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European Mathematical Society Publishing House
2024-03-29 01:30:26
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=RLM&vol=19&iss=2&update_since=2024-03-29
Rendiconti Lincei - Matematica e Applicazioni
Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.
RLM
1120-6330
1720-0768
General
10.4171/RLM
http://www.ems-ph.org/doi/10.4171/RLM
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2006)
19
2008
2
Vanishing and conservativeness of harmonic forms of a non-compact CR manifold
Jun
Masamune
Worchester Polytechnic Institute, WORCHESTER, UNITED STATES
CR manifold, sub-Laplacian, Kohn-Rossi Laplacian, essentially self-adjoint, vanishing theorem, conservative
The self-adjointness of a sublaplacian and Kohn-Rossi laplacians on a non-compact strictly pseudoconvex CR manifold is proved. As applications to geometry, the vanishing and the conservativeness of harmonic forms are obtained.
Differential geometry
General
79
102
10.4171/RLM/510
http://www.ems-ph.org/doi/10.4171/RLM/510
Homoclinic solutions to invariant tori in a center manifold
Vittorio
Coti Zelati
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Marta
Macrì
Università degli Studi di Napoli Federico II, NAPOLI, ITALY
Heteroclinic orbits, critical point theory, invariant tori, center manifold
Ordinary differential equations
General
103
134
10.4171/RLM/511
http://www.ems-ph.org/doi/10.4171/RLM/511
On a theorem of Schmid
Francesco
Esposito
Università di Padova, PADOVA, ITALY
Andrea
Maffei
Università di Roma La Sapienza, ROMA, ITALY
Semisimple algebraic groups, homogeneous spaces, symmetric spaces, regular functions
We establish for which parabolic subgroups $P$ of a simply connected and semisimple algebraic group $G$ with unipotent radical $U$ and Levi factor $H$ the two rings $\mk[G/H]^{U}$ and $\mk[U^-]$ are isomorphic as $H$ algebras. We show the relation of this problem with a Theorem of Schmid and we compare the multiplications in the rings $\mk[U^-]$ and $\mk[G/H]$.
Group theory and generalizations
General
135
140
10.4171/RLM/512
http://www.ems-ph.org/doi/10.4171/RLM/512
Correlation inequalities for spin glass in one dimension
Pierluigi
Contucci
Università di Bologna, BOLOGNA, ITALY
Francesco
Unguendoli
Università Modena e Reggio Emilia, MODENA, ITALY
Spin glasses, correlation inequalities, one-dimensional systems
We prove two inequalities for the direct and truncated correlation functions for the nearest-neighboor one-dimensional Edwards-Anderson model with symmetric quenched disorder. The second inequality has the opposite sign of the GKS inequality of type II. In the non symmetric case with positive average we show that while the direct correlation keeps its sign the truncated one changes sign when crossing a suitable line in the parameter space. That line separates the regions satisfying the GKS second inequality and the one proved here.
Partial differential equations
General
141
147
10.4171/RLM/513
http://www.ems-ph.org/doi/10.4171/RLM/513
Rational points in periodic analytic sets and the Manin–Mumford conjecture
Jonathan
Pila
University of Bristol, BRISTOL, UNITED KINGDOM
Umberto
Zannier
Scuola Normale Superiore, PISA, ITALY
Torsion points on algebraic varieties, rational points on analytic varieties, conjecture of Manin–Mumford
We present a new proof of the Manin–Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion points as rational points on a complex torus and then compare (i) upper bounds for the number of rational points on a transcendental analytic variety (Bombieri–Pila–Wilkie) and (ii) lower bounds for the degree of a torsion point (Masser), after taking conjugates. In order to be able to deal with (i), we discuss (Thm. 2.1) the semi-algebraic curves contained in an analytic variety supposed invariant for translations by a full lattice, which is a topic with some independent motivation.
Algebraic geometry
General
149
162
10.4171/RLM/514
http://www.ems-ph.org/doi/10.4171/RLM/514
A criterion for the reality of the spectrum of PT-symmetric Schrödinger operators with complex-valued periodic potentials
Emanuela
Caliceti
Università di Bologna, BOLOGNA, ITALY
Sandro
Graffi
Università di Bologna, BOLOGNA, ITALY
PT-symmetry, real spectrum, periodic potentials
Consider in $L^2(\R)$ the \Sc\ operator family $H(g):=-d^2_x+V_g(x)$ depending on the real parameter $g$, where $V_g(x)$ is a complex-valued but $PT$ symmetric periodic potential. An explicit condition on $V$ is obtained which ensures that the spectrum of $H(g)$ is purely real and band shaped; furthermore, a further condition is obtained which ensures that the spectrum contains at least a pair of complex analytic arcs.
Quantum theory
General
163
173
10.4171/RLM/515
http://www.ems-ph.org/doi/10.4171/RLM/515