* ISBN-13: 978-1041358596 / 9781041358596
* Publisher: CRC Press (Taylor & Francis Group)
* Publication Date: October 2026
* Binding: Hardcover
* Language: English
* Pages: Approx. 200 pages
This textbook is designed to offer a visual and geometric approach to functional analysis. Functional analysis can often feel abstract and highly analytical, but this book bridges the gap by providing strong geometric intuition behind complex mathematical structures. Aimed at college and university-level students, it helps readers visualize concepts in infinite-dimensional spaces, making fundamental topics like normed spaces, operators, and Hilbert spaces much more accessible and intuitive.
Chapter 1: Geometric Intuition in Linear Spaces
From finite to infinite dimensions
Visualizing vector spaces and norms
Chapter 2: Normed and Banach Spaces
Convergence and completeness
Classic examples viewed geometrically
Chapter 3: Inner Product and Hilbert Spaces
Orthogonality and projections
The geometry of Hilbert space approximations
Chapter 4: Linear Operators and Functionals
Bounded linear mappings and their visual meaning
Dual spaces and the Hahn-Banach theorem
Chapter 5: Fundamental Theorems of Functional Analysis
The Open Mapping Theorem and Closed Graph Theorem
Geometric interpretations of operator bounds
Chapter 6: Introduction to Spectral Theory
Eigenvalues in infinite dimensions
Self-adjoint operators and their geometric impact
ISBN-13: 978-1032894249 / 9781032894249
Publisher: Chapman & Hall / CRC (Taylor & Francis Group)
Series: Chapman & Hall/CRC Numerical Analysis and Scientific Computing
Series
Publication Date: October/November 2026
Binding: Hardcover / Paperback
Language: English
Pages: 528 pages
This book provides a thorough and comprehensive guide to mathematical modelling using ordinary differential equations (ODEs). It is designed for students and researchers in applied mathematics, engineering, and scientific computing. This updated second edition introduces various numerical methods alongside fundamental theory, while expanding on modern topics such as neural networks, residual networks, and reservoir computing applied to dynamical systems. It emphasizes the full pipeline of modelling-from formulating equations to analysis, simulation, optimal control, and inverse problems.
Chapter 1: Introduction
Chapter 2: Elementary solution methods for simple ODEs
Chapter 3: Theory of ordinary differential equations
Chapter 4: Systems of ordinary differential equations
Chapter 5: Higher-order ordinary differential equations
Chapter 6: Mechanics and second-order ODEs
Chapter 7: Numerical solution of ODE problems
Chapter 8: Stability of ODE systems
Chapter 9: ODEs and the calculus of variations
Chapter 10: Optimal control of ODE models
Chapter 11: Inverse problems with ODE models
Chapter 12: Differential games
Chapter 13: Stochastic differential equations
Chapter 14: Neural networks and ODE problems
ISBN-13: 978-3111372419 / 9783111372419
Publisher: De Gruyter
Publication Date: October 19, 2026
Binding: Hardcover
Language: English
Pages: 268 pages
This advanced mathematical work provides a systematic study of Fourier expansions of automorphic forms on GL(2) at arbitrary cusps, with a primary focus on behaviors at finite cusps. While classical modular form theory predominantly emphasizes expansions and behaviors at infinity, this book centers finite cusps within the arithmetic theory. By fusing classical modular form theory with modern adelic and representation-theoretic methods, it details both classical and adelic approaches to Fourier expansions.
Key mathematical developments in the book include the use of local Whittaker models to explain Fourier coefficients at finite cusps, alongside metaplectic and Weil representation frameworks for half-integral weight modular forms and theta functions. It is tailored for graduate-level students and researchers specializing in analytic number theory, representation theory, and arithmetic geometry.
