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• 1.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Combinatorial invariance of Kazhdan-Lusztig-Vogan polynomials for fixed point free involutions2018In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 47, no 4, p. 543-560Article in journal (Refereed)

When acts on the flag variety of , the orbits are in bijection with fixed point free involutions in the symmetric group . In this case, the associated Kazhdan-Lusztig-Vogan polynomials can be indexed by pairs of fixed point free involutions , where denotes the Bruhat order on . We prove that these polynomials are combinatorial invariants in the sense that if is a poset isomorphism of upper intervals in the Bruhat order on fixed point free involutions, then for all v amp;gt;= u.

• 2.
Albert Einstein Institute, Golm, Germany..
Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden .
Affine transformation crossed product type algebras and noncommutative surfaces2009In: Operator structures and dynamical systems :: July 21-25 2008, Lorentz Center, Leiden, the Netherlands, satellite conference of the fifth European Congress of Mathematics, American Mathematical Society (AMS), 2009, 503, p. 1-25Chapter in book (Refereed)

Several classes of *-algebras associated to teh action of an affine transformation are considered, and an investigation of the interplay between the different classes is initiated. Connections are established that relate representations of *-algebras, geometry of algebraic surfaces, dynamics of affine transformations, graphs and algebras coming from a quantization procedure of Poisson structures. In particular, algebras related to surgaced being inverse images of fourth order polynomials (in $\mathbb{R}^3$) are studied in detail, and a close link between representation theory and geometric properties is established for compact as well as non-compact surfaces.

• 3.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Complutense University of Madrid. Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Interpolation of Closed Subspaces and Invertibility of Operators2015In: Journal of Analysis and its Applications, ISSN 0232-2064, Vol. 34, no 2015, p. 1-15Article in journal (Refereed)

Let (Y0,Y1) be a Banach couple and let Xj be a closed complemented subspace of Yj, (j = 0,1). We present several results for the general problem of finding necessary and sufficient conditions on the parameters (θ, q) such that the real interpolation space (X0,X1)θ,q is a closed subspace of (Y0,Y1)θ,q. In particular, we establish conditions which are necessary and sufficient for the equality (X0,X1)θ,q = (Y0,Y1)θ,q, with the proof based on a previous result by Asekritova and Kruglyak on invertibility of operators. We also generalize the theorem by Ivanov and Kalton where this problem was solved under several rather restrictive conditions, such as that X1 = Y1 and X0 is a subspace of codimension one in Y0.

• 4.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
On the Branch Loci of Moduli Spaces of Riemann Surfaces2012Doctoral thesis, comprehensive summary (Other academic)

The spaces of conformally equivalent Riemann surfaces, Mg where g ≥ 1, are not manifolds. However the spaces of the weaker Teichmüller equivalence, Tg are known to be manifolds. The Teichmüller space Tg is the universal covering of Mg and Mg is the quotient space by the action of the modular group. This gives Mg an orbifold structure with a branch locus Bg. The branch loci Bg can be identified with Riemann surfaces admitting non-trivial automorphisms for surfaces of genus g ≥ 3. In this thesis we consider the topological structure of Bg. We study the connectedness of the branch loci in general by considering families of isolated strata and we we establish that connectedness is a phenomenon for low genera. Further, we give the orbifold structure of the branch locus of surfaces of genus 4 and genus 5 in particular, by studying the equisymmetric stratification of the branch locus.

Paper 1. In this paper we show that the strata corresponding to actions of order 2 and 3 belong to the same connected component for arbitrary genera. Further we show that the branch locus is connected with the exception of one isolated point for genera 5 and 6, it is connected for genus 7 and it is connected with the exception of two isolated points for genus 8.

Paper 2. This paper contains a collection of results regarding components of the branch loci, some of them proved in detail in other papers. It is shown that for any integer d if p is a prime such that p > (d + 2)2, there there exist isolated strata of dimension d in the moduli space of Riemann surfaces of genus (d + 1)(p − 1)/2. It is also shown that if we consider Riemann surfaces as Klein surfaces, the branch loci are connected for every genera due to reflections.

Paper 3. Here we consider surfaces of genus 4 and 5. Here we study the automorphism groups of Riemann surfaces of genus 4 and 5 up to topological equivalence and determine the complete structure of the equisymmetric stratification of the branch locus.

Paper 4. In this paper we establish that the connectedness of the branch loci is a phenomenon for low genera. More precisely we prove that the only genera g where Bg is connected are g = 3, 4, 13, 17, 19, 59.

