Author: Aiman S. Gannous

Graph Theory: Connectivity, Software Engineering and Bioinformatics

Publisher: De Gruyter
Series: De Gruyter Textbook
Publication Date: April 8, 2026
Language: English
Pages: approximately 158–168 pages
Format: Paperback / Softcover

DETAILS

This book is an introductory-to-intermediate textbook on graph theory with strong applications in:

• computer science,
• software engineering,
• networks,
• algorithms,
• and bioinformatics.

The author emphasizes a gradual and accessible presentation of graph theory concepts for:

• undergraduate students,
• graduate students,
• software engineers,
• and applied researchers.

The subtitle highlights three major application domains:

1. Connectivity theory
2. Software engineering
3. Bioinformatics

The text reportedly includes:

• step-by-step examples,
• graph modeling problems,
• practical applications,
• and algorithmic implementations.

EXPLANATION (What the Book Is About)

The book introduces graph theory as a mathematical framework for modeling relationships and connections.

A graph is typically represented as:

$G=(V,E)$

where:

• $V$ is a set of vertices (nodes),
• $E$ is a set of edges (connections).

The text begins with foundational graph concepts and gradually develops more advanced topics such as:

• graph connectivity,
• Eulerian and Hamiltonian paths,
• spanning trees,
• shortest paths,
• network flow,
• and graph matching.

The author especially focuses on how graph theory solves real-world problems in:

• internet search systems,
• communication networks,
• dependency analysis,
• software architecture,
• biological interaction networks,
• and computational biology.

For example, shortest-path algorithms are central to:

• routing systems,
• navigation,
• and network optimization.

A classical shortest-path formulation is:

$d(u,v)=\min\{\text{path lengths from }u\text{ to }v\}$

The book also covers algorithmic efficiency and computational complexity, helping readers analyze how graph algorithms scale.

TABLE OF CONTENTS

A publicly available contents listing from the publisher includes:

1. Introduction
2. Basic Graph Terminology
3. Graph Properties
4. Graph Representation
5. Graph Applications
6. Graph Algorithms
7. Appendix A: Algorithm Complexity
8. Bibliography
9. Index
   
   

   Author: Svetlin G. Georgiev

   Tensor Calculus on Time Scales: Dynamic Calculus and Riemannian Spaces

   
   Publisher: De Gruyter
   Series: De Gruyter Textbook
   Language: English
   Format: Paperback
   Pages: about 308
   Publication Date: February 2026

   ---

   DETAILS

   This book is an advanced mathematics textbook combining:

   • tensor calculus,
   • differential geometry,
   • Riemannian geometry,
   • relativistic mechanics,
   • and time-scale calculus.

   The book develops tensor calculus in the setting of time scales, a mathematical framework that unifies:

   • continuous calculus,
   • discrete calculus,
   • and hybrid dynamical systems.

   A central object is the tensor formulation on generalized spaces:

   $T^i_{\ j},\quad g_{ij},\quad R^i_{\ jkl}$

   The text reportedly includes:

   • coordinate transformations,
   • covariant and contravariant tensors,
   • Christoffel symbols,
   • covariant derivatives,
   • curvature tensors,
   • relativistic dynamics,
   • Lorentz transformations on arbitrary time scales,
   • and applications in physics.

   The intended audience includes:

   • graduate students,
   • mathematics researchers,
   • theoretical physicists,
   • and advanced engineering students.

   ---

   EXPLANATION (What the Book Is About)

   The book begins with the geometric foundations of tensor analysis in $N$-dimensional spaces.

   It studies vectors that transform differently under coordinate changes:

   $x^{i'},\quad A^i,\quad A_i$

   where:

   • $A^i$ are contravariant components,
   • $A_i$ are covariant components.

   The text then develops full tensor algebra:

   • tensor products,
   • contractions,
   • quotient laws,
   • and coordinate invariance.

   A major focus is Riemannian geometry, where geometry is encoded through the metric tensor:

   $ds^2=g_{ij}dx^i dx^j$

   The book introduces:

   • affine coordinates,
   • curvilinear coordinates,
   • intrinsic differentiation,
   • and covariant differentiation using Christoffel symbols:

   $\Gamma^k_{ij}$

   Curvature theory is then developed through:

   • the Riemann-Christoffel tensor,
   • Ricci tensor,
   • Einstein tensor,
   • and curvature operators.

