152,283 research outputs found

    Regularity of Kobayashi metric

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    We review some recent results on existence and regularity of Monge-Amp\`ere exhaustions on the smoothly bounded strongly pseudoconvex domains, which admit at least one such exhaustion of sufficiently high regularity. A main consequence of our results is the fact that the Kobayashi pseudo-metric k on an appropriare open subset of each of the above domains is actually a smooth Finsler metric. The class of domains to which our result apply is very large. It includes for instance all smoothly bounded strongly pseudoconvex complete circular domains and all their sufficiently small deformations.Comment: 14 pages, 8 figures - The previously announced main result had a gap. In this new version the corrected statement is given. To appear on the volume "Geometric Complex Analysis - Proceedings of KSCV 12 Symposium

    Two-dimensional Stokes flow driven by elliptical paddles

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    A fast and accurate numerical technique is developed for solving the biharmonic equation in a multiply connected domain, in two dimensions. We apply the technique to the computation of slow viscous flow (Stokes flow) driven by multiple stirring rods. Previously, the technique has been restricted to stirring rods of circular cross section; we show here how the prior method fails for noncircular rods and how it may be adapted to accommodate general rod cross sections, provided only that for each there exists a conformal mapping to a circle. Corresponding simulations of the flow are described, and their stirring properties and energy requirements are discussed briefly. In particular the method allows an accurate calculation of the flow when flat paddles are used to stir a fluid chaotically

    Numerical range for random matrices

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    We analyze the numerical range of high-dimensional random matrices, obtaining limit results and corresponding quantitative estimates in the non-limit case. For a large class of random matrices their numerical range is shown to converge to a disc. In particular, numerical range of complex Ginibre matrix almost surely converges to the disk of radius 2\sqrt{2}. Since the spectrum of non-hermitian random matrices from the Ginibre ensemble lives asymptotically in a neighborhood of the unit disk, it follows that the outer belt of width 2−1\sqrt{2}-1 containing no eigenvalues can be seen as a quantification the non-normality of the complex Ginibre random matrix. We also show that the numerical range of upper triangular Gaussian matrices converges to the same disk of radius 2\sqrt{2}, while all eigenvalues are equal to zero and we prove that the operator norm of such matrices converges to 2e\sqrt{2e}.Comment: 23 pages, 4 figure

    Relative CC"-Numerical Ranges for Applications in Quantum Control and Quantum Information

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    Motivated by applications in quantum information and quantum control, a new type of CC"-numerical range, the relative CC"-numerical range denoted WK(C,A)W_K(C,A), is introduced. It arises upon replacing the unitary group U(N) in the definition of the classical CC"-numerical range by any of its compact and connected subgroups K⊂U(N)K \subset U(N). The geometric properties of the relative CC"-numerical range are analysed in detail. Counterexamples prove its geometry is more intricate than in the classical case: e.g. WK(C,A)W_K(C,A) is neither star-shaped nor simply-connected. Yet, a well-known result on the rotational symmetry of the classical CC"-numerical range extends to WK(C,A)W_K(C,A), as shown by a new approach based on Lie theory. Furthermore, we concentrate on the subgroup SUloc(2n):=SU(2)⊗...⊗SU(2)SU_{\rm loc}(2^n) := SU(2)\otimes ... \otimes SU(2), i.e. the nn-fold tensor product of SU(2), which is of particular interest in applications. In this case, sufficient conditions are derived for WK(C,A)W_{K}(C,A) being a circular disc centered at origin of the complex plane. Finally, the previous results are illustrated in detail for SU(2)⊗SU(2)SU(2) \otimes SU(2).Comment: accompanying paper to math-ph/070103

    Second Order Freeness and Fluctuations of Random Matrices: I. Gaussian and Wishart matrices and Cyclic Fock spaces

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    We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We introduce the concept of "second order freeness" and derive the global fluctuations of Gaussian and Wishart random matrices by a general limit theorem for second order freeness. By introducing cyclic Fock space, we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation, annihilation, and preservation operators. We show that orthogonal families of Gaussian and Wishart random matrices are asymptotically free of second order.Comment: 46 pages, 13 figures, second revision adds explanations, figures, and reference
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