context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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$ F $ | is the number of frequency bins and each complex spectrum has real and imaginary parts . % | F | |
-fold bond-orientational order parameter for each particle | $ j $ | , | j |
well at timestep | $ t $ | of reverse process , which is important for the next editing stage . The algorithm of the inversion and tuning stages is provided in Algorithm~ | t |
does not provide a precise forecast for the measurement of the specific coefficients | $ \delta^ { GW } _ { \ell m } $ | , but it estimates the level of their detectability under the assumption of statistical isotropy . Under this assumption , ref . | delta raised to the GW, sub-ell, em. |
For the | $ t\bar { t } $ | process , there will be no apparent tendency . However , in the signal process , because both the charged Higgs boson and the neutral Higgs boson are light , | t bar t |
for all | $ x , y \in V ( G ) $ | . So finding a global parameter , connectivity of | x, y belong to the set of vertices of graph G. |
$ A_ { ij } $ | , | A sub i j. | |
interacting Bose atoms of mass | $ m $ | , rotating with angular velocity | m |
in | $ \Omega_ { \mathrm { nfSf } , \sigma , R } $ | and in terms of | Omega sub n f S f, sigma, R |
of | $ x_i $ | . | x sub i. |
is the Hamilton vector field of | $ p $ | , which is defined using the standard symplectic form on | p |
and take | $ k_ * $ | as the fitted functional form in their Tab.~2 . | k star |
= samples between cluster | $ i $ | and | i |
where | $ H $ | is a positive definite fixed matrix . | H |
for | $ E^2 < 1 $ | , and | E squared is less than one. |
and | $ N=4 $ | Fourier coefficients , alongside the exact optimal control derived from the solution of the Euler-Lagrange system of equations . | N equals four. |
$ O ( \epsilon ) $ | , | O of epsilon. | |
means that | $ G_m ( x ) \rightarrow G ( x ) $ | as | G sub m of x goes to G of x. |
with the fiber | $ K_2 $ | as shown in Example 5.29 of | K sub 2. |
$ \Delta x $ | =1 | Delta x. | |
& viscoelastic damping parameters & 50 | $ v_s $ | , 2 | v s |
where~ | $ \mathcal { V } = L^2/2 $ | is the volume of the magnetic lattice . Further , we have defined the coupling constants | V equals L squared divided by 2. |
is centered at | $ o_i $ | and | o sub i. |
$ d ( v ) $ | is the degree of | d of v | |
-uniform hypergraph on | $ n $ | vertices . A permutation | n |
& | $ -1.72_ { -1.85 } ^ { +1.82 } $ | & | Negative 1.72 with a lower uncertainty bound of negative 1.85 and an upper uncertainty bound of positive 1.82. |
An array | $ \mathcal { M } $ | of length | M |
& | $ 0.40 $ | & | 0.40: None |
$ b_k/q_k $ | is an approximation by continued fraction , for which | b sub k divided by q sub k | |
is reachable at every step | $ h \in [ H ] $ | under the target policy . If there exists an CCE | h belongs to the set containing H. |
of | $ f $ | passing through the point of the generic fiber of | f |
into a genus three handlebody | $ H_1 $ | and a genus two handlebody | H sub 1. |
where | $ \psi ( x ) $ | is the concave increasing function given by | psi of x. |
-S/N | $ > $ | 5 & 1.42 | greater than |
the entangling layers | $ \widetilde { L } $ | , defined as the number of layers needed by an architecture to reach | L tilde. |
on the | $ \mu_2 $ | -part . We write from now on | mu sub 2 |
and integrate over | $ \CF_ { Y_- ( \bfm ) } $ | with | the set F sub Y sub negative m |
versus those calculated using the mean posterior estimates of the | $ \boldsymbol { \alpha } $ | , | Alpha. |
and any | $ \varepsilon > 0 $ | , | Epsilon is greater than zero. |
as the subgroup fixing pointwise the curve | $ K_1 $ | we obtain the following subnormal series | K sub 1. |
$ g $ | is the determinant of the metric tensor | g | |
Let | $ A^k $ | be the set of authorized colors for tour | A raised to the power of k. |
consider | $ \lambda \neq 0 $ | . The support of | Lambda is not equal to zero. |
A coherent dynamics of | $ \varphi $ | is governed by the action | phi. |
Let | $ H \in ( 0,1 ) $ | . Then a continuous stochastic process | H belongs to the open interval from 0 to 1. |
\hline | $ n $ | & | n |
A canonical form of the operator | $ \rho $ | and its inverse should be constructed from the right eigenvector and left eigenvector | rho. |
DNA-MU first initialises the residual supply | $ m'=m $ | . For each agent | m prime equals m |
in | $ X $ | ( see proposition | X |
If the coefficient | $ \alpha_ { 1 } $ | of executive ownership is positive , it means that executive ownership can increase INV . | Alpha sub 1. |
and | $ R_ { 2\omega } $ | under | R sub 2 omega |
