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• 51.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
On certain minimax problems and the Pontryagin maximum principle2008Report (Other academic)

• 52.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Simulering av formfyllnadsförlopp vid formsprutning av plast enligt avståndsmodellen2004In: Workshop i tillämpad matematik,2004, 2004Conference paper (Other academic)
• 53.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Department of Mathematics, Univ. of California, Santa Barbara, Santa Barbara, CA 93106. Department of Mathematics, P.O. Box 35, FIN-40014 Jyväskylä, Finland.
A tour of the theory of absolutely minimizing functions2004In: Bulletin of the American Mathematical Society, ISSN 0273-0979, E-ISSN 1088-9485, Vol. 41, no 4, p. 439-505Article in journal (Refereed)

These notes are intended to be a rather complete and self-contained exposition of the theory of absolutely minimizing Lipschitz extensions, presented in detail and in a form accessible to readers without any prior knowledge of the subject. In particular, we improve known results regarding existence via arguments that are simpler than those that can be found in the literature. We present a proof of the main known uniqueness result which is largely self-contained and does not rely on the theory of viscosity solutions. A unifying idea in our approach is the use of cone functions. This elementary geometric device renders the theory versatile and transparent. A number of tools and issues routinely encountered in the theory of elliptic partial differential equations are illustrated here in an especially clean manner, free from burdensome technicalities - indeed, usually free from partial differential equations themselves. These include a priori continuity estimates, the Harnack inequality, Perron's method for proving existence results, uniqueness and regularity questions, and some basic tools of viscosity solution theory. We believe that our presentation provides a unified summary of the existing theory as well as new results of interest to experts and researchers and, at the same time, a source which can be used for introducing students to some significant analytical tools.

• 54.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
A Tour of the Theory of Absolutely Minimizing Functions2004Report (Other academic)
• 55.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Mathematics University of California. Mathematics University of Jyväskylä.
A TOUR OF THE THEORY OF ABSOLUTELY MINIMIZING FUNCTIONS2004In: Bulletin of the American Mathematical Society, ISSN 0273-0979, E-ISSN 1088-9485, Vol. 41, no 4, p. 439-505Article in journal (Refereed)

A detailed analysis of the class of absolutely minimizing functions in Euclidean spaces and the relationship to the infinity Laplace equation

• 56.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Linkoping Univ, Dept Math, Linkoping, Sweden Univ Calif Berkeley, Dept Math, Berkeley, CA 94720 USA.
An asymptotic model for compression molding2002In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 51, no 1Article in journal (Refereed)

We discuss an idealized model for compression molding, had by taking an asymptotic limit for highly non-Newtonian materials. We interpret the changing pressure distributions as being dictated by a Monge-Kantorovich mass transfer on a fast time scale, and thereby derive a nonlocal geometric law of motion for the air/plastic interface.

• 57.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
A note on transformations of edge colorings of bipartite graphs2009In: JOURNAL OF COMBINATORIAL THEORY SERIES B, ISSN 0095-8956 , Vol. 99, no 5, p. 814-818Article in journal (Refereed)

The author and A. Mirumian proved the following theorem: Let G be a bipartite graph with maximum degree Delta and let t, n be integers, t andgt;= n andgt;= Delta. Then it is possible to obtain, from one proper edge t-coloring of G, any proper edge n-coloring of G using only transformations of 2-colored and 3-colored subgraphs such that the intermediate colorings are also proper. In this note we show that if t andgt; Delta then we can transform f to g using only transformations of 2-colored subgraphs. We also correct the algorithm suggested in [A.S. Asratian, Short solution of Kotzigs problem for bipartite graphs, J. Combin. Theory Set. B 74 (1998) 160-168] for transformation of f to g in the case when t = n = Delta and G is regular.

