32,832 research outputs found

### The formation of young B/PS bulges in edge-on barred galaxies

We report about the fact that the stellar population that is born in the gas
inflowing towards the central regions can be vertically unstable leading to a
B/PS feature remarkably bluer that the surrounding bulge. Using new
chemodynamical simulations we show that this young population does not remain
as flat as the gaseous nuclear disc and buckles out of the plane to form a new
boxy bulge. We show that such a young B/PS bulge can be detected in colour
maps.Comment: 2 pages, 5 figures, to appear in IAU Symposium 245, Formation and
Evolution of Galaxy Bulges, M. Bureau, E. Athanassoula, and B. Barbuy (eds.),
Oxford, 16-20 July 200

### Riemann Zeroes and Phase Transitions via the Spectral Operator on Fractal Strings

The spectral operator was introduced by M. L. Lapidus and M. van
Frankenhuijsen [La-vF3] in their reinterpretation of the earlier work of M. L.
Lapidus and H. Maier [LaMa2] on inverse spectral problems and the Riemann
hypothesis. In essence, it is a map that sends the geometry of a fractal string
onto its spectrum. In this survey paper, we present the rigorous functional
analytic framework given by the authors in [HerLa1] and within which to study
the spectral operator. Furthermore, we also give a necessary and sufficient
condition for the invertibility of the spectral operator (in the critical
strip) and therefore obtain a new spectral and operator-theoretic reformulation
of the Riemann hypothesis. More specifically, we show that the spectral
operator is invertible (or equivalently, that zero does not belong to its
spectrum) if and only if the Riemann zeta function zeta(s) does not have any
zeroes on the vertical line Re(s)=c. Hence, it is not invertible in the
mid-fractal case when c=1/2, and it is invertible everywhere else (i.e., for
all c in(0,1) with c not equal to 1/2) if and only if the Riemann hypothesis is
true. We also show the existence of four types of (mathematical) phase
transitions occurring for the spectral operator at the critical fractal
dimension c=1/2 and c=1 concerning the shape of the spectrum, its boundedness,
its invertibility as well as its quasi-invertibility

### Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator

A spectral reformulation of the Riemann hypothesis was obtained in [LaMa2] by
the second author and H. Maier in terms of an inverse spectral problem for
fractal strings. This problem is related to the question "Can one hear the
shape of a fractal drum?" and was shown in [LaMa2] to have a positive answer
for fractal strings whose dimension is c\in(0,1)-\{1/2} if and only if the
Riemann hypothesis is true. Later on, the spectral operator was introduced
heuristically by M. L. Lapidus and M. van Frankenhuijsen in their theory of
complex fractal dimensions [La-vF2, La-vF3] as a map that sends the geometry of
a fractal string onto its spectrum. We focus here on presenting the rigorous
results obtained by the authors in [HerLa1] about the invertibility of the
spectral operator. We show that given any $c\geq0$, the spectral operator
$\mathfrak{a}=\mathfrak{a}_{c}$, now precisely defined as an unbounded normal
operator acting in a Hilbert space $\mathbb{H}_{c}$, is `quasi-invertible'
(i.e., its truncations are invertible) if and only if the Riemann zeta function
$\zeta=\zeta(s)$ does not have any zeroes on the line $Re(s)=c$. It follows
that the associated inverse spectral problem has a positive answer for all
possible dimensions $c\in (0,1)$, other than the mid-fractal case when $c=1/2$,
if and only if the Riemann hypothesis is true.Comment: To appear in: "Fractal Geometry and Dynamical Systems in Pure and
Applied Mathematics", Part 1 (D. Carfi, M. L. Lapidus, E. P. J. Pearse and M.
van Frankenhuijsen, eds.), Contemporary Mathematics, Amer. Math. Soc.,
Providence, RI, 2013. arXiv admin note: substantial text overlap with
arXiv:1203.482

### Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

We survey some of the universality properties of the Riemann zeta function
$\zeta(s)$ and then explain how to obtain a natural quantization of Voronin's
universality theorem (and of its various extensions). Our work builds on the
theory of complex fractal dimensions for fractal strings developed by the
second author and M. van Frankenhuijsen in \cite{La-vF4}. It also makes an
essential use of the functional analytic framework developed by the authors in
\cite{HerLa1} for rigorously studying the spectral operator $\mathfrak{a}$
(mapping the geometry onto the spectrum of generalized fractal strings), and
the associated infinitesimal shift $\partial$ of the real line:
$\mathfrak{a}=\zeta(\partial)$. In the quantization (or operator-valued)
version of the universality theorem for the Riemann zeta function $\zeta(s)$
proposed here, the role played by the complex variable $s$ in the classical
universality theorem is now played by the family of `truncated infinitesimal
shifts' introduced in \cite{HerLa1} to study the invertibility of the spectral
operator in connection with a spectral reformulation of the Riemann hypothesis
as an inverse spectral problem for fractal strings. This latter work provided
an operator-theoretic version of the spectral reformulation obtained by the
second author and H. Maier in \cite{LaMa2}. In the long term, our work (along
with \cite{La5, La6}), is aimed in part at providing a natural quantization of
various aspects of analytic number theory and arithmetic geometry

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