Electronic Research Announcements
http://aimsciences.org/
Desingularization of surface maps
http://aimsciences.org//article/id/9ae00a92-5942-4a24-92b4-5886bd60a8e3
2017-02-24We prove a result for maps of surfaces that illustrates how singularhyperbolic flows can be desingularized if a global section can be collapsed to a surface along stable leaves.]]>24019
Erica Clay, Boris Hasselblatt, Enrique Pujals
Equational theories of unstable involution semigroups
http://aimsciences.org//article/id/5c95b808-8041-4c0e-8191-654f87d0875a
2017-02-24It is long known that with respect to the property of having a finitely axiomatizable equational theory, there is no relationship between a general involution semigroup and its semigroup reduct. The present article establishes such a relationship within the class of involution semigroups that are unstable in the sense that the varieties they generate contain semilattices with nontrivial involution. Specifically, it is shown that the equational theory of an unstable involution semigroup is not finitely axiomatizable whenever the equational theory of its semigroup reduct satisfies the same property. Consequently, many results on equational properties of semigroups can be converted into results applicable to involution semigroups.]]>2401020
Edmond W. H. Lee
The orbifold Langer-Miyaoka-Yau Inequality and Hirzebruch-type inequalities
http://aimsciences.org//article/id/39c7fde3-2b9e-4d91-ac15-df0d32cb1778
2017-02-24Using Langer's variation on the Bogomolov-Miyaoka-Yau inequality, we provide some Hirzebruch-type inequalities for curve arrangements in the complex projective plane.]]>2402127
Piotr Pokora
Rigidity of the <inline-formula><tex-math id="M1">\begin{document}${{L}^{p}}$\end{document}</tex-math></inline-formula>-norm of the Poisson bracket on surfaces
http://aimsciences.org//article/id/4315a7d6-170b-4216-8b34-81621ee09000
2017-05-13For a symplectic manifold \begin{document}$(M,ω)$\end{document}, let \begin{document}$\{·,·\}$\end{document} be the corresponding Poisson bracket. In this note we prove that the functional \begin{document}$ (F,G) \mapsto \|\{F,G\}\|_{L^p(M)} $\end{document} is lower-semicontinuous with respect to the \begin{document}$C^0$\end{document}-norm on \begin{document}$C^∞_c(M)$\end{document} when \begin{document}$\dim M = 2$\end{document} and \begin{document}$p < ∞$\end{document}, extending previous rigidity results for \begin{document}$p = ∞$\end{document} in arbitrary dimension.]]>2402837
Karina Samvelyan, Frol Zapolsky
Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations
http://aimsciences.org//article/id/1e4462e5-0a7a-4b62-8623-f56c63ad85be
2017-06-01In this paper, we study the Dirichlet boundary value problem of a class of nonlinear parabolic equations. By a priori estimates, difference and variation techniques, we establish the existence and uniqueness of weak solutions of this problem.]]>2403852
Peiying Chen
Sharpness of the Brascamp–Lieb inequality in Lorentz spaces
http://aimsciences.org//article/id/d3519f28-5f8c-44f1-a2b5-dbf3a91c4ce8
2017-06-09We provide necessary conditions for the refined version of the Brascamp-Lieb inequality where the input functions are allowed to belong to Lorentz spaces, thereby establishing the sharpness of the range of Lorentz exponents in the subcritical case. Using similar considerations, some sharp refinements of the Strichartz estimates for the kinetic transport equation are established.]]>2405363
Neal Bez, Sanghyuk Lee, Shohei Nakamura, Yoshihiro Sawano
A note on parallelizable dynamical systems
http://aimsciences.org//article/id/c295577c-6595-4cc1-9328-7591a970f568
2017-06-09Hájek [3] showed that a dynamical system on a Tychonoff space with paracompact orbit space is parallelizable if and only if its corresponding bundle is a locally trivial fiber bundle with fiber \begin{document}$\mathbb{R}$\end{document}. The present paper provides an enhancement for this classical theorem by omitting all topological hypotheses.]]>2406467
Josiney A. Souza, Tiago A. Pacifico, Hélio V. M. Tozatti
Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras
http://aimsciences.org//article/id/e7c052bf-3b5b-4188-8b93-ca42954df833
2017-08-18We characterize the groupoids for which an operator is Fredholm if and only if its principal symbol and all its boundary restrictions are invertible. A groupoid with this property is called Fredholm. Using results on the Effros-Hahn conjecture, we show that an almost amenable, Hausdorff, second countable groupoid is Fredholm. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. In particular, one can use these results to characterize the Fredholm operators on manifolds with cylindrical and poly-cylindrical ends, on manifolds that are asymptotically Euclidean or asymptotically hyperbolic, on products of such manifolds, and on many other non-compact manifolds. Moreover, we show that the desingularization of groupoids preserves the class of Fredholm groupoids.]]>2406877
Catarina Carvalho, Victor Nistor, Yu Qiao
On matrix wreath products of algebras
http://aimsciences.org//article/id/cd0d100f-f7af-4da9-8258-ae99a8a4c094
2017-08-18We introduce a new construction of matrix wreath products of algebras that is similar to the construction of wreath products of groups introduced by L. Kaloujnine and M. Krasner [17]. We then illustrate its usefulness by proving embedding theorems into finitely generated algebras and constructing nil algebras with prescribed Gelfand-Kirillov dimension.]]>2407886
Adel Alahmadi, Hamed Alsulami, S.K. Jain, Efim Zelmanov
Real orientations, real Gromov-Witten theory, and real enumerative geometry
http://aimsciences.org//article/id/441b2d01-adb3-4eef-bcfd-47c792e076a9
2017-09-06The present note overviews our recent construction of real Gromov-Witten theory in arbitrary genera for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold, its properties, and its connections with real enumerative geometry. Our construction introduces the principle of orienting the determinant of a differential operator relative to a suitable base operator and a real setting analogue of the (relative) spin structure of open Gromov-Witten theory. Orienting the relative determinant, which in the now-standard cases is canonically equivalent to orienting the usual determinant, is naturally related to the topology of vector bundles in the relevant category. This principle and its applications allow us to endow the uncompactified moduli spaces of real maps from symmetric surfaces of all topological types with natural orientations and to verify that they extend across the codimension-one boundaries of these spaces, thus implementing a far-reaching proposal from C.-C. Liu's thesis.]]>2408788
Penka Georgieva, Aleksey Zinger
Bounds between Laplace and Steklov eigenvalues on nonnegatively curved manifolds
http://aimsciences.org//article/id/311033d7-ebd8-4105-a2cd-c58488eba350
2017-09-07Consider a compact Riemannian manifold with boundary. In this short note we prove that under certain positive curvature assumptions on the manifold and its boundary the Steklov eigenvalues of the manifold are controlled by the Laplace eigenvalues of the boundary. Additionally, in two dimensions we obtain an upper bound for Steklov eigenvalues in terms of topology of the surface without any curvature restrictions.]]>240100109
Mikhail Karpukhin
Central limit theorems in the geometry of numbers
http://aimsciences.org//article/id/d702d111-12b1-4b92-a108-4bb31c4276f9
2017-10-01We investigate in this paper the distribution of the discrepancy of various lattice counting functions. In particular, we prove that the number of lattice points contained in certain domains defined by products of linear forms satisfies a Central Limit Theorem. Furthermore, we show that the Central Limit Theorem holds for the number of rational approximants for weighted Diophantine approximation in $\mathbb{R}^d$. Our arguments exploit chaotic properties of the Cartan flow on the space of lattices.]]>240110122
Michael Björklund, Alexander Gorodnik