• 58.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
A new local condition for a graph to be hamiltonian2004In: CTW04 Workshop on graph theory and combinatorial optimization,2004, 2004, p. 47-49Conference paper (Other academic)
• 59.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Interval edge colorings of graphs and related problems2008In: Operations research symposium in the honour of T. Liebling and D. de Werra,2008, 2008Conference paper (Other academic)
• 60.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
New local conditions for a graph to be hamiltonian2006In: Graphs and Combinatorics, ISSN 0911-0119, E-ISSN 1435-5914, Vol. 22, no 2, p. 153-160Article in journal (Refereed)

For a vertex w of a graph G the ball of radius 2 centered at w is the subgraph of G induced by the set M 2(w) of all vertices whose distance from w does not exceed 2. We prove the following theorem: Let G be a connected graph where every ball of radius 2 is 2-connected and d(u)+d(v)|M 2(w)|-1 for every induced path uwv. Then either G is hamiltonian or [InlineMediaObject not available: see fulltext.] for some p2 where denotes join. As a corollary we obtain the following local analogue of a theorem of Nash-Williams: A connected r-regular graph G is hamiltonian if every ball of radius 2 is 2-connected and [InlineMediaObject not available: see fulltext.] for each vertex w of G. © Springer-Verlag 2006.

• 61.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
New localization theorems on Hamilton cycles2004In: 8th Nordic Combinatorial conference,2004, 2004Conference paper (Other academic)
• 62.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
On cyclic properties of some infinite, locally finite graphs2011Conference paper (Other academic)
• 63.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Umeå University.
On Path Factors of (3,4)-Biregular Bigraphs2008In: Graphs and Combinatorics, ISSN 0911-0119, E-ISSN 1435-5914, Vol. 24, no 5, p. 405-411Article in journal (Refereed)

A (3, 4)-biregular bigraph G is a bipartite graph where all vertices in one part have degree 3 and all vertices in the other part have degree 4. A path factor of G is a spanning subgraph whose components are nontrivial paths. We prove that a simple (3,4)-biregular bigraph always has a path factor such that the endpoints of each path have degree three. Moreover we suggest a polynomial algorithm for the construction of such a path factor.

• 64.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
A sufficient condition for interval edge colorings of (4,3)-biregular bipartite graphs2006Report (Other academic)
• 65.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Some results on interval edge colorings of (a, B)-biregular bipartitie graphs2006Report (Other academic)
• 66.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Umeå University. Polytech State University, Marietta, GA. University of Illinois.
Proper Path-Factors and Interval Edge-Coloring of (3,4)-Biregular Bigraphs2009In: JOURNAL OF GRAPH THEORY, ISSN 0364-9024 , Vol. 61, no 2, p. 88-97Article in journal (Refereed)

An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring using 6 colors when G is a (3,4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2, 4, 6, 8}. We provide several sufficient conditions for the existence of such a subgraph.

• 67.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Department of Mathematics, UmeÅ University, SE-901 87 UmeÅ, Sweden.
On interval edge colorings of (a, ß)-biregular bipartite graphs2007In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 307, no 15, p. 1951-1956Article in journal (Refereed)

A bipartite graph G is called (a, ß)-biregular if all vertices in one part of G have degree a and all vertices in the other part have degree ß. An edge coloring of a graph G with colors 1, 2, 3, ..., t is called an interval t-coloring if the colors received by the edges incident with each vertex of G are distinct and form an interval of integers and at least one edge of G is colored i, for i = 1, ..., t. We show that the problem to determine whether an (a, ß)-biregular bipartite graph G has an interval t-coloring is NP-complete in the case when a = 6, ß = 3 and t = 6. It is known that if an (a, ß)-biregular bipartite graph G on m vertices has an interval t-coloring then a + ß - gcd (a, ß) = t = m - 1, where gcd (a, ß) is the greatest common divisor of a and ß. We prove that if an (a, ß)-biregular bipartite graph has m = 2 (a + ß) vertices then the upper bound can be improved to m - 3. © 2006 Elsevier B.V. All rights reserved.

• 68.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
On Local Nature of Some Classical Theorems on Hamilton Cycles2005Report (Other academic)
• 69.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
On the local nature of some classical theorems on Hamilton cycles2007In: Australasian journal of combinatorics, ISSN 1034-4942, Vol. 38, p. 77-86Article in journal (Refereed)

The following result gives a flavour of what is in this paper. Ore's theorem is that if $d(u)+d(v)\ge|G|$ for all non-adjacent $u,v\in G$ then $G$ is Hamiltonian. The authors show that this is equivalent to specifying that $d(u)+d(v)\ge |B(x)|$ for every $x\in G$ and all non-adjacent $u,v\in B(x)$, where $B(x)$ is the ball of radius three centred at $x$. The reason is that the condition implies $G$ has diameter at most two, and so $B(x)=G$.

