341 research outputs found
On the structure of subsets of an orderable group with some small doubling properties
The aim of this paper is to present a complete description of the structure
of subsets S of an orderable group G satisfying |S^2| = 3|S|-2 and is
non-abelian
The Lagrange and Markov spectra from the dynamical point of view
This text grew out of my lecture notes for a 4-hours minicourse delivered on
October 17 \& 19, 2016 during the research school "Applications of Ergodic
Theory in Number Theory" -- an activity related to the Jean-Molet Chair project
of Mariusz Lema\'nczyk and S\'ebastien Ferenczi -- realized at CIRM, Marseille,
France. The subject of this text is the same of my minicourse, namely, the
structure of the so-called Lagrange and Markov spectra (with an special
emphasis on a recent theorem of C. G. Moreira).Comment: 27 pages, 6 figures. Survey articl
Inverse Additive Problems for Minkowski Sumsets II
The Brunn-Minkowski Theorem asserts that for convex bodies , where
denotes the -dimensional Lebesgue measure. It is well-known that
equality holds if and only if and are homothetic, but few
characterizations of equality in other related bounds are known. Let be a
hyperplane. Bonnesen later strengthened this bound by showing where
and
. Standard
compression arguments show that the above bound also holds when
and , where denotes a
projection of onto , which gives an alternative generalization
of the Brunn-Minkowski bound. In this paper, we characterize the cases of
equality in this later bound, showing that equality holds if and only if
and are obtained from a pair of homothetic convex bodies by `stretching'
along the direction of the projection, which is made formal in the paper. When
, we characterize the case of equality in the former bound as well
Low temperature heat capacity of fullerite C60 doped with nitrogen
The heat capacity Cm of polycrystalline fullerite C60 doped with nitrogen has
been measured in the temperature interval 2 - 13 K. The contributions to heat
capacity from translational lattice vibrations (Debye contribution),
orientational vibrations of the C60 molecules (Einstein contribution) and from
the motion of the N2 molecules in the octahedral cavities of the C60 lattice
have been estimated. However, we could not find (beyond the experimental error
limits) any indications of the first - order phase transformation that had been
detected earlier in the dilatometric investigation of the orientational N2-C60
glass. A possible explanation of this fact is proposed.Comment: 4 pages, 2 figures, to be published in Fiz. Nizk. Temp. (Low Temp.
Phys.
Quantum Fluctuations Driven Orientational Disordering: A Finite-Size Scaling Study
The orientational ordering transition is investigated in the quantum
generalization of the anisotropic-planar-rotor model in the low temperature
regime. The phase diagram of the model is first analyzed within the mean-field
approximation. This predicts at a phase transition from the ordered to
the disordered state when the strength of quantum fluctuations, characterized
by the rotational constant , exceeds a critical value . As a function of temperature, mean-field theory predicts a range of
values of where the system develops long-range order upon cooling, but
enters again into a disordered state at sufficiently low temperatures
(reentrance). The model is further studied by means of path integral Monte
Carlo simulations in combination with finite-size scaling techniques,
concentrating on the region of parameter space where reentrance is predicted to
occur. The phase diagram determined from the simulations does not seem to
exhibit reentrant behavior; at intermediate temperatures a pronounced increase
of short-range order is observed rather than a genuine long-range order.Comment: 27 pages, 8 figures, RevTe
Small doubling in groups
Let A be a subset of a group G = (G,.). We will survey the theory of sets A
with the property that |A.A| <= K|A|, where A.A = {a_1 a_2 : a_1, a_2 in A}.
The case G = (Z,+) is the famous Freiman--Ruzsa theorem.Comment: 23 pages, survey article submitted to Proceedings of the Erdos
Centenary conferenc
Small ball probability, Inverse theorems, and applications
Let be a real random variable with mean zero and variance one and
be a multi-set in . The random sum
where are iid copies of
is of fundamental importance in probability and its applications.
We discuss the small ball problem, the aim of which is to estimate the
maximum probability that belongs to a ball with given small radius,
following the discovery made by Littlewood-Offord and Erdos almost 70 years
ago. We will mainly focus on recent developments that characterize the
structure of those sets where the small ball probability is relatively
large. Applications of these results include full solutions or significant
progresses of many open problems in different areas.Comment: 47 page
Microwave Inter-Connections and Switching by means of Carbon Nano-tubes
In this work, carbon nanotube (CNT) based
interconnections and switches will be reviewed,
discussing the possibility to use nanotubes as potential
building blocks for signal routing in microwave
networks. In particular, theoretical design of coplanar
waveguide (CPW), micro‐strip single‐pole‐single‐throw
(SPST) and single‐pole‐double‐throw (SPDT) devices has
been performed to predict the electrical performances of
CNT‐based RF switching configurations. Actually, by
using the semiconductor‐conductor transition obtained
by properly biasing the CNTs, an isolation better than 30
dB can be obtained between the ON and OFF states of the
switch for very wide bandwidth applications. This
happens owing to the shape deformation and consequent
change in the band‐gap due to the external pressure
caused by the electric field. State‐of‐art for other
switching techniques based on CNTs and their use for RF
nano‐interconnections is also discussed, together with
current issues in measurement techniques
Third order dielectric susceptibility in a model quantum paraelectric
In the context of perovskite quantum paraelectrics, we study the effects of a
quadrupolar interaction , in addition to the standard dipolar one .
We concentrate here on the nonlinear dielectric response , as
the main response function sensitive to quadrupolar (in our case
antiquadrupolar) interactions. We employ a 3D quantum four-state lattice model
and mean-field theory. The results show that inclusion of quadrupolar coupling
of moderate strength () is clearly accompanied by a
double change of sign of from negative to positive, near the
quantum temperature where the quantum paraelectric behaviour sets in. We
fit our to recent experimental data for SrTiO, where the
sign change is identified close to .Comment: 22 page
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