context_before stringlengths 0 199 | equation stringlengths 5 80 | context_after stringlengths 0 249 | spoken_English stringlengths 0 119 |
|---|---|---|---|
$ s $ | . In the center of mass frame it can be checked that the momentum of the exchanged particle , in the | s | |
under the rotation | $ { \cal R } \in { \cal SO } ( 3 ) $ | on a 2-sphere is given as : | R belongs to S O of 3. |
If we have | $ \F { b } ^\ # \in \Junk ( A_ { x } ) $ | , then we also have | F b raised to the power of hash belongs to the kernel of A sub x. |
distance between the associated state trajectory~ | $ y $ | and a given desired state~ | y |
The most dominant contribution to the | $ \omega\pi $ | channels is the | omega pi. |
Initially there are | $ Q = 8 $ | panels on each side with | Q equals 8. |
The results indicate that the proposed FA converges with all number of antenna setups as all the observed curves level off after the maximum size of population of | $ N=50 $ | . However , the higher of the antenna elements is , the larger the size of the population is required for a converged transmit power . For example , with | N equals 50. |
we only focus on the case | $ r+s\leq K $ | . By Remark | r plus s less than or equal to K |
for | $ k $ | large enough assumption | k |
Such integration thimbles are composed of steepest descent paths flowing out from critical points of | $ \mathcal { W } ( w , z ) $ | . So for a critical point | W of w comma z. |
Let | $ X $ | be a topological space , then a simplicial diagram | X |
and | $ b\in B $ | , if | b belongs to B |
$ _ { 1-x } $ | RhPd | One minus x. | |
during the | $ n $ | -th time slot as | n |
for all | $ \varepsilon_1 $ | and | Epsilon one. |
with | $ \ < x_i^2\ > / 2\varepsilon_i^2 ( t ) $ | leads to the equivalent form | Angle bracket, x sub i squared, close angle bracket, divided by 2 epsilon sub i squared of t. |
adopt the | $ N $ | -way | N |
is a constant reference density and | $ b $ | is the `` buoyancy '' . If , for example , the fluid is stratified by temperature variations then | b |
and | $ \sigma $ | profiles obtained with GRaTeR or similar codes should be less reliable than fits of the SB , as well as more challenging to interpret in terms of what the fitted | Sigma. |
we say that the space | $ ( X , d , \mu ) $ | satisfies the | X, d, mu. |
is the hitting-time of~ | $ \partial G_m $ | for the random walk~ | partial G sub m. |
and decreasing | $ \beta_2 $ | at the same rate , the length is non-decreasing . That is , we want to show that | Beta two. |
consider the curve corresponding to | $ w=10 $ | and | w equals 10. |
93 | $ \pm $ | 0.03 : 1 | Plus or minus. |
$ \psi_ { \rm pol } $ | in Eq.~ | psi sub pol | |
if | $ i < q $ | , then let | i less than q. |
where | $ N $ | denotes the number of sites . The mass spectrum of such models would consist of a band of closely separated heavy neutral fermions in addition to the light neutrinos . | N |
Let | $ \gamma : [ 0,1 ] \rightarrow\mathcal\ { M $ | be a geodesic connecting | Gamma: Open bracket 0, 1 close bracket, right arrow, script M. |
if a uniform Lyapunov function | $ \mu ( \cdot ) $ | exists , then | mu of dot. |
converges at a rate depending on | $ a_n $ | . % % -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- | a sub n. |
and | $ A_ { 22 } $ | are Schur . With | A sub 22. |
Since | $ G $ | vanishes on projective | G |
$ ^\textrm { \scriptsize 27b } $ | , | 27b | |
Then | $ \Gamma_1 = \ { 0,1 , \infty\ } $ | with | Gamma sub 1 equals open brace 0, 1, infinity close brace. |
$ \Delta_2 $ | and | Delta 2. | |
shows that only the implementation of large values of | $ L_\star $ | , | L sub star. |
CFL condition of | $ 0.5 $ | , and a simple TVD Lax-Friedrichs Riemann solver | 0.5: None |
the width | $ d $ | of the layer is reconstructed , and secondly the refractive index . We delay the justification of our arguments to | d |
and | $ d $ | ; and | d |
-user code for | $ B $ | bits at block-length | B |
The formulas in | $ \mbox\ { MRL $ | are defined as follows : | MRL. |
$ , $ | A | comma | |
or any sets or quantities marked with a | $ ' $ | symbol ) are considered within the auxiliary graph | apostrophe |
& ~~~ | $ P $ | ~~~ & | P |
be the length of | $ \xi $ | . If | xi. |
at the | $ n_ { \rm c } $ | number of contacts between the feet and the ground . The dynamics are governed by the Euler-Lagrange equations : | n sub c. |
equation and | $ H_n $ | is a coefficient chosen to simplify the recursion relations for the polynomials | H sub n. |
for all | $ i $ | , and | i |
the weighting | $ \beta $ | introduces in the second term is a function only of the labels present in the client data | Beta. |
as the set of all | $ \gamma $ | -ML mechanisms satisfying the one-bit communication constraint . Also , we will simply write | Gamma. |
equation where | $ \zeta^ * $ | is defined by Eq. | zeta star. |
where | $ H_e $ | and | H sub e. |
