Mathematics and Statistics

This book focuses on a large class of multi-valued variational differential inequalities and inclusions of stationary and evolutionary types with constraints reflected by subdifferentials of convex functionals. Its main goal is to provide a systematic, unified, and relatively self-contained exposition of existence, comparison and enclosure principles, together with other qualitative properties of multi-valued variational inequalities and inclusions. The problems under consideration are studied in different function spaces such as Sobolev spaces, Orlicz-Sobolev spaces, Sobolev spaces with variable exponents, and Beppo-Levi spaces.

This is the first in-depth treatise on the sub-supersolution (lattice) method for multi-valued variational inequalities without any variational structures, together with related topics. The choice of the included materials and their organization in the book also makes it useful and accessible to a large audience consisting of graduate students and researchers in various areas of Mathematical Analysis and Theoretical Physics.

This text lays the foundation for understanding the beauty and power of discrete-time models. It covers rich mathematical modeling landscapes, each offering deep insights into the dynamics of biological systems. A harmonious balance is achieved between theoretical principles, mathematical rigor, and practical applications. Illustrative examples, numerical simulations, and empirical case studies are provided to enhance mastery of the subject and facilitate the translation of discrete-time mathematical biology into real-world challenges. Mainly geared to upper undergraduates, the text may also be used in graduate courses focusing on discrete-time modeling.  

Chapters 1–4 constitute the core of the text. Instructors will find the dependence chart quite useful when designing their particular course. This invaluable resource begins with an exploration of single-species models where frameworks for discrete-time modeling are established. Competition models and Predator-prey interactions are examined next followed by evolutionary models, structured population models, and models of infectious diseases. The consequences of periodic variations, seasonal changes, and cyclic environmental factors on population dynamics and ecological interactions are investigated within the realm of periodically forced biological models.

This indispensable resource is structured to support educational settings:

• A first course in biomathematics, introducing students to the fundamental mathematical techniques essential for biological research.
• A modeling course with a concentration on developing and analyzing mathematical models that encapsulate biological phenomena.
• An advanced mathematical biology course that offers an in-depth exploration of complex models and sophisticated mathematical frameworks designed to tackle advanced problems in biology.

With its clear exposition and methodical approach, this text educates and inspires students and professionals to apply mathematical biology to real-world situations.  While minimal knowledge of calculus is required, the reader should have a solid mathematical background in linear algebra.

This book is the second volume in a two-part series on the theory of ordinary differential equations and their applications to population dynamics. 

While the first volume provides an introduction to the topic, this second volume presents advanced mathematical tools for analyzing such problems. 

Part I focuses on refined techniques for describing the long-term behavior of these systems. It includes a detailed discussion of dissipative dynamical systems, omega and alpha limit sets, global attractors, bifurcations, the construction of smooth center manifolds, and normal form theory. 

Part II introduces new perspectives on predator-prey systems by applying theoretical results to derive oscillating solutions through Hopf bifurcation, traveling invasion waves using global attractor theory, and a description of long-term dynamics in competitive interactions between predator variants. 

Throughout the book, concepts are illustrated with numerical examples, and MATLAB codes are provided. 

Bridging an interdisciplinary gap, this book will be valuable to graduate students and researchers studying mathematical models in population dynamics.

This book is open access. The COVID-19 pandemic has highlighted many adaptation issues for health and social services establishments. In both European and worldwide health care systems, managers must more than ever undergo major transformations and face challenges to respond to public concerns. Innovative solutions are presented and analyzed through a very detailed study of specific contexts. The sharing of knowledge and experiences to better understand the multiple facets of these challenges is strategically relevant for healthcare managers, especially in learning healthcare systems logic. To address it, the second and the third editions of the Francophone Health Services Management Summer School focus on innovation in a post-pandemic health and social services system under the lens of artificial intelligence and operations research, aimed at disseminating and enriching knowledge between managers and providing useful tools to healthcare managers. Several topics were treated within the summer school, such as the reorganization of health care and services, the innovation in care pathways, the digital transformation and artificial intelligence in health and social services, the operationalization and evaluation of learning health systems, the new trends in hospital management and the patient-centered care pathways.

This graduate text presents the elegant and profound theory of continuous parameter Markov processes and many of its applications.  The authors focus on developing context and intuition before formalizing the theory of each topic, illustrated with examples.

After a review of some background material, the reader is introduced to semigroup theory, including the Hille–Yosida Theorem,  used to construct continuous parameter Markov processes.  Illustrated with examples, it is a cornerstone of Feller’s seminal theory of the most general one-dimensional diffusions studied in a later chapter. This is followed by two chapters with probabilistic constructions of jump Markov processes,  and   processes with independent increments, or Lévy processes. The greater part of the book is devoted to  Itô’s fascinating theory of stochastic differential equations,  and to the study of  asymptotic properties of diffusions  in all dimensions, such as   explosion, transience, recurrence,  existence of steady states,  and the speed of convergence to equilibrium.  A broadly applicable functional central limit theorem for ergodic Markov processes is presented with important examples. Intimate connections between diffusions  and linear second order elliptic and parabolic partial differential equations are laid out in two chapters, and are used for computational purposes.  Among Special Topics chapters, two study anomalous diffusions: one on  skew Brownian motion, and the other on an intriguing multi-phase homogenization of solute transport in porous media.