1. On the connectedness of the branch locus of the moduli space of Riemann surfaces
Open this publication in new window or tab >>On the connectedness of the branch locus of the moduli space of Riemann surfaces
2010 (English)In: REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS, ISSN 1578-7303, Vol. 104, no 1, p. 81-86Article in journal (Refereed) Published
Abstract [en]

The moduli space M-g of compact Riemann surfaces of genus g has the structure of an orbifold and the set of singular points of such orbifold is the branch locus B-g. In this article we present some results related with the topology of B-g. We study the connectedness of B-g for g andlt;= 8, the existence of isolated equisymmetric strata in the branch loci and finally we stablish the connectedness of the branch locus of the moduli space of Riemann surfaces considered as Klein surfaces. We just sketch the proof of some of the results; complete proofs will be published elsewhere.

Place, publisher, year, edition, pages
REAL ACAD CIENCIAS EXACTAS FISICAS and NATURALES, CALLE VALVERDE 22, MADRID, 28004, SPAIN, 2010
Keywords
Riemann surface, moduli space, automorphism
Mathematics
Identifiers
urn:nbn:se:liu:diva-54848 (URN)10.5052/RACSAM.2010.08 (DOI)000276304700008 ()
Note
Original Publication: Gabriel Bartolini, Antonio F Costa, Milagros Izquierdo and Ana M Porto, On the connectedness of the branch locus of the moduli space of Riemann surfaces, 2010, REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS, (104), 1, 81-86. http://dx.doi.org/10.5052/RACSAM.2010.08 Copyright: Real Academia de Ciencias, Espana Available from: 2010-04-16 Created: 2010-04-16 Last updated: 2015-03-09
2. On the connectedness of the branch locus of the moduli space of Riemann surfaces of low genus
Open this publication in new window or tab >>On the connectedness of the branch locus of the moduli space of Riemann surfaces of low genus
2012 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 140, no 1, p. 35-45Article in journal (Refereed) Published
Abstract [en]

Let be an integer and let , where denotes the moduli space of compact Riemann surfaces of genus . Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space, we prove that the subloci corresponding to Riemann surfaces with automorphism groups isomorphic to cyclic groups of order 2 and 3 belong to the same connected component. We also prove the connectedness of for and with the exception of the isolated points given by Kulkarni.

Place, publisher, year, edition, pages
American Mathematical Society, 2012
Keywords
Moduli spaces, Teichmüller modular group, automorphism group
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-73196 (URN)10.1090/S0002-9939-2011-10881-5 (DOI)000299596000004 ()
Funder
Swedish Research Council, 621-2007-6240 Available from: 2011-12-21 Created: 2011-12-21 Last updated: 2018-09-01
3. On the Orbifold Structure of the Moduli Space of Riemann Surfaces of Genera Four and Five
Open this publication in new window or tab >>On the Orbifold Structure of the Moduli Space of Riemann Surfaces of Genera Four and Five
2014 (English)In: REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS, ISSN 1578-7303, Vol. 108, no 2, p. 769-793Article in journal (Refereed) Published
Abstract [en]

The moduli space Mg, of compact Riemann surfaces of genus g has orbifold structure since Mg is the quotient space of the Tiechmüller space by the action of the mapping class group. Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space we find the orbifold structure of the moduli spaces of Riemann surfaces of genera 4 and 5.

Place, publisher, year, edition, pages
Springer, 2014
Mathematics
Identifiers
urn:nbn:se:liu:diva-78016 (URN)10.1007/s13398-013-0140-8 (DOI)000340875100032 ()
Available from: 2012-06-04 Created: 2012-06-04 Last updated: 2017-04-10Bibliographically approved
4. On the connected branch loci of moduli spaces
Open this publication in new window or tab >>On the connected branch loci of moduli spaces
Abstract [en]

The moduli space Mg of compact Riemann surfaces of genus g has orbifold structure and the set of singular points of such orbifold is the branch locus Bg. In this article we show that Bg is connected exactly for genera three, four, thirteen, seventeen, nineteen and fitfynine by the use automorphisms of order 5 and 7 of Riemann surfaces, and calculations with GAP for some small genera.

Mathematics
Identifiers
urn:nbn:se:liu:diva-78018 (URN)
Available from: 2012-06-04 Created: 2012-06-04 Last updated: 2012-06-05Bibliographically approved
• 5.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Departamento Matematicas Fundamentales, UNED, Madrid, Spain. Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
On automorphisms groups of cyclic p-gonal Riemann surfaces2013In: Journal of symbolic computation, ISSN 0747-7171, E-ISSN 1095-855X, Vol. 57, p. 61-69Article in journal (Refereed)

In this work we obtain the group of conformal and anticonformal automorphisms of real cyclic p-gonal Riemann surfaces, where p⩾3p⩾3 is a prime integer and the genus of the surfaces is at least (p−1)2+1(p−1)2+1. We use Fuchsian and NEC groups, and cohomology of finite groups.

