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• 1.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. Jagiellonian University, Faculty of Mathematics and Computer ScienceInstitute of Mathematics, Institute of Mathematics..
Attractors of compactly generated semigroups of regular polynomial mappings.Manuscript (preprint) (Other academic)
• 2.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Mathematics..
Bergman Functions and Nearly Orthonormal PolynomialsManuscript (preprint) (Other academic)
• 3.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Om oändliga tal2015Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 4.
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
The Symmetric Meixner-Pollaczek polynomials2003Doctoral thesis, comprehensive summary (Other academic)

The Symmetric Meixner-Pollaczek polynomials are considered. We denote these polynomials in this thesis by pn(λ)(x) instead of the standard notation pn(λ) (x/2, π/2), where λ > 0. The limiting case of these sequences of polynomials pn(0) (x) =limλ→0 pn(λ)(x), is obtained, and is shown to be an orthogonal sequence in the strip, S = {z ∈ ℂ : −1≤ℭ (z)≤1}.

From the point of view of Umbral Calculus, this sequence has a special property that makes it unique in the Symmetric Meixner-Pollaczek class of polynomials: it is of convolution type. A convolution type sequence of polynomials has a unique associated operator called a delta operator. Such an operator is found for pn(0) (x), and its integral representation is developed. A convolution type sequence of polynomials may have associated Sheffer sequences of polynomials. The set of associated Sheffer sequences of the sequence pn(0)(x) is obtained, and is found

to be ℙ = {{pn(λ) (x)} =0 : λ ∈ R}. The major properties of these sequences of polynomials are studied.

The polynomials {pn(λ) (x)}n=0, λ < 0, are not orthogonal polynomials on the real line with respect to any positive real measure for failing to satisfy Favard’s three term recurrence relation condition. For every λ ≤ 0, an associated nonstandard inner product is defined with respect to which pn(λ)(x) is orthogonal.

Finally, the connection and linearization problems for the Symmetric Meixner-Pollaczek polynomials are solved. In solving the connection problem the convolution property of the polynomials is exploited, which in turn helps to solve the general linearization problem.

1. The Meixner-Pollaczek polynomials and and a system of orthogonal polynomials in a strip
Open this publication in new window or tab >>The Meixner-Pollaczek polynomials and and a system of orthogonal polynomials in a strip
In: Journal of computational and applied MathematicsArticle in journal (Refereed) Submitted
##### Identifiers
urn:nbn:se:uu:diva-90614 (URN)
Available from: 2003-05-16 Created: 2003-05-16Bibliographically approved
2. Umbral calculus and the Symmetric Meixner-Pollaczek polynomials
Open this publication in new window or tab >>Umbral calculus and the Symmetric Meixner-Pollaczek polynomials
Manuscript (Other academic)
##### Identifiers
urn:nbn:se:uu:diva-90615 (URN)
Available from: 2003-05-16 Created: 2003-05-16 Last updated: 2010-01-13Bibliographically approved
3. The symmetric Meixner-Pollaczek polynomials with real parameter
Open this publication in new window or tab >>The symmetric Meixner-Pollaczek polynomials with real parameter
Manuscript (Other academic)
##### Identifiers
urn:nbn:se:uu:diva-90616 (URN)
Available from: 2003-05-16 Created: 2003-05-16 Last updated: 2010-01-13Bibliographically approved
4. Linearization and connection problems for the Symmetric Meixner-Pollaczek polynomials
Open this publication in new window or tab >>Linearization and connection problems for the Symmetric Meixner-Pollaczek polynomials
Manuscript (Other academic)
##### Identifiers
urn:nbn:se:uu:diva-90617 (URN)
Available from: 2003-05-16 Created: 2003-05-16 Last updated: 2010-01-13Bibliographically approved
• 5.
Univ Hamburg, Inst Theoret Phys, Luruper Chaussee 149, D-22761 Hamburg, Germany.;Univ Hamburg, Zentrum Math Phys, Bundesstr 55, D-20146 Hamburg, Germany.;Steklov Math Inst, Moscow, Russia..
Univ Hamburg, Inst Theoret Phys, Luruper Chaussee 149, D-22761 Hamburg, Germany.;Univ Hamburg, Zentrum Math Phys, Bundesstr 55, D-20146 Hamburg, Germany.. Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy. KTH Royal Inst Technol, NORDITA, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden.;Stockholm Univ, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden..
Superintegrability of geodesic motion on the sausage model2017In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 50, no 24, article id 244002Article in journal (Refereed)

Reduction of the η-deformed sigma model on AdS5 x S5 to the two-dimensional squashed sphere (S2)η can be viewed as a special case of the Fateev sausage model where the coupling constant v is imaginary. We show that geodesic motion in this model is described by a certain superintegrable mechanical system with four-dimensional phase space. This is done by means of explicitly constructing three integrals of motion which satisfy the sl(2) Poisson algebra relations, albeit being non-polynomial in momenta. Further, we find a canonical transformation which transforms the Hamiltonian of this mechanical system to the one describing the geodesic motion on the usual two-sphere. By inverting this transformation we map geodesics on this auxiliary two-sphere back to the sausage model.

• 6.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Pathological functions and the Baire category theorem2017Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 7.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Information Technology, Division of Computing Science.
A practical comparison between algebraic and statistical attacks on the lightweight cipher SIMON2016Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis

In the summer of 2013 NSA released a new family of lightweight block ciphers called SIMON. However they did not publish any assessment of the security of SIMON. Since then only a few papers on this topic have been released and none of them have included an algebraic analysis. Moreover only one paper described a practical implementation of the attack. This master thesis aims to implement a practical attack, both algebraic and differential, on SIMON. In doing so we are able to make a comparison between the two different attack methods. The algebraic attack was executed with SAT-solver CryptoMiniSat2 and could break 7 rounds. The differential attack was implemented in three steps. First we created a difference distribution table (DDT) and then we identified a differential by a search algorithm for the DDT. In the last step we designed a key recovery attack to recover the last round key. The attack could break 13 rounds for a 9 round differential. With a simple expansion on the key recovery attack it has the potential to break even more rounds for the same 9 round differential. This indicate that algebraic cryptanalysis might not be such a strong tool since it could only break 7 rounds. Furthermore, if a generic algebraic attack does not work on SIMON it has little or no chance of being successful on a more complex cipher. In other words this algebraic attack may serve as a benchmark for the efficiency of generic algebraic attacks.