   One distinctive aspect is the incorporation of time-scale calculus.

   Time-scale theory attempts to unify:

   • differential equations,
   • difference equations,
   • and mixed discrete-continuous systems.

   Symbolically:

   $\mathbb{T}\subseteq\mathbb{R}$

   where $\mathbb{T}$ is a general time scale. This allows physical laws to be formulated simultaneously for:

   • continuous time,
   • discrete time,
   • and hybrid models.

   The final chapters apply tensor methods to:

   • relativistic kinematics,
   • relativistic dynamics,
   • Lorentz transformations,
   • energy-momentum conservation,
   • and Lagrangian systems.

   ---

   TABLE OF CONTENTS

   Based on publisher descriptions, the book is organized approximately as follows:

   1. N-Dimensional Spaces and Coordinate Transformations
   2. Tensor Algebra
   3. Tensor Calculus in Riemannian Spaces
   4. Christoffel Symbols and Covariant Derivatives
   5. Curvature Tensors and Einstein Tensor
   6. Applications to Relativistic Dynamics and Kinematics
   7. Lorentz Transformations on Time Scales
   8. Velocity and Acceleration Vectors
   9. Lagrange Equations
   10. Conservation Laws for Energy-Momentum and Angular Momentum
       
       
   
   

Authors: Nikolai Khots and Boris Khots

Observability and Mathematics Modeling:
Lie Groups, Polynomial Equations, and Fundamental Theorems of Arithmetic

Publisher: De Gruyter
Publication Date: April 13, 2026
Language: English
Format: Hardcover
Pages: approximately 280 pages

DETAILS

This book presents a highly unconventional mathematical framework called “Observability in Mathematics.”

The core idea is that:

• arithmetic,
• algebra,
• and mathematical structures

should depend on an Observer, rather than being absolute and observer-independent.

The theory is reportedly built on:

• rejecting or modifying the classical notion of infinity,
• introducing observer-dependent arithmetic,
• redefining algebraic structures probabilistically,
• and extending these ideas into:
  • Lie groups,
  • polynomial equations,
  • quaternions,
  • and mathematical physics.

The authors claim that:

• many classical arithmetic and algebraic theorems remain approximately valid,
• but now with probabilities strictly less than 1.

The book sits at the intersection of:

• abstract algebra,
• foundations of mathematics,
• mathematical logic,
• and theoretical physics.

EXPLANATION (What the Book Is About)

Classical mathematics usually assumes:

• numbers and operations are absolute,
• mathematical truth is observer-independent,
• and infinity is a legitimate mathematical concept.

For example:

$1+1=2$

is treated as universally true.

This book challenges that viewpoint.

The authors introduce the idea that arithmetic depends on an Observer:

$A_{\mathcal O}$

where $\mathcal O$ denotes an observer-dependent arithmetic system.

Under this framework:

• algebraic operations may become probabilistic,
• classical identities may only hold with certain probabilities,
• and mathematical structures can vary depending on observational constraints.

The book then explores how this impacts:

• polynomial equations,
• Lie groups,
• quaternion algebras,
• and symmetry theory.

For example, Lie groups are central objects in geometry and physics:

$G$

especially in:

• quantum mechanics,
• relativity,
• gauge theory,
• and particle physics.

The authors reportedly reinterpret these structures using their observer-dependent framework.

Another recurring theme is the reinterpretation of classical algebraic theorems such as:

• the Fundamental Theorem of Arithmetic,
• polynomial factorization,
• and algebraic closure.

The mathematical philosophy resembles a mix of:

• constructivism,
• finitism,
• nonclassical logic,
• and alternative foundational mathematics.

The book also claims relevance to:

• relativity theory,
• quantum mechanics,
• and physical observability.

TABLE OF CONTENTS

A complete official TOC is not yet publicly available online.

