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• 1.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Department of Mathematics, Faculty of Science, Albaha University, Al Baha, Saudi Arabia.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Jagiellonian University, Faculty of Mathematics and Computer ScienceInstitute of Mathematics, Institute of Mathematics..
Attractors of compactly generated semigroups of regular polynomial mappings.2018Inngår i: Complexity, ISSN 1076-2787, E-ISSN 1099-0526, artikkel-id 5698021Artikkel i tidsskrift (Fagfellevurdert)

We investigate the metric space of pluriregular sets as well as the contractions on that space induced by infinite compact families of proper polynomial mappings of several complex variables. The topological semigroups generated by such families, with composition as the semigroup operation, lead to the constructions of a variety of Julia-type pluriregular sets. The generating families can also be viewed as infinite iterated function systems with compact attractors. We show that such attractors can be approximated both deterministically and probabilistically in a manner of the classic chaos game.

• 2.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Mathematics..
Bergman Functions and Nearly Orthonormal PolynomialsManuskript (preprint) (Annet vitenskapelig)
• 3.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Cavalieris indivisibler2018Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
• 4.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Om oändliga tal2015Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
• 5.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
The Symmetric Meixner-Pollaczek polynomials2003Doktoravhandling, med artikler (Annet vitenskapelig)

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
Åpne denne publikasjonen i ny fane eller vindu >>The Meixner-Pollaczek polynomials and and a system of orthogonal polynomials in a strip
Inngår i: Journal of computational and applied MathematicsArtikkel i tidsskrift (Fagfellevurdert) Submitted
##### Identifikatorer
urn:nbn:se:uu:diva-90614 (URN)
Tilgjengelig fra: 2003-05-16 Laget: 2003-05-16bibliografisk kontrollert
2. Umbral calculus and the Symmetric Meixner-Pollaczek polynomials
Åpne denne publikasjonen i ny fane eller vindu >>Umbral calculus and the Symmetric Meixner-Pollaczek polynomials
Manuskript (Annet vitenskapelig)
##### Identifikatorer
urn:nbn:se:uu:diva-90615 (URN)
Tilgjengelig fra: 2003-05-16 Laget: 2003-05-16 Sist oppdatert: 2010-01-13bibliografisk kontrollert
3. The symmetric Meixner-Pollaczek polynomials with real parameter
Åpne denne publikasjonen i ny fane eller vindu >>The symmetric Meixner-Pollaczek polynomials with real parameter
Manuskript (Annet vitenskapelig)
##### Identifikatorer
urn:nbn:se:uu:diva-90616 (URN)
Tilgjengelig fra: 2003-05-16 Laget: 2003-05-16 Sist oppdatert: 2010-01-13bibliografisk kontrollert
4. Linearization and connection problems for the Symmetric Meixner-Pollaczek polynomials
Åpne denne publikasjonen i ny fane eller vindu >>Linearization and connection problems for the Symmetric Meixner-Pollaczek polynomials
Manuskript (Annet vitenskapelig)
##### Identifikatorer
urn:nbn:se:uu:diva-90617 (URN)
Tilgjengelig fra: 2003-05-16 Laget: 2003-05-16 Sist oppdatert: 2010-01-13bibliografisk kontrollert
• 6.
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 universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Fysiska sektionen, Institutionen för fysik och astronomi. 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 model2017Inngår i: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 50, nr 24, artikkel-id 244002Artikkel i tidsskrift (Fagfellevurdert)

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.

• 7.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Pathological functions and the Baire category theorem2017Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
• 8.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Avdelningen för datalogi.
A practical comparison between algebraic and statistical attacks on the lightweight cipher SIMON2016Independent thesis Advanced level (professional degree), 20 poäng / 30 hpOppgave

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.