• 8.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Boundary Behavior of p-Laplace Type Equations2013Doctoral thesis, comprehensive summary (Other academic)

This thesis consists of six scientific papers, an introduction and a summary. All six papers concern the boundary behavior of non-negative solutions to partial differential equations.

Paper I concerns solutions to certain p-Laplace type operators with variable coefficients. Suppose that u is a non-negative solution that vanishes on a part Γ of an Ahlfors regular NTA-domain. We prove among other things that the gradient Du of u has non-tangential limits almost everywhere on the boundary piece Γ, and that log|Du| is a BMO function on the boundary.  Furthermore, for Ahlfors regular NTA-domains that are uniformly (N,δ,r0)-approximable by Lipschitz graph domains we prove a boundary Harnack inequality provided that δ is small enough.

Paper II concerns solutions to a p-Laplace type operator with lower order terms in δ-Reifenberg flat domains. We prove that the ratio of two non-negative solutions vanishing on a part of the boundary is Hölder continuous provided that δ is small enough. Furthermore we solve the Martin boundary problem provided δ is small enough.

In Paper III we prove that the boundary type Riesz measure associated to an A-capacitary function in a Reifenberg flat domain with vanishing constant is asymptotically optimal doubling.

Paper IV concerns the boundary behavior of solutions to certain parabolic equations of p-Laplace type in Lipschitz cylinders. Among other things, we prove an intrinsic Carleson type estimate for the degenerate case and a weak intrinsic Carleson type estimate in the singular supercritical case.

In Paper V we are concerned with equations of p-Laplace type structured on Hörmander vector fields. We prove that the boundary type Riesz measure associated to a non-negative solution that vanishes on a part Γ of an X-NTA-domain, is doubling on Γ.

Paper VI concerns a one-phase free boundary problem for linear elliptic equations of non-divergence type. Assume that we know that the positivity set is an NTA-domain and that the free boundary is a graph. Furthermore assume that our solution is monotone in the graph direction and that the coefficients of the equation are constant in the graph direction. We prove that the graph giving the free boundary is Lipschitz continuous.

1. Estimates for Solutions to Equations of p-Laplace type in Ahlfors regular NTA-domains
Open this publication in new window or tab >>Estimates for Solutions to Equations of p-Laplace type in Ahlfors regular NTA-domains
2014 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 266, no 9, p. 5955-6005Article in journal (Refereed) Published
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-163517 (URN)10.1016/j.jfa.2014.02.027 (DOI)000334652000018 ()
Available from: 2011-12-13 Created: 2011-12-12 Last updated: 2017-12-08Bibliographically approved
2. Boundary estimates for solutions to operators of $p$-Laplace type with lower order terms
Open this publication in new window or tab >>Boundary estimates for solutions to operators of $p$-Laplace type with lower order terms
2011 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 250, no 1, p. 264-291Article in journal (Refereed) Published
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-163370 (URN)10.1016/j.jde.2010.09.011 (DOI)
Available from: 2011-12-12 Created: 2011-12-12 Last updated: 2017-12-08Bibliographically approved
3. Optimal doubling, Reifenberg flatness and operators of p-Laplace type
Open this publication in new window or tab >>Optimal doubling, Reifenberg flatness and operators of p-Laplace type
2011 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, no 17, p. 5943-5955Article in journal (Refereed) Published
##### Abstract [en]

In this paper we consider operators of p-Laplace type of the form ∇·A(x,∇u) = 0. ConcerningA we assume, for p ∈ (1,∞) fixed, an appropriate ellipticity type condition, H¨older continuityin x and that A(x, ) = ||p−1A(x, /||) whenever x ∈ Rn and ∈ Rn \ {0}. Let  ⊂ Rn be abounded domain, let D be a compact subset of . We say that ˆu = ˆup,D, is the A-capacitaryfunction for D in  if ˆu ≡ 1 on D, ˆu ≡ 0 on @ in the sense of W1,p0 () and ∇·A(x,∇ˆu) = 0 in \D in the weak sense. We extend ˆu to Rn \  by putting ˆu ≡ 0 on Rn \ . Then there existsa unique finite positive Borel measure ˆμ on Rn, with support in @, such thatZ hA(x,∇ˆu),∇i dx = −Z dˆμ whenever ∈ C∞0 (Rn \ D).In this paper we prove that if  is Reifenberg flat with vanishing constant, thenlimr→0infw∈∂ˆμ(B(w, r))ˆμ(B(w, r))= limr→0supw∈∂ˆμ(B(w, r))ˆμ(B(w, r))= n−1,for every , 0 < ≤ 1. In particular, we prove that ˆμ is an asymptotically optimal doublingmeasure on @.

Mathematics
##### Identifiers
urn:nbn:se:uu:diva-163435 (URN)10.1016/j.na.2011.05.061 (DOI)
Available from: 2012-11-30 Created: 2011-12-12 Last updated: 2017-12-08Bibliographically approved
4. Boundary Estimates for Certain Degenerate and Singular Parabolic Equations
Open this publication in new window or tab >>Boundary Estimates for Certain Degenerate and Singular Parabolic Equations
2016 (English)In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 18, no 2, p. 381-424Article in journal (Refereed) Published
##### Abstract [en]

We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic p-Laplace equation. Assuming that such solutions continuously vanish on some distinguished part of the lateral part S-T of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations of porous medium type.

##### Keywords
Degenerate and singular parabolic equations; Harnack estimates; boundary Harnack inequality; Carleson estimate
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:uu:diva-186267 (URN)10.4171/JEMS/593 (DOI)000370249100005 ()
Available from: 2013-02-12 Created: 2012-11-28 Last updated: 2017-12-07Bibliographically approved
5. Wolff-Potential Estimates and Doubling of Subelliptic p-harmonic measures
Open this publication in new window or tab >>Wolff-Potential Estimates and Doubling of Subelliptic p-harmonic measures
2013 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 85, p. 149-159Article in journal (Refereed) Published
##### Abstract [en]

Let be a system of C vector fields in Rn satisfying Hörmander’s finite rank condition and let Ω be a non-tangentially accessible domain with respect to the Carnot–Carathéodory distance d induced by X. We prove the doubling property of certain boundary measures associated to non-negative solutions, which vanish on a portion of Ω, to the equation

Given p, 1<p<, fixed, we impose conditions on the function A=(A1,…,Am):Rn×RmRm, which imply that the equation is a quasi-linear partial differential equation of p-Laplace type structured on vector fields satisfying the classical Hörmander condition. In the case p=2 and for linear equations, our result coincides with the doubling property of associated elliptic measures. To prove our result we establish, and this is of independent interest, a Wolff potential estimate for subelliptic equations of p-Laplace type.

