context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
|---|---|---|---|
and | $ \beta $ | are constants. | Beta. |
the first one is | $ n ( t ) $ | , the probability for a given site to be occupied at time | n of t |
with | $ \alpha $ | -enhanced chemical composition and without the inclusion of the approximate overshooting in the calculation of the convective flux . | Alpha. |
covers values between | $ \sim 0.08-0.1 $ | during core helium burning , we adopt | sim zero point zero eight minus zero point one. |
as well as that between | $ V $ | and | V |
which provides a measure for the amount of potential progress that can be made at iteration | $ k $ | ) in either non-convex or strongly convex settings for the algorithm . While the random variable | k |
em or | $ v_i = \A [ F , y , K ] $ | with | v sub i equals A times F, y, K |
Conjugation by | $ u $ | defines an automorphism of | u |
is convex and | $ V ( u ) $ | has a unique minimum of | V of u. |
for various choices | $ \Delta t $ | we lower-bound the fidelity of | Delta t. |
for | $ f $ | if | f |
$ ~ ( here $ | y | not provided. | |
$ f_ { \theta } ( . ) $ | f sub theta of dot | ||
is | $ c_ { \delta } $ | -dense and has no | c sub delta |
The energetics of this machine is analyzed by studying the work | $ W $ | , heat | W |
& | $ 0.2 $ | & | 0.2: None |
is the channel coefficient for device | $ i $ | in the | i |
For two distributions | $ F_1 $ | and | F sub 1. |
fixed points of | $ G $ | and | G |
$ 10 $ | and | Ten. | |
& | $ 0.7247 $ | zero point seven two four seven. | |
is called a retraction in | $ x $ | , if | x |
We observe that | $ n^T_ { s , s\p } $ | is bounded from above by the total number of jumps up to time~ | en to the power of tee sub ess backslash p |
trials and success-probability | $ p = \frac { \eps } { 1+\eps } $ | : | p equals the fraction of epsilon divided by one plus eps. |
where | $ \beta_t $ | represents a predefined sequence of variances . The denoising process | Beta sub t. |
represented as | $ I ( \theta/2 ) $ | , is not equal to the average of the pixel values at the individual angles , represented as | eye of theta over 2 |
$ \tilde { F } ( \tilde { \omega } ) $ | , as a function of | tilde F of tilde omega | |
$ & $ | 0.4128 | and | |
is a semisimple | $ \C $ | -tensor category with countably many isomorphism classes of simple objects acting on a | C: None |
-open set | $ U $ | with | U |
and in addition by | $ \gamma ' ( 0 ) \cdot ( P_1-P_0 ) > 0 $ | we have | gamma prime of zero dot product P sub 1 minus P sub 0 is greater than zero. |
and | $ \ket { e } $ | denote the ground and excited states , respectively . Here , the qubit | ket e. |
as the set of configurations in | $ \cdl $ | where all tokens are valid and correct , and | dee ell. |
we model the topology by a symmetric strongly-connected simple directed graph | $ G = ( V , E ) $ | , where | G is equal to open parenthesis V, comma E, close parenthesis. |
The following result shows that | $ \ubar \SSS_ * $ | is a subset of | u bar S S star |
and | $ n\geqslant 2 $ | . On the other hand , for a potential which is dominated by a negative self-interaction , the boson star can be larger than the orbital radius leading to precession | n greater than or equal to 2. |
$ l $ | , | l | |
here we are just doing the Mellin transformation with respect to | $ \omega $ | . This is enough for studying the conformal soft theorem , but one can also write the conformal operators with dependence only on the celestial variable . It requires writing | omega. |
$ k\ne l $ | Then the matrix | k not equal to l | |
We find that the time evolution of | $ R $ | is | R |
the lock-in is configured to take in an input of | $ \pm $ | 2000 mV . For a 100 mV signal if 16x gain is turned on , that makes the signal 1600 mV and hence the full 4000 mV range of lock-in | Plus or minus. |
which extend smoothly to the closure | $ \bar { B } $ | , and such that there exists | B bar. |
-M | $ 2 $ | and Top-Popular baselines , respectively . While | two |
and | $ \ { \alpha_j\ } _ { j\in S } \subset \R $ | such that | Alpha sub j for j in S is a subset of R. |
the trajectory | $ \Gamma_0 $ | connects | Gamma zero. |
into chains on the loop space of | $ Y $ | using | Y |
$ h_3 $ | , | h sub 3 | |
-torsor over | $ \bA^1_A $ | trivializes over the punctured formal neighborhood | B A raised to the power of 1 subscript A. |
be the set of controller which are involved in a CVC solution | $ y $ | , | y |
with a probability | $ x/\ell ( t ) $ | , contains the abrupt jump of size | ex divided by ell of tee |
