- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 00:45:09
5
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JEMS&vol=20&iss=6&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
20
2018
6
The onset of instability in first-order systems
Nicolas
Lerner
Université Pierre et Marie Curie, Paris, France
Toan
Nguyen
Pennsylvania State University, University Park, USA
Benjamin
Texier
Université Paris-Diderot, France
Cauchy problem, hyperbolicity, ellipticity
We study the Cauchy problem for first-order quasi-linear systems of partial differential equations. When the spectrum of the initial principal symbol is not included in the real line, i.e., hyperbolicity is violated at initial time, the Cauchy problem is strongly unstable in the sense of Hadamard. This phenomenon, which extends the linear Lax–Mizohata theorem, was explained by G. Métivier [Contemp. Math. 368, 2005]. In the present paper, we are interested in the transition from hyperbolicity to non-hyperbolicity, that is, the limiting case where hyperbolicity holds at initial time, but is violated at positive times: under that hypothesis, we generalize a recent work by N. Lerner, Y. Morimoto and C.-J. Xu [Amer. J. Math. 132 (2010)] on complex scalar systems, as we prove that even a weak defect of hyperbolicity implies a strong Hadamard instability. Our examples include Burgers systems, Van der Waals gas dynamics, and Klein–Gordon–Zakharov systems. Our analysis relies on an approximation result for pseudo-differential flows, proved by B. Texier [Indiana Univ. Math. J. 65 (2016)].
Partial differential equations
1303
1373
10.4171/JEMS/788
http://www.ems-ph.org/doi/10.4171/JEMS/788
4
16
2018
A stable, polynomial-time algorithm for the eigenpair problem
Diego
Armentano
Universidad de la República, Montevideo, Uruguay
Carlos
Beltrán
Universidad de Cantabria, Santander, Spain
Peter
Bürgisser
Technische Universität Berlin, Germany
Felipe
Cucker
City University of Hong Kong, Kowloon Tong, Hong Kong
Michael
Shub
City University of New York, USA
Eigenvalue computations, homotopy methods
We describe algorithms for computing eigenpairs (eigenvalue-eigenvector pairs) of a complex $n \times n$ matrix $A$. These algorithms are numerically stable, strongly accurate, and theoretically efficient (i.e., polynomial-time). We do not believe they outperform in practice the algorithms currently used for this computational problem. The merit of our paper is to give a positive answer to a long-standing open problem in numerical linear algebra.
Numerical analysis
Linear and multilinear algebra; matrix theory
1375
1437
10.4171/JEMS/789
http://www.ems-ph.org/doi/10.4171/JEMS/789
4
16
2018
Finitely related algebras in congruence modular varieties have few subpowers
Libor
Barto
Charles University, Prague, Czechia
Finitely related algebra, congruence modular variety, Gumm terms, few subpowers, cube terms
We show that every finite algebra which is finitely related and lies in a congruence modular variety has few subpowers. This result, combined with other theorems, has interesting consequences for the complexity of several computational problems associated to finite relational structures: the constraint satisfaction problem, the primitive positive formula comparison problem, and the learnability problem for primitive positive formulas. Another corollary is that it is decidable whether an algebra given by a set of relations has few subpowers.
General algebraic systems
Computer science
1439
1471
10.4171/JEMS/790
http://www.ems-ph.org/doi/10.4171/JEMS/790
4
23
2018
Triangulated surfaces in triangulated categories
Tobias
Dyckerhoff
University of Bonn, Germany
Mikhail
Kapranov
IPMU, Kashiwa, Japan
Triangulated categories, ribbon graphs, topological Fukaya categories, mapping class groups
For a triangulated category $\mathcal A$ with a 2-periodic dg-enhancement and a triangulated oriented marked surface $S$, we introduce a dg-category $F(S,\mathcal A)$ parametrizing systems of exact triangles in $\mathcal A$ labelled by triangles of $S$. Our main result is that $\mathcal F(S,\mathcal A)$ is independent of the choice of a triangulation of $S$ up to essentially unique Morita equivalence. In particular, it admits a canonical action of the mapping class group. The proof is based on general properties of cyclic 2-Segal spaces. In the simplest case, where $\mathcal A$ is the category of 2-periodic complexes of vector spaces, $\mathcal F(S,\mathcal A)$ turns out to be a purely topological model for the Fukaya category of the surface $S$. Therefore, our construction can be seen as implementing a 2-dimensional instance of Kontsevich's program on localizing the Fukaya category along a singular Lagrangian spine.
Category theory; homological algebra
Algebraic geometry
1473
1524
10.4171/JEMS/791
http://www.ems-ph.org/doi/10.4171/JEMS/791
4
30
2018
Recognizing PGL$_3$ via generic 4-transitivity
Tuna
Altınel
Université Claude Bernard Lyon 1, France
Joshua
Wiscons
California State University, Sacramento, USA
Primitive permutation groups, multiple transitivity, finite Morley rank
We show that the only transitive and generically 4-transitive action of a group of finite Morley rank on a set of Morley rank 2 is the natural action of PGL$_3$ on the projective plane.
Group theory and generalizations
Mathematical logic and foundations
1525
1559
10.4171/JEMS/792
http://www.ems-ph.org/doi/10.4171/JEMS/792
5
7
2018