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Commutants, Reducing Subspaces and von Neumann Algebras Associated with Multiplication Operators
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
After a careful study in [GSZZ] of the case when the order of B is equal to 3, Guo, Sun, Zheng and Zhong formulated a more delicate conjecture: the number of minimal reducing subspaces of MBonLa2(D)equals the number of connected components of the Riemann surfaceSBof B−1 ∘ B on the unit disk [GSZZ, DSZ]. Here by a Riemann surface we mean a complex manifold of complex dimension 1, not necessarily connected. The amazing part of this modified conjecture is that the operator-theoretic quantity (the number of minimal reducing subspaces of MB) is accurately linked to a geometric quantity.
Integral Transforms and Complex-valued Functions
Published in Dingyü Xue, YangQuan Chen, Scientific Computing with MATLAB®, 2018
The limitations of function are that, it can only be used to draw Riemann surfaces of root functions, and cannot be used to draw other multi-valued functions. The existing function can be extended and saved as a new function . Then, delete the and statement. Then, it can be used to draw Riemann surfaces of multi-valued complex functions.
Algebraic Geometry
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
Riemann surfaces can be thought of as ‘deformed copies’ of the complex plane. Locally near every point, they look like patches of the complex plane. Every algebraic curve with coefficients in ℂ is a compact Riemann surface. Moreover, every compact Riemann surface is a sphere with some handles attached (see Figure 3.3).
Dubrovin equation for periodic Dirac operator on the half-line
Published in Applicable Analysis, 2022
Evgeny Korotyaev, Dmitrii Mokeev
In the theorem, any real power of supposed to be positive. We formulate the theorem in neighborhoods of 0. Shifting V by any , one can obtain Theorem 2.6 in a neighborhoods of .Identities (12) and (13) can be interpreted as derivatives of asymptotic estimates (4) and (5), where is determined from (11). From this point of view, one can differentiate these estimates because the conditions of theorem are satisfied. See discussion and results about the problem of differentiating an asymptotic expansion in [30]. In order to prove the theorem, we use the Monotone Density Theorem (see Theorem 1.7.2b in [27]).As we noted above, it follows from Lemma 3.5 that the position of on the Riemann surface is related to the value of . Thus, we can determine the sign of in (13) by the position of on the Riemann surface.
H ∞ bounded real lemma for singular fractional-order systems
Published in International Journal of Systems Science, 2021
Yuman Li, Yiheng Wei, Chen Yuquan, Yong Wang
Let , and be given. is defined in (20) and is defined as the null space of . Suppose Λ represents the region , and s belongs to the principal Riemann surface. Then the following statements are equivalent:;There exists such that and
Symmetric surface complex waves in Goubau Line
Published in Cogent Engineering, 2018
E.A. Kuzmina, Y.V. Shestopalov
Note that the differential operator of boundary eigenvalue problem (4)–(5) is not defined at: (i) when the interval supporting transmission conditions in (4) vanishes and the differential operator degenerates; (ii) the points which are branch points on the Riemann surface . may be thus classified as singular spectral set and as a singular value for the differential operator of the boundary eigenvalue problem. At the singular spectral points , the potential spectral function in (6) is not defined. Below, it will be shown that the points belonging to are singularities in the dependence of eigenvalues of (4)–(5) on the problem parameters , , and .