Bøger udgivet af Birkhauser

This monograph offers a self-contained introduction to the regularity theory for integro-differential elliptic equations, mostly developed in the 21st century. This class of equations finds relevance in fields such as analysis, probability theory, mathematical physics, and in several contexts in the applied sciences. The work gives a detailed presentation of all the necessary techniques, with a primary focus on the main ideas rather than on proving all the results in their greatest generality.The basic building blocks are presented first, with the study of the square root of the Laplacian, and weak solutions to linear equations. Subsequently, the theory of viscosity solutions to nonlinear equations is developed, and proofs are provided for the main known results in this context. The analysis finishes with the investigation of obstacle problems for integro-differential operators and establishes the regularity of solutions and free boundaries.A distinctive feature of this work lies in its presentation of nearly all covered material in a monographic format for the first time, and several proofs streamline, and often simplify, those in the original papers. Furthermore, various open problems are listed throughout the chapters.

The aim of this book is to introduce the reader to different topics of the theory of elliptic partial differential equations by avoiding technicalities and complicated refinements. Apart from the basic theory of equations in divergence form, it includes subjects as singular perturbations, homogenization, computations, asymptotic behavior of problems in cylinders, elliptic systems, nonlinear problems, regularity theory, Navier-Stokes systems, p-Laplace type operators, large solutions, and mountain pass techniques. Just a minimum on Sobolev spaces has been introduced and work on integration on the boundary has been carefully avoided to keep the reader attention focused on the beauty and variety of these issues. The chapters are relatively independent of each other and can be read or taught separately. Numerous results presented here are original, and have not been published elsewhere. The book will be of interest to graduate students and researchers specializing in partial differential equations.

This monograph presents a mathematically rigorous and accessible treatment of the interaction between information, decision, control, and probability in single-agent and multi-agent systems. The book provides a comprehensive and unified theory of information structures for stochastic control, stochastic teams, stochastic games, and networked control systems.Part I of the text is concerned with a general mathematical theory of information structures for stochastic teams, leading to systematic characterizations and classifications, geometric and topological properties, implications on existence, approximations and relaxations, their comparison, and regularity of optimal solutions in information. Information structures in stochastic games are then considered in Part II, and the dependence of equilibrium solutions and behavior on information is demonstrated. Part III studies information design through information theory in networked control systems - both linear and nonlinear - and discusses optimality and stability criteria. Finally, Part IV introduces information and signaling games under several solution concepts, with applications to prior mismatch, cost mismatch and privacy, reputation games and jamming. This text will be a valuable resource for researchers and graduate students interested in control theory, information theory, statistics, game theory, and applied mathematics. Readers should be familiar with the basics of linear systems theory, stochastic processes, and Markov chains.

Athanasius Kircher, the eminent 17th-century German Jesuit professor of mathematics at the Roman College emerges as a captivating figure within the pages of this monograph by Agustín Udías. Aptly deemed 'the man who knew everything, ' Kircher's thirty-two comprehensive works, spanning an array of subjects, provide a unique lens into his visionary perspectives. This book delves into three selected works where Kircher unveils his conceptualization of the Earth, termed the 'Geocosmos, ' treated magnetism as a cosmic and spiritual force, and embarks on a cosmic exploration from Earth to the stars. From his groundbreaking speculations on the Earth's interior, attributing earthquakes and volcanoes to intricate channels of air, water, and fire, to his cosmic journey accompanied by the ethereal spirit Cosmiel, Kircher's enduring allure persists. Despite variance from contemporary knowledge, situated at the beginning of modern science, Kircher's proposals of the structure of the Earth's interior, cosmic magnetic theories, and space journey to the stars offer a compelling glimpse into the intellectual landscape of a bygone era, making this book an essential exploration for scholars seeking a nuanced understanding of Kircher's profound influence.