• 9.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och tillämpad matematik.
Boundary Behavior of p-Laplace Type Equations2013Doktoravhandling, med artikler (Annet vitenskapelig)

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
Åpne denne publikasjonen i ny fane eller vindu >>Estimates for Solutions to Equations of p-Laplace type in Ahlfors regular NTA-domains
2014 (engelsk)Inngår i: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 266, nr 9, s. 5955-6005Artikkel i tidsskrift (Fagfellevurdert) Published
##### Identifikatorer
urn:nbn:se:uu:diva-163517 (URN)10.1016/j.jfa.2014.02.027 (DOI)000334652000018 ()
Tilgjengelig fra: 2011-12-13 Laget: 2011-12-12 Sist oppdatert: 2017-12-08bibliografisk kontrollert
2. Boundary estimates for solutions to operators of $p$-Laplace type with lower order terms
Åpne denne publikasjonen i ny fane eller vindu >>Boundary estimates for solutions to operators of $p$-Laplace type with lower order terms
2011 (engelsk)Inngår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 250, nr 1, s. 264-291Artikkel i tidsskrift (Fagfellevurdert) Published
##### Identifikatorer
urn:nbn:se:uu:diva-163370 (URN)10.1016/j.jde.2010.09.011 (DOI)
Tilgjengelig fra: 2011-12-12 Laget: 2011-12-12 Sist oppdatert: 2017-12-08bibliografisk kontrollert
3. Optimal doubling, Reifenberg flatness and operators of p-Laplace type
Åpne denne publikasjonen i ny fane eller vindu >>Optimal doubling, Reifenberg flatness and operators of p-Laplace type
2011 (engelsk)Inngår i: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, nr 17, s. 5943-5955Artikkel i tidsskrift (Fagfellevurdert) 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 @.

##### Identifikatorer
urn:nbn:se:uu:diva-163435 (URN)10.1016/j.na.2011.05.061 (DOI)
Tilgjengelig fra: 2012-11-30 Laget: 2011-12-12 Sist oppdatert: 2017-12-08bibliografisk kontrollert
4. Boundary Estimates for Certain Degenerate and Singular Parabolic Equations
Åpne denne publikasjonen i ny fane eller vindu >>Boundary Estimates for Certain Degenerate and Singular Parabolic Equations
2016 (engelsk)Inngår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 18, nr 2, s. 381-424Artikkel i tidsskrift (Fagfellevurdert) 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.

##### Emneord
Degenerate and singular parabolic equations; Harnack estimates; boundary Harnack inequality; Carleson estimate
##### Identifikatorer
urn:nbn:se:uu:diva-186267 (URN)10.4171/JEMS/593 (DOI)000370249100005 ()
Tilgjengelig fra: 2013-02-12 Laget: 2012-11-28 Sist oppdatert: 2017-12-07bibliografisk kontrollert
5. Wolff-Potential Estimates and Doubling of Subelliptic p-harmonic measures
Åpne denne publikasjonen i ny fane eller vindu >>Wolff-Potential Estimates and Doubling of Subelliptic p-harmonic measures
2013 (engelsk)Inngår i: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 85, s. 149-159Artikkel i tidsskrift (Fagfellevurdert) 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.

##### Identifikatorer
urn:nbn:se:uu:diva-186268 (URN)10.1016/j.na.2013.02.023 (DOI)000318378700013 ()
Tilgjengelig fra: 2013-03-26 Laget: 2012-11-28 Sist oppdatert: 2017-12-07bibliografisk kontrollert
6. On a one-phase free boundary problem
Åpne denne publikasjonen i ny fane eller vindu >>On a one-phase free boundary problem
2013 (engelsk)Inngår i: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, nr 1, s. 181-191Artikkel i tidsskrift (Annet vitenskapelig) 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.

##### Emneord
One-phase, free boundary, NTA, non-divergence, linear
Matematik
##### Identifikatorer
urn:nbn:se:uu:diva-186265 (URN)10.5186/aasfm.2013.3815 (DOI)000316239200009 ()
Tilgjengelig fra: 2012-11-30 Laget: 2012-11-28 Sist oppdatert: 2017-12-07bibliografisk kontrollert
• 10.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
On a one-phase free boundary problem2013Inngår i: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, nr 1, s. 181-191Artikkel i tidsskrift (Annet vitenskapelig)

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.

