Searcharxiv⌕ Search

arXiv subjects

Bernhelm Booss-Bavnbek

Publications and source records attributed to Bernhelm Booss-Bavnbek.

At least 19 recordsLinked to original sources
arXiv2020

Continuity of family of Calderón projections

We consider a continuous family of linear elliptic differential operators of arbitrary order over a smooth compact manifold with boundary. Assuming constant dimension of the spaces of inner solutions, we prove that the orthogonalized Calderón projections of the underlying family of elliptic operators form a continuous family of projections. Hence, its images (the Cauchy data spaces) form a continuous family of closed subspaces in the relevant Sobolev spaces. We use only elementary tools and classical results: basic manipulations of operator graphs and other closed subspaces in Banach spaces; elliptic regularity; Green's formula and trace theorems for Sobolev spaces; well-posed boundary conditions; duality of spaces and operators in Hilbert space; and the interpolation theorem for operators in Sobolev spaces. \keywords{Calder{ó}n projection\and Cauchy data spaces \and Elliptic differential operators \and Green's formula\and Interpolation theorem\and Manifolds with boundary\and Parameter dependence \and Trace theorem \and Variational properties

math.AP↗
arXiv2020

Dynamics and Control of Covid-19: Comments by Two Mathematicians

We are asking: why are the dynamics and control of Covid-19 most interesting for mathematicians and why are mathematicians urgently needed for controlling the pandemic? First we present our comments in a Bottom-up approach, i.e., following the events from their beginning as they evolved through time. They happened differently in different countries, and the main objective of the first part is to compare these evolutions in a few selected countries with each other. The second part of the article is not "country-oriented" but "problem-oriented". From a given problem we go Top-down to its solutions and their applications in concrete situations. We have organized this part by the mathematical methods that play a role in their solution. We give an overview of the main branches of mathematics that play a role and sketch the most frequent applications, emphasising mathematical pattern analysis in laboratory work and statistical-mathematical models in judging the quality of tests; demographic methods in the collection of data; different ways to model the evolution of the pandemic mathematically; and clinical epidemiology in attempts to develop a vaccine.

math.HO↗
arXiv2020

Multiplicity of time scales in climate, matter, life, and economy

This topic review communicates working experiences regarding interaction of a multiplicity of processes. Our experiences come from climate change modelling, materials science, cell physiology and public health, and macroeconomic modelling. We look at the astonishing advances of recent years in broad-band temporal frequency sampling, multiscale modelling and fast large-scale numerical simulation of complex systems, but also the continuing uncertainty of many science-based results. We describe and analyse properties that depend on the time scale of the measurement; structural instability; tipping points; thresholds; hysteresis; feedback mechanisms with runaways or stabilizations or delays. We point to grave disorientation in statistical sampling, the interpretation of observations and the design of control when neglecting the presence or emergence of multiple characteristic times. We explain what these working experiences can demonstrate for environmental research.

econ.GN↗
arXiv2018

The Maslov index in symplectic Banach spaces

We consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with continuously varying (weak) symplectic structures. Assuming vanishing index, we obtain intrinsically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions we define the curve's Maslov index by symplectic reduction to the classical finite-dimensional case. We prove the transitivity of repeated symplectic reductions and obtain the invariance of the Maslov index under symplectic reduction, while recovering all the standard properties of the Maslov index. As an application, we consider curves of elliptic operators which have varying principal symbol, varying maximal domain and are not necessarily of Dirac type. For this class of operator curves, we derive a desuspension spectral flow formula for varying well-posed boundary conditions on manifolds with boundary and obtain the splitting of the spectral flow on partitioned manifolds.

math.SG↗
arXiv2013

On the difficulties of acquiring mathematical experience: Case rural education

Based on a variety of philosophical approaches and my own work for decades in pure and applied mathematics teaching and research, I explain my view upon the basic difficulties of acquiring the Mathematical Experience (in the sense of P.J. Davis and R. Hersh 1981) and submit a list of claims how these difficulties can and should be confronted.

math.HO↗
arXiv2013

Unity and Disunity in Mathematics

The unity of mathematics has its power to compactify experiences in a form capable of being transferred and modified or adapted to new mathematical situations. Yet, we believe that the phrase "Unity of Mathematics" expresses a dream, an ideal that doesn't exist. We shall point to diachronic and cross cultural disunities, to semantic, semiotic and philosophic ambiguities and to the non-acceptance of certain mathematical texts by some practitioners of the subject.

math.HO↗
arXiv2013

The Maslov index in weak symplectic functional analysis

We recall the Chernoff-Marsden definition of weak symplectic structure and give a rigorous treatment of the functional analysis and geometry of weak symplectic Banach spaces. We define the Maslov index of a continuous path of Fredholm pairs of Lagrangian subspaces in continuously varying Banach spaces. We derive basic properties of this Maslov index and emphasize the new features appearing.

math.DG↗
arXiv2012

Unity in Major Themes - Convergence vs. Arbitrariness in the Development of Mathematics

