context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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follows from the unicity of | $ \beta. $ | Beta. | |
as | $ E_ { 11 } $ | commutes with | E subscript 11 |
and | $ \mu $ | are the variable density and variable viscosity of the mixture , | mu. |
A noble set | $ U $ | } % | U |
damping from the same result for | $ C^ { \infty } $ | damping . | C to the power of infinity. |
which emerges near the | $ 2\rho_0 $ | mark . The charge neutrality condition imposed on the neutron star matter suppresses the presence of positively charged | Two rho zero. |
& asymptotic equivalence denoted | $ A \sim B $ | means | A is similar to B. |
$ \mc G_i $ | contains no more information than | m c G sub i | |
$ { \cal S } _ { \chi } $ | , and | cal S sub chi | |
& | $ 868 $ | [ MHz ] | Eight hundred sixty-eight. |
smoothly decreasing with growth of | $ n_r $ | functions with maximum at | n sub r |
expressions for the partition functions of free fields in | $ 1+1 $ | dimensions , emphasizing the interpretations of the various sectors in terms of gauging discrete symmetries . For a free scalar in | One plus one. |
% of a Gaussian of width | $ W $ | modulated by one of a | W |
is | $ \phi^ { * } $ | and one gets the average of measurement outcomes | phi star. |
with respect to | $ \mathbf { x } $ | . The measurement function | ex |
$ B=X_t $ | , and | B equals X sub t. | |
and | $ H $ | , } | H |
where | $ \balpha_ { Rgj } = ( \alpha_ { Rgj1 } , \ldots , \alpha_ { RgjR } ) $ | and | Alpha sub Rgj equals open parenthesis alpha sub Rgj1, ellipsis, alpha sub RgjR close parenthesis. |
and | $ R_3 $ | plane , | R sub 3. |
arises as a deformation of | $ \delta_0 $ | , to specify signs in | Delta zero. |
namely the sequence | $ ( v , v_1 , v_2 , \dots , v_k , u ) $ | . Due to the | v, v sub 1, v sub 2, dot dot dot, v sub k, u |
The function | $ f $ | is said to be | f |
then there is some vector | $ v $ | such that | v |
is a simple and straight arc on | $ S $ | . By ( | S |
can be obtained exactly by the choice of | $ S_+=0 $ | or | S plus equals zero. |
4 | $ \pm $ | 0.1 & 0.414 | Plus or minus. |
in | $ \mathbb { R } ^n $ | , we mean an orthogonal set of vectors with unit length . In light of | R to the power of n. |
0 | $ \pm $ | Plus or minus. | |
from | $ \sf Q_0 = Q $ | . % | Q sub zero equals Q. |
$ p $ | , and | p | |
but localizes in a nearly two dimension `` fault '' at some particular value of | $ z $ | . This is qualitatively observed in the localization of the avalanches as observed in the lateral view in Fig . | z |
$ } $ | denote the facet of the reduced Bruhat-Tits building of | closing brace | |
$ b_\Gamma $ | , and | b sub gamma | |
For any | $ 0 < \lambda_1 < \lambda_2 , $ | this is a subspace of | Zero is less than lambda one, which is less than lambda two. |
Then we construct the | $ 4 $ | th-order polynomial equation as | four |
$ \tilde { T } \to 0 $ | on zero-mean functions . It is interesting to note that the right-hand side in | tilde T goes to zero. | |
and | $ p_ { \bot } / ( \rho|H| ) $ | respectively . Using | p sub bot divided by open parenthesis rho absolute value of H close parenthesis. |
$ & $ | d_ { rel } | and | |
be the first unreleased request in | $ \pi $ | at time | pi. |
so that the Noether charge derived from | $ \widetilde { S } $ | can be conserved without any flux on the boundary . From the modified Lagrangian density : | S tilde. |
denotes the map taking each element of | $ J $ | to the identity of | J |
for the decay | $ B_c^+\to \eta_c ( 3S ) \pi^+ $ | , its branching ratio was predicted as | B sub c plus goes to eta sub c in the 3S state plus pi plus. |
$ [ n ] $ | denotes the set | n in brackets. | |
where | $ \deg ( F ) =m $ | . In addition , | Degree of F equals m. |
The external input | $ u $ | is the external force acting on the first mass | u |
eigenstates may not even exist in the thermodynamic limit and more broadly an inner product on the corresponding Hilbert space | $ \mathcal { H } $ | is not well-defined | H: None |
as | $ [ B , X ] =0 $ | . We have thus found | Bracket B comma X equals zero. |
turning on the | $ \Omega $ | -background on | Omega. |
averaging these stratified bounds weighted by | $ p ( s ) $ | . Clearly , these bounds reduce to those in Equation | p of s |
because the mean detection rate | $ R $ | is fixed for any given | R |
since the scattering rate | $ W $ | derives from an underlying hamiltonian dynamics , also the kinetic energy and the momentum are summational invariants : | W |
such that | $ \theta|_L=\id_L $ | and | Theta, given L, equals identity sub L. |
we let | $ G $ | be the set of nodes in a network . Each node represents an individual , with a single heritable type . As a simple model of collective action , suppose | G |
