Related papers: A general splitting formula for the spectral flow

We propose a variational splitting technique for the generalized-$\alpha$ method to solve hyperbolic partial differential equations. We use tensor-product meshes to develop the splitting method, which has a computational cost that grows…

This Part establishes the geometric theory of uniformly hyperbolic sets with explicit quantitative bounds throughout, and contains five main theorems. The Stable Manifold Theorem is proved via the backward graph transform, with a complete…

The single-source unsplittable flow (SSUF) problem asks to send flow from a common source to different terminals with unrelated demands, each terminal being served through a single path. One of the most heavily studied SSUF objectives is to…

Let Y be a compact, oriented 3-manifold with a contact form a. For any Dirac operator D, we study the asymptotic behavior of the spectral flow between D and D+cl(-ira) as r very large. If a is the Thurston-Winkelnkemper contact form whose…

We consider a smooth submanifold $N$ with a smooth boundary in an ambient closed manifold $M$ and assign a spectral invariant $c(\alpha,H)$ to every singular homological class $\alpha\in H_*(N)$ and a Hamiltonian $H$ defined on the…

We present a systematic approach to deriving normal forms and related amplitude equations for flows and discrete dynamics on the center manifold of a dynamical system at local bifurcations and unfoldings of these. We derive a general,…

An odd deformation of a super Riemann surface $\mathcal S$ is a deformation of $\mathcal S$ by variables of odd parity. In this article we study the obstruction theory of these odd deformations $\mathcal X$ of $\mathcal S$. We view…

We apply pseudo-spectral methods to integrate functional flow equations with high accuracy, extending earlier work on functional fixed point equations \cite{Borchardt:2015rxa}. The advantages of our method are illustrated with the help of…

The purpose of this note is to describe a unified approach to the fundamental results in the spectral theory of boundary value problems, restricted to the case of Dirac type operators. Even though many facts are known and well presented in…

This paper introduces a novel approach for the construction of bulk--surface splitting schemes for semi-linear parabolic partial differential equations with dynamic boundary conditions. The proposed construction is based on a reformulation…

We study enhancement of diffusive mixing on a compact Riemannian manifold by a fast incompressible flow. Our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small…

The Hochschild and cyclic homology groups are computed for the algebra of `cusp' pseudodifferential operators on any compact manifold with boundary. The index functional for this algebra is interpreted as a Hochschild 1-cocycle and…

Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. We show that the spectrum of $A$ decomposes,…

Let M be a complete n-dimensional Riemannian spin manifold, partitioned by q two-sided hypersurfaces which have a compact transverse intersection N and which in addition satisfy a certain coarse transversality condition. Let E be a…

We show that the spectrum of a flow field can be extracted within a local region by straightforward filtering in physical space. We find that for a flow with a certain level of regularity, the filtering kernel must have a sufficient number…

Consider a manifold $M$ endowed locally with a pair of complementary distributions $\Delta^H \oplus \Delta^V=TM$ and let $\text{Diff}(\Delta^H, M)$ and $\text{Diff}(\Delta^V, M)$ be the corresponding Lie subgroups generated by vector fields…

We consider homoclinic solutions for Hamiltonian systems in symplectic Hilbert spaces and generalise spectral flow formulas that were proved by Pejsachowicz and the author in finite dimensions some years ago. Roughly speaking, our main…

We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that almost every such locally Hamiltonian flow with only simple saddles has singular…

We consider general (not necessarily Hamiltonian) first-order symmetric system $J y'-B(t)y=\D(t) f(t)$ on an interval $\cI=[a,b) $ with the regular endpoint $a$. A distribution matrix-valued function $\Si(s), \; s\in\bR,$ is called a…

In this survey article, we investigate the spectral properties of fractal differential operators on self-similar fractals. In particular, we discuss the decimation method, which introduces a renormalization map whose dynamics describes the…