• 11.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Dipartimento di Matematica "F. Casorati", Università di Pavia. Dipartimento di Matematica "F. Brioschi", Politecnico di Milano.
Boundary Estimates for Certain Degenerate and Singular Parabolic Equations2016Inngår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 18, nr 2, s. 381-424Artikkel i tidsskrift (Fagfellevurdert)

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.

• 12.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
A note on the hyperconvexity of pseudoconvex domains beyond Lipschitz regularity2015Inngår i: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, nr 3, s. 531-545Artikkel i tidsskrift (Fagfellevurdert)

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.

• 13.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
Department of mathematics and mathematical statistics, Umeå University. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
Approximation and Bounded Plurisubharmonic Exhaustion Functions Beyond Lipschitz DomainsManuskript (preprint) (Annet vitenskapelig)

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.

• 14.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori. Univ Jyvaskyla, Dept Math & Stat, Jyvaskyla 40014, Finland.
Umeå University. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Approximation of plurisubharmonic functions2016Inngår i: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 61, nr 1, s. 23-28Artikkel i tidsskrift (Fagfellevurdert)

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.

• 15.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori. Aalto University, Institute of Mathematics, P.O. Box 11100, FI-00076 Aalto, Finland.
A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term2017Inngår i: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 272, nr 8, s. 3176-3215Artikkel i tidsskrift (Fagfellevurdert)

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.

• 16.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Boundary behavior of solutions to the parabolic p-Laplace equation2019Inngår i: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 12, nr 1, s. 1-42Artikkel i tidsskrift (Fagfellevurdert)

We establish boundary estimates for non-negative solutions to the $p$-parabolic equation in the degenerate range $p>2$. Our main results include new parabolic intrinsic Harnack chains in cylindrical NTA-domains together with sharp boundary decay estimates. If the underlying domain is $C^{1,1}$-regular, we establish a relatively complete theory of the boundary behavior, including boundary Harnack principles and Hölder continuity of the ratios of two solutions, as well as fine properties of associated boundary measures. There is an intrinsic waiting time phenomena present which plays a fundamental role throughout the paper. In particular, conditions on these waiting times rule out well-known examples of explicit solutions violating the boundary Harnack principle.

• 17.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Aalto Univ, Finland.
A comparison principle for the porous medium equation and its consequences2017Inngår i: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616, Vol. 33, nr 2, s. 573-594Artikkel i tidsskrift (Fagfellevurdert)

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.

• 18.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Neural ODEs as the Deep Limit of ResNets with constant weights2019Inngår i: Artikkel i tidsskrift (Fagfellevurdert)

In this paper we prove that, in the deep limit, the stochastic gradient descent on a ResNet type deep neural network, where each layer share the same weight matrix, converges to the stochastic gradient descent for a Neural ODE and that the corresponding value/loss functions converge. Our result gives, in the context of minimization by stochastic gradient descent, a theoretical foundation for considering Neural ODEs as the deep limit of ResNets. Our proof is based on certain decay estimates for associated Fokker-Planck equations.

• 19.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
Wolff-Potential Estimates and Doubling of Subelliptic p-harmonic measures2013Inngår i: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 85, s. 149-159Artikkel i tidsskrift (Fagfellevurdert)

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.

• 20.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori. 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 equation2017Inngår i: Journal of evolution equations (Printed ed.), ISSN 1424-3199, E-ISSN 1424-3202, Vol. 17, nr 2, s. 827-848Artikkel i tidsskrift (Fagfellevurdert)

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.

• 21.
University of Washington, Seattle, USA.
University of Missouri, Columbia, USA. Instituto de Ciencias Matematicas, Madrid, Spain. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori. University of Washington, Seattle, USA.
A new characterization of chord-arc domains2017Inngår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 19, nr 4, s. 967-981Artikkel i tidsskrift (Fagfellevurdert)

We show that if Ω⊂Rn+1, n≥1, is a uniform domain (also known as a 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of Ω implies the existence of exterior corkscrew points at all scales, so that in fact, Ω 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.