##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:uu:diva-186268 (URN)10.1016/j.na.2013.02.023 (DOI)000318378700013 ()
Available from: 2013-03-26 Created: 2012-11-28 Last updated: 2017-12-07Bibliographically approved
6. On a one-phase free boundary problem
Open this publication in new window or tab >>On a one-phase free boundary problem
2013 (English)In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, no 1, p. 181-191Article in journal (Other academic) Published
##### Abstract [en]

In this paper we extend a result regarding the free boundary regularity in a one-phaseproblem, by De Silva and Jerison [DJ], to non-divergence linear equations of second order.Roughly speaking we prove that the free boundary is given by a Lipschitz graph.

##### Keywords
One-phase, free boundary, NTA, non-divergence, linear
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-186265 (URN)10.5186/aasfm.2013.3815 (DOI)000316239200009 ()
Available from: 2012-11-30 Created: 2012-11-28 Last updated: 2017-12-07Bibliographically approved
• 9.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
On a one-phase free boundary problem2013In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, no 1, p. 181-191Article in journal (Other academic)

In this paper we extend a result regarding the free boundary regularity in a one-phaseproblem, by De Silva and Jerison [DJ], to non-divergence linear equations of second order.Roughly speaking we prove that the free boundary is given by a Lipschitz graph.

• 10.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Dipartimento di Matematica "F. Casorati", Università di Pavia. Dipartimento di Matematica "F. Brioschi", Politecnico di Milano.
Boundary Estimates for Certain Degenerate and Singular Parabolic Equations2016In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 18, no 2, p. 381-424Article in journal (Refereed)

We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic p-Laplace equation. Assuming that such solutions continuously vanish on some distinguished part of the lateral part S-T of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations of porous medium type.

• 11.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
A note on the hyperconvexity of pseudoconvex domains beyond Lipschitz regularity2015In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, no 3, p. 531-545Article in journal (Refereed)

We show that bounded pseudoconvex domains that are Hölder continuous for all α < 1 are hyperconvex, extending the well-known result by Demailly (Math. Z. 184 1987) beyond Lipschitz regularity.

• 12.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Department of mathematics and mathematical statistics, Umeå University. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Approximation and Bounded Plurisubharmonic Exhaustion Functions Beyond Lipschitz DomainsManuscript (preprint) (Other academic)

Using techniques from the analysis of PDEs to studythe boundary behaviour of functions on domains with low boundaryregularity, we extend results by Fornaæss-Wiegerinck (1989)on plurisubharmonic approximation and by Demailly (1987) onthe existence on bounded plurisubharmonic exhaustion functionsto domains beyond Lipschitz boundary regularity.

• 13.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Univ Jyvaskyla, Dept Math & Stat, Jyvaskyla 40014, Finland.
Umeå University. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Approximation of plurisubharmonic functions2016In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 61, no 1, p. 23-28Article in journal (Refereed)

We extend a result by Fornaaess and Wiegerinck [Ark. Mat. 1989;27:257-272] on plurisubharmonic Mergelyan type approximation to domains with boundaries locally given by graphs of continuous functions.

• 14.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Aalto University, Institute of Mathematics, P.O. Box 11100, FI-00076 Aalto, Finland.
Univ Jyvaskyla, Dept Math & Stat, POB 35, Jyvaskyla 40014, Finland..
A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term2017In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 272, no 8, p. 3176-3215Article in journal (Refereed)

This paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term. We elaborate on the non-homogeneous generalized Harnack inequality proved by the second author in [26], to prove a generalized Carleson estimate. We also prove boundary Holder continuity and a boundary Harnack type inequality.

• 15.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Aalto Univ, Finland.
A comparison principle for the porous medium equation and its consequences2017In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616, Vol. 33, no 2, p. 573-594Article in journal (Refereed)

We prove a comparison principle for the porous medium equation in more general open sets in Rn+1 than space-time cylinders. We apply this result in two related contexts: we establish a connection between a potential theoretic notion of the obstacle problem and a notion based on a variational inequality. We also prove the basic properties of the PME capacity, in particular that there exists a capacitary extremal which gives the capacity for compact sets.

• 16.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Wolff-Potential Estimates and Doubling of Subelliptic p-harmonic measures2013In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 85, p. 149-159Article in journal (Refereed)

Let be a system of C vector fields in Rn satisfying Hörmander’s finite rank condition and let Ω be a non-tangentially accessible domain with respect to the Carnot–Carathéodory distance d induced by X. We prove the doubling property of certain boundary measures associated to non-negative solutions, which vanish on a portion of Ω, to the equation

Given p, 1<p<, fixed, we impose conditions on the function A=(A1,…,Am):Rn×RmRm, which imply that the equation is a quasi-linear partial differential equation of p-Laplace type structured on vector fields satisfying the classical Hörmander condition. In the case p=2 and for linear equations, our result coincides with the doubling property of associated elliptic measures. To prove our result we establish, and this is of independent interest, a Wolff potential estimate for subelliptic equations of p-Laplace type.

• 17.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Aalto Univ, Dept Math & Syst Anal, Sch Sci, Aalto 00076, Finland..
Aalto Univ, Dept Math & Syst Anal, Sch Sci, Aalto 00076, Finland..
Characterizations of interior polar sets for the degenerate p-parabolic equation2017In: Journal of evolution equations (Printed ed.), ISSN 1424-3199, E-ISSN 1424-3202, Vol. 17, no 2, p. 827-848Article in journal (Refereed)

This paper deals with different characterizations of sets of nonlinear parabolic capacity zero, with respect to the parabolic p-Laplace equation. Specifically we prove that certain interior polar sets can be characterized by sets of zero nonlinear parabolic capacity. Furthermore we prove that zero capacity sets are removable for bounded supersolutions and that sets of zero capacity have a relation to a certain parabolic Hausdorff measure.

