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Gustafsson, B. & Putinar, M. (2019). A field theoretic operator model and Cowen-Douglas class. Banach Journal of Mathematical Analysis, 13(2), 338-358
Open this publication in new window or tab >>A field theoretic operator model and Cowen-Douglas class
2019 (English)In: Banach Journal of Mathematical Analysis, ISSN 1735-8787, Vol. 13, no 2, p. 338-358Article in journal (Refereed) Published
Abstract [en]

In resonance with a recent geometric framework proposed by Douglas and Yang, we develop a functional model for certain linear bounded operators with rank-one self-commutator acting on a Hilbert space. By taking advantage of the refined existing theory of the principal function of a hyponormal operator, we transfer the whole action outside the spectrum, on the resolvent of the underlying operator, localized at a distinguished vector. The whole construction turns out to rely on an elementary algebra body involving analytic multipliers and Cauchy transforms. We propose a natural field theory interpretation of the resulting resolvent functional model.

Place, publisher, year, edition, pages
Duke University Press, 2019
Keywords
hyponormal operator, exponential transform, Cauchy transform, ideal fluid flow
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-249866 (URN)10.1215/17358787-2018-0041 (DOI)000462021700005 ()
Note

QC 20190426

Available from: 2019-04-26 Created: 2019-04-26 Last updated: 2019-04-26Bibliographically approved
Gustafsson, B. (2019). Vortex motion and geometric function theory: the role of connections. Philosophical Transactions. Series A: Mathematical, physical, and engineering science, 377(2158), Article ID 20180341.
Open this publication in new window or tab >>Vortex motion and geometric function theory: the role of connections
2019 (English)In: Philosophical Transactions. Series A: Mathematical, physical, and engineering science, ISSN 1364-503X, E-ISSN 1471-2962, Vol. 377, no 2158, article id 20180341Article in journal (Refereed) Published
Abstract [en]

We formulate the equations for point vortex dynamics on a closed two-dimensional Riemannian manifold in the language of affine and other kinds of connections. This can be viewed as a relaxation of standard approaches, using the Riemannian metric directly, to an approach based more on local coordinates provided with a minimal amount of extra structure. The speed of a vortex is then expressed in terms of the difference between an affine connection derived from the coordinate Robin function and the Levi-Civita connection associated with the Riemannian metric. A Hamiltonian formulation of the same dynamics is also given. The relevant Hamiltonian function consists of two main terms. One of the terms is the well-known quadratic form based on a matrix whose entries are Green and Robin functions, while the other term describes the energy contribution from those circulating flows which are not implicit in the Green functions. One main issue of the paper is a detailed analysis of the somewhat intricate exchanges of energy between these two terms of the Hamiltonian. This analysis confirms the mentioned dynamical equations formulated in terms of connections. This article is part of the theme issue 'Topological and geometrical aspects ofmass and vortex dynamics'.

Place, publisher, year, edition, pages
ROYAL SOC, 2019
Keywords
point vortex motion, affine connection, projective connection, Robin function, renormalization, Hamiltonian
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-262761 (URN)10.1098/rsta.2018.0341 (DOI)000488279300001 ()2-s2.0-85073260708 (Scopus ID)
Note

QC 20191022

Available from: 2019-10-22 Created: 2019-10-22 Last updated: 2019-10-22Bibliographically approved
Gustafsson, B. & Putinar, M. (2018). Line bundles defined by the Schwarz function. Analysis and Mathematical Physics, 8(2), 171-183
Open this publication in new window or tab >>Line bundles defined by the Schwarz function
2018 (English)In: Analysis and Mathematical Physics, ISSN 1664-2368, E-ISSN 1664-235X, Vol. 8, no 2, p. 171-183Article in journal (Refereed) Published
Abstract [en]

Cauchy and exponential transforms are characterized, and constructed, as canonical holomorphic sections of certain line bundles on the Riemann sphere defined in terms of the Schwarz function. A well known natural connection between Schwarz reflection and line bundles defined on the Schottky double of a planar domain is briefly discussed in the same context.

