https://mklab.gr/ Mon, 19 Oct 2026 10:35:05 +0000 MyBB https://mklab.gr/showthread.php?tid=2104 Wed, 23 Sep 2026 22:41:40 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2104 Introduction to the Mathematics of Medical Imaging: Second Edition, published by SIAM, develops the mathematical foundations behind modern medical imaging, using X-ray computed tomography (CT) as its main example. The book begins by modeling how X-rays interact with matter and then connects the resulting measurements to linear algebra, integral equations, the Radon transform, Fourier analysis, convolution, sampling theory, and noise. A central theme is the reconstruction of a three-dimensional object from two-dimensional measurements: the Radon transform represents the measured line integrals, while the central slice theorem connects these measurements to the Fourier transform.

The later chapters move from mathematical theory toward practical reconstruction algorithms. In particular, the book derives the filtered back-projection formula, an exact inversion of the Radon transform that underlies many CT reconstruction methods, and then examines the complications introduced by discrete sampling, filtering, measurement errors, and noise. 

The second edition also adds material on magnetic resonance imaging (MRI), the relationship between continuous and discrete Fourier transforms, gridding, and MRI noise analysis. Overall, it is a bridge between mathematical analysis and computational medical imaging, showing how abstract tools such as Fourier transforms and integral transforms become concrete algorithms for producing diagnostic images. 

BOOK]]> Introduction to the Mathematics of Medical Imaging: Second Edition, published by SIAM, develops the mathematical foundations behind modern medical imaging, using X-ray computed tomography (CT) as its main example. The book begins by modeling how X-rays interact with matter and then connects the resulting measurements to linear algebra, integral equations, the Radon transform, Fourier analysis, convolution, sampling theory, and noise. A central theme is the reconstruction of a three-dimensional object from two-dimensional measurements: the Radon transform represents the measured line integrals, while the central slice theorem connects these measurements to the Fourier transform.

The later chapters move from mathematical theory toward practical reconstruction algorithms. In particular, the book derives the filtered back-projection formula, an exact inversion of the Radon transform that underlies many CT reconstruction methods, and then examines the complications introduced by discrete sampling, filtering, measurement errors, and noise. 

The second edition also adds material on magnetic resonance imaging (MRI), the relationship between continuous and discrete Fourier transforms, gridding, and MRI noise analysis. Overall, it is a bridge between mathematical analysis and computational medical imaging, showing how abstract tools such as Fourier transforms and integral transforms become concrete algorithms for producing diagnostic images. 

BOOK]]> https://mklab.gr/showthread.php?tid=2045 Sat, 19 Sep 2026 00:33:23 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=2045 Mathematics of Epidemics on Networks: From Exact to Approximate Models by István Z. Kiss, Joel C. Miller, Grzegorz A. Rempała and Péter L. Simon is a comprehensive mathematical treatment of how infectious diseases—and, more broadly, contagion processes—spread through networks. The second edition, published in 2026, expands the original framework to 15 chapters and connects graph theory, probability, stochastic processes, dynamical systems and epidemiology. 

A central theme is that traditional epidemic models such as SIR often treat populations as homogeneous, whereas real populations have structure: individuals have different numbers of contacts, belong to communities, change their contacts over time, and may interact in groups rather than simply in pairs. The book therefore develops models ranging from exact stochastic descriptions of individual-level transmission to increasingly tractable approximations such as mean-field, pairwise and edge-based models. 

The book's major contribution is its hierarchy of models: rather than presenting mathematical approximations as isolated formulas, it explains how simpler models arise from more detailed network descriptions and what assumptions are introduced at each step. It covers percolation methods, stochastic trajectories, statistical and Bayesian inference, simple versus complex contagion, higher-order networks, adaptive networks, non-Markovian epidemics, and PDE limits for very large networks. 

In this way, the authors show how mathematical structure determines what an epidemic model can capture—and what it necessarily ignores. The book also emphasizes computation, providing simulation algorithms and software alongside the theory, making it useful not only as a research reference but also as an advanced undergraduate/graduate textbook.

