Artificial Intelligence
Mathematical Foundations and Concepts
A comprehensive exploration of the mathematical and theoretical underpinnings of artificial intelligence. This insightful book breaks down the fundamental principles of AI, including logic, algorithms, probability, calculus and linear algebra, providing readers with a deep and thorough understanding of the essential mathematical frameworks that drive advancements in AI. This work bridges the gap between cognitive neuroscience and artificial intelligence, examining topics from biophysical constraints to emergent intelligence through hierarchical predictive processing. It provides a comparative analysis of processing paradigms in biological and artificial systems, making it an essential resource for researchers, students, and professionals in the fields of artificial intelligence, mathematics, and computer science.
Medical AI
A Mathematical, Clinical, and Decision-Theoretic Treatise
This treatise presents a rigorous mathematical and clinical framework for the application of artificial intelligence in medicine. Covering Bayesian inference, neural network architectures for diagnostic imaging, decision-theoretic models for clinical pathways, and the ethical dimensions of algorithmic medicine, the work provides a comprehensive bridge between mathematical theory and clinical practice. An essential reference for physicians, medical informaticists, and AI researchers working at the frontier of healthcare technology.
Treatise on Systems Biology
A Comprehensive Study of Biological Organisation from Molecules to Ecosystems
A comprehensive study of biological organisation spanning every level of complexity, from molecular interactions and gene regulatory networks to cellular dynamics, organ systems, and ecosystem-level processes. This treatise integrates mathematical modelling, network theory, and computational biology to provide a unified framework for understanding how living systems are organised, regulated, and maintained. Essential reading for biologists, bioinformaticians, and anyone seeking a rigorous, systems-level understanding of life.
History of Evolutionary Thought in the Nineteenth Century
From Lamarck and Paley to Darwin, Wallace, and the First Defenders
This comprehensive scholarly work examines the historical development of evolutionary thought from antiquity through the nineteenth century, tracing the intellectual journey from early naturalistic philosophies to Darwin's revolutionary theory. Montgomery explores how Western conceptions of biological diversity evolved through distinct historical phases, from Presocratic Greek atomism and Aristotelian classification, through the medieval Great Chain of Being, to the transformative contributions of Buffon, Lamarck, Paley, Darwin, Wallace, and the first defenders of Darwinism including Huxley, Haeckel, Hooker, and Lyell. Originally presented as a livre docência thesis at the Universidade de São Paulo, this work demonstrates that evolutionary thinking has deep historical roots extending far beyond the nineteenth century.
Physics and Its Mathematical Foundations, Volume 4
A Comprehensive Treatment from Classical to Quantum Physics
Volume 4 of Physics and Mathematical Foundations presents a rigorous exploration of modern physics, from quantum mechanics to advanced mathematical methods. This volume covers Lagrangian and Hamiltonian mechanics, electrodynamics, statistical mechanics, and quantum field theory, providing a unified mathematical framework that connects classical and modern physics. Designed for advanced undergraduates, graduate students, and researchers seeking a thorough, mathematically grounded treatment of the physical sciences.
Calculus I
A Complete Course in Single-Variable Calculus
A thorough and self-contained course in single-variable calculus, covering limits, continuity, differentiation, integration, sequences, and series with full mathematical rigour. Each concept is developed from first principles with detailed proofs, worked examples, and applications drawn from physics, engineering, and the natural sciences. The treatment progresses from the foundations of the real number system through the Fundamental Theorem of Calculus and beyond, making it suitable for university students and self-directed learners alike.
Partial Differential Equations
Theory, Methods, and Applications — Core Concepts and Its Stochastic Uses
A comprehensive textbook on Partial Differential Equations covering fundamental theory, analytical and numerical methods, and real-world applications. This book provides a rigorous yet accessible treatment of PDEs, including first-order equations, the heat equation, wave equation, Laplace equation, and advanced topics in mathematical physics and engineering. The work extends into stochastic partial differential equations, bridging pure mathematics with modern applications in finance, physics, and biology.
A Treatise on Functional Analysis
Structures, Dualities, and Spectra
A rigorous and comprehensive treatment of functional analysis, covering normed and Banach spaces, Hilbert spaces, duality theory, spectral theory, and operator algebras. The work develops the subject from its topological and algebraic foundations through to advanced topics including the Hahn–Banach theorem, the open mapping theorem, compact operators, and the spectral theorem for self-adjoint operators. Designed for graduate students and researchers in mathematics and mathematical physics.
A Treatise on Real Analysis
Foundations, Structure, and the Architecture of the Continuum
A thorough and formally rigorous treatise on real analysis, developing the subject from the construction of the real numbers through measure theory, Lebesgue integration, and the topology of metric spaces. The work examines the architecture of the continuum with particular attention to completeness, compactness, convergence, and the interplay between algebraic and topological structures. An indispensable reference for graduate students and researchers in pure mathematics.