context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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This result represents the modular conjugation for a thermal state of the fermion on the line reduced to | $ n $ | intervals . For | n |
where 'significant ' refers to a pre-specified threshold | $ \gamma $ | . This means that a | Gamma. |
$ 0.5,18 $ | ] ~TeV as well as integrating over the time | 0.5,18: None | |
and ten are | $ t $ | -distributed random variables with | t |
We endow | $ X $ | with the product topology , which makes it a compact metric space . Let | X |
then showing that if | $ T $ | is positive and | T |
the VSS network 's encoder | $ E $ | extracts feature representations for each frame : | E |
for all | $ x $ | and | x |
are dimensionless coefficients and | $ \phi=1+\theta^2 m_ { 3/2 } $ | is the conformal compensator . While the operators in | Phi equals one plus theta squared m sub three halves. |
We use | $ b $ | to keep track of the configuration of the nodes in the previous time step and we only change the values for the current time step while iterating over the nodes | b |
at large axion masses the sensitivity boost primarily comes from having a large number of layers and is limited by the constraint | $ N \leq N_\text { max } $ | . For polycrystalline ferrite , one can only use this many layers if the setup is operated well away from the resonance . Numerically , we find | N less than or equal to N max |
Fetch the relative time | $ \xi $ | from the Sharer , and sleep for | xi. |
is defined by at most | $ d_ { \text { comb } } $ | constraints . The first statement of Theorem | d sub comb. |
and otherwise depend only on | $ \xi $ | . Then , the conservation equation , | xi. |
& | $ 447 $ | & | Four hundred forty-seven. |
Unrolling the automaton once allows us to consider the triplet | $ ( q_2 , q'_3 , q'_2 ) $ | whose edge labels correspond to the pair of letters | q sub 2, q prime sub 3, q prime sub 2 |
Measurement of the | $ W $ | -boson mass in pp collisions at | W |
also approximately equals to | $ \Delta\tau_ * $ | for the dyonic Kerr-Sen-AdS | Delta tau star |
is an | $ r $ | -stacking partition of | r |
indicate the number of items in group | $ g $ | , and let | g |
$ \omega_g $ | is independent of the topological index. | omega sub g. | |
The label | $ ( B , g , \emptyset ) $ | is white , because | Open bracket B, comma g, comma empty set. |
and | $ V $ | are the intra-orbital and inter-orbital interaction parameters , respectively , and | V |
to | $ P $ | , and therefore we obtain a quasi-isomorphism | P |
the terms in | $ \hlaplace $ | only involve a constant number of spins , which allows for its implementation in near-term experiments . To develop some intuition , let us first analyze the modified QAA when | ell-ah. |
allows to construct | $ \tilde { \ell_2 } $ | on | tilde ell two. |
$ B1 $ | , | B1: None | |
4 | $ \pm $ | 1.5 & 50.5 & 18.3 | Plus or minus. |
for | $ s \geq 2 $ | . If | s is greater than or equal to 2. |
and denote by | $ \mathfrak { F } ' \to \mathfrak { S } ' $ | the corresponding family . | F prime maps to S prime. |
25 | $ \pm $ | 9.81 | Plus or minus. |
we expect | $ S $ | and | S |
the full set of information available to agents | $ t $ | and | t |
we plot for which | $ n $ | it holds that | n |
Our goal is to study the effective behavior or the relationship | $ v^\text { eff } ( F ) $ | . However , this relationship is non-smooth at | v eff of F |
where | $ \mathbf { B } $ | is the causal strength matrix , | B |
with | $ J^P = 5/2^- $ | and mass of 11125.0 MeV , | J to the power of P equals 5 divided by 2 minus. |
for some suitable | $ C_3 > 0 $ | . Pick any | C sub 3 is greater than zero. |
We next demonstrate that the bi-Higgs-frequency sidebands observed for strong driving fields and large | $ U $ | are generated by non-adiabatic pseudo-spin canting that is parametrically driven by relative phase dynamics at frequency | U |
Suppose | $ F_J^C=\emptyset $ | . Then by Lemma | F sub J raised to the power of C equals empty set. |
the topology of these lines depends on the constant of integration | $ \mathcal { C } _w $ | . Moreover , along these lines | C sub w. |
and | $ g ( 1 ) =0 $ | , the map | g of 1 equals 0. |
-linear | $ \infty $ | -representation of | Infinity. |
for every child | $ t ' $ | of | t prime |
model that at | $ N=\infty $ | , its standard implementation misses in all dimensions below four some fixed points of the renormalization group . These new fixed points show singularities under the form of cusps at | N equals infinity. |
in external magnetic field | $ \mathbf { h } $ | is defined via the Hamiltonian | h |
we neglect terms proportional to | $ ( 1/\mu ) ^n $ | with | One divided by mu raised to the power of n. |
and patient | $ j $ | j | |
but with the same | $ p $ | ) to be scheduled on processors of type | p |
