context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
|---|---|---|---|
For the vectored Zak phase | $ ( \pi , \pi ) $ | , there exist imaginary solutions of the wave number for both | pi comma pi |
Assume that IRS 1 and IRS 2 have | $ N=100 $ | REs each ; but we will test different values of | N equals one hundred. |
by means of a Rabi frequency | $ \Omega $ | ~ | Omega. |
$ J $ | , and | J | |
at date | $ t $ | , | t |
$ C $ | is the classifier to output | C | |
and the policy | $ \pi $ | itself require further expensive data processing . There are exceptions such as when the data stems from an optimal expert and the behaviour policy is already the optimal , but | pi. |
top- | $ k $ | ) : | k |
with | $ \mathcal { Z } ( \bar { \theta } _0 , \Sigma_0 ) $ | yields : | Z of theta bar sub zero, Sigma sub zero. |
to denote the Angenent torus that is rotationally symmetric around the | $ y $ | -axis , with its closest point to the | y |
^T | $ , $ | comma | |
The angular power spectrum determines the covariance of the modes with numbers~ | $ l , m $ | and~ | l, m: None |
is decreasing with the evolution of the system for all initial states starting from | $ X_k $ | . The stated properties in | X sub k. |
$ C_k $ | is the Fourier coefficient of order | C sub k. | |
we pattern-match on | $ l_ { 1 } $ | , which introduces | l sub 1 |
for all | $ A $ | . In order to show it is cofiltered , we need to prove that it is nonempty , that for any two objects there is another object above them , | A |
performs the following verification for | $ m $ | consecutive rounds : First , | m |
in which case the verifiers catch an attack with high probability by means of a different commit | $ c_A \neq c_B $ | showing up , or it is | c sub A not equal to c sub B. |
thus surviving a strong suppression of the | $ d_ { xy } $ | -component . Notably , such a state is still fully gapped in the bulk , with a Chern number | d sub x y. |
in terms of the redshift | $ z $ | , this may be written as : | z |
with | $ \bar { q } _ { \min } $ | as the multiplicative action value function : % | bar q sub min |
assign | $ j $ | th swapped % element to row vector % reconstruct new data | j |
on | $ \bP ( \cA ) $ | , and a morphism of vector bundles | bee pee open parenthesis see ay close parenthesis |
the symbolic program we develop in this paper is based on both the | $ L $ | -multilinear map and Bell polynomials , which can be used efficiently for generating the integral equations | L |
$ 0 < \gamma < 1 $ | , and | 0 is less than gamma which is less than 1. | |
we define | $ x^i $ | as a top eigenvector | x raised to the power of i. |
and then reconstructs its own image | $ \mathbf { s } _2 $ | based on | s sub 2. |
For | $ i=1 , \ldots , d $ | , set | i equals one, dot dot dot, d. |
The change of basis matrix is written with the column labeled | $ a $ | referring to | a |
91 | $ \pm $ | 0.02 | Plus or minus. |
Here size of | $ \theta $ | takes into account the degrees of the polynomials in addition to the number of terms . | Theta. |
for any given | $ v \in [ 0,1 ] $ | , we construct four population joint symmetry test functions : | v belongs to the interval from 0 to 1. |
we have that | $ \hat E $ | is stronger than any | E-hat. |
line | $ K_n $ | -INGARCH in table | K sub n. |
Let | $ X = \mathbb { R } ^d $ | . The function | X equals the set of real numbers raised to the power of d. |
becomes extremely small for the second doublet | $ n=1 $ | at | n equals one. |
We chose to use the general- | $ l $ | ansatz expansion to extract the large- | l |
-vertex digraph | $ D $ | such that | D |
is the space of vector fields on | $ M $ | . Our Lie algebra consists of elements of the form | M |
and | $ B $ | 's personalized price | B |
$ L_ { i , j } =L_ { j , i } $ | , and its element | L sub i comma j equals L sub j comma i. | |
03 | $ \pm $ | 2.1 & 8.63 | Plus or minus. |
possibly depending on | $ A $ | . Further work is needed to show that the characterization holds for the entire law of | A |
$ B $ | and | B | |
function prior to down-converting the signal with the earlier mentioned multiplied initial transmit signal to receive the uninterrupted Doppler response as shown by | $ f_ { D } $ | . % | f sub D |
when | $ \theta $ | is broken , we may have instead odd | Theta. |
and the measured | $ \mathbf { k } _2 $ | is almost zero no matter what phase is added . We use the strict approach to illustrate this point for a single photon at the initial state | k sub 2 |
& | $ a $ | & | a |
$ k $ | , | k | |
getting | $ M $ | we can determine the complex coefficients of transmission | M |
and | $ j_\nu $ | are chosen wisely to have the shears mechanism starts at | j sub nu. |
