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
attached to	$ e_j $	if	e sub j
in terms of	$ h_ { \ { AB\ } } $	,	h sub A B
between	$ ( L_ { I } ^ { ( q ) } ) _ { I \subseteq [ 3 ] } $	and	L sub I superscript q, sub I subset equal to the set containing the numbers 1, 2, and 3.
where	$ x_0 $	is the middle intersection point . When	x sub zero.
and	$ I_ { ij , \ell\ell } $	is given by	I subscript i j, ell ell
	$ m=5 $	,	m equals 5
We prove a Solomon-Tits theorem when	$ R $	either satisfies a stable range condition , or is the ring of	R
in	$ \cH\oplus\cH ' $	with oracles in	H tensor direct sum H prime.
say	$ B $	that is a word in terms or	B
Genuinely multipartite concurrence of	$ N $	-qubit	N
the height function at time	$ t $	is not depending on the fact that the initial condition starts with empty initial condition on	t
For	$ a = 0,1 $	,	a equals zero point one
in the definition of	$ \mathcal { R } ^1_i $	with the	R one sub i
	$ & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0.032\\ $		You: One, one, one, one, one, one, one, zero, zero, zero point zero three two.
The peak burst rate can reach 122 hr	$ ^ { -1 } $	with the FAST observation , with the majority of the burst	to the power of negative one.
We attempted to determine	$ \epsilon_6 $	directly through XRD and compare to the value extracted from the FMR data . To do this the lattice constants of each sample where determined through reciprocal space mapping with	Epsilon sub 6.
/	$ M_4 $		M sub 4.
for arbitrary \Star-continuous Kleene algebras	$ K $	. Every element	K
given	$ x $	'' in terms of the ratio %	x
if	$ r $	is clear from context . A candidate	r
we simply write	$ 0 $	, as dimensions are easily understood from context .	zero
Let	$ \rho_C $	denote the Gibbs state at inverse temperature	rho sub C.
Shuo Yang	$ ^ { 2 } $	, Ziyan An	squared.
and	$ y $	are the actual ETH and USDT deposits in that tick range , respectively . Virtual reserves are just a mathematical artifact : they extend the physical ( real ) deposits	y
at the minimal value point	$ x_ { 0 } $	of	x sub zero.
we show that for any ideal	$ I $	, the map	I
Let	$ \sigma $	be a stability condition on a triangulated category	Sigma.
and ignore the contribution of the other triples in	$ \mathbf { E } $	eq.	E bold.
and	$ \mu_2 $	-mass at least	mu sub 2
and	$ T $	commute . Hence	T
honeycomb lattice	$ \rm \alpha $	-	Alpha.
	$ \Omega_m=0.315 $	,	Omega m equals zero point three one five.
is natural in	$ \sigma $	. Let us fix a morphism	Sigma.
46	$ ^ { +0.05 } _ { -0.05 } $	& 37.3 & 0.24 & -0.03 & 0.17 & I	plus 0.05 with a superscript of positive 0.05 and a subscript of negative 0.05.
It is left to show that we can always find a LRQ rule for sensor	$ k $	that makes the final performance at the FC , i.e . ,	k
	$ \lambda $	12.132	Lambda.
of a simple binomial random graph of size	$ n $	- i.e . a single block of the SBM . Let	n
&	$ 4 $	&	four
the discontinuation time	$ D_i ( 1 ) $	and the covariates , %	D sub i of 1.
in order to avoid the problem of overestimation of	$ Q $	caused by the distribution shift of the actions in offline learning , we use BRAC ~	Q
	$ x $	is then on the whole	x
	$ ^\textrm { \scriptsize 19 } $	,	19
for the surjective function given by	$ a \mapsto [ a ] $		a maps to the closed interval from a.
Now if there is another periodic cycle for these values of	$ n $	and	n
The symmetric group	$ S_n $	acts faithfully and smoothly on the orientable cover	S sub n.
Since	$ e $	is the unit of	e
suppose	$ x $	is a self-adjoint element in	x
conjectured that if	$ M $	is hyperbolic , with volume	M
	$ \mu_P ( S_0 ) =1 $	. For	mu sub P of S sub 0 equals 1.
Since	$ \mbox { Br } ( Z\to \ell^+\ell^- ) = ( 3.3658\pm 0.0023 ) $		Br of Z to ell plus ell minus equals 3.3658 plus or minus 0.0023.
and	$ R_T $	is arc connected if and only if	R sub T.
% For each	$ i\geq 0 $	we consider a weight %	i is greater than or equal to zero.
This implies that	$ v $	is adjacent to at least two neighbours of	v
on the repetition quiver	$ \ZZ Q_ { \BA } $	. See	Z Z Q sub BA.