• 22. Babaoglu, Ceni
Department of Mathematics, Stockholm University.
Some properties of two-phase quadrature domains2011Inngår i: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, nr 10, s. 3386-3396Artikkel i tidsskrift (Fagfellevurdert)
• 23.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Gdansk Univ Technol, Dept Probabil & Biomath, Ul Narutowicza 11-12, PL-80233 Gdansk, Poland..
A Noether theorem for stochastic operators on Schatten classes2017Inngår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 452, nr 2, s. 1395-1412Artikkel i tidsskrift (Fagfellevurdert)

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.

• 24.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Tillämpad matematik och statistik.
Gdansk University of Technology.
Asymptotic properties of quadratic stochastic operators acting on the L1 space2015Inngår i: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 114, s. 26-39Artikkel i tidsskrift (Fagfellevurdert)

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.

• 25.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och tillämpad matematik.
Free Boundary Problems of Obstacle Type, a Numerical and Theoretical Study2012Doktoravhandling, med artikler (Annet vitenskapelig)

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
Åpne denne publikasjonen i ny fane eller vindu >>Some properties of two-phase quadrature domains
2011 (engelsk)Inngår i: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, nr 10, s. 3386-3396Artikkel i tidsskrift (Fagfellevurdert) Published
##### Identifikatorer
urn:nbn:se:uu:diva-163099 (URN)10.1016/j.na.2011.02.014 (DOI)
Tilgjengelig fra: 2012-02-07 Laget: 2011-12-07 Sist oppdatert: 2017-12-08bibliografisk kontrollert
2. Numerical Approximation of One Phase Quadrature Domains
Åpne denne publikasjonen i ny fane eller vindu >>Numerical Approximation of One Phase Quadrature Domains
2013 (engelsk)Inngår i: Numerical Methods for Partial Differential Equations, ISSN 0749-159X, E-ISSN 1098-2426, Vol. 29, nr 5, s. 1709-1728Artikkel i tidsskrift (Annet vitenskapelig) 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.

##### Identifikatorer
urn:nbn:se:uu:diva-170221 (URN)10.1002/num.21773 (DOI)000322203200013 ()
Tilgjengelig fra: 2012-03-13 Laget: 2012-03-09 Sist oppdatert: 2017-12-07bibliografisk kontrollert
3. Numerical Schemes for Multi Phase Quadrature Domains
Åpne denne publikasjonen i ny fane eller vindu >>Numerical Schemes for Multi Phase Quadrature Domains
2014 (engelsk)Inngår i: International Journal of Numerical Analysis & Modeling, ISSN 1705-5105, Vol. 11, nr 4, s. 726-744Artikkel i tidsskrift (Fagfellevurdert) 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.

##### Emneord
Quadrature domain; Free boundary problem; Finite difference method; Degenerate elliptic equation
##### Forskningsprogram
Numerisk analys
##### Identifikatorer
urn:nbn:se:uu:diva-183391 (URN)000343624500004 ()
Tilgjengelig fra: 2012-12-07 Laget: 2012-10-25 Sist oppdatert: 2017-12-07bibliografisk kontrollert
4. Quadrature domains in a subdomain of R^n, theory and a numerical approach
Åpne denne publikasjonen i ny fane eller vindu >>Quadrature domains in a subdomain of R^n, theory and a numerical approach
(engelsk)Manuskript (preprint) (Annet vitenskapelig)
##### Emneord
free boundary problems, quadrature domain, level set method
##### Forskningsprogram
Matematik med inriktning mot tillämpad matematik; Numerisk analys
##### Identifikatorer
urn:nbn:se:uu:diva-183392 (URN)
Tilgjengelig fra: 2012-12-07 Laget: 2012-10-25 Sist oppdatert: 2012-12-07bibliografisk kontrollert
5. Tangential Touch between the Free and the Fixed Boundary in a Semilinear Free Boundary Problem in Two Dimensions
Åpne denne publikasjonen i ny fane eller vindu >>Tangential Touch between the Free and the Fixed Boundary in a Semilinear Free Boundary Problem in Two Dimensions
2014 (engelsk)Inngår i: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 52, nr 1, s. 21-42Artikkel i tidsskrift (Fagfellevurdert) 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}.