In this captivating historical novel, Daniele Struppa skillfully weaves a fictional autobiography, bringing Fibonacci to life with vivid details of his upbringing and adult years in Medieval Europe. As we explore the historical context of Fibonacci's time, we delve into the intriguing aspects of a bygone era, painting a compelling picture of a man whose contributions to mathematics continue to resonate today. From his groundbreaking work on congruent numbers to the famous numerical sequence that bears his name, the author invites readers to imagine the creative sparks that ignited Fibonacci's mathematical innovations. When historical evidence is elusive, accuracy and passion are seamlessly combined, offering plausible scenarios grounded in documented facts. A meticulously crafted apparatus of notes distinguishes fact from fiction, providing readers with a clear guide to navigate this enthralling reconstruction of Fibonacci's life. Step into the medieval world ofLeonardo Fibonacci, one of the most celebrated mathematicians in history, and discover the man behind the mathematical genius. Mathematicians and curious readers alike will appreciate the allure of Fibonacci's mathematical brilliance.

The NLAGA's Biennial International Research Symposium (NLAGA-BIRS) is intended to gather African expertises in Nonlinear Analysis, Geometry and their Applications with their international partners in a four days conference where new mathematical results are presented and discussed. This book features the best papers presented during this Biennial. The different topics addressed are related to Partial Differential Equations, Differential inclusions, Geometrical Analysis of Optimal Shapes, Complex Analysis, Geometric Structures, Algebraic Geometry, Algebraic, Optimization, Optimal Control and Mathematical modeling. The main focus of the NLAGA project is to deepen and consolidate the development in West and Center Africa of Nonlinear Analysis, Geometry and their Applications, aimed at solving in particular real-world problems such as coastal erosion, urban network, pollution problems, and population dynamics.

Felix Hausdorff is a singular phenomenon in the history of science. As a mathematician, he played a major role in shaping the development of modern mathematics in the 20th century. He founded general topology as an independent mathematical discipline, while enriching set theory with a number of fundamental concepts and results. His general approach to measure and dimension led to profound developments in numerous mathematical disciplines, and today Hausdorff dimension plays a central role in fractal theory with its many fascinating applications by means of computer graphics. Hausdorff 's remarkable mathematical versatility is reflected in his published work: today, no fewer than thirteen concepts, theorems and procedures carry his name. Yet he was not only a creative mathematician - Hausdorff was also an original philosophical thinker, a poet, essayist and man of letters. Under the pseudonym Paul Mongré, he published a volume of aphorisms, an epistemological study, abook of poetry, an oft-performed play, and a number of notable essays in leading literary journals. As a Jew, Felix Hausdorff was increasingly persecuted and humiliated under the National Socialist dictatorship. When deportation to a concentration camp was imminent, he, along with his wife and sister-in law, decided to take their own lives. This book will be of interest to historians and mathematicians already fascinated by the rich life of Felix Hausdorff, as well as to those readers who wish to immerse themselves in the intricate web of intellectual and political transformations during this pivotal period in European history.

This book meticulously examines over one hundred documents of research notes by Albert Einstein, many of which were previously unidentified, held in the archives of The Hebrew University of Jerusalem and the Einstein Papers Project at Caltech. Focused on Einstein's quest for a five-dimensional unified field theory of gravitation and electromagnetism, the analysis provides unique insights into his mathematical skills, thinking, and modus operandi. This academic exploration also investigates the role of mathematics in Einstein's theorizing with a special focus on projective geometry and delta functions.

This contributed volume presents recent advances as well as new directions in number theory and its applications. Algebraic and analytic number theory are the main focus with chapters showing how these areas are rapidly evolving. By gathering authors from over seven countries, readers will gain an international perspective on the current state of research as well as potential avenues to explore. Specific topics covered include: Algebraic Number TheoryElliptic curves and CryptographyHopf Galois theoryAnalytic and elementary number theory and applicationsNew Frontiers in Number Theory and Applications will appeal to researchers interested in gaining a global view of current research in number theory.

This book is devoted to the least gradient problem and its variants. The least gradient problem concerns minimization of the total variation of a function with prescribed values on the boundary of a Lipschitz domain. It is the model problem for studying minimization problems involving functionals with linear growth. Functions which solve the least gradient problem for their own boundary data, which arise naturally in the study of minimal surfaces, are called functions of least gradient.The main part of the book is dedicated to presenting the recent advances in this theory. Among others are presented an Euler-Lagrange characterization of least gradient functions, an anisotropic counterpart of the least gradient problem motivated by an inverse problem in medical imaging, and state-of-the-art results concerning existence, regularity, and structure of solutions. Moreover, the authors present a surprising connection between the least gradient problem and the Monge-Kantorovich optimal transport problem and some of its consequences, and discuss formulations of the least gradient problem in the nonlocal and metric settings. Each chapter is followed by a discussion section concerning other research directions, generalizations of presented results, and presentation of some open problems. The book is intended as an introduction to the theory of least gradient functions and a reference tool for a general audience in analysis and PDEs. The readers are assumed to have a basic understanding of functional analysis and partial differential equations. Apart from this, the text is self-contained, and the book ends with five appendices on functions of bounded variation, geometric measure theory, convex analysis, optimal transport, and analysis in metric spaces.

