231 research outputs found
A negative mass theorem for surfaces of positive genus
We define the "sum of squares of the wavelengths" of a Riemannian surface
(M,g) to be the regularized trace of the inverse of the Laplacian. We normalize
by scaling and adding a constant, to obtain a "mass", which is scale invariant
and vanishes at the round sphere. This is an anlaog for closed surfaces of the
ADM mass from general relativity. We show that if M has positive genus then on
each conformal class, the mass attains a negative minimum. For the minimizing
metric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and a
Moser-Trudinger-Onofri type inequality.Comment: 8 page
Quantum information entropies of the eigenstates and the coherent state of the P\"oschl-Teller potential
The position and momentum space information entropies, of the ground state of
the P\"oschl-Teller potential, are exactly evaluated and are found to satisfy
the bound, obtained by Beckner, Bialynicki-Birula and Mycielski. These
entropies for the first excited state, for different strengths of the potential
well, are then numerically obtained. Interesting features of the entropy
densities, owing their origin to the excited nature of the wave functions, are
graphically demonstrated. We then compute the position space entropies of the
coherent state of the P\"oschl-Teller potential, which is known to show revival
and fractional revival. Time evolution of the coherent state reveals many
interesting patterns in the space-time flow of information entropy.Comment: Revtex4, 11 pages, 11 eps figures and a tabl
Almost-Euclidean subspaces of via tensor products: a simple approach to randomness reduction
It has been known since 1970's that the N-dimensional -space contains
nearly Euclidean subspaces whose dimension is . However, proofs of
existence of such subspaces were probabilistic, hence non-constructive, which
made the results not-quite-suitable for subsequently discovered applications to
high-dimensional nearest neighbor search, error-correcting codes over the
reals, compressive sensing and other computational problems. In this paper we
present a "low-tech" scheme which, for any , allows to exhibit nearly
Euclidean -dimensional subspaces of while using only
random bits. Our results extend and complement (particularly) recent work
by Guruswami-Lee-Wigderson. Characteristic features of our approach include (1)
simplicity (we use only tensor products) and (2) yielding "almost Euclidean"
subspaces with arbitrarily small distortions.Comment: 11 pages; title change, abstract and references added, other minor
change
Closure properties of solutions to heat inequalities
We prove that if are
sufficiently well-behaved solutions to certain heat inequalities on then
the function given by
also satisfies a heat inequality of a
similar type provided . On
iterating, this result leads to an analogous statement concerning -fold
convolutions. As a corollary, we give a direct heat-flow proof of the sharp
-fold Young convolution inequality and its reverse form.Comment: 12 page
Entropic Uncertainty Relations in Quantum Physics
Uncertainty relations have become the trademark of quantum theory since they
were formulated by Bohr and Heisenberg. This review covers various
generalizations and extensions of the uncertainty relations in quantum theory
that involve the R\'enyi and the Shannon entropies. The advantages of these
entropic uncertainty relations are pointed out and their more direct connection
to the observed phenomena is emphasized. Several remaining open problems are
mentionedComment: 35 pages, review pape
Nonlinear spectral calculus and super-expanders
Nonlinear spectral gaps with respect to uniformly convex normed spaces are
shown to satisfy a spectral calculus inequality that establishes their decay
along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to
behave sub-multiplicatively under zigzag products. These results yield a
combinatorial construction of super-expanders, i.e., a sequence of 3-regular
graphs that does not admit a coarse embedding into any uniformly convex normed
space.Comment: Typos fixed based on referee comments. Some of the results of this
paper were announced in arXiv:0910.2041. The corresponding parts of
arXiv:0910.2041 are subsumed by the current pape
Towards a unified theory of Sobolev inequalities
We discuss our work on pointwise inequalities for the gradient which are
connected with the isoperimetric profile associated to a given geometry. We
show how they can be used to unify certain aspects of the theory of Sobolev
inequalities. In particular, we discuss our recent papers on fractional order
inequalities, Coulhon type inequalities, transference and dimensionless
inequalities and our forthcoming work on sharp higher order Sobolev
inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1
Assessing the quality of reports of randomized trials in pediatric complementary and alternative medicine
OBJECTIVE: To evaluate the quality of reports of complementary and alternative medicine (CAM) randomized controlled trials (RCTs) in the pediatric population. We also examined whether there was a change in the quality of reporting over time. METHODS: We used a systematic sample of 251 reports of RCTs that used a CAM intervention. The quality of each report was assessed using the number of CONSORT checklist items included, the frequency of unclear allocation concealment, and a 5-point quality assessment instrument. RESULTS: Nearly half (40%) of the CONSORT checklist items were included in the reports, with an increase in the number of items included. The majority (81.3%) of RCTs reported unclear allocation concealment with no significant change over time. The quality of reports achieved approximately 40% of their maximum possible total score as assessed with the Jadad scale with no change over time. Information regarding adverse events was reported in less than one quarter of the RCTs (22%) and information regarding costs was mentioned in only a minority of reports (4%). CONCLUSIONS: RCTs are an important tool for evidence based health care decisions. If these studies are to be relevant in the evaluation of CAM interventions it is important that they are conducted and reported with the highest possible standards. There is a need to redouble efforts to ensure that children and their families are participating in RCTs that are conducted and reported with minimal bias. Such studies will increase their usefulness to a board spectrum of interested stakeholders
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