context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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For the hypergraph family | $ \A^ * $ | we have | A conjugate. |
or | $ g_2 ( \bm x ) = x_3 $ | . However , unlike the previous examples , the HR model remains locally unobservable at | g sub 2 of x equals x sub 3. |
заканчивающееся образованием | $ T $ | -симметрического хаотического аттрактора ( Рис . | T |
ensembles with non-perturbatively | $ \mathcal { O } ( a ) $ | -improved Wilson fermions and tree-level improved L | big O of a. |
shows that as | $ \beta $ | decreases , attention entropy increases , resulting in a wider context being attended to . As | Beta. |
is the weight of the quality dimension | $ i $ | for the concept | i |
equation in chiral magnets in a typical time step-size of | $ 1 $ | ps , markedly exceeding the limit subjected by existing numerical methods of typically | one |
and satisfies | $ f ( \pi ' ) =\pi $ | . Even if the original path | f of pi prime equals pi |
for the | $ q $ | -deformation and | q |
Note that when | $ \Pi_ { n } $ | is visualized as the coordinates of an | Pi sub n. |
with respect to some scalar weight | $ \myW_i $ | is : | W sub i |
where | $ S_g $ | is the gauge part of the action | S sub g |
well-defined on | $ B $ | . | B |
and | $ \rho\colon G\to U^0 ( \AG ) $ | a homomorphism . Then there is a | Rho colon G to U zero of A |
for all | $ k\in I $ | and that there exists a positive constant | k belongs to I. |
& | $ l $ | real-valued phenotypic traits of an individual . | l |
are constants and | $ f ( k ) $ | is a second-order polynomial in | f of k |
to learn the correction function | $ \hat { g } $ | from paired RANS and DNS simulation estimated kinetic energy | g-hat. |
are interpreted by | $ 0 $ | . Therefore the sequent | zero |
is a topological generator of | $ G $ | . We can choose | G |
The dataset was divided into | $ 70\ % $ | for training , | 70 percent. |
suppose that | $ x = x_i $ | is a design point . Starting with the bias , we have , | x equals x sub i. |
$ \alpha=1 $ | and | Alpha equals one. | |
and | $ \Gamma_- $ | is the set | Gamma minus. |
The | $ k $ | -th variation regularization term penalizes the | k |
that are offsprings of type-A active individuals in generation | $ 0 $ | ; | zero |
and energy outflow rate | $ \dot { E } $ | are the principal physical quantities we measure . The mass outflow rate is given by | E dot. |
in a truncation defined by 16-cell and in a simplified case where the dynamics is defined by | $ SU ( 2 ) $ | BF theory . We compute mean geometry , fluctuations and correlations . The results are consistent with the hypothesis that refining the triangulation does not affect the global physical picture | S U two. |
into | $ v_1 $ | and | v sub 1 |
for any | $ t \geq 0 $ | , | t is greater than or equal to zero. |
If | $ c $ | and | c |
and | $ t_2 $ | strongly normalise . We prove , by induction on | t sub 2 |
Writing | $ e_ { t_1 } $ | for the positive generator of the fiber of | e sub t sub 1 |
Here R | $ _0 $ | is the normal Hall coefficient and R | Zero. |
is determined by an invertible sheaf | $ \calL $ | over | script L. |
by | $ \hat { \bb { \theta } } _ { \mathrm { A } } $ | from main study , we can substitute the effect size of | hat B theta sub A |
and hence | $ \beta $ | is a fixed point for | Beta. |
the vector potential relevant for the electric field undergoes a Wick rotation so that | $ A_4=iA_0 $ | , similarly to the case of a chemical potential | A sub 4 equals i times A sub 0. |
& | $ 85.4 $ | 85.4: Eighty-five point four | |
$ \Delta x=\sqrt { 2 } \cdot 2^ { -5 } \approx 0.0044 $ | , | Delta x equals square root of 2 times 2 to the power of negative 5, approximately 0.0044. | |
One approach to handling the varying | $ \tau $ | in the inverse transform method is to first sample a uniform random variable | Tau. |
This figure gives the results for weak and strong adversaries and the case where the function | $ g ( \cdot ) $ | is not applied . The following points are evident from Fig . | g of dot. |
the restriction of | $ H $ | to the interval | H |
real Wilson coefficients & of the EFT and UV amplitude \\ \hline Causality & \tabitem No resolvable time advance & \tabitem Analyticity of amplitude \\ & & in the complex | $ s $ | plane for fixed | s |
We reuse the first stage summarization method based on BART and continual fine-tune it to generate the target sequence | $ T_Y $ | of the gold summary based on the input representations on the training set . Note that we adopt the partial casual masking in the decoder to perform auto-aggressive generation , | T Y. |