• 18.
University of Washington, Seattle, USA.
University of Missouri, Columbia, USA. Instituto de Ciencias Matematicas, Madrid, Spain. Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. University of Washington, Seattle, USA.
A new characterization of chord-arc domains2015In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863Article in journal (Refereed)

We show that if $\om \subset \ree$, $n\geq 1$, is a uniform domain (aka 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of $\om$ implies the existence of exterior Corkscrew points at all scales, so that in fact, $\om$ is a chord-arc domain,i.e., a domain with an Ahlfors-David regular boundary which satisfies both interior and exterior Corkscrew conditions, and an interior Harnack Chain condition.  We discuss some implications of this result, for theorems of F. and M. Riesz type, and for certain free boundary problems.

• 19. Babaoglu, Ceni
Department of Mathematics, Stockholm University.
Some properties of two-phase quadrature domains2011In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, no 10, p. 3386-3396Article in journal (Refereed)
• 20.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Gdansk Univ Technol, Dept Probabil & Biomath, Ul Narutowicza 11-12, PL-80233 Gdansk, Poland..
A Noether theorem for stochastic operators on Schatten classes2017In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 452, no 2, p. 1395-1412Article in journal (Refereed)

We show that a stochastic (Markov) operator S acting on a Schatten class C-1 satisfies the Noether condition (i.e. S' (A) = A and S' (A(2)) = A(2), where A is an element of C-infinity is a Hermitian and bounded operator on a fixed separable and complex Hilbert space (H, <.,.>)), if and only if S(E-A(G)XEA(G)) = E-A (G)S(X)E-A (G) for any state X is an element of C-1 and all Borel sets G subset of R, where E-A (G) denotes the orthogonal projection coming from the spectral resolution A = integral(sigma(A)) zE(A)(dz). Similar results are obtained for stochastic one-parameter continuous semigroups.

• 21.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.
Gdansk University of Technology.
Asymptotic properties of quadratic stochastic operators acting on the L1 space2015In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 114, p. 26-39Article in journal (Refereed)

Quadratic stochastic operators can exhibit a wide variety of asymptotic behaviours andthese have been introduced and studied recently in the l1 space. It turns out that inprinciple most of the results can be carried over to the L1 space. However, due to topologicalproperties of this space one has to restrict in some situations to kernel quadratic stochasticoperators. In this article we study the uniform and strong asymptotic stability of quadratic stochastic operators acting on the L1 space in terms of convergence of the associated (linear)nonhomogeneous Markov chains.

• 22.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Free Boundary Problems of Obstacle Type, a Numerical and Theoretical Study2012Doctoral thesis, comprehensive summary (Other academic)

This thesis consists of five papers and it mainly addresses the theory and schemes to approximate the quadrature domains, QDs. The first deals with the uniqueness and some qualitative properties of the two QDs. The concept of two phase QDs, is more complicated than its one counterpart and consequently introduces significant and interesting open.

We present two numerical schemes to approach the one phase QDs in the paper. The first method is based on the properties of the free boundary the level set techniques. We use shape optimization analysis to construct second method. We illustrate the efficiency of the schemes on a variety of experiments.

In the third paper we design two finite difference methods for the approximation of the multi phase QDs. We prove that the second method enjoys monotonicity, consistency and stability and consequently it is a convergent scheme by Barles-Souganidis theorem. We also present various numerical simulations in the case of Dirac measures.

We introduce the QDs in a sub domain of and Rn study the existence and uniqueness along with a numerical scheme based on the level set method in the fourth paper.

In the last paper we study the tangential touch for a semi-linear problem. We prove that there is just one phase free boundary points on the flat part of the fixed boundary and it is also shown that the free boundary is a uniform C1-graph up to that part.

1. Some properties of two-phase quadrature domains
Open this publication in new window or tab >>Some properties of two-phase quadrature domains
2011 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, no 10, p. 3386-3396Article in journal (Refereed) Published
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:uu:diva-163099 (URN)10.1016/j.na.2011.02.014 (DOI)
Available from: 2012-02-07 Created: 2011-12-07 Last updated: 2017-12-08Bibliographically approved
2. Numerical Approximation of One Phase Quadrature Domains
Open this publication in new window or tab >>Numerical Approximation of One Phase Quadrature Domains
2013 (English)In: Numerical Methods for Partial Differential Equations, ISSN 0749-159X, E-ISSN 1098-2426, Vol. 29, no 5, p. 1709-1728Article in journal (Other academic) Published
##### Abstract [en]

In this work, we present two numerical schemes for a free boundary problem called one phase quadrature domain. In the first method by applying the proprieties of given free boundary problem, we derive a method that leads to a fast iterative solver. The iteration procedure is adapted in order to work in the case when topology changes. The second method is based on shape reconstruction to establish an efficient Shape-Quasi-Newton-Method. Various numerical experiments confirm the efficiency of the derived numerical methods.

##### National Category
Computational Mathematics
##### Identifiers
urn:nbn:se:uu:diva-170221 (URN)10.1002/num.21773 (DOI)000322203200013 ()
Available from: 2012-03-13 Created: 2012-03-09 Last updated: 2017-12-07Bibliographically approved
3. Numerical Schemes for Multi Phase Quadrature Domains
Open this publication in new window or tab >>Numerical Schemes for Multi Phase Quadrature Domains
2014 (English)In: International Journal of Numerical Analysis & Modeling, ISSN 1705-5105, Vol. 11, no 4, p. 726-744Article in journal (Refereed) Published
##### Abstract [en]

In this work, numerical schemes to approximate the solution of one and multi phase quadrature domains are presented. We shall construct a monotone, stable and consistent finite difference method for both one and two phase cases, which converges to the viscosity solution of the partial differential equation arising from the corresponding quadrature domain theory. Moreover, we will discuss the numerical implementation of the resulting approach and present computational tests.

