context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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displays one way of coupling | $ D $ | scalar multiplets in a way consistent with invariance under SO ( | D |
is in the conformal class | $ X $ | , independently of | X |
-hardness for recursively enumerable graph classes | $ \mathcal { H } $ | of unbounded treewidth~ | H: None |
and | $ \theta=80^\circ $ | with the mass and the angular momentum of the CO chosen as | Theta equals 80 degrees. |
and is connected to the EF wavefunction by | $ \pi_ { k } =\Re ( \hat { p } _ { k } \psi/\psi ) -\hbar A_ { k } $ | , where | Pi sub k equals the real part of the hat p sub k psi divided by psi minus h-bar A sub k. |
% with MLPs denoted as | $ f_T $ | for prompt | f sub T |
We compute the source current by calculating the charge | $ Q_S $ | emanating from | Q sub S. |
We first recall that if | $ Y $ | is a random vector distributed according to the Dirichlet distribution with scale parameter | Y |
cm | $ ^ { -1 } $ | , with | to the power of negative one. |
the local degree of | $ 0 $ | is the same for | zero |
closure of | $ \mathcal { Q } $ | capital Q | |
up to linear order in | $ \phi $ | we can say that all of the components in | phi. |
08 | $ \pm $ | 0.28 & | Plus or minus. |
is a fixed partial configuration on | $ \Lambda $ | ; i.e . , a spin assignment from | Lambda. |
and | $ t_2 $ | entries from | t sub 2 |
% where the implicit constant depends on | $ q $ | only . This implies % | q |
by substituting | $ i=0 $ | above , while the corresponding future input matrix is defined as | i equals zero. |
say | $ \BigO { n^ { -2 } } $ | or | Big O of n to the power of negative two. |
corresponds to a | $ B^- $ | -invariant prime divisor | B to the power of negative. |
$ x $ | ) | x | |
annihilate one atom in the atomic level | $ \omega_k $ | and create one atom in the atomic level | omega sub k. |
Sale takes place if | $ v_i\ge s^i $ | for some | v sub i greater than or equal to s superscript i |
and | $ x_0^i \in \ { 0,1 \ } $ | , for | x sub zero to the power of i belongs to the set containing zero and one. |
60 & | $ -2.02 $ | & | Negative two point zero two. |
denote the probability of | $ h $ | appearing as this histogram from | h |
and | $ \ell^\infty $ | norm . | ell infinity. |
$ D $ | is a continuous distribution . % Equation | D | |
is a finite subset of | $ \mathcal { A } _k $ | such that | A sub k. |
is a small perturbation of | $ X $ | that does not satisfy ( Adep ) . | X |
which is however always shorter than the recovery time | $ T $ | ( see Fig.~ | T |
is a linear combination of | $ D A^T B a_j $ | and | D A transpose B times a sub j. |
as the varying of | $ k_ { x } $ | under | k sub x |
-regular slice for | $ G $ | on | G |
is the quotient of | $ S^ { n+1 } $ | by the antipodal map , it is also natural to study minimal hypersurfaces in | S to the power of n plus one. |
If | $ x_3 = \frac { J_1 J_3 } { c_1 } $ | , then we obtain the following three families of solutions : | $x_3$ equals the product of $J_1$ and $J_3$ divided by $c_1$. |
& | $ ERR_1 $ | & 27.43 | Ee-R-R one. |
can not produce a CNG since the hypersurfaces | $ \mathscr { H } =\ { r=r_ { \pm } \ } $ | are closed totally geodesic submaninfolds by Prop . | The set H is equal to the set containing r which is equal to r sub plus or minus. |
we do not compute the vector | $ \bm { b } $ | but rather we infer the transformation matrix | b |
larger | $ \ell $ | means longer inflation , with the total number of inflationary e-folds given by | lowercase L. |
est fidèlement plate sur | $ R $ | modulo | R |
in the large | $ \ell $ | limit , we get | lowercase L. |
where | $ o_i $ | signifies the | o sub i. |
-300 | $ \leq $ | Less than or equal to. | |
Given a congruence | $ \equiv $ | over | is equivalent to. |
and that | $ D ( \bar\tau ) \neq \phi^ { \bar\tau } ( C_ * ) $ | . The limit | D of tau is not equal to phi to the power of tau of C star. |
It follows that the length of | $ \eta ' $ | is at least | Eta prime. |
each with mean | $ \mu_u $ | and transformed variance | mu sub u. |
the Hugoniot curve intersects | $ \mathcal { C } $ | at two points , while if | see |
022 | $ \pm $ | 0.0037 & 0.017 | Plus or minus. |
em | $ c = 1 $ | , | c equals one |
is approximately | $ 1.95 $ | K. Notably , this temperature range aligns with the feasible mass spectrum of neutrino eigenstates , thus rendering neutrinos in their current non-relativistic state . The transition from relativistic to | 1.95: One point nine five. |
asymptotic bounds in | $ d , t , n $ | for all | dee, t, en |
Details of TRA operation in stage | $ j $ | can be formulated as follows : | j |
