context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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Rawlins | $ ^ { 3 } $ | , Z. Rechav | Three to the power of three. |
is smaller than | $ \gamma $ | . Otherwise , when the lip target instant is away from the sentence end , we propose a linear combination of | Gamma. |
Note that | $ \Lcal^l $ | and | L to the power of l. |
vanishes for a conserved charge with | $ q_ { tot } = 0 $ | . This can be resolved , though , by treating the positive and negative charge density separately : % | q sub tot equals 0. |
we probe the precise behavior as | $ \lambda $ | approaches the critical points , demonstrating that each of heights | Lambda. |
for two graphs | $ G $ | and | G |
such that approaching it leads to a quickly growing | $ \mu_I $ | . We find in particular that the error ellipsoid corresponding to a 95 | mu sub I. |
This in turn reduced the temperature dependent recombination rate which leads to higher residual | $ X_e $ | . A higher residual | X sub e. |
would be a parent of | $ \prt $ | in | p-t |
Li+ | $ d $ | threshold , and in | d |
be a Jaffard family of | $ D $ | . Let | D |
and | $ \sigma \in \mathbb { R } ^ { m } $ | . In order to compute | Sigma belongs to the set of real numbers to the power of m. |
the sets | $ W $ | , | W |
has a constant coefficient | $ \mu $ | and results in | mu. |
Let | $ g ( x ) \in \Z [ x ] $ | be a monic irreducible polynomial of degree | g of x belongs to the set of polynomials with integer coefficients in x. |
the potential well is a half of what it was for the BP profile as a function of | $ R $ | . The frequency of the ` slow ' degree of freedom is however smaller , due to the additional anisotropy of the potential . Just as for the BP skyrmion | R |
M | $ -cylinders . One has $ | minus s dot one. | |
is second-order differentiation of | $ \bm { u } $ | with respect to the subscript | u |
it has been extremely valuable in better understanding the | $ k=1 $ | model~ | k equals one. |
for all | $ 1\leq i\leq r $ | and the ideal of | One is less than or equal to i, which is less than or equal to r. |
Then the following holds true for every | $ f\in L^2 ( \R^d ) $ | f belongs to L two of R to the power of d. | |
of~ | $ ( x , u ) $ | if it satisfies the adjoint equation~ | open parenthesis x comma u close parenthesis. |
we show that if | $ \Gamma $ | is an | Gamma. |
-electrons and | $ c $ | -electrons , which has an energy scale much smaller the direct exchanges , hence resulting the rapid decreasing of the | c |
is a | $ ( k-2 ) $ | -linear path . Any intersection between two edges of | (k minus 2) |
and therefore the stress tensor in the definition of | $ K $ | is quadratic in the fields , whereas | K |
where | $ \varepsilon $ | is the spatial average dissipation rate of kinetic energy . The total turbulent kinetic energy | epsilon. |
02~ps | $ ^ { 2 } $ | /rad , including the corresponding maximal | squared. |
Observe that | $ \phi $ | essentially restricts attention to that subteam | phi. |
all the derivatives of | $ u $ | on | u |
and define | $ \rvu_1 = \frac { \sqrt { 2 } } { \sqrt { q } } \sigma ( W\rvv_1 ) $ | and | r v sub 1 equals the square root of 2 divided by the square root of q times sigma of W r v sub 1. |
is a | $ 2 $ | -norm-scaled random normal vector . Taking | two |
and | $ b $ | , then obtain the scaling factor | b |
we note that even the ` good ' transformations produced by | $ \hat { S } _x^ { ( C \rightarrow A ) } $ | and | Hat S sub x superscript open parenthesis C right arrow A close parenthesis |
regions '' and `` very bright | $ \rm H\RomNumCap { 2 } $ | regions '' in the article . The radiation field model is stronger as a result of infrared radiation components scaling linearly with gas column densities . In the absence of | H naught cap 2. |
for the convex hull of the set | $ A $ | . | A |
ensures that | $ \mathcal { S } $ | correctly behaves as a representation rather than an anti-representation , i.e . , | ess |
and | $ V_i $ | with dimensions | V sub i |
corresponds to computing function | $ f $ | as well . | f |
This yields the average translational dynamics in the | $ x $ | - | x |
The differential on | $ \priddy { V , W } $ | is zero by | mole V, W |
and | $ k=1 $ | by executing the code , we obtain the following wave functions and their corresponding derivatives within both Region 1 and Region 3 . This outcome reflects the results of the | k equals one. |
It is tempting to ask if our strategy can be applied to other fillings as | $ \nu=n/ ( 2pn+1 ) $ | . The wave function | Nu equals n divided by open parenthesis two p n plus n close parenthesis. |
denote the vertex set of a graph | $ G $ | , and | G |
Since | $ v $ | must appear in equal quantities in both terms , | v |
For every | $ n\in\mathds { N } $ | there exists a sequence | n belongs to the set of natural numbers. |
