context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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These effective contributions arise on both sides of the interface and indexes | $ L , R $ | denote the left or the right side . We consider an interface between two n-type semiconductors with a simple zone structure . We assume the bottom of the conduction band | L, R: None |
for a large interelectronic distance | $ r $ | , where | r |
particles should in some sense be close to the | $ N $ | -BMP , where selection keeps the population size exactly equal to | N |
The eventual goal is to solve for the completed tensor | $ \mathcal { T } ^ * $ | , which is calculated from the following : | T star |
D topological boundary states is referred to as an | $ n $ | th-order topological insulator . Unlike the Benalcazar-Bernevig-Hughes model , characterized by a bulk quadrupole moment , the 2D SSH model works as a HOTI without a multipole moment~ | n |
set of active consumers connected at bus | $ b $ | } | b |
We only consider those | $ \beta $ | such that | Beta. |
to a | $ p $ | -divisible group | p |
are given in terms of the coefficients | $ T_m $ | , | T sub m. |
it follows that | $ \chi_ { \ep } \varphi \to 0 $ | in | Chi sub e p, phi, goes to zero. |
is | $ ( k , \ell ) $ | -cup-cap-free , the point | k, ell |
where | $ N $ | represents the total number of simulated GW data , | N |
is a subset of | $ \mathcal { C } $ | and | see |
we can again use the observation that each | $ R $ | matrix is fully characterised by a rotation matrix | R |
fits for | $ H\beta $ | , [ OIII ] and [ NI ] , but allow H | H beta. |
% Consider now an | $ x\in B $ | visited by the random walk | x belongs to B. |
$ B=js^ { 3 } $ | , | B equals j times s cubed. | |
uniformly and divides the support into roughly | $ m $ | levels | m |
When | $ X $ | is a Hadamard manifold with pinched negative curvature , the geometrically finite subgroups of | X |
$ \mathscr C $ | -totally real with respect to the | see | |
we assume sharp interfaces and approximate the volume fraction profile~ | $ \phi ( x ) $ | as a box function , | phi of x. |
the model can be used for generating solutions | $ \widetilde { f } _ { \theta } ( z ) $ | , satisfying the non-linear constraints | tilde f sub theta of z |
$ , $ | comma | ||
The translation part of the extrinsic parameter between the frame | $ \ { V\ } $ | and the frame | V curly brace |
$ t_1 $ | in Figure | t sub 1 | |
in the singular cohomology groups of | $ X $ | . % For more details , we refer to | X |
so that we can have | $ \omega\rightarrow\infty $ | if needed . This constant is only used in Algorithms | Omega tends to infinity. |
corresponds to the trivial coloured cone | $ 0^c : = ( 0 , \varnothing ) $ | . We let | Zero raised to the power of c is equal to open parenthesis zero comma empty set close parenthesis. |
scale up and can exceed the electron mean free path at | $ B\rightarrow 0 $ | , in Fig . | B implies zero. |
$ s $ | -sparse | s | |
will have a copy that blocks | $ M ' $ | . % First observe , that for any blocking edge | M prime. |
is the X-ray flux in given energy band in ergs cm | $ ^ { -2 } $ | s | Negative two. |
solving this optimal search % Solving | $ ( P_ { \text { NN } } ) $ | is equivalent to identifying the nearest neighbor of | P sub NN. |
through the coupling operator | $ B $ | , this also restricts the regularity we can achieve for the wavefunction . The primary objective is to derive time-independent a priori bounds on | B |
where | $ N_a $ | is the number of parameters in | N sub a. |
based on the initial and final Hamiltonians | $ H $ | and | H |
depends on | $ \Delta $ | in a similar way as | Delta. |
Red | $ ( f_1 , f_2 ) $ | contains at least two independent singular conics . % The second assertion of the proposition follows from Lemma~ | f sub 1, f sub 2. |
% For | $ j=1,2 $ | , consider hypergraph | j equals one comma two. |
$ \pm $ | 0.6 & | Plus or minus. | |
em For | $ k = 0 , 1 , 2 , ... $ | , compute | k equals zero, one, two, star. |
especially Ge | $ _1 $ | , contribute to the reduction in energy and thus stabilize the 2 | One. |
shows that | $ K ( 0 ) $ | is Seifert fibered , | K of zero. |
$ E_2 $ | is already proportional to | E sub 2. | |
in the presence of other users of an arbitrary class | $ j $ | is such that : | j |
or | $ M $ | , the bound in Theorem~ | M |
every parallel switch is linked to | $ N $ | source servers and | N |
becomes very large for small values of | $ \theta_\textrm { p } $ | . However , it is not realistic for the star to have closed field lines that extend to hundreds of stellar radii , as the wind of the star will | theta sub p |
where | $ 1 $ | , | one |
one obtains integrals with up to rank | $ L $ | numerators and up to | L |
-almost all | $ x $ | . Now for all | x |
as the convergence rate for | $ \hat\pi ( \bX ) $ | to establish the asymptotic distribution of | hat pi of b X |
-directed colimits of objects of | $ \Ccal $ | , then | script C |
$ \alpha $ | , | Alpha. | |
m wide slits and | $ 40\ , \mu $ | m pitch in a distance of | 40 mu. |
& | $ 2 $ | & | two |
Construct vertex set | $ A $ | by including | A |
is a leaf or a vertex of degree | $ 2 $ | in | two |
The second expression after flipping | $ X $ | and | X |
and velocity dispersion | $ \sigma $ | Sigma. | |
the R-squarks are made much heavier | $ M_ { \widetilde { q } _ { R } } > 5 $ | TeV and hence decoupled . | M sub tilde q sub R greater than 5 |
with | $ U $ | , so that | U |
periodically by | $ r $ | as well as the second | r |
At % weak % interference | $ \alpha=0.5 $ | , the minimum distance at | Alpha equals zero point five. |
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