Chapter 1: Classical Modular Forms and Fourier Expansions
Foundations of modular forms on GL(2)
Classic Fourier expansions at the cusp at infinity
Chapter 2: The Geometry of Cusps on Congruence Subgroups
Defining finite and infinite cusps classically and adelically
Coset decompositions and scaling matrices
Chapter 3: Adelic Formulations of Automorphic Forms
Transitioning from classical to adelic representations on GL(2)
Adelic Schwartz-Bruhat spaces
Chapter 4: Local Whittaker Models and Finite Cusps
Whittaker functions and their local properties
Evaluating local Whittaker vectors at non-trivial cusps
Chapter 5: Metaplectic Groups and Weil Representations
The metaplectic cover of SL_2 and GL_2
The Weil representation and half-integral weight forms
Chapter 6: Theta Functions and Fourier Coefficients at Finite Cusps
Explicit formulas for theta coefficients at arbitrary cusps
Transformation laws and twisted theta series
Chapter 7: Applications and Arithmetic Conjectures
Connections to the work of Goldfeld and Gunnells
L-functions and asymptotic behaviors
ISBN-13: 978-0262057271 / 9780262057271
Publisher: MIT Press
Publication Date: October 2026 (Paperback edition; Hardcover originally
published in 2024)
Binding: Paperback
Language: English
Pages: 240 pages
In Beautiful Math, author Chris Bernhardt (known for Quantum Computing for Everyone) explores the mathematical foundations at the absolute core of the modern digital information age. The book is written for general readers and math enthusiasts alike, explaining complex technological concepts with surprisingly simple mathematical models.
Bernhardt structures his exploration around four major pillars: information, communication, computation, and learning. The text clarifies how everyday digital systems operate, exploring topics such as:
* The transition from analog to digital data.
* The mechanics behind algorithms and universal computing.
* How data is compressed, encrypted, and protected from digital noise.
* The fundamental principles behind artificial intelligence and machine learning.
Introduction: A Roadmap
Chapter 1: Digital Revolutions
Chapter 2: Information
Chapter 3: Information, Redundancy, and Compression
Chapter 4: Error Correction and Noise
Chapter 5: Encryption
Chapter 6: Analog-to-Digital Conversion
Chapter 7: Computation
Chapter 8: Machine Learning
Chapter 9: Neural Networks
Appendix A, B, & C
ISBN-13: 978-0197917053 / 9780197917053
Publisher: Oxford University Press
Publication Date: January 2027 (Expected / Pre-order available in late
2026)
Binding: Hardcover
Language: English
Pages: 432 pages
This textbook provides an accessible, visually intuitive, yet mathematically rigorous introduction to Fourier series. Designed to be suitable for both undergraduate and graduate courses, the book bridges the gap between pure mathematics and real-world utility.
It stands out by weaving significant applications from physics, engineering, data analysis, and differential equations directly into the theoretical narrative. Rather than presenting Fourier analysis purely as abstract machinery, the authors demonstrate its direct impact on diverse modern technologies and scientific fields-ranging from signal processing and digital filtering to predicting ocean tides and medical imaging. The book is highly practical and includes a vast collection of exercises with full solutions provided at the back of the book.
Where to Start
Chapter 1: Fourier Series: Development of the Theory
1.1 Periodic functions
1.2 How to "optimally" approximate periodic functions
1.3 Some further observations in L^2 ((0,1))
1.4 Decay of Fourier coefficients
1.5 The quest for convergence
1.6 Pointwise convergence of Fourier Series "all in one breath"
1.7 Pointwise convergence of Fourier Series: a refined version
1.8 Uniqueness results
1.9 More on the decay of Fourier coefficients
1.10 Uniform convergence results
1.11 Convergence in L^2 ((0,1))
1.12 Convergence in L^p ((0,1))
1.13 Averaging procedures for Fourier Series
1.14 Persistent overshooting phenomena
1.15 Convergence issues
1.16 Functions of arbitrary periods
1.17 Fourier Series in any dimension
Chapter 2: Fourier Series: Applications
2.1 The old-fashioned epicycle theory
2.2 Does anyone need a computer?
2.3 Predicting tides
2.4 Partial differential equations
2.5 Fourier's cellar
2.6 The Dirichlet problem on the two-dimensional disk
2.7 The Dirichlet problem on a square and the behaviour near corners
2.8 Calculating the exact value of a series
2.9 Number theory
2.10 Inequalities of analytic flavour
2.11 Inequalities of geometric flavor
2.12 The Weierstrass Approximation Theorem
2.13 The Radon Transform
2.14 Filters
2.15 Edge detection
2.16 Denoising
2.17 Who listens to the radio?
2.18 Overdrives, clipping, distortion, and ringing artifacts
2.19 Linear motions on tori
2.20 Weyl's Equidistribution Theorem
2.21 Minkowski's Theorem on convex sets
2.22 The Central Limit Theorem
What Comes Next?
Appendix A: One more example confirming Theorem 1.15.1
Solutions to selected exercises