and | $ j=j ' $ | . From this we get | j equals j prime. |
made up of the subspace | $ \ { \ket { e , g } , \ket { g , e } \ } $ | , in which | The set containing ket e comma g, ket g comma e. |
from | $ \vectorl\ { x $ | to | vector x |
and | $ X + Y $ | to indicate | X plus Y |
The probability that | $ G [ X ] $ | contains a copy of | G of X |
$ \Pi $ | is the collection of rules of the Rules~ | Pi. | |
and this classical information is transferred to an agent Bob at | $ N $ | , then Bob knows the state is already the honest qubit Alice at site | N |
holds : take | $ \bar { \ttt } _j^+ = \ttt_j^+ $ | for | Bar t t j plus equals t j plus |
for simulations with | $ \alpha_ { 4 } $ | , | Alpha sub 4. |
given the input | $ ( x , y ) $ | , by appropriately setting parameters in Eq . | open parenthesis x comma y close parenthesis. |
$ & $ | [ 20 , n ] | and | |
and given that | $ \bar { d } $ | is arbitrary , this holds for any | d bar |
the | $ X $ | -type logical operators are non-trivial closed | X |
and | $ b ' $ | . In particular , by state-operator correspondence , it corresponds to a state of the Hamiltonian | b prime |
is the genus of the % surface and | $ d $ | the number of singularities ) and such that the % application | d |
Here | $ P_ { t } $ | denotes the Ornstein-Uhlenbeck semigroup . Then , | P sub t. |
Without an asymmetry in the | $ \psi $ | population , the halos will annihilate after the initial stage of collapse . Annihilations will begin when the average distance between particles within the halo is less than the Compton | Psi. |
vector models at finite | $ \mu $ | and their relation to the large charge expansion have appeared , for instance , in Refs.~ | mu. |
gives the piece of the metric corresponding to the five-sphere | $ S^5 $ | , while the first two terms correspond to the | S to the power of 5. |
When | $ \alpha $ | is 0 , | Alpha. |
56 | $ \\ GroupDRO~\cite { sagawa2019distributionally } & \bfseries54.31 & $ | 34.95 | Group DRO tilde sigma raised to the power of ag raised to the power of iota, bold series raised to the power of 54.31. |
of the integral | $ J $ | satisfies | J |
and | $ j $ | , or 0 if | j |
and | $ m $ | are two positive integers such that | m |
for hexadecane at | $ T = 500 $ | ~K and a hydrostatic pressure of | T equals 500. |
~km~s | $ ^ { -1 } $ | to the power of negative one. | |
obtained by deleting all rows containing a variable | $ X_i $ | with | X sub i. |
is said to be a | $ p $ | -basis for | p |
and | $ - $ | components correspond to the | minus |
The average of | $ t $ | copies of a Haar-random state is the maximally mixed state over the symmetric subspace | t |
It follows that | $ \Delta_ { i } -2\leq \Delta_ { H } \leq \Delta_ { i } +2 $ | . | Delta sub i minus 2 is less than or equal to Delta sub H, which is less than or equal to Delta sub i plus 2. |
to neuron | $ i $ | is given by | i |
at | $ x $ | . The approximate derivative will be denoted by | x |
be two filtrations in | $ R_1 $ | and | R sub 1. |
the UAV path | $ \pi_a $ | will be defined in the following way : % | Pi subscript a. |
be the restriction of | $ \cnats $ | to sets of the form | c-n-a-t |
and | $ n_0 \in \N $ | such that | n sub zero belongs to the set of natural numbers. |
The index gradient | $ b $ | is defined as | b |
the | $ \chi-e $ | and | chi minus e. |
is the reward received when taking action | $ a_t $ | in state | a sub t. |
The | $ \Lambda $ | -grading is given by | Lambda. |
we are measuring quantifies the rate of KE being transferred across scale | $ \ell $ | and whether it is upscale or downscale . The above findings allow us to make an important conclusion about the length-scales at which the cascade is depositing energy . Using | lowercase L. |
at scales ranging between | $ \sim1\ , $ | Gauss to | Approximation of 1. |
and | $ \eta $ | are lighter than SM Higgs . Such decays and hence the corresponding couplings are constrained from the observation of the invisible Higgs decay . Considering the current constraint on the | eta. |
& Voltage angle difference between buses | $ i $ | and | i |
If we compute the first Walsh-Hadamard coefficient of the function | $ g_i $ | we get the following result : | g sub i. |
$ \lambda_2 $ | = 0.9 , | Lambda 2. | |
and the detection output of the fusion feature is represented by | $ \bm { O } _i $ | . The objective of collaborative 3D object detection is to minimize the detection loss | O sub i. |
$ b $ | } ( W2 ) ; | b | |
Since | $ L $ | is a Markov process , | L |
of size | $ |A|\le Z $ | that | The absolute value of A is less than or equal to Z. |
: Let | $ ( X , \ast ) $ | be a based | X, star. |
for the field content of | $ \mathcal { N } =2 $ | supergravity is given by | N equals two. |
$ p= { p_\mathrm { d } } =2 $ | , | p equals p sub d equals 2. |
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