11 research outputs found
Self-similar solutions of R\'enyi's entropy and the concavity of its entropy power
We study the class of self-similar probability density functions with finite
mean and variance which maximize R\'{e}nyi's entropy. The investigation is
restricted in the Schwartz space and in the space of
-differentiable compactly supported functions .
Interestingly the solutions of this optimization problem do not coincide with
the solutions of the usual porous medium equation with a Dirac point source, as
it occurs in the optimization of Shannon's entropy. We also study the concavity
of the entropy power in with respect to time using two different
methods. The first one takes advantage of the solutions determined earlier
while the second one is based on a setting that could be used for Riemannian
manifolds.Comment: 18 pages, 2 figure
The weakly coupled fractional one-dimensional Schr\"{o}dinger operator with index
We study fundamental properties of the fractional, one-dimensional Weyl
operator densely defined on the Hilbert space
and determine the asymptotic behaviour of
both the free Green's function and its variation with respect to energy for
bound states. In the sequel we specify the Birman-Schwinger representation for
the Schr\"{o}dinger operator
and extract the finite-rank portion which is essential for the asymptotic
expansion of the ground state. Finally, we determine necessary and sufficient
conditions for there to be a bound state for small coupling constant .Comment: 16 pages, 1 figur
The Partition Function of the Dirichlet Operator D2s=∑i=1d(-∂i2)s on a d-Dimensional Rectangle Cavity
We study the asymptotic behavior of the free partition function in the t→0+ limit for a diffusion process which consists of d-independent, one-dimensional, symmetric, 2s-stable processes in a hyperrectangular cavity K⊂Rd with an absorbing boundary. Each term of the partition function for this polyhedron in d-dimensions can be represented by a quermassintegral and the geometrical information conveyed by the eigenvalues of the fractional Dirichlet Laplacian for this solvable model is now transparent. We also utilize the intriguing method of images to solve the same problem, in one and two dimensions, and recover identical results to those derived in the previous analysis
The Partition Function of the Dirichlet Operator D 2 = ∑ =1 ( − 2 ) on a -Dimensional Rectangle Cavity
We study the asymptotic behavior of the free partition function in the → 0 + limit for a diffusion process which consists of -independent, one-dimensional, symmetric, 2 -stable processes in a hyperrectangular cavity ⊂ R with an absorbing boundary. Each term of the partition function for this polyhedron in -dimensions can be represented by a quermassintegral and the geometrical information conveyed by the eigenvalues of the fractional Dirichlet Laplacian for this solvable model is now transparent. We also utilize the intriguing method of images to solve the same problem, in one and two dimensions, and recover identical results to those derived in the previous analysis