Research Article
Minimal Reducing Subspaces of 3-order Slant Toeplitz Operator on Hardy Space over the Disc
Yang Zou*
Issue:
Volume 11, Issue 1, March 2025
Pages:
1-9
Received:
9 January 2025
Accepted:
26 January 2025
Published:
17 February 2025
Abstract: The reducing subspace problem and the invariant subspace problem of an operator are two core problems in operator theory. There are lots of works on reducing subspaces and invariant subspaces of Toeplitz operators in recent years. A slant Toeplitz operator is a generalization of Toeplitz operator. In this paper, we study minimal reducing subspaces of the third-order slant Toeplitz operator with the symbol zN. By classifying N into three cases, we give a complete description of minimal reducing subspaces. Finally, all minimal reducing subspaces of the third-order slant Toeplitz operator with the symbol zN on the Hardy space of the disc in the complex plane are given. This paper generalizes the relevant results on reducing subspaces of second-order slant Toeplitz Operators, enriches the study of reducing subspaces of slant Toeplitz Operators on Lebesgue spaces, and of the structure of slant Toeplitz Operators.
Abstract: The reducing subspace problem and the invariant subspace problem of an operator are two core problems in operator theory. There are lots of works on reducing subspaces and invariant subspaces of Toeplitz operators in recent years. A slant Toeplitz operator is a generalization of Toeplitz operator. In this paper, we study minimal reducing subspaces ...
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Research Article
A Circular Spatial-diffusion Mathematical Model to Analysis Hopf-Turing Bifurcation in Plankton Population Under the Toxin Control Variation in 2D
Issue:
Volume 11, Issue 1, March 2025
Pages:
10-31
Received:
23 February 2025
Accepted:
6 March 2025
Published:
24 March 2025
DOI:
10.11648/j.ml.20251101.12
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Abstract: In a mathematical model of a system with two reaction-diffusion equations with Neumann-Dirichlet boundary conditions, we formulated zooplankton-phytoplankton in the aquatic environment on the circular domain. The attention has been focused on the toxin producing role of the space in explaining heterogeneity, the distribution of the species and the influence of the spatial structure on their abundance. The key idea of the model formulation is based on a nonlinear equations systems version with Holling II functional response. We base our mathematical analysis on the search for local and global solution with spatial diffusion. We present some mathematical results concerning the solution existence, the stability of the model equilibria. We have obtained important mathematical results for model equilibria stability at long time. Under certain mathematical conditions, the model without diffusion is locally asymptotically stable. Mathematical analysis also shows that the Hopf bifurcation breaks the time symmetry of the system and leads to uniform oscillations in space and periodic oscillations. The Turing bifurcation breaks the space symmetry and leads to the formation of stationary patterns in time and oscillatory patterns in space. A series of structured numerical simulations highlighted the formation of patterns and allowed to identify critical threshold of toxin released by phytoplankton leading to phytoplankton blooms.
Abstract: In a mathematical model of a system with two reaction-diffusion equations with Neumann-Dirichlet boundary conditions, we formulated zooplankton-phytoplankton in the aquatic environment on the circular domain. The attention has been focused on the toxin producing role of the space in explaining heterogeneity, the distribution of the species and the ...
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