##### Identifikatorer
urn:nbn:se:uu:diva-170218 (URN)10.1007/s11512-012-0179-3 (DOI)000332797200003 ()
Tilgjengelig fra: 2012-03-13 Laget: 2012-03-09 Sist oppdatert: 2017-12-07bibliografisk kontrollert
• 26.
Department of Mathematics, Stockholm University.
Some properties of one and twophase quadrature domains2010Licentiatavhandling, monografi (Annet vitenskapelig)
• 27.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och tillämpad matematik.
Tangential Touch between the Free and the Fixed Boundary in a Semilinear Free Boundary Problem in Two Dimensions2014Inngår i: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 52, nr 1, s. 21-42Artikkel i tidsskrift (Fagfellevurdert)

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}.

• 28.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
Quadratic irrationals, generating functions and Lévy constants.Manuskript (preprint) (Annet vitenskapelig)
• 29.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Data-driven discovery of PDEs in complex datasets2019Inngår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 384, s. 239-252Artikkel i tidsskrift (Fagfellevurdert)

Many processes in science and engineering can be described by partial differential equations (PDEs). Traditionally, PDEs are derived by considering first principles of physics to derive the relations between the involved physical quantities of interest. A different approach is to measure the quantities of interest and use deep learning to reverse engineer the PDEs which are describing the physical process. In this paper we use machine learning, and deep learning in particular, to discover PDEs hidden in complex data sets from measurement data. We include examples of data from a known model problem, and real data from weather station measurements. We show how necessary transformations of the input data amounts to coordinate transformations in the discovered PDE, and we elaborate on feature and model selection. It is shown that the dynamics of a non-linear, second order PDE can be accurately described by an ordinary differential equation which is automatically discovered by our deep learning algorithm. Even more interestingly, we show that similar results apply in the context of more complex simulations of the Swedish temperature distribution

• 30.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Pricing American Options using Lévy Processes and Monte Carlo Simulations2015Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
• 31.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Pricing the American Option Using Itô’s Formula and Optimal Stopping Theory2014Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
• 32. Betancor, Jorge J.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Solutions of Weinstein equations representable by Bessel Poisson integrals of BMO functions2015Inngår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 431, nr 1, s. 440-470Artikkel i tidsskrift (Fagfellevurdert)

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.

• 33.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Financial Modeling Under Incomplete Information2015Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
• 34.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Scuola Normale Super Pisa, Piazza Cavalieri 7, I-56126 Pisa, PI, Italy;Univ Trento, Via Sommar 14, I-38123 Povo, TN, Italy.
Scuola Normale Super Pisa, Piazza Cavalieri 7, I-56126 Pisa, PI, Italy;Univ Libre Bruxelles, Dept Math, Ave Franklin Roosevelt 50, B-1050 Brussels, Belgium. Univ Trento, Via Sommar 14, I-38123 Povo, TN, Italy.
Some Sphere Theorems In Linear Potential Theory2019Inngår i: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 371, nr 11, s. 7757-7790Artikkel i tidsskrift (Fagfellevurdert)

In this paper we analyze the capacitary potential due to a charged body in order to deduce sharp analytic and geometric inequalities, whose equality cases are saturated by domains with spherical symmetry. In particular, for a regular bounded domain Omega subset of R-n, n >= 3, we prove that if the mean curvature H of the boundary obeys the condition -[1/Cap(Omega)](1/n-2) <= H/n-1 <= [1/Cap(Omega)](1/n-2), then Omega is a round ball.

• 35.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Optimal exercise of an American Option under drift uncertainty2016Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
• 36.
Nazarbayev Univ, Dept Math, Astana 010000, Kazakhstan.
Stockholm Univ, Dept Math, SE-10691 Stockholm, Sweden. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Transference of local to global L-2 maximal estimates for dispersive partial differential equations2019Inngår i: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 471, nr 1-2, s. 411-422Artikkel i tidsskrift (Fagfellevurdert)

In this paper we give an elementary proof for transference of local to global maximal estimates for dispersive PDEs. This is done by transferring local L-2 estimates for certain oscillatory integrals with rough phase functions, to the corresponding global estimates. The elementary feature of our approach is that it entirely avoids the use of the wave packet techniques which are quite common in this context, and instead is based on scalings and classical oscillatory integral estimates.

