context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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approaches the critical value | $ b_ { c } $ | , which represents the impact parameter for photon trajectories on the photon sphere at | b sub c |
$ \omega_2 $ | as purely imaginary . If | Omega sub 2. | |
Alice generates a syndrome | $ \mathbf { s } _A $ | and sends it to Bob . The syndrome size varies depending on the code rate , i.e . , low code rates require longer syndrome . This gives better chances | s sub A |
introduced by the HP transformation whose | $ z $ | -component reads | z |
where | $ \mathcal { P } $ | and | P in calligraphic font. |
where | $ \mathbf { v } _2 $ | , | v sub 2. |
with respect to | $ g $ | } and we denote the number by | g |
one can also immediately write down bounds limiting the allowed anomalous dimensions of the renormalised fields | $ \phi $ | and | phi. |
& | $ -0.202 $ | & | negative zero point two zero two. |
When | $ r $ | is even , similarly , we can prove | r |
The reason why we take this Gaussian operator is that it is normalizable and thus well-defined : | $ ( { \cal O } _0| { \cal O } _0 ) = 1 $ | where the inner-product is the standard trace in the quantum mechanics Hilbert space , with | Open curly brace, calligraphic O, subscript 0, vertical bar, calligraphic O, subscript 0, equals 1. |
and for | $ i \geq 1 $ | , | i is greater than or equal to 1. |
32 | $ \pm $ | 2.0 & 68.46 | Plus or minus. |
the generalized | $ L_p $ | -discrepancy suffers from the curse of dimensionality . | L sub p |
if | $ B\subseteq N_G [ X ] $ | . Thus , | B is a subset of N sub G of X. |
and a stability condition | $ \alpha $ | on | Alpha. |
\\ & IRS & | $ 0 $ | & | zero |
then | $ \beta $ | is contained in | Beta. |
rather than solving for | $ P ( r ) $ | and | P of r |
$ \frac { d\Phi } { dE } $ | , | dee phi over dee e | |
omitting the indices | $ n $ | and | n |
& | $ \text { SA } ( 1 , 2 , 3 ) $ | & 47.00 & 61.50 & 0.059 & 5.540 | S A one, two, three. |
-softness is still maintained with | $ \kappa=0.2 $ | , where the quasi-degenerate | kappa equals zero point two. |
by | $ -1 $ | -homogeneity we have | Minus one. |
em For any | $ R > 0 $ | consider a set | R is greater than zero. |
in | $ W^ { 1 , \infty } ( \Omega ) $ | , and smooth convergence of , | W raised to the power of one comma infinity, open parenthesis Omega, close parenthesis. |
as | $ r \to \infty $ | where | r goes to infinity. |
-independent constant | $ C > 0 $ | such that | C is greater than zero. |
are denoted | $ \rodProfSf { n } $ | and | rodProfSf n |
and dropping the monomial corresponding to the exponent vector | $ ( 0,0 ) $ | from the result . The similarity of two objects is rather superficial and this leads to the problem of finding the explicit form of a polynomial given its amoeba . | zero comma zero. |
24 & | $ \pm $ | & 0.39 & 0.46 & | Plus or minus. |
and | $ D_ { eff } $ | of the ensemble of the particles along the | D sub effective. |
and | $ Q $ | is an arbitrarily chosen symmetric positive definite matrix . Given strictly stable | Q |
increases from 159 to 917 and | $ \chi $ | decreases from 0.354 to 0.147 , the stretching speed of the early liquid film becomes faster , the average time to complete the swing backup becomes shorter , and the | chi. |
Is it necessary to speak about the third rectangle | $ R $ | , or is it equivalent to require that | R |
1 | $ \pm $ | 1.1 & 63.7 | Plus or minus. |
if for any scheme | $ S $ | and morphism | S |
-th channel paths for the | $ k $ | -th user , respectively , and follow the probability density function | k |
of unknown length | $ n $ | over an alphabet | n |
is the space of all | $ 1 $ | -dimensional vector subspaces of | one |
$ C < \infty $ | so that for every | C is less than infinity. | |
for | $ i\in [ n-2 ] $ | and | i belongs to the set of integers from n minus 2. |
so there is usually no loss of generality in assuming that | $ X $ | is some specific space such as | X |
form with its | $ T $ | ( | T |
th tensor component unchanged and being either Pauli | $ X $ | or Pauli | X |
implies that | $ \Aut ( X ) $ | is second countable and locally compact . Furthermore , since every element of | Automorphism of X. |
trained on training data | $ X_ { trn } = U\Sigma_ { trn } V_ { trn } ^T $ | with | X sub t r n equals U Sigma sub t r n V sub t r n transpose. |
tuples reduced to a single~ | $ 1 $ | before the last cut ( hence | one |
and the value of | $ \lambda $ | is between 0.01 to 1 in all four datasets . Model performance degenerates significantly when | Lambda. |
