context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
|---|---|---|---|
di- | $ J/\psi $ | , | J psi. |
% % % A mixed spin- | $ j $ | state | j |
of matrix | $ B $ | are deleted , otherwise the rows and columns are repeated | B |
We define | $ B $ | and | B |
with | $ \Delta_0 ( t ) =0 $ | , see Fig.~ | Delta sub zero of t equals zero. |
and | $ \Omega $ | , representing the total angular momentum of the metal ions and its projection onto the inter-atomic axis , respectively . In particular , we labelled the electronic states using the | Omega. |
$ \epsilon_i $ | , | epsilon sub i. | |
for | $ r < r_N $ | . The normalization constants | r is less than r sub N. |
% Also by Taylor 's Theorem there exists | $ \overline { x } _2 $ | such that % | x-bar sub 2. |
for small | $ W $ | , it is better to use the triangle inequality than Lemma~ | W |
in lieu of | $ \gamma ( t ) $ | . The space | Gamma of t. |
$ in a Young diagram with respect to the partition $ | in a new diagram with respect to the xi. | ||
has no pole in the upper complex | $ p_1'^- $ | plane , the CLFQM used here and the standard light-front formulism where the covariance property is not satisfied , will lead to identical results at the one-loop level . We | p sub 1 prime |
$ G $ | ) over | G | |
% These correlations are revealed only at low | $ Q^2 $ | , close to the pseudothreshold , and can not be ignored in the parametrizations of the data . % The relevance of the pseudothreshold constraints is manifest very close to | Q squared. |
Experiments on the CCPP dataset with | $ N=4 $ | are depicted in Figure~ | N equals four. |
We say that | $ X $ | is a Frolik space if | X |
we provide the calculations for | $ e_X $ | as : | e sub X |
that is recursively defined for each subtree | $ T_H $ | of | T sub H |
does not depend on | $ \ell $ | and are the same for both | lowercase L. |
where | $ \mathcal { C } $ | is a level , | see |
Nuclear modification factor | $ R_ { AA } $ | and anisotropic flows | R sub A A. |
We know that each | $ X_l $ | is a | X sub l. |
and the | $ \mathcal { J } $ | J | |
we think of a | $ \varphi $ | -dependent function | phi. |
equation % Each | $ \sigma_i $ | is summed over | Sigma i. |
there exists | $ i $ | with | i |
satisfying the conditions in Proposition | $ 7.2 $ | , with | Seven point two. |
as the ratio of the final and the initial pericenter distance is of order of | $ \sim 10^ { 0 } $ | in most scenarios . | sim one zero to the power of zero. |
where | $ ( . ) ^H $ | , | open parenthesis dot close parenthesis raised to the H power. |
evenly located on the line of | $ h_2=0 $ | from | h sub 2 equals 0 |
$ \pm $ | 0.00 & | Plus or minus. | |
up to the limit of | $ 0.8 $ | we considered . For inclination smaller than | 0.8: None |
is the Burgers tensor for slip system | $ k $ | . | k |
be a fuzzy set in | $ X $ | and let | X |
it is noted that | $ C_4 $ | has some effects on the value of mass of gravastar . Strickingly , with the determination of the mass of the gravastar , we have found that the viable condition | C sub 4. |
the non-linearity of the operator lies in | $ \mathcal { H } _ { g } $ | and | H sub g. |
per unit time injected into the outflow versus the luminosity of the central source | $ L $ | L | |
with | $ B-L $ | charge | B minus L |
and we disregard the second term that contains the contributions from all the cavity modes different from the initial cavity mode at | $ \tau=0. $ | So , in this case the coherent coupling between different cavity modes due to the movement of the membrane is negligible . We assume that at | Tau equals zero. |
we compute the local degree of | $ f^k $ | at | f to the power of k. |
for | $ \tau \in [ 0,1 ] $ | . Each map | Tau belongs to the interval from 0 to 1. |
let the number of instances where | $ F $ | was applied on any dataset ( as observed in query logs ) be | F |
independent of | $ n $ | . % % Then we have , for all | n |
even at the physical value of | $ m_c $ | . At N | mc: None |
The method permits ion localization more than an order of magnitude below the ion spacing and delivers about | $ 94\ % $ | state-readout fidelity . The techniques employed here apply to large ion crystals in Paul and Penning traps and neutral atom arrays in optical lattices . One exciting opportunity pertains to | 94 percent. |
The parameters of | $ f_t $ | are updated using the exponential moving average of the parameters of | f sub t |
means the set | $ \ { 1 , \ldots , n\ } $ | . We write | One through n. |
