| 1. |
- Eggers, Jens, et al.
(author)
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Singularities of relativistic membranes
- 2015
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In: Geometric Flows. - : Portico. - 2353-3382. ; 1:1
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Journal article (peer-reviewed)abstract
- Pointing out a crucial relation with caustics of the eikonal equation we discuss the singularity formation of 2-dimensional surfaces that sweep out 3-manifolds of zero mean curvature in R3,1.
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| 2. |
- Arnlind, Joakim, et al.
(author)
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Discrete minimal surface algebras
- 2010
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In: Symmetry, Integrability and Geometry. - 1815-0659. ; 6, s. Paper 042,18-
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Journal article (peer-reviewed)abstract
- We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sl(n) (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d <= 4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is ( generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.
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| 3. |
- Arnlind, Joakim, et al.
(author)
-
Eigenvalue dynamics, Follytons and large N limits of matrices
- 2006
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In: Applications of Random Matrices in Physics. - DORDRECHT : SPRINGER. - 1402045298 - 9781402045295 - 9781402045318 ; , s. 89-94
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Conference paper (peer-reviewed)abstract
- How do the eigenvalues of a “free” hermitian N × N matrix X(t) evolve in time? The answer is provided by the rational Calogero-Moser systems [5, 13] if (!) the initial conditions are chosen such that i[X(0),Ẋ(0)] has a non-zero eigenvalue of multiplicity N–1; for generic X(0),Ẋ(0) the question remained unanswered for 30 years.
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| 4. |
- Arnlind, Joakim, et al.
(author)
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Eigenvalue Dynamics off the Calogero-Moser system
- 2004
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In: Letters in Mathematical Physics. - : Springer. - 0377-9017 .- 1573-0530. ; 68:2, s. 121-129
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Journal article (peer-reviewed)abstract
- By finding N(N-1)/2 suitable conserved quantities, free motions of real symmetric N x N matrices X(t), with arbitrary initial conditions, are reduced to nonlinear equations involving only the eigenvalues of X-in contrast to the rational Calogero-Moser system, for which [X(0), X(0)] has to be purely imaginary, of rank one.
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| 5. |
- Arnlind, Joakim, et al.
(author)
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Fuzzy Riemann surfaces
- 2009
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In: Journal of High Energy Physics (JHEP). - : Springer Science and Business Media LLC. - 1126-6708 .- 1029-8479. ; :6, s. 047-
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Journal article (peer-reviewed)abstract
- We introduce C-Algebras (quantum analogues of compact Riemann surfaces), defined by polynomial relations in non-commutative variables and containing a real parameter that, when taken to zero, provides a classical non-linear, Poisson-bracket, obtainable from a single polynomial C (onstraint) function. For a continuous class of quartic constraints, we explicitly work out finite dimensional representations of the corresponding C-Algebras.
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| 6. |
- Arnlind, Joakim, et al.
(author)
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Goldfish geodesics and hamiltonian reduction of matrix dynamics
- 2008
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In: Letters in Mathematical Physics. - : Springer Science and Business Media LLC. - 0377-9017 .- 1573-0530. ; 84:1, s. 89-98
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Journal article (peer-reviewed)abstract
- We describe the Hamiltonian reduction of a time-dependent real-symmetric NxN matrix system to free vector dynamics, and also provide a geodesic interpretation of Ruijsenaars-Schneider systems. The simplest of the latter, the goldfish equation, is found to represent a flat-space geodesic in curvilinear coordinates.
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| 7. |
- Arnlind, Joakim, 1979-
(author)
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Graph Techniques for Matrix Equations and Eigenvalue Dynamics
- 2008
-
Doctoral thesis (other academic/artistic)abstract
- One way to construct noncommutative analogues of a Riemannian manifold Σ is to make use of the Toeplitz quantization procedure. In Paper III and IV, we construct C-algebras for a continuously deformable class of spheres and tori, and by introducing the directed graph of a representation, we can completely characterize the representation theory of these algebras in terms of the corresponding graphs. It turns out that the irreducible representations are indexed by the periodic orbits and N-strings of an iterated map s:(reals) 2→(reals)2 associated to the algebra. As our construction allows for transitions between spheres and tori (passing through a singular surface), one easily sees how the structure of the matrices changes as the topology changes. In Paper II, noncommutative analogues of minimal surface and membrane equations are constructed and new solutions are presented -- some of which correspond to minimal tori embedded in S7. Paper I is concerned with the problem of finding differential equations for the eigenvalues of a symmetric N × N matrix satisfying Xdd=0. Namely, by finding N(N-1)/2 suitable conserved quantities, the time-evolution of X (with arbitrary initial conditions), is reduced to non-linear equations involving only the eigenvalues of Χ.
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| 8. |
- Arnlind, Joakim, et al.
(author)
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Multi-linear Formulation of Differential Geometry and Matris Regularizations
- 2012
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In: Journal of differential geometry. - : International Press. - 0022-040X .- 1945-743X. ; 91:1, s. 1-39
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Journal article (peer-reviewed)abstract
- We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi-linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingartens formula, the Ricci curvature, and the Codazzi-Mainardi equations. For matrix analogues of embedded surfaces, we define discrete curvatures and Euler characteristics, and a non-commutative Gauss-Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and explicit examples are provided. Furthermore, we illustrate the fact that techniques from differential geometry can carry over to matrix analogues by proving that a bound on the discrete Gauss curvature implies a bound on the eigenvalues of the discrete Laplace operator.
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| 9. |
- Arnlind, Joakim, et al.
(author)
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Noncommutative Minimal Surfaces
- 2016
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In: Letters in Mathematical Physics. - : Springer Science and Business Media LLC. - 0377-9017 .- 1573-0530. ; 106:8, s. 1109-1129
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Journal article (peer-reviewed)abstract
- We define noncommutative minimal surfaces in the Weyl algebra, and give a method to construct them by generalizing the well-known Weierstrass representation.
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| 10. |
- Arnlind, Joakim, et al.
(author)
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Noncommutative Riemann Surfaces by Embeddings in R-3
- 2009
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In: Communications in Mathematical Physics. - : Springer Science and Business Media LLC. - 0010-3616 .- 1432-0916. ; 288:2, s. 403-429
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Journal article (peer-reviewed)abstract
- We introduce C-Algebras of compact Riemann surfaces ∑ as non-commutative analogues of the Poisson algebra of smooth functions on ∑. Representations of these algebrasgive rise to sequences of matrix-algebras for which matrix-commutators converge to Poisson-brackets as N → ∞. For a particular class of surfaces, interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras.
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