Datasets:
Last-Bullet
/
MathBridge_Splitted

Dataset card Data Studio Files
xet
 Community
1
context_before
stringlengths
0
199
equation
stringlengths
5
80
context_after
stringlengths
0
249
spoken_English
stringlengths
0
119
and
$ \left\ { \hat { h } _2 ( k ) \right\ } _ { k\in\Z } $
are two linearly independent eigenfunctions of
Left brace, hat h sub 2 open parenthesis k close parenthesis, right brace, sub k in Z.
then it computes
$ \bar r [ 0 : c\log ( n ) -1 ] $
from this advice in time
r bar of open square bracket zero colon c log open parenthesis n close parenthesis minus one close square bracket.
block is a diagonal matrix of length
$ p $
for
p
the dynamical decomposition is a diagnostic tool to measure the SST-driven component of
$ Q $
and consequently , establish how patterns of SST variability affect atmospheric circulation .
Q
with values of
$ V_0 $
compatible with the observed scalar power spectrum
V sub zero
$ \mathcal { H } $
is a finite-dimensional Hilbert space with some basis
H: None
and common distribution
$ H $
, independent of
H
The broadband flux values were extinction corrected using the A
$ _V $
value given by
V
we can extend the alternating form on
$ C $
to an alternating form on
C
does not divide over
$ C $
.
C
$ ^\textrm { \scriptsize 110 } $
,
One hundred ten.
and
$ b $
, such that
b
and
$ w $
,
w
The general idea is to use the analogy to finite-size scaling in statistical mechanics : In a system with length
$ L $
an order parameter
L
depending on whether the underlying model is
$ M $
or
M
This deviation attains a maximum at
$ \Delta\gamma \approx \Delta\gamma_c $
for all
Delta gamma approximately delta gamma sub c.
Suppose that
$ \rho $
is an
rho.
$ u^n $
has no zero components with indices in
u raised to the power of n.
Note that
$ \mathbf { d } $
has the same functional form as
d bold.
-flag
$ \flag { V_1\subset V_2\subset \cdots\subset V_ { n } \subset V_ { n+1 } } $
represents
Flag V sub 1 subset V sub 2 subset dot dot dot subset V sub n subset V sub n plus 1
In terms of
$ A_t $
, that is
A at t.
agrees with
$ \iota ( W ) $
defined in
iota of W
since the maps
$ \rho_ { m , n } $
are all c.p.c . , and so by
rho sub m comma n
implies that for any
$ i=0 , \cdots , d $
, the matrix sequence
i equals zero, dot dot dot, d.
Let
$ \MM_\ZZ $
be the subring of
M sub Z Z.
near
$ \ell = 0 $
. Using
ell equals zero.
Given a subset
$ J $
, and rounded flows
J
and
$ \lambda=4g $
, Fig .
lambda equals four g.
mediated by effective insertions of the
$ g_8 $
vertex : The total amplitude
g sub 8.
The disks of radius
$ \tfrac12 $
centered in the points corresponding to
One-half.
From now on we will write
$ \br { \mc { C } } ( \Delta ) $
for
br of m c C of Delta
by successively passing to subsequences of
$ A_k $
and using a diagonal sequence argument , the last property can be strengthened to hold with
A sub k.
in terms of the ordinary partial derivative of
$ L $
with respect to the inertial time derivative
L
and
$ k ' = |A| $
. Now relabel
The derivative of k with respect to time equals the magnitude of A.
: The argument of
$ \tan $
in the second line of
tangent.
for some effective Weil divisor
$ E\geq ( p^e-1 ) \Delta^- $
. Indeed , working locally and letting
E greater than or equal to the quantity p to the power of e minus 1, multiplied by Delta to the power of negative.
and
$ r $
are the radial distances in the retinal and original images , respectively ,
r
and is rooted at
$ k $
. The second component
k
it means that implementing
$ B=B_1 C B_2 $
, or implementing instead
B equals B sub 1 C B sub 2.
$ A $
open and dense in
A
there is an integer
$ d $
such that
d
chronological ages of
$ m $
subjects .
m
of
$ x ( t ) $
, which , for the process
x of t.
it is observed at the beginning of phase 2 and more generally before the infected population is too close to
$ N $
. For more quantitative results on this point for the SIR model , we refer to Proposition 5.1 of
N
&
$ 1.0 $
&
One point zero.
and provides an information of the environment deemed to be a random source providing instances which may match with the reference
$ R $
. This formulation captures ergodicity in terms of normalized counting , giving a measure of
R
&
$ 0.29 $
&
0.29: Zero point two nine.
is the unique diameter of
$ K_u $
. We claim that
K sub u.
from the derivatives of
$ H $
:
H
be the vertices of
$ G $
. Suppose that
G
to uniquely determine the ejecta opacity on the basis of the input
$ Y_e $
.