Place, publisher, year, edition, pages
SPRINGER BASEL AG, 2018
Keywords
Line bundle, Schwarz function, Cauchy transform, Exponential transform, Schottky double
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-232268 (URN)10.1007/s13324-017-0201-9 (DOI)000436304600002 ()
Note

QC 20180719

Available from: 2018-07-19 Created: 2018-07-19 Last updated: 2018-07-19Bibliographically approved
Gustafsson, B. (2018). The string equation for polynomials. Analysis and Mathematical Physics, 8(4), 637-653
Open this publication in new window or tab >>The string equation for polynomials
2018 (English)In: Analysis and Mathematical Physics, ISSN 1664-2368, E-ISSN 1664-235X, Vol. 8, no 4, p. 637-653Article in journal (Refereed) Published
Abstract [en]

For conformal maps defined in the unit disk one can define a certain Poisson bracket that involves the harmonic moments of the image domain. When this bracket is applied to the conformal map itself together with its conformally reflected map the result is identically one. This is called the string equation, and it is closely connected to the governing equation, the Polubarinova–Galin equation, for the evolution of a Hele-Shaw blob of a viscous fluid (or, by another name, Laplacian growth). In the present paper we show that the string equation makes sense and holds for general polynomials.

Place, publisher, year, edition, pages
Springer Basel, 2018
Keywords
harmonic moment, Hele-Shaw flow, Laplacian growth, Poisson bracket, Polubarinova–Galin equation, resultant, String equation
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-247089 (URN)10.1007/s13324-018-0239-3 (DOI)000451394300010 ()2-s2.0-85057449868 (Scopus ID)
Note

QC 20190404

Available from: 2019-04-04 Created: 2019-04-04 Last updated: 2019-04-04Bibliographically approved
Gustafsson, B., Teodorescu, R. & Vasiliev, A. (2014). Classical and Stochastic Laplacian Growth (1ed.). Cham: Birkhäuser Verlag
Open this publication in new window or tab >>Classical and Stochastic Laplacian Growth
2014 (English)Book (Refereed)
Place, publisher, year, edition, pages
Cham: Birkhäuser Verlag, 2014. p. 317 Edition: 1
Series
Advances in Mathematical Fluid Mechanics, ISSN 2297-0320
Keywords
Hele-Shaw flow
National Category
Natural Sciences
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-164027 (URN)10.1007/978-3-319-08287-5 (DOI)978-3-319-08286-8 (ISBN)
Note

QC 20150414

Available from: 2015-04-13 Created: 2015-04-13 Last updated: 2015-04-14Bibliographically approved
Gustafsson, B. & Lin, Y.-L. (2013). On the dynamics of roots and poles for solutions of the polubarinova-galin equation. Annales Academiae Scientiarum Fennicae Mathematica, 38(1), 259-286
Open this publication in new window or tab >>On the dynamics of roots and poles for solutions of the polubarinova-galin equation
2013 (English)In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, no 1, p. 259-286Article in journal (Refereed) Published
Abstract [en]

We study the dynamics of roots of f'(zeta, t), where F(zeta, t) is a locally univalent polynomial solution of the Polubarinova-Galin equation for the evolution of the conformal map onto a Hele-Shaw blob subject to injection at one point. We give examples of the sometimes complicated motion of roots, but show also that the asymptotic behavior is simple. More generally we allow f'(zeta, t) to be a rational function and give sharp estimates for the motion of poles and for the decay of the Taylor coefficients. We also prove that any global in time locally univalent solution actually has to be univalent.

Keywords
Hele-Shaw flow, Laplacian growth, Polubarinova-Galin equation, Lowner-Kufarev equation, root dynamics, pole dynamics
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-121142 (URN)10.5186/aasfm.2013.3802 (DOI)000316239200014 ()2-s2.0-84877662477 (Scopus ID)
Note

QC 20130419

Available from: 2013-04-19 Created: 2013-04-19 Last updated: 2017-12-06Bibliographically approved
Gustafsson, B. & Sebbar, A. (2012). Critical Points of Green's Function and Geometric Function Theory. Indiana University Mathematics Journal, 61(3), 939-1017
Open this publication in new window or tab >>Critical Points of Green's Function and Geometric Function Theory
2012 (English)In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 61, no 3, p. 939-1017Article in journal (Refereed) Published
Abstract [en]

We study questions related to critical points of the Green's function of a bounded multiply connected domain in the complex plane. The motion of critical points, their limiting positions as the pole approaches the boundary and the differential geometry of the level lines of the Green's function are main themes in the paper. A unifying role is played by various affine and projective connections and corresponding Mobius invariant differential operators. In the doubly connected case the three Eisenstein series E-2, E-4, E-6 are used. A specific result is that a doubly connected domain is the disjoint union of the set of critical points of the Green's function, the set of zeros of the Bergman kernel and the separating boundary limit positions for these.

