context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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em the complement of the orbit of | $ \Lambda_1 \cup \dots \cup \Lambda_k $ | is a non-empty set of isolated vertices . | Lambda sub 1 union dot dot dot union Lambda sub k |
$ 3 $ | } ; | three | |
$ k_ * $ | is the pivot scale corresponding to | k star | |
$ \vtheta_0^v $ | and | v-theta-sub-0 to the power of v | |
$ while $ | Y | While. | |
$ \mathsf { AL } $ | parameters , trained SNR value , and epoch number . We found that the best performance versus complexity on both examined | A L. | |
Assume that there are | $ N $ | samples and | N |
Taylor series truncation of | $ e^x $ | ] | e to the power of x. |
c'est à dire les | $ [ r , s ] $ | avec | the commutator of r and s. |
we can characterize the pdf of | $ \mathbf { z } _l $ | as Gaussian distribution . First , define the estimated mean | z sub l. |
In the | $ J/\psi $ | case , the GPD modeling given by the Shuvaev transform is surmised to have an even smaller effect in comparison to the | J psi. |
but absent in | $ \textrm { KV } _ { 3 } \textrm { Sb } _ { 5 } $ | KV 3 Sb 5: None | |
is replaced either by | $ 2 $ | , with probability | two |
$ 0 $ | } } | zero | |
in which the initial state | $ |i\rangle $ | is coupled to the excited state | The absolute value of i. |
The log-likelihood function of training data | $ \mathcal { T } $ | under the OSS setup is | tee |
Let | $ \xi , \tilde { \xi } \in \Xi^0 $ | and | Xi, tilde xi belong to Xi to the power of zero. |
Let us begin with the | $ nD $ | nD: None | |
for the gravitons to be non-ghostly and the stability of the vector sector requires | $ J_ { \rm I } \xi ' > 0 $ | , we see that the scalar sector can not be ghost-free , since we have | J sub I xi prime greater than zero. |
If | $ l=0 $ | , the desired estimate is Corollary | l equals zero. |
The first provides control of the nearest neighbour distances to a point | $ x $ | in the support of | x |
in terms of functions of | $ \chi $ | . Substituting the result into | chi. |
and | $ W $ | couplings , and the Yukawa and trilinear and quartic self-couplings of the Higgs boson are modified in the presence of non-zero coefficients for the dimension-6 and -8 Higgs operators . | W |
of vertices which contains the initially infected vertex and all vertices that were ever adjacent to infected vertices before time | $ t $ | . Note that | t |
and | $ L $ | , | L |
-dependent denominator factors in | $ \S_n $ | coincides with the one in | S sub n. |
A function | $ f $ | is | f |
$ w_1 $ | and | w sub 1. | |
and 1 more PMU on side | $ B $ | . Then the condition that | B |
be a linear operator and | $ 1 \leq p , q \leq \infty $ | . Let | One is less than or equal to p, comma, q, comma, less than or equal to infinity. |
K | $ _u $ | = 0.73 MJ/m | Underscore u. |
and show that it goes to zero as | $ N\to+\infty $ | . That term can be rewritten as | N tends to positive infinity. |
The only explicit use of the lattice spacing is made to convert the Fermi constant | $ G_F $ | as an external scale to lattice units . We use the value | G sub F |
is a non-empty family of | $ k $ | -bisimulations | k |
and | $ 2 $ | of energies | two |
for | $ % q=8,4,2,1 $ | , | q equals 8, 4, 2, 1. |
An infinite two-sided sequence of zeros and ones will be called | $ k $ | - { | k |
and infected excess degree of | $ s $ | . For this task , we leverage the properties of probability generating functions ( PGFs ) . If we take a random | s |
at a confidence level lower than | $ 90\ % $ | Ninety percent. | |
\boldmath | $ r $ | } } } | r |
For arbitrary | $ j $ | , we prove this by induction . For | j |
curves of bidegree | $ ( 1,0 ) $ | , the curves | one comma zero. |
: The step size | $ \epsilon $ | in | epsilon. |
and | $ s > 0 $ | we have | s is greater than zero. |
of the standardized kernel | $ \hat { \kappa } = ( 1/\sigma_ { \theta } ^2 ) \kappa $ | to draw | Hat kappa equals open parenthesis one over sigma theta squared close parenthesis kappa. |
independent of | $ \rho $ | and it does not affect the estimate in Proposition | rho. |
is the shared weights that are for calculating output for all pixel positions centered by | $ ( i , j ) $ | . The computational cost is | open parenthesis i, comma j, close parenthesis. |
yr | $ ^ { -1 } $ | ~ | to the power of negative one. |
we obtain the contradiction | $ \WW ( \palt + ( \vrt , \dots , t ) ) \strictpo \WW ( \p ) $ | . | W of t plus v r t, dot dot dot, t, multiplied by the strict value of W of p. |