Key takeaways

• Networks matter: Epidemic dynamics depend strongly on who is connected to whom, not merely on the total population.
  
• SIR is a starting point, not the whole story: Network structure can substantially change epidemic behavior.
  
• Exact models are mathematically powerful but often difficult to solve, motivating carefully derived approximations.
  
• Mean-field models trade detail for tractability by replacing individual network interactions with population-level averages.
  
• Heterogeneous networks require richer mathematics because individuals can have very different numbers of contacts.
  
• Model hierarchy is crucial: simpler epidemic equations can often be understood as approximations of more fundamental stochastic network models.
  
• Stochasticity matters: Two epidemics with identical parameters can follow different trajectories because transmission is inherently random.
  
• Percolation theory provides another perspective: epidemic outbreaks can be related to connectivity and cluster formation in networks.
  
• Complex contagion differs from simple contagion: transmission may require exposure to multiple contacts rather than a single infectious neighbor.
  
• Higher-order networks capture group interactions that ordinary graphs cannot represent adequately.
  
• Networks can change during an epidemic: people may alter their contacts in response to disease, creating feedback between behavior and transmission.
  
• Non-Markovian models allow realistic waiting-time distributions, rather than assuming exponentially distributed infection/recovery times.
  
• Statistical inference connects theory with data: likelihood-based and Bayesian methods can estimate epidemic parameters from observations.
  
• Large networks can lead to continuum/PDE descriptions, providing a bridge from discrete networks to macroscopic mathematical models.
  
• The overarching lesson: the right epidemic model is determined by the question being asked, the available data, and the level of network detail that is scientifically important. 
  

For a mathematics teacher, the particularly interesting aspect is that the book provides a natural progression from graphs → probability → differential equations → stochastic processes → simulation, making epidemic spread an excellent real-world context for connecting several areas of mathematics.

BOOK]]> Mathematics of Epidemics on Networks: From Exact to Approximate Models by István Z. Kiss, Joel C. Miller, Grzegorz A. Rempała and Péter L. Simon is a comprehensive mathematical treatment of how infectious diseases—and, more broadly, contagion processes—spread through networks. The second edition, published in 2026, expands the original framework to 15 chapters and connects graph theory, probability, stochastic processes, dynamical systems and epidemiology. 

A central theme is that traditional epidemic models such as SIR often treat populations as homogeneous, whereas real populations have structure: individuals have different numbers of contacts, belong to communities, change their contacts over time, and may interact in groups rather than simply in pairs. The book therefore develops models ranging from exact stochastic descriptions of individual-level transmission to increasingly tractable approximations such as mean-field, pairwise and edge-based models. 

The book's major contribution is its hierarchy of models: rather than presenting mathematical approximations as isolated formulas, it explains how simpler models arise from more detailed network descriptions and what assumptions are introduced at each step. It covers percolation methods, stochastic trajectories, statistical and Bayesian inference, simple versus complex contagion, higher-order networks, adaptive networks, non-Markovian epidemics, and PDE limits for very large networks. 

In this way, the authors show how mathematical structure determines what an epidemic model can capture—and what it necessarily ignores. The book also emphasizes computation, providing simulation algorithms and software alongside the theory, making it useful not only as a research reference but also as an advanced undergraduate/graduate textbook.

Key takeaways

• Networks matter: Epidemic dynamics depend strongly on who is connected to whom, not merely on the total population.
  
• SIR is a starting point, not the whole story: Network structure can substantially change epidemic behavior.
  
• Exact models are mathematically powerful but often difficult to solve, motivating carefully derived approximations.
  
• Mean-field models trade detail for tractability by replacing individual network interactions with population-level averages.
  
• Heterogeneous networks require richer mathematics because individuals can have very different numbers of contacts.
  
• Model hierarchy is crucial: simpler epidemic equations can often be understood as approximations of more fundamental stochastic network models.
  
• Stochasticity matters: Two epidemics with identical parameters can follow different trajectories because transmission is inherently random.
  
