Denjoy-Carleman differentiable perturbation of polynomials and unbounded operators

Let $t\mapsto A(t)$ for $t\in T$ be a $C^M$-mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here $C^M$ stands for C^\om (real analytic), a quasianalytic or non-quasianalytic Denjoy-Carleman class, $C^\infty$, or a H\"older continuity class C^{0,\al}. The parameter domain $T$ is either $\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector space. We prove and review results on $C^M$-dependence on $t$ of the eigenvalues and eigenvectors of $A(t)$.Comment: 8 page

Many parameter Hoelder perturbation of unbounded operators

If $u\mapsto A(u)$ is a $C^{0,\alpha}$-mapping, for $0< \alpha \le 1$, having as values unbounded self-adjoint operators with compact resolvents and common domain of definition, parametrized by $u$ in an (even infinite dimensional) space, then any continuous (in $u$) arrangement of the eigenvalues of $A(u)$ is indeed $C^{0,\alpha}$ in $u$.Comment: LaTeX, 4 pages; The result is generalized from Lipschitz to Hoelder. Title change

Sobolev Metrics on Diffeomorphism Groups and the Derived Geometry of Spaces of Submanifolds

Given a finite dimensional manifold $N$, the group $\operatorname{Diff}_{\mathcal S}(N)$ of diffeomorphism of $N$ which fall suitably rapidly to the identity, acts on the manifold $B(M,N)$ of submanifolds on $N$ of diffeomorphism type $M$ where $M$ is a compact manifold with $\dim M<\dim N$. For a right invariant weak Riemannian metric on $\operatorname{Diff}_{\mathcal S}(N)$ induced by a quite general operator $L:\frak X_{\mathcal S}(N)\to \Gamma(T^*N\otimes\operatorname{vol}(N))$, we consider the induced weak Riemannian metric on $B(M,N)$ and we compute its geodesics and sectional curvature. For that we derive a covariant formula for curvature in finite and infinite dimensions, we show how it makes O'Neill's formula very transparent, and we use it finally to compute sectional curvature on $B(M,N)$.Comment: 28 pages. In this version some misprints correcte

Lectures on mathematical aspects of (twisted) supersymmetric gauge theories

Supersymmetric gauge theories have played a central role in applications of quantum field theory to mathematics. Topologically twisted supersymmetric gauge theories often admit a rigorous mathematical description: for example, the Donaldson invariants of a 4-manifold can be interpreted as the correlation functions of a topologically twisted N=2 gauge theory. The aim of these lectures is to describe a mathematical formulation of partially-twisted supersymmetric gauge theories (in perturbation theory). These partially twisted theories are intermediate in complexity between the physical theory and the topologically twisted theories. Moreover, we will sketch how the operators of such a theory form a two complex dimensional analog of a vertex algebra. Finally, we will consider a deformation of the N=1 theory and discuss its relation to the Yangian, as explained in arXiv:1308.0370 and arXiv:1303.2632.Comment: Notes from a lecture series by the first author at the Les Houches Winter School on Mathematical Physics in 2012. To appear in the proceedings of this conference. Related to papers arXiv:1308.0370, arXiv:1303.2632, and arXiv:1111.423

Real-time depth sectioning: Isolating the effect of stress on structure development in pressure-driven flow

Transient structure development at a specific distance from the channel wall in a pressure-driven flow is obtained from a set of real-time measurements that integrate contributions throughout the thickness of a rectangular channel. This “depth sectioning method” retains the advantages of pressure-driven flow while revealing flow-induced structures as a function of stress. The method is illustrated by applying it to isothermal shear-induced crystallization of an isotactic polypropylene using both synchrotron x-ray scattering and optical retardance. Real-time, depth-resolved information about the development of oriented precursors reveals features that cannot be extracted from ex-situ observation of the final morphology and that are obscured in the depth-averaged in-situ measurements. For example, at 137 °C and at the highest shear stress examined (65 kPa), oriented thread-like nuclei formed rapidly, saturated within the first 7 s of flow, developed significant crystalline overgrowth during flow and did not relax after cessation of shear. At lower stresses, threads formed later and increased at a slower rate. The depth sectioning method can be applied to the flow-induced structure development in diverse complex fluids, including block copolymers, colloidal systems, and liquid-crystalline polymers