em The synthetic light curves made in the | $ J $ | - , Water Absorption , and | J |
and | $ U $ | be Euclidean spaces , i.e . , | U |
em The total weight on | $ 4 $ | -cycles with vertices | four |
\ee where | $ z $ | is the distance from the Galactic plane , | z |
in the case where | $ \forall t\geq 0 $ | , | Spoken English: For all t greater than or equal to zero. |
Then the | $ r $ | -th sequential parametrized topological complexity | r |
and | $ \langle \mathcal { O } _ { 19 } ^2 \rangle $ | . | Angle O sub 19 squared |
\small | $ b $ | } ; | b |
the dendrite rearrangement group | $ G_n $ | contains isomorphic copies of the alternating group | G sub n |
$ \sigma^ { n } $ | is denoted as the number of whole proposals multiplied with a factor | sigma raised to the power of n. | |
a sequence of conformal Willmore immersion | $ ( \Sigma , h_k ) \to \R^3 $ | satisfying | Open parenthesis, capital sigma, comma, h subscript k, close parenthesis, arrow, capital R superscript 3. |
$ \theta $ | and | Theta. | |
Tuning | $ d_c $ | gives | d c. |
for small | $ \varepsilon > 0 $ | , | Epsilon is greater than zero. |
at round | $ t $ | . The goal is to design algorithms that perform as well as the best fixed set of | t |
assume that | $ T $ | is NIP . Thus on every indiscernible segment , for any increasing or decreasing sequence | T |
While the overall | $ d $ | -dependent trends are in agreement , the simulated values for | d |
In small- | $ x $ | formalism , the initial TMD quark distribution can be seen from dipole-nucleus multiple scattering and small- | x |
\lambda | $ } $ | and | closing brace |
but does not know | $ \theta_0 $ | , could estimate | Theta zero. |
$ \cos ( \cdot ) $ | function and the combination | cosine of something. | |
torsion-free sheaf | $ F $ | of degree | F |
Let | $ \Delta $ | be an effective | Delta. |
& | $ 3 $ | constraints | three |
or we could realize the set of | $ \on { SL } _3 ( \ { \mathbb\ Z ) $ | -orbits on | L S sub 3 of Z |
we can therefore assume that there is some | $ n \in \N $ | and a finite subset | n belongs to the set of natural numbers. |
$ o_y $ | , | o sub y. | |
every existential variable in | $ C $ | plays the same as | C |
aims to estimate the noise | $ \theta $ | in the latent space with the aid of timestep | Theta. |
all procedures | $ \mathcal { P } $ | , and all medications | P in calligraphic font. |
was reserved for the specification of the value of | $ N $ | . Here , since we always assume | N |
as a function of the reflection angle | $ \theta_ { \rm r } $ | is shown in | Theta sub r. |
the weights | $ \beta_1 $ | , | Beta one. |
Although the recurrence in this definition takes | $ \bigO ( N^2 ) $ | time to evaluate for each | Big O of N squared. |
of | $ S_F $ | , itself corresponding to a face | S sub F. |
$ \mathscr { Q } $ | will always denote the closed unit cube | cue | |
The presence of H | $ \alpha $ | emission can be reddened sources | Alpha. |
When | $ d = \mathcal { O } ( T^ { \alpha } ) $ | with | d equals big O of T to the power of alpha |
51 & WRAY 16 | $ - $ | 92 & [ WC 4 ] & & & & & & & & & & 69 | minus |
such that for every | $ \varepsilon > 0 $ | it holds that % | Epsilon is greater than zero. |
& 737 | $ \pm $ | 6 & 736 | Plus or minus. |
and | $ b \colon B \mono X $ | of | b, colon, B, monomotive, X |
& | $ O ( d ) $ | O of d. | |
into sets | $ J_1 , \ldots , J_n $ | such that the pattern induced by | J sub 1, dot dot dot, J sub n. |
Up to multiplying both sides by | $ x^ { -1 } $ | on the left and re-labelling , it suffices to study intersections | x to the power of negative one. |
and | $ L_i $ | . How to fix | L sub i. |
and | $ p $ | are predicted and ground truth within-class points , | p |
& | $ -1.5990156188560014468 $ | & | Negative one point five nine one six eight five six one four six six. |
the simulations for each field had to be run multiple times with different random seeds to generate | $ \mathcal { O } ( 10 ) $ | detectable events in total . The final numbers shown below were then scaled down by a factor of | big O of ten. |
We thus have that | $ \bar\vv_ { \pi ( B ) } $ | is bijective . It is a diffeomorphism because | v bar sub pi open parenthesis B close parenthesis |
the agent gets a bonus cost of | $ -100 $ | for one time step . If the distance rises above | Negative one hundred. |
we can obtained the autonomous equations for | $ u $ | and | u |
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