• Percolation theory provides another perspective: epidemic outbreaks can be related to connectivity and cluster formation in networks.
  
• Complex contagion differs from simple contagion: transmission may require exposure to multiple contacts rather than a single infectious neighbor.
  
• Higher-order networks capture group interactions that ordinary graphs cannot represent adequately.
  
• Networks can change during an epidemic: people may alter their contacts in response to disease, creating feedback between behavior and transmission.
  
• Non-Markovian models allow realistic waiting-time distributions, rather than assuming exponentially distributed infection/recovery times.
  
• Statistical inference connects theory with data: likelihood-based and Bayesian methods can estimate epidemic parameters from observations.
  
• Large networks can lead to continuum/PDE descriptions, providing a bridge from discrete networks to macroscopic mathematical models.
  
• The overarching lesson: the right epidemic model is determined by the question being asked, the available data, and the level of network detail that is scientifically important. 
  

For a mathematics teacher, the particularly interesting aspect is that the book provides a natural progression from graphs → probability → differential equations → stochastic processes → simulation, making epidemic spread an excellent real-world context for connecting several areas of mathematics.

BOOK]]> https://mklab.gr/showthread.php?tid=1636 Mon, 17 Aug 2026 18:05:29 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=1636 Mathematical Biology: I. An Introduction
Author: James D. Murray
Publication: 3rd edition, 2002
Publisher: Springer
Series:Interdisciplinary Applied Mathematics, Vol. 17

James D. Murray’s Mathematical Biology is one of the classic introductions to the use of mathematical modelling in biology. Rather than presenting mathematics as an abstract collection of techniques, Murray begins with biological questions and shows how equations can be constructed to describe and investigate them. The book ranges across population dynamics, interacting species, reaction kinetics, biological oscillators and switches, infectious-disease models, reaction–diffusion systems, chemotaxis, biological waves and pattern generation. The principal mathematical machinery is based on ordinary differential equations, supplemented by nonlinear dynamics and introductory spatial modelling, making the book particularly appropriate for advanced undergraduate and graduate students with a solid applied-mathematics background. 

A major strength of the book is the close relationship between mathematical structure and biological interpretation. Murray repeatedly demonstrates how relatively simple equations can generate surprisingly complicated biological behaviour: stable populations, oscillations, epidemics, travelling waves and spatial patterns. Models are not treated simply as equations to solve; assumptions are identified, parameters are interpreted biologically, and the resulting predictions are connected back to real phenomena. Examples come from population ecology, developmental biology, physiology, epidemiology and evolution, while exercises throughout encourage the reader to construct and analyze models rather than merely reproduce calculations. 

The book is especially valuable because it illustrates the broader philosophy of mathematical modelling: mathematics provides a simplified representation of biological reality whose usefulness depends on whether it reveals mechanisms, produces testable predictions or explains observed behaviour. Murray therefore offers more than a textbook of mathematical techniques; he provides an introduction to how an applied mathematician approaches biological research. Readers interested particularly in spatial phenomena can continue with the companion Mathematical Biology II: Spatial Models and Biomedical Applications, which develops reaction–diffusion systems and applications including animal coat patterns, bacterial patterns, wound healing, brain tumours, epidemics and animal territoriality. 

Key Takeaways

• Biology provides rich mathematical problems. Population growth, epidemics, biological rhythms and pattern formation can all be investigated quantitatively.
  
• Modelling is more important than calculation alone. Choosing assumptions and translating biological mechanisms into equations are central skills.
  
• Simple nonlinear equations can produce complex behaviour, including oscillations, thresholds, waves and spatial patterns.
  
• Murray's book remains an excellent bridge between applied mathematics and biology, particularly for mathematicians who want to see how differential equations become tools for studying real biological systems. 
  

