context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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then the output of Alice and Bob may be either~ | $ 0 $ | or~ | zero |
$ \ { ( A_ { L_t } ^ * , B_ { L_t } ^ * ) \ } $ | for all | Open brace, A sub L sub t superscript asterisk, comma, B sub L sub t superscript asterisk, close brace. | |
For | $ i\in I_2 $ | , we say that | i belongs to the set I sub 2. |
$ ^\textrm { \scriptsize 41 } $ | , | 41 | |
$ O $ | , which are independent of the state of the system | O | |
which determine the basal Ni~ | $ 3d_ { x^2-y^2 } $ | -O~ | Three times the derivative of the function \(x\) squared minus \(y\) squared with respect to \(d\). |
For large | $ d $ | , the onset times of scale invariance become long , so that the time interval of scale invariance in finite systems shrinks , see the shaded regions in Figs.~ | d |
the summand | $ d_i^2 $ | appears exactly | dee sub i squared |
is exponentially stable for some values of | $ \lambda $ | and unstable for others . Second , the spectrum of | Lambda. |
there is no trivial | $ \mathcal\ X ' $ | -- | X prime |
there exist | $ \beta $ | -H | Beta. |
& | $ - $ | minus | |
em If | $ \alpha_1 = \alpha_2 = 0 $ | , | Alpha one equals alpha two equals zero. |
98 | $ \pm0.01 $ | & 0.21 & 2.17 & - & - | Plus or minus zero point zero one. |
especially if | $ K $ | is large , which might decrease the relative energy gap . | K |
If | $ \beta\ , Y ( T_ * ) \sim 1 $ | , the annihilation effect is important . As | Beta times Y times T star is approximately equal to 1. |
and hence are replaced with the momenta | $ P_i $ | . Therefore , we may write the scalar constraint in terms of our phase space variables~ | P sub i. |
and | $ A $ | , we replace the scalar products in the matrices ' entries by sums of | A |
93 | $ \pm $ | 0.26 & 34.70 | Plus or minus. |
4 | $ d^ { 10 } $ | d raised to the power of ten. | |
$ R $ | -subgaussian , | R | |
8124 | $ \pm $ | .0191 & .7904 | Plus or minus. |
for all | $ j\in \N $ | . Then , under the assukmkption % | j belongs to the set of natural numbers. |
and for all sufficiently large | $ x $ | let | x |
since | $ \omega_L $ | increases with | omega sub L. |
the inclusion or exclusion of the second lag of | $ C_ { i , t } $ | does not affect the classification results , suggesting that residual autocorrelation does not have an impact for clustering purposes . | C sub i, t. |
and | $ \mu $ | , which spelled out means % | mu. |
and | $ B $ | a generating set of | B |
labeled by some finite index set~ | $ \Omega $ | , such that~ | Omega. |
of a continuous function | $ f $ | is the closure of the open set | f |
indicates whether | $ y_ { st } $ | is in the detected group . We model the parameters as follows : % We assume the logistic linear regression model for | y sub s t |
L_1^- | $ + $ | L_2^- | plus |
where the last inequality is due to the quadratic growth condition of | $ \mu $ | -PL functions , i.e . , | mu. |
for a constant | $ C > 0 $ | independent of | C is greater than zero. |
is the | $ i $ | -th row of | i |
where | $ m_j : = \mathop\ { \rm\ rank\E ( \lambda_j ) $ | and the natural embedding | m sub j is defined as the rank of E of lambda sub j. |
with the radial pressure | $ p_r $ | , i.e . | p sub r. |
or | $ C_i $ | with | C sub i. |
Here we recall that | $ C_1 $ | is the optimal value of | C sub 1. |
and | $ w\in L^ { \infty } ( 0 , \infty ; L^2 ) $ | . Then , for any | w belongs to L infinity of 0, infinity, L squared. |
provide us new examples of | $ 2 $ | -Calabi-Yau categories ? | two |
005 & | $ - $ | 0.004 | minus |
for | $ x $ | and for | x |
to a 2-group global symmetry by fusing with | $ \text\ { U^ { [ 1 ] } _\text { top } $ | . We turn on a background 2-form gauge field | U superscript 1 subscript top |
$ N $ | ) representation matrix of | N | |
Both | $ \psi $ | and | Psi. |
When | $ \psi ( z ) $ | is large , the neural subsystem manifests three fixed points by means of a saddle-node bifurcation . In other words , at the bifurcation point , there is a fundamental | Psi of z. |
between the vacuum and the torus core leads to walls with high energy and the passage near the torus core leads to an almost discontinuous change of the argument of | $ \phi $ | -field , resulting in the sharpness of the walls . In the weakly interacting regime , since | phi. |
and | $ R_ { \mu\nu\rho\sigma } $ | represents the Riemann tensor . | R sub mu nu rho sigma. |
We will show that the numerator on the RHS is | $ 0 $ | . By the definition of the coupling it suffices to show that for | zero |