using special factoring properties of | $ \ { n\ } $ | . Recently , Garret and Killpatrick~ | n |
of | $ N_S $ | configurations | N sub S. |
holds for all | $ i $ | , but we prefer to write it in the above shape in order to highlight the similarities with the argument in the previous section . As in Section~ | i |
$ \PPhi_ { 13 } $ | , and | P i sub 13 | |
satisfying | $ a_j > 0 $ | for all | a sub j is greater than zero. |
and hence intersect | $ \del M $ | in at most one component . As the triangulation of | delta M |
\EndIf \EndWhile \Return | $ N $ | , | N |
we are going to present and discuss the quotients that appear in the spectral sequence construction for each | $ p=0 , \ldots,4 $ | . In order to make the computations more understandable , we are also going to summarise the overall information at each page in a more pictorial way with the use | p equals zero, dot dot dot, four. |
Each | $ BD_ { n } $ | proposes | D B sub n |
that shows a typical value of | $ 1.5 $ | for | 1.5: One point five. |
in | $ T $ | . Moreover , we study matches between substrings of | T |
We denote the encoder and its trainable parameters as | $ E_ { \text { 3D } } $ | , and its output feature as | E sub 3D |
& | $ 0.927 $ | & | zero point nine two seven. |
em if | $ x\xtransto { ~a~ } x ' $ | in | x transits to a prime, where x is mapped to a by a function or transformation. |
$ \delta=0 $ | , the resulting statistical field theory obeys a fluctuation -- dissipation relation~ | delta equals zero. | |
Now we choose | $ s_0 $ | so that | s sub zero. |
with polynomial growing number of parameters | $ P ( a ) $ | , parameter magnitude | P of a. |
while the scattering length | $ a $ | can have an effect on the velocity-dependence of the cross section at low energy . The self-interacting cross section can be written as | a |
from the best fit of the stacked spectrum inside | $ 1 R_ { \rm e } $ | of UDG11 we obtained a velocity dispersion of | one R sub r m e |
and | $ i = 1 $ | . After computing a period matrix | i equals one. |
is the first positive root of the Bessel function | $ J_0 ( z ) $ | . % | J zero of z. |
where both inputs | $ x_n $ | 's and outputs | x sub n. |
As | $ \psi $ | is an even function with unit integral we get from | Psi. |
& | $ 10 $ | & | Ten. |
has the form | $ \oc B $ | , and | oh-k B |
A factorization | $ \mathcal { D } $ | -module is | D |
that ensure consistency of the associated sequence of CGVI % densities | $ \ { \bar { p } _N\ } $ | . % | the bar p sub N. |
$ for all $ | [ t : u ] | For all. | |
denote the mapping of an input dataset | $ \bm { X } $ | to a lower dimensional representation via algorithm | capital X. |
$ L_1 $ | is positive definite . As a result , | L sub 1. | |
the pairformer has features for all | $ \binom { n } { 2 } $ | pairs of nodes , whether they correspond to existing edges or not . We believe that considering non-existing edges is crucial for the pairformer to inform the policy to decide | n choose 2. |
and | $ c_i $ | and | c sub i |
with | $ \gamma $ | the gyromagnetic ratio , | Gamma. |
Then | $ T $ | has at least a fixed point | T |
% The phase difference between | $ \lambda_ { AU } $ | and | Lambda sub AU. |
for every | $ t \in I $ | , then maximal parabolic regularity of | t belongs to I. |
of dimension | $ n $ | ) that takes the form : | n |
parameterized reduction from | $ L $ | to | L |
$ U \subseteq \mathcal { U } $ | and | U is a subset of U. | |
we first compute an estimate | $ \hat { \mathcal { S } } $ | for the support | S hat. |
can be represented by a harmonic form | $ ( u_ { 1 } , u_ { 2 } ) $ | . % Here we used % | u sub 1, u sub 2 |
are concentrated in homological degrees | $ 0 $ | , | zero |
We observe that SAL and SAFL improve AUC ROC and yield up to | $ 48.9\ % $ | relative improvement in mean latency . % | 48.9 percent. |
& | $ -1 $ | & | Minus one. |
% and the fact that the unitary | $ U_ { \rho \sigma } $ | in the polar decomposition % of | U sub rho sigma |
and~ | $ r ( \tree , i ) $ | respectively denote the numbers of leaves in the left and right subtrees of the | r of tree i |
$ ] $ | respectively yield that | closing bracket | |
Suppose first that | $ t $ | is non-locally-modular . Then both | t |
is the number of odd zeros of | $ q $ | . Even for the top stratum | q |
stands for the measured cross-section in the | $ s $ | - ( or | s |
to cancel the divergent | $ 1/\delta $ | . For the | One divided by delta. |
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