This text presents a collection of mathematical exercises with the aim of guiding readers to study topics in statistical physics, equilibrium thermodynamics, information theory, and their various connections. It explores essential tools from linear algebra, elementary functional analysis, and probability theory in detail and demonstrates their applications in topics such as entropy, machine learning, error-correcting codes, and quantum channels. The theory of communication and signal theory are also in the background, and many exercises have been chosen from the theory of wavelets and machine learning. Exercises are selected from a number of different domains, both theoretical and more applied. Notes and other remarks provide motivation for the exercises, and hints and full solutions are given for many. For senior undergraduate and beginning graduate students majoring in mathematics, physics, or engineering, this text will serve as a valuable guide as theymove on to more advanced work.

Metric algebraic geometry combines concepts from algebraic geometry and differential geometry. Building on classical foundations, it offers practical tools for the 21st century. Many applied problems center around metric questions, such as optimization with respect to distances.After a short dive into 19th-century geometry of plane curves, we turn to problems expressed by polynomial equations over the real numbers. The solution sets are real algebraic varieties. Many of our metric problems arise in data science, optimization and statistics. These include minimizing Wasserstein distances in machine learning, maximum likelihood estimation, computing curvature, or minimizing the Euclidean distance to a variety.This book addresses a wide audience of researchers and students and can be used for a one-semester course at the graduate level. The key prerequisite is a solid foundation in undergraduate mathematics, especially in algebra and geometry. This is an openaccess book.

This book provides a successful solution to one of the central problems of mathematical fluid mechanics: the Leray's problem on existence of a solution to the boundary value problem for the stationary Navier--Stokes system in bounded domains under sole condition of zero total flux. This marks the culmination of the authors' work over the past few years on this under-explored topic within the study of the Navier--Stokes equations. This book will be the first major work on the Navier--Stokes equations to explore Leray's problem in detail. The results are presented with detailed proofs, as are the history of the problem and the previous approaches to finding a solution to it. In addition, for the reader's convenience and for the self-sufficiency of the text, the foundations of the mathematical theory for incompressible fluid flows described by the steady state Stokes and Navier--Stokes systems are presented. For researchers in this active area, this book will be a valuable resource.

This volume features presentations from the International Workshop on Operator Theory and its Applications that was held in Kraków, Poland, September 6-10, 2022. The volume reflects the wide interests of the participants and contains original research papers in the active areas of Operator Theory. These interests include weighted Hardy spaces, geometry of Banach spaces, dilations of the tetrablock contractions, Toeplitz and Hankel operators, symplectic Dirac operator, pseudodifferential and differential operators, singular integral operators, non-commutative probability, quasi multipliers, Hilbert transform, small rank perturbations, spectral constants, Banach-Lie groupoids, reproducing kernels, and the Kippenhahn curve. The volume includes contributions by a number of the world's leading experts and can therefore be used as an introduction to the currently active research areas in operator theory.

This monograph explores the use of mathematical modeling and control theory in a variety of contemporary challenges in mathematical biology and environmental sciences. Emphasizing an approach of learning by doing, the authors focus on a set of significant case studies emerging from real-world problems and illustrate how mathematical techniques and computational experiments can be employed in the search for sustainable solutions.The following topics are extensively discussed:Eradicability and control of a paradigmatic epidemic model, with a view to the existence of endemic states, their stability, and the existence of travelling wavesA spatially structured epidemic model concerning malaria as an example of vector-borne epidemicsOptimal harvesting problems for space-structured and age-structured population dynamicsControlling epidemics in agriculture due to pest insectsThe role of predators as a possible biocontrol agent of epidemics in agricultureControl by taxation of the environmental pollution produced by human activitiesThe originality of this text is in its leitmotif ¿ regional control ¿ along the principle of ¿Think Globally, Act Locally.¿ Indeed, for example, in many real spatially structured ecosystems, it is practically impossible to control the relevant system by global interventions in the whole habitat.Proofs are given whenever they may serve as a guide to the introduction of new concepts. Each chapter includes a comprehensive description of the numerical methods used for the computational experiments, and MATLAB© codes for many of the numerical simulations are available for download. Several challenging open problems are also provided to stimulate future research.This text is aimed at mathematicians, engineers, and other scientists working in areas such as biology, medicine, and economics. Graduate and advanced undergraduate students of a quantitative subject related to the analysis and applications of dynamical systems and their control will also find it to be a valuable resource.