Keywords
critical point, Green's function, Neumann function, Bergman kernel, Schiffer kernel, Schottky-Klein prime form, Schottky double, weighted Bergman space, Poincare metric, Martin boundary, projective structure, projective connection, affine connection, Eisenstein series
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-125796 (URN)10.1512/iumj.2012.61.4621 (DOI)000321231000003 ()2-s2.0-84880872132 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20130814

Available from: 2013-08-14 Created: 2013-08-13 Last updated: 2017-12-06Bibliographically approved
Gustafsson, B. & Sakai, M. (2012). On the curvature of some free boundaries in higher dimensions. Analysis and Mathematical Physics, 2, 247-275
Open this publication in new window or tab >>On the curvature of some free boundaries in higher dimensions
2012 (English)In: Analysis and Mathematical Physics, ISSN 1664-2368, E-ISSN 1664-235X, Vol. 2, p. 247-275Article in journal (Refereed) Published
Abstract [en]

It is known that any subharmonic quadrature domain in two dimensions satisfies a natural inner ball condition, in other words there is a specific upper bound on the curvature of the boundary. This result directly applies to free boundaries appearing in obstacle type problems and in Hele-Shaw flow. In the present paper we make partial progress on the corresponding question in higher dimensions. Specifically, we prove the equivalence between several different ways to formulate the inner ball condition, and we compute the Brouwer degree for a geometrically important mapping related to the Schwarz potential of the boundary. The latter gives in particular a new proof in the two dimensional case.

Place, publisher, year, edition, pages
Springer, 2012
Keywords
Quadrature domain, inner ball condition, Schwarz potential, Brouwer degree
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-139040 (URN)10.1007/s13324-012-0032-7 (DOI)000209055100003 ()
Note

QC 20140219

Available from: 2013-12-29 Created: 2013-12-29 Last updated: 2017-12-06Bibliographically approved
Gustafsson, B. & Tkachev, V. (2011). On the exponential transform of lemniscates. In: P. Bränden, M. Passare, M. Putinar (Ed.), Notions of Positivity and the Geometry of Polynomials: (pp. 239-257). Basel: Springer
Open this publication in new window or tab >>On the exponential transform of lemniscates
2011 (English)In: Notions of Positivity and the Geometry of Polynomials / [ed] P. Bränden, M. Passare, M. Putinar, Basel: Springer, 2011, p. 239-257Chapter in book (Refereed)
Abstract [en]

It is known that the exponential transform of a quadrature domain is a rational function for which the denominator has a certain separable form. In the present paper we show that the exponential transform of lemniscate domains in general are not rational functions, of any form. Several examples are given to illustrate the general picture. The main tool used is that of polynomial and meromorphic resultants.

Place, publisher, year, edition, pages
Basel: Springer, 2011
Series
Trends in Mathematics
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-139038 (URN)10.1007/978-3-0348-0142-3_13 (DOI)978-3-0348-0141-6 (ISBN)
Note

QC 20140113

Available from: 2013-12-29 Created: 2013-12-29 Last updated: 2014-04-11Bibliographically approved
Gustafsson, B. & Tkachev, V. (2011). On the exponential transform of multi-sheeted algebraic domains. Computational methods in Function Theory, 11(2), 591-615
Open this publication in new window or tab >>On the exponential transform of multi-sheeted algebraic domains
2011 (English)In: Computational methods in Function Theory, ISSN 1617-9447, E-ISSN 2195-3724, Vol. 11, no 2, p. 591-615Article in journal (Refereed) Published
Abstract [en]

We introduce multi-sheeted versions of algebraic domains and quadrature domains, allowing them to be branched covering surfaces over the Riemann sphere. The two classes of domains turn out to be the same, and the main result states that the extended exponential transform of such a domain agrees, apart from some simple factors, with the extended elimination function for a generating pair of functions. In an example we discuss the algebraic curves associated to level curves of the Neumann oval, and determine which of these give rise to multi-sheeted algebraic domains.

Keywords
algebraic domain, quadrature domain, exponential transform, elimination function, Riemann surface, Klein surface, Neumann’s oval
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-78806 (URN)000311620300010 ()2-s2.0-84871178114 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20120412

Available from: 2012-02-08 Created: 2012-02-08 Last updated: 2017-12-08Bibliographically approved
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-3125-3030

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