- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 07:01:35
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https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=10&iss=1&update_since=2024-03-29
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
10
2008
1
Quasi-periodic solutions of nonlinear random Schrödinger equations
Jean
Bourgain
Institute for Advanced Study, PRINCETON, UNITED STATES
Wei-Min
Wang
University of Massachusetts, AMHERST, UNITED STATES
Compact convex hypersurfaces, closed characteristics, Hamiltonian systems, Morse theory, mean index identity, stability
In this paper, let $\Sigma\subset\R^{6}$ be a compact convex hypersurface. We prove that if $\Sigma$ carries only finitely many geometrically distinct closed characteristics, then at least two of them must possess irrational mean indices. Moreover, if $\Sg$ carries exactly three geometrically distinct closed characteristics, then at least two of them must be elliptic.
Global analysis, analysis on manifolds
Ordinary differential equations
Dynamical systems and ergodic theory
General
1
45
10.4171/JEMS/102
http://www.ems-ph.org/doi/10.4171/JEMS/102
Semiclassical states for weakly coupled nonlinear Schrödinger systems
Eugenio
Montefusco
Università di Roma La Sapienza, ROMA, ITALY
Benedetta
Pellacci
Università degli Studi di Napoli 'Parthenope', NAPOLI, ITALY
Marco
Squassina
Università degli Studi di Verona, VERONA, ITALY
Weakly coupled nonlinear Schrödinger systems, concentration phenomena, semiclassical limit, ground states, critical point theory, Clarke's subdifferential
We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and we investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.
Ordinary differential equations
Partial differential equations
General
47
71
10.4171/JEMS/103
http://www.ems-ph.org/doi/10.4171/JEMS/103
Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions
Catherine
Bandle
, AESCH/BL, SWITZERLAND
Joachim
von Below
Université du Littoral Côte d'Opale, CALAIS, CEDEX, FRANCE
Wolfgang
Reichel
Universität Basel, BASEL, SWITZERLAND
Positivity principle, anti-maximum principle, eigenvalues, Harnack inequality
Partial differential equations
Computer science
General
73
104
10.4171/JEMS/104
http://www.ems-ph.org/doi/10.4171/JEMS/104
Single-point blow-up on the boundary where the zero Dirichlet boundary condition is imposed
Marek
Fila
Comenius University, BRATISLAVA, SLOVAK REPUBLIC
Michael
Winkler
Universität Paderborn, PADERBORN, GERMANY
Reaction-diffusion-convection equation, selfsimilar solution, blow-up on the boundary
We consider a reaction-diffusion-convection equation on the halfline $(0,\infty)$ with the zero Dirichlet boundary condition at $x=0$. We find a positive selfsimilar solution $u$ which blows up in a finite time $T$ at $x=0$ while $u(x,T)$ remains bounded for $x>0$.
Partial differential equations
Ordinary differential equations
General
105
132
10.4171/JEMS/105
http://www.ems-ph.org/doi/10.4171/JEMS/105
Giant component and vacant set for random walk on a discrete torus
Itai
Benjamini
Weizmann Institute of Science, REHOVOT, ISRAEL
Alain-Sol
Sznitman
ETH Zürich, ZÜRICH, SWITZERLAND
We consider random walk on a discrete torus $E$ of side-length $N$, in sufficiently high dimension $d$. We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time $uN^d$. We show that when $u$ is chosen small, as $N$ tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const $\log N$. Moreover, this connected component occupies a non-degenerate fraction of the total number of sites $N^d$ of $E$, and any point of $E$ lies within distance $N^\beta$ of this component, with $\beta$ an arbitrary positive number.