• 70.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Institute for System Programming Russian Acad. Sci..
Analysis of randomized rounding for integer programs2004In: Discrete mathematics and applications, ISSN 0924-9265, Vol. 14, no 6Article in journal (Refereed)
• 71.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Institute for System Programming Russian Acad. Sci..
Two sensitivity theorems in fuzzy integer programming2004In: Discrete Applied Mathematics, ISSN 0166-218X, E-ISSN 1872-6771, Vol. 134, no 1-3, p. 129-140Article in journal (Refereed)

We consider the problem of estimating optima of covering integer linear programs with 0-1 variables under the following conditions: we do not know exact values of elements in the constraint matrix A but we know what elements of A are zero and what are nonzero, and also know minimal and maximal values of nonzero elements. We find bounds for variation of the optima of such programs in the worst and average cases. We also find some conditions guaranteeing that the variation of the optimum of such programs in the average case is close to 1 as the number of variables tends to infinity. This means that the values of nonzero elements in A can vary without significantly affecting the value of the optimum of the integer program.

• 72.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Linköping University, Department of Mathematics, Applied Mathematics.
Pk+1- decomposition of eulerian graphs: complexity and some solvable cases2003Report (Other academic)
• 73.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Pk+1-Decompositions of Eulerian Graphs: Complexity and Some Solvable Cases2003In: Electronic Notes in Discrete Mathematics, ISSN 1571-0653, E-ISSN 1571-0653, Vol. 13Article in journal (Refereed)

We consider the problem of PMk+1-decomposition of a simple eulerian graph G, that is, decomposition of G into edge disjoint paths of length k. We show that the problem of deciding whether there exists a Pk+1 - decomposition of an eulerian simple graph is NP-complete, for every k = 3. However we find some new classes of graphs where the problem of P4-decomposition can be solved polynomially. We show that an eulerian simple graph G on 3m = 6 edges admits a P4-decomposition if G has no cut vertex v such that exactly one of the components in the graph G - ? has two vertices. In particular, this implies that a 2-connected eulerian simple graph G on 3m = 6 edges admits a P4 -decomposition. © 2003.

• 74.
Université Paris-Sud 11, France.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Remarks on maximal regularity2011In: Parabolic Problems: The Herbert Amann Festschrift / [ed] Joachim Escher, Patrick Guidotti, Matthias Hieber, Piotr Mucha, Jan W. Prüss, Yoshihiro Shibata, Gieri Simonett, Christoph Walker, Wojciech Zajaczkowski, Springer, 2011, p. 45-56Chapter in book (Other academic)

We prove weighted estimates for the maximal regularity operator. Such estimates were motivated by boundary value problems. We take this opportunity to study a class of weak solutions to the abstract Cauchy problem. We also give  a new proof of maximal regularity for closed and maximal accretive operators following from Kato's inequality for fractional powers and almost orthogonality arguments.

• 75.
Université Paris-Sud.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I2011In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 184, no 1, p. 47-115Article in journal (Refereed)

We develop new solvability methods for divergence form second order, real and complex, elliptic systems  above Lipschitz graphs, with $L_2$ boundary data.    The coefficients $A$ may depend on all variables, but are assumed to be close to coefficients $A_0$ that are independent of the coordinate transversal to the boundary, in the Carleson sense $\|A-A_0\|_C$ defined by Dahlberg.  We obtain a number of {\em a priori} estimates and boundary behaviour results under finiteness of $\|A-A_0\|_C$.  Our methods yield full characterization of weak solutions, whose gradients have $L_2$ estimates of a non-tangential maximal function or of the square function, via an integral representation acting on the conormal gradient, with a singular operator-valued kernel.   Also, the non-tangential maximal function of a weak solution is controlled in $L_2$ by the square function of its   gradient. This estimate is new for systems in such generality, and even for real non-symmetric equations in dimension $3$  or higher. The existence of a proof {\em a priori} to well-posedness, is also a new fact.  As corollaries, we obtain well-posedness of the Dirichlet, Neumann and Dirichlet regularity problems under   smallness of $\|A-A_0\|_C$ and well-posedness for $A_0$, improving earlier results for real symmetric equations.  Our methods build on an algebraic reduction to a first order system first made for coefficients $A_0$ by the two authors   and A. McIntosh in order to use functional calculus related to the Kato conjecture solution,   and the main analytic tool for coefficients $A$ is an operational calculus to prove weighted maximal regularity estimates.

