context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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using the undamped BCS gap for various values of | $ T_B $ | at which there is a discontinuity in its derivative as it merges the zero gap curve , i.e . the derivative is positive infinite for | T subscript B |
the diquark spin is | $ \vec { s } _d = 0 $ | . The total angular momentum | Vector s sub d equals zero. |
and let | $ M $ | and | M |
with | $ \psi_q $ | being the target state . % of a set of initial state | Psi sub q. |
let 's say | $ t $ | . A pair of transformations convert this equation into an integrable form and are introduced as ( provided | t |
and the sigmoid function is represented by | $ \varsigma $ | In summary , Figure | varsigma. |
% the quadruply vortices can be well prepared by the topological phase imprinting which is also used to create the | $ n=2 $ | vortices , while the topological phase imprinting can not be used to create the | n equals 2 |
: Let | $ I $ | be category-recursion free and let | I |
as | $ C_g $ | increases , the system shifts to periodic behavior , characterized by one zero exponent and 29 negative Lyapunov exponents , before switching back to chaos . This is illustrated in | C sub g. |
implying for | $ n $ | large that | n |
Analysis by varying the parameter | $ \alpha $ | . The | Alpha. |
- | $ 6_0 $ | & 24.7 & 0.285 & 3.1501 & & & & | six sub zero |
For right | $ A $ | -modules , this super trace comes from the composition | A |
in image | $ \alpha $ | is a high-dimensional vector that encodes % with the keypoint 's position and visual descriptor . The keypoint position is embedded into | Alpha. |
to the case of | $ \eta $ | -unambiguous ensemble exclusion with | eta. |
only has | $ \x_j^i $ | terms , thus , its corresponding time evolution matrix is block diagonal , i.e . | x sub j superscript i |
so that the result | $ \hat { \Delta } ( X ) = \ { 12356 \mid 4 \ } $ | was found to be quite robust to the choice of threshold . Since only one pattern of dichotomic independence was found to hold , | The hat delta of X equals the set 12356 modulo 4. |
is not attached to any | $ d $ | -dimensional topological defect of | d |
Without priors on both | $ \Delta m^2_ { 21 } $ | and | Delta m squared sub 21 |
\\ | $ [ $ | 2 & 0.001 & 0.292 | opening bracket |
The arc-melted mixture was placed in a | $ 2 $ | two | |
$ 5 $ | -bit example , carried out over its largest vocabulary size of | five | |
if | $ f $ | and | f |
which has to contain a | $ g $ | -invariant bi-infinite | g |
& | $ 1 $ | & - | one |
$ U_ { 2 } $ | and the seeing . The optical turbulence can be roughly characterized by the Richardson number | U subscript 2 | |
to | $ 1.5 $ | times the solar metal mass fraction of | 1.5: One point five. |
on | $ t $ | = +81.90 hours . % Similarly , the { | t |
contains all | $ l $ | -adic places . | l |
where | $ \partial_i $ | refers to the derivative with respect to the | Partial i. |
too large values for | $ L_n $ | do not accelerate the transient , returning to purely isolated local estimates in the limit | L sub n. |
and outputs | $ 0 $ | otherwise . | zero |
of | $ \mathcal { C } _1 $ | on a finite integral is always finite . For | C one. |
regularized generator | $ G ( \cdot ) $ | . | G of center dot. |
We estimate the reconnection rate | $ \beta_ { \rm rec } $ | based on the upstream velocities near the aforementioned fiducial X-point . As visible in Figure | beta sub rec |
regret considers the suboptimality as | $ \Delta $ | . Next , we consider the effect of random noise . For any algorithm running on this parameter set | Delta. |
: = | $ ( z_1 , \ldots , z_k ) $ | is a submersion , but | Open parenthesis z sub 1, ellipsis, z sub k, close parenthesis. |
Case 3 : | $ \eta_ { k+1 } ^ { \sigma } ( x+1 ) =1 $ | for some | Eta sub k plus 1 to the power of sigma, open parenthesis x plus 1, close parenthesis, equals 1. |
and | $ H_ { \rm mat } $ | the unbroken groups on the vacuum . The same groups are denoted in | H sub mat |
When | $ S=3 $ | the simulations never returned disjoint winner sets , showing that elections like what are constructed in Proposition | S equals three. |
They found that each photon carries an entropy of the order | $ k $ | . % NP Entropy and information are related . The physics of information was studied already by Brillouin . In his influential book | k |
the upper bound of | $ \gamma $ | is | Gamma. |
$ \sigma $ | , | Sigma. | |
be the set of equicontinuity points of | $ F $ | . We call | F |
In each coordinate | $ j \in [ d ] $ | , | j belongs to the set of integers from 1 to d. |