+128M @ | $ 224^2 $ | & 628 & 67.8 & 60.4 & 38.3 & 81.2 & 52.8 & 56.4 & 40.1 & 58.4 & 64.0 & 81.5 | The equation you provided is: 224 squared. |
where | $ V $ | and | V |
$ ^\textrm { \scriptsize 163 } $ | , | One hundred sixty-three. | |
whereas | $ \alpha $ | varies for different curves in the left and middle plots for fixed values of | Alpha. |
i | $ ) $ | ] The set of all column vectors of | closing parenthesis |
The values of the resistors chosen at 295 K and 35 K are motivated by the | $ RT $ | and indicated by open circles in fig . | R T |
The | $ \Phi_ { ab } $ | are kernels given by the logarithmic derivatives of the phase shifts , and | Phi sub a b. |
$ r^j $ | is the | r raised to the power of j. | |
We still refer to | $ P $ | as pressure , since it is the thermodynamic dual of the stretch . To have an integrable Boltzmann weight , | P |
em | $ f $ | is | f |
of | $ H $ | and find that | H |
in the | $ ( t , s ) $ | -adaptive method seem to be located equidistantly on both sides of the limit of | t comma s. |
algebra | $ V $ | is called | V |
on the | $ k $ | th-order perturbatively diagonalized Floquet Hamiltonian | k |
the characterisation gives the impression that there is a dependence on the lag | $ h $ | . % Completely analogously to | h |
Let | $ m $ | denote the smallest weight of | m |
proton collisions at | $ \sqrt { s } = 13 $ | Square root of s equals 13. | |
$ \xi_2 $ | , | xi sub 2 | |
983 & | $ p_ { ulcl } $ | & 0.07 & | p sub l l |
are determined by analyzing | $ L $ | -wave | L |
and | $ A_1^+ $ | contain all the other leaving arcs of | A one superscript plus. |
Expanding in powers of the fields one obtains an exact hierarchy of flow equations for the | $ n $ | -particle irreducible vertex functions . In practical implementations , this infinite hierarchy is truncated at the two-particle level : neglecting the renormalization of three- and higher order particle vertices yields | n |
only determines the magnitude that the control action | $ a_i $ | and its derivatives changes at the junction . In particular , setting | a sub i. |
is obtained by interpreting | $ \widehat\ B ( x , z ) $ | as counting binary trees with a distinguished leaf | wide hat B of x and z |
$ X $ | is | X | |
5~ | $ \mu $ | m . % Another comparison between CH | mu. |
with boundary at infinity equal to | $ C_ { 1,2 } $ | . For each triple of components | C sub one comma two. |
It follows from the formula for | $ \alpha_\ell $ | established above that | Alpha sub l. |
High-T | $ _C $ | surface superconductivity in topological Weyl semimetal t-PtBi | C |
in sentence | $ d $ | in tf†“ idf means that the word might be more important in the sentence . | d |
em an affine connection on the vector bundle | $ K $ | . | K |
$ H $ | denotes the depth of the hierarchy . Each level | H | |
be a graph and | $ W\subseteq V ( G ) $ | . We call a set | W is a subset of or equal to V of G. |
denote the grid formed by | $ k $ | horizontal straight-line segments crossing | k |
in | $ H_n $ | . Suppose towards contradiction that | H sub n. |
X_S \times Y_S | $ , $ | X_S | comma |
D_i | $ , number of airborne droplets , and $ | C_i | Number of air droplets and |
in | $ \mathbb R $ | , we have | the set of real numbers. |
The adjacency matrix of a graph | $ G=\left ( V , E\right ) $ | with | G equals open parenthesis V comma E close parenthesis. |
The functional | $ F $ | and the initialization measure | F |
$ j $ | is Klein 's modular invariant function , | j | |
Together with the help of Chebyshev polynomials of the second kind | $ U_n ( x ) $ | U n of x | |
denote column | $ j $ | of | j |
& & & | $ H^ { \dagger } $ | & | H dagger. |
setting | $ k_y=0 $ | , | k sub y equals zero. |
s | $ . % \red { Hast du hier $ | E [ | percent red hast du |
instead of | $ A=2 $ | as we worked through Example 10 | A equals 2. |
away from | $ z^ { * } $ | , we obtain that | z star |
is a fundamental domain for | $ E_q ( K ) $ | . Then it holds true that | E sub q of K |
factor | $ g $ | and the external field | g |
In the language of that literature one says that `` | $ ( \mcL , \widehat { \varphi } ^ { -1 } ) $ | is bounded by a bound | Open parenthesis, m c L, comma, hat phi inverse, close parenthesis. |
then the operator | $ \mathcal { L } $ | is Hermitian with respect to the inner product | ell |
east-steps of | $ P $ | on the line | P |
and since all level sets of | $ y $ | have measure zero by | y |
Hetero-epitaxial growth of | $ \alpha $ | -W thin films on the other hand , may be enabled through a lattice match between in-plane atomic distances in the film and substrate . The crystal structure of bcc | Alpha. |
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