represent the dialogue context and response at step	$ t $	, and	t
	$ I $	, so we show by induction on	I
First we consider	$ u_a $	following a path starting on the positive pole ,	u sub a.
in the case where	$ N $	is small , we need some alternative argument which does not rely on convergence in the group algebra . Indeed , we will show that although	N
the Lipschitz constant of a margin-based surrogate loss functions is unbounded when	$ u $	runs over	u
is conditionally independent of	$ C_i $	given	C sub i.
where	$ g_ { Y } $	is a metric on	g sub Y
where	$ l_\mathrm { max } = r \sqrt { E_0^2 - m_0^2 } $	. The additional factor 2 appearing in the above formula comes from the fact that a given set of values	l subscript max equals r square root of E subscript 0 squared minus m subscript 0 squared
be an	$ n $	-dimensional smooth (	n
the	$ k $	-th element 's norm vector can be written as	k
	$ \ket { 101 } $	,	ket 101
with	$ L $	chosen sufficiently large as needed . Note that any subsequence of neighboring terms is also an	L
Note that Bartnik uses the weaker	$ L^ { 1 } $	-decay condition on	L to the power of 1.
The number of qubits we have to introduce for this layer is	$ k_ { pen } \cdot \delta $	where	k subscript pen dot delta
is of the order of	$ 10^5 $	m . The width of our smallest cell is	Ten to the power of five.
are expressed by	$ 2d+1 $	free parameters . If in addition to	Two d plus one.
is a balanced	$ r $	-partite graph with each vertex class of size	r
such that	$ \gamma ( x_1 ) =c $	,	Gamma of x sub 1 equals c.
Ra	$ & $	Re_z	and
& 1 &	$ 0.02 $	&	0.02: None
the position	$ i $	of	i
He and SrTi	$ _ { 3-x } $	Nb	three minus x.
&	$ x^ { \prime } _3 $	&	x prime sub 3
they are due to the relaxation of the hypothesis on	$ w $	compared to	w
which consists of	$ 17 $	entities across	Seventeen.
% % The maximum of relative	$ l_2 $	errors over time , in terms of both the mean and standard deviation of the model 's predictions , are presented in Table	l sub 2
the string-breaking transition rate	$ g $	can be obtained by evaluating the time derivative with an appropriate normalization coefficient after some relaxation time for which higher excited light QCD configurations have decayed ( see Ref.~	g
is set to 10	$ ^ { -6 } $	and 10	Negative six.
For each	$ j\in [ k ] $	, we arbitrarily fix	j belongs to the set of integers from 1 to k.
supports hypothesis	$ H^ { ' } $	, for case 2 ,	H prime
this integral can be trivially computed from the chains just by binning	$ \theta_1 $	and counting the number of points in the chain that fall in each bin . This can be done pretty fast . The resulting histogram is an approximation of the	Theta one.
	$ \mathcal { S } $	is defined to store the nodes waiting to be visited ,	ess
the orbital speed is determined by the formula	$ v=Z\alpha/n $	. Therefore , for a meson in circular Rydberg states it is suppressed by the factor	v equals Z alpha divided by n.
and	$ M_4 $	computed from this perspective should include a family of decorated Donaldson-type invariants of the four manifold	M sub 4.
the line in	$ \Pi $	is	Pi.
for each of the	$ N $	neighbors with the highest peak	N
This implies that the diagrams that include	$ \sigma_1 $	or	Sigma one.
and	$ e^ * \colon \mathbb { B } ( d ) ^ { \mathrm { red } } _v\to \mathbb { B } $	.	e star, colon, B of d raised to the power of red, sub v, arrow, B
is sharper for	$ \rho \ge 8 $	. The result in fact holds for all	rho greater than or equal to 8.
that Poincaré transformations preserve surfaces of constant	$ \sigma $	. We have now	Sigma.
only NS EOSs with	$ \cmax=0.75 $	can potentially be ruled out , a condition tightens to	C max equals zero point seven five
where it was noted that the potential is finite at	$ r=0 $	. % In~	r equals zero.
the parameter	$ d $	in the exponent can generally be much smaller than	d
we have	$ \zeta ( v ) = \zeta ( \eta ) \zeta^ * ( v ) $	. Since vertex	Zeta of v equals Zeta of eta times Zeta star of v.
On the	$ i $	-th day (	i
	$ E ( M ) $	denotes an injective hull of	E of M

