q
3	$ \pm $	0.5 & 16.1	Plus or minus.
Our next goal is to prove an adjunction theorem between the	$ \hat { \theta } _ { t } $	.	theta hat sub t.
let	$ u_t $	be the	u subscript t
There exists a universal constant	$ c > 0 $	such that if	c greater than zero
with	$ \epsilon $	arbitrary . As noted , it will be convenient to take	epsilon.
We prove the result by induction on	$ n . $	If	n
18 \rbrack	$ , $	n=500 , k=15	comma
irreducible representation of the Whitehead link lies in the diagonal	$ D ' $	iff the curve	D prime.
is	$ 2L_d $	-Lipschitz , hence	two L sub d
& 0 & & & & \\ \hline	$ 5 $	& 0 &	five
	$ B $	in	B
as well as the NH and IH cases of	$ \nu $	Axi-EDE and	nu.
we additionally need the inclusions of	$ x $	and	x
by recalling that	$ 2k_1^2 $	corresponds to second order derivatives in Fourier space while the terms	two times k sub one squared.
the high-energy spectral slope increases from	$ s\simeq 2 $	to	S approximately equal to 2
Let	$ G $	be an	G
and	$ -0.27 $	, which results in CEMP-no , CEMP-	Negative 0.27.
V & F	$ _ { \rm eff } $	& B	eff sub.
\\ % The driving force introduces a spatial anisotropy leaving the	$ y $	-components of	y
is the phase field of	$ \psi ( x , y ) $	and the phase differences between sites are taken to be	Psi of x, y.
and if	$ e $	the atom of	e
it is known that the coefficients are either	$ 0 $	or	zero
where	$ k $	is the total number of classes : e.g . , the Emotion data set has labels	k
for	$ j=1,2 $	, hence	j equals one comma two.
and	$ \lambda $	is a hyperparameter used to control the sparsity level . We employ two additional networks ,	Lambda.
outputs a	$ ( d+2 ) $	-point witness of non-convexity and reject	d plus two.
and has a dimension of	$ 66 $	which can be decreased by need . We follow	66
and	$ l'=l_ { k ' } $	otherwise .	l prime equals l sub k prime
-norm of the vector	$ \vec { b } = ( b_i ) $	of individual absolute bounds .	Vector b equals open parenthesis b sub i close parenthesis.
56 &	$ - $	0.49 & 8.9 & 1.3 & 18.7 & 6.20E+20 & 1	minus
if	$ \mathbf { S } $	is an optimal alignment i.e . a global minimum then so is	S
&	$ N_\mathrm { H } $	=	N sub H
as it has experienced contraction along	$ S^ { \textnormal { a } } $	. % defined as %	S to the power of a
6 M	$ _J $	, R = 1.2 R	J
and	$ \Mod ( \Sigma_g ) [ G_2 ] $	of	Mod of Sigma g, square brackets G sub 2.
and	$ m $	and	m
	$ g\in G $	and	g belongs to G
When an evolved	$ i $	-th generator architecture is more complex ,	i
we observe that odd operators under the	$ \mathbb { Z } _2 $	spin transformation relax towards the equilibrium value at relatively short time scales that grow only polynomially with the disorder strength . On the other hand , the dynamics of even	Z sub 2.
The physical meaning of	$ \sigma_0 $	and	Sigma sub zero.
and	$ \sigma $	two-row MSTs is treated in Appendix	Sigma.
at this index and a frequency	$ \nu $	, the phase is given by	nu.
then clearly	$ U_1 $	has prefix	U sub 1.
is obtained from the constraint	$ \theta ( 0 ) = \theta_0 $	. Hence , the	Theta of 0 equals theta sub 0.
can be thought as a generalized dipole array	$ \mathbf { d } _k $	that , via the interaction tensor	dee sub k.
65	$ \pm $	0.057	Plus or minus.
