context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
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the global minimum will be shifted toward the spatial infinity and decreased with the increasing of | $ \phi_H $ | . Fig . | phi sub H. |
rescale all time variables with | $ \frac { 1 } { T } $ | and all frequency variables with | One over T. |
Also note for almost every | $ \theta\in \R $ | , | theta belongs to the set of real numbers. |
is evolutionarily stable against | $ \theta $ | . Hence , type | Theta. |
in the graph defined by | $ A $ | . We call such a product the | A |
w \in \CC | $ , $ | z | comma |
at scale | $ \frac { 1 } { p } $ | . Let | One divided by p. |
The obtained field dependence of the spin gaps at | $ T = 50 $ | T equals fifty. | |
Suppose that there is a conformally compact end and an | $ f $ | -complete end . Then , one constructs a globally minimizing | f |
of | $ G $ | , | G |
on | $ W $ | such that | W |
and | $ 7 $ | are fixed , and so are the edges between them . By the Finite Order Theorem , there exists a re-embedding of | seven |
Note the fixed points near the intersection between the Dehn twist region | $ N $ | and the Morse region | N |
illustrating how each `` band '' for a given value of | $ \lambda $ | would shift on the vertical scale as | Lambda. |
is a pair | $ ( X , D ) $ | where | X comma D. |
when | $ n $ | tends to infinity , as shown in | n |
$ P $ | the pressure , | P | |
\boldmath | $ V $ | } } | V |
extending | $ \iota|_ { \O } $ | , where | iota restricted to the empty set. |
and | $ P\cap\bet=Q\cap\bet $ | . If | P intersection bet equals Q intersection bet. |
0^+ | $ & $ | 99 | and |
-function of policy | $ \pi $ | in | pi. |
is indeed divisible by | $ 12 $ | since by assumption , | Twelve. |
remains contained in | $ \overline\Omega $ | . | Omega bar. |
EGDM is incapable to describe | $ \mu ( F ) $ | even on a cubic lattice . % % EGDM is sometimes considered as an extension of equations published previously in order to fit numerical data obtained on cubic lattices . | mu of F. |
& | $ -1 $ | & | Minus one. |
Po experiences | $ \alpha $ | -decay , through which the atomic number drops by two , hence cycling the element back to | Alpha. |
& | $ 0.24 ( 0.08 , 0.41 ) $ | & | zero point two four, open parenthesis, zero point zero eight, comma, zero point one four, close parenthesis. |
Let | $ { p_i^ { obs } = [ x_i^ { obs } , y_i^ { obs } , z_i^ { obs } ] ^T } $ | and | p sub i obs equals open square bracket x sub i obs, y sub i obs, z sub i obs close square bracket transpose |
our method is able to generate the entire path of the process up to | $ t=T $ | and hence information about the dynamics within the interval | t equals T |
we compute the difference of | $ \ell $ | at each lookback time with the value today , but now we normalise this difference by | lowercase L. |
in | $ C_K\ { M_k \ } $ | for a compact interval | C sub K, open brace, M sub k, close brace. |
865 & 1 & | $ D^ { * 0 } $ | & 2.007 & 1 | D raised to the power of star zero. |
054 when the first two-hundreds | $ 1^ { + } $ | states of the virtual intermediate nucleus | One plus to the power of positive. |
94 \\ SAOP/TZ2P & | $ - $ | 9.51 & | minus |
Of course we can switch the roles of | $ X $ | and | X |
be an | $ ( i+1 ) $ | -simplex of | i plus one. |
in the boundary of | $ \sigma $ | is its | Sigma. |
and | $ ( \cdot ) ^ { \text { H } } $ | denote the matrix transpose and the Hermitian transpose , respectively . | center dot raised to the power of H. |
representation may be obtained by considering `` natural representations of | $ X $ | '' when the coarse structure is too small . For example , if | X |
and | $ C > 0 $ | independent of | C is greater than zero. |
denotes the estimated center of the | $ k^ { th } $ | class by pseudo labels , | k to the power of th |
and thus | $ ( \ID-\pmi_1 ) \rvw^ * _T = 0 $ | , | Open parenthesis, ID minus plus or minus i sub 1, close parenthesis, v w star sub T, equals zero. |
$ ( \tau=1,2,3 ) $ | for propagating modes . They are given by | tau equals one, two, three. | |
% maximises the average contribution of the | $ n-k-1 $ | smallest inputs . % | n minus k minus 1 |
location parameter | $ \alpha $ | , proposal distrition | Alpha. |
using the color indexes | $ V-J $ | and | V minus J |
for a given | $ L $ | . Although | L |
it would most likely be negligible in the limit | $ \varepsilon \to 0 $ | . However , at the | Epsilon tends to zero. |
is set to 5 for all three cases of size- | $ N $ | Ising problems , so that quantum solvers with 5 qubits can be employed . The | N |
are unitary operators satisfying | $ uv = e^ { i2\pi\theta } vu $ | , and where | u v equals e to the power of i 2 pi theta v u |