• 76.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology. University of Alaska at Fairbanks. University of Alaska at Fairbanks.
Iterative methods for solving a nonlinear boundary inverse problem in glaciology2009In: JOURNAL OF INVERSE AND ILL-POSED PROBLEMS, ISSN 0928-0219 , Vol. 17, no 3, p. 239-258Article in journal (Refereed)

We address a Cauchy problem for a nonlinear elliptic PDE arising in glaciology. After recasting the Cauchy problem as an ill-posed operator equation, we prove (for values of a certain parameter allowing Hilbert space techniques) differentiability properties of the associated operator. We also suggest iterative methods which can be applied to solve the operator problem.

• 77.
Stockholms University.
Stockholms University. University of Seville. Stockholms University. Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Pentagrams and Paradoxes2011In: FOUNDATIONS OF PHYSICS, ISSN 0015-9018, Vol. 41, no 3, p. 414-423Article in journal (Refereed)

Klyachko and coworkers consider an orthogonality graph in the form of a pentagram, and in this way derive a Kochen-Specker inequality for spin 1 systems. In some low-dimensional situations Hilbert spaces are naturally organised, by a magical choice of basis, into SO(N) orbits. Combining these ideas some very elegant results emerge. We give a careful discussion of the pentagram operator, and then show how the pentagram underlies a number of other quantum "paradoxes", such as that of Hardy.

• 78.
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
• 79.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
On the Branch Loci of Moduli Spaces of Riemann Surfaces of Low Genera2009Licentiate thesis, comprehensive summary (Other academic)

Compact Riemann surfaces of genus greater than 1 can be realized as quotient spaces of the hyperbolic plane by the action of Fuchsian groups. The Teichmüller space is the set of all complex structures of Riemann surfaces and the moduli space the set of conformal equivalence classes of Riemann surfaces. For genus greater than two the branch locus of the covering of the moduli space by the Teichmüller space can be identified wi the set of Riemann surfaces admitting non-trivial automorphisms. Here we give the orbifold structure of the branch locus of surfaces of genus 5 by studying the equisymmetric stratification of the branch locus. This gives the orbifold structure of the moduli space.

We also show that the strata corresponding to surfaces with automorphisms of order 2 and 3 belong to the same connected component for every genus. 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.

1. 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
• 80.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
On the connected branch loci of moduli spacesManuscript (preprint) (Other academic)

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.

• 81.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Dept. Matemáticas, UNED, 28040 Madrid, Spain. Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
On the Orbifold Structure of the Moduli Space of Riemann Surfaces of Genera Four and Five2014In: 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)

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.

• 82.
On the connectedness of the branch locus of the moduli space of Riemann surfaces2010In: 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)

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.

• 83.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Lie algebras and classification of partial differential equations2004In: Symmetry in Nonlinear Mathematical Physics,2004, 2004Conference paper (Other academic)
• 84.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Group classification of the general quasi-linear wave equation: invance under low-dimensional Lie algebras2004Report (Other academic)
• 85.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
The structure of Lie algebras and the classification problem for partial differential equations2004In: Proceedings of Institute of Mathematics of NAS of Ukraine.Mathematics and its Applications, Vol. 50, p. 40-46Article in journal (Refereed)
• 86.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Classifying evolution equations2001In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 47, no 8, p. 5135-5144Article in journal (Refereed)

A Lie point symmetry classification of evolution equations in 1+1 time-space dimensions was presented. A combination of the standard Lie algorithm for point symmetry and the equivalence group of the given type of equation was used for the classification. For each canonical evolution the maximal symmetry algebra was calculated and related theorems were proved.