for	$ \tauhadvis $	candidates with both one and three associated charged-particle tracks . For this method to be accurate , it is important that the fail-	tau tau tau
variables that evolve over time	$ t $	,	t
k=1	$ , $		comma
the verifier verifies the signature	$ \omega $	with the received RAW even photo and public key	omega.
	$ Z $	is the number of protons in a nucleus , and	Z
In addition to	$ \alpha_l $	, the variable	Alpha sub l.
&	$ 5 $	&	five
the definition of global representation allows the user to actively control the bounds over the representation of different groups in the top-	$ k $	ranked items , even if their representation in the overall data is low/high . For instance , consider a ranking algorithm for job applicants in fields that are dominated by	k
If	$ G $	contains a	G
g_0	$ & $	i_0	and
42	$ \pm $	0.09 & 0.45	Plus or minus.
and	$ b $	[	b
% Here	$ j $	is the	j
&	$ \SI { 0.1 } { \micro Nms/rad } $		sub I 0.1, micro N m s over rad.
is in the adjoint representation of its corresponding	$ H $	-symmetry indices	H
on	$ X $	as	X
approach	$ \infty $	as	Infinity.
and in particular in the classical limit	$ \hbar\to 0 $	, having a single maximum at	h-bar tends to zero.
Free parameters were the time of periastron passage	$ T_0 $	, eccentricity	T sub zero.
for	$ s=0 , 1 , ... $	, where	s equals zero, one, dot dot dot.
of	$ G $	but outside	G
	$ \theta $	=	Theta.
	$ T $	, and	T
&	$ -1.608 \pm 0.032 $	&	Negative 1.608 plus or minus 0.032.
denote the anomalous label of the	$ i $	th disaster image , and let	i
resampling	$ 10^ { 5 } $	times . % With the randomly reseeded models , we randomly sample over all the models , as well as sampling over all the examples in the development set .	Ten to the power of five.
76	$ \pm $	0.14 & 0.46 & 2534	Plus or minus.
where	$ a ( t ) $	and	a function of t
Writing an element	$ m\in \sigma $	as	m belongs to sigma.
consists of two circles	$ D_1 $	and	D one.
and we can determine the energy exactly as	$ \mean { E } =0 $	. On the contrary , if the state is aligned with the z-axis , implying	the mean of E equals zero.
and at about	$ -x_0 $	at the same time will collapse to	Negative x sub zero.
as long as	$ \alpha \ge O ( \eps \sqrt { \log ( 1/\eps ) } ) $	.	Alpha greater than or equal to O of epsilon times the square root of log of one over epsilon.
	$ G $	has at least six vertices and	G
if and only if for any	$ a\in\mathbb\ R^q $	, the conditional characteristic function of	a belongs to the set of real numbers raised to the power of q.
-graphs	$ \Gamma_j $	in the sense of	Gamma sub j.
Such results further illustrate the necessity of optimizing	$ \theta $	.	Theta.
Therefore	$ \mathcal { C } $	has to be equal to the unique closed geodesic	see
and	$ V ( M_1 ) \cup \ { x_2 , x_3 , x_4\ } $	are colored red ; all edges between	V of M sub 1 union x sub 2, x sub 3, x sub 4
and	$ M ' $	are connected by a chain of invariant curves , the same is true for	M prime.
: We assume that it is possible to draw samples from the measure	$ \rho $	in Step 1 . We use the uniform rejection sampling method to draw samples . We end the rejection sampler once we have enough sample points .	rho.

