context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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& | $ x_1 $ | & | x sub 1. |
the set of all the DSMs of order | $ n $ | . It is easy to see that any permutation matrix is also a DSM but not vice versa , thus | n |
04 | $ \pm $ | 10.60 & 63.55 | Plus or minus. |
$ 0 $ | , | zero | |
so | $ \mu ( S ( h ) ) =\mu ( S ( g ) ) $ | and | Mu of S of h equals mu of S of g |
then there is a point | $ x\in E $ | such that | x belongs to E. |
4 | $ \pm $ | 0.4 & 98.4 | Plus or minus. |
For | $ \zeta_ { j , r , i } ^ { ( t ) } $ | , recall from | zeta sub j, r, i superscript t |
% This definition can be extended to | $ \sigma $ | -finite measures by defining | Sigma. |
is closed in | $ G_0 $ | , so that | G zero. |
trajectories over | $ 5 $ | seeds in the standard state-action setting . | five |
$ ^\textrm { \scriptsize 101,98 } $ | , | 101 point 98 | |
A token | $ \theta $ | can be a word or , for example , or a number . A word embedding of a token | Theta. |
where | $ L ( \cdot ) $ | denotes length . We observe | L of dot. |
for the Lorentz Lie algebra | $ \mathfrak { so } ( 3,1 ) $ | and | s-o open parenthesis three comma one close parenthesis. |
we know that the function | $ \boldsymbol { \tilde { X } } _ { oh } $ | , defined from | Bold tilde X sub o h |
For | $ n = 2 $ | , the graph of | n equals 2 |
The sums involved in | $ E_0 $ | are : | E zero. |
given a target geometry | $ S $ | and the frozen weights | S |
yielding the same sets | $ \bar { X } $ | and | X bar. |
such that the asymptotic spread dimension of | $ U $ | is | U |
% One simple way of fusing is averaging all image embeddings from the same object : | $ E^f_j = \frac { 1 } { L } \sum_ { l = 1 } ^L e_ { j , l } $ | . But this can not ensure that the norm of | E raised to the power of f sub j equals one over L, the sum from l equals 1 to L of e sub j, l. |
states that if the dimension of | $ \mathcal { N } $ | is greater than 1 , then | capital N |
idempotents | $ E_A $ | and | E sub A. |
$ . Furthermore , the probability density $ | dot plus the probability | ||
0136 | $ + $ | 0438 & 26.4417 & 1.3583 & 3 , 5 , 6 & | plus |
for the meet operation of | $ L $ | and the meet operation of | L |
denote the steering vectors at the Rx and Tx for the | $ k $ | -th target , respectively , and are given by | k |
311 \\ & R | $ k $ | NN & 0.176 & 0.181 & 0.185 & 0.202 & 0.214 & 0.221 & 0.183 & 0.189 & 0.194 & | k |
are finite multi-sets for | $ 0 \le j \le r $ | . Xin et al . provided an algorithm , called Algorithm | Zero less than or equal to j less than or equal to r. |
The quantity | $ \alpha $ | is the fraction of transmitted energy going into lidar block , and | Alpha. |
$ \aseq_1 = \aseq $ | , | asterisk sub 1 equals asterisk. | |
The important point here is that the drift is evaluated at the left boundary | $ t $ | of any time interval | t |
is lower than that of | $ s_j $ | if and only if | s subscript j |
03 & | $ \pm $ | 0.07 & | Plus or minus. |
there exists a constant | $ c > 0 $ | such that | c greater than zero |
at | $ \bm { \beta } $ | can be computed as | beta. |
$ n=3 $ | , | n equals three. | |
for the existence of perfect matchings in | $ r $ | -uniform | r |
24 | $ \pm $ | 0.15 & 1.76 | Plus or minus. |
is a graph with degree at most | $ K $ | , if the LERRW is recurrent on | K |
stands for the | $ \ell^\text { th } $ | level of the NPA hierarchy . The result of this optimization is zero iff the point is on the boundary of the NPA relaxation . | ell to the power of th. |
& | $ 13.56 \pm 0.05 $ | & | thirteen point six six plus or minus zero point zero five. |
which we use with a bias of | $ -1 $ | : | Minus one. |
We screen candidates : the negative | $ \lambda $ | is invalid so we are left with the first two candidates . The first | Lambda. |
each client | $ c_ { p } $ | must also consider the heterogeneity of other clients | c sub p. |
and | $ \Delta $ | V are adopted , resulting in a total of 14 modeling parameters . To properly account for uncertainty and covariance between model parameters , we use Affine-invariant Markov Chain Monte | Delta. |
then | $ \pi ' $ | has cycle structure on List~ | pi prime. |
Now we let | $ C\to 0+ $ | . Fixing a frequency cutoff | C goes to zero plus. |
a problem that arises is that the incorporation of an increased dimensionality | $ \mathbf { e } $ | in the validation module design conflicts with the requirement of finding an acceptable | e bold. |
we consider attacks on the system output | $ y $ | . In particular , we consider an attack where a subset of the components of the system output is compromised . Under such an attack model , the | y |