##### Keywords
Quadrature domain; Free boundary problem; Finite difference method; Degenerate elliptic equation
Mathematics
##### Research subject
Numerical Analysis
##### Identifiers
urn:nbn:se:uu:diva-183391 (URN)000343624500004 ()
Available from: 2012-12-07 Created: 2012-10-25 Last updated: 2017-12-07Bibliographically approved
4. Quadrature domains in a subdomain of R^n, theory and a numerical approach
Open this publication in new window or tab >>Quadrature domains in a subdomain of R^n, theory and a numerical approach
(English)Manuscript (preprint) (Other academic)
##### Keywords
free boundary problems, quadrature domain, level set method
Mathematics
##### Research subject
Mathematics with specialization in Applied Mathematics; Numerical Analysis
##### Identifiers
urn:nbn:se:uu:diva-183392 (URN)
Available from: 2012-12-07 Created: 2012-10-25 Last updated: 2012-12-07Bibliographically approved
5. Tangential Touch between the Free and the Fixed Boundary in a Semilinear Free Boundary Problem in Two Dimensions
Open this publication in new window or tab >>Tangential Touch between the Free and the Fixed Boundary in a Semilinear Free Boundary Problem in Two Dimensions
2014 (English)In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 52, no 1, p. 21-42Article in journal (Refereed) Published
##### Abstract [en]

We study minimizers of the functional where B_{1}^{{\mathchoice {\raise .17ex\hbox {\scriptstyle +}} {\raise .17ex\hbox {\scriptstyle +}} {\raise .1ex\hbox {\scriptscriptstyle +}} {\scriptscriptstyle +}}}=\{x\in B_{1}: x_{1}>0\} , u=0 on {xB 1:x 1=0}, \lambda^{{\mathchoice {\raise .17ex\hbox {\scriptstyle \pm }} {\raise .17ex\hbox {\scriptstyle \pm }} {\raise .1ex\hbox {\scriptscriptstyle \pm }} {\scriptscriptstyle \pm }}} are two positive constants and 0<p<1. In two dimensions, we prove that the free boundary is a uniform C 1 graph up to the flat part of the fixed boundary and also that two-phase points cannot occur on this part of the fixed boundary. Here, the free boundary refers to the union of the boundaries of the sets {xu(x)>0}.

##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:uu:diva-170218 (URN)10.1007/s11512-012-0179-3 (DOI)000332797200003 ()
Available from: 2012-03-13 Created: 2012-03-09 Last updated: 2017-12-07Bibliographically approved
• 23.
Department of Mathematics, Stockholm University.
Some properties of one and twophase quadrature domains2010Licentiate thesis, monograph (Other academic)
• 24.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Tangential Touch between the Free and the Fixed Boundary in a Semilinear Free Boundary Problem in Two Dimensions2014In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 52, no 1, p. 21-42Article in journal (Refereed)

We study minimizers of the functional where B_{1}^{{\mathchoice {\raise .17ex\hbox {\scriptstyle +}} {\raise .17ex\hbox {\scriptstyle +}} {\raise .1ex\hbox {\scriptscriptstyle +}} {\scriptscriptstyle +}}}=\{x\in B_{1}: x_{1}>0\} , u=0 on {xB 1:x 1=0}, \lambda^{{\mathchoice {\raise .17ex\hbox {\scriptstyle \pm }} {\raise .17ex\hbox {\scriptstyle \pm }} {\raise .1ex\hbox {\scriptscriptstyle \pm }} {\scriptscriptstyle \pm }}} are two positive constants and 0<p<1. In two dimensions, we prove that the free boundary is a uniform C 1 graph up to the flat part of the fixed boundary and also that two-phase points cannot occur on this part of the fixed boundary. Here, the free boundary refers to the union of the boundaries of the sets {xu(x)>0}.

• 25.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
Quadratic irrationals, generating functions and Lévy constants.Manuscript (preprint) (Other academic)
• 26.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Pricing American Options using Lévy Processes and Monte Carlo Simulations2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 27.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Pricing the American Option Using Itô’s Formula and Optimal Stopping Theory2014Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 28. Betancor, Jorge J.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Solutions of Weinstein equations representable by Bessel Poisson integrals of BMO functions2015In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 431, no 1, p. 440-470Article in journal (Refereed)

We consider the Weinstein type equation L(lambda)u = 0 on (0, infinity) X (0, infinity), where L-lambda= delta(2)(t) + delta-lambda(lambda-1)/x(2), In this paper we characterize the solutions of L(lambda)u = = 0 on (0, infinity) x (0, infinity) representable by Bessel-Poisson integrals of BMO-functions as the ones satisfying certain Carleson properties.

• 29.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Financial Modeling Under Incomplete Information2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 30.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Optimal exercise of an American Option under drift uncertainty2016Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 31.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
L2 Solvability of the Dirichlet, Neumann and the regularity problems for parabolic equations with time-independent Hölder-continuous coefficientsIn: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850Article in journal (Refereed)

We establish the $L^2$-solvability of Dirichlet, Neumann and regularity problems for divergence-form heat (or diffusion) equations with time-independent H\"older-continuous diffusion coefficients, on bounded Lipschitz domains in $\mathbb{R}^n$. This is achieved through the demonstration of invertibility of the relevant layer-potentials which is in turn based on Fredholm theory and a systematic transference scheme which yields suitable parabolic Rellich-type estimates.

• 32.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Homogenization of a parabolic Dirichlet problem by a method of DahlbergIn: Article in journal (Refereed)
• 33.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Micro payments: Viable technical platforms and models for a bankto provide payments on micro amounts2018Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 34.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Univ Edinburgh, Edinburgh EH8 9YL, Midlothian, Scotland..
Local L-infinity-estimates, weak Harnack inequality, and stochastic continuity of solutions of SPDEs2017In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 262, no 1, p. 615-632Article in journal (Refereed)

We consider stochastic partial differential equations under minimal assumptions: the coefficients are merely bounded and measurable and satisfy the stochastic parabolicity condition. In particular, the diffusion term is allowed to be scaling-critical. We derive local supremum estimates with a stochastic adaptation of De Giorgi's iteration and establish a weak Harnack inequality for the solutions. The latter is then used to obtain pointwise almost sure continuity.

• 35.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Valuation of American put options with exercise restrictions2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 36.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Option Pricing in Jump-to-Default Models2015Licentiate thesis, comprehensive summary (Other academic)
• 37.
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
Pluripolar Sets and Pluripolar Hulls2005Doctoral thesis, comprehensive summary (Other academic)

For many questions of complex analysis of several variables classical potential theory does not provide suitable tools and is replaced by pluripotential theory. The latter got many important applications within complex analysis and related fields. Pluripolar sets play a special role in pluripotential theory. These are the exceptional sets this theory. Complete pluripolar sets are especially important. In the thesis we study complete pluripolar sets and pluripolar hulls. We show that in some sense there are many complete pluripolar sets. We show that on each closed subset of the complex plane there is continuous function whose graph is complete pluripolar. On the other hand we study the propagation of pluripolar sets, equivalently we study pluripolar hulls. We relate the pluripolar hull of a graph to fine analytic continuation of the function. Fine analytic continuation of an analytic function over the unit disk is related to the fine topology introduced by Cartan and to the previously known notion of finely analytic functions. We show that fine analytic continuation implies non-triviality of the pluripolar hull. Concerning the inverse direction, we show that the projection of the pluripolar hull is finely open. The difficulty to judge from non-triviality of the pluripolar hull about fine analytic continuation lies in possible multi-sheetedness. If however the pluripolar hull contains the graph of a smooth extension of the function over a fine neighborhood of a boundary point we indeed obtain fine analytic continuation.

