context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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Let | $ e $ | be the number of edges in the complete bipartite graph on | e |
and analogously for an equivalence class of | $ \sim_ { r } $ | ; | tilde sub r |
For a specific location | $ \bold { p } $ | on the ground , we use an area of 4 m | pee |
then | $ \rho $ | is non-acyclic . | rho. |
% % % % % % % % % If we specialize to the order- | $ \alpha_s $ | case , then the collinear perturbative expression from | Alpha s. |
be a finite dimensional vector space over a field | $ F $ | with basis | F |
The rank | $ s+1 $ | subdivision parts to which | s plus one. |
The expressive power of NNs refers to the gap between the two spaces | $ \mathcal { F } $ | and | F |
is of type | $ A $ | . | A |
Thus | $ C_q\cap lk_0 ( v , K ) =\emptyset $ | . Assume next that | C sub q intersection with the link of k sub 0, open parenthesis v, comma, K, close parenthesis, equals empty set. |
is calculated in each step of the procedure for all the chosen parameters up to the | $ j $ | -th step based on the projection of each HFM solution | j |
where | $ N $ | denotes the system size . It was confirmed that there is no significant size-dependent effect . All calculations were carried out using the ALPS application~ | N |
The second-rank term is nonvanishing only if the total angular momentum of the state | $ \Psi_0 $ | is at least 1 . Although the ground state of an alkali-metal atom has a nuclear spin in addition to the electron spin 1/2 so that the total angular momentum | Psi sub zero. |
we have | $ \gamma_r ( x , \ker L ) \Jac^0 ( L ) = \gamma ( x ) \jac { L } $ | . | Gamma sub r of x, kernel L, J c superscript 0 of L, equals gamma of x, jac L. |
321 & | $ - $ | 3.062 | minus |
& | $ 0.70 $ | Zero point seven. | |
Most of these methods are included in the broad family of projection-based methods : the internal state | $ x $ | is approximated by a vector of smaller dimension | x |
We apply this procedure | $ r $ | times , for a total cost of | r |
so that the average value of | $ D ( r ) $ | over the spherical capsule is constant : | D of r |
the coverage requirement for target point | $ j+1 $ | should be that of target point | j plus one |
& | $ 6.009535667649 [ 7 ] $ | & | six point zero five three six six four, open bracket seven, close bracket. |
we only consider the | $ \nabla \mathcal\ { U $ | -term . The | Nabla U |
or properties of | $ B $ | ( comfort with children , flammability , et cetera ) . | B |
dynamics of the tracer particles exploring the ramifications of non-zero | $ r $ | on them . Due to resetting , we anticipate the tracer particles to attain a non-equilibrium steady state at long times . But the question is - how does the | r |
and | $ -4 $ | because | Negative four. |
Letting | $ h ( x , m ) $ | denote | h of x, m |
tries to find a reward function | $ r $ | that assigns a low cost to the expert policy and a high cost to any other policy | r |
& 2 & 2 & Y & | $ ( -100 , -100 ) $ | & | Negative one hundred, negative one hundred. |
to estimate the integral of | $ f $ | , where | f |
from | $ D $ | to | D |
is the residue at | $ s=1 $ | of the Dedekind zeta-function | s equals one |
% By translating | $ Y $ | we may assume | Y |
as in | $ \mathbf { P } _ { Z|W } $ | . We index the columns by | P sub Z given W |
For | $ L $ | -smooth function | L |
& | $ 4 , -5,1,6,2,3 $ | four minus negative five, one, six, two, three. | |
and | $ \phi^ { \mathrm { G } } $ | denote the azimuth in the | phi raised to the power of G. |
with no rainbow | $ K_4 $ | . This solves a problem by Erd | K sub 4. |
direction of weights | $ ( k_1 , k_2 ) $ | satisfying | k sub 1, k sub 2. |
be a finite index subgroup and let | $ U_1 , \ldots , U_r \subset \R^d $ | be affine subspaces of dimension strictly smaller than | U sub 1, ellipsis, U sub r is a subset of R to the power of d. |
We can see from the figure that the proposed method has low localization accuracy when | $ T=3 $ | and | T equals three. |
for | $ a\geq b $ | is bounded as | a is greater than or equal to b. |
$ \eta^i_ { t } $ | is the mask term at | eta to the power of i sub t. | |
is a martingale in | $ m $ | . So its convex function is a submartingale and hence by Jensen 's | m |
are just | $ E^i $ | . This motivates % | E raised to the power of i. |
symmetric | $ \mathcal { U } $ | random variable s.t . | U |
at large | $ x $ | , this term converges and integrates to a constant , which we absorb into a constant within the logarithm | x |
the first subleading contribution to | $ Z ( \beta ) $ | at large | Z of beta. |
This tells us that | $ T_n $ | can be written as a linear combination of | T sub n. |
and perimeter | $ p_i $ | for | p sub i. |
Note that on each circle | $ C $ | , defined by | C |
