context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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$ \Delta t $ | can increase by the factor of | Delta t. | |
& | $ 25.4 $ | & | 25.4: Two five point four |
The comparison of two triplets | $ a_1=\langle a_ { s1 } , a_ { p1 } , a_ { o1 } \rangle $ | and | a sub 1 equals the inner product of a sub s1 and a sub p1 and a sub o1 |
there will be a more significant mismatch between actual jet | $ p_T $ | and | p sub T |
be the underlying undirected cycle of | $ \overrightarrow { C } $ | . We define a new perfect matching | vector C |
over the energy gap | $ w $ | . The aim is to find the largest possible | w |
% The strong Bi/Pb 6 | $ s $ | -O 2 | s |
on a | $ C^ * $ | -algebra as the smallest number | C star. |
which is a pointed metric space whose elements are equivalence classes of sequences | $ ( p_n ) $ | which are bounded relative to | p sub n. |
and | $ ( A^ { \Phi } , \fbs { B_W } ) $ | on | A raised to the power of Phi, dot dot dot, B sub W. |
for any | $ C > 0 , \alpha > -1 $ | . However it is not automatically verified , as for instance | C greater than 0, alpha greater than negative 1. |
-rack of | $ K $ | . These relations are equivalent to the relations | K |
& & | $ \beta_2 $ | & -0.083 & 0.513 & 0.270 & 94 & 2.04 & -0.061 & 0.492 & 0.245 & 93 & 1.89 & -0.057 & 0.545 & 0.300 & 92 & 2.09 | Beta two. |
as well as a torsion section | $ r $ | which always lies in the correct component of the fiber . ( Per Remark~ | r |
- | $ d^9 $ | configuration . On the other hand , if one manages to push | d raised to the power of 9. |
$ a $ | from | a | |
If the sequence | $ \ { f_k\ } $ | in Lemma | f sub k. |
simply returns the expectation value | $ \langle p_T \rangle $ | with no observable dependence . % Deviations of the model output from | the average value of p sub T. |
in | $ \Delta \mathcal { H } _1 $ | is nonnegative . For | Delta H sub 1 |
As | $ C $ | is a | C |
smaller values of | $ \lambda $ | allow Alice and Bob to transfer entanglement to a larger number of pairs . This trend is evident from Fig . | Lambda. |
morphisms whose both ranks equal | $ r $ | . | r |
there exist a subsequence of | $ \varepsilon $ | and | epsilon. |
for all | $ \xi\in \mathfrak { g } $ | , where | Xi belongs to the set of g. |
$ f $ | is node feature . | f | |
to a single terminal | $ r $ | of degree | r |
The attractive force between | $ \Lambda_c $ | and | Lambda sub c. |
Note that the second Stokes pulse | $ \Omega_ { s2_0 } ( t ) $ | completely overlaps with the pump pulse | Omega sub s 2 sub 0 of t |
The vertical lines indicate the range of | $ \alpha $ | with which the experimental data can be well reproduced . In Fig.~ | Alpha. |
$ x_ { \text { quasi-periodic } } \simeq 0.00186 Re_ { b } + 0.164 $ | and | x sub negative periodic is approximately 0.00186 Re sub b plus 0.164. | |
-formula~ | $ \varphi ( x ) $ | . A sentence expressing that the problem defined by~ | phi of x. |
and | $ \psi $ | . This implies that | Psi. |
we aim to design a control signal | $ u ( k ) $ | for the aforementioned setups . To accomplish this , we propose an online switched controller composed of an exciting signal | u at k |
and choosing | $ h $ | by cross -- validation . A precise control of the discretisation error is left for future research . | h |
the features used as inputs and the target SGS stresses are all obtained from the three components of the velocity field | $ \boldsymbol { u } $ | . Therefore , in the first stage of preparing the features , components of the instantaneous velocity fields obtained from DNSs listed in Tab . | u |
the neighborhood | $ N ( X ) $ | can be chosen small enough so that its boundary sphere does n't intersect any components of | N of X |
while systems with | $ \rm { SFR } \gtrsim 5 $ | SFR is greater than or equal to 5 | |
denote the | $ i $ | th largest eigenvalue of | i |
which is derived from the time-dependent Hamiltonian | $ H ( t ) $ | defined in Eq.~ | H of t. |
and | $ { w } ( u_j^n ) $ | . | w of u sub j superscript n |
& | $ - $ | minus | |
and | $ F $ | is square-free in | F |
for | $ k\geq 8 $ | , there exist maps | k greater than or equal to 8 |
$ n^\alpha $ | is future-directed , normal to each point of a given | n raised to the power of alpha. | |
the ultimate bound of | $ V $ | for ETC and ETPC are also higher than the designed | V |
we make use of clustering-based approaches : Riemmanian | $ K $ | -means ( | K |
for each | $ s \ ( 1\leq s\leq k ) $ | . Observe that | s, where s ranges from 1 to k. |
implying that the absolute value of | $ f_ { \mathrm { NL } } $ | is less than | f sub N L |
since this would lead to a zero in the denominator of | $ C_i $ | . Note that in cases where the parameters of the Heun functions are such that they reduce to polynomials , or to solutions valid at more than one singularity , | C sub i. |
