context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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Let | $ X = ( X_u ) _ { u \in U } $ | be a centered Gaussian process indexed by a semi-metric space | X equals X subscript u, subscript u in U. |
This can only happen for a small value of | $ F $ | . For example , in order to have | F |
Since | $ U $ | is an orthogonal matrix , we have that | U |
that the statement holds for | $ k=2 $ | as well . Now fix | k equals 2. |
then | $ \hat \mu $ | converges to the set of correlated equilibria | mu hat. |
defined by | $ \ad_\ast x \in \cB ( \GL ( \Lie G ) , \bbF ) $ | is given , for | ad star x belongs to cB of GL of LieG, bbF |
We mention in passing that the upper bound on the size of a positive | $ k $ | -basis has yet to be determined , though it is known to exceed | k |
These indices are steeper than the power-law | $ - $ | 0.37 predicted by Kolmogorov Spectrum , { and proves the value of long-term monitoring of high DM pulsars for studies of RISS . We tested the model using chi-square and | minus |
Let | $ k= $ | k equals None | |
The configuration of UAV is defined by the position | $ x \in \mathbb { R } ^3 $ | of the center of mass and attitude | x belongs to the set of real numbers raised to the power of 3. |
and | $ \sLpi $ | . | slash i |
for some | $ 1\leq i\leq n $ | . On the other hand , we had | One is less than or equal to i, which is less than or equal to n. |
If | $ G $ | contains elements | G |
Let | $ K $ | be a global field and | K |
Let | $ s > 0 $ | be some free parameter that we will later fix to optimize our bound . The event | s is greater than zero. |
unitary representations of | $ G $ | . The 2-morphisms | G |
$ WD = 17.5mm , NA= 0.30 $ | , the beam size on the SSE device : | WD equals 17 millimeters, NA equals 0.30. | |
-matrix with each row containing exactly two | $ 1 $ | 's ; | one |
or | $ -1 $ | and the remaining other | Minus one. |
where the transition frequency is denoted by | $ \omega _0 $ | , | omega sub zero. |
3 | $ \pm $ | 2.8 | Plus or minus. |
Assume now that | $ t \ge t_0 $ | . Throughout , let | t greater than or equal to t sub zero. |
% which can be seen in the following way : for | $ \rho \geq 1 $ | , we have | rho is greater than or equal to one. |
by repeating the above computation | $ \log ( d/\epsilon ) $ | times , the | log of d divided by epsilon. |
This equation obviously holds for | $ k=0 $ | by | k equals zero. |
for | $ H^\ast_ { \mathrm { alg } } ( X ) $ | , where | H star alpha g of X |
bA = 2\pi | $ ( $ | l = 0 | opening parenthesis |
made at time | $ t $ | . A suitable terminal cost | t |
it is easy to see the distances for points in | $ G^1 $ | are not changed , the distances for points in | G raised to the power of 1. |
but insensitive to | $ \gamma $ | for MBHs with larger masses . In triaxial galaxy potentials , the phase space of the loss cone described for spherical galaxies can be replaced by the phase space of | Gamma. |
is the least such | $ N $ | . We denote these by | N |
and keep | $ \boldsymbol { A } $ | constant . Without loss of generality , if we know that our input IMF has constant amplitude | A. |
with log link | $ g $ | and known dispersion . | g |
then only the variables | $ \mathrm { C } ^ { \delta } _ { 1 } $ | and | C delta one. |
and the triplet | $ t_ { i x } $ | operators , we consider the Hamiltonian % | tee sub eye eks |
$ \text { AN } $ | , | A G | |
between the polaritonic ground state | $ \ket { P_0 } $ | and a given | ket P sub 0 |
The only rules in | $ \mathcal { TE } $ | for which all profiles are dictatorial are precisely | L-T |
such inaccessible cardinals exist unboundedly in | $ \lambda $ | . | Lambda. |
connected graph | $ G $ | , we introduce the definition of the fractional Laplacian on a subgraph with Dirichlet boundary condition . We recall the setting of graphs . A locally finite , simple , | G |
is a sequence in | $ [ 0,1 ] ^n $ | such that | Zero to one raised to the power of n. |
or | $ \wt S_2 $ | , so | wt S 2 |
is the covering map | $ P ( p , v ) = ( \pi ( p ) , d\pi ( v ) ) $ | , induced by the covering map | P of p, v equals open parenthesis, pi of p, comma, d pi of v, close parenthesis. |
Steps 3 to 6 ensure that the matrix | $ \mathcal { S } _ { \text { final } } $ | is rational , Hermitian , belongs to the subspace | S final |
on the compact set | $ \psi_\Omega^ { -1 } ( K ) $ | . Therefore | Psi sub omega to the power of negative one, open parenthesis K, close parenthesis. |
that on the wing the probability distribution function | $ P ( x_ { \rm out } ) $ | one obtais from Eq.~ | P of x sub rm out |
be the smallest nonnegative integer | $ m $ | such that | m |
An explicit example is shown for the | $ n=3 $ | case in Fig.~ | n equals three. |
It also avoids running complex EKT simulations at various coupling strengths by providing the rescaling of key medium profiles like the time | $ \tau $ | and the temperature | Tau. |