$ \psi_\text { vac } $ | , is assumed to be a plane wave in the vacuum . The interaction term | psi sub vac | |
and thus | $ d $ | is isotone . | d |
% If | $ y \in \om [ \K ] $ | , then there sequences | y belongs to the set of homomorphisms from K |
em For each | $ f $ | , Algorithm | f |
At other | $ R $ | values , | R |
It is easy to see that there has to be at least one | $ \beta $ | -player and one | Beta. |
-structure satisfying the Scott sentence of | $ \mcM $ | ; | M-calligraphic. |
s | $ ^ { -1 } $ | to the power of negative one. | |
and the cylindrical almost complex structure | $ \widetilde J = J \oplus J_0 $ | which is | Widetilde J equals J direct sum J sub zero |
from | $ \vX_i $ | , thus allowing for the incorporation of flexible neural network architectures . We summarize our method in Algorithm~ | X sub i. |
and correspondingly the vacuum and free field module for | $ \V ( Y , S_0 ) =\Pi ( Y , S_0 ) $ | are by definition the same , so that the two aforementioned conjectures coincide in this case . Recall that | V of Y, S sub 0 equals Pi of Y, S sub 0. |
The latent vector | $ z_T $ | is sampled from the standard normal distribution | z sub T |
we can simply take | $ P=A_ { \G } $ | and there is nothing to prove . Suppose instead that , for some | P equals A sub G. |
time on an unweighted undirected planar graph with diameter | $ D $ | . We first present a randomized | D |
Then | $ \Sigma_ { 1 } $ | and | sigma sub one. |
denotes hyperbolic lattice of rank | $ 2 $ | , and | two |
eigenvalue for the bilinear form specified in the that row must lie between the | $ i $ | i | |
the ratio | $ \kappa $ | required for one-bit massive MIMO cellular system to achieve the sum average rate of conventional massive MIMO cellular system with | kappa. |
this is like imposing Dirichlet boundary conditions for | $ q_ { AB } $ | . Note | q sub A B. |
The fermion parity fluctuation attributed to the complex fermion decoration under the | $ F $ | -move is | F |
The standard flow on the | $ n $ | -simplex } Let | n |
where | $ P $ | is the binary orbital period . We solve Equations 5.6 and 5.7 in | P |
The parameters | $ f_\ell $ | and | f sub l |
suppose all external momenta except | $ k_s $ | are massive , with equal mass | k sub s. |
there there exists at most one | $ Q $ | which is a Q-structure for | Q |
illustrate | $ \mathcal { KP } $ | shows a stable and high correlation with the ranking metric , confirming the robustness of | K P |
these magnifications start to deviate from the SSS case if the | $ \delta\theta $ | and | delta theta. |
is an eigenvalue of | $ A $ | , usually called its | A |
gates simultaneously on qubits | $ Q_ { i , j } $ | and | Q sub i comma j. |
with | $ \kappa $ | the wave number. | kappa. |
$ s_ { 28 } $ | , | s sub 28 | |
we have | $ S ( x , z ; D ) =\emptyset $ | by Lemma | S of x, z, D equals empty set. |
manifestly defined | $ a $ | in a gauge invariant way . Finally , in | a |
be a | $ T $ | -torsor over | T |
and the Bernoulli 's quantity | $ B $ | : | B |
is known apriori for all | $ x\in \cX $ | . The task of generalization to satisfy the constraint on unseen data is well-defined in this setting . Furthermore , if a surrogate loss | x belongs to the set X. |
we measure the weight of the tails of distribution | $ P $ | by | P |
It can be easily seen that these extrema are maxima if | $ \beta_x = \pm \pi , \beta_y = \pm \pi $ | . Based on numerical studies of the function , the conjecture that these maxima are the ones relevant for the stability bound seems reasonable . Then , at these maxima | Beta sub x equals plus or minus pi, beta sub y equals plus or minus pi. |
if | $ G= ( V , E ) $ | is a simple hypergraph , we can construct a graph | G equals open parenthesis V comma E close parenthesis. |
to | $ \boldsymbol { b } $ | , then we have : % | b |
the parameters behave asymmetrically with respect to | $ \phi $ | . Thus , the symmetry of the thermoelectric parameters can be used as a probe to detect the presence and nature of MBS in the setup . The reasons for | phi. |
$ \beta $ | and H | Beta. | |
is also consistent with the interpretation of identifying | $ \tau_p $ | with the modelled distribution | tau sub p. |
we performed a regression of the | $ G $ | magnitude on the Pan-STARRS colors | G |
Observe that for each | $ k $ | -set | k |
For | $ m \in \N $ | , we denote by | m belongs to the set of natural numbers. |
there exists a positive | $ H^1 $ | solution | H raised to the power of 1. |
on | $ W_2 $ | it can always be translated to a vector field on | W sub 2. |
-th element of the vector | $ \tilde { \mathbf { h } } $ | . We approximate the | h tilde bold h |
we use the notation | $ b \triangleq \gamma/ ( 2v_ { \mathrm { lr } } ) $ | . First , we show the existence of the constants utilized in Definition~ | b is defined as gamma divided by 2 times v sub lr. |
Suppose that | $ J $ | is otherwise , so that | J |
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