is a solution to | $ ( G , S ) $ | . Consider any terminal pair | G comma S. |
to an intermediate skeleton image | $ s $ | , | s |
for optimization algorithms on | $ 100 $ | -dimensional test functions . See Figure~ | One hundred. |
is a | $ T $ | -periodic , | T |
where | $ \gamma $ | is the cooling rate ; let us denote two constants that express the effect of the scheduler : | Gamma. |
in | $ \Omega $ | . | Omega. |
for all | $ x\in V $ | and | x belongs to V |
the same still yield proper | $ H $ | -colorings of | H |
of | $ [ \eta ] $ | with | eta in square brackets. |
as | $ \alpha^ { -1 } $ | . | Alpha to the power of negative one. |
to complete a scan over a mimicking tubular structure of | $ 160~mm $ | length . % % % % % % % % % % % % % % % % % % % % % % % % % % % % | 160 millimeters. |
form factor obtained from the | $ T^ { ij } $ | component of the EMT current from the IF mass form factor | T raised to the power of i j. |
are independent of | $ \tau $ | , | Tau. |
but | $ \sigma $ | -polystable . So we do not have generic indecomposability in | Sigma. |
this is because we made no assumptions on the regularity of the distribution of | $ ( X_t , Y_t ) $ | ; for a smooth density , it seems plausible that the estimator for conditional expectations is better behaved for small | X sub t, Y sub t. |
denotes summing over lattice neighbors laying inside and outside of domain | $ \mathcal { D } $ | , respectively . Thus , Eq.~ | D |
whatever the deme | $ i $ | where it started . | i |
$ f ( X , V , Z , Y , R ; t ) \vert_ { t=0 } = f ( X , V , Z , Y , R ) $ | . We use | f of X, V, Z, Y, R; t, evaluated at t equals 0, equals f of X, V, Z, Y, R. | |
one obtains the usual second Friedmann equation as : | $ \dot H = - 4\pi G \left ( \rho + p \right ) $ | . Integrating this immediately leads to the first Friedmann equation , | dot H |
Such a change is possible in the large- | $ N $ | limit | N |
and | $ \rho_ * $ | is absolutely continuous with respect to | rho star. |
Multiplying | $ S ( \pmb { v_i } ) $ | by a vector which represents a mass distribution , we can derive a new mass distribution transported by the velocity | S of v sub i. |
are projection matrices which are different for every layer | $ l $ | . % We refer to this as Token based attention since the attention scores are computed based on transformer output in the previous layer which in turn are dependent on | l |
400 | $ \pm $ | Plus or minus. | |
over the last | $ 5 $ | trials yields | five |
$ \eta=0.01 $ | . Let us recall that the hyper-parameter search based on F1~score does not include | Eta equals zero point zero one. | |
$ for all $ | 1 | For all. | |
in the second family because | $ z_2 $ | vanishes in this case . | z sub 2 |
that can be transformed via a suitable permutation | $ \textbf { P } $ | to the form of | P |
and label | $ y\in\ { -1,1\ } $ | . Let | y belongs to the set of negative one, one. |
$ \mathbf { I } $ | and | I. | |
and consider the | $ d $ | -dimensional zero mean Gaussian distribution at each timestep | d |
augmenting | $ \beta $ | proves conducive to enhancing the neural network 's ability to capture non-linearities , albeit occasionally at the expense of generating non-zero predictions for Boundary 1 . | Beta. |
$ \theta_n $ | , and | Theta sub n. | |
The irrational | $ \alpha_x $ | and | Alpha sub x. |
the minimum in PVTEC for the GITM simulation occurred earlier at | $ 23 $ | : | Twenty-three. |
where | $ q_1 , q_2 $ | and | q sub 1, q sub 2 |
-path in | $ H $ | has cost at most | H |
and | $ p ( \cdot ) $ | is a marginal density of covariates | p of dot. |
and | $ V_\perp $ | , which are huge in the frequently encountered case where | V sub perpendicular. |
-simplices of | $ B_d $ | are of the form | B sub d. |
To characterise the growth of the critical-line firehose when | $ \beta_i \gg 10^ { 6 } $ | , we set | Beta sub i much greater than 10 to the power of 6. |
and | $ B_2 $ | . For convenience , we call it | B sub 2. |
$ E_i $ | and | E sub i. | |
with | $ \bar\zeta = \bar\beta m $ | , | The bar zeta equals bar beta m. |
but also requiring smallness or structural conditions on the terminal cost | $ G $ | . The current work uses the fully probabilistic approach based on forward-backward propagation of chaos to derive quantitative mean field game convergence results . The main novelties of our results | G |
the degree- | $ 8 $ | manifold covers discussed above ) . Then the systole of | eight |
$ B\rightarrow C $ | and | B goes to C | |
We also confirm the same rule for | $ \hat { v } _ { \pm } $ | as | v hat sub plus or minus |
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