Structural Additive Theory

​Nestled between number theory, combinatorics, algebra and analysis lies a rapidly developing subject in mathematics variously known as additive combinatorics, additive number theory, additive group theory, and combinatorial number theory. Its main objects of study are not abelian groups themselves, but rather the additive structure of subsets and subsequences of an abelian group, i.e., sumsets and subsequence sums. This text is a hybrid of a research monograph and an introductory graduate textbook. With few exceptions, all results presented are self-contained, written in great detail, and only reliant upon material covered in an advanced undergraduate curriculum supplemented with some additional Algebra, rendering this book usable as an entry-level text. However, it will perhaps be of even more interest to researchers already in the field. The majority of material is not found in book form and includes many new results as well. Even classical results, when included, are given in greater generality or using new proof variations. The text has a particular focus on results of a more exact and precise nature, results with strong hypotheses and yet stronger conclusions, and on fundamental aspects of the theory. Also included are intricate results often neglected in other texts owing to their complexity. Highlights include an extensive treatment of Freiman Homomorphisms and the Universal Ambient Group of sumsets A+B, an entire chapter devoted to Hamidoune’s Isoperimetric Method, a novel generalization allowing infinite summands in finite sumset questions, weighted zero-sum problems treated in the general context of viewing homomorphisms as weights, and simplified proofs of the Kemperman Structure Theorem and the Partition Theorem for setpartitions.

​Nestled between number theory, combinatorics, algebra and analysis lies a rapidly developing subject in mathematics variously known as additive combinatorics, additive number theory, additive group theory, and combinatorial number theory. Its main objects of study are not abelian groups themselves, but rather the additive structure of subsets and subsequences of an abelian group, i.e., sumsets and subsequence sums. This text is a hybrid of a research monograph and an introductory graduate textbook. With few exceptions, all results presented are self-contained, written in great detail, and only reliant upon material covered in an advanced undergraduate curriculum supplemented with some additional Algebra, rendering this book usable as an entry-level text. However, it will perhaps be of even more interest to researchers already in the field. The majority of material is not found in book form and includes many new results as well. Even classical results, when included, are given in greater generality or using new proof variations. The text has a particular focus on results of a more exact and precise nature, results with strong hypotheses and yet stronger conclusions, and on fundamental aspects of the theory. Also included are intricate results often neglected in other texts owing to their complexity. Highlights include an extensive treatment of Freiman Homomorphisms and the Universal Ambient Group of sumsets A+B, an entire chapter devoted to Hamidoune’s Isoperimetric Method, a novel generalization allowing infinite summands in finite sumset questions, weighted zero-sum problems treated in the general context of viewing homomorphisms as weights, and simplified proofs of the Kemperman Structure Theorem and the Partition Theorem for setpartitions.

This is the fourth in a series of proceedings of the Combinatorial and Additive Number Theory (CANT) conferences, based on talks from the 2019 and 2020 workshops at the City University of New York. The latter was held online due to the COVID-19 pandemic, and featured speakers from North and South America, Europe, and Asia. The 2020 Zoom conference was the largest CANT conference in terms of the number of both lectures and participants. These proceedings contain 25 peer-reviewed and edited papers on current topics in number theory. Held every year since 2003 at the CUNY Graduate Center, the workshop surveys state-of-the-art open problems in combinatorial and additive number theory and related parts of mathematics. Topics featured in this volume include sumsets, zero-sum sequences, minimal complements, analytic and prime number theory, Hausdorff dimension, combinatorial and discrete geometry, and Ramsey theory. This selection of articles will be of relevance to both researchers and graduate students interested in current progress in number theory.

Additive Combinatorics: A Menu of Research Problems is the first book of its kind to provide readers with an opportunity to actively explore the relatively new field of additive combinatorics. The author has written the book specifically for students of any background and proficiency level, from beginners to advanced researchers. It features an extensive menu of research projects that are challenging and engaging at many different levels. The questions are new and unsolved, incrementally attainable, and designed to be approachable with various methods. The book is divided into five parts which are compared to a meal. The first part is called Ingredients and includes relevant background information about number theory, combinatorics, and group theory. The second part, Appetizers, introduces readers to the book’s main subject through samples. The third part, Sides, covers auxiliary functions that appear throughout different chapters. The book’s main course, so to speak, is Entrees: it thoroughly investigates a large variety of questions in additive combinatorics by discussing what is already known about them and what remains unsolved. These include maximum and minimum sumset size, spanning sets, critical numbers, and so on. The final part is Pudding and features numerous proofs and results, many of which have never been published. Features: The first book of its kind to explore the subject Students of any level can use the book as the basis for research projects The text moves gradually through five distinct parts, which is suitable both for beginners without prerequisites and for more advanced students Includes extensive proofs of propositions and theorems Each of the introductory chapters contains numerous exercises to help readers

