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In this article introduction of technical
terminology was rather systematically
avoided, and therefore a
precise formulation of open problems
is a somewhat difficult task.
The existence of many problems
should, however, already be clear.
Continuing in the same spirit of
vagueness, we can make them a little
bit more precise. Some of them
are present already in most basic
settings, and their solutions may be
elementary. Other problems involve extending the
existing knowledge to more complicated situations.
We may ask for:
• Global lower bounds of the form (11) on the
number of resonances; at the moment very few
unconditional bounds are known. To put this
in perspective, until the work of Plamen Stefanov
in 1998, the sphere was the only obstacle
for which the optimal lower bound was
known; it is still unknown for an arbitrary obstacle.
• Local lower bounds related to finer aspects of
the dynamical structure: a modification of
(10). At present only the “one hyperbolic orbit”
examples and their extensions provide lower
bounds corresponding to finer upper bounds.
• The modified Lax-Phillips conjecture of Ikawa
stating that there should be a strip with infinitely
many resonances for the Dirichlet Laplacian
on the exterior of several convex bodies.
Ikawa proved this for the Neumann Laplacian.
• Understanding of meromorphic continuation
of the resolvent on manifolds. In addition to
manifolds with simple structure at infinity
(some of which were discussed above), the
best understood general class consists of “conformally
compact manifolds” studied by Rafe
Mazzeo and Richard Melrose. They generalize
surfaces of the type shown in Figure 6. Even
there, the method of complex scaling is not
properly understood, nor are the upper
bounds. For other natural classes of manifolds
the situation is even less clear.
• Generalization of existing methods and results
(upper bounds, Poisson formulæ) to situations
where there are singularities at infinity.
The natural directions are provided by
higher-rank symmetric spaces and by the quantum
The Riemann hypothesis could also have been
added, since it can be formulated in terms of resonances
(see Figure 5). It should, however, be remembered
that in their book on automorphic scattering,
Laoox and Phillips had a chapter titled “How
Not to Prove the Riemann Hypothesis”. So it is better
to leave it out.
Originally posted by tgidkp
reply to post by Bleeeeep
there is recent evidence of bio-molecular systems being able to generate their own unique reckoning of time. it was stated that this unique timeframe is what creates an "identity" at the molecular level. it is also gives the system the ability to generate its own quantized coherence, independent from the restraints of decoherence typically associated with large (noisy) systems.
I do not like the idea that time doesn't exist or is just a human construction. I do, however, think that its definition is far more broad than 'c' (absolute).
When you measure time you measure change...
Originally posted by Bleeeeep
reply to post by tgidkp
How can you not see that time is the energy's reactionary speed of change to information? We all generate our own reckoning of time based on our perception of the change of information. When you're having fun you lose track of change - thus time "flies".
And to carry that over to the physical self, "time" slows the faster you move, but what is a better way of viewing it is that the faster something moves, the slower it reacts to change. Think about inertia or friction or think about "hot" or "cold". It is the change of information/force that is the cause of these things - so why should time be any different?
So what I'm saying is... both our perspective and our actual physical selves change based on the reaction. That change is literally time. So don't think time is the cause, time is the effect. When you measure time you measure change - time is nothing.
see my elegance
dining on the periodic table crawl for elements
the universe designs my intelligence
drop science down a bottomless pit
one swift, doing handstands on pyramid tips
the sun splits the waterfront causing prismatic effects
butterflies come alive and have sex
birds fly out of my top hat slow
to join the brilliance of wilderness and soar through the congo
speak the convo to colors and shapes
my wordchoice is turquoise I love to create
my art hurdles over the clouds in dark purple
red mixes yellow and blue in sharp circles
paint splashes over your conscious like canvas
molecules rule unseen by deep glances
trace your physical form to that of a mantis
think of all varieties of life on the planet
glaciers and avalanches, they have a dance with
the gods put their lips on the galaxy
man interrupts true love and acts savagely
juice from the fingers of zeus cause calamity