& | $ 55.12 $ | & | 55.12: Fifty-five point one two. |
and berthing end time of assignment | $ i $ | , respectively . | i |
& | $ 3 $ | & | three |
and | $ L > 0 $ | . Let | L is greater than zero. |
$ ( AH\vee BK \vee \no { H } K ) | ( H \vee K ) $ | is true , while | (AH or BK or not HK) or (H or K) : None | |
The RBF network is used to learn the | $ i , j $ | -th entry of | i, j |
has two components each satisfying | $ E=2V-1 $ | and having no triple bond . By Proposition | E equals two V minus one. |
under the action of | $ L $ | . However , recall that for non-degenerate | L |
The remaining portion of the parameter space | $ \Theta $ | is the set of all regular points . | Theta. |
01 | $ \pm $ | 0.18 & 19 & 5.71 | Plus or minus. |
it suffices to consider the matching equation obtained at quadratic order in the | $ u $ | -component , which is given by | u |
$ \Gamma $ | diverges when approaching an ideal emitter -- resonator system with no decay of the emitter into the environment , i.e . , for | Gamma. | |
$ a_1 , a_2 > 0 $ | , such that | a sub 1, a sub 2 greater than zero. | |
to be finite | $ \P_0 $ | -a.s.~for | P zero. |
for | $ q \geq 3 $ | . We also have the following inequality for every | q is greater than or equal to 3. |
and | $ \beta $ | . | Beta. |
$ p $ | =5 mTorr ) with an elevated substrate temperature of 400 | p | |
and another in which the spin is aligned with the | $ y $ | axis . In~ | y |
in | $ [ 0,1 ] $ | , and let | the interval from zero to one, inclusive. |
do not depend on the metric function | $ g ( r ) $ | , while the potentials in Fig.~ | g of r |
The ionized gas traced by the H | $ \alpha $ | emission shows a spatial structure similar to that of the rest-frame UV . The peak emission coincides with that of the HST F160W and MIRIM F560W images , clump S | Alpha. |
$ \xi $ | is a space-time white noise so that | xi. | |
the physical value of the magnetic Reynolds ' number in the observed dwarf nova discs is around | $ \sim 10^ { 10 } $ | . Thus , while the lack of agreement between the shearing box simulations and the observational data may yet be due to a discrepancy in magnetic diffusivity , it may | Approximately ten to the power of ten. |
$ i $ | ) % | i | |
$ \phi\in C $ | , as | phi belongs to C. | |
9 & | $ { \bm { 83.5 } } $ | & 82.4 & 76.4 & 91.7 & | B times m times 83.5 |
the number of false positives is bounded by | $ \min ( M , N ) $ | . Hence , we have | minimum of M and N |
We sample uniformly from a range of pseudo-PSNR numbers and then compute the corresponding value for | $ p $ | accordingly . The process is described in section~ | p |
We first observe that no algorithm can guarantee acceptance probability greater than | $ 1/2 $ | . | One-half. |
Let | $ r $ | and | r |
-absolute and | $ ( n , m ) $ | -relative cliques : | n comma m. |
player | $ l $ | , i.e . , if and only if | l |
and | $ v_4 $ | where payments of 1 can emerge on each edge in the cycle . This network is a global optimum with respect to total assets . | v sub 4 |
to | $ L^1 $ | L one. | |
there exists | $ r\in ( s , s ' ) $ | such that | r is an element of s, s prime. |
that can be chosen from an | $ n $ | -element set ? ( b ) * * * Problem 126 * * | n |
The set of vertices | $ V $ | is as follows : | V |
We assume flat | $ \Lambda $ | CDM cosmology throughout this paper with | Lambda. |
method aims to predict shim values for the | $ X , Y $ | , and | X, Y |
and | $ \theta $ | is close to zero . Under this condition , we have | Theta. |
-- -As an application of our proposed | $ \bm { r } $ | -space formula , we explore the Euler topology in quasicrystals and amorphous lattices . Specifically , we consider the 2D Ammann-Beenker-tiling quasicrystal , which possesses 8-fold rotational symmetry but lacks | r |
let | $ \epsilon_1=0 $ | . Let | Epsilon sub one equals zero. |
and | $ \text { d } \Xi=\text { d } u\text { d } v $ | . Based on the moments of the gas distribution function , the flow variables and their fluxes can be obtained . Due to the similar equations for different layers , | dee Xi equals dee u dee v |
Let | $ V_ { F , \infty } \subset S\subset V_F $ | be a finite set such that all objects | V sub F, infinity subset S subset V sub F. |
& | $ c_j $ | not satisfied by the | c sub j. |
- | $ v $ | -path in | v |
We can take much larger steps if we instead expand the EBV as a Taylor series in | $ g $ | to obtain a guess at the new | g |
with | $ |a| \leq 2 N ( J ) ^ { 1/n } $ | . | Absolute value of a is less than or equal to 2 times N times J raised to the power of one over n. |
em either | $ G_B $ | is a | G sub B. |
$ H_ { \mathcal F } $ | is the Rado graph . Otherwise , there exists a smallest | H sub F | |
and | $ \ { Z_ { t } \ } $ | in | Z subscript t |
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