context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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using | $ \gamma=\sqrt { 1-\frac { 1 } { 2\chi } ( 1-\lambda_2 ) } $ | and | Gamma equals the square root of one minus one over two chi times the quantity one minus lambda sub 2. |
has a canonical divisor | $ K_R $ | , | K sub R. |
$ l_\lambda^0 [ a\eta ] $ | , and | L sub lambda superscript 0 open square bracket a eta close square bracket | |
This also implies that the optimal parameter | $ \eta $ | is directly proportional to the wavenumber | eta. |
where | $ q $ | is the gain in a single pass through the crystal , | q |
the information in | $ z^t $ | does not include such regularization . Due to the computational intractability of directly penalizing | z raised to the power of t. |
The concept of composite particle can be generalized to the entangled states for | $ n\pi $ | -particles . Under `` field '' description , the function for unified elementary particle is given by | n pi |
can be written in terms of the amplitude | $ \psi_j $ | as | Psi sub j. |
for different | $ l $ | and with respect to the parameter | l |
exceeding 2 cts s | $ ^ { -1 } $ | . The good time intervals of each observation were further divided into multiple continuous data segments based on the orbit of the ISS . % % % % % % | to the power of negative one. |
carried by | $ \tau $ | with | Tau. |
a | $ ( 1-\varepsilon ) $ | -approximate solution . To that end , we make the following observation . | One minus epsilon. |
and | $ h^ { F } $ | be a Hermitian metric on | h to the power of F |
and | $ B $ | are parameters used to generate models with different thermal histories . As the universe evolves towards lower redshifts , the thermal state of the IGM tends to stabilize due to | B |
the dual opinions monotonically converge to | $ \frac { 1 } { 2 } $ | . As the bias parameter decreases , i.e . , when | One-half. |
em | $ \bm x $ | : the state vector in model space . | x |
and choose to work only with the field | $ \mathbb { C } $ | from now on . By abuse of notation , we will write simply | see |
polar | $ T_ { \rm eff } $ | at | T sub e-f-f. |
$ \lambda ( t ) =\lambda t^2 $ | and | Lambda of t equals lambda t squared. | |
0 | $ \pm $ | 5.2 & 53.2 | Plus or minus. |
9267 | $ \pm $ | .0320 & .9558 | Plus or minus. |
shows that our PHD-based classifier outperforms all baselines with a large margin : | $ +10\ % $ | for GPT- | Plus ten percent. |
and | $ \mathcal { C } _ { \xi } $ | . We have the following claims : | C sub xi. |
for | $ x\le\sup ( A ) $ | , and construct sets reaching this bound for any given | ex less than or equal to sup of A |
Let | $ \lambda_1 \leq \lambda_2 \leq \dots \leq \lambda_n $ | be the ordered eigenvalues of | Lambda one is less than or equal to lambda two is less than or equal to dot dot dot is less than or equal to lambda n. |
where | $ g=\Omega ( f ) $ | means | g equals big omega of f |
The semantics of STL are evaluated over trajectories | $ x ( t ) $ | : % and defined recursively as : | x of t. |
it is essential to reconfirm that the qualitative insights gained in the large- | $ \beta_0 $ | approximation also apply in full QCD . This is what we address in Sec.~ | Beta sub zero. |
while the smaller | $ K $ | values approach the desired values rapidly . The | K |
-homogeneous elements in | $ S $ | . | S |
of finite cardinality that is in the essential image of | $ \mathbf { G } $ | , we define the Fontaine-Laffaille local finite/singular structure | G |
The metric | $ g $ | induces a | g |
97E | $ + $ | 00 & 1.55E | plus |
of two graphs | $ F $ | and | F |
generated by mapping | $ g $ | ( or by | g |
Let | $ E_1 , \ldots , E_n $ | be i.i.d . standard exponential with | E sub 1, dot dot dot, E sub n. |
$ Q $ | -stars become too compact and unstable , forming BHs . Here , we have studied the dynamical robustness of dipolar | Q | |
-representation in this case is given by matrix | $ A^t $ | in Table~ | A raised to the power of t. |
and the auxiliary variables | $ \xi_k ( t ) $ | , | ksi sub k of t |
At this value of | $ \lambda $ | , by considering a large value of | Lambda. |
becomes more feasible if the Gr\ '' obner basis of the presentation ideal | $ J $ | exists ; see | J |
otherwise there will be | $ i $ | such that | i |
let | $ p_j $ | be the leftmost position in | p sub j |
$ C_ { 2 } ^ { * } $ | , and | C sub 2 superscript asterisk. | |
where | $ * $ | corresponds to the conjugate transpose operator . % For a matrix | asterisk |
behaves if the matrices | $ D $ | and | D |
$ has small steps at the integer dopings . The derivative of $ | The equation has small steps a t the integer dot s dot. | ||
be of finite order and let | $ k\in\N $ | so that | k belongs to the set of natural numbers. |