containing

$ C $

classes .

C

is a short Gorenstein

$ k $

-algebra . Moreover , prescribing

k

Note that if

$ \lambda = 0 $

i.e . for Sutherland potential the stable liquid branch of the binodal does not exist at

Lambda equals zero.

Each of the determinants admits an integral representation in terms of a special real solution to the

$ n $

-th member of the Painlev

n

C_D

$ & $

C_L

and

no edge of

$ \Gamma_e $

has positive vertical slope . By balancing , it follows that no edge of

Gamma sub e.

if

$ M/\mathfrak { m } _RM $

is ramified , then the

M divided by the R-sub-M times M.

We consider a photonic system with

$ m $

modes that is transformed by a Haar-random linear optical unitary

m

and

$ E_k > 0 $

,

E sub k is greater than zero.

65

$ ^\circ $

and +90

degree.

&

$ 0.07_ { -0.03 } ^ { +0.04 } $

& & & 12821.9/11368

0.07 with a lower uncertainty of 0.03 and an upper uncertainty of 0.04.

is the

$ c $

-th channel of

c

The proposed dark fermion can be produced through the process of pair annihilation of two quarks

$ q\bar { q } $

mediated by the heavy vector boson Z

q bar q

3

$ \pm $

0.5 & 16.1

Plus or minus.

Our next goal is to prove an adjunction theorem between the

$ \hat { \theta } _ { t } $

.

theta hat sub t.

let

$ u_t $

be the

u subscript t

There exists a universal constant

$ c > 0 $

such that if

c greater than zero

with

$ \epsilon $

arbitrary . As noted , it will be convenient to take

epsilon.

We prove the result by induction on

$ n . $

If

n

18 \rbrack

$ , $

n=500 , k=15

comma

irreducible representation of the Whitehead link lies in the diagonal

$ D ' $

iff the curve

D prime.

is

$ 2L_d $

-Lipschitz , hence

two L sub d

& 0 & & & & \\ \hline

$ 5 $

& 0 &

five

$ B $

in

B

as well as the NH and IH cases of

$ \nu $

Axi-EDE and

nu.

we additionally need the inclusions of

$ x $

and

x

by recalling that

$ 2k_1^2 $

corresponds to second order derivatives in Fourier space while the terms

two times k sub one squared.

the high-energy spectral slope increases from

$ s\simeq 2 $

to

S approximately equal to 2

Let

$ G $

be an

G

and

$ -0.27 $

, which results in CEMP-no , CEMP-

Negative 0.27.

V & F

$ _ { \rm eff } $

& B

eff sub.

\\ % The driving force introduces a spatial anisotropy leaving the

$ y $

-components of

y

is the phase field of

$ \psi ( x , y ) $

and the phase differences between sites are taken to be

Psi of x, y.

and if

$ e $

the atom of

e

it is known that the coefficients are either

$ 0 $

or

zero

where

$ k $

is the total number of classes : e.g . , the Emotion data set has labels

k

for

$ j=1,2 $

, hence

j equals one comma two.

and

$ \lambda $

is a hyperparameter used to control the sparsity level . We employ two additional networks ,

Lambda.

outputs a

$ ( d+2 ) $

-point witness of non-convexity and reject

d plus two.

and has a dimension of

$ 66 $

which can be decreased by need . We follow

66

and

$ l'=l_ { k ' } $

otherwise .

l prime equals l sub k prime

-norm of the vector

$ \vec { b } = ( b_i ) $

of individual absolute bounds .

Vector b equals open parenthesis b sub i close parenthesis.

56 &

$ - $

0.49 & 8.9 & 1.3 & 18.7 & 6.20E+20 & 1

minus

if

$ \mathbf { S } $

is an optimal alignment i.e . a global minimum then so is

S

&

$ N_\mathrm { H } $

=

N sub H

as it has experienced contraction along

$ S^ { \textnormal { a } } $

. % defined as %

S to the power of a

6 M

$ _J $

, R = 1.2 R

J

and

$ \Mod ( \Sigma_g ) [ G_2 ] $

of

Mod of Sigma g, square brackets G sub 2.

and

$ m $

and

m

$ g\in G $

and

g belongs to G

When an evolved

$ i $

-th generator architecture is more complex ,

i

we observe that odd operators under the

$ \mathbb { Z } _2 $

spin transformation relax towards the equilibrium value at relatively short time scales that grow only polynomially with the disorder strength . On the other hand , the dynamics of even

Z sub 2.

