MATHEMATICS: CONCEPTS AND FOUNDATIONS

MATRICES, VECTORS, DETERