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
and	$ j=1,2 $	:	j equals one comma two.
	$ t $	in	t
From the row containing	$ n_ { \fpos } $	we observe that the glow persists for a long time , and also extends downwards by about	n sub fpos
The probability density for an incoming photon to be found at position	$ x $	on the imaging screen will be calculated just as in section	x
&	$ 60.5_ { \pm 0.9 } $	&	six point five, sub plus or minus zero point nine.
The tensor product of evaluation vectors of polynomials over the field	$ \mathbb { F } _2 $	is an evaluation vector of a polynomial from a larger ring of the same field . In particular , for positive integers	F two.
such that	$ \om_k + \ddc_k u_k \geq \delta \om_k $	for all	Om sub k plus d c sub k u sub k greater than or equal to delta Om sub k.
a choice of line bundle on	$ \CP^3 $	is equivalent to a first Chern class . So , to generalize topological Horrocks ' addition to rank	complex projective three.
closest match	$ ( W1-W2 ) $	and the	W1 minus W2.
For	$ s\in T_1^F $	, from the proof of Corollary~	s belongs to T sub 1 F.
index	$ s $	labels time . That is	s
for any	$ f $	-divergence	f
&	$ -2m/M $	&	Negative two m over M.
and the force vector	$ \bm { F } $	is given by	capital F
All selections in	$ \mathcal { P } _3 $	are admissible .	P sub 3.
quantified by	$ \mathscr { C } $	, the cooling rate per unit area	C script.
The torsion-free derivative operator of	$ g_ { ab } $	is denoted by	g sub a b.
first suppose that	$ \mu^ { max } ( \mathcal { E } ) \geq 2g $	. Let	mu to the power of max, open parenthesis, E, close parenthesis, is greater than or equal to 2g.
applied to	$ f_\gamma $	now reads	f sub gamma
we can find distinct fixed points	$ x $	and	x
where	$ c_1 $	in the integrating constant . Even though the model of traversable wormholes is one of various solutions of Einstein 's general relativity , it suffers the violation of the energy	c sub one.
74	$ \pm $	0.01 & 1.65	Plus or minus.
individuals of opinion	$ A $	in the population , and the environment in state	A
since the right-hand side is always either	$ 0 $	, negative or at least double the value of	zero
for some	$ \xi $	and	xi.
and can be seen as a quantitative measure of the	$ d $	-rigidity of	d
and leads to the very sharp peak in the	$ \beta \gamma $	distribution . An increase in	Beta gamma.
the sum of the second and sixth columns of	$ M $	is zero . By	M
	$ H_S=\omega_m a^ { \dagger } a $	, where	H sub S equals omega sub m a dagger a
and a	$ k $	-point mesh of	k
the associated integrand is the product of an ERI and an amplitude entry where these two terms are both smooth everywhere except at their zero momentum transfer points with order	$ -1 $	and	Minus one.
and is distinct from the subleading logarithmic scaling of	$ S $	in a subsystem with sharp corners~	S
be an arbitrary F\o lner sequence in	$ G $	, and use Theorem	G
and	$ n $	denote the number of posts and agents , respectively .	n
	$ G ' $	contains a cycle	the derivative of G.
is a stochastic process whose cumulants are defined in terms of the moments of the stochastic process	$ x ( t ) $	, namely , for any	x of t.
is invariant under replacing	$ h $	with	h
resulting in an algebra with generators in	$ \CC^N $	for some	C to the power of N.
gives an edge-coloring of~	$ H $	( with the same set of colors ) without a monochromatic~	H
we give the full parameters	$ \phi $	in the appendix	phi.
thereby resulting in stronger dependence of	$ T_c $	as expected under conventional BCS theory	T c.
to make it induce linear-in-	$ T $	regret , i.e . ,	T
or the parallel propagator along the edge	$ \gamma_i $	,	Gamma i.
through replacing every edge in	$ \mathcal { G } _ { q } ( g ) $	by a	G sub q of g.
of length	$ n $	over	n
where the bottom row is a rearrangement of	$ \Omega $	.	Omega.
There are only finitely many such pseudoquads in	$ \Delta $	, so this proves the rest of the lemma .	Delta.
assuming that	$ a $	and	a
and	$ \mathbf { y } $	in the proposed hopping-based OTFS-SCMA system is given by	y
-dependence of the GF is the result of the operator	$ d_\sigma $	placed at time	d sub sigma.
number	$ M=20 $	of quantization levels , number	M equals 20.
and thus require adapted weight distributions	$ \pi_j $	to be captured in a rainbow model .	Pi sub j.
we calculate the numbers of neighbours the vertices of	$ G_n $	can have . % %	G sub n
and the set of edges	$ E $	. The former set	E
function of	$ J $	-th section } of	J
we assume the value of 670	$ \pm $	67 Myr	Plus or minus.
# 1	$ \pm $	# 2	Plus or minus.
	$ H_ { \rm end } $	and	H sub end.
For each	$ n \in \mathbb { N } $	, a matrix of type	n belongs to the set of natural numbers.
em	$ j=1 $	, vertices	j equals one.
there exists some	$ r\geq 0 $	such that Equation	r is greater than or equal to zero.
with	$ y_j : =\alpha_j+J_j\beta_j $	,	why sub j equals alpha sub j plus J sub j beta sub j.
Effect of Truss Support Parameter	$ k $	: } Figure~	k
since by definition the	$ \phi $	-dependence of	phi.
Now we construct the absorbing set	$ A $	by the following lemma .	A
by choosing mollifiers	$ \rho_n $	,	rho sub n.
	$ 2 $	of which are trees . If~	two
is a fixed polynomial function in	$ \sum_ { i=1 } ^nW_i $	variables of	Sum from i equals 1 to n of W sub i.
% The first term now converges weakly to	$ \cN ( 0,1 ) $	by Slutsky 's theorem , Theorem~	N zero comma one.
can be bounded from above by 0 as	$ N \to \infty $	, and thus can be ignored in this case with respect to our overarching bound on	N tends to infinity.
each of which has depth	$ 2d $	and acts on	Two d.
and absolute	$ M_ { B } $	magnitudes , i.e .	M sub B.
be the walls of	$ P_k $	forming the exceptional dihedral angle of	P subscript k.
is	$ F = \sum\limits_ { v\in V ( G ) } f ( s , v ) $	.	F equals the sum over v in V of G of f of s, v.
5	$ \pm $	1.9 & 988	Plus or minus.
Now we can divide	$ \Gamma $	into	Gamma.
Minimizing this loss results in weights	$ \Theta_0 $	. Upon training the detector in this way , we observe that it can identify more objects than those in our original proposals . Critically , this includes smaller and	Theta zero.
There are at least two reasons why the probability of failure without the requirement that	$ u_1 , u_2 , \ldots , u_k $	are linearly independent , which we denoted	u sub 1, u sub 2, ellipsis, u sub k
nodes in the network receive connections from	$ M $	input nodes , and project to a set of	M
so that	$ f $	is a mapping from	f
as the maximum absolute value of	$ \mathbf { R } $	,	R
time	$ t $	, and patient population constitutes a task . In this section , we present a definition for when a task is affected by temporal shift .	t
and renormalization scale	$ \mu_R $	. We set	mu sub R.
we fix the size of a subset sampled from	$ S_c $	as 135,000 to include as many positive samples as possible . % == Then we control this subset to contain a certain proportion of positive samples , i.e . ,	S sub c.
can be done easily after writing it as a one-dimensional integral over	$ \|\vec k\| $	. In the minimal subtraction scheme , we choose the counterterms to be	the absolute value of vector k.
If	$ r $	is a multiple root of	r
as we do not have direct access to	$ p $	and	p
since the bounds established on	$ r_ { t+1 } $	and	r sub t plus one.
	$ ( X_1 , X_2 ) $	.	X sub 1, comma, X sub 2.
since we call the replicable heavy hitters algorithm	$ k $	times with replicability parameter	k
	$ \bar { n } $	, are related by	n bar.
-	$ 1 $	and	one
near	$ p\bar { p } $	threshold in this process , but has also discovered the heavier	p bar p
is valid : If	$ k=j_0 $	, then in order to have	k equals j sub 0.
In the density model we require	$ d < \frac { 1 } { 5 } $	. Cayley representations turn out to be important when studying regular representations .	d is less than one-fifth.
The angle	$ i $	between the line of sight and the	i
to	$ \Sigma_e $	near infinity . More precisely ,	Sigma sub e.
and its preimage in	$ A $	. As usual , we identify	A
to	$ t $	. We say that	t
with	$ \sim 75\ % $	of the total mismatches below	Approximately 75 percent.

and

$ j=1,2 $

:

j equals one comma two.

$ t $

in

t

From the row containing

$ n_ { \fpos } $

we observe that the glow persists for a long time , and also extends downwards by about

n sub fpos

The probability density for an incoming photon to be found at position

$ x $

on the imaging screen will be calculated just as in section

x

&

$ 60.5_ { \pm 0.9 } $

&

six point five, sub plus or minus zero point nine.

The tensor product of evaluation vectors of polynomials over the field

$ \mathbb { F } _2 $

is an evaluation vector of a polynomial from a larger ring of the same field . In particular , for positive integers

F two.

such that

$ \om_k + \ddc_k u_k \geq \delta \om_k $

for all

Om sub k plus d c sub k u sub k greater than or equal to delta Om sub k.

a choice of line bundle on

$ \CP^3 $

is equivalent to a first Chern class . So , to generalize topological Horrocks ' addition to rank

complex projective three.

closest match

$ ( W1-W2 ) $

and the

W1 minus W2.

For

$ s\in T_1^F $

, from the proof of Corollary~

s belongs to T sub 1 F.

index

$ s $

labels time . That is

s

for any

$ f $

-divergence

f

&

$ -2m/M $

&

Negative two m over M.

and the force vector

$ \bm { F } $

is given by

capital F

All selections in

$ \mathcal { P } _3 $

are admissible .

P sub 3.

quantified by

$ \mathscr { C } $

, the cooling rate per unit area

C script.

The torsion-free derivative operator of

$ g_ { ab } $

is denoted by

g sub a b.

first suppose that

$ \mu^ { max } ( \mathcal { E } ) \geq 2g $

. Let

mu to the power of max, open parenthesis, E, close parenthesis, is greater than or equal to 2g.

applied to

$ f_\gamma $

now reads

f sub gamma

we can find distinct fixed points

$ x $

and

x

where

$ c_1 $

in the integrating constant . Even though the model of traversable wormholes is one of various solutions of Einstein 's general relativity , it suffers the violation of the energy

c sub one.

74

$ \pm $

0.01 & 1.65

Plus or minus.

individuals of opinion

$ A $

in the population , and the environment in state

A

since the right-hand side is always either

$ 0 $

, negative or at least double the value of

zero

for some

$ \xi $

and

xi.

and can be seen as a quantitative measure of the

$ d $

-rigidity of

d

and leads to the very sharp peak in the

$ \beta \gamma $

distribution . An increase in

Beta gamma.

the sum of the second and sixth columns of

$ M $

is zero . By

M

$ H_S=\omega_m a^ { \dagger } a $

, where

H sub S equals omega sub m a dagger a

and a

$ k $

-point mesh of

k

the associated integrand is the product of an ERI and an amplitude entry where these two terms are both smooth everywhere except at their zero momentum transfer points with order

$ -1 $

and

Minus one.

and is distinct from the subleading logarithmic scaling of

$ S $

in a subsystem with sharp corners~

S

be an arbitrary F\o lner sequence in

$ G $

, and use Theorem

G

and

$ n $

denote the number of posts and agents , respectively .

n

$ G ' $

contains a cycle

the derivative of G.

is a stochastic process whose cumulants are defined in terms of the moments of the stochastic process

$ x ( t ) $

, namely , for any

x of t.

is invariant under replacing

$ h $

with

h

resulting in an algebra with generators in

$ \CC^N $

for some

C to the power of N.

gives an edge-coloring of~

$ H $

( with the same set of colors ) without a monochromatic~

H

we give the full parameters

$ \phi $

in the appendix

phi.

thereby resulting in stronger dependence of

$ T_c $

as expected under conventional BCS theory

T c.

to make it induce linear-in-

$ T $

regret , i.e . ,

T

or the parallel propagator along the edge

$ \gamma_i $

,

Gamma i.

through replacing every edge in

$ \mathcal { G } _ { q } ( g ) $

by a

G sub q of g.

of length

$ n $

over

n

where the bottom row is a rearrangement of

$ \Omega $

.

Omega.

There are only finitely many such pseudoquads in

$ \Delta $

, so this proves the rest of the lemma .

Delta.

assuming that

$ a $

and

a

and

$ \mathbf { y } $

in the proposed hopping-based OTFS-SCMA system is given by

y

-dependence of the GF is the result of the operator

$ d_\sigma $

placed at time

d sub sigma.

number

$ M=20 $

of quantization levels , number

M equals 20.

and thus require adapted weight distributions

$ \pi_j $

to be captured in a rainbow model .