• 6.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Departamento de Matematicas Fundamentales, UNED, Madrid, Spain. Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
On isolated strata of pentagonal Riemann surfaces in the branch locus of moduli spaces2012In: Contemporary Mathematics, ISSN 0271-4132, E-ISSN 1098-3627, Vol. 572, p. 19-24Article in journal (Refereed)
• 7.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Departamento de Matematicas Fundamentales, UNED, Madrid, Spain. Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
On isolated strata of p-gonal Riemann surfaces in the branch locus of moduli spaces2012In: Albanian Journal of Mathematics, ISSN 1930-1235, E-ISSN 1930-1235, Vol. 6, p. 11-19Article in journal (Refereed)
• 8.
Linköping University, Department of Science and Technology, Communications and Transport Systems. Linköping University, Faculty of Science & Engineering. Polish Academic Science, Poland.
Separable quantizations of Stackel systems2016In: Annals of Physics, ISSN 0003-4916, E-ISSN 1096-035X, Vol. 371, p. 460-477Article in journal (Refereed)

In this article we prove that many Hamiltonian systems that cannot be separably quantized in the classical approach of Robertson and Eisenhart can be separably quantized if we extend the class of admissible quantizations through a suitable choice of Riemann space adapted to the Poisson geometry of the system. Actually, in this article we prove that for every quadratic in momenta Stackel system (defined on 2n dimensional Poisson manifold) for which Stackel matrix consists of monomials in position coordinates there exist infinitely many quantizations - parametrized by n arbitrary functions - that turn this system into a quantum separable Stackel system. (C) 2016 Elsevier Inc. All rights reserved.

• 9.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Strain Energy of Bézier Surfaces2018Independent thesis Basic level (degree of Bachelor), 10,5 credits / 16 HE creditsStudent thesis

Bézier curves and surfaces are used to great success in computer-aided design and finite element modelling, among other things, due to their tendency of being mathematically convenient to use. This thesis explores the different properties that make Bézier surfaces the strong tool that it is. This requires a closer look at Bernstein polynomials and the de Castiljau algorithm. To illustrate some of these properties, the strain energy of a Bézier surface is calculated. This demands an understanding of what a surface is, which is why this thesis also covers some elementary theory regarding parametrized curves and surface geometry, including the first and second fundamental forms.

• 10.
Departamento de Matematicas Fundamentales, UNED.
Departamento de Matematicas Fundamentales, UNED. Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
On Riemann surfaces with 4g automorphisms2017In: Topology and its Applications, ISSN 0166-8641, E-ISSN 1879-3207, Vol. 218, no 1, article id 18Article in journal (Refereed)

We determine, for all genus g≥2g≥2 the Riemann surfaces of genus g with exactly 4g automorphisms. For g  ≠ 3,6,12,153,6,12,15 or 30, these surfaces form a real Riemann surface FgFg in the moduli space MgMg: the Riemann sphere with three punctures. We obtain the automorphism groups and extended automorphism groups of the surfaces in the family. Furthermore we determine the topological types of the real forms of real Riemann surfaces in FgFg. The set of real Riemann surfaces in FgFg consists of three intervals its closure in the Deligne–Mumford compactification of MgMg is a closed Jordan curve. We describe the nodal surfaces that are limits of real Riemann surfaces in Fg

• 11.
Departamento de Matematicas Fundamentales, UNED.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. Department of Mathematics, University of Fribourg, Switzerland.
Connecting p-gonal loci in the compactification of moduli space2015In: Revista Matemática Complutense, ISSN 1139-1138, Vol. 28, no 2, p. 469-486Article in journal (Refereed)

Consider the moduli space M g of Riemann surfaces of genusg≥2 and its Deligne-Munford compactification M g ¯ . We are interested in the branch locus B g for g>2 , i.e., the subset of M g consisting of surfaces with automorphisms. It is well-known that the set of hyperelliptic surfaces (the hyperelliptic locus) is connected in M g but the set of (cyclic) trigonal surfaces is not. By contrast, we show that for g≥5 the set of (cyclic) trigonal surfaces is connected in M g ¯ . To do so we exhibit an explicit nodal surface that lies in the completion of every equisymmetric set of 3-gonal Riemann surfaces. For p>3 the connectivity of the p -gonal loci becomes more involved. We show that for p≥11 prime and genus g=p−1 there are one-dimensional strata of cyclic p -gonal surfaces that are completely isolated in the completion B g ¯ of the branch locus in M g ¯ .