Additive combinatorics is a relatively recent term coined to comprehend the developments of the more classical additive number theory, mainly focussed on problems related to the addition of integers. Some classical problems like the Waring problem on the sum of k-th powers or the Goldbach conjecture are genuine examples of the original questions addressed in the area. One of the features of contemporary additive combinatorics is the interplay of a great variety of mathematical techniques, including combinatorics, harmonic analysis, convex geometry, graph theory, probability theory, algebraic geometry or ergodic theory. This book gathers the contributions of many of the leading researchers in the area and is divided into three parts. The two first parts correspond to the material of the main courses delivered, Additive combinatorics and non-unique factorizations, by Alfred Geroldinger, and Sumsets and structure, by Imre Z. Ruzsa. The third part collects the notes of most of the seminars which accompanied the main courses, and which cover a reasonably large part of the methods, techniques and problems of contemporary additive combinatorics.

Integrating fundamental research with the technical applications of this rapidly evolving field, Structure and Functional Properties of Colloidal Systems clearly presents the connections between structure and functional aspects in colloid and interface science. It explores the physical fundamentals of colloid science, new developments of synthesis and conditioning, and many possible applications. Theory Divided into three parts, the book begins with a discussion of the theoretical side of colloid dynamics. It then transitions to dynamically arrested states and capillary forces in colloidal systems at fluid interfaces. Structure Covering the structural aspects of different colloidal systems, the second section examines electric double layers and effective interactions as well as the structure of extremely bimodal suspensions and filaments made up of microsized magnetic particles. The contributors analyze the role played by the attractive interaction, confinement, and external fields on the structure of colloidal systems. They also discuss structural aspects in food emulsions and the rheological properties of structured fluids. Functional Materials The last part focuses on examples of functional colloids. These include polymer colloids, protein-functionalized colloidal particles, magnetic particles, metallic nanoparticles, micro- and nanogels, responsive microgels, colloidal photonic crystals, microfluidics, gel-glass dispersed liquid crystals (GDLCs) devices, and nanoemulsions. This volume provides a sound understanding of the link between the structure and functional properties in two- and three-dimensional colloidal systems. It describes techniques to functionalize colloids, characterization methods, the physical fundamentals of structure formation, diffusion dynamics, transport properties in equilibrium, the physical fundamentals of nonequilibrium systems, the measuring principles to exploit properties in applications, the differences in designing lab experiments and devices, and several application examples.

"Advances in Mathematical Chemistry and Applications, Volume 1" highlights the emerging discipline of mathematical chemistry, or, more precisely, discrete mathematical chemistry. This Volume is written by internationally renowned experts in the field. It comprises of a wise integration of mathematical and chemical concepts and covers numerous applications in the field of drug discovery, bioinformatics, chemoinformatics, computational biology and ecological health. The contents of this book include chapters on mathematical structural descriptors of molecules and biomolecules, topological representation of molecular structure, connectivity matrices, use of weighted 2D Fingerprints in similarity-based virtual screening and much more. This ebook is a valuable resource for MSc and PhD students, academic personnel and researchers seeking updated and critically important information on the fundamental concepts of mathematical chemistry and their applications.

Control of Noise and Structural Vibration presents a MATLAB®-based approach to solving the problems of undesirable noise generation and transmission by structures and of undesirable vibration within structures in response to environmental or operational forces. The fundamentals of acoustics, vibration and coupling between vibrating structures and the sound fields they generate are introduced including a discussion of the finite element method for vibration analysis. Following this, the treatment of sound and vibration control begins, illustrated by example systems such as beams, plates and double walls. Sensor and actuator placement is explained as is the idea of modal sensor–actuators. The design of appropriate feedback systems includes consideration of basic stability criteria and robust active structural acoustic control. Positive position feedback (PPF) and multimode control are also described in the context of loudspeaker–duct and loudspeaker–microphone models. The design of various components is detailed including the analog circuit for PPF, adaptive (semi-active) Helmholtz resonators and shunt piezoelectric circuits for noise and vibration suppression. The text makes extensive use of MATLAB® examples and these can be simulated using files available for download from the book’s webpage at springer.com. End-of-chapter exercises will help readers to assimilate the material as they progress through the book. Control of Noise and Structural Vibration will be of considerable interest to the student of vibration and noise control and also to academic researchers working in the field. It’s tutorial features will help practitioners who wish to update their knowledge with self-study.