1. Complete pluripolar curves and graphs
Open this publication in new window or tab >>Complete pluripolar curves and graphs
2004 In: Annales Polonici Mathematici, ISSN 0066-2216, Vol. 84, no 1, p. 75-86Article in journal (Refereed) Published
##### Identifiers
urn:nbn:se:uu:diva-93257 (URN)
Available from: 2005-09-01 Created: 2005-09-01Bibliographically approved
2. The pluripolar hull of a graph and fine analytic continuation
Open this publication in new window or tab >>The pluripolar hull of a graph and fine analytic continuation
2006 (English)In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 44, no 1, p. 39-60Article in journal (Refereed) Published
##### Abstract [en]

We show that if the graph of an analytic function in the unit disk D is not complete pluripolar in C-2 then the projection of its pluripolar hull contains a fine neighborhood of a point p is an element of partial derivative D. Moreover the projection of the pluripolar hull is always finely open. On the other hand we show that if an analytic function f in D extends to a function T which is defined on a fine neighborhood of a point p is an element of partial derivative D and is finely analytic at p then the pluripolar hull of the graph of f contains the graph of F over a smaller fine neighborhood of p. We give several examples of functions with this property of fine analytic continuation. As a corollary we obtain new classes of analytic functions in the disk which have non-trivial pluripolar hulls, among them C-infinity functions on the closed unit disk which are nowhere analytically extendible and have infinitely-sheeted pluripolar hulls. Previous examples of functions with non-trivial pluripolar hull of the graph have fine analytic continuation.

Mathematics
##### Identifiers
urn:nbn:se:uu:diva-93258 (URN)10.1007/s11512-005-0004-3 (DOI)000240662300003 ()
Available from: 2005-09-01 Created: 2005-09-01 Last updated: 2017-12-14Bibliographically approved
3. Pluripolar hulls of graphs and extension to fine neighborhoods
Open this publication in new window or tab >>Pluripolar hulls of graphs and extension to fine neighborhoods
Manuscript (Other academic)
##### Identifiers
urn:nbn:se:uu:diva-93259 (URN)
Available from: 2005-09-01 Created: 2005-09-01 Last updated: 2010-01-13Bibliographically approved
4. A note on the pluripolar hull of a graph and analytic structure
Open this publication in new window or tab >>A note on the pluripolar hull of a graph and analytic structure
Manuscript (Other academic)
##### Identifiers
urn:nbn:se:uu:diva-93260 (URN)
Available from: 2005-09-01 Created: 2005-09-01 Last updated: 2010-01-13Bibliographically approved
• 38.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Complex Absorbing Potential Method: theory and implementation2011Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
• 39.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Effective equidistribution of horospheres in infinite-volume quotients of SO(n, 1) by geometrically finite groupsManuscript (preprint) (Other academic)

We use the method of Burger to study the rate of equidistribution for translates of pieces of horospheres in Γ\ SO0(n, 1) for geometrically finite discrete subgroups Γ < SO0(n, 1) with infinite covolume.

• 40.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
On the equidistribution of translates of orbits of symmetric subgroups in Γ\GManuscript (preprint) (Other academic)

We use the method of Burger to study the rate of equidistribution for translates of orbits of symmetric subgroups in homogeneous spaces Γ\G for semisimple Lie groups G and lattices Γ.

• 41.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
On the rate of equidistribution of expanding translates of horospheres in finite-volume quotients of SL(2,C)2017In: Journal of Modern Dynamics, ISSN 1930-5311, E-ISSN 1930-532X, Vol. 11, p. 155-188Article in journal (Refereed)

Let Gamma be a lattice in G = SL(2, C). We give an effective equidistribution result with precise error terms for expanding translates of pieces of horospherical orbits in Gamma\G. Our method of proof relies on the theory of unitary representations.

• 42.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
On the rate of equidistribution of expanding translates of horospheres in Γ\GManuscript (preprint) (Other academic)

Let G be a semisimple Lie group and Γ a lattice in G. We generalize a method of Burger to prove precise effective equidistribution results for translates of pieces of horospheres in the homogeneous space Γ\G.

• 43.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Renormalization of integrals of products of Eisenstein series and analytic continuation of representationsManuscript (preprint) (Other academic)

We combine Zagier’s theory of renormalizable automorphic functions on the hyperbolic plane with the analytic continuation of representations of SL(2, R) due to Bernstein and Reznikov to study triple products of Eisenstein series of generic (in particular, non-arithmetic) non-compact finite-volume hyperbolic surfaces.

• 44.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
The Rate of Mixing for Diagonal Flows on Spaces of Affine Lattices2013Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
• 45.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Some applications of representation theory in homogeneous dynamics and automorphic functions2018Doctoral thesis, comprehensive summary (Other academic)

This thesis consists of an introduction and five papers in the general area of dynamics and functions on homogeneous spaces. A common feature is that representation theory plays a key role in all articles.

Papers I-IV are concerned with the effective equidistribution of translates of pieces of subgroup orbits in quotient spaces of semisimple Lie groups by discrete subgroups. In Paper I we focus on finite-volume quotients of SL(2,C) and study the speed of equdistribution for expanding translates orbits of horospherical subgroups. Paper II also studies the effective equidistribution of translates of horospherical orbits, though now in the setting of a quotient of a general semisimple Lie group by a lattice subgroup. Like Paper II, Paper III considers effective equidistribution in quotients of general semisimple Lie groups, but now studies translates of orbits of symmetric subgroups. In all these papers we show that the translates equidistribute with the same exponential rate as for the decay of the corresponding matrix coefficients of the translating subgroup. In Paper IV we consider the effective equidistribution of translates of pieces of horospheres in infinite-volume quotients of groups SO(n,1) by geometrically finite subgroups, and improve the dependency on the spectral gap for certain known effective equidistribution results.