the predictor-rejector function involves selecting a predictor | $ h $ | out of a collection of real-valued functions defined over | h |
's and | $ \omega_u $ | 's . As can be observed , the challenges of solving | omega sub u. |
This follows by applying | $ e $ | to the left output in | e |
pairs in | $ \mathbf { a } $ | . Symmetrically , Charlie knows the positions of almost all | a |
our strategy will consist in making a local random complete basis measurement onto each | $ C_i $ | , and a random projective measurement of rank- | C sub i. |
the relevant multi-particle states for our analysis are the | $ \varphi \bar { \varphi } $ | asymptotic two-particle states | phi bar phi |
for all | $ j $ | in Eq.~ | j |
if there exists | $ t_r^ + \ge 0 $ | , | t sub r superscript plus greater than or equal to zero. |
is the profinite | $ R $ | -algebra obtained by extracting all the | R |
3 | $ \pm $ | 16.2 & | Plus or minus. |
then the corresponding loading | $ l_ { ga } =0 $ | ; otherwise the loading is drawn from the prior distribution | l sub g equals zero |
& | $ \text { Spin } \left ( 8\right ) $ | Spin of 8 | |
It was established that below 4BLs the film is a semiconductor with the direct gap open at the | $ \Gamma $ | point and the positive indirect band gap leading to nontrivial topology peculiar for an intrinsic 2D TI . Above 4BLs the indirect band gap becomes negative resulting in a semiconductor- | Gamma. |
actually achieves an approximation ratio of | $ 5/6 $ | . Moreover , we will give a tight example for this algorithm . | Five-sixths. |
other than | $ W $ | . Then | W |
is node-based length of the taxonomy and | $ l_ { 1 } $ | indicates the leaves of the taxonomy , which is associated with the class labels . Each node in taxonomy except nodes in the leaves is denoted as | l sub 1 |
for any | $ \delta\in ( 0,1 ) $ | , since | delta belongs to the open interval from zero to one. |
-cycle | $ C=\Sigma_ { i } \sigma^n_i \in \mathbb\ { Z BG_n $ | , denote the desired | C equals the sum over i of sigma to the power of n sub i, belonging to the set of integers, BG sub n. |
and | $ c_2 $ | against the two mass paramters | c sub 2 |
is given by the restrictions to the lines | $ \ell $ | through | lowercase L. |
$ K ( n ) $ | -local Picard group functor | K of n. | |
em Parameter | $ b $ | of truncated Pareto distributions is sampled form a Beta distribution with parameters | b |
it can be argued that users of | $ F $ | and users | F |
and | $ L $ | is defined as the characteristic length or the length of the cavity in this case . % In this work , we asses the plug phenomenology and asymmetry of the | L |
for large | $ t $ | . Translation symmetry implies the law of | t |
and | $ q $ | are called cousin primes if and only if | q |
For | $ x\in [ 0,1 ] $ | , when | x belongs to the interval from 0 to 1. |
does not contain | $ \zeta_5 $ | . | zeta sub 5. |
$ x \in { } ^ * G $ | , for convenience . | x belongs to the closure of G. | |
$ X_1 $ | and | X sub 1. | |
and both | $ p=1 $ | and | p equals one. |
Measuring the right qubit of | $ \ket { \psi_5 } $ | Eve obtains the outcome | ket psi sub 5 |
The fixed parameter | $ \alpha\in\left [ 0,1\right ] $ | allows us to select the degree of serial dependence . When | Alpha belongs to the closed interval from 0 to 1. |
is the final embedding vector of candidate column | $ v_j $ | . The probability to select | v sub j. |
& | $ -2.1^ { +0.2 } _ { -0.2 } $ | & | Negative 2.1 raised to the power of positive 0.2, sub negative 0.2. |
although | $ \beta $ | can be infinitesimally close to unity , the NSE can only be solved for | Beta. |
We vary | $ f $ | between | f |
for some ring | $ R $ | if and only if it is of type | R |
is either identical to | $ \mathbf { F } $ | or a nontrivial CM extension of | F |
Note that if | $ |S|-k = \gamma_P\left ( G\right ) $ | , then | The absolute value of S minus k equals gamma sub P of G. |
would function identically to the Schrodinger equation on a lattice with regards to | $ A $ | , except instead of oscillating eigenstates we have decaying heat modes . Alternatively , we could add a non-Hermitian decay term | A |
Thus the limit function | $ f ( u ) $ | is uniquely determined by | f of u |
holds for | $ s \geq 6 $ | . The property | s is greater than or equal to 6. |
and | $ V_ { \Lambda } ^ { ( 3 ) } ( \rho ) \propto\rho^2 $ | . A large-scale SHF fit~ | V sub Lambda, superscript 3, open parenthesis rho, close parenthesis, proportional to rho squared. |
7 | $ \pm $ | 5.8 & 99.9 | Plus or minus. |
and | $ \kappa $ | , where | kappa. |
at each day | $ t\in [ T ] $ | . Let | t belongs to the set of T. |
50 & | $ \pm $ | 1.93 & | Plus or minus. |
$ R $ | , and | R |
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