The book gives the basic results of the theory of the spaces Ap¿ of functions holomorphic in the unit disc, halfplane and in the finite complex plane, which depend on functional weights ¿ permitting any rate of growth of a function near the boundary of the domain. This continues and essentially improves M.M. Djrbashian's theory of spaces Ap¿ (1945) of functions holomorphic in the unit disc, the English translation of the detailed and complemented version of which (1948) is given in Addendum to the book. Besides, the book gives the ¿-extensions of M. M. Djrbashian's two factorization theories of functions meromorphic in the unit disc of 1945-1948 and 1966-1975 to classes of functions delta-subharmonic in the unit disc and in the half-plane.The book can be useful for a wide range of readers. It can be a good handbook for Master, PhD students and Postdoctoral Researchers for enlarging their knowledge and analytical methods, as well as a useful resource for scientists who want to extend their investigation fields.

Many philosophers, physicists, and mathematicians have wondered about the remarkable relationship between mathematics with its abstract, pure, independent structures on one side, and the wilderness of natural phenomena on the other. Famously, Wigner found the "effectiveness" of mathematics in defining and supporting physical theories to be unreasonable, for how incredibly well it worked. Why, in fact, should these mathematical structures be so well-fitting, and even heuristic in the scientific exploration and discovery of nature? This book argues that the effectiveness of mathematics in physics is reasonable. The author builds on useful analogies of prime numbers and elementary particles, elementary structure kinship and the structure of systems of particles, spectra and symmetries, and for example, mathematical limits and physical situations. The two-dimensional Ising model of a permanent magnet and the proofs of the stability of everyday matter exemplify such effectiveness, and the power of rigorous mathematical physics. Newton is our original model, with Galileo earlier suggesting that mathematics is the language of Nature.

This volume presents modern developments in analysis, PDEs and geometric analysis by some of the leading worldwide experts, prominent junior and senior researchers who were invited to be part of the Ghent Analysis & PDE Center Methusalem Seminars from 2021 to 2022. The contributions are from the speakers of the Methusalem Colloquium, Methusalem Junior Seminar and Geometric Analysis Seminar. The volume has two main topics: 1. Analysis and PDEs. The volume presents recent results in fundamental problems for solving partial integro-differential equations in different settings such as Euclidean spaces, manifolds, Banach spaces, and many others. Discussions about the global and local solvability using micro-local and harmonic analysis methods, studies of new techniques and approaches arising from a physical perspective or the mathematical point of view have also been included. Several connected branches arising in this regard are shown. 2. Geometric analysis. The volume presents studies of modern techniques for elliptic and subelliptic PDEs that in recent times have been used to establish new results in differential geometry and differential topology. These topics involve the intrinsic research in microlocal analysis, geometric analysis, and harmonic analysis abroad. Different problems having relevant geometric information for different applications in mathematical physics and other problems of classification have been considered.

This volume explores state-of-the-art developments in theoretical and applied fluid mechanics with a focus on stabilization and control. Chapters are based on lectures given at the summer school "Fluids under Control", held in Prague from August 23-27, 2021. With its accessible and flexible presentation, readers will be motivated to deepen their understanding of how mathematics and physics are connected. Specific topics covered include: Stabilization of the 3D Navier-Stokes systemFlutter stabilization of flow-state systemsTurbulence controlDesign through analysis Fluids Under Control will appeal to graduate students and researchers in both mathematics and physics. Because of the applications presented, it will also be of interest to engineers working on environmental and industrial issues.

