9,339 research outputs found
On sound ranging in Hilbert space
We consider the sound ranging problem, which is to find the position of the
source-point from the moments when the wave-sphere of linearly, with time,
increasing radius reaches the sensor-points, in the infinite-dimensional
separable Euclidean space H, and describe the solving methods, for entire space
and for its unit sphere. In the former case, we give some sufficient conditions
for uniqueness of the solution. We also provide two examples with the sets of
sensors being a basis of H: 1st, when sound ranging problem and so-called dual
problem both have single solutions, and 2nd, when sound ranging problem has two
distinct solutions.Comment: 15 pages; added prop. 3, ex. 3,4; removed refs; minor text cor
Cluster ensembles, quantization and the dilogarithm
Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A ->
X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related
to the A-space. We develope general properties of cluster ensembles, including
its group of symmetries - the cluster modular group, and a relation with the
motivic dilogarithm. We define a q-deformation of the X-space. Formulate
general duality conjectures regarding canonical bases in the cluster ensemble
context. We support them by constructing the canonical pairing in the finite
type case.
Interesting examples of cluster ensembles are provided the higher Teichmuller
theory, that is by the pair of moduli spaces corresponding to a split reductive
group G and a surface S defined in math.AG/0311149.
We suggest that cluster ensembles provide a natural framework for higher
quantum Teichmuller theory.Comment: Version 7: Final version. To appear in Ann. Sci. Ecole Normale. Sup.
New material in Section 5. 58 pages, 11 picture
Black hole physics, confining solutions of SU(3)-Yang-Mills equations and relativistic models of mesons
The black hole physics techniques and results are applied to find the set of
the exact solutions of the SU(3)-Yang-Mills equations in Minkowski spacetime in
the Lorentz gauge. All the solutions contain only the Coulomb-like or linear in
components of SU(3)-connection. This allows one to obtain some possible
exact and approximate solutions of the corresponding Dirac equation that can
describe the relativistic bound states. Possible application to the
relativistic models of mesons is also outlined.Comment: 13 pages, LaTeX with using the mpla1.sty file from the package of
World Scientific Publishing C
Limits of structure stability of simple liquids revealed by study of relative fluctuations
We analyse the inverse reduced fluctuations (inverse ratio of relative volume
fluctuation to its value in the hypothetical case where the substance acts an
ideal gas for the same temperature-volume parameters) for simple liquids from
experimental acoustic and thermophysical data along a coexistence line for both
liquid and vapour phases. It has been determined that this quantity has a
universal exponential character within the region close to the melting point.
This behaviour satisfies the predictions of the mean-field (grand canonical
ensemble) lattice fluid model and relates to the constant average structure of
a fluid, i.e. redistribution of the free volume complementary to a number of
vapour particles. The interconnection between experiment-based fluctuational
parameters and self-diffusion characteristics is discussed. These results may
suggest experimental methods for determination of self-diffusion and structural
properties of real substances.Comment: 5 pages, 4 figure
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