• 12.
Departamento de Matematicas Fundamentales, UNED.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. Departamento de Matematicas Fundamentales, UNED.
Maximal and non-maximal NEC and Fuchsian groups uniformizing Klein and Riemann surfaces2014In: Riemann and Klein Surfaces, Automorphisms, Symmetries and Moduli Spaces / [ed] Milagros Izquierdo, S. Allen Broughton, Antonio F. Costa, Rubí E. Rodríguez, American Mathematical Society (AMS), 2014, Vol. 629, p. 107-118Chapter in book (Refereed)
• 13.
Departamento de Matematicas Fundamentales, UNED.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. Departamento de Matematicas Fundamentales, UNED.
On the connectedness of the branch loci of moduli spaces of orientable Klein surfaces2015In: Geometriae Dedicata, ISSN 0046-5755, E-ISSN 1572-9168, Vol. 177, no 1, p. 149-164Article in journal (Refereed)

Let M K (g,+,k) be the moduli space of orientable Klein surfaces of genus g with k boundary components (see Alling and Greenleaf in Lecture notes in mathematics, vol 219. Springer, Berlin, 1971; Natanzon in Russ Math Surv 45(6):53–108, 1990). The space M K (g,+,k) has a natural orbifold structure with singular locus B K (g,+,k) . If g>2 or k>0 and 2g+k>3 the set B K (g,+,k) consists of the Klein surfaces admitting non-trivial symmetries and we prove that, in this case, the singular locus is connected.

• 14.
Departamento de Matematicas Fundamentales, UNED.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. Departamento de Matematicas Fundamentales, UNED.
On the connectedness of the branch locus of moduli space of hyperelliptic Klein surfaces with one boundary2017In: International Journal of Mathematics, ISSN 0129-167X, ISSN 0129-167X, Vol. 28, no 5, article id 1750038Article in journal (Refereed)
• 15. Eriksson-Bique, sylvester
Optimal Geometric Flows via Dual Programs2014Conference paper (Refereed)
• 16.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Supersolvability and the Koszul property of root ideal arrangements2016In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 144, p. 1401-1413Article in journal (Refereed)

A root ideal arrangement A_I is the set of reflecting hyperplanes corresponding to the roots in an order ideal I of the root poset on the positive roots of a finite crystallographic root system. A characterisation of supersolvable root ideal arrangements is obtained. Namely, A_I is supersolvable if and only if I is chain peelable, meaning that it is possible to reach the empty poset from I by in each step removing a maximal chain which is also an order filter. In particular, supersolvability is preserved under taking subideals. We identify the maximal ideals that correspond to non-supersolvable arrangements. There are essentially two such ideals, one in type D_4 and one in type F_4. By showing that A_I is not line-closed if I contains one of these, we deduce that the Orlik-Solomon algebra OS(A_I) has the Koszul property if and only if A_I is supersolvable.

• 17.
Broughton, S AllenRose-Hulman Institute of Technology, USA.Costa, Antonio F.Departamento de Matematicas Fundamentales, Universidad Nacional de Educación a Distancia, Madrid, Spain.Rodriguez, Rubi E.Departamento de Matemática, Universidad de la Frontera, Temuco, Chile.
Riemann and Klein surfaces, automorphisms, symmetries and moduli spaces: conference in honor of Emilio Bujalance on Riemann and Klein surfaces, symmetries and moduli spaces, June 24-28, 2013, Linköping University, Linköping, Sweden2014Collection (editor) (Refereed)
• 18.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Högskolan i Skövde.
Isometric Point-Circle Configurations on Surfaces from Uniform Maps2016In: Springer Proceedings in Mathematics and Statistics, ISSN 2194-1009, Vol. 159, p. 201-212Article in journal (Refereed)

We embed neighborhood geometries of graphs on surfaces as point-circle configurations. We give examples coming from regular maps on surfaces with a maximum number of automorphisms for their genus, and survey geometric realization of pentagonal geometries coming from Moore graphs. An infinite family of point-circle v4'>v4v4 configurations on p-gonal surfaces with two p-gonal morphisms is given. The image of these configurations on the sphere under the two p-gonal morphisms is also described.

• 19.
Diskret krökning, en jämförelse2012Independent thesis Basic level (degree of Bachelor), 10,5 credits / 16 HE creditsStudent thesis

In this thesis we analyze and compare two different methods for approximating the Gauss and mean curvature on a surface, which is given as a set of points. It is important to find a method that agrees well with the analytic Gauss and mean curvatures and guarantees robust estimations. There is a great interest in Gauss and mean curvature since these two curvatures give information about the local geometry of the surface around the point at which these curvatures are calculated.

The thesis begins with a short overview of differential theory and then the methods are explained and described. The reason for this is to give the reader an understanding of the theory before explaining the methods.

The first method is called Bézier surfaces, which interpolates the given points. These surfaces are differentiable which makes it possible to approximate the Gauss and mean curvature, and are therefore very well suited for our problem.

The second method comes from the research article ``Discrete Differential-Geometry Operators for Triangulated 2-Manifolds'' by Mark Meyer, Mathieu Desbrun, Peter Schröder and Alan H. Barr. Their algorithm requires a triangulated surface, which itself is a hard problem to solve (at least if one has requirements on the triangulation). Their approximations of the Gauss and mean curvatures use a well chosen area around the point, and the Gauss curvature also makes use of the Gauss-Bonnet theorem.