• 37.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
L2 Solvability of the Dirichlet, Neumann and the regularity problems for parabolic equations with time-independent Hölder-continuous coefficientsInngår i: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850Artikkel i tidsskrift (Fagfellevurdert)

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.

• 38.
Nazarbayev Univ, Dept Math, Astana 010000, Kazakhstan.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Homogenization of a parabolic Dirichlet problem by a method of Dahlberg2018Inngår i: Publicacions matemàtiques, ISSN 0214-1493, E-ISSN 2014-4350, Vol. 62, nr 2, s. 439-473Artikkel i tidsskrift (Fagfellevurdert)

Consider the linear parabolic operator in divergence form: Hu := partial derivative(t)u (X, t) - div(A (X) del u (X, t)). We employ a method of Dahlberg to show that the Dirichlet problem for H in the upper half plane is well-posed for boundary data in L-p, for any elliptic matrix of coef-ficients A which is periodic and satisfies a Dini-type condition. This result allows us to treat a homogenization problem for the equation partial derivative(t)u(epsilon) (X, t) - div(A (X/epsilon) del u(epsilon) (X, t)) in Lipschitz domains with L-p-boundary data.

• 39.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Tekniska sektionen, Institutionen för teknikvetenskaper, Elektricitetslära.
Karolinska Inst, Stockholm, Sweden.
A Novel Interpretation of the Electromagnetic Fields of Lightning Return Strokes2019Inngår i: Atmosphere, ISSN 2073-4433, E-ISSN 2073-4433, Vol. 10, nr 1, artikkel-id 22Artikkel i tidsskrift (Fagfellevurdert)

Electric and/or magnetic fields are generated by stationary charges, uniformly moving charges and accelerating charges. These field components are described in the literature as static fields, velocity fields (or generalized Coulomb field) and radiation fields (or acceleration fields), respectively. In the literature, the electromagnetic fields generated by lightning return strokes are presented using the field components associated with short dipoles, and in this description the one-to-one association of the electromagnetic field terms with the physical process that gives rise to them is lost. In this paper, we have derived expressions for the electromagnetic fields using field equations associated with accelerating (and moving) charges and separated the resulting fields into static, velocity and radiation fields. The results illustrate how the radiation fields emanating from the lightning channel give rise to field terms varying as <mml:semantics>1/r</mml:semantics> and <mml:semantics>1/r2</mml:semantics>, the velocity fields generating field terms varying as <mml:semantics>1/r2</mml:semantics>, and the static fields generating field components varying as <mml:semantics>1/r2</mml:semantics> and <mml:semantics>1/r3</mml:semantics>. These field components depend explicitly on the speed of propagation of the current pulse. However, the total field does not depend explicitly on the speed of propagation of the current pulse. It is shown that these field components can be combined to generate the field components pertinent to the dipole technique. However, in this conversion process the connection of the field components to the physical processes taking place at the source that generate these fields (i.e., static charges, uniformly moving charges and accelerating charges) is lost.

• 40.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Micro payments: Viable technical platforms and models for a bankto provide payments on micro amounts2018Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
• 41.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Univ Edinburgh, Edinburgh EH8 9YL, Midlothian, Scotland..
Local L-infinity-estimates, weak Harnack inequality, and stochastic continuity of solutions of SPDEs2017Inngår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 262, nr 1, s. 615-632Artikkel i tidsskrift (Fagfellevurdert)

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.