However, publisher descriptions and subtitle references strongly suggest coverage of the following themes:

1. Foundations of Observability in Mathematics
2. Observer-Dependent Arithmetic
3. New Algebra with Observers
4. Polynomial Equations in Observable Mathematics
5. Lie Groups and Symmetry Structures
6. Quaternion Algebras
7. Probabilistic Arithmetic Theorems
8. Fundamental Theorems of Arithmetic Reconsidered
9. Applications to Relativity Theory
10. Applications to Quantum Mechanics
11. Logical Structures and Observer Frameworks
12. Mathematical Modeling Beyond Classical Infinity
    
    

    Authors: Alexander Leonidovich Skubachevskii, Leonid Efimovich Rossovskii

Partial Differential Equations Theory:
Sobolev Space, Weak Solution, Semigroup Theory, Fourier and Galerkin Methods

Details

• Publisher: De Gruyter
• Series: De Gruyter Textbook Series
• Language: English
• Format: Paperback / eBook
• Publication Date: March 16, 2026
• ISBN-13: 9783112229620
• ISBN-10: 3112229622

Explanation / Overview

This textbook introduces the modern theory of partial differential equations (PDEs).
The book focuses on the mathematical foundations and analytical methods used to study PDEs, including:

• Sobolev spaces and functional analysis
• Weak solutions of differential equations
• Semigroup theory for evolution equations
• Fourier methods
• Galerkin approximation methods

It is aimed at advanced undergraduate students, graduate students, and researchers in mathematics, physics, and engineering. The text emphasizes rigorous theory together with analytical techniques commonly used in mathematical physics and applied mathematics.

Table of Contents (available information)

A complete official table of contents was not publicly available in the search results, but the main thematic structure described by the publisher includes:

1. Sobolev Spaces
2. Weak Solutions of PDEs
3. Semigroup Theory
4. Fourier Methods
5. Galerkin Methods
6. Applications to Partial Differential Equations

These topics form the core framework of the book.

You can also check the publisher/product page here:

Author: Keng Hao Ooi

Capacitary Calculus:
With Special Attention to Sobolev Multiplier Spaces and Their Preduals

Publisher: De Gruyter
Series: Advances in Analysis and Geometry (Vol. 13)
Publication Date: July 2, 2026
Language: English
Format: Hardcover
Pages: approximately 200–214 pages

DETAILS

This is an advanced research monograph in:

• functional analysis,
• harmonic analysis,
• nonlinear potential theory,
• and partial differential equations (PDEs).

The book develops a unified theory of function spaces associated with capacities and non-additive measures. A central theme is the use of:

• Choquet integration,
• Bessel capacities,
• Lorentz-type spaces,
• Sobolev multiplier spaces,
• and maximal operator estimates.

The work focuses especially on analytical structures arising in:

• supercritical PDEs,
• nonlinear potential theory,
• and fine properties of Sobolev functions.

The author, Keng Hao Ooi, has worked on:

• capacities,
• Choquet integral spaces,
• Sobolev multipliers,
• weighted inequalities,
• and nonlinear harmonic analysis.

EXPLANATION (What the Book Is About)

Classical analysis often studies functions using ordinary measures such as Lebesgue measure. This book instead studies spaces built from capacities, which are generalized set functions important in potential theory.

A capacity is typically non-additive:

$\operatorname{Cap}(A\cup B)\neq \operatorname{Cap}(A)+\operatorname{Cap}(B)$

This allows analysts to measure “thinness,” singularity structure, and exceptional sets more precisely than standard measure theory.

The book develops the theory of Choquet integration, which replaces ordinary integration with integration relative to capacities:

$\int f\, d\operatorname{Cap}$

These tools are particularly useful in:

• nonlinear PDEs,
• fine regularity theory,
• and singular potential estimates.

A major focus is on Bessel capacities, which are deeply connected to Sobolev spaces:

$H^{s,p}(\mathbb{R}^n)$

The book studies how capacities characterize:

• fine properties of Sobolev functions,
• embeddings,
• trace phenomena,
• and multiplier spaces.

Another major topic is Sobolev multiplier spaces, which describe functions $m(x)$ such that multiplication preserves Sobolev regularity:

$u\mapsto mu$

These spaces are important in:

• PDE estimates,
• Schrödinger operators,
• nonlinear analysis,
• and operator theory.