• 87.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Linkoping Univ, S-58183 Linkoping, Sweden Pedag Univ, UA-314000 Poltava, Ukraine Inst Math, UA-252004 Kiev, Ukraine. Linkoping Univ, S-58183 Linkoping, Sweden Pedag Univ, UA-314000 Poltava, Ukraine Inst Math, UA-252004 Kiev, Ukraine.
The structure of lie algebras and the classification problem for partial differential equations2001In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, Vol. 69, no 1, p. 43-94Article in journal (Refereed)

The present paper solves completely the problem of the group classification of nonlinear heat-conductivity equations of the form u(t)=F(t,x,u,u(x))u(xx)+G(t,x,u,u(x)). We have proved, in particular, that the above class contains no nonlinear equations whose invariance algebra has dimension more than five. Furthermore, we have proved that there are two, thirty-four, thirty-five, and six inequivalent equations admitting one-, two-, three-, four- and five-dimensional Lie algebras, respectively. Since the procedure which we use relies heavily upon the theory of abstract Lie algebras of low dimension, we give a detailed account of the necessary facts. This material is dispersed in the literature and is not fully available in English. After this algebraic part we give a detailed description of the method and then we derive the forms of inequivalent invariant evolution equations, and compute the corresponding maximal symmetry algebras. The list of invariant equations obtained in this way contains (up to a local change of variables) all the previously-known invariant evolution equations belonging to the class of partial differential equations under study.

• 88.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Institute of Mathematics, 3 Tereshchenkivska Street, 252004 Kyiv, Ukraine.
Initial-value problems for evolutionary partial differential equations and higher-order conditional symmetries2001In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 42, no 1, p. 376-389Article in journal (Refereed)

We suggest a new approach to the problem of dimensional reduction of initial/ boundary value problems for evolution equations in one spatial variable. The approach is based on higher-order (generalized) conditional symmetries of the equations involved. It is shown that reducibility of an initial value problem for an evolution equation to a Cauchy problem for a system of ordinary differential equations can be fully characterized in terms of conditional symmetries which leave invariant the equation in question. We also give some examples of the solution of initial value problems for second- and third-order nonlinear differential equations by reduction by their conditional symmetries. We give a systematic classification of general second-order partial differential equations admitting second-order conditional symmetries, based on Lie's classification of invariant second-order ordinary differential equations. This yields five classes of principally new initial value problems for nonlinear evolution equations which admit no Lie symmetries and are reducible via second-order conditional symmetries. © 2001 American Institute of Physics.

• 89.
Stockholms Universitet, AlbaNova, Fysikum, Stockholm, Sweden.
Instytut Fizyki im. Smoluchowskiego, Uniwersytet Jagielloński, Kraków, Poland. Stockholms Universitet, AlbaNova, Fysikum, Stockholm, Sweden. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics. Wydział Matematyczno-Przyrodniczy, Szkoła Nauk Ścisłych, Universytet Kardynała Stefana Wyszyńskiego, Warszawa, Poland. Instytut Fizyki im. Smoluchowskiego, Uniwersytet Jagielloński, Kraków, Poland and Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Warszawa, Poland.
Mutually unbiased bases and Hadamard matrices of order six2007In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 48, no 5, p. 052106-1-052106-21Article in journal (Refereed)

We report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.

• 90.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Conformal Einstein spaces and Bach tensor generalization in n dimensions2004Licentiate thesis, monograph (Other academic)

In this thesis we investigate necessary and su±cient conditions for an n-dimensional space, n ≥ 4, to be locally conformal to an Einstein space. After reviewing the classical results derived in tensors we consider the four-dimensional spinor result of Kozameh, Newman and Tod. The involvement of the four-dimensional Bach tensor (which is divergence-free and conformally well-behaved) in their result motivates a search for an n-dimensional generalization of the Bach tensor Bab with the same properties. We strengthen a theorem due to Belfagón and Jaén and give a basis (Uab, V ab and Wab) for all n-dimensional symmetric, divergence-free 2-index tensors quadratic in the Riemann curvature tensor. We discover the simple relationship Bab = 1/2Uab + 1/6V ab and show that the Bach tensor is the unique tensor with these properties in four dimensions. Unfortunately we have to conclude, in general that there is no direct analogue in higher dimension with all these properties.

Nevertheless, we are able to generalize the our-dimensional results due to Kozameh, Newman and Tod to n dimensions. We show that a generic space is conformal to an Einstein space if and only if there exists a vector field satisfying two conditions. The explicit use of dimensionally dependent identities (some of which are newly derived in this thesis) is also exploited in order to make the two conditions as simple as possible; explicit examples are given in five and six dimensions using these tensor identities. For n dimensions, we define the tensors babc and Bab, and we show that their vanishing is a conformal invariant property which guarantees that the space with non-degenerate Weyl tensor is a conformal Einstein space.

