Research Article
Long Time Behavior of Solution to Equal Mitosis PDE
Meas Len*
Issue:
Volume 11, Issue 5, October 2025
Pages:
71-77
Received:
17 September 2025
Accepted:
9 October 2025
Published:
19 December 2025
DOI:
10.11648/j.ijtam.20251105.11
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Abstract: In this work, we are interested in the long time behavior of a solution to equal mitosis partial differential equation with positive and periodic coefficients. First, we prove the existence and uniqueness of solution of Floquet eigenvalue and its adjoint eigenvalue problem to the equal mitosis equation by using the fixed point theorem in the suitable L1 weighted space under general division rate hypotheses. Let us recall that the Floquet exponent measures the growth rates of the population and understanding an eigenfunction is crucial for proving the long run behavior of the Cauchy problem. Then we apply the generalized relative entropy method to derive such long time asymptotic behavior of the population density.
Abstract: In this work, we are interested in the long time behavior of a solution to equal mitosis partial differential equation with positive and periodic coefficients. First, we prove the existence and uniqueness of solution of Floquet eigenvalue and its adjoint eigenvalue problem to the equal mitosis equation by using the fixed point theorem in the suitab...
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Research Article
Visualization of Polynomial Maps from the Complex Plane to Itself
Mark Daniel Meyerson*
Issue:
Volume 11, Issue 5, October 2025
Pages:
78-85
Received:
5 November 2025
Accepted:
24 November 2025
Published:
20 December 2025
DOI:
10.11648/j.ijtam.20251105.12
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Abstract: We show a way to cut up the complex plane into regions that are mapped 1-1 onto the complex plane by a polynomial. This is done for any finite number of ramification points with any multiplicity. For any polynomial map with only one or two ramification points we can do this explicitly (with minor adjustments). Most of the figures are drawn by approximating solutions to polynomial equations using Newton’s method. However, some of the special cases are computed exactly. At ramification points the plane is cut up by equally spaced arcs and the mapping there acts as if it is a hinge which opens to map to the full plane. In order to show the full extent of possibilities, our last example is a degree 12 polynomial with 5 ramification points of varying degrees.
Abstract: We show a way to cut up the complex plane into regions that are mapped 1-1 onto the complex plane by a polynomial. This is done for any finite number of ramification points with any multiplicity. For any polynomial map with only one or two ramification points we can do this explicitly (with minor adjustments). Most of the figures are drawn by appro...
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