The monograph also analyzes boundedness properties of maximal operators using nonlinear potential methods and proves vector-valued inequalities in this framework.

TABLE OF CONTENTS

A complete official TOC is not yet publicly available online, but publisher descriptions strongly indicate coverage of:

1. Capacities and Non-Additive Set Functions
2. Choquet Integration
3. Lorentz-Type Spaces Associated with Capacities
4. Bessel Capacities
5. Fine Properties of Sobolev Functions
6. Normability of Choquet Integral Spaces
7. Sobolev Multiplier Spaces
8. Preduals of Multiplier Spaces
9. Embedding Theorems
10. Maximal Operators in Capacity Settings
11. Vector-Valued Inequalities
12. Applications to Nonlinear Potential Theory and PDEs
    
    *
    

    Author: C. Bryan Dawson

    Multivariable Calculus Set Free: Infinitesimals Ride Again

    
    Publisher: Oxford University Press
    Publication Date: May 1, 2026
    Format: Paperback
    Pages: 752
    Language: English

    ---

    DETAILS

    This book is a large undergraduate-level textbook on multivariable calculus, but with a highly distinctive approach based on infinitesimal analysis rather than the standard epsilon-delta framework.

    It is the sequel to:

    • Calculus Set Free: Infinitesimals to the Rescue

    and continues the author’s project of teaching calculus using:

    • infinitesimals,
    • approximation relations,
    • geometric intuition,
    • and visual reasoning.

    The book covers standard multivariable-calculus topics such as:

    • partial derivatives,
    • multiple integrals,
    • vector fields,
    • line integrals,
    • surface integrals,
    • and multivariable applications,

    but reformulates them using infinitesimal methods.

    Publisher descriptions emphasize:

    • abundant diagrams,
    • exploratory exercises,
    • narrative explanations,
    • and line-by-line worked examples.

    The style is intended to remain mathematically rigorous while being more intuitive and accessible than many conventional calculus texts.

    ---

    EXPLANATION (What the Book Is About)

    Most traditional calculus textbooks define limits using epsilon-delta arguments.

    This book instead develops multivariable calculus using infinitesimals:
    quantities that are infinitely small but nonzero.

    A central approximation relation used throughout the text is:

    $f(x+dx)\approx f(x)+f'(x)dx$

    The author argues that infinitesimal reasoning:

    • simplifies computations,
    • improves geometric intuition,
    • and makes advanced concepts easier to visualize.

    The book develops multivariable calculus geometrically through local linearity:

    $f(x,y)\approx f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$

    This interpretation helps explain:

    • tangent planes,
    • differential approximations,
    • Jacobians,
    • and integration formulas.

    The text reportedly reinterprets:

    • double integrals,
    • triple integrals,
    • line integrals,
    • and surface integrals

    as sums over infinitely many infinitesimal regions.

    For example, a double integral is viewed as:

    $\iint_R f(x,y)\,dA$

    where the region is partitioned into infinitesimal subregions.

    The book also emphasizes the geometric idea that curved objects can locally be approximated by linear ones:

    $\Delta z\approx f_x\Delta x+f_y\Delta y$

    According to publisher summaries, the text attempts to:

    • demystify multivariable calculus,
    • reduce technical overhead,
    • and make difficult ideas more intuitive without sacrificing rigor.

    A special review chapter is included for readers unfamiliar with infinitesimal methods, covering:

    • single-variable calculus,
    • infinitesimal notation,
    • and approximation principles.

    ---

    TABLE OF CONTENTS

    A complete official TOC is not yet publicly available online.

    However, based on publisher descriptions and the stated scope, the book likely includes chapters such as:

    1. Review of Infinitesimals and Approximation
    2. Geometry of Multivariable Functions
    3. Partial Derivatives
    4. Tangent Planes and Local Linearity
    5. Directional Derivatives and Gradients
    6. Multiple Integrals
    7. Coordinate Transformations
    8. Vector Fields
    9. Line Integrals
    10. Surface Integrals
    11. Divergence and Curl
    12. Integral Theorems
    13. Applications to Geometry and Physics
    14. Advanced Approximation Methods
    15. Worked Examples and Exploratory Exercises