finite graph | $ G $ | such that | G |
and | $ 52\ % $ | in the same order . This observation suggests that improvement in matching the magnitude of | 52 percent. |
whereas the lack of experimental information prevented us from analysing the pseudoscalar-vector modes | $ { \bar B } _ { d , s } \to K^ { 0 } \bar { K } ^ { * 0 } $ | and | The average of B sub d comma s goes to K naught times the conjugate of K star naught. |
a high photon density and hence a high-intensity laser is needed for efficient excitation and subsequent ionization of | $ H $ | atoms . The most efficient realization is a pulsed laser . Such a laser system has been set up at | H |
Note that the third term on the right-hand side of the preceding inequality can be decreased by either diminishing the stepsize | $ \alpha_k $ | or increasing the batch size | Alpha sub k. |
and | $ a $ | , right ? } % | a |
$ I_ { c } $ | exhibits oscillations around zero as a function of | c, | |
in | $ G $ | , we conclude that | G |
The | $ \mu $ | -independence of scattering parameters essentially comes from this property . However , the HAL QCD potential , Eq.~ | mu. |
which results in | $ C $ | 's robust generalizability over UAEs ( i.e . the adversarial robustness ) . As proved later , the KL-divergence of | C |
and the | $ \Delta a_ { z } \approx 0.05 $ | term avoids pathological behaviors by allowing this additional term to `` smoothly '' ramp on . Note we have varied | Delta a sub z approximately equals zero point zero five. |
we define banded unlink diagrams for surfaces embedded in | $ S^4 $ | or | S to the power of 4. |
& | $ 10^ { 9 } $ | & | Ten to the power of nine. |
reuse the sample | $ x = D_ { i-1 } ( t ) $ | and set | x equals D sub i minus 1 of t. |
which will help us show that | $ H $ | -tubes foliate | H |
$ -0.10 $ | & | Negative zero point one zero. | |
the integral spans the visible area | $ S $ | of the target in | S |
of which corresponds to | $ \alpha_i $ | such that | Alpha sub i. |
and unit costs | $ c_ { ij } ^k $ | for using any arc | c sub i j superscript k. |
and a continuous | $ 2 $ | -cocycle | two |
stands for the curl of a vector field and | $ k $ | is a scalar field . If there exists a solution | k |
We thus assume that | $ f ( S ) $ | is uncountable . Since | f of S |
$ & $ | and | ||
at each lattice site are related to the discretized version of | $ \hat { \phi } $ | and | phi hat. |
The Defence Budget is also a function of the year and is expressed as | $ b_t $ | . We then need to make sure the options in the portfolio will satisfy the budget constraint . To do so we add to the ILP the | b sub t |
$ 6 $ | ) } & & & & & & & | six | |
which educates our choice of | $ \rho $ | . Consider some Pauli observable | rho. |
By | $ R $ | -identifiability , there exists a bijection | R |
path in | $ G $ | , and by | G |
and any | $ t\geq0 $ | . In particular , particles never `` collide '' . | t greater than or equal to zero. |
has a neighbor | $ w\in W_t $ | , it induces | w belongs to the set W sub t. |
note that we are implicitly assuming that | $ n\geq 2 $ | . For two sets | n greater than or equal to 2. |
We regard | $ Y $ | as a degree | Y |
are defined over | $ \Omega \subset \R $ | . We also just saw that it is optimal when | Omega is a subset of the set of real numbers. |
It follows that | $ \mC_ { I_1\cdots I_m } $ | can be chosen as a symmetric tensor of order | C sub I 1 dot dot dot I m. |
has a neighbor | $ v ' $ | outside of | v prime |
66 | $ : $ | 05 | colon |
Let | $ \mathcal { C } $ | denote the set | see |
-divisor of | $ C^0 $ | -type on | C raised to the power of 0. |
$ n $ | the order of the periodic map , and the valency data is | n | |
We show that if | $ F $ | has short elements | F |
The finite-size effect pushed the regime crossovers to smaller | $ \eta $ | values from the phase boundaries in the thermodynamic limit , a feature that can be systematically analyzed using finite size scaling~ | eta. |
90 | $ & $ | 7.73 | and |
denotes the Jacobian of | $ \alpha $ | evaluated at | Alpha. |
$ \chi=\pi/2 $ | corresponds to the pure radial and | chi equals pi divided by two. | |
to vertex | $ v_ { i+1 } $ | labeled with an update symbol . For each pair of vertices | v sub i plus one. |
if | $ xy=0 $ | and | x times y equals zero. |
em | $ n_- $ | is the normal vector pointing downwards . | n minus |
and they converge to the following 4 points as | $ \epsilon $ | becomes sufficiently small , which are : | epsilon. |
which is determined by | $ \epsilon_\text { s } $ | . These uncertainties are considered separately for the signal and decoy states | epsilon sub s |
are integers we see that there exists | $ \eps > 0 $ | such that for | Epsilon is greater than zero. |
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