John Corcoran was a very well-known logician who worked on several areas of logic. He produced decisive works giving a better understanding of two major figures in the history of logic, Aristotle and Boole. Corcoran had a close association with Alfred Tarski, a prominent 20th-century logician. This collaboration manifested in Corcoran's substantial introduction to Tarski's seminal book, Logic, Semantics, Metamathematics (1956). Additionally, Corcoran's posthumous editorial involvement in 'What are logical notions?' (1986) breathed new life into this seminal paper authored by Tarski. His scholarly pursuits extended to the intricate explication of fundamental concepts in modern logic, including variables, propositions, truth, consequences, and categoricity. Corcoran's academic curiosity extended further to the intersection of ethics and logic, reflecting his contemplation of their interrelation. Beyond these theoretical contributions, Corcoran was deeply engaged in the pedagogical dimensions of logic instruction. This volume serves as a compilation of articles contributed by Corcoran's students, colleagues, and international peers. By encompassing a diverse range of subjects, this collection aptly mirrors Corcoran's wide-ranging interests, offering insights that not only deepen our understanding of his work but also advance the theoretical frameworks he explored.

Das in Zurich 1996 gegrundete Architekturburo von Mateja Vehovar (*1961, Absolventin der Columbia University New York und der ETH Zurich) und Stefan Jauslin (*1966, Studium an der ETH Zurich) zeichnet sich durch grosse Interdisziplinaritat aus. Neben Architekturaufgaben widmen sie sich der Landschaftsarchitektur, Aussenraumgestaltungen, Ausstellungsdesign und Rauminstallationen. 1999 wurden Vehovar & Jauslin mit dem Eidgenossischen Preis fur freie Kunst ausgezeichnet. Die Publikation dokumentiert 16 ausgewahlte Arbeiten und stellt sie in den aktuellen Kontext experimenteller Raumgestaltungen. Jacqueline Burckhardt ist Prasidentin der Eidg. Kunstkommission und Herausgeberin von "Parkett"; Jakob Hotz studierte an der Architectural Association in London und arbeitet seit 2001 als Architekt in Zurich; Cary Siress studierte in New York, seit 1996 ist er Assistenzprofessor an der Architekturabteilung der ETH Zurich."

One of the fundamental considerations in landscape architecture is whether to create something which appears to be natural, or to design something deliberately artificial. Always moving between nature and artifice the art of landscape architecture expresses itself in a sensitive awareness of time and place. If the design is not to become a mere ornament, then it must be supported by concepts and visions, daring to create something new from the existing surroundings. This collection of essays which have been published over the last decade in Topos European Landscape Magazine, is a valuable contribution to the literature in the specialised field of landscape architecture. The authors include Paolo Burgi who writes on dimensions of memory, Joachim W. Jacobs who investigates the Bauhaus and the theory of space, and Kathinka Schreiber who takes a critical look at landscape in film."

A central principle in the design of large-scale distributed systems is that components should be organized to place those that interact frequently close together. This is essentially a basic clustering problem, but the context creates new challenges. Traditional clustering algorithms are designed to work on relatively simple units of information stored in a centralized database. This work explores the consequences of clustering autonomous entities, each with individual, possibly different, criteria defining similarity and cluster composition requirements. In this setting clustering is transformed from being mainly a catagorization task, into a problem of discovering similarity criteria and classification categories. Original research results define a general model of decentralized clustering of autonomous entities, and present simulations investigating key process, from matchmaking, to catagorization, to learning behaviors needed for adaptive cluster discovery.

Photovoltaik ist im Kontext von Energiekrise und Klimawandel eine der vielversprechendsten Technologien zur globalen Energiegewinnung. Photovoltaikmodule stehen inzwischen in einer solch grossen Bandbreite zur Verfügung, dass nahezu alle üblichen flächigen Bauteile mit photovoltaischer Funktion versehen werden können. Neben der Energiegewinnung bieten die Module zahlreiche Synergieeffekte, indem sie zunehmend wie Verglasungsbauteile eingesetzt werden und weitere Funktionen wie Witterungs-, Sonnen-, und Sichtschutz übernehmen. Für spezielle Anwendungen stehen Sondermodule wie z.B. Solardachziegel und Solarmembranen zur Verfügung. Dieses Buch erklärt die Technologie, zeigt die verfügbaren Produkte und vermittelt anschaulich deren Einsatz am Gebäude mit Schwerpunkt auf großmaßstäblichen Bauten. Es liefert Architekten alles notwendige Know-how, um ein neues oder bestehendes Gebäude mit einer Photovoltaikanlage auszustatten - Planung und Ausführung verbindend. Nicht zuletzt ist es ein wertvolles Arbeitsinstrument, um sich auf die Kommunikation mit den relevanten Herstellern und den Bauherren vorzubereiten.