Note that | $ \beta\geq 0 $ | due to the parametrization in | Beta is greater than or equal to zero. |
with | $ s $ | larger . In the particular case when | s |
$ L_2 $ | and | L sub 2. | |
being applied to halos with low | $ W $ | . Equation | W |
Then there exists a smooth | $ U ( 1 ) $ | -action on | U of 1. None. |
which possesses several stable solutions for certain positions in the | $ f_d $ | - | f d |
$ in the DFM setting is exactly the $ | i | in DFM set is exactly | |
we split the vertex | $ a $ | and the vertex | a |
group | $ G $ | is called { | G |
and | $ \xi_ { g_ni_f } $ | , which appear at different scales and do not sum to zero as in the contaminated correlation function . Additionally , they now share the same sign , another consequence | ksi sub g sub n i sub f. |
extends | $ X $ | in | X |
since | $ f=\max\ { 0 , \cdot\ } $ | in this case , while an additional initial datum can be chosen so that | f equals the maximum of 0, dot |
for all | $ n $ | . The approximation in Eq . ( | n |
on the MLC benchmark datasets measured by P @ | $ 5 $ | . | five |
the chiral effective theory is dominated by the axial | $ U ( 1 ) $ | anomaly for the | U of 1. None. |
but for | $ \beta $ | Dor . Effective gravities below zero are unphysical and are plotted with a cross at the lowest observed | Beta. |
with | $ \varPhi_n ( s_n ) =x_n $ | . By | The variable phi sub n of s sub n equals x sub n. |
is monotone increasing with respect to | $ z $ | . Since monotonicity implies Borel measurability , | z |
and | $ K_3 $ | for each of the three two-particle channels | K sub 3. |
and choosing | $ \gamma^\ell , C^\ell , \mu^\ell \sim \epsilon^\ell $ | , it follows that | gamma to the power of ell, C to the power of ell, mu to the power of ell similar to epsilon to the power of ell. |
of | $ e $ | -folds . | e |
The sigma and | $ \omega $ | meson coupling has been introduced to obtain a reasonable value of incompressibility coefficient | omega. |
The wavefront correction for large apertures and large stellar separations amounts to canceling the | $ x $ | - and | x |
the limit | $ p\to\infty $ | should be related to West/East strips operators . We thus give an explicit solution to the harmonic problem in a simple case , see Propositions | p implies infinity. |
have rank | $ 1 $ | . Hence , we can write it as a direct product of two spinors , | one |
then | $ \Short ( m ) =\ { 1\ } $ | , and therefore | S hort of m equals the set containing 1. |
is a normalized version of | $ f_j $ | obtained by subtracting the mean on the train set and dividing by the standard deviation . The adaptation is obtained assuming that a state-space property is satisfied . Precisely , | f sub j. |
of decision rules from | $ S $ | satisfying the following conditions : | S |
It then sorts the test anomaly scores and selects a threshold | $ \delta $ | such that a proportion | delta. |
: We train our merging-refinement subnet | $ \mathcal { M } $ | , with both the pre-trained PST network and | M |
The peak in the causal kernel at | $ t=0 $ | is to compensate for the small bit of information that lies in the non-causal half . The existence of such sharp peaks in the time-domain kernel implies an increase in | t equals zero. |
the microscopic mechanism of the polarization-dependent photocurrent contributions proportional to | $ P_ { \rm lin } $ | and | P sub lin |
is | $ \mu $ | -strongly-convex w.r.t . | mu. |
$ 2785_ { ~\pm~75 } $ | 2785 sub plus or minus 75. | ||
and is no greater than | $ -3 $ | for any | Negative three. |
is then a complex bundle on | $ X $ | of rank | X |
For any MA | $ \vec { B } $ | , | vector B |
conditioned on an indicator | $ i $ | . | i |
where | $ \pi $ | is the equilibrium Boltzmann distribution . We can notice that | pi. |
the power spectrum approaches | $ O ( 1 ) $ | , whereas for the large-scale modes with smaller | O of 1. |
Therefore we have for any | $ z\in \sZ $ | z belongs to the set of integers. | |
is representing the average clustering coefficient of graph | $ G $ | , then numerically we candefine : | G |
The theorem states that when the size of the dataset | $ | { \cal X } | $ | and the number of required labels | The equation you provided is: the cardinality of set 𝒳. |
for non P-ideals and | $ \adds ( \I ) \geq \omega_1 $ | for P-ideals . % % % % % % % % % % % % % % % % % % % % % % % % % % % | Adds of I is greater than or equal to omega sub 1. |
Let the channel connecting | $ v_i $ | and | v sub i |
we are now ready to establish | $ \underset { \gamma \to \infty } { \lim } \chi ( \gamma ) $ | as well as the convergence rate : | The limit as gamma approaches infinity of the limit of chi of gamma. |
$ A $ | and | A |
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