In Paper V we study the Fourier coefficients of Eisenstein series for generic non-cocompact cofinite Fuchsian groups. We use Zagier's renormalization of certain divergent integrals to enable use of the so-called triple product method, and then combine this with the analytic continuation of irreducible representations of SL(2,R) due to Bernstein and Reznikov.

1. On the rate of equidistribution of expanding translates of horospheres in finite-volume quotients of SL(2,C)
Open this publication in new window or tab >>On the rate of equidistribution of expanding translates of horospheres in finite-volume quotients of SL(2,C)
2017 (English)In: Journal of Modern Dynamics, ISSN 1930-5311, E-ISSN 1930-532X, Vol. 11, p. 155-188Article in journal (Refereed) Published
##### Abstract [en]

Let Gamma be a lattice in G = SL(2, C). We give an effective equidistribution result with precise error terms for expanding translates of pieces of horospherical orbits in Gamma\G. Our method of proof relies on the theory of unitary representations.

##### Keywords
Effective equidistribution, translates, horospheres
##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:uu:diva-315688 (URN)10.3934/jmd.2017008 (DOI)000396538600008 ()
##### Funder
Swedish Research Council, 621-2011-3629Göran Gustafsson Foundation for promotion of scientific research at Uppala University and Royal Institute of Technology Available from: 2017-02-18 Created: 2017-02-18 Last updated: 2018-04-11Bibliographically approved
2. On the rate of equidistribution of expanding translates of horospheres in Γ\G
Open this publication in new window or tab >>On the rate of equidistribution of expanding translates of horospheres in Γ\G
(English)Manuscript (preprint) (Other academic)
##### Abstract [en]

Let G be a semisimple Lie group and Γ a lattice in G. We generalize a method of Burger to prove precise effective equidistribution results for translates of pieces of horospheres in the homogeneous space Γ\G.

##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:uu:diva-347856 (URN)
Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-11
3. On the equidistribution of translates of orbits of symmetric subgroups in Γ\G
Open this publication in new window or tab >>On the equidistribution of translates of orbits of symmetric subgroups in Γ\G
(English)Manuscript (preprint) (Other academic)
##### Abstract [en]

We use the method of Burger to study the rate of equidistribution for translates of orbits of symmetric subgroups in homogeneous spaces Γ\G for semisimple Lie groups G and lattices Γ.

##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:uu:diva-347857 (URN)
Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-11
4. Effective equidistribution of horospheres in infinite-volume quotients of SO(n, 1) by geometrically finite groups
Open this publication in new window or tab >>Effective equidistribution of horospheres in infinite-volume quotients of SO(n, 1) by geometrically finite groups
(English)Manuscript (preprint) (Other academic)
##### Abstract [en]

We use the method of Burger to study the rate of equidistribution for translates of pieces of horospheres in Γ\ SO0(n, 1) for geometrically finite discrete subgroups Γ < SO0(n, 1) with infinite covolume.

##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:uu:diva-347858 (URN)
Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-11
5. Renormalization of integrals of products of Eisenstein series and analytic continuation of representations
Open this publication in new window or tab >>Renormalization of integrals of products of Eisenstein series and analytic continuation of representations
(English)Manuscript (preprint) (Other academic)
##### Abstract [en]

We combine Zagier’s theory of renormalizable automorphic functions on the hyperbolic plane with the analytic continuation of representations of SL(2, R) due to Bernstein and Reznikov to study triple products of Eisenstein series of generic (in particular, non-arithmetic) non-compact finite-volume hyperbolic surfaces.

##### National Category
Mathematical Analysis
##### Identifiers
urn:nbn:se:uu:diva-347859 (URN)
Available from: 2018-04-09 Created: 2018-04-09 Last updated: 2018-04-24
• 46.
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
Selected Problems in Financial Mathematics2004Doctoral thesis, comprehensive summary (Other academic)

This thesis, consisting of six papers and a summary, studies the area of continuous time financial mathematics. A unifying theme for many of the problems studied is the implications of possible mis-specifications of models. Intimately connected with this question is, perhaps surprisingly, convexity properties of option prices. We also study qualitative behavior of different optimal stopping boundaries appearing in option pricing.

In Paper I a new condition on the contract function of an American option is provided under which the option price increases monotonically in the volatility. It is also shown that American option prices are continuous in the volatility.

In Paper II an explicit pricing formula for the perpetual American put option in the Constant Elasticity of Variance model is derived. Moreover, different properties of this price are studied.

Paper III deals with the Russian option with a finite time horizon. It is shown that the value of the Russian option solves a certain free boundary problem. This information is used to analyze the optimal stopping boundary.

A study of perpetual game options is performed in Paper IV. One of the main results provides a condition under which the value of the option is increasing in the volatility.

In Paper V options written on several underlying assets are considered. It is shown that, within a large class of models, the only model for the stock prices that assigns convex option prices to all convex contract functions is geometric Brownian motion.

Finally, in Paper VI it is shown that the optimal stopping boundary for the American put option is convex in the standard Black-Scholes model.

1. Properties of American option prices
Open this publication in new window or tab >>Properties of American option prices
2004 (English)In: Stochastic Processes and their Applications, ISSN 0304-4149, E-ISSN 1879-209X, Vol. 114, no 2, p. 265-278Article in journal (Refereed) Published
##### Abstract [en]

We investigate some properties of American option prices in the setting of time- and level-dependent volatility. The properties under consideration are convexity in the underlying stock price, monotonicity and continuity in the volatility and time decay. Some properties are direct consequences of the corresponding properties of European option prices that are already known, and some follow by writing solutions of different stochastic differential equations as time changes of the same Brownian motion.

