Issues of unboundedness on degenerate states, as in , then do not arise.
Degenerate Configurations, Singularities and the Non-Abelian Nature of Loop Quantum Gravity
In many cases one can reduce this to a single determinant using multi-linearity and gauge invariance, but in general this is not possible and there are explicit examples with unboundedness .
Degenerate Configurations, Singularities and the Non-Abelian Nature of Loop Quantum Gravity
It is, however, diﬃcult to ﬁnd a geometrical reason for unboundedness since many different possibilities are realized.
Degenerate Configurations, Singularities and the Non-Abelian Nature of Loop Quantum Gravity
In models, as discussed before, one can then see that unboundedness, even if it occurs such as in anisotropic models, is no obstruction to non-singular evolution.
Degenerate Configurations, Singularities and the Non-Abelian Nature of Loop Quantum Gravity
So far the observed unboundedness of inverse volume on tri-valent vertices of the full theory has no direct implications for the singularity removal mechanism of loop quantum cosmology.
Degenerate Configurations, Singularities and the Non-Abelian Nature of Loop Quantum Gravity
The behavior is thus closer to that in the full theory and provides an explicit example for the role of non-Abelian effects in unboundedness as discussed in general in Sec. 3.1.
Degenerate Configurations, Singularities and the Non-Abelian Nature of Loop Quantum Gravity
Kinematically, it does not matter which kinds of states are used to compute expectation values of inverse volume, and also in coherent states does one generally still have unboundedness.
Degenerate Configurations, Singularities and the Non-Abelian Nature of Loop Quantum Gravity
There are some inhomogeneous models which are not necessarily more complicated as far as unboundedness is concerned, but dynamical properties are more diﬃcult to disentangle.
Degenerate Configurations, Singularities and the Non-Abelian Nature of Loop Quantum Gravity
Here, inhomogeneity does not play any role whatsoever and thus cannot be responsible for unboundedness.
Degenerate Configurations, Singularities and the Non-Abelian Nature of Loop Quantum Gravity
Otherwise, unboundedness properties of curvature already arise when introducing anisotropy, not even inhomogeneities.
Degenerate Configurations, Singularities and the Non-Abelian Nature of Loop Quantum Gravity
Unboundedness of triad – like operators in loop quantum gravity.
Loop Quantum Gravity: An Inside View
The rescaling by the factor n/(n + 1), avoids diﬃculties arising from potential unboundedness of cθ (u) when one of ui ’s tends to 1.
New estimates and tests of independence in some copula models
This crucial property (valid for diffusions with natural boundaries as well) allows to extend the theory of stochastic differential equations and integrals to diffusions , whose drifts show up a bad (unboundedness or divergence to inﬁnity) behaviour when approaching the boundaries.
Natural boundaries for the Smoluchowski equation and affiliated diffusion processes
We caution the reader not to confuse ios with the notion named input/output to state stability (ioss) in (also called “detectability” in , and “strong unboundedness observability” in ).
Notions of Input to Output Stability
In Fig, the unboundedness of the solution is shown for Ω = 2.
Lotka-Volterra dynamics under periodic influence
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