35,855 research outputs found
Bounds on the heat kernel of the Schroedinger operator in a random electromagnetic field
We obtain lower and upper bounds on the heat kernel and Green functions of
the Schroedinger operator in a random Gaussian magnetic field and a fixed
scalar potential. We apply stochastic Feynman-Kac representation, diamagnetic
upper bounds and the Jensen inequality for the lower bound. We show that if the
covariance of the electromagnetic (vector) potential is increasing at large
distances then the lower bound is decreasing exponentially fast for large
distances and a large time.Comment: some technical improvements, new references, to appear in
Journ.Phys.
Upper and lower bounds on the mean square radius and criteria for occurrence of quantum halo states
In the context of non-relativistic quantum mechanics, we obtain several upper
and lower limits on the mean square radius applicable to systems composed by
two-body bound by a central potential. A lower limit on the mean square radius
is used to obtain a simple criteria for the occurrence of S-wave quantum halo
sates.Comment: 12 pages, 2 figure
Exact upper and lower bounds on the difference between the arithmetic and geometric means
Let denote a nonnegative random variable with .
Upper and lower bounds on are obtained,
which are exact, in terms of and for the upper bound and in terms
of and for the lower bound, where ,
,
, , , and is the support set of the distribution of . Note that, if
takes each of distinct real values with probability ,
then and are, respectively, the arithmetic
and geometric means of .Comment: 8 pages; to appear in the Bulletin of the Australian Mathematical
Society. Version 2: the condition that the random variable X is nonnegative
was missing in the abstrac
Fractional Power Control for Decentralized Wireless Networks
We consider a new approach to power control in decentralized wireless
networks, termed fractional power control (FPC). Transmission power is chosen
as the current channel quality raised to an exponent -s, where s is a constant
between 0 and 1. The choices s = 1 and s = 0 correspond to the familiar cases
of channel inversion and constant power transmission, respectively. Choosing s
in (0,1) allows all intermediate policies between these two extremes to be
evaluated, and we see that usually neither extreme is ideal. We derive
closed-form approximations for the outage probability relative to a target SINR
in a decentralized (ad hoc or unlicensed) network as well as for the resulting
transmission capacity, which is the number of users/m^2 that can achieve this
SINR on average. Using these approximations, which are quite accurate over
typical system parameter values, we prove that using an exponent of 1/2
minimizes the outage probability, meaning that the inverse square root of the
channel strength is a sensible transmit power scaling for networks with a
relatively low density of interferers. We also show numerically that this
choice of s is robust to a wide range of variations in the network parameters.
Intuitively, s=1/2 balances between helping disadvantaged users while making
sure they do not flood the network with interference.Comment: 16 pages, in revision for IEEE Trans. on Wireless Communicatio
Properties of Classical and Quantum Jensen-Shannon Divergence
Jensen-Shannon divergence (JD) is a symmetrized and smoothed version of the
most important divergence measure of information theory, Kullback divergence.
As opposed to Kullback divergence it determines in a very direct way a metric;
indeed, it is the square of a metric. We consider a family of divergence
measures (JD_alpha for alpha>0), the Jensen divergences of order alpha, which
generalize JD as JD_1=JD. Using a result of Schoenberg, we prove that JD_alpha
is the square of a metric for alpha lies in the interval (0,2], and that the
resulting metric space of probability distributions can be isometrically
embedded in a real Hilbert space. Quantum Jensen-Shannon divergence (QJD) is a
symmetrized and smoothed version of quantum relative entropy and can be
extended to a family of quantum Jensen divergences of order alpha (QJD_alpha).
We strengthen results by Lamberti et al. by proving that for qubits and pure
states, QJD_alpha^1/2 is a metric space which can be isometrically embedded in
a real Hilbert space when alpha lies in the interval (0,2]. In analogy with
Burbea and Rao's generalization of JD, we also define general QJD by
associating a Jensen-type quantity to any weighted family of states.
Appropriate interpretations of quantities introduced are discussed and bounds
are derived in terms of the total variation and trace distance.Comment: 13 pages, LaTeX, expanded contents, added references and corrected
typo
Balanced distribution-energy inequalities and related entropy bounds
Let be a self-adjoint operator acting over a space endowed with a
partition. We give lower bounds on the energy of a mixed state from its
distribution in the partition and the spectral density of . These bounds
improve with the refinement of the partition, and generalize inequalities by
Li-Yau and Lieb--Thirring for the Laplacian in . They imply an
uncertainty principle, giving a lower bound on the sum of the spatial entropy
of , as seen from , and some spectral entropy, with respect to its
energy distribution. On , this yields lower bounds on the sum of the
entropy of the densities of and its Fourier transform. A general
log-Sobolev inequality is also shown. It holds on mixed states, without
Markovian or positivity assumption on .Comment: 21 page
Modulus of convexity for operator convex functions
Given an operator convex function , we obtain an operator-valued lower
bound for , . The lower bound
is expressed in terms of the matrix Bregman divergence. A similar inequality is
shown to be false for functions that are convex but not operator convex.Comment: 5 pages, change of title. The new version shows that the main result
of the original paper cannot be extended to convex functions that are not
operator convex
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