##### National Category
Natural Sciences
##### Identifiers
urn:nbn:se:uu:diva-92190 (URN)10.1016/j.spa.2004.05.002 (DOI)
Available from: 2004-10-08 Created: 2004-10-08 Last updated: 2017-12-14Bibliographically approved
2. The perpetual American put option in a level-dependent volatility model
Open this publication in new window or tab >>The perpetual American put option in a level-dependent volatility model
2003 In: Journal of Applied Probability, ISSN 0021-9002, Vol. 40, no 3, p. 783-789Article in journal (Refereed) Published
##### Identifiers
urn:nbn:se:uu:diva-92191 (URN)
Available from: 2004-10-08 Created: 2004-10-08Bibliographically approved
3. Russian options with a finite time horizon
Open this publication in new window or tab >>Russian options with a finite time horizon
2004 In: Journal of Applied Probability, ISSN 0021-9002, Vol. 41, no 2, p. 313-326Article in journal (Refereed) Published
##### Identifiers
urn:nbn:se:uu:diva-92192 (URN)
Available from: 2004-10-08 Created: 2004-10-08Bibliographically approved
4. Properties of game options
Open this publication in new window or tab >>Properties of game options
Article in journal (Refereed) Submitted
##### Identifiers
urn:nbn:se:uu:diva-92193 (URN)
Available from: 2004-10-08 Created: 2004-10-08Bibliographically approved
5. Superreplication of options on several underlying assets
Open this publication in new window or tab >>Superreplication of options on several underlying assets
2005 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 42, no 1, p. 27-38Article in journal (Refereed) Published
##### National Category
Natural Sciences
##### Identifiers
urn:nbn:se:uu:diva-92194 (URN)
Available from: 2004-10-08 Created: 2004-10-08 Last updated: 2017-12-14Bibliographically approved
6. Convexity of the optimal stopping boundary for the American put option
Open this publication in new window or tab >>Convexity of the optimal stopping boundary for the American put option
2004 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 299, no 1, p. 147-156Article in journal (Refereed) Published
##### Abstract [en]

We show that the optimal stopping boundary for the American put option is convex in the standard Black–Scholes model. The methods are adapted from ice-melting problems and rely upon studying the behavior of level curves of solutions to certain parabolic differential equations.

##### National Category
Natural Sciences
##### Identifiers
urn:nbn:se:uu:diva-92195 (URN)10.1016/j.jmaa.2004.06.018 (DOI)
Available from: 2004-10-08 Created: 2004-10-08 Last updated: 2017-12-14Bibliographically approved
• 47.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
The Black-Scholes equation in stochastic volatility models2010In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 368, no 2, p. 498-507Article in journal (Refereed)

We study the Black-Scholes equation in stochastic volatility models. In particular, we show that the option price is the unique classical solution to a parabolic differential equation with a certain boundary behaviour for vanishing values of the volatility. If the boundary is attainable, then this boundary behaviour serves as a boundary condition and guarantees uniqueness in appropriate function spaces. On the other hand, if the boundary is non-attainable, then the boundary behaviour is not needed to guarantee uniqueness, but is nevertheless very useful for instance from a numerical perspective.

• 48.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
On the existence and non-existence of a solution to the Hartree-Fock equations with an external magnetic field2007Licentiate thesis, monograph (Other academic)
• 49.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
Existence of infinitely many distinct solutions to the quasi-relativistic Hartree-Fock equations2009Report (Other academic)
• 50.
Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics.
On the pricing equations of some path-dependent options2006Doctoral thesis, comprehensive summary (Other academic)

This thesis consists of four papers and a summary. The common topic of the included papers are the pricing equations of path-dependent options. Various properties of barrier options and American options are studied, such as convexity of option prices, the size of the continuation region in American option pricing and pricing formulas for turbo warrants. In Paper I we study the effect of model misspecification on barrier option pricing. It turns out that, as in the case of ordinary European and American options, this is closely related to convexity properties of the option prices. We show that barrier option prices are convex under certain conditions on the contract function and on the relation between the risk-free rate of return and the dividend rate. In Paper II a new condition is given to ensure that the early exercise feature in American option pricing has a positive value. We give necessary and sufficient conditions for the American option price to coincide with the corresponding European option price in at least one diffusion model. In Paper III we study parabolic obstacle problems related to American option pricing and in particular the size of the non-coincidence set. The main result is that if the boundary of the set of points where the obstacle is a strict subsolution to the differential equation is C1-Dini in space and Lipschitz in time, there is a positive distance, which is uniform in space, between the boundary of this set and the boundary of the non-coincidence set. In Paper IV we derive explicit pricing formulas for turbo warrants under the classical Black-Scholes assumptions.

1. Monotonicity in the volatility of single-barrier option prices
Open this publication in new window or tab >>Monotonicity in the volatility of single-barrier option prices
2006 (English)In: International Journal of Theoretical and Applied Finance, ISSN 0219-0249, Vol. 9, no 6, p. 987-996Article in journal (Refereed) Published
##### Abstract [en]

We generalize earlier results on barrier options for puts and calls and log-normal stock processes to general local volatility models and convex contracts. We show that Γ ≥ 0, that Δ has a unique sign and that the option price is increasing with the volatility for convex contracts in the following cases:

If the risk-free rate of return dominates the dividend rate, then it holds for up-and-out options if the contract function is zero at the barrier and for down-and-in options in general.

If the risk-free rate of return is dominated by the dividend rate, then it holds for down-and-out options if the contract function is zero at the barrier and for up-and-in options in general.

We apply our results to show that a hedger who misspecifies the volatility using a time-and-level dependent volatility will super-replicate any claim satisfying the above conditions if the misspecified volatility dominates the true (possibly stochastic) volatility almost surely.

##### Keywords
Barrier option, convexity, volatility, parabolic equation
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-93999 (URN)10.1142/S0219024906003822 (DOI)
Available from: 2006-02-17 Created: 2006-02-17 Last updated: 2017-12-14Bibliographically approved
2. When American options are European
Open this publication in new window or tab >>When American options are European
(English)In: Decisions in Economics and Finance , ISSN 1593-8883Article in journal (Refereed) Submitted
##### Identifiers
urn:nbn:se:uu:diva-94000 (URN)
Available from: 2006-02-17 Created: 2006-02-17 Last updated: 2009-02-26Bibliographically approved
3. On the size of the non-coincidence set of parabolic obstacle problems with applications to American option pricing
Open this publication in new window or tab >>On the size of the non-coincidence set of parabolic obstacle problems with applications to American option pricing
2007 (English)In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 101, no 1, p. 148-160Article in journal (Refereed) Published
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-94001 (URN)
Available from: 2006-02-17 Created: 2006-03-19 Last updated: 2017-12-14Bibliographically approved
4. Explicit pricing formulas for turbo warrants
Open this publication in new window or tab >>Explicit pricing formulas for turbo warrants
(English)In: Risk magazine, ISSN 1082-6394Article in journal (Refereed) Submitted
##### Identifiers
urn:nbn:se:uu:diva-94002 (URN)
Available from: 2006-02-17 Created: 2006-02-17 Last updated: 2009-02-26Bibliographically approved
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