We describe and explain the desire, common among mathematicians, both for unity and independence in its major themes. In the dialogue that follows, we express our spontaneous and considered judgment and reservations by contrasting the development of mathematics as a goal-driven process as opposed to one that often seems to possess considerable arbitrariness.

math.HO↗
arXiv2012

Towards a Nano Geometry? Geometry and Dynamics on Nano Scale

This paper applies I.M. Gelfand's distinction between adequate and non-adequate use of mathematical language in different contexts to the newly opened window of model-based measurements of intracellular dynamics. The specifics of geometry and dynamics on the mesoscale of cell physiology are elaborated - in contrast to the familiar Newtonian mechanics and the more recent, but by now also rather well established quantum field theories. Examples are given originating from the systems biology of insulin secreting pancreatic beta-cells and the mathematical challenges of an envisioned non-invasive control of magnetic nanoparticles.

math-ph↗
arXiv2011

Perturbation of Sectorial Projections of Elliptic Pseudo-differential Operators

Over a closed manifold, we consider the sectorial projection of an elliptic pseudo-differential operator A of positive order with two rays of minimal growth. We show that it depends continuously on A when the space of pseudo-differential operators is equipped with a certain topology which we explicitly describe. Our main application deals with a continuous curve of arbitrary first order linear elliptic differential operators over a compact manifold with boundary. Under the additional assumption of the weak inner unique continuation property, we derive the continuity of a related curve of Calderon projections and hence of the Cauchy data spaces of the original operator curve. In the Appendix, we describe a topological obstruction against a verbatim use of R. Seeley's original argument for the complex powers, which was seemingly overlooked in previous studies of the sectorial projection.

math.SP↗
arXiv2010

Geometric and Electromagnetic Aspects of Fusion Pore Making

For regulated exocytosis, we model the morphology and dynamics of the making of the fusion pore or porosome as a cup-shaped lipoprotein structure (a dimple or pit) on the cytosol side of the plasma membrane. We describe the forming of the dimple by a free boundary problem. We discuss the various forces acting and analyze the magnetic character of the wandering electromagnetic field wave produced by intracellular spatially distributed pulsating (and well observed) release and binding of calcium ions anteceding the bilayer membrane vesicle fusion of exocytosis. Our approach explains the energy efficiency of the observed dimple forming prior to hemifusion and fusion pore, and the observed flickering in secretion. It provides a frame to relate characteristic time length of exocytosis to the frequency, amplitude and direction of propagation of the underlying electromagnetic field wave.

math.AP↗
arXiv2009

Steps towards "Quantum Gravity" and the practice of science: will the merger of mathematics and physics work?

The author recalls general tendencies of the "mathematization" of the sciences and derives challenges and tentative obstructions for a successful merger of mathematics and physics on fancied steps towards "Quantum Gravity". This is an edited version of the author's opening words to an international workshop "Quantum Gravity: An Assessment", Denmark, May 17-18, 2008. It followed immediately after the Quantum Gravity Summer School 2008, see http://QuantumGravity.ruc.dk/

math-ph↗
arXiv2009

The Calderon Projection: New Definition and Applications

We consider an arbitrary linear elliptic first--order differential operator A with smooth coefficients acting between sections of complex vector bundles E,F over a compact smooth manifold M with smooth boundary N. We describe the analytic and topological properties of A in a collar neighborhood U of N and analyze various ways of writing A|U in product form. We discuss the sectorial projections of the corresponding tangential operator, construct various invertible doubles of A by suitable local boundary conditions, obtain Poisson type operators with different mapping properties, and provide a canonical construction of the Calderon projection. We apply our construction to generalize the Cobordism Theorem and to determine sufficient conditions for continuous variation of the Calderon projection and of well--posed selfadjoint Fredholm extensions under continuous variation of the data.

math.DG↗
arXiv2009

The mathematization of the individual sciences - revisited

We recall major findings of a systematic investigation of the mathematization of the individual sciences, conducted by the author in Bielefeld some 35 years ago under the direction of Klaus Krickeberg, and confront them with recent developments in physics, medicine, economics, and spectral geometry.

math.HO↗
arXiv2009

The invertible double of elliptic operators

First, we review the Dirac operator folklore about basic analytic and geometrical properties of operators of Dirac type on compact manifolds with smooth boundary and on closed partitioned manifolds and show how these properties depend on the construction of a canonical invertible double and are related to the concept of the Calderon projection. Then we summarize a recent construction of a canonical invertible double for general first order elliptic differential operators over smooth compact manifolds with boundary. We derive a natural formula for the Calderon projection which yields a generalization of the famous Cobordism Theorem. We provide a list of assumptions to obtain a continuous variation of the Calderon projection under smooth variation of the coefficients. That yields various new spectral flow theorems. Finally, we sketch a research program for confining, respectively closing, the last re- maining gaps between the geometric Dirac operator type situation and the general linear elliptic case.

math.DG↗