Y sub e.
for
$ m=120 $
orbitals and
m equals 120
is a rational number and is 0 if
$ x $
is irrational
x
This is because a higher MUI is experienced as
$ U $
increases . More specifically , it can be observed that the four-user system equipped with
U
the set of strongly continuous bounded
$ \mathcal { B } ( \rmL^p , \rmL^q ) $
-valued functions in the interval
B of L raised to the power of p, L raised to the power of q.
the SFR --
$ M_ { \rm HI } $
correlation that has been modelled for the continuum catalogue shown in Fig .
M sub HI
up to entries
$ i $
) is
i
independently on both
$ a $
and
a
And integral representation for
$ \sum_ { k=0 } ^ { n } \binom { n } { k } x^ { \overline { k } } $
:
The sum from k equals 0 to n of the binomial coefficient n choose k times x to the power of k.
$ we arrive at $
We arrive at None.
the positive character of both
$ EoS $
components ,
Ee oh S
we can replace the
$ \liminf $
on the left-hand side with
limit inferior.
are more significant than in
$ y $
. The implications of these findings for future observations are that determining the membership of a UDG in the NGC1052 group within the sequence of DMDGs is a challenging task
y
beginning at timestamp
$ i $
, it can be decomposed into components of seasonality
i
and every element
$ a \in A $
, the set
a belongs to A
Assign the indeterminant
$ z $
to the first equation and the indeterminants
z
with minimal
$ \|\cdot\|_Y $
norm . In other words :
norm of x, subscript Y.
including the commonly-used
$ L^2 $
norm , the Sobolev norms
L squared.
delivers a message
$ m $
from
m
is independent of
$ m $
and hence also applies to the case of the ` original ' multi-copy estimators
m
we show that a stable hypersurface with a
$ T^2 $
-invariant integrable region is isotopic to a
T squared.
The colored lines represent isochrones for brown dwarfs at ages
$ 0.1 $
,
0.1: None
simple zeroes
$ z_1 , z_2 , z_3 $
and that
z sub 1, z sub 2, z sub 3
where
$ d_ { l } $
and
dee sub ell.
Due to the
$ Z_2 $
-odd nature ,
Z sub 2.
0438 & 34270
$ \pm $
592 & 5.77
Plus or minus.
-map
$ \psi : P\to Q $
between
Psi colon P arrow Q
rather than
$ L^2 $
inevitably conjugates the potential by
L squared.
so that
$ Z=X-U_0 $
is a closed analytic subset of codimension
Z equals X minus U sub zero.
the SM-like Higgs field is a linear combination of
$ \phi^0_ { 1,2 } $
and , instead of a factor of
phi superscript zero subscript one comma two
we express our metrics as a bipartite matching between
$ \mathbf { G } $
and
G
&
$ M_B $
&
M sub B.
we inherently enforce a KL-constraint on the optimal policy
$ \pi^ * $
and the behavior policy
pi star.
in
$ A_i $
. Moreover , the sets
A sub i.
we can check that we have a homomorphism in
$ \End_ { \widetilde { A } [ 1/p ] } ( \prod_i V_i^0 ) $
, where it follows from the fact that
End of the tilde A, indexed by 1 over p, acting on the product over i of V sub i superscript 0.
was observed to scale as
$ ( n-\delta ) ^ { 3.5 ( 3 ) } $
(n minus delta) to the power of 3.5 times 3
we study the immediate herding events
$ \bar a=h $
and
a bar equals h.
Let
$ \overline { L } \subset \overline { Q } _0 $
be the set of loops of the quiver
L bar is a subset of Q bar sub zero.
and every vertex
$ x\in V ( G ) $
it holds that
x belongs to V of G.
$ l^\alpha $
generated by this procedure form the Mori cone of the ambient variety which may be different from the Mori cone of the Calabi-Yau hypersurface . To determine the Mori cone
l to the power of alpha
of class
$ C^2 $
satisfying
C squared.
under
$ \Delta ( 27 ) $
. In the case of
Delta 27.
we collect the first few batches of data frames into
$ \mathcal { B } $
to initialize
bee
be the von Mangoldt arithmetical function and
$ w ( t ) $
be a real valued function defined for
w of t
in the GH-subtracted spectra at
$ kd_i \leq 10^ { -3 } $
. Also displayed in Figure~
K times d sub i is less than or equal to 10 to the power of negative 3.
This holds for
$ a\ge 100 $
, proving the result .
a greater than or equal to 100
If
$ k $
is odd , then
k
$ [ x_1 , x_2 ] $
? If the interval is too wide ,
The interval from x sub 1 to x sub 2.
through a hook
$ \pi ' $
, and
pi prime.
A
$ k $
-algebraic torus
k