My simulations show that Bézier surfaces approximate both Gauss and mean curvature well, and the approximations seem to converge to the analytic value when the information gets better. The articles algorithm also works well for approximating both curvatures, though this method seems to depend somewhat on the triangulation. This gives some requirements on the triangulation and will therefore be a harder problem to solve. The approximations do not converge when given a triangulation with obtuse triangles, though it shows signs to do so.

• 20.
Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Idempotent Geometry in Generic Algebras2018In: Advances in Applied Clifford Algebras, ISSN 0188-7009, E-ISSN 1661-4909, Vol. 28, no 5, article id UNSP 84Article in journal (Refereed)

Using the syzygy method, established in our earlier paper (Krasnov and Tkachev, 2018), we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.

• 21.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Admissible transformations and the group classification of Schrödinger equations2017Doctoral thesis, comprehensive summary (Other academic)

We study admissible transformations and solve group classification problems for various classes of linear and nonlinear Schrödinger equations with an arbitrary number n of space variables.

The aim of the thesis is twofold. The first is the construction of the new theory of uniform seminormalized classes of differential equations and its application to solving group classification problems for these classes. Point transformations connecting two equations (source and target) from the class under study may have special properties of semi-normalization. This makes the group classification of that class using the algebraic method more involved. To extend this method we introduce the new notion of uniformly semi-normalized classes. Various types of uniform semi-normalization are studied: with respect to the corresponding equivalence group, with respect to a proper subgroup of the equivalence group as well as the corresponding types of weak uniform semi-normalization. An important kind of uniform semi-normalization is given by classes of homogeneous linear differential equations, which we call uniform semi-normalization with respect to linear superposition of solutions.

The class of linear Schrödinger equations with complex potentials is of this type and its group classification can be effectively carried out within the framework of the uniform semi-normalization. Computing the equivalence groupoid and the equivalence group of this class, we show that it is uniformly seminormalized with respect to linear superposition of solutions. This allow us to apply the version of the algebraic method for uniformly semi-normalized classes and to reduce the group classification of this class to the classification of appropriate subalgebras of its equivalence algebra. To single out the classification cases, integers that are invariant under equivalence transformations are introduced. The complete group classification of linear Schrödinger equations is carried out for the cases n = 1 and n = 2.

The second aim is to study group classification problem for classes of generalized nonlinear Schrödinger equations which are not uniformly semi-normalized. We find their equivalence groupoids and their equivalence groups and then conclude whether these classes are normalized or not. The most appealing classes are the class of nonlinear Schrödinger equations with potentials and modular nonlinearities and the class of generalized Schrödinger equations with complex-valued and, in general, coefficients of Laplacian term. Both these classes are not normalized. The first is partitioned into an infinite number of disjoint normalized subclasses of three kinds: logarithmic nonlinearity, power nonlinearity and general modular nonlinearity. The properties of the Lie invariance algebras of equations from each subclass are studied for arbitrary space dimension n, and the complete group classification is carried out for each subclass in dimension (1+2). The second class is successively reduced into subclasses until we reach the subclass of (1+1)-dimensional linear Schrödinger equations with variable mass, which also turns out to be non-normalized. We prove that this class is mapped by a family of point transformations to the class of (1+1)-dimensional linear Schrödinger equations with unique constant mass.

1. Equivalence groupoid for (1+2)-dimensional linear Schrodinger equations with complex potentials
Open this publication in new window or tab >>Equivalence groupoid for (1+2)-dimensional linear Schrodinger equations with complex potentials
2015 (English)In: SEVENTH INTERNATIONAL WORKSHOP: GROUP ANALYSIS OF DIFFERENTIAL EQUATIONS AND INTEGRABLE SYSTEMS (GADEISVII), IOP Publishing: Conference Series / Institute of Physics (IoP) , 2015, Vol. 621, no UNSP 012008, p. UNSP 012008-Conference paper, Published paper (Refereed)
Abstract [en]

We describe admissible point transformations in the class of (1+2)-dimensional linear Schrodinger equations with complex potentials. We prove that any point transformation connecting two equations from this class is the composition of a linear superposition transformation of the corresponding initial equation and an equivalence transformation of the class. This shows that the class under study is semi-normalized.

