4,261 research outputs found
Insecurity for compact surfaces of positive genus
A pair of points in a riemannian manifold is secure if the geodesics
between the points can be blocked by a finite number of point obstacles;
otherwise the pair of points is insecure. A manifold is secure if all pairs of
points in are secure. A manifold is insecure if there exists an insecure
point pair, and totally insecure if all point pairs are insecure.
Compact, flat manifolds are secure. A standing conjecture says that these are
the only secure, compact riemannian manifolds. We prove this for surfaces of
genus greater than zero. We also prove that a closed surface of genus greater
than one with any riemannian metric and a closed surface of genus one with
generic metric are totally insecure.Comment: 37 pages, 11 figure
Zeta function regularization for a scalar field in a compact domain
We express the zeta function associated to the Laplacian operator on
in terms of the zeta function associated to the Laplacian on
, where is a compact connected Riemannian manifold. This gives formulas
for the partition function of the associated physical model at low and high
temperature for any compact domain . Furthermore, we provide an exact
formula for the zeta function at any value of when is a -dimensional
box or a -dimensional torus; this allows a rigorous calculation of the zeta
invariants and the analysis of the main thermodynamic functions associated to
the physical models at finite temperature.Comment: 19 pages, no figures, to appear in J. Phys.
The Berry-Keating operator on L^2(\rz_>, x) and on compact quantum graphs with general self-adjoint realizations
The Berry-Keating operator H_{\mathrm{BK}}:=
-\ui\hbar(x\frac{
\phantom{x}}{
x}+{1/2}) [M. V. Berry and J. P. Keating,
SIAM Rev. 41 (1999) 236] governing the Schr\"odinger dynamics is discussed in
the Hilbert space L^2(\rz_>,
x) and on compact quantum graphs. It is
proved that the spectrum of defined on L^2(\rz_>,
x) is
purely continuous and thus this quantization of cannot yield
the hypothetical Hilbert-Polya operator possessing as eigenvalues the
nontrivial zeros of the Riemann zeta function. A complete classification of all
self-adjoint extensions of acting on compact quantum graphs
is given together with the corresponding secular equation in form of a
determinant whose zeros determine the discrete spectrum of .
In addition, an exact trace formula and the Weyl asymptotics of the eigenvalue
counting function are derived. Furthermore, we introduce the "squared"
Berry-Keating operator which is a special case of the
Black-Scholes operator used in financial theory of option pricing. Again, all
self-adjoint extensions, the corresponding secular equation, the trace formula
and the Weyl asymptotics are derived for on compact quantum
graphs. While the spectra of both and on
any compact quantum graph are discrete, their Weyl asymptotics demonstrate that
neither nor can yield as eigenvalues the
nontrivial Riemann zeros. Some simple examples are worked out in detail.Comment: 33p
Kronecker's Double Series and Exact Asymptotic Expansion for Free Models of Statistical Mechanics on Torus
For the free models of statistical mechanics on torus, exact asymptotic
expansions of the free energy, the internal energy and the specific heat in the
vicinity of the critical point are found. It is shown that there is direct
relation between the terms of the expansion and the Kronecker's double series.
The latter can be expressed in terms of the elliptic theta-functions in all
orders of the asymptotic expansion.Comment: REVTeX, 22 pages, this is expanded version which includes exact
asymptotic expansions of the free energy, the internal energy and the
specific hea
High cooperativity coupling of electron-spin ensembles to superconducting cavities
Electron spins in solids are promising candidates for quantum memories for
superconducting qubits because they can have long coherence times, large
collective couplings, and many quantum bits can be encoded into the spin-waves
of a single ensemble. We demonstrate the coupling of electron spin ensembles to
a superconducting transmission-line resonator at coupling strengths greatly
exceeding the cavity decay rate and comparable to spin linewidth. We also use
the enhanced coupling afforded by the small cross-section of the transmission
line to perform broadband spectroscopy of ruby at millikelvin temperatures at
low powers. In addition, we observe hyperfine structure in diamond P1 centers
and time domain saturation-relaxation of the spins.Comment: 4pgs, 4 figure
CIRENE Air-Sea Interactions in the Seychelles-Chagos Thermocline Ridge Region
A field experiment in the southwestern Indian Ocean provides new insights into ocean-atmosphere interactions in a key climatic region
Looking backward: From Euler to Riemann
We survey the main ideas in the early history of the subjects on which
Riemann worked and that led to some of his most important discoveries. The
subjects discussed include the theory of functions of a complex variable,
elliptic and Abelian integrals, the hypergeometric series, the zeta function,
topology, differential geometry, integration, and the notion of space. We shall
see that among Riemann's predecessors in all these fields, one name occupies a
prominent place, this is Leonhard Euler. The final version of this paper will
appear in the book \emph{From Riemann to differential geometry and relativity}
(L. Ji, A. Papadopoulos and S. Yamada, ed.) Berlin: Springer, 2017
Structural and magnetic properties of Mn3-xCdxTeO6 (x = 0, 1, 1.5 and 2)
Mn3TeO6 exhibits a corundum-related A3TeO6 structure and a complex magnetic
structure involving two magnetic orbits for the Mn atoms [*]. Mn3-xCdxTeO6
(x=0, 1, 1.5 and 2) ceramics were synthesized by solid state reaction and
investigated using X-ray powder diffraction, electron microscopy, calorimetric
and magnetic measurements. Cd2+ replaces Mn2+ cations without greatly affecting
the structure of the compound. The Mn and Cd cations were found to be randomly
distributed over the A-site. Magnetization measurements indicated that the
samples order antiferromagnetically at low temperature with a transition
temperature that decreases with increasing Cd doping. The nuclear and magnetic
structure of one specially prepared 114Cd containing sample:
Mn1.5(114Cd)1.5TeO6, was studied using neutron powder diffraction over the
temperature range 2 to 295 K. Mn1.5(114Cd)1.5TeO6 was found to order in an
incommensurate helical magnetic structure, very similar to that of Mn3TeO6 [*].
However, with a lower transition temperature and the extension of the ordered
structure confined to order 240(10) {\AA}. [*] S. A. Ivanov et al. Mater. Res.
Bull. 46 (2011) 1870.Comment: 20 pages, 8 figure
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