The physical meaning of

$ \sigma_0 $

and

Sigma sub zero.

and

$ \sigma $

two-row MSTs is treated in Appendix

Sigma.

at this index and a frequency

$ \nu $

, the phase is given by

nu.

then clearly

$ U_1 $

has prefix

U sub 1.

is obtained from the constraint

$ \theta ( 0 ) = \theta_0 $

. Hence , the

Theta of 0 equals theta sub 0.

can be thought as a generalized dipole array

$ \mathbf { d } _k $

that , via the interaction tensor

dee sub k.

65

$ \pm $

0.057

Plus or minus.

for

$ \tauhadvis $

candidates with both one and three associated charged-particle tracks . For this method to be accurate , it is important that the fail-

tau tau tau

variables that evolve over time

$ t $

,

t

k=1

$ , $

comma

the verifier verifies the signature

$ \omega $

with the received RAW even photo and public key

omega.

$ Z $

is the number of protons in a nucleus , and

Z

In addition to

$ \alpha_l $

, the variable

Alpha sub l.

&

$ 5 $

&

five

the definition of global representation allows the user to actively control the bounds over the representation of different groups in the top-

$ k $

ranked items , even if their representation in the overall data is low/high . For instance , consider a ranking algorithm for job applicants in fields that are dominated by

k

If

$ G $

contains a

G

g_0

$ & $

i_0

and

42

$ \pm $

0.09 & 0.45

Plus or minus.

and

$ b $

[

b

% Here

$ j $

is the

j

&

$ \SI { 0.1 } { \micro Nms/rad } $

sub I 0.1, micro N m s over rad.

is in the adjoint representation of its corresponding

$ H $

-symmetry indices

H

on

$ X $

as

X

approach

$ \infty $

as

Infinity.

and in particular in the classical limit

$ \hbar\to 0 $

, having a single maximum at

h-bar tends to zero.

Free parameters were the time of periastron passage

$ T_0 $

, eccentricity

T sub zero.

for

$ s=0 , 1 , ... $

, where

s equals zero, one, dot dot dot.

of

$ G $

but outside

G

$ \theta $

=

Theta.

$ T $

, and

T

&

$ -1.608 \pm 0.032 $

&

Negative 1.608 plus or minus 0.032.

denote the anomalous label of the

$ i $

th disaster image , and let

i

resampling

$ 10^ { 5 } $

times . % With the randomly reseeded models , we randomly sample over all the models , as well as sampling over all the examples in the development set .

Ten to the power of five.

76

$ \pm $

0.14 & 0.46 & 2534

Plus or minus.

where

$ a ( t ) $

and

a function of t

Writing an element

$ m\in \sigma $

as

m belongs to sigma.

consists of two circles

$ D_1 $

and

D one.

and we can determine the energy exactly as

$ \mean { E } =0 $

. On the contrary , if the state is aligned with the z-axis , implying

the mean of E equals zero.

and at about

$ -x_0 $

at the same time will collapse to

Negative x sub zero.

as long as

$ \alpha \ge O ( \eps \sqrt { \log ( 1/\eps ) } ) $

.

Alpha greater than or equal to O of epsilon times the square root of log of one over epsilon.

$ G $

has at least six vertices and

G

if and only if for any

$ a\in\mathbb\ R^q $

, the conditional characteristic function of

a belongs to the set of real numbers raised to the power of q.

-graphs

$ \Gamma_j $

in the sense of

Gamma sub j.

Such results further illustrate the necessity of optimizing

$ \theta $

.

Theta.

Therefore

$ \mathcal { C } $

has to be equal to the unique closed geodesic

see

and

$ V ( M_1 ) \cup \ { x_2 , x_3 , x_4\ } $

are colored red ; all edges between

V of M sub 1 union x sub 2, x sub 3, x sub 4

and

$ M ' $

are connected by a chain of invariant curves , the same is true for

M prime.

: We assume that it is possible to draw samples from the measure

$ \rho $

in Step 1 . We use the uniform rejection sampling method to draw samples . We end the rejection sampler once we have enough sample points .

rho.