On Z-gradations of twisted loop Lie algebras of complex simple Lie algebras

We define the twisted loop Lie algebra of a finite dimensional Lie algebra $\mathfrak g$ as the Fr\'echet space of all twisted periodic smooth mappings from $\mathbb R$ to $\mathfrak g$. Here the Lie algebra operation is continuous. We call such Lie algebras Fr\'echet Lie algebras. We introduce the notion of an integrable $\mathbb Z$-gradation of a Fr\'echet Lie algebra, and find all inequivalent integrable $\mathbb Z$-gradations with finite dimensional grading subspaces of twisted loop Lie algebras of complex simple Lie algebras.Comment: 26 page

Molecular Simulation of Flow-Enhanced Nucleation in n-Eicosane Melts Under Steady Shear and Uniaxial Extension

Non-equilibrium molecular dynamics is used to study crystal nucleation of n-eicosane under planar shear and, for the first time, uniaxial extension. A method of analysis based on the mean first-passage time is applied to the simulation results in order to determine the effect of the applied flow field type and strain rate on the steady-state nucleation rate and a characteristic growth rate, as well as the effects on kinetic parameters associated with nucleation: the free energy barrier, critical nucleus size, and monomer attachment pre-factor. The onset of flow-enhanced nucleation (FEN) occurs at a smaller critical strain rate in extension as compared to shear. For strain rates larger than the critical rate, a rapid increase in the nucleation rate is accompanied by decreases in the free energy barrier and critical nucleus size, as well as an increase in chain extension. These observations accord with a mechanism in which FEN is caused by an increase in the driving force for crystallization due to flow-induced entropy reduction. At high applied strain rates, the free energy barrier, critical nucleus size, and degree of stretching saturate, while the monomer attachment pre-factor and degree of orientational order increase steadily. This trend is indicative of a significant diffusive contribution to the nucleation rate under intense flows that is correlated with the degree of global orientational order in a nucleating system. Both flow fields give similar results for all kinetic quantities with respect to the reduced strain rate, which we define as the ratio of the applied strain rate to the critical rate. The characteristic growth rate increases with increasing strain rate, and shows a correspondence with the nucleation rate that does not depend on the type of flow field applied. Additionally, a structural analysis of the crystalline clusters indicates that the flow field suppresses the compaction and crystalline ordering of clusters, leading to the formation of large articulated clusters under strong flow fields, and compact well-ordered clusters under weak flow fields

On Differential Structure for Projective Limits of Manifolds

We investigate the differential calculus defined by Ashtekar and Lewandowski on projective limits of manifolds by means of cylindrical smooth functions and compare it with the C^infty calculus proposed by Froehlicher and Kriegl in more general context. For products of connected manifolds, a Boman theorem is proved, showing the equivalence of the two calculi in this particular case. Several examples of projective limits of manifolds are discussed, arising in String Theory and in loop quantization of Gauge Theories.Comment: 38 pages, Latex 2e, to be published on J. Geom. Phys minor misprints corrected, reference adde

Shape analysis on homogeneous spaces: a generalised SRVT framework

Shape analysis is ubiquitous in problems of pattern and object recognition and has developed considerably in the last decade. The use of shapes is natural in applications where one wants to compare curves independently of their parametrisation. One computationally efficient approach to shape analysis is based on the Square Root Velocity Transform (SRVT). In this paper we propose a generalised SRVT framework for shapes on homogeneous manifolds. The method opens up for a variety of possibilities based on different choices of Lie group action and giving rise to different Riemannian metrics.Comment: 28 pages; 4 figures, 30 subfigures; notes for proceedings of the Abel Symposium 2016: "Computation and Combinatorics in Dynamics, Stochastics and Control". v3: amended the text to improve readability and clarify some points; updated and added some references; added pseudocode for the dynamic programming algorithm used. The main results remain unchange