one additionally needs that each | $ G/G_i $ | belongs to | G over G sub i. |
The | $ \delta $ | -pulse leads to mathematically ill-defined expressions , but these obstacles can be overcome by making use of the so called Colombeau theory of generalized functions | delta. |
Let | $ B $ | be a positive domain in | B |
only the operator | $ H_\nu^ { ( 0 ) } $ | with | H sub nu superscript 0 |
such that for any conjugacy class | $ \alpha $ | not carried by | Alpha. |
& | $ L_ { \sun } $ | L sub sun. | |
The information on the composition of and how rapidly these heavier states decay is contained in the density of states | $ \rho ( m ) $ | that arises as the solution to the bootstrap condition , | rho of m. |
we can safely conclude that NC is equavelent to the energy score | $ E $ | in this situation . | E |
where | $ \eta $ | is again the conformal time while | eta. |
and | $ \mu=0.1 $ | the models span the range of | mu equals zero point one. |
we let | $ \pi=0.1 $ | , and we estimated the number of common factors | pi equals zero point one. |
Whenever the current policy | $ \pi_i $ | executes an action | pi sub i. |
After replacing | $ \overline { S } $ | with a smooth finite covering we may reduce to the analogous problem for a | the bar over S. |
which obey | $ \gamma \sigma \gamma^ { -1 } = \sigma $ | for any | Gamma sigma gamma to the power of negative one equals sigma. |
& | $ 462 $ | & | 462: None |
a decay rate in strain energy of | $ r^ { -4 } $ | , predicted from linear elasticity theory . Inset : strain energy decay with radial distance in Network 1 for disordered network with | r to the power of negative four |
The interaction term can be decomposed into three parts according to the spherical tensor components of | $ \tensorial { J } _ { ij } $ | , namely | tensor product J sub i j |
for | $ G $ | as | G |
to the case where the image of the singular locus | $ S^ { \sing } $ | of | S raised to the power of sine g. |
the elements with prices higher than | $ q_ { \max } $ | . Then | q sub max |
with | $ x_r \in C $ | , but no Pareto-optimal search point . % % The proof of this theorem is not completely identical to the proof for theorem 2 since an individual | x sub r belongs to C. |
% As Casas-Ibarra parametrization involves recasting the Yukawa couplings in the Dirac mass term | $ M_D $ | using the PMNS matrix , the diagonal mass matrix for both light neutrinos and heavy Majorana right-handed neutrinos , and an orthogonal matrix | M subscript D. |
with respect to | $ L $ | and | L |
it is found that the | $ \Lambda $ | hyperon is | Lambda. |
Suppose that the first | $ n_ { * * } $ | interjump times are within | n sub star star. |
09127493 | $ i $ | i | |
when the question is | $ q $ | , which can be calculated as | q |
then | $ T\leq G \leq \mathrm { Aut } ( T ) $ | for some non-abelian simple group | T less than or equal to G less than or equal to Automorphism of T. |
such that | $ \aut ( P ( G , S ) ) $ | is a Cayley representation of | Automorphism of P of G and S |
847 | $ \pm $ | 0.29 | Plus or minus. |
in | $ N $ | must be at least 2 . Hence , | N |
the reflection matrix of the | $ m $ | th RIS is defined by | m |
and subgroup | $ H $ | the multiplicative induction functor | H |
is a copy of | $ X $ | consisting of `` primed '' variables ) . | X |
is satisfied when | $ p_i $ | r-delivers its own broadcast at | p sub i. |
belonging to class | $ k $ | which are classified as belonging to class | k |
and | $ y $ | directions are | y |
is smaller than a threshold value | $ \epsilon_\text { thr } $ | , where | epsilon sub thr |
-local algorithm for independent sets is the function that returns | $ 1 $ | if the label of the root is strictly smaller than the label of each of its neighbors . If | one |
as a function of | $ k $ | , and | k |
a stacked RNN architecture passes hidden state outputs from the layer | $ l $ | , | l |
\beta^+ | $ ) $ | ^ { 26 } | closing parenthesis |
For | $ i=0 $ | , this follows from Lemma | i equals zero. |
% introduce event camera Once the change in intensity | $ \mathcal { I } $ | at a pixel between the current moment and the moment of the last generated event at that pixel surpasses the contrast threshold | capital I. |
is strong in | $ L^2 $ | on | L squared. |
The attention scores | $ \alpha_i $ | for each catalog entity are calculated using the scaled dot-product attention mechanism | Alpha sub i. |
We demonstrate how to quickly compute a matrix of maximal rank for which | $ \varphi $ | has a positive multigrading . Then in each graded component we compute the minimal generators of the kernel in that multidegree with linear algebra . We have implemented our techniques | phi. |
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