first by adding extra dice at fixed | $ M $ | , second by increasing | M |
and the 3D displacement | $ \mathbf { d } $ | . We define all possible changes that can be fitted by the camera sampler as extrinsic changes , and design | d bold. |
$ , $ | comma | ||
we denote by | $ H_B ( A ) $ | the subgroup where matrices are chosen in | H sub B of A |
Since | $ E_2^ { p , q } =0 $ | for | E subscript 2 superscript p comma q equals 0. |
the line generated by | $ D $ | . We are going to prove that | D |
be a closure space and let | $ A $ | be a subspace of | A |
and drift | $ 2\mu $ | . The present article does not rely on any detailed information about this process . The only input used is the invariance and uniqueness of the SH under the KPZ | Two mu. |
we now have three ways of representing | $ D $ | : | D |
-channel of the planar amplitude and for the | $ s $ | - and | s |
10 | $ ^ { -6 } $ | & 2.4 | Negative six. |
can be decomposed into a tensor product of | $ M $ | finite-dimensional Hilbert spaces | M |
the interior angles of | $ \mathcal { D } $ | at the vertices | D |
Where | $ \delta $ | represents the declination , | delta. |
independent of the local parametrization | $ a $ | of the boundary such that | a |
is the | $ i $ | -th column vector of | i |
depending on | $ N $ | and on the seminorm in | N |
on | $ X $ | agree , and the induced | X |
for any | $ 0\le t_1 < t_2\le N $ | , | Zero less than or equal to t sub 1 less than t sub 2 less than or equal to N. |
and since every hyperedge in | $ H ' $ | has size | H prime. |
to | $ k $ | and write | k |
the number of | $ v_j $ | in | v sub j. |
$ \rho > 0 $ | , implies the log-Sobolev inequality , our result deal with less general situations than in | rho is greater than zero. | |
105 | $ \pm $ | 0.006 & 5.922 | Plus or minus. |
independent of time | $ T $ | , such that with probability 1 , | T |
% The reason for incorporating the reflection map | $ J $ | into the definition will become apparent in Section 4.2 ; it is a convenient way to ensure that % | J |
Let | $ u $ | be an infinite ground state for some fixed | u |
To save space we will use in the sequel | $ t $ | instead of | t |
is a symmetry of the effective 1D Hamiltonian | $ \tilde { H } $ | . | H tilde. |
in the document | $ j $ | is calculated with the following formula : | j |
- | $ \Delta h $ | feature indicates that the large-bias limit is valid when | Delta h |
% % In this last approximation the profile is written as the % integral of the potential of interparticle interaction % | $ \nu ( r_ { 12 } ) $ | % | nu of r sub 12. |
is uniformly sampled on the interval | $ f\in [ f_1 , f_2 ] $ | where | f belongs to the interval from f sub 1 to f sub 2. |
17 | $ \pm $ | 0.03 & 0.37 | Plus or minus. |
fixed at | $ \vr_0 $ | . The displacement of | v r sub zero |
0084 | $ \pm $ | 0.0000 & | Plus or minus. |
is free of rank | $ n $ | with generators | n |
at length scale | $ \lambda^ { -\tau } $ | to get | lambda to the power of negative tau. |
For constructing the interpolated PSF at a given distance below | $ 0.5 $ | 0.5: None | |
which links to a pair of clients | $ c_\alpha $ | and | c sub alpha |
$ b $ | and | b | |
is called the syndrome where every | $ +1 $ | in the | Plus one. |
% Yet under this scenario we could have % | $ % \frac { 1 } { 6 } \Delta > \delta . % $ | The most effort in Case 2 is devoted to dealing with this issue , and we now start the process . The graph | Zero divided by six, Delta is greater than delta dot percent. |
so it equal to | $ cf ( h_1 ) f ( h_2 ) $ | for some constant | c f of h sub 1 f of h sub 2 |
is an edge in | $ \widetilde { G } $ | . If | G tilde |
43 | $ \pm $ | 0.21 & 17.44 | Plus or minus. |
be the sharp reverse Hölder exponent of | $ w $ | . Fix | w |
For a binary function | $ f^ * : X\rightarrow\ { 0,1\ } $ | , the support-access object | f star: X to the set containing 0 and 1. |
we obtain that as soon as | $ \eps $ | is small enough , there holds | epsilon. |
we split | $ R_ { i } $ | into several segments with equal length , and then utilize OmniVL to caption them with the prompt `` What is the | R sub i. |
as | $ i \to \infty $ | . This means that for all hyperbolic elements | i goes to infinity |
we describe and compare the photo- | $ z $ | for our sample of | z |
& 0 & 1 & | $ \frac { 3 } { 2 } $ | & | Three divided by two. |
in a magnetic field | $ B $ | moves along a spiral trajectory with the synchrotron frequency | B |
has radius | $ 1 $ | ) . The Casimir energy | one |
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