I: Genetics and conservation biology.- Introductory remarks: Genetics and conservation biology.- Global issues of genetic diversity.- II: Genetic variation and fitness.- Introductory remarks.- Genetic variation and fitness: Conservation lessons from pines.- Genetic diversity and fitness in small populations.- Mutation load depending on variance in reproductive success and mating system.- Extinction risk by mutational meltdown: Synergistic effects between population regulation and genetic drift.- III: Inbreeding, population and social structure.- Introductory remarks.- Inbreeding: One word, several meanings, much confusion.- The genetic structure of metapopulations and conservation biology.- Effects of inbreeding in small plant populations: Expectations and implications for conservation.- The interaction of inbreeding depression and environmental stochasticity in the risk of extinction of small populations.- Genetic structure of a population with social structure and migration.- Guidelines in conservation genetics and the use of the population cage experiments with butterflies to investigate the effects of genetic drift and inbreeding.- IV: Molecular approaches to conservation.- Introductory remarks.- Rare alleles, MHC and captive breeding.- Andean tapaculos of the genus Scytalopus (Aves, Rhinocryptidae): A study of speciation using DNA sequence data.- Genetic distances and the setting of conservation priorities.- Multi-species risk analysis, species evaluation and biodiversity conservation.- V: Case studies.- Introductory remarks.- On genetic erosion and population extinction in plants: A case study in Scabiosa columbaria and Salvia pratensis.- Effects of releasing hatchery-reared brown trout to wild trout populations.- Genetics and demography of rare plants and patchily distributed colonizing species.- Response to environmental change: Genetic variation and fitness in Drosophila buzzatii following temperature stress.- Alternative life histories and genetic conservation.- The principles of population monitoring for conservation genetics.- VI: Genetic resource conservation.- Introductory remarks.- Optimal sampling strategies for core collections of plant genetic resources.- Conservation genetics and the role of botanical gardens.- Animal breeding and conservation genetics.- Scenarios.- Introductory remarks.- A: The genetic monitoring of primate populations for their conservation.- B: Heavy metal tolerance, plant evolution and restoration ecology.- C: Genetic conservation and plant agriculture.- D: Fragmented plant populations and their lost interactions.- E: Host-pathogen coevolution under in situ conservation.- Concluding remarks.

For the first time, this book documents the extensive work of Emmy Zweybruck and her school. Zweybruck was an applied artist who made innovative contributions in graphic design, book design and in toy production in Austria and the USA, and who started her own workshop early on in her career. Outstandingly illustrated and with a rich documentation, this publication provides the basis for all further research."

Parallel Completion Techniques.- The Computation of Gröbner Bases Using an Alternative Algorithm.- Symmetrization Based Completion.- On the Reduction of G-invariant Polynomials for an Arbitrary Permutation Groups G.- The Non-Commutaive Gröbner Freaks.- Alternatives in Implementing Noncommutative Gröbner Basis Systems.- String Rewriting and Gröbner Bases - A General Approach to Monoid and Group Rings.- Gröbner Fans and Projective Schemes.- Normalized Rewriting: A Unified View of Knuth-Bendix Completion and Gröbner Bases Computation.- New Directions for Syntactic Termination Orderings.- Two-sided Gröbner Bases in Iterated Ore Extensions.- Computing the Torsion Group of Elliptic Curves by the Method of Gröbner Bases.- Finding a Finite Group presentation Using Rewriting.- Deciding Degree-Four-Identities for Alternative Rings by Rewriting.

Structural optimization - a survey.- Mathematical optimization: an introduction.- Design optimization with the finite element program ANSYSR.- B&B: a FE-program for cost minimization in concrete design.- The CAOS system.- Shape optimization with program CARAT.- DYNOPT: a program system for structural optimization weight minimum design with respect to various constraints.- MBB-Lagrange: a computer aided structural design system.- The OASIS-ALADDIN structural optimization system.- The structural optimization system OPTSYS.- SAPOP: an optimization procedure for multicriteria structural design.- SHAPE: a structural shape optimization program.- STARS: mathematical foundations.