context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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the objective function | $ J ( \wb , b ) $ | may have a better minimum on the larger set | J of wb and b |
under the braid group and assume that the special branch points are still the first | $ \ell $ | . For any | lowercase L. |
If | $ k $ | is complete , then | k |
where | $ \ell $ | is the index such that | lowercase L. |
; as we discussed | $ \epsilon $ | can be much smaller than | epsilon. |
if it never goes back to the | $ x $ | -axis in the middle steps . Similarity , we claim that a | x |
; % \State Sample the corresponding ground-truth of | $ X $ | for object detection : | X |
where | $ \ { h_ { ij } \ } $ | are bipartite interaction operators , | h sub i j |
and | $ A_t ( P ) $ | is free over | A sub t of P |
be the singular locus of | $ \mathcal { C } \to B $ | . By | C to B |
let | $ \cC_ { \cH } : = \ { c_h\ } _ { h \in \cH } $ | and | c C sub c H is equal to the set of c sub h, sub h in c H. |
We obtain the pressure | $ P $ | from the diagonal components of the general expression for the stress tensor arising from a Yukawa-like screened electrostatic interaction of finite-size ions with the Ewald construction . More in particular | P |
the distance between any two points is given by the sum of their | $ x $ | - and | x |
where | $ \bm h $ | s are hidden states . Training a symbolic executor without step-by-step supervision signals is non-trivial . A typical training method is reinforcement learning in a trial-and-error fashion . However , | h |
to be | $ \alpha $ | . Then conditional on the fitted candidate models | Alpha. |
I shall imply but omit below the | $ O ( N ) $ | -vector indices . Thus working with the Pauli-Villars regulators is apparently equivalent to accounting for the higher-derivative terms in the action for | O of N |
for some | $ s\in\ { 1,2\ } ^\mathbf { N } $ | distinct from the constant | s belongs to the set of 1, 2 raised to the power of N. |
phase factor : in this non-equilibrium case the | $ \pm $ | modes are not at equilibrium . However , we can still use Eq.~ | Plus or minus. |
Let | $ k\in\N $ | and | k belongs to the set of natural numbers. |
The main source of the errors is the deviation between the PDFs | $ p_ { \hat { e } _i } ( \cdot ) $ | and their estimations | p sub e hat i of dot |
and | $ t_1 > t_0 $ | , i.e. | t sub 1 is greater than t sub 0. |
with a constant | $ C $ | depending only on | C |
If there are only finitely many entries into | $ \mathscr D $ | of the | script D. |
we have for each fixed | $ k \ge 1 $ | , | k is greater than or equal to 1. |
if for all | $ |z| < \delta_1 ( d , C ) $ | Absolute value of z is less than delta sub 1 of d comma C. | |
and | $ \varphi $ | is the unknown . We denoted by | phi. |
$ Q_0 $ | , and | Q zero. | |
in the system and for a range of intermediate values | $ 0\leq a \leq 1 $ | such that the parametric integral over | Zero is less than or equal to a, and a is less than or equal to one. |
7 | $ \pm $ | 7.6 & 0.04 | Plus or minus. |
we could allow | $ X $ | to be non-compact in | X |
and let | $ m \geq 1 $ | be a given integer . The classical Whitney problem is the question % How can we decide whether | m greater than or equal to 1 |
Exact determination of | $ \xi_c $ | is becoming exceedingly important due to Josephson junctions created by twisted Bi2212 crystals | Xi sub c. |
& BART & | $ 0.034 $ | & | 0.034: None |
This is not the case for the constraints on bias parameters as discussed above and we thus repeat this analysis setting | $ b_ { s^ { 2 } } =0 $ | in both cases . The ratio of the 1- | b sub s squared equals zero. |
because | $ G $ | is locally | G |
there exists some | $ y \in X $ | such that | y belongs to X. |
satisfies | $ [ X ( t ) , H ( t ) ] = 0 $ | , the expectation value of the operator | X of t, H of t, equals zero. |
The timing uncertainties of | $ \delta t_f=0.005\ , \si { \milli \s } $ | and | milli seconds. |
is | $ k $ | -mobile , by inductive hypothesis on statement ( | k |
we recommend that users report the 2- | $ \sigma $ | uncertainty on | Sigma. |
we define a graph | $ F_i $ | on | F sub i |
Consider the | $ n^ { \rm th } $ | -order ODE | n to the power of nth. |
-perturbations | $ \phi_n $ | of | phi sub n. |
by taking | $ \ell = 2 $ | , provided that we can use the additional lower bound | ell equals two. |
Small values of | $ d $ | and | d |
we can greedily partition | $ Q $ | into | Q |
which involves a summation over logarithmic terms involving powers of | $ \nu=-1/\ln \epsilon $ | Nu equals negative one divided by the natural logarithm of epsilon. | |
