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Inherent Numerical Instability in Computing Invariant Measures of Markov Chains

Applied Mathematics 2017年第9期8卷 1367-1385页

作者： hendrik baumann Thomas Hanschke Clausthal University of Technology Clausthal-Zellerfeld Germany

Invariant measures of Markov chains in discrete or continuous time with a countable set of states are characterized by its steady state recurrence relations. Exemplarily, we consider transition matrices and Q-matrices with upper bandwidth n and lower bandwidth 1 where the invariant measures satisfy an (n + 1)-order linear difference equation. Markov chains of this type arise from applications to queueing problems and population dynamics. It is the purpose of this paper to point out that the forward use of this difference equation is subject to some hitherto unobserved aspects. By means of the concept of generalized continued fractions (GCFs), we prove that each invariant measure is a dominated solution of the difference equation such that forward computation becomes numerically unstable. Furthermore, the GCF-based approach provides a decoupled recursion in which the phenomenon of numerical instability does not appear. The procedure results in an iteration scheme for successively computing approximants of the desired invariant measure depending on some truncation level N. Increasing N leads to the desired solution. A comparison study of forward computation and the GCF-based approach is given for Q-matrices with upper bandwidth 1 and 2.

关键词： Invariant Measures of Markov Chains Inherent Numerical Instability of Linear Difference Equations Generalized Continued Fractions Convergence Criteria for Generalized Continued Fractions Truncation Procedures for Infinite Matrices

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Cancer: Education and Primary Prevention Starts in Childhood and Adolescence

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Journal of Cancer Therapy 2016年第12期7卷 851-856页

作者： Nadja Seidel Friederike Stoelzel Sandra Herrmann Cornelia Zimmermann hendrik Berth Melanie Glausch Kristina Loewe Michael baumann Gerhard Ehninger University Cancer Center University Hospital Carl Gustav Carus Dresden Dresden Germany Department of Child and Adolescent Psychiatry University Hospital Carl Gustav Carus Dresden Dresden Germany Department of Medical Psychology Medical Faculty of the Technical University Dresden Dresden Germany

At least one third of all newly diagnosed cancers could be prevented if lifestyle factors were changed. The University Cancer Center Dresden initiated two programs aiming at cancer awareness and intentions to engage in health-promoting behavior among children and adolescents. Study 1 examined sun protection knowledge of 80 preschool children in a non-randomized design and Study 2 inspected 235 7th grade students’ knowledge of cancer and its behavioral risk-factors as well as intentions on health-promoting behavior using a randomized pre-post design on group-level. Study 1 showed significant improvement of sun protection knowledge in preschool children (p 0.05). Study 2 was effective in increasing knowledge about risk factors for cancer (p 0.001) and in increasing intentions to engage in health-promoting behavior (p 0.001). Communicating health-related behavior in preschools as well as school-based programs targeting multiple cancer-related risk factors are promising tools for primary prevention of cancer.

关键词： Cancer Prevention Cancer Education Health Behavior School-Based Health Promotion Literacy

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