we get that | $ z $ | and | z |
and the first | $ m $ | entries in | m |
represents | $ i $ | -th record of charging behavior in | i |
/s or larger ; this means that | $ \underline { \underline { D } } _\text { h } \geq D $ | for the timescales considered~ | Spoken English: Double underline D subscript h is greater than or equal to D. |
is equivalent to representation | $ \varrho_ { \mu_ { \imath } } $ | for some | Rho sub mu sub i |
We distinguish four types of edges of | $ P $ | ; see part~ ( a ) of Figure~ | P |
for evaluating the complete | $ q\bar { q } $ | component | q bar q |
for some | $ n $ | and ideal | n |
Introduction of the comoving coordinates | $ y $ | provides a geometric meaning to the gauge transformations on | y |
22 | $ \pm $ | 0.56 & - & 59.55 | Plus or minus. |
are positive numbers such that | $ \alpha \beta=\pi^2 $ | , and | Alpha beta equals pi squared. |
A | $ 4 $ | -oriented SOD with alignment | four |
from the wall as compared with that in the bulk liquid | $ D_b $ | , Since , as we has shown in Fig . 2 , the difference between the liquid structure close to the was is not significantly different from that in the | D sub b. |
then all open | $ r $ | -spin correlators can be calculated via TRRs . | r |
be the Perron root of | $ A $ | , and | A |
Note that in the cases of Hf through Os the | $ 5s $ | and | Five times s. |
though we caution that a more careful comparison of | $ D $ | requires a detailed treatment of the anisotropy of the dielectric environment in R5G/hBN . | D |
that have signs | $ + $ | and | plus |
then it is upper semicontinous with respect to | $ \tilde { d } $ | , since | d tilde. |
$ f ' ( 0 ) =0 $ | and | f prime of 0 equals 0 | |
in the | $ N $ | -boson system or in the full | N |
remoteness | $ \rho $ | and exactly two centroidal vertices of equal degrees . Let | rho. |
algorithm aims to find the normalized weights | $ \ { w_ { i } \ } _ { i=1 } ^ { n } $ | , i.e . , | Curly brace, w sub i, end curly brace, sub i equals 1 to n. |
with a larger | $ n_i $ | , we can leverage mini-batches | n sub i. |
& | $ 5 $ | & | five |
must be matched with itself in | $ M_k $ | . | M sub k. |
In order for this to be less than | $ 0 $ | , we see that | zero |
and | $ 4.2 $ | X faster than their non-reduced versions . Similarly , their inference time is also significantly faster after reduction . For instance , the inference time for the reduced MLP , | 4.2: None |
we denote by | $ \sigma ( E ) $ | the | sigma of E. |
with | $ \deg P \leq k $ | such that | Degree of P is less than or equal to k. |
where | $ \alpha $ | is an intercept term and | Alpha. |
for all | $ i $ | ; | i |
of entries ranging from | $ 50\ % $ | to | Fifty percent. |
reducing | $ m $ | may involve selecting different training examples rather than a subset of the original | m |
& | $ 0.77 \pm 0.04 $ | & | zero point seven seven plus or minus zero point four four. |
assigned \\ & to the | $ Z $ | boson based on their invariant mass being closest to the | Z |
one of singular points of the boundary of a stratum of the form | $ { \mathcal R } _1 $ | is a semicubical cusp . The set | R sub 1. |
the tagging of | $ \alpha $ | is plain at that endpoint . Then | Alpha. |
and is not contained in any | $ F $ | -cycle or splitting | F |
there are at least | $ \frac { c |T_L| } { ( 2C_0 ) ^2 } $ | edges | Fraction c absolute value of T sub L divided by open parenthesis 2 C sub 0 close parenthesis squared. |
and | $ d_ { yz } $ | orbitals instead of the expected | d yz. |
is the same quantity that is used in the Galois conjugation of the | $ S $ | -matrix , and is always a signed permutation matrix | S |
the EDMD operator converges to | $ \mathcal { K } $ | in SOT . Therefore , it may be judicious in the autonomous setting to add as many feature maps and data points as possible . This will eventually render the | kay |
9 to | $ z $ | z | |
for subsystems | $ e\mu $ | , | e mu (None) |
that the feature vector | $ \mathcal { X } ^ { \langle t^\prime\rangle } $ | and the previous hidden state | X to the angle bracket t prime angle bracket |
embedded in | $ \bbR^2 $ | . It is the planar graph with set of vertices | the set of real numbers squared. |
we follow the use of fixed | $ \alpha $ | suggested in the original SAC paper to reduce irrelevant factors . | Alpha. |
for all | $ t , n\geq 5 $ | . | t, n greater than or equal to 5 |
there are only a finite number of | $ t_i $ | such that | t sub i. |
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