• 42.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Valuation of American put options with exercise restrictions2014Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
• 43.
Natl Res Univ, Fac Math, Higher Sch Econ, Usacheva 6, Moscow 119048, Russia;Inst Theoret & Expt Phys, Moscow 117218, Russia.
Univ Amsterdam, Korteweg Vriesinst Wiskunde, Postbus 94248, NL-1090 GE Amsterdam, Netherlands. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Fysiska sektionen, Institutionen för fysik och astronomi, Teoretisk fysik. Inst Theoret & Expt Phys, Moscow 117218, Russia;Inst Informat Transmiss Problems, Moscow 127994, Russia. Univ Amsterdam, Korteweg Vriesinst Wiskunde, Postbus 94248, NL-1090 GE Amsterdam, Netherlands.
Cut-and-join equation for monotone Hurwitz numbers revisited2019Inngår i: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 137, s. 1-6Artikkel i tidsskrift (Fagfellevurdert)

We give a new proof of the cut-and-join equation for the monotone Hurwitz numbers, derived first by Goulden, Guay-Paquet, and Novak. The main interest in this particular equation is its close relation to the quadratic loop equation in the theory of spectral curve topological recursion, and we recall this motivation giving a new proof of the topological recursion for monotone Hurwitz numbers, obtained first by Do, Dyer, and Mathews.

• 44.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Option Pricing in Jump-to-Default Models2015Licentiatavhandling, med artikler (Annet vitenskapelig)
• 45.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
Pluripolar Sets and Pluripolar Hulls2005Doktoravhandling, med artikler (Annet vitenskapelig)

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
Åpne denne publikasjonen i ny fane eller vindu >>Complete pluripolar curves and graphs
2004 Inngår i: Annales Polonici Mathematici, ISSN 0066-2216, Vol. 84, nr 1, s. 75-86Artikkel i tidsskrift (Fagfellevurdert) Published
##### Identifikatorer
urn:nbn:se:uu:diva-93257 (URN)
Tilgjengelig fra: 2005-09-01 Laget: 2005-09-01bibliografisk kontrollert
2. The pluripolar hull of a graph and fine analytic continuation
Åpne denne publikasjonen i ny fane eller vindu >>The pluripolar hull of a graph and fine analytic continuation
2006 (engelsk)Inngår i: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 44, nr 1, s. 39-60Artikkel i tidsskrift (Fagfellevurdert) 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.

##### Identifikatorer
urn:nbn:se:uu:diva-93258 (URN)10.1007/s11512-005-0004-3 (DOI)000240662300003 ()
Tilgjengelig fra: 2005-09-01 Laget: 2005-09-01 Sist oppdatert: 2017-12-14bibliografisk kontrollert
3. Pluripolar hulls of graphs and extension to fine neighborhoods
Åpne denne publikasjonen i ny fane eller vindu >>Pluripolar hulls of graphs and extension to fine neighborhoods
Manuskript (Annet vitenskapelig)
##### Identifikatorer
urn:nbn:se:uu:diva-93259 (URN)
Tilgjengelig fra: 2005-09-01 Laget: 2005-09-01 Sist oppdatert: 2010-01-13bibliografisk kontrollert
4. A note on the pluripolar hull of a graph and analytic structure
Åpne denne publikasjonen i ny fane eller vindu >>A note on the pluripolar hull of a graph and analytic structure
Manuskript (Annet vitenskapelig)
##### Identifikatorer
urn:nbn:se:uu:diva-93260 (URN)
Tilgjengelig fra: 2005-09-01 Laget: 2005-09-01 Sist oppdatert: 2010-01-13bibliografisk kontrollert
• 46.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och tillämpad matematik.
Complex Absorbing Potential Method: theory and implementation2011Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
• 47.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
Effective equidistribution of horospheres in infinite-volume quotients of SO(n, 1) by geometrically finite groupsManuskript (preprint) (Annet vitenskapelig)

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.

• 48.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
On the equidistribution of translates of orbits of symmetric subgroups in Γ\GManuskript (preprint) (Annet vitenskapelig)

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.

• 49.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
On the rate of equidistribution of expanding translates of horospheres in finite-volume quotients of SL(2,C)2017Inngår i: Journal of Modern Dynamics, ISSN 1930-5311, E-ISSN 1930-532X, Vol. 11, s. 155-188Artikkel i tidsskrift (Fagfellevurdert)

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.

• 50.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
On the rate of equidistribution of expanding translates of horospheres in Γ\GManuskript (preprint) (Annet vitenskapelig)

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.

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