• 91.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Mathematical modelling in upper secondary mathematics education in Sweden2010Doctoral thesis, comprehensive summary (Other academic)

The aim of this thesis is to investigate and enhance our understanding of the notions of mathematical models and modelling at the Swedish upper secondary school level. Focus is on how mathematical models and modelling are viewed by the different actors in the school system, and what characterises the collaborative process of a didactician and a group of teachers engaged in designing and developing, implementing and evaluating teaching modules (so called modelling modules) exposing students to mathematical modelling in line with the present mathematics curriculum. The thesis consists of five papers and reports, along with a summary introduction, addressing both theoretical and empirical aspects of mathematical modelling.

The thesis uses both qualitative and quantitative methods and draws partly on design-based research methodology and cultural-historical activity theory (CHAT). The results of the thesis are presented using the structure of the three curriculum levels of the intended, potentially implemented, and attained curriculum respectively.

The results show that since 1965 and to the present day, gradually more and more explicit emphasis has been put on mathematical models and modelling in the syllabuses at this school level. However, no explicit definitions of these notions are provided but described only implicitly, opening up for a diversity of interpretations.

From the collaborative work case study it is concluded that the participating teachers could not express a clear conception of the notions mathematical models or modelling, that the designing process often was restrained by constraints originating from the local school context, and that working with modelling highlights many systemic tensions in the established school practice. In addition, meta-results in form of suggestions of how to resolve different kinds of tensions in order to improve the study design are reported.

In a questionnaire study with 381 participating students it is concluded that only one out of four students stated that they had heard about or used mathematical models or modelling in their education before, and the expressed overall attitudes towards working with mathematical modelling as represented in the test items were negative. Students’ modelling proficiency was positively affected by the students’ grade, last taken mathematics course, and if they thought the problems in the tests were easy or interesting. In addition empirical findings indicate that so-called realistic Fermi problems given to students working in groups inherently evoke modelling activities.

1. Towards understanding teachers’ beliefs and affects about mathematical modelling
Open this publication in new window or tab >>Towards understanding teachers’ beliefs and affects about mathematical modelling
2009 (English)Conference paper, Published paper (Other academic)
##### Abstract [en]

Work in progress on a framework aiming at capturing teachers’ beliefs about mathematical models and modelling is presented. It is suggested that the belief structure of mathematical models and modelling as perceived by teachers fruitfully might be explored as partly constituted of the teachers’ beliefs about the real world, the nature of mathematics, school mathematics, and applying and applications of mathematics. Some aspects of the suggested framework are explored using two case study interviews. It is found that the two teachers do not have any well formed beliefs about mathematical models and modelling, and that the interpreted beliefs structure of the teachers contain inconsistencies which are made explicit within the framework. The empiric findings also suggest some modifications of the framework.

Pedagogy
##### Identifiers
urn:nbn:se:liu:diva-54316 (URN)
##### Conference
CERME 6 - Sixth Conference of European Research in Mathematics Education Lyon, France, January 28 - February1
Available from: 2010-03-09 Created: 2010-03-09 Last updated: 2018-09-01
2. On the use of realistic Fermi problems for introducingmathematical modelling in school
Open this publication in new window or tab >>On the use of realistic Fermi problems for introducingmathematical modelling in school
2009 (English)In: The Montana Mathematics Enthusiast, ISSN 1551-3440, E-ISSN 1551-3440, Vol. 6, no 3, p. 331-364Article in journal (Refereed) Published
##### Abstract [en]

In this paper an analytical tool referred to as the MAD (Modelling Activity Diagram) framework adapted from Schoenfeld’s parsing protocol coding scheme is used to address the issues of how to introduce mathematical modelling to upper secondary students. The work of three groups of students engaged in solving so called realistic Fermi problems were analysed using this framework, and it was observed that the processes involved in a typical mathematical modelling cycle were richly represented in the groups’ solving processes. The importance of the social interactions within the groups was noted, as well as the extensive use of extra-mathematical knowledge used by the students during the problem solving session.

