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Related Experiment Video

Updated: May 24, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Resolvent positive linear operators exhibit the reduction phenomenon.

Lee Altenberg¹

¹BioSystems, 2605 Lioholo Place, Kihei, HI 96753-7118, USA. altenber@hawaii.edu

Proceedings of the National Academy of Sciences of the United States of America

|February 24, 2012

Summary

This summary is machine-generated.

This study reveals that the spectral bound of combined operators is convex in α, implying greater mixing reduces growth in diffusion models. This finding has implications for understanding the evolution of dispersal strategies.

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Area of Science:

Functional Analysis
Mathematical Biology
Operator Theory

Background:

Kato previously established the convexity of the spectral bound s(αA + βV) with respect to β.
Resolvent positive operators and multiplication operators are fundamental in various mathematical and physical models.

Purpose of the Study:

To demonstrate the convexity of the spectral bound s(αA + βV) with respect to α > 0.
To explore the implications of this convexity for diffusion processes and the evolution of dispersal.

Main Methods:

Utilizing Kato's result on the convexity in β.
Applying an elementary 'dual convexity' lemma.
Analyzing the derivative ∂s(αA + βV)/∂α in relation to s(A).

Main Results:

The spectral bound s(αA + βV) is shown to be convex in α > 0.
It is proven that ∂s(αA + βV)/∂α ≤ s(A).
For diffusions with s(A) ≤ 0, increased mixing leads to reduced growth.

Conclusions:

The convexity in α provides a general framework for understanding phenomena like selection for reduced dispersal in mathematical models.
This unified approach encompasses various diffusion operators, including Laplacian and nonlocal types.
The findings offer a broad explanation for the 'reduction' phenomenon observed in dispersal evolution models.