Place, publisher, year, edition, pages
IOP Publishing: Conference Series / Institute of Physics (IoP), 2015
Series
Journal of Physics Conference Series, ISSN 1742-6588 ; 621
Mathematics
Identifiers
urn:nbn:se:liu:diva-120668 (URN)10.1088/1742-6596/621/1/012008 (DOI)000357939100008 ()
Conference
7th International Workshop on Group Analysis of Differential Equations and Integrable Systems (GADEIS)
Available from: 2015-08-20 Created: 2015-08-20 Last updated: 2017-05-15
• 22.
Univ Minnesota, MN 55455 USA.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Polyharmonic capacity and Wiener test of higher order2018In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 211, no 2, p. 779-853Article in journal (Refereed)

In the present paper we establish the Wiener test for boundary regularity of the solutions to the polyharmonic operator. We introduce a new notion of polyharmonic capacity and demonstrate necessary and sufficient conditions on the capacity of the domain responsible for the regularity of a polyharmonic function near a boundary point. In the case of the Laplacian the test for regularity of a boundary point is the celebrated Wiener criterion of 1924. It was extended to the biharmonic case in dimension three by Mayboroda and Mazya (Invent Math 175(2):287-334, 2009). As a preliminary stage of this work, in Mayboroda and Mazya (Invent Math 196(1):168, 2014) we demonstrated boundedness of the appropriate derivatives of solutions to the polyharmonic problem in arbitrary domains, accompanied by sharp estimates on the Green function. The present work pioneers a new version of capacity and establishes the Wiener test in the full generality of the polyharmonic equation of arbitrary order.

• 23.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Some properties of tube minimal surfaces of arbitrary codimension1989In: Sbornik. Mathematics, ISSN 1064-5616, E-ISSN 1468-4802, Vol. 180, p. 1278-1295Article in journal (Refereed)

We establish the finiteness of life-time of a tubular minimal surface of an arbitrary codimension.

• 24.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
The structure in the large of externally complete minimal surfaces in \$R^3\$1987In: Izvestiia vysshikh uchebnykh zavedenii. Matematika (Izvestija Vuzov. Mathematics), ISSN 0021-3446, E-ISSN 2076-4626, Vol. 31, no 7, p. 37-35Article in journal (Refereed)

We establish the absence of nontrivial noncompact parametric minimal surfaces in the Euclidean 3-space satisfying some additional geometrical properties.

• 25.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Sets with the Baire property in topologies formed from a given topology and ideals of sets2017In: Questions and answers in General Topology, ISSN 0918-4732, Vol. 35, no 1, p. 59-76Article in journal (Refereed)

Let X be a set, τ1, τ2 topologies on X and Bp(X, τi) the family of all subsets of X possessing the Baire property in (X, τi), i = 1, 2. In this paper we study conditions on τ1 and τ2 that imply a relationship (for example, inclusion or equality) between the families Bp(X, τ1) and Bp(X, τ2). We are mostly interested in the case where the topology τ2 is formed with the help of a local function defined by the topology τ1 and an ideal of sets I on X. We also consider several applications of the local function defined by the usual topology on the reals and the ideal of all meager sets there, for proving some known facts.

• 26.
Department of Mathematics, Central Connecticut State University, New Britain, USA.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Algebraic CMC hypersurfaces of order 3 in Euclidean spaces2019In: Journal of Geometry, ISSN 0047-2468, E-ISSN 1420-8997, Vol. 110, no 1, article id 110:6Article in journal (Refereed)

Understanding and finding of general algebraic constant mean curvature surfaces in the Euclidean spaces is a hard open problem. The basic examples are the standard spheres and the round cylinders, all defined by a polynomial of degree 2. In this paper, we prove that thereare no algebraic hypersurfaces of degree 3 in higherdimensional (n>2) Euclidean spaces, with nonzero constant mean curvature.

• 27.
University of Skovde, Sweden.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Geometric point-circle pentagonal geometries from Moore graphs2016In: Ars Mathematica Contemporanea, ISSN 1855-3966, Vol. 11, no 1, p. 215-229Article in journal (Refereed)

We construct isometric point-circle configurations on surfaces from uniform maps. This gives one geometric realisation in terms of points and circles of the Desargues configuration in the real projective plane, and three distinct geometric realisations of the pentagonal geometry with seven points on each line and seven lines through each point on three distinct dianalytic surfaces of genus 57. We also give a geometric realisation of the latter pentagonal geometry in terms of points and hyperspheres in 24 dimensional Euclidean space. From these, we also obtain geometric realisations in terms of points and circles (or hyperspheres) of pentagonal geometries with k circles (hyperspheres) through each point and k 1 points on each circle (hypersphere).

• 28.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Chonbuk National University, South Korea. Shimane University, Japan.
On hereditarily reversible spaces2017In: Topology and its Applications, ISSN 0166-8641, E-ISSN 1879-3207, Vol. 225, p. 53-66Article in journal (Refereed)

In [15] Rajagopalan and Wilansky called a space reversible if each continuous bijection of the space onto itself is a homeomorphism. They called also a space hereditarily reversible if each its subspace is reversible. We characterize the hereditarily reversible spaces in several classes of topologicals spaces, in particular, in the class of Hausdorff spaces of the first countability and in some subclass of the class of locally finite T-0-spaces relevant to digital topology. Besides we suggest different examples of (non-)reversible and (non-)hereditarily reversible spaces. (C) 2017 Elsevier B.V. All rights reserved.

