context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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where | $ \gamma $ | is a curve parametrized by the arc length and | Gamma. |
We have extended | $ m $ | to be an even function of | m |
at the grid | $ \Omega $ | the | Omega. |
edges for | $ c\in C $ | . For | c belongs to C |
and | $ \Ycal_0^\square $ | , however , the formulations introduced below can be adapted easily . We first define the direct optimization problem to find design | Y sub 0 raised to the square. |
We complete the definition of | $ A $ | by defining | A |
The source with an | $ I $ | -band baseline magnitude | I |
-1 | $ ( $ | opening parenthesis | |
& | $ * $ | & 39 | asterisk |
$ { \cal A } $ | is the Lebesgue measure . Then , | A calligraphic A. | |
specifies that region | $ y_3 $ | should be eventually visited , as in Fig . | y sub 3. |
$ \theta $ | with respect to the intrinsic orientation | Theta. | |
and | $ e^ { -\frac { \alpha \pi } { 2 } i } $ | . | e raised to the power of negative alpha pi divided by 2 times i |
with unique solutions and identified as the | $ 2_2^+ $ | and | Two subscript two superscript plus. |
is localised at | $ g $ | . | g |
Although the proposed scheme requires the knowledge of | $ \alpha $ | when each human sensor updates the belief regarding the behavioral identity of the corresponding physical sensors , this knowledge is not necessarily needed to guarantee a good performance . Choosing | Alpha. |
for a given level | $ n_1 $ | , | n sub 1 |
Measurement of | $ \tau $ | decay products is a way to measure polarization , which in turn makes it possible to measure | Tau. |
higher graph | $ 4 $ | -manifold | four |
$ \gamma $ | of the RBF kernel are optimized via grid search . Correspondingly , the | Gamma. | |
When | $ G\to\infty $ | , these estimators are again , uniformly consistent . Finally , construct a grid in | G goes to infinity. |
the CDW solution admits | $ M\to 0 $ | even if | M goes to 0. |
trivial braids and | $ 1000 $ | non-trivial braids . The line corresponding to CES1-AUT is only marginally higher than the line corresponding to CES2-CEP-AUT . This is because CES1 is equivalent to CES2 | One thousand. |
Let | $ \pi $ | be a disjunctive logic program and let | pi. |
for | $ \IP ( \Omega_2 ) $ | . For any independence polytope , | IP of Omega 2 |
we say that~ | $ u $ | is~ | u |
is defined geometrically via the area of the two-spheres coordinatized by | $ \theta , \phi $ | . For the Schwarzschild geometry , the two metric functions | Theta, phi. |
$ x_t $ | is sampled from a Gaussian distribution with mean | x sub t. | |
and~ | $ c_2 $ | % in the parallel composition | c sub 2 |
A ` typical ' copy of | $ F_2^ { 2N-3 } < \ { \rm\ { Aut ( F_N ) $ | has one direct factor consisting of inner automorphisms from a rank two free factor | F sub 2 to the power of 2N minus 3 is less than the automorphism of F sub N. |
-norm ; % therefore | $ ( u_ { 1 } , u_ { 2 } ) $ | can be seen as the best representation for | u sub 1, u sub 2 |
by contractibility of | $ { f } $ | we obtain | f. |
be the set of functions in~ | $ \Omega $ | with range~ | Omega. |
the higher | $ \eta $ | value is , the more chemically stable the molecule is and , therefore , harder the rearrangement of its electrons~ | eta. |
we expect that the height of the peak for | $ T=T_c $ | is reduced by the presence of a higher noise that induces another maximum of the | T equals T sub c. |
to | $ 0 $ | implying that | zero |
palindrome-pattern on | $ u $ | } . Let | u |
remains almost unaltered with | $ \alpha $ | while its increase is drastic when | Alpha. |
is a volume of lossy dielectric objects described by | $ \epsilon ( \mathbf { r } , \omega ) $ | , | Epsilon of r, omega |
$ \widehat { \Theta } $ | wide hat theta | ||
we first set the element at position | $ j $ | , | j |
dominates for large | $ N $ | . Our goal is to study the localisation features of this system for large | N |
and | $ u $ | , respectively , forming edges | u |
20 | $ & $ | 24 | and |
e | $ 0.8 $ | , indicadas entre parênteses . Observa-se que o desempenho da técnica Simultaneous Learning , utilizando os dois datasets auxiliares , foi significativamente superior ao baseline e ao dropout , apresentando | 0.8: None |
is the | $ p $ | -value of the | p |
on | $ X $ | is nondegenerate . | X |
and | $ \mathcal { E } _1 \cap \mathcal { E } _2 = \emptyset $ | , | E one intersection E two equals empty set. |
$ ^ { 2 } $ | and Third Author | squared. | |
and | $ 0 $ | otherwise . Then | zero |
where each vector has dimension | $ m $ | . Each vector | m |
-strongly compact cardinal for any | $ \kappa \in \mathcal { K } $ | , and there is no strongly compact cardinal . | kappa belongs to the set K. |
Then there are no more possible pairs of | $ \pm $ | ahead of the exponent and squared momenta so we have to write down the term for the propagator with momentum | Plus or minus. |