attached to

$ e_j $

if

e sub j

in terms of

$ h_ { \ { AB\ } } $

,

h sub A B

between

$ ( L_ { I } ^ { ( q ) } ) _ { I \subseteq [ 3 ] } $

and

L sub I superscript q, sub I subset equal to the set containing the numbers 1, 2, and 3.

where

$ x_0 $

is the middle intersection point . When

x sub zero.

and

$ I_ { ij , \ell\ell } $

is given by

I subscript i j, ell ell

$ m=5 $

,

m equals 5

We prove a Solomon-Tits theorem when

$ R $

either satisfies a stable range condition , or is the ring of

R

in

$ \cH\oplus\cH ' $

with oracles in

H tensor direct sum H prime.

say

$ B $

that is a word in terms or

B

Genuinely multipartite concurrence of

$ N $

-qubit

N

the height function at time

$ t $

is not depending on the fact that the initial condition starts with empty initial condition on

t

For

$ a = 0,1 $

,

a equals zero point one

in the definition of

$ \mathcal { R } ^1_i $

with the

R one sub i

$ & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0.032\\ $

You: One, one, one, one, one, one, one, zero, zero, zero point zero three two.

The peak burst rate can reach 122 hr

$ ^ { -1 } $

with the FAST observation , with the majority of the burst

to the power of negative one.

We attempted to determine

$ \epsilon_6 $

directly through XRD and compare to the value extracted from the FMR data . To do this the lattice constants of each sample where determined through reciprocal space mapping with

Epsilon sub 6.

/

$ M_4 $

M sub 4.

for arbitrary \Star-continuous Kleene algebras

$ K $

. Every element

K

given

$ x $

'' in terms of the ratio %

x

if

$ r $

is clear from context . A candidate

r

we simply write

$ 0 $

, as dimensions are easily understood from context .

zero

Let

$ \rho_C $

denote the Gibbs state at inverse temperature

rho sub C.

Shuo Yang

$ ^ { 2 } $

, Ziyan An

squared.

and

$ y $

are the actual ETH and USDT deposits in that tick range , respectively . Virtual reserves are just a mathematical artifact : they extend the physical ( real ) deposits

y

at the minimal value point

$ x_ { 0 } $

of

x sub zero.

we show that for any ideal

$ I $

, the map

I

Let

$ \sigma $

be a stability condition on a triangulated category

Sigma.

and ignore the contribution of the other triples in

$ \mathbf { E } $

eq.

E bold.

and

$ \mu_2 $

-mass at least

mu sub 2

and

$ T $

commute . Hence

T

honeycomb lattice

$ \rm \alpha $

-

Alpha.

$ \Omega_m=0.315 $

,

Omega m equals zero point three one five.

is natural in

$ \sigma $

. Let us fix a morphism

Sigma.

46

$ ^ { +0.05 } _ { -0.05 } $

& 37.3 & 0.24 & -0.03 & 0.17 & I

plus 0.05 with a superscript of positive 0.05 and a subscript of negative 0.05.

It is left to show that we can always find a LRQ rule for sensor

$ k $

that makes the final performance at the FC , i.e . ,

k

$ \lambda $

12.132

Lambda.

of a simple binomial random graph of size

$ n $

- i.e . a single block of the SBM . Let

n

&

$ 4 $

&

four

the discontinuation time

$ D_i ( 1 ) $

and the covariates , %

D sub i of 1.

in order to avoid the problem of overestimation of

$ Q $

caused by the distribution shift of the actions in offline learning , we use BRAC ~

Q

$ x $

is then on the whole

x

$ ^\textrm { \scriptsize 19 } $

,

19

for the surjective function given by

$ a \mapsto [ a ] $

a maps to the closed interval from a.

Now if there is another periodic cycle for these values of

$ n $

and

n

The symmetric group

$ S_n $

acts faithfully and smoothly on the orientable cover

S sub n.

Since

$ e $

is the unit of

e

suppose

$ x $

is a self-adjoint element in

x

conjectured that if

$ M $

is hyperbolic , with volume

M

$ \mu_P ( S_0 ) =1 $

. For

mu sub P of S sub 0 equals 1.

Since

$ \mbox { Br } ( Z\to \ell^+\ell^- ) = ( 3.3658\pm 0.0023 ) $

Br of Z to ell plus ell minus equals 3.3658 plus or minus 0.0023.

and

$ R_T $

is arc connected if and only if

R sub T.

% For each

$ i\geq 0 $

we consider a weight %

i is greater than or equal to zero.

