context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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large | $ \alpha $ | & | Alpha. |
is then the smallest | $ \delta ' $ | such that | Delta prime. |
of | $ V_0 $ | . Moreover , from the equation above , the importance of avoiding zeroes of | V sub zero |
all equations which do not affect | $ P_ { \ConsSet ' } $ | can be removed from the model to reduce its overall complexity . In particular we point out that | P sub ConsSet prime |
$ S $ | runs over all subsets of the orbitals that contain as many elements as the number of spin-up electrons , | S | |
and | $ A^ { \prime } _ { 4,2 } $ | , other four quantum states | A prime sub 4, 2. |
or | $ j=k=l=m $ | , which we consider separately . Without loss of generality , we find the following integrals | j equals k equals ell equals em |
\\ | $ \ $ | +DSL & 42.1 & 68.5 & 79.8 & | backslash |
are invariant under the following | $ CP $ | transformation : | CP (None) |
expression can be later found out by subtracting the sum of variational profiles of | $ n_ { \pm 1 } $ | from the total number density profile , i.e . , | n sub plus or minus 1. |
Then | $ \psi $ | solves | Psi. |
-block | $ B $ | . More precisely , by Lemma | B |
as the pack of optimal examples to learners at | $ t $ | -th iteration , | t |
and | $ j , k =2 , \dots , N $ | , | j, k equals 2, dot dot dot, N |
statistical independence of | $ x $ | and | x |
In case | $ B=0 $ | and | B equals zero. |
& | $ 4.7 $ | & | 4.7: None |
the value | $ v_i $ | is an eigenvector of | v sub i |
Observe that the obtained trigraph is | $ G ' $ | . Thus , | the derivative of G. |
for any field of characteristic | $ p > N $ | . In particular , we see that | p greater than N |
with | $ a < c < d < b $ | , so the images of these elements under an order-preserving | a is less than c is less than d is less than b |
If | $ D_\gamma $ | is clopen for all | D sub gamma |
~when | $ c < d $ | . Moreover , the right-hand side is also well defined even when | c is less than d. |
A comprehensive analysis of the LHC potential for the | $ W_R $ | search , with the emphasis on the KS process , was performed in~ | W sub R. |
We can rewrite | $ r $ | as follows | r |
-contraction at the edge | $ AA ' $ | ; see Figures | AA prime. |
We also restrict the area from which the nodes can choose their new vectors compared to the convex hull of | $ \barSet $ | in | bar set |
the variables | $ p_c $ | and | p sub c. |
when | $ n $ | has prime factors larger than | n |
For | $ n\ge 3 $ | , let | n greater than or equal to 3. |
we compute the intersection points with the given mask and rotate their coordinates with the respect to the | $ I_1 $ | , | I sub one. |
we may have that | $ S=0 $ | . If this is the case , we assume that | S equals zero. |
Note that BDF | $ k $ | methods are | k |
decreases whereas | $ \rho ( r=0 , t = 0 ) $ | and | Rho, open parenthesis, r equals zero, comma, t equals zero, close parenthesis. |
where we have defined the pull of an observable | $ O $ | in a theoretical model | O |
we can notice a clear correspondence between our MED model and the case of | $ \alpha=1.9 $ | in | Alpha equals 1.9. |
is a pair of dominating distribution for individual mechanism | $ \M_i $ | . In the following text , we denote the product distribution | M sub i |
for | $ \alpha\ge 1 $ | , then | Alpha is greater than or equal to one. |
to show that there are convex hulls strictly on both sides of the segment | $ e $ | . Let us modify slightly the argument from there . Now we work with a convex hull thrackle with | e |
$ & $ | e^ { f ( 1 ) } G | and | |
The initial values | $ X_0 $ | are drawn from a uniform distribution on | X sub zero. |
for the projections and | $ \langle - , - \rangle $ | for the pairing operation which is induced by the universal property . | angle bracket minus comma minus angle bracket. |
to three variational GP approximations for simulated data with Gaussian noise at | $ n=200 $ | randomly sampled training inputs on | n equals 200. |
for integer | $ d\ge 0 $ | , and thus this holds as | d greater than or equal to zero |
for | $ n\geq 2 $ | . Using the Ritz-projection , the two terms can be bounded by | n greater than or equal to 2. |
as permuting | $ a $ | and | a |
Although any odd prime that is inert in | $ K $ | is not inert in | K |
or whether or not this fact is independent of | $ D $ | . Moreover , | D |
to obtain the edited image | $ I_e $ | , where | I sub e |
Assume for contradiction that the geometric generic fibre of | $ \tilde { \mathcal { X } } $ | admits a decomposition of the diagonal . Then the map | tilde X with a calligraphic X. |