##### Place, publisher, year, edition, pages
Montana Council of Teachers of Mathematics & Information Age Publishing, 2009
##### Keywords
Fermi problems, Fermi estimates, Modelling cycles, Mathematical modelling, Modelling activity diagram
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-54664 (URN)
Available from: 2010-03-30 Created: 2010-03-30 Last updated: 2017-12-12
3. First results from a study investigating Swedish upper secondary students’ mathematical modelling competencies
Open this publication in new window or tab >>First results from a study investigating Swedish upper secondary students’ mathematical modelling competencies
2009 (English)Conference paper, Published paper (Other academic)
##### Abstract [en]

This paper reports on the first results from a study investigating Swedish upper secondary students’ (11th – 12th grade) mathematical modelling competency. Using non-parametric statistical methods the data from 381 students are analysed and the students’ modelling competency is described in terms of seven subcompetencies. Possible factors affecting the students’ mathematical competency such as attitudes toward modelling, previous experiences, last taken mathematics course, grade, class and gender were also investigated.

Mathematics
##### Identifiers
urn:nbn:se:liu:diva-54317 (URN)
##### Conference
14th International Conference on the Teaching of Mathematical Modelling and Applications (ICTMA 14), University of Hamburg, 27th to 31st July, Germany
Available from: 2010-03-09 Created: 2010-03-09 Last updated: 2014-06-23Bibliographically approved
4. Matematisk modellering i svenska gymnasieskolans kursplaner i matematik 1965-2000
Open this publication in new window or tab >>Matematisk modellering i svenska gymnasieskolans kursplaner i matematik 1965-2000
##### Abstract [en]

Denna rapport studerar hur begreppen matematisk modell och matematisk modellering behandlats i kursplanerna i matematik för motsvarande svenska gymnasieskolans naturvetenskapliga program mellan åren 1965 och 2000. En översikt över sex kursplaners olika form och innehåll ges och tre olika analyser på tre olika nivåer genomförs; en innehållsanalys, en kvalitativ analys, och en analys som beaktar relationen mellan begreppen modellering, tillämpningar och problemlösning. Analysen visar att matematiska modeller och modellering förekommit implicit sedan kursplanen från 1965, men att begreppen främst i de två senaste kursplanerna lyfts fram explicit och fått en central position i kursplanerna. I den nuvarande kursplanen Gy2000 framgår att matematiska modeller kan tolkas både som ett undervisningsmål i sig och som ett didaktiskt hjälpmedel att lära matematik.

##### Series
LiTH-MAT-R, ISSN 0348-2960 ; 2009:8
##### National Category
Mathematics Computational Mathematics
##### Identifiers
urn:nbn:se:liu:diva-54267 (URN)LiTH-MAT-R-2009-8 (ISRN)
Available from: 2010-03-05 Created: 2010-03-05 Last updated: 2014-06-23
5. Designing, implementing and evaluating mathematical modelling modules at the upper secondary level
Open this publication in new window or tab >>Designing, implementing and evaluating mathematical modelling modules at the upper secondary level
##### Abstract [en]

This study provides an account of how it is possible to work with mathematical models and modelling at the Swedish upper secondary level. It shows how a researcher and two teachers, departing from what is written in the mathematics curriculum document about mathematical models and modelling and the teachers’ mathematics teaching practices, collaborate to design and develop two so called modelling modules. These two modules designed for, and integrated in, the Mathematics C and Mathematics D course respectively, were implemented in the teachers’ classes and the whole process was evaluated. The students’ experiences of working with the modules were also investigated.

The conceptual framework used to study this process of design, implementation, and evaluation is built up of design-based research methodology, cultural historical activity theory (CHAT), and co-learning agreement between the researcher and the participants.

The teachers as well as the students expressed that they enjoyed working on the modules and found the experience positive and rewarding. However, some doubt concerning the learning outcome on behalf of the students were raised among the teachers, and the wish to incorporate more scheduled time into the modules was expressed by the students.

It is concluded that the participating teachers could had difficulties in expressing a clear conception of the notions of mathematical models or modelling and no affirmative conclusions could be drawn that the participation of the teachers changed their attitudes towards mathematical models and modelling. The designing process was often was restrained by constraints originating from the local school context, and working with modelling highlights many systemic tensions in the established school practice. Also, meta-results in form of suggestions of how to resolve different kinds of tensions in order to improve the study design are reported.