and | $ \bm { A } $ | are an arbitrary scalar and vector fields respectively . Similarly , for vanishing | capital A |
performing the following : It replaces all | $ x_i $ | in a subset of | x sub i. |
$ c $ | is the underwater speed of sound , | c | |
_3 | $ Ge $ | _3 | Ge (None) |
Assume the interior of | $ \Theta $ | is not empty , so that is a constant | Theta. |
in | $ \mathcal\ H $ | as | H: None |
& | $ { 22.21 } \pm { 0.87 } $ | Twenty-two point two one plus or minus zero point eight seven. | |
$ p $ | < | p | |
where | $ g^ { ( 0 ) } _ { ij } $ | is the boundary metric . The subleading component | g raised to the power of zero, subscript i j. |
in a snark | $ G $ | with | G |
ensures that there is some computation path which terminates after this happens | $ O ( \log k ) $ | times . Recall that we chose | O of log k. |
We have a good definition now of what it means for a projector | $ P $ | to be `` effectively finite dimensional '' with respect to | P |
and | $ V $ | is partitioned into subsets | V |
then we construct | $ G_ { p , q , r } $ | as follows : start with a complete graph | G sub p, q, r |
is a set of i-vectors and | $ C $ | is the number of different speakers . Differently in this paper , we define | C |
and | $ C_ { L , jk } ^ { \alpha\beta } $ | defined by : | C sub L, jk, superscript alpha beta. |
depending only on | $ d $ | ( and not on the particular instance | d |
which alternates gradient steps with respect to the differentiable term | $ f $ | , and proximal steps with respect to the nonsmooth term | f |
be induced by the inclusion | $ X\hookrightarrow Y $ | . | X hookrightarrow Y |
as its violation leads to energetically unfavorable | $ S=0 $ | local defects , which are thermally suppressed at low temperatures | S equals zero. |
Inserting this into the expression for | $ C_ { xy } $ | , we get | C sub xy. |
Note that the associated simple graph of an involutive graph that is a tree | $ T $ | , is a simple graph that is also a tree as a simple graph , i.e . , every two vertices are connected by a unique simple path . | T |
$ k = 2 $ | and | k equals 2 | |
& | $ 38.40 $ | & | thirty-eight point four zero. |
we review the main results about the projection onto the intersection of an | $ \ell_1 $ | ball/sphere and an | ell sub one. |
where constructive criteria for | $ \bm { \pi } $ | the mathematical constant pi. | |
58 | $ \pm $ | 6.73 & 64.80 | Plus or minus. |
and learning rates of | $ 0.0025 $ | and | Zero point zero zero two five. |
Sb | $ _ { 0.5 } $ | Te | subscript 0.5. |
if | $ \delta $ | is an ordinal , we will write | delta. |
& | $ 20.0022 \pm 1.9933 $ | & | twenty point zero zero two plus or minus one point nine nine three three. |
where | $ r $ | is the probability of target existence , and | r |
we will compute the self-energies of both | $ \varphi $ | and | phi. |
For each decision node | $ n \in N $ | and each active player | n belongs to the set of natural numbers. |
with the radiative corrections giving a | $ { \sim } 10 $ | Approximately ten. | |
$ \models_S^V B \to A $ | and | B implies A in the model S under the valuation V. | |
and address | $ \sigma $ | starting in Section 20 | Sigma. |
We say | $ \boldsymbol { x } $ | is stationary for | x. |
come with two quantum parameters | $ \epsilon_1 , \epsilon_2 $ | which are related to | Epsilon one, epsilon two. |
$ \ell = 1,2 $ | , such that % | ell equals one, two. | |
05mm2 & C | $ _1 $ | & A & U & 1.9 | One. |
-variate circular density | $ f $ | on | f |
contains a positive proportion of irreducible polynomials over | $ K $ | if : | K |
to | $ \omega^ * $ | . As it is reasonable to expect , the ergodization time associated with | omega raised to the power of asterisk. |
and | $ \hat { y } $ | are the transverse components . We use the Disambiguated Lambert Cylindrical Equal-Area Projection vector field , which gives | y-hat. |
and a | $ \kappa^ * $ | such that the following holds . For any | kappa star |
em latent-space age uncertainty | $ \sigma_ { \tau } $ | < 40 | sigma sub tau. |
the data | $ \bm y $ | and | why |
by using induction over | $ k $ | , | k |
problems indeed exist under certain conditions on | $ A $ | . They do not have to exist in general . To illustrate this fact , let us consider for the moment the case | A |
be the | $ \ell $ | -th moment of | lowercase L. |
the | $ a $ | -contraction method has been extended to composite waves made of a shock and a rarefaction in | a |
00 | $ \pm $ | 0.53 & 0.42 | Plus or minus. |
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