This implies that

$ v $

is adjacent to at least two neighbours of

v

on the repetition quiver

$ \ZZ Q_ { \BA } $

. See

Z Z Q sub BA.

represent the dialogue context and response at step

$ t $

, and

t

$ I $

, so we show by induction on

I

First we consider

$ u_a $

following a path starting on the positive pole ,

u sub a.

in the case where

$ N $

is small , we need some alternative argument which does not rely on convergence in the group algebra . Indeed , we will show that although

N

the Lipschitz constant of a margin-based surrogate loss functions is unbounded when

$ u $

runs over

u

is conditionally independent of

$ C_i $

given

C sub i.

where

$ g_ { Y } $

is a metric on

g sub Y

where

$ l_\mathrm { max } = r \sqrt { E_0^2 - m_0^2 } $

. The additional factor 2 appearing in the above formula comes from the fact that a given set of values

l subscript max equals r square root of E subscript 0 squared minus m subscript 0 squared

be an

$ n $

-dimensional smooth (

n

the

$ k $

-th element 's norm vector can be written as

k

$ \ket { 101 } $

,

ket 101

with

$ L $

chosen sufficiently large as needed . Note that any subsequence of neighboring terms is also an

L

Note that Bartnik uses the weaker

$ L^ { 1 } $

-decay condition on

L to the power of 1.

The number of qubits we have to introduce for this layer is

$ k_ { pen } \cdot \delta $

where

k subscript pen dot delta

is of the order of

$ 10^5 $

m . The width of our smallest cell is

Ten to the power of five.

are expressed by

$ 2d+1 $

free parameters . If in addition to

Two d plus one.

is a balanced

$ r $

-partite graph with each vertex class of size

r

such that

$ \gamma ( x_1 ) =c $

,

Gamma of x sub 1 equals c.

Ra

$ & $

Re_z

and

& 1 &

$ 0.02 $

&

0.02: None

the position

$ i $

of

i

He and SrTi

$ _ { 3-x } $

Nb

three minus x.

&

$ x^ { \prime } _3 $

&

x prime sub 3

they are due to the relaxation of the hypothesis on

$ w $

compared to

w

which consists of

$ 17 $

entities across

Seventeen.

% % The maximum of relative

$ l_2 $

errors over time , in terms of both the mean and standard deviation of the model 's predictions , are presented in Table

l sub 2

the string-breaking transition rate

$ g $

can be obtained by evaluating the time derivative with an appropriate normalization coefficient after some relaxation time for which higher excited light QCD configurations have decayed ( see Ref.~

g

is set to 10

$ ^ { -6 } $

and 10

Negative six.

For each

$ j\in [ k ] $

, we arbitrarily fix

j belongs to the set of integers from 1 to k.

supports hypothesis

$ H^ { ' } $

, for case 2 ,

H prime

this integral can be trivially computed from the chains just by binning

$ \theta_1 $

and counting the number of points in the chain that fall in each bin . This can be done pretty fast . The resulting histogram is an approximation of the

Theta one.

$ \mathcal { S } $

is defined to store the nodes waiting to be visited ,

ess

the orbital speed is determined by the formula

$ v=Z\alpha/n $

. Therefore , for a meson in circular Rydberg states it is suppressed by the factor

v equals Z alpha divided by n.

and

$ M_4 $

computed from this perspective should include a family of decorated Donaldson-type invariants of the four manifold

M sub 4.

the line in

$ \Pi $

is

Pi.

for each of the

$ N $

neighbors with the highest peak

N

This implies that the diagrams that include

$ \sigma_1 $

or

Sigma one.

and

$ e^ * \colon \mathbb { B } ( d ) ^ { \mathrm { red } } _v\to \mathbb { B } $

.

e star, colon, B of d raised to the power of red, sub v, arrow, B

is sharper for

$ \rho \ge 8 $

. The result in fact holds for all

rho greater than or equal to 8.

that Poincaré transformations preserve surfaces of constant

$ \sigma $

. We have now

Sigma.

only NS EOSs with

$ \cmax=0.75 $

can potentially be ruled out , a condition tightens to

C max equals zero point seven five

where it was noted that the potential is finite at

$ r=0 $

. % In~

r equals zero.

the parameter

$ d $

in the exponent can generally be much smaller than

d

we have

$ \zeta ( v ) = \zeta ( \eta ) \zeta^ * ( v ) $

. Since vertex

Zeta of v equals Zeta of eta times Zeta star of v.

On the

$ i $

-th day (

i

$ E ( M ) $

denotes an injective hull of

E of M