Pi sub j.

we calculate the numbers of neighbours the vertices of

$ G_n $

can have . % %

G sub n

and the set of edges

$ E $

. The former set

E

function of

$ J $

-th section } of

J

we assume the value of 670

$ \pm $

67 Myr

Plus or minus.

# 1

$ \pm $

# 2

Plus or minus.

$ H_ { \rm end } $

and

H sub end.

For each

$ n \in \mathbb { N } $

, a matrix of type

n belongs to the set of natural numbers.

em

$ j=1 $

, vertices

j equals one.

there exists some

$ r\geq 0 $

such that Equation

r is greater than or equal to zero.

with

$ y_j : =\alpha_j+J_j\beta_j $

,

why sub j equals alpha sub j plus J sub j beta sub j.

Effect of Truss Support Parameter

$ k $

: } Figure~

k

since by definition the

$ \phi $

-dependence of

phi.

Now we construct the absorbing set

$ A $

by the following lemma .

A

by choosing mollifiers

$ \rho_n $

,

rho sub n.

$ 2 $

of which are trees . If~

two

is a fixed polynomial function in

$ \sum_ { i=1 } ^nW_i $

variables of

Sum from i equals 1 to n of W sub i.

% The first term now converges weakly to

$ \cN ( 0,1 ) $

by Slutsky 's theorem , Theorem~

N zero comma one.

can be bounded from above by 0 as

$ N \to \infty $

, and thus can be ignored in this case with respect to our overarching bound on

N tends to infinity.

each of which has depth

$ 2d $

and acts on

Two d.

and absolute

$ M_ { B } $

magnitudes , i.e .

M sub B.

be the walls of

$ P_k $

forming the exceptional dihedral angle of

P subscript k.

is

$ F = \sum\limits_ { v\in V ( G ) } f ( s , v ) $

.

F equals the sum over v in V of G of f of s, v.

5

$ \pm $

1.9 & 988

Plus or minus.

Now we can divide

$ \Gamma $

into

Gamma.

Minimizing this loss results in weights

$ \Theta_0 $

. Upon training the detector in this way , we observe that it can identify more objects than those in our original proposals . Critically , this includes smaller and

Theta zero.

There are at least two reasons why the probability of failure without the requirement that

$ u_1 , u_2 , \ldots , u_k $

are linearly independent , which we denoted

u sub 1, u sub 2, ellipsis, u sub k

nodes in the network receive connections from

$ M $

input nodes , and project to a set of

M

so that

$ f $

is a mapping from

f

as the maximum absolute value of

$ \mathbf { R } $

,

R

time

$ t $

, and patient population constitutes a task . In this section , we present a definition for when a task is affected by temporal shift .

t

and renormalization scale

$ \mu_R $

. We set

mu sub R.

we fix the size of a subset sampled from

$ S_c $

as 135,000 to include as many positive samples as possible . % == Then we control this subset to contain a certain proportion of positive samples , i.e . ,

S sub c.

can be done easily after writing it as a one-dimensional integral over

$ |\vec k| $

. In the minimal subtraction scheme , we choose the counterterms to be

the absolute value of vector k.

If

$ r $

is a multiple root of

r

as we do not have direct access to

$ p $

and

p

since the bounds established on

$ r_ { t+1 } $

and

r sub t plus one.

$ ( X_1 , X_2 ) $

.

X sub 1, comma, X sub 2.

since we call the replicable heavy hitters algorithm

$ k $

times with replicability parameter

k

$ \bar { n } $

, are related by

n bar.

-

$ 1 $

and

one

near

$ p\bar { p } $

threshold in this process , but has also discovered the heavier

p bar p

is valid : If

$ k=j_0 $

, then in order to have

k equals j sub 0.

In the density model we require

$ d < \frac { 1 } { 5 } $

. Cayley representations turn out to be important when studying regular representations .

d is less than one-fifth.

The angle

$ i $

between the line of sight and the

i

to

$ \Sigma_e $

near infinity . More precisely ,

Sigma sub e.

and its preimage in

$ A $

. As usual , we identify

A

to

$ t $

. We say that

t

with

$ \sim 75\ % $

of the total mismatches below

Approximately 75 percent.