$ j\in\N $ | , and such that | j belongs to the set of natural numbers. | |
and | $ \eta_0 $ | are not reported in | Eta sub zero. |
plane to another flat state on the | $ YZ $ | plane , as illustrated in Fig. | YZ: None |
shows the temporal development of the integral volume | $ V_R $ | of the reconstruction relative to the ground truth volume measured from the images in the experiment | V sub R |
in which | $ - $ | 1124-I split into four pieces , | minus |
the proposed IEFSF can achieve a higher CSI feedback accuracy than existing CSI feedback methods but with | $ 16 $ | times less feedback overhead . | Sixteen. |
to | $ m+1 $ | given by | m plus one |
Choose an adapted chart | $ ( U , x ) $ | around | U comma x. |
so that | $ b_t = B ( a_t ) $ | . The noninvertibility of % | b sub t equals B open parenthesis a sub t close parenthesis |
whereas we are able to perform it in | $ \sim2 $ | days . We employ the minima hopping algorithm~ | Approximately two. |
divides the | $ ( x , y ) $ | -plane into two parts near 0 : the part below the graph corresponding to the projection of the surface | open parenthesis x comma y close parenthesis. |
So we may assume | $ \nu=\mu+1 $ | , | Nu equals mu plus one. |
shows that the probability of the sum of discordant pairs being greater than that of the concordant pairs vanishes for any pair as | $ m $ | approaches infinity . Thus , the Kemeny-optimal ranking will always approach | m |
must be strictly less than all | $ \alpha_j $ | and | Alpha sub j. |
and | $ j $ | such that | j |
% the NLE generated to explain the model 's decision-making process for the prediction of | $ \widehat { y_i ' } $ | . First , to produce causal interventions that change the prediction to a given label | y hat sub i prime |
at | $ t=0 $ | and then slowly drives it with the Lindblad equation for a time-dependent hopping amplitude at a fixed value of | t equals zero. |
is equivalent to the optimal regressor | $ h^ { \star } $ | , which is formally defined as follows . On the other hand , we define the regressor-consistency as follows . | h star. |
is likely to be close to the prior | $ \pi_i $ | by the law of large numbers . But in general | pi sub i. |
$ ^\textrm { \scriptsize 24 } $ | , | 24 | |
and both rates are equal to | $ \epsilon_n $ | . Recall that | Epsilon sub n. |
and | $ v $ | . We say that | v |
& | $ 1 $ | & | one |
our selection of hyperparameter | $ k $ | may not exactly match the actual fraction of removals in the future . However , we note that , in Figure~ | k |
$ E_\mathrm { g } /k_\mathrm { B } T_\mathrm { E } \geq 10 $ | , see Fig . | E sub g divided by k sub B T sub E greater than or equal to 10. | |
while the star formation of the ones with the large | $ R_ { \rm max } $ | can be quenched until the gas is recovered . The quenched time scale can be | R sub max. |
at time | $ t $ | . | t |
$ 0 $ | , and | zero | |
& & & & | $ \ , \xbar { 27 } \ , ( 1 ) $ | & | x bar 27 of 1. |
then the function | $ \pi : G\to G/H $ | defined by | Pi maps G to G mod H. |
and | $ E^s $ | is nontrivial . | E raised to the power of s. |
The cases treated in this section are very similar to the | $ A_1 $ | case ; we will keep the same notation. | A sub one. |
Adversarial training is then performed to converge on a fully robust network | $ ( f_a^ * , f_b^ * ) $ | . The adversarial accuracy of | f sub a star, f sub b star |
if the true value of | $ r $ | is | r |
72 | $ \pm $ | 0.21 & 88.77 | Plus or minus. |
The fidelity between the two states is directly related to the tolerance | $ \varepsilon_C $ | chosen for the variational compiler . Let | epsilon sub C. |
The assumption | $ \omega|_m \geq \rho \geq 0 $ | implies that | Omega, bar, m, greater than or equal to, rho, greater than or equal to, zero. |
We denote the number of ascents in | $ \mathcal { T } $ | by | tee |
For | $ 1 \leq i \leq 3 $ | , let | One is less than or equal to i, which is less than or equal to three. |
$ \pm $ | 1.2 & | Plus or minus. | |
and | $ T_z $ | . % Here , we restrict ourselves to even | T sub z. |
for some | $ i $ | . In the former case , | i |
Nearly | $ 80 $ | Eighty. | |
36 | $ \pm $ | 2.02 & | Plus or minus. |
which is determined from the global state | $ s_t \in \mathcal { S } $ | according to the observation probability | s sub t belongs to the set S. |
35 | $ m $ | and configured the safety system to activate the impeller upon crossing this distance . Before the experiment , a training session was conducted in which we demonstrated HAD to the | m |
For each | $ N $ | in this set , we determine the value of | N |
the ground instance of | $ \Pi $ | , that is the program obtained from~ | Pi. |
it follows that for | $ n $ | sufficiently large we have | n |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.