and for | $ N\leq55 $ | particles~ | N less than or equal to 55 |
q profile with | $ q_0=1.2 $ | , | q sub 0 equals 1.2 |
assigns | $ T_X $ | to be the | T sub X. |
the general gauging procedure for the conformal factor is actually one where both | $ \sigma $ | and | Sigma. |
$ . \begin { lemma } \label { lemma : robustness fix phi } Given $ | Dot, less than or equal to a, colon, less than or equal to a, colon, robustness fix phi. | ||
& | $ 54 $ | 54 | |
and | $ C_K = ( C_ { kl } ) $ | , and | C sub K equals open parenthesis C sub k l close parenthesis. |
with | $ \theta=\frac { \pi } { 2 } $ | , thus | Theta equals pi divided by 2. |
it 's crucial to consider the directed edges when aggregating messages | $ \operatorname { Agg } ( \cdot ) $ | . Thus , current node | operator name A much greater than open parenthesis dot close parenthesis |
of players | $ \subS \subseteq \allS $ | . The | subset S is a subset of or equal to all S. |
We define the confidence vector of a sequence | $ \boldsymbol { s } $ | as the corresponding predictive probability , written as | s. |
if | $ \vec x $ | satisfies the constraints . We build the Hamiltonians for the presented problem instances and Trotterize them using the quantum simulation library Qibo | vector x. |
Put | $ F ' : = \bigcup_ { i=1 } ^M F_i $ | . % Further , let | the union from i equals 1 to M F sub i |
for a generic variable | $ Q $ | , we denote | Q |
$ 100 $ | , | One hundred. | |
Let | $ G $ | be a graph with no induced subgraph isomorphic to a subdivision of | G |
500 | $ \mathrm { km } \ , \mathrm { s } ^ { -1 } $ | of the AD as measured by | kilometers per second to the power of negative one |
For | $ i=0 $ | the statement follows from | i equals zero. |
that moves to the right with probability | $ 1/2 $ | and stays where it is with probability | One-half. |
the task that | $ j $ | RUs processed in the VFC system leaves , the vehicles arrive at or depart the VFC system , the situation is similar . % % So under different system state | j |
coincide in the limit | $ y_\alpha \to \infty $ | . Then it follows that the connection between the components of the `` physical '' S-matrix | y sub alpha goes to infinity |
uniformly in | $ \eps $ | , and converge a.e . to | epsilon. |
for any given | $ \Delta $ | , which means the | Delta. |
% proving new theorems which count the number of connected components of these spaces in terms of cohomology % of the underlying manifold~ | $ M $ | and which , more deeply , connect these structures to the problem of lifting % | M |
and the subset | $ K_0 $ | of | K zero. |
of finite sequences | $ \boldsymbol { s } $ | and | s. |
The MTSR is calculated as the ratio of the change in | $ \kappa $ | between the presence and absence of a magnetic field . The MTSR is expressed as [ | kappa. |
Do we use this notion | $ C $ | in the paper ? -- QY } This gives a Borel subgroup | C |
and unitary matrix | $ U $ | , one gets the representation of interface conditions on | U |
and | $ R_ { B/A } \coloneqq R_ { X/Y } $ | . | R sub B over A is defined as R sub X over Y. |
we compare FTLM and QMC data for | $ J_t=0.5J_d $ | and | J sub t equals zero point five J sub d. |
and adding an edge from | $ p $ | to | p |
with | $ \text { val } _ { \mathfrak { p } } ( f_ { \alpha } ) =1 $ | . Thus , the action on the opposite one-parameter unipotent subgroup | Val sub p of f sub alpha equals one. |
we define | $ \mathcal { Z } ^ { \gamma } $ | as the set of distributions | Z raised to the power of gamma. |
$ & $ | 399 | and | |
after applying at most | $ k $ | vertex splits to the vertices in | k |
is the unit inward normal vector of | $ D $ | at | D |
gradients for positive sample | $ l $ | can rewritten as follows ; | l |
and | $ \epsilon $ | with standard choice | epsilon. |
there exists a channel completion | $ \Phi $ | of the Uhlmann transformation | Phi. |
è globale e non partecipa in | $ E $ | } % % % % % % % % % % % % % % % % % % % % % % % % % % % % % | E |
two additional parameters are needed to scale and translate the spatial argument | $ \bm { x } $ | , a scaling factor | x |
converges to | $ ( \tau^ * , \tau^ * ) $ | for some stationary point | tau star, tau star |
It was shown that if the 5/3~MMR is approached with eccentricities of both satellites smaller than | $ \sim 0.01 $ | , a long-term capture in resonance is certain . Therefore , in order to evade or skip this resonance , at least one of the eccentricities must have been close | Approximately zero point zero one. |
the systematic decrease of the SG-like phase with | $ T_s $ | should dominate , leading to the reduction of the SEB effect with the decrease of surface-to-core ratio until it vanishes for the single crystal . Here , it is worth | T sub s. |
$ 7.7 $ | seven point seven. | ||
$ \omega ( H ) =0 $ | if | omega of H equals zero. | |
$ T > 0 $ | is a fixed time horizon , the function | T is greater than zero. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.