and the Planck scale | $ \ell_p $ | , in which case we ca n't say much about the dictionary : basically just the generators of | ell sub p. |
that there exists a second component of | $ N ( v ) \cap U $ | containing a vertex | N of v intersect U |
The function | $ f $ | reaches its local maximum at | f |
Note that | $ t=0 $ | in the simulation time corresponds to the time of the protoplanetary gas dispersal some | t equals zero. |
and | $ NA ( = 0.275 ) $ | , the value of V becomes 80 - signifying that the multi-mode fiber can support around 1200 LP modes . Thus , due to the superposition of all those LP | N times A equals 0.275. |
and some | $ R $ | such that | R |
In the following we specify for each case whether the first neighbor of | $ v_2 $ | in the new instance is prescribed or not , and how to incorporate the neighbors of | v sub 2 |
its language context | $ X_L $ | , the options | X sub L. |
16 | $ \mu $ | m . However , with sufficiently high formamide concentrations , it may still be possible to identify formamide in these spectral regions , as these matrix bands are relatively weak | mu. |
and | $ z=0,1 $ | . The second derivatives of | z equals zero point one. |
and | $ b/c=2.33 $ | and pole orientation | b divided by c equals 3. |
for all | $ g\in G $ | and all | g belongs to G |
is a projection matrix of rank | $ p $ | . Therefore , we have | p |
7 & | $ b $ | b | |
and is invoked as an argument against the reality of the | $ \sim 9 $ | s pulsation | Approximately nine. |
of connected non-null subgraphs of | $ G $ | such that | G |
Let | $ x $ | and | x |
that maps any initial condition | $ u_0 $ | in the support of | u sub zero. |
-- -The AKIM and resulting circuit with generalized dual-unitarity can be straightforwardly extended to chains of qudits of arbitrary local Hilbert-space dimension | $ q $ | , and similar statements of solvable entanglement dynamics derived . The key is to observe that all diagrammatic relations of the basic tensors , which underlie the contractability of the | q |
$ K $ | , | K | |
We do not have a generic parameterization of | $ n^ { ( 2 ) } $ | that is suitable for this purpose and , therefore , resort to approximations , that is , rather than minimizing over | n to the power of 2. |
equations become intractable for large | $ N $ | , thus making it necessary to make additional approximations . The matrices | N |
We will work with the explicit sum and then deal later with the multiplicative factor | $ 1+o ( 1 ) $ | . The leading-order contribution to the explicit sum will come from the terms where | One plus little-o of one. |
are dimensionful couplings involved in different four-fermion operators involving | $ \nu_R $ | and | nu R. |
and | $ K^ { \prime } $ | . Chirality | K prime. |
of | $ \Z^n $ | associated to the system | Z to the power of n. |
to | $ 200 $ | . As a consequence , central charge | Two hundred. |
at each inner node | $ v $ | of | v |
is a | $ ( p+1 ) $ | -form . Similar to the electromagnetic case , there are only two kinds of Bargmann invariant actions : | p plus one. |
7 | $ \pm $ | 2.5 & | Plus or minus. |
Let us call | $ C $ | a | C |
we perform an intensity-clip operation to limit these peaks with a given ratio threshold | $ \theta $ | , and then redistribute the sum of extra peak values | Theta. |
and time | $ t $ | , we will define an integer | t |
This Hilbert space has a dimension | $ 4 $ | consisting of the basis vectors | four |
containing | $ o $ | . Moreover , this | o |
and | $ U_r\subset\widetilde { E } ^u_A $ | is a ball centered in | U sub r is a subset of E tilde to the power of u sub A. |
If | $ m $ | consecutive sentences describe a scene , we argue that those sentences are reasonable within logic . If | m |
$ \beta_3=D+i+j $ | , | Beta sub 3 equals D plus i plus j. | |
and | $ p\not=q $ | , | p not equals q |
% Therefore | $ \widehat { q } $ | is decreasing in | wide hat q |
2 | $ \pm $ | 0.2 & 62.9 | Plus or minus. |
where we require | $ u $ | to vanish when | u |
The Hamiltonian is | $ H =h_j ( \theta , t ) X_j $ | , for scalar | H equals h sub j open parenthesis theta, t close parenthesis X sub j. |
Note that | $ U $ | naturally inherits a smooth structure from | U |
at the endpoints of the corresponding dual edge | $ \hat { e } $ | . | e-hat |
modes that are even in | $ x $ | ) can be relatively quite large compared to the lowest-order mode , especially when the waist is larger . Adding an offset in | x |
and its truncated expansion | $ \mathcal { K } ^ { H , \rm NLL } $ | actually does not decrease from one-loop to two-loop accuracy . The crucial reason is that we evaluate the value of | NLL |
for each individual in the protected class under the null hypothesis | $ H_0 $ | that | H sub zero. |
Note that the matrices | $ B $ | and | B |
& | $ O ( 2 ) $ | & | O of 2. |
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