Probability theory and stochastic processes
General
133
172
10.4171/JEMS/106
http://www.ems-ph.org/doi/10.4171/JEMS/106
Finite projective planes, Fermat curves, and Gaussian periods
Koen
Thas
Universiteit Gent, GENT, BELGIUM
Don
Zagier
, BONN, GERMANY
Flag-transitive projective plane, Gauss sum, Jacobi sum, Fermat surface, prime
One of the oldest and most fundamental problems in the theory of finite projective planes is to classify those having a group which acts transitively on the incident point-line pairs (flags). The conjecture is that the only ones are the Desarguesian projective planes (over a finite field). In this paper, we show that non-Desarguesian finite flag-transitive projective planes exist if and only if certain Fermat surfaces have no nontrivial rational points, and formulate several other equivalences involving Fermat curves and Gaussian periods. In particular, we show that a non-Desarguesian flag-transitive projective plane of order $n$ exists if and only if $n>8$, the number $p=n^2+n+1$ is prime, and the square of the absolute value of the Gaussian period $\,\sum_{a\in\D_n}\z^a\,$ ($\z\,=$ primitive $p$th root of unity, $\D_n\,=$ group of $n$th powers in $\Fm$) belongs to $\Z$. We also formulate a conjectural classification of all pairs $(p,n)$ with $p$ prime and $n\mid p-1$ having this latter property, and give an application to the construction of symmetric designs with flag-transitive automorphism groups. Numerical computations are described verifying the first conjecture for $p
Geometry
Combinatorics
Number theory
Algebraic geometry
173
190
10.4171/JEMS/107
http://www.ems-ph.org/doi/10.4171/JEMS/107
Further characterizations of Sobolev spaces
Hoai-Minh
Nguyen
EPFL SB CAMA, LAUSANNE, SWITZERLAND
Sobolev spaces
Functional analysis
General
191
229
10.4171/JEMS/108
http://www.ems-ph.org/doi/10.4171/JEMS/108
A remark on the homotopical dimension of some moduli spaces of stable Riemann surfaces
Gabriele
Mondello
Università di Roma “La Sapienza”, Roma, ITALY
Reaction-diffusion-convection equation, selfsimilar solution, blow-up on the boundary
Using a result of Harer, we prove certain upper bounds for the homotopical/cohomological dimension of the moduli spaces of Riemann surfaces of compact type, of Riemann surfaces with rational tails and of Riemann surfaces with at most $k$ rational components. These bounds would follow from conjectures of Looijenga and Roth-Vakil.
Several complex variables and analytic spaces
Algebraic topology
General
231
241
10.4171/JEMS/109
http://www.ems-ph.org/doi/10.4171/JEMS/109
Homological category weights and estimates for cat1(X,ξ)
Mikhail
Belolipetsky
University of Durham, DURHAM, UNITED KINGDOM
Michael
Farber
Queen Mary University of London, LONDON, UNITED KINGDOM
Lusternik - Schnirelmann theory, category weight, topology of closed 1-form, homology classes movable to infinity, asymptotic cycle
In this paper we study a new notion of category weight of homology classes developing further the ideas of E. Fadell and S. Husseini. In the case of closed smooth manifolds the homological category weight is equivalent to the cohomological category weight of E. Fadell and S. Husseini but these two notions are distinct already for Poincar\\'e complexes. An important advantage of the homological category weight is its homotopy invariance. We use the notion of homological category weight to study various generalizations of the Lusternik - Schnirelmann category which appeared in the theory of closed one-forms and have applications in dynamics. Our primary goal is to compare two such invariants $\\cat(X,\\xi)$ and $\\cat^1(X,\\xi)$ which are defined similarly with reversion of the order of quantifiers. We compute these invariants explicitly for products of surfaces and show that they may differ by an arbitrarily large quantity. The proof of one of our main results uses an algebraic characterization of homology classes $z\\in H_i(\\tilde X;\\Z)$ (where $\\tilde X\\to X$ is a free abelian covering) which are movable to infinity of $\\tilde X$ with respect to a prescribed cohomology class $\\xi\\in H^1(X;\\R)$. This result is established in Part II which can be read independently of the rest of the paper.
Algebraic topology
General
243
266
10.4171/JEMS/110
http://www.ems-ph.org/doi/10.4171/JEMS/110