##### National Category
Mathematics Computational Mathematics
##### Identifiers
urn:nbn:se:liu:diva-54269 (URN)LiTH-MAT-R-2009-9 (Local ID)LiTH-MAT-R-2009-9 (Archive number)LiTH-MAT-R-2009-9 (OAI)
Available from: 2010-03-05 Created: 2010-03-05 Last updated: 2014-06-23
• 92.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Causal tensors and simple forms2002In: International Journal of Modern Physics A, ISSN 0217-751X, E-ISSN 1793-656X, Vol. 17, no 20, p. 2748-2748Article in journal (Refereed)
• 93.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Causal tensors in Lorentzian geometry2003In: Publications of the Spanish mathematical society, Vol. 5, p. 73-83Article in journal (Refereed)
• 94.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Monotone quantities for the Penrose inequality2003In: Penrose Inequalities,2003, 2003Conference paper (Other academic)
• 95.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Spinors and conformal curvature2005In: Publications of the Spanish mathematical societyArticle in journal (Refereed)
• 96.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
The Penrose inequality and null hypersurfaces2003In: Joint meeting of the American Mathematical Society and the Spanish Mathematical Society,2003, 2003Conference paper (Other academic)
• 97.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
The Chevreton tensor and Einstein-Maxwell spacetimes conformal to Einstein spaces2007In: Classical and Quantum Gravity, ISSN 0264-9381, Vol. 24, no 13, p. 3437-3455Article in journal (Refereed)

In this paper, we characterize the source-free Einstein–Maxwell spacetimes which have a trace-free Chevreton tensor. We show that this is equivalent to the Chevreton tensor being of pure radiation type and that it restricts the spacetimes to Petrov type N or O. We prove that the trace of the Chevreton tensor is related to the Bach tensor and use this to find all Einstein–Maxwell spacetimes with a zero cosmological constant that have a vanishing Bach tensor. Among these spacetimes we then look for those which are conformal to Einstein spaces. We find that the electromagnetic field and the Weyl tensor must be aligned, and in the case that the electromagnetic field is null, the spacetime must be conformally Ricci-flat and all such solutions are known. In the non-null case, since the general solution is not known on a closed form, we settle by giving the integrability conditions in the general case, but we do give new explicit examples of Einstein–Maxwell spacetimes that are conformal to Einstein spaces, and we also find examples where the vanishing of the Bach tensor does not imply that the spacetime is conformal to a C-space. The non-aligned Einstein–Maxwell spacetimes with vanishing Bach tensor are conformally C-spaces, but none of them are conformal to Einstein spaces.

• 98.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology. Departamento de Física Teórica, Universidad del País Vasco, Bilbao, Spain.
New electromagnetic conservation laws2003In: Classical and Quantum Gravity, ISSN 0264-9381, Vol. 20, no 13, p. 2663-2668Article in journal (Refereed)

The Chevreton superenergy tensor was introduced in 1964 as a counterpart, for electromagnetic fields, of the well-known Bel–Robinson tensor of the gravitational field. We here prove the unnoticed facts that, in the absence of electromagnetic currents, Chevreton's tensor (i) is completely symmetric, and (ii) has a trace-free divergence if the Einstein–Maxwell equations hold. It follows that the trace of the Chevreton tensor is a rank-2, symmetric, trace-free, conserved tensor, which is different from the energy–momentum tensor, and nonetheless can be constructed for any test Maxwell field or any Einstein–Maxwell spacetime.

• 99.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
New conservation laws in Einstein-Maxwell spacetimes2004In: 17th International Conference on General Relativity and Gravitation,2004, 2004Conference paper (Other academic)
• 100.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
University of Melbourne, Victoria, Australia.
Rank probabilities for real random NxNx2 tensors2011In: Electronic Communications in Probability, ISSN 1083-589X, E-ISSN 1083-589X, Vol. 16, p. 630-637Article in journal (Refereed)

We prove that the probability P_N for a real random Gaussian NxNx2 tensor to be of real rank N is P_N=(Gamma((N+1)/2))^N/G(N+1), where Gamma(x) and G(x) denote the gamma and the Barnes G-functions respectively. This is a rational number for N odd and a rational number multiplied by pi^{N/2} for N even. The probability to be of rank N+1 is 1-P_N. The proof makes use of recent results on the probability of having k real generalized eigenvalues for real random Gaussian N x N matrices. We also prove that log P_N= (N^2/4)log (e/4)+(log N-1)/12-zeta'(-1)+O(1/N) for large N, where zeta is the Riemann zeta function.

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