Comparison of the characteristics length | $ \lambda_ { 1,2 } $ | with the universal curve , Eqn. | Lambda sub one comma two. |
using the coordinates | $ ( z_1 , z_2 ) \in \mathbb { H } $ | , i.e . the transform is given by | Open parenthesis z sub 1 comma z sub 2 close parenthesis belongs to the set of quaternions. |
The instanton piece is the usual product of Nekrasov partition functions on | $ \mathbb { C } ^2_ { \epsilon^i_1 , \epsilon^i_2 } $ | over the fixed points , applying the appropriate shifts to | C two sub epsilon superscript i sub 1, epsilon superscript i sub 2 |
vs | $ < $ | dog , on , bed | less than |
In a Cartesian coordinate system~ | $ x^i= ( x , y , z ) $ | with origin located at the center-of-mass of the body~ | x to the power of i equals the ordered triple (x, y, z). |
We know that there is no | $ U/L $ | flippers in the reduced form of | U over L. |
& | $ < 11 $ | & -- & | less than eleven. |
the mean and variance are given by | $ \mu_ { \rm Pois } : =\lambda $ | and | mu sub Pois equals lambda |
for all~ | $ \theta $ | and under sufficient regularity conditions on~ | Theta. |
These eigenstates are the eigenstates of the quadrature operator | $ \hat { x } $ | : | x-hat. |
In this case we write | $ \widehat { \Gamma } _ { \underline x } : =\widehat { \Delta } $ | for the formal completion of | Gamma hat sub x equals Delta hat. |
and fixes a locus of codimension | $ 1 $ | in | one |
We estimate the total spatiotemporal agent density from each simulation and average over all | $ R $ | simulations to obtain | R |
then we discard all intervals | $ [ c_i , d_i ] $ | with | the interval between c sub i and d sub i. |
51 | $ \pm $ | 0.22 & | Plus or minus. |
is equivalent to | $ \eps = o_ { \omega } ( 1 ) $ | . The subset | epsilon equals little-o of omega of one. |
The radio array will cover a surface of about | $ 500 $ | Five hundred. | |
Then for all | $ n $ | , we have | n |
$ and a point $ | and a point | ||
% where | $ \pi ( . ) $ | is the target density and | pi of dot. |
$ \Omega_k $ | , and | Omega sub k. | |
A clearly asymmetric 3D surface shows a deep trench at negative | $ r $ | ( gaps of the opposite signs ) , corresponding to a suppression of | r |
remains to be | $ k $ | and with probability | k |
for every | $ \theta $ | , the excess risk | Theta. |
This result can only be obtained for | $ v\neq 0 $ | . The Carroll regime now yields an energy-momentum tensor of the form of a spacelike Carroll fluid as presented in | v not equal to zero |
& | $ 0.370 $ | & | 0.370: None |
is the | $ n $ | -fold degenerate eigenvalue of | n |
Note that the results obtained from | $ \delta_g $ | , | delta g. |
we choose | $ S $ | such that | S |
Select | $ \theta_1 \notin \Theta_0 $ | for power calculation . | Theta sub 1 is not in Theta sub 0. |
random variables | $ Z_k $ | in the normal domain of attraction of an | Z sub k. |
~Olivera-Nieto | $ ^ { 2 } $ | , E.~de~Ona~Wilhelmi | squared. |
and | $ \lambda_ { r } $ | denoting the combination weights . We use the BERT-base model as our encoder | Lambda sub r. |
where | $ G_i $ | is of real rank one . Let | G sub i. |
and | $ h $ | . Figure~ | h |
denotes the modified exit time to indicate that | $ V $ | has been changed to | V |
$ k $ | -point mesh for | k | |
and | $ G \leq H $ | if Left wins moving second in | G is less than or equal to H. |
is either | $ f ( t ) $ | or | f of t |
& Max depth & | $ [ 3,10 ] $ | & | three ten |
degree-blocking gadget to each terminal node and | $ \Delta-1 $ | degree-blocking gadgets to the remaining leaf . Finally , we combine the vertex gadgets for all vertices in | Delta minus one. |
and the time-reversal operation | $ K $ | , on the highest weight state as well as the generalized highest weight states , we only need to consider the singular value decomposition for such a degenerate ground state | K |
~~ & ~~ | $ \phi_A ( \bm { k } ) $ | ~~ | phi A of k. |
em write down the address of | $ d $ | to identify | d |
$ ( $ | opening parenthesis | ||
would be met if | $ \Pi_m $ | . The proposition | Pi sub m. |
as above and any odd rank | $ r $ | , we have | r |
and that the bad points introduce extra squared-error of order | $ \sigma\Delta $ | . Given our threat model , this is exactly expected . The radius around | sigma delta. |
Combining this with the scaled overlaps concentrations gives for | $ n\rightarrow\infty $ | n tends to infinity. | |
The canonical decomposition of a dimension vector | $ \dd\in\BoN^ { Q_0 } $ | can be determined algorithmically . Indeed , for a given | d d belongs to the set of integers raised to the power of Q sub 0. |
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