the constant | $ \delta\in ( 0,1 ) $ | only depends on the aforementioned properties of the shear flow | delta belongs to the open interval from zero to one. |
an | $ \epsilon $ | sequence | epsilon. |
decay amplitude we work consistently to first order in | $ F/f $ | , so the above mass corrections can be ignored . We will also treat isospin-breaking effects in the quark-mass difference | eff over eff. |
where | $ B_c $ | depends on | B sub c. |
An internal discontinuity | $ \Gamma $ | is in the body and the time-dependent Dirichlet boundary condition | Gamma. |
$ \mu $ | T for the same value % of | mu. | |
under the same hypothesis there exist | $ \gamma_1 , \gamma_2 > 0 $ | such that | Gamma one, gamma two greater than zero. |
has | $ \im ( \rho_k ) = \Mod_ { ( k ) } ( \Sigma_1 ) $ | . We then use the higher Johnson homomorphisms of Morita to show that the group | The imaginary part of rho sub k equals the modulus of k times Sigma sub 1. |
This means if | $ p\leq p_0 $ | , all predictors being used are informative , whereas | p less than or equal to p sub zero. |
Alice 's expected payoff in this MAID-CE is | $ 3.5 $ | , and Bob 's is | 3.5: None |
The convergence conditions are set at | $ 1e-6 $ | for all residuals . The discretized momentum equations and pressure equations relative to the set of control volumes are solved with an implicit solver that ensured stability of the numerical | one e minus six |
and | $ a_ { ji } $ | . This definition enables us to assign multiple graphs to the same adjacency matrix , as can be seen from figures | a sub j i. |
and use the previous lemmas to write recursive inequalities for the quantities | $ u_k $ | and | u subscript k. |
so that | $ \abs { \d w_ { n } ( z_ { n } ) } $ | is unbounded , then we will add a point to | Abs of d w sub n of z sub n |
are two | $ \mathfrak { sl } ( 2 ) $ | generators , | s l two. |
& | $ { \color { gray } [ -0.01 , \ , 0.003 ] } $ | & | Model gray, negative zero point zero one, comma zero point zero zero three. |
and for all | $ t\geq0 $ | , where | t greater than or equal to zero. |
$ \delta \tilde D_ { ij } ( l ) $ | and | Delta tilde D sub i j, in parentheses l. | |
and | $ \lambda_ { \rm eff } $ | , there is exactly one critical value for the combination | Lambda sub e f f. |
can turn into | $ f=R B_ { \phi } $ | which is related to the poloidal plasma current . | f equals R times B sub phi. |
-bit string | $ y $ | to Bob , any arbitrary binary function | y |
where the curvature vanishes is a closed subset of | $ M $ | . It is called the { | M |
$ \sym { \ell } \mid \sym { \ell+1 } $ | Sym ell, given sym ell plus one. | ||
and let | $ C $ | be the constant from Proposition | C |
Since | $ q_1 $ | is a constant coefficient operator in the region where the splitting | q sub 1. |
for large values of | $ d $ | , leading to the approximation for | d |
By letting | $ H'=H_ { t } $ | , we have another Hermitian operator | H prime equals H sub t. |
where | $ H $ | and | H |
be an | $ r $ | -graph with | r |
by | $ A $ | if there exists an accepting run for it , viz.~a sequence of configurations | A |
and eigenvalue | $ E^ { \mathbf { k } } _ { n } $ | . The amplitudes of the SC pairing and the orbital current order , as well as the electron densities , are obtained through the following self-consistent equations , | E raised to the power of k sub n |
$ y_i $ | denotes the ground truth value , and | y sub i. | |
& | $ 0.259 $ | 0.259: None | |
# # # # Eliminating an unstructured matrix We first consider the case when no structure in | $ A_ { 11 } $ | is exploited | A sub eleven. |
and | $ - $ | 7 days relative to maximum light , respectively . The high degree of continuum polarization can be explained by a global asphericity of as much as 15 | minus |
3 | $ \text { cm } ^ { -1 } $ | , which changes to a week Raman vibration of frequency 1584.08 | cm to the power of negative one. |
and | $ N^- $ | . Since | N to the power of negative. |
can be made into an | $ \infty $ | -functor | Infinity. |
whose cokernel is killed by | $ E $ | . | E |
em | $ O_2 $ | saturation in hemoglobin | O sub 2. |
for a given tolerance | $ \epsilon > 0 $ | , we have from | Epsilon is greater than zero. |
% | $ t_0 $ | being the starting time of the analysis in GPS time , | t sub zero |
the result could resemble | $ \epsilon $ | ~Cha or I~385 . If | epsilon. |
\\ \hline Fe & | $ < $ | 0.0002 & 4 & 2 & 4 & 1 & 187 | less than |
As an application we compute the asymptotic probability that a random simple closed geodesic on a hyperbolic surface of genus | $ g $ | with | g |
and for the four-manifold | $ M_4=S_0^ { \textup { an } } $ | underlying the analytification of | M sub 4 equals S sub 0 superscript a n. |
and | $ \ell_i=0 $ | . Hence , there holds % | ell sub i equals zero. |
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