• 29.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Chonbuk National University, South Korea. Shimane University, Japan.
The Small Inductive Dimension of Subsets of Alexandroff Spaces2016In: Filomat, ISSN 0354-5180, Vol. 30, no 11, p. 3007-3014Article in journal (Refereed)

We describe the small inductive dimension ind in the class of Alexandroff spaces by the use of some standard spaces. Then for ind we suggest decomposition, sum and product theorems in the class. The sum and product theorems there we prove even for the small transfinite inductive dimension trind. As an application of that, for each positive integers k, n such that k amp;lt;= n we get a simple description in terms of even and odd numbers of the family S(k, n) = {S subset of K-n : vertical bar S vertical bar = k + 1 and ind S = k}, where K is the Khalimsky line.

• 30.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
A note on isoparametric polynomials2014In: Analysis and Mathematical Physics, ISSN 1664-2368, E-ISSN 1664-235X, Vol. 4, no 3, p. 237-245Article in journal (Refereed)

We show that any homogeneous polynomial solution of the eiconal type equation |∇F(x)|2 = m2|x|2m-2, m ≥ 1, is either a radially symmetric polynomial F(x) = ±|x|m (for even values of m’s) or it is a composition of a Chebychev polynomial and a Cartan–Münzner polynomial.

• 31.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
A remark on the Jörgens-Calabi-Pogorelov theorem1995In: Doklady. Mathematics, ISSN 1064-5624, E-ISSN 1531-8362, Vol. 340, p. 317-318Article in journal (Refereed)
• 32.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics.
A sharp lower bound for the first eigenvalue on a minimal surface1993In: Mathematical notes of the Academy of Sciences of the USSR, ISSN 0001-4346, E-ISSN 1573-8876, Vol. 54, p. 835-840Article in journal (Refereed)
• 33.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
A theorem on the clearance radius for minimal surfaces1996In: Mathematical notes of the Academy of Sciences of the USSR, ISSN 0001-4346, E-ISSN 1573-8876, Vol. 59, no 5-6, p. 657-660Article in journal (Refereed)

A least upper bound for the inner radius of an opening in a complete minimal hypersurface contained in a parallel layer is given.

• 34.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics.
Disjoint minimal graphs2009In: Annals of Global Analysis and Geometry, ISSN 0232-704X, E-ISSN 1572-9060, Vol. 35, no 2, p. 139-155Article in journal (Refereed)

We prove that the number s(n) of disjoint minimal graphs supported on domains in R^n is bounded by e(n+1)2. In the two-dimensional case we show that s(2)=2 or 3 which (the well-known Meeks conjecture asserts that this number must be 2).

• 35.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
External geometry of p-minimal surfaces1997In: Geometry from Pacific Rim / [ed] Berrick, A. Jon / Loo, Bonaventure / Wang, Hong-Yu, Walter de Gruyter, 1997, , p. 363–375p. 363-376Conference paper (Refereed)

We introduce a new class of p-minimal surfaces. We shaw that the Gauss map of a two-dimensional p-minimal surface in R3 is quasiconformal . A counterpart to Bernstein's celebrated result about entire solutions of the minimal surface equation is obtained. A study of tubular p-minimal hypersurfaces is included.

• 36.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Finiteness of the number of ends of minimal submanifolds in Euclidean space1994In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 82, p. 313-330Article in journal (Refereed)

We prove a version of the well-known Denjoy-Ahlfors theorem about the number of asymptotic values of an entire function for properly immersed minimal surfaces of arbitrary codimension in ℝN. The finiteness of the number of ends is proved for minimal submanifolds with finite projective volume. We show, as a corollary, that a minimal surface of codimensionn meeting anyn-plane passing through the origin in at mostk points has no morec(n, N)k ends.

• 37.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Life-time of minimal tubes and coefficients of univalent functions in a circular ring1995In: Bulletin de la Société des sciences et des lettres de Lódz, ISSN 0459-6854, Vol. 20, p. 19-26Article in journal (Refereed)

We obtain various estimates of the life-time of two-dimensional min-imal tubes inR3by potential theory methods.

• 38.
Royal Institute of Technology, Stockholm, Sweden.
Minimal cubic cones via Clifford algebras2010In: Complex Analysis and Operator Theory, ISSN 1661-8254, E-ISSN 1661-8262, Vol. 4, no 3, p. 685-700Article in journal (Refereed)

In this paper, we construct two infinite families of algebraic minimal cones in R n . The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes of congruent minimal cubics are in one to one correspondence with those of geometrically equivalent Clifford systems. As a byproduct, we prove that for any n ≥ 4, n ≠ 16k + 1, there is at least one minimal cone in R n given by an irreducible homogeneous cubic polynomial. The second family consists of minimal cones in R m 2 , m ≥ 2, defined by an irreducible homogeneous polynomial of degree m. These examples provide particular answers to the questions on algebraic minimal cones in R n posed by Wu-Yi Hsiang in the 1960s.