and | $ B $ | mode polarizations are not correlated . However , if CMB photons experience some sort of interaction that violates parity symmetry and the Lorentz symmetry , the phase velocities of the | B |
$ R $ | ) and these terms become | R | |
and | $ \varepsilon > 0 $ | such that | Epsilon is greater than zero. |
\scriptsize | $ / $ | } } ( 4mm,0mm ) ; } } } | slash |
Suppose | $ a_1 , \ldots , a_n $ | are points in the open unit disk | a sub 1, dot dot dot, a sub n |
falloff conditions of each coefficient of | $ h_ { AB } $ | . The first is based on the linearised analysis performed above the second is on the consistency of the solution space under BMS diffeomorphisms . | h sub A B |
and the new spectral tilt angle | $ \theta ' $ | is related to the initial angle | Theta prime. |
have the same color in | $ F $ | . Hence , each | F |
/sub\_q | $ > $ | greater than | |
is | $ \C $ | -linear for any | C: None |
where | $ p_i $ | is the amount of money given to agent | p sub i. |
Compute | $ L_fh ( x ) $ | and | L sub f h of x |
$ k $ | is the number of these labels ; | k | |
and assign its | $ 1\sigma $ | scatter among CDM subhalos to be the range of typical | One sigma. |
cm | $ ^ { -1 } $ | in this study . A Semrock RazorEdge filter was used to block the elastically scattered laser light , where the cutoff energy is 79 | to the power of negative one. |
-th task are | $ k $ | -nearest neighbors of each other , then | k |
and | $ h $ | . Equip | h |
em If two maps of dg algebras are | $ n $ | -homotopic then they are chain homotopic and hence induce the same map on homology . If two dg algebras are | n |
and | $ G $ | respectively . Since | G |
& | $ 1\cdot 10^ { -14 } $ | & | One times ten to the power of negative fourteen. |
em Let | $ { x } _0 , { x } _1\in \ { 0,1\ } ^n $ | be any two distinct vectors not in the span of | x sub 0, x sub 1 belongs to the set of 0,1 raised to the power of n. |
89E+20 & 1 \\ 2097 & 05941 | $ + $ | 0064 | plus |
and can be done in time | $ O ( k\log k ) $ | . Thus , unconditional inference can be done in | Big O of k log k |
is | $ 1 $ | -Lipschitz . Consequently , we extend | one |
we conduct experiments with various values of | $ \alpha $ | in Eq. | Alpha. |
in which case she does not yield any randomness to Alice and Bob on a trial where | $ E $ | takes that value. | E |
It is the percentage of subjects for which | $ y_i = 1 $ | and | y sub i equals 1. |
$ \pm $ | Plus or minus. | ||
the SINR of the | $ k $ | -th user ’ s signal recovered is given by | k |
and | $ y $ | . | y |
and | $ \varepsilon_3 $ | , we now show that the scheme converges to a weak solution as the mesh is refined . | Epsilon sub 3. |
$ \eta_ { S } $ | prefers | Eta sub S. | |
253e-06 Jy % GDC fluxes at 2300 and 2550:61 | $ \pm $ | 12 & 89 | Plus or minus. |
is an increasing sequence of compact subsets of | $ \calD $ | such that | capital D |
namely when transport robots carry enough oxygen to provide each main robot with | $ 100\ , \picowatt $ | for | One hundred picowatt. |
or modify some | $ f_ { \ell } $ | for more transparency - see | f sub l |
denote the point | $ m_1 $ | and let | m sub 1 |
while | $ \zeta_z $ | moves from | zeta sub z. |
The redshift of the Fe K | $ \alpha $ | line is unaffected by the inclination selected for the fits . We allowed the emissivity index , | Alpha. |
but the uncertainty in F2 depends strongly on the choice of | $ f_\mathrm { c } $ | . Given the additional computational cost of the PG model , especially for small values of | f sub c |
$ \Rightarrow $ | # # # 4 | leads to. | |
we input it into the motion decoder | $ { TCN_ { enc } } $ | to obtain the human motion sequence | T N N sub c |
0078 & | $ \pm $ | 0.0081 & | Plus or minus. |
which carries a unit vector denoted | $ \hat { \mu } _1 $ | in Fig. | mu hat sub one. |
we construct an example defined over an arithmetic field of cohomological dimension | $ 3 $ | , such as | three |
Free energies are related to the finite-region pinned expansions | $ \Sigma^\Lambda_ { \B_0 } ( \bbd { \rho } ) $ | which are defined by restricting sums to polymers | Sigma to the power of Lambda sub B sub 0 of rho |
larger intervals | $ [ t_ { k-1 } , t_ { k } ] $ | likely require more quadrature points to achieve a high accuracy . Knowing how these quantities should affect the number of terms , we propose the heuristic % | t sub k minus 1, comma, t sub k |
hence | $ f_T $ | and | f sub T |
denotes the | $ n $ | -th derivative spectrum of | n |
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