and | $ \mathcal { B } _2 $ | testing procedures , we usually observe the reverse behavior , which can be explained by their extreme conservative character . Furthermore , these two tests are the least powerful among | B sub 2. |
mod | $ \varphi $ | coloring . | phi. |
are | $ C ' $ | -contracting for some | C prime. |
It has the following parameters that need to be tuned : | $ m $ | representing the number of VI updates , the Lipschitz constant | m |
where | $ \overline { Y } $ | denotes the complement of | Y bar. |
$ y=t_2 $ | , | y equals t sub 2. | |
$ n $ | is the refractive index , | n | |
& -- & | $ 0.27 $ | & | 0.27: None |
and then | $ j\to\infty $ | , we obtain that | j tends to infinity. |
has | $ \tau $ | data points . This guarantees that with more than | Tau. |
of size at most | $ k $ | such that | k |
P | $ _1 $ | 8 , H | One. |
around | $ \zeta $ | . | zeta. |
In the parallel configuration | $ \theta=0 $ | , | Theta equals zero. |
we look at the following decomposition of | $ V $ | as a direct sum of | V |
the signs of the vertices | $ s_i $ | and | s sub i. |
Let | $ G $ | be any finite group of automorphisms of a finite poset | G |
is thus achieved for the first time at | $ t $ | , i.e . , % | t |
If | $ f $ | is a nonzero polynomial , then Lemma~ | f |
& | $ \pm $ | & | Plus or minus. |
holds also for all | $ s , t \in Q $ | and all | s and t belong to the set of rational numbers. |
This persists until | $ \mu=1 $ | , where a PT/localization transition occurs as both | mu equals one. |
& | $ d=3 $ | & | d equals three. |
it is worth mentioning that due to the more complex form of the radiation condition for | $ k\in ( 2\sqrt { 2 } , 3 ) $ | , cf . | k belongs to the interval between 2 times the square root of 2 and 3. |
& Polymerization parameter & | $ 10^ { 3 } $ | & | Ten to the power of three. |
of | $ ( \omega_1 , \omega_2 ) $ | with standard Gaussian | omega one comma omega two. |
-count and number of logical qubits required to implement the | $ \mathcal { Q } $ | operator with | capital Q |
Bob randomly produces a photon polarized in one of the four states | $ \ { |0\rangle , |1\rangle , |+\rangle , |-\rangle\ } $ | and he transmits it to Alice . When this particle reaches Alice , she measures it with a probability of | ket minus |
bound states and their spin | $ J $ | should be the total angular momentum | J |
for all | $ i \in [ m ] , $ | we see that | i belongs to the set of integers from 1 to m. |
is carried by | $ \mathcal { K } $ | for each | kay |
This algorithm fits the APOGEE spectrum using models with two different | $ T_ { \rm { eff } } $ | to estimate a temperature difference and a spot fraction | T sub e-f-f. |
with | $ i < j < k $ | , then we have | i is less than j is less than k |
$ \mu ( x , v ) $ | denotes a local Maxwellian which will be specified below and | mu of x and v | |
and | $ C_3 $ | are independent with | C sub 3. |
$ - $ | 6.45 } & { 9.10 } | minus | |
follows from the assumption that | $ T_ { \sfc [ i ] } $ | has | T sub i, left square bracket i, right square bracket. |
Clusters with an average cluster density of 100 pc | $ ^ { -3 } $ | to a few 10 | Negative three. |
by taking | $ F ( \cdot , u ) = - R_ { g } u + Su^ { p-1 } $ | and | F of open parenthesis dot comma u close parenthesis equals negative R sub g u plus S u to the power of p minus 1. |
We call | $ \alpha_2 $ | the image of the path | Alpha sub 2. |
$ c $ | ) , realized at sufficiently high energies~ | c | |
and the multi-index | $ \beta $ | , but is uniform in | Beta. |
40 | $ & $ | 0.23 | and |
for some large constant | $ K > 0 $ | . We then present a sketch of how to bound | K is greater than zero. |
The resulting | $ ( \min , + ) $ | -circuit model of computation is somewhat more general than the class of relaxation algorithms , because the sequence of operations performed in this model does not need to come from | min, plus |
the lowest set of excited states we are going to consider has one electron with spin | $ \sigma $ | in the excited level , and the second electron of the same ion , with spin | Sigma. |
Train an VAE | $ ( p , q ) $ | by maximizing | p, q |
We are thus unsure of an optimal form of the | $ + $ | -Heegner point main conjecture . We draw our viewpoint from the general philosophy that the bound arising from a Kolyvagin system over a complete local Noetherian ring is optimal if | plus |
is one of the neighboring cells of | $ i $ | . The interface between them is labeled as | i |
be a unitary representation of | $ G $ | that is of positive energy . Set | G |
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