• 39.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Minimal tubes of finite integral curvature1998In: Siberian mathematical journal, ISSN 0037-4466, E-ISSN 1573-9260, Vol. 39, no 1, p. 159-167Article in journal (Refereed)

The author defines a tube to be an immersed submanifold u:Mp→Rn+1 and the interval of existence τ(Mp) to be the interval of those t for which the intersection Σt of u(Mp) with the hyperplane xn+1=t in Rn+1 is nonempty and compact. The length of τ(Mp) is called the time of existence of the tube. The tube is minimal if u is a minimal immersion. Denote by vT an orthogonal projection of v into the tangent space of M, ν=eTn+1/∥eTn+1∥, and introduce a vector J, called a vector-flow with coordinates, Jk=∫Σt((ek)T,ν),1≤k≤n+1. The angle between J and en+1 is denoted by α. The main result of the article under review is the following estimate: |τ(M)|≤G(M)∥J∥cos(α)16α2, where G(M) denotes the absolute integral Gauss curvature of M.

• 40.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Some estimates for the mean curvature of graphs over domains in \$R^n\$1991In: Doklady. Mathematics, ISSN 1064-5624, E-ISSN 1531-8362, Vol. 42, no 2, p. 387-390Article in journal (Refereed)

Let f be a C2-solution in a domain ΩRn of the equation

where H is a nondecreasing function of a real variable. In the particular case H(t)=a+bt, b>0, the preceding differential equation describes the capillarity in a tube of liquid [cf. R. Finn, Z. Angew. Math. Mech. 61 (1981), no. 3, 165–173; MR0626020]. The author establishes that P|H[f(x)]|pdxcap(P,Ω;Ω), where Ω is the boundary of Ω and cap(P,Ω,Ω) is the conformal capacity of a condenser (P,Ω;Ω) [cf., e.g., Yu. G. Reshetnyak, Spatial mappings with bounded distortion (Russian), see p. 52, "Nauka'' Sibirsk. Otdel., Novosibirsk, 1982; MR0665590]. The author also obtains estimates of the form supxΩ{|H[f(x)]|⋅dist(x,Ω)}p/n (1pn) for some categories of domains.

• 41.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Some estimates for the mean curvature of nonparametric surfaces defined over domains in \$R^n\$1994In: Journal of Mathematical Sciences, ISSN 1072-3374, E-ISSN 1573-8795, Vol. 72, no 4, p. 3250-3260Article in journal (Refereed)
• 42.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Denjoy-Ahlfors theorem for harmonic functions on Riemannian manifolds and external structure of minimal surfaces1996In: Communications in analysis and geometry, ISSN 1019-8385, E-ISSN 1944-9992, Vol. 4, no 4, p. 547-587Article in journal (Refereed)

We extend the well-known Denjoy-Ahlfors theorem about the number of different asymptotic tracts of a holomorphic function to subharmonic functions on arbitrary Riemannian manifolds. We obtain some versions of the Liouville theorem for Aa-harmonic functions without the geodesic completeness requirement on a manifold. Moreover, the upper estimate of the topological index of the height function of a minimal surface in Rn has been established and, as a consequence, a new prove of the Bernstein's theorem has been derived. Other applications to the theory of minimal surfaces are also discussed.

• 43.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
On the Gauss map of embedded minimal tubes1999In: Note di Matematica, ISSN 1123-2536, E-ISSN 1590-0932, Vol. 19, no 1, p. 7-17Article in journal (Refereed)

A surface is called a tube if its level-sets with respect to some coordinate function (the axis of the surface) are compact. Any tube of zero mean curvature has an invariant, the so-called flow vector. We study how the geometry of the Gaussian image of a higher-dimensional minimal tube M is controlled by the angle alpha(M) between the axis and the flow vector of M. We prove that the diameter of the Gauss image of M is at least 2alpha(M). As a consequence we derive an estimate on the length of a two-dimensional minimal tube M in terms of alpha(\M) and the total Gaussian curvature of M.

• 44.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Doubly periodic maximal surfaces with singularities2002In: Siberian Advances in Mathematics, ISSN 1055-1344, Vol. 12, no 1, p. 77-91Article in journal (Refereed)

We construct and study a family of double-periodic almost entire solutions of the maximal surface equation. The solutions are parameterized by a submanifold of 3×3-matrices (the so-called generating matrices). We show that the constructed solutions are either space-like or of mixed type with the light-cone type isolated singularities.

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