BOOK]]> Mathematical Biology: I. An Introduction
Author: James D. Murray
Publication: 3rd edition, 2002
Publisher: Springer
Series:Interdisciplinary Applied Mathematics, Vol. 17

James D. Murray’s Mathematical Biology is one of the classic introductions to the use of mathematical modelling in biology. Rather than presenting mathematics as an abstract collection of techniques, Murray begins with biological questions and shows how equations can be constructed to describe and investigate them. The book ranges across population dynamics, interacting species, reaction kinetics, biological oscillators and switches, infectious-disease models, reaction–diffusion systems, chemotaxis, biological waves and pattern generation. The principal mathematical machinery is based on ordinary differential equations, supplemented by nonlinear dynamics and introductory spatial modelling, making the book particularly appropriate for advanced undergraduate and graduate students with a solid applied-mathematics background. 

A major strength of the book is the close relationship between mathematical structure and biological interpretation. Murray repeatedly demonstrates how relatively simple equations can generate surprisingly complicated biological behaviour: stable populations, oscillations, epidemics, travelling waves and spatial patterns. Models are not treated simply as equations to solve; assumptions are identified, parameters are interpreted biologically, and the resulting predictions are connected back to real phenomena. Examples come from population ecology, developmental biology, physiology, epidemiology and evolution, while exercises throughout encourage the reader to construct and analyze models rather than merely reproduce calculations. 

The book is especially valuable because it illustrates the broader philosophy of mathematical modelling: mathematics provides a simplified representation of biological reality whose usefulness depends on whether it reveals mechanisms, produces testable predictions or explains observed behaviour. Murray therefore offers more than a textbook of mathematical techniques; he provides an introduction to how an applied mathematician approaches biological research. Readers interested particularly in spatial phenomena can continue with the companion Mathematical Biology II: Spatial Models and Biomedical Applications, which develops reaction–diffusion systems and applications including animal coat patterns, bacterial patterns, wound healing, brain tumours, epidemics and animal territoriality. 

Key Takeaways

• Biology provides rich mathematical problems. Population growth, epidemics, biological rhythms and pattern formation can all be investigated quantitatively.
  
• Modelling is more important than calculation alone. Choosing assumptions and translating biological mechanisms into equations are central skills.
  
• Simple nonlinear equations can produce complex behaviour, including oscillations, thresholds, waves and spatial patterns.
  
• Murray's book remains an excellent bridge between applied mathematics and biology, particularly for mathematicians who want to see how differential equations become tools for studying real biological systems. 
  

BOOK]]> https://mklab.gr/showthread.php?tid=839 Sat, 04 Jul 2026 15:18:28 +0300 mklabgr]]> https://mklab.gr/showthread.php?tid=839 Proving Darwin: Making Biology Mathematical 
by Gregory Chaitin

Summary

Proving Darwin: Making Biology Mathematical is a short, provocative book where mathematician Gregory Chaitin tries to reframe Darwin’s theory of evolution in purely mathematical terms. Instead of treating evolution only as a biological process, he explores the idea that DNA can be thought of like software, and that evolution itself can be modeled as a kind of “random walk” through a vast space of possible programs. 

He calls this approach “metabiology,” aiming to show how simple computational rules and randomness might generate increasingly complex life forms over time. Along the way, he draws connections to figures like Turing and von Neumann, and reflects on what mathematics can (and maybe cannot) ultimately explain about life and creativity. The book is less a standard biology text and more a philosophical and mathematical thought experiment about whether life can be fully captured by computation.

BOOK]]> Proving Darwin: Making Biology Mathematical 
by Gregory Chaitin

Summary

Proving Darwin: Making Biology Mathematical is a short, provocative book where mathematician Gregory Chaitin tries to reframe Darwin’s theory of evolution in purely mathematical terms. Instead of treating evolution only as a biological process, he explores the idea that DNA can be thought of like software, and that evolution itself can be modeled as a kind of “random walk” through a vast space of possible programs. 

He calls this approach “metabiology,” aiming to show how simple computational rules and randomness might generate increasingly complex life forms over time. Along the way, he draws connections to figures like Turing and von Neumann, and reflects on what mathematics can (and maybe cannot) ultimately explain about life and creativity. The book is less a standard biology text and more a philosophical and mathematical thought experiment about whether life can be fully captured by computation.

BOOK]]>