act as multiplication by the Fourier coefficient | $ a_\ell ( f ) $ | . It is a | a sub ell of f |
since we only have the gradient of | $ v_0 $ | and not of all species ... % % % PROOFS WITH NONLOCAL TERMS % | v sub zero. |
and | $ a ' $ | in | a prime. |
it is assumed that the function | $ f $ | belongs to a set of continuous , monotonically increasing , and globally concave functions called | f |
% By the fact that | $ G^ { -1 } $ | is decreasing on | G inverse. |
and evidence from 3 | $ \mu $ | m OH+H | mu. |
interacting electrons under the influence of a time-dependent scalar potential | $ V ( \bfr , t ) $ | and a time-dependent magnetic field | V of r, t. |
Let | $ A\in F_r\mathbf W_ { -p } , B\in F_s\mathbf W_ { -q } $ | , then there exists a unique element | A belongs to F sub r W sub negative p, B belongs to F sub s W sub negative q. |
11 ; 79 | $ \pm $ | 80 & 551 & 127 | Plus or minus. |
Other disks that we have not modeled explicitly and where giant impacts are thought to have occurred on Kuiper-belt-like scales include AU Mic and | $ \beta $ | Pic . In AU Mic , a collision point at an orbital radius of | Beta. |
which gives | $ E_ { \rm I } ( q ) =0 $ | . | E sub I of q equals zero. |
it is unexpected that the energy difference continues to grow without saturating even at | $ x $ | = 1.00 . This behavior significantly deviates from the CDE picture in the sense that there is no optimal doping concentration ( | x |
and | $ a $ | . In trapped ions system , these operators can be used to cool the harmonic oscillators to ground states | a |
3 | $ \mu $ | m | mu. |
In both MaNGA and iMaNGA we find a clear positive correlation between | $ \Sigma_ * $ | and [ Z/H ] . Hence , surface mass density is identified as a significant driver of stellar metallicity in MaNGA , and this is well reproduced in the simulations | Sigma star. |
still requires | $ O ( N^2 ) $ | cost . To reduce the overhead , we apply the regularization on the | O of N squared. |
investigated the properties of stellar structures by considering some positive values of | $ x $ | for embedding class-I solutions of different models of compact stars . Later , Bhar et al . | x |
The AHE due to the | $ X $ | and | X |
fails to dominate | $ X $ | and we enlarge | X |
$ M $ | _ | M | |
& | $ 1.30 $ | & | 1.30: None |
mask | $ _ { \ ! } $ | subscript | |
& | $ - \tfrac1n ( n+2 ) $ | & | Negative one over n times the quantity n plus two. |
be the positive phase of | $ u $ | . We will identify | u |
-term is relevant which is true for | $ T/\pi < 8/3 $ | . Below | T divided by pi is less than 8 divided by 3. |
with respect to | $ \bm { \theta } $ | : | theta |
and | $ g_ { 3 } $ | . For the Robin boundary condition , the on-shell action does not diverge as long as the initial or final Hubble ratio is taken to finite . For these boundary | g sub 3 |
$ x\in { \mathbb R } ^n , $ | and | x belongs to the set of real numbers raised to the power of n. | |
if | $ \delta $ | chosen small enough , then | delta. |
% Using | $ G $ | ( equation | G |
and | $ \mathcal { H } _0 $ | given in the main text . It is useful to remind ourselves that the vertices of this | H naught. |
and is most sensitive to an extremely small | $ \gamma $ | value . The estimation of | Gamma. |
For each | $ \Delta_i $ | , the algorithm from Lemma~ | delta i. |
9 arcmin | $ ^ { -2 } $ | . The deep imaging data enable a comprehensive redshift coverage ranging from 0 to 3 , and the calibrated bias residual shows no dependence on the redshift . There are | Negative two. |
\\ \hline | $ C $ | & 0.02 | C |
thus | $ u_1 \longrightarrow\^\ * \Lambda X.u_1 ' $ | and | u sub 1 goes to star Lambda X dot u sub 1 prime |
where | $ v $ | is the convective velocity , | v |
This vanishing is controlled by the pullback of | $ \mathsf { m } $ | : | m. |
is the predicted classification confidence and | $ IoU $ | is the rotated IoU between the predicted location and its corresponding | IoU: None |
that bound this face | $ \phi^ * $ | . There is a point in the above algorithm where all | Phi star. |
We say | $ ( M , P ) $ | M comma P. | |
and the last integral vanishes since | $ 0 $ | is the centroid of | zero |
operators in | $ E $ | . Equation~ ( | E |
& | $ 0.785 $ | & | 0.785: None |
when | $ \epsilon $ | does , i.e . if | epsilon. |
U | $ ( N ) _ { k_a } $ | . In this work , we focus on the case where | N sub k sub a |
Suppose | $ g $ | is an Einstein metric on | g |
As | $ Y $ | is connected , there is a unique { | Y |
which implies | $ 3k^ * \ge 12 $ | . Similarly , if | Three k star greater than or equal to twelve. |
0 | $ \pm $ | 0.4 & 51.9 | Plus or minus. |
of a nonconvex potential | $ \phi $ | . A PnP scheme with this | phi. |
probability larger than | $ 1-N^ { -D } $ | for any | One minus N to the power of negative D. |
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