are connected and | $ p $ | -finite . This completes the proof ( by induction on the finite number of connected components ) . | p |
affecting only the | $ ( \ell,0 ) $ | modes , and the oscillatory memory . For both the memory and tail contributions , we compute the integrands for generic orbits , i.e . , in terms of | ell comma zero. |
already if | $ |\Gamma|\geq 2 $ | . Assume | Absolute value of Gamma, greater than or equal to 2. |
is aliased with | $ p $ | . In this case , all paths rooted in | p |
98 | $ \pm $ | 0.01 & | Plus or minus. |
785 | $ \pm $ | 0.002 & 9.806 | Plus or minus. |
is simply some | $ \pi \in \Pi $ | for which | Pi belongs to capital pi. |
gradient scale | $ s $ | ) . % | s |
where | $ \boldsymbol { pc1 } $ | is shown to have a | p c 1 |
% and | $ 2 $ | `` external '' vertices | two |
; Number of iterations | $ S $ | S | |
we orthogonalize | $ { \bf u } ^l_ { H , I } $ | against | u raised to the power of l sub H, I. |
with | $ x_s $ | being a point source location . | x sub s. |
at a time from homes with IDs | $ 20 $ | - | Twenty. |
The structure of Wilson loops in ABJM theory is richer than in | $ { \cal N } =4 $ | SYM~ | N equals four. |
and | $ m_ { ij } $ | are bounded variables and the two Donsker classes are the same . The asymptotic properties of | m sub i j |
and | $ \Tilde { T } $ | stands for | Tilde T |
$ & ( \text { A5 } ) \\ $ | A 5 | ||
At the point | $ \alpha_i=0 $ | , the first derivative | Alpha sub i equals zero. |
1 | $ \pm $ | 4.0 & | Plus or minus. |
gives a solution with less than | $ 1\ % $ | of the constraint violation , such that the empirical | One percent. |
When | $ 2k_0 $ | becomes equal to the | Two k sub zero. |
is weakly between | $ a $ | and | a |
& | $ \log { } $ | & | log. |
on a graph | $ G $ | with initial set | G |
-pigmented words | $ \PWord $ | such that | pee-mew |
let | $ T $ | be large enough so that | T |
& | $ 0.018 $ | & & | Zero point zero one eight. |
problem | $ \mathcal { P } _1 $ | is reduced to the conventional radar-only beamforming design , determining the optimal beampattern for radar detection . Explicitly , any improvements of the communication QoS and PLS level are attained | P sub 1. |
contours of 5 | $ \sigma $ | and 20 | Sigma. |
for | $ i = 1 , \dots , k $ | and | i equals 1, dot dot dot, k. |
% The following weighting certifies that the value of | $ DLP ( T_ { 32 } , T_ { 33 } ) $ | is less than | D P L open parenthesis T sub 32 comma T sub 33 close parenthesis |
should be specified for each | $ t $ | ; that is , | t |
maximum stable sets and | $ c $ | -optimal stable sets in a bipartite graph ; as well as antichains , maximum antichains , | c |
-ary linear code with a generator matrix | $ G= ( \boldsymbol { g } _0^T , \cdots , \boldsymbol { g } _ { n-1 } ^T ) $ | , where | G equals open parenthesis, g sub zero transpose, comma, ellipsis, g sub n minus one transpose, close parenthesis. |
requires us to choose a vector-valued function | $ U_0 $ | in | U sub zero. |
than in the case of the standard | $ g ( r ) $ | -like formula with | g of r |
Let | $ \bar y\in\mathbb\ R^m $ | be fixed , | y bar belongs to the set of real numbers raised to the power of m. |
orbits along the | $ s $ | , | s |
where | $ \mathbf { O } $ | is the observed spectrum , | capital O in bold. |
with negative | $ J $ | since there is a one-to-one mapping by simply replacing | J |
in terms of | $ \mathscr { C } $ | . % We stress at this point that it is crucial for us to assume some bound on | C script. |
of | $ M $ | under the adjoint isomorphism . Let | M |
this is an example of a purely imperative proof that involves no propositional formulas other than | $ \top $ | . As we will see in Example | Top. |
and | $ \gamma ' $ | . Consider the arc obtained by following | Gamma prime. |
and | $ S_2 $ | will never have non-zero entries for the same column . As shown in~ | S sub 2. |
We look for optimized fabrication parameters such that we can observe a strong coupling between | $ s $ | and | s |
$ { \vec R } _b $ | is the lattice vector along | R with a vector arrow, subscript b. | |
are observed before a fixed time | $ t $ | ( where | t |
Number of agents | $ n $ | } We define the agent to be the operator for a single asset , and each agent would be responsible for the execution of the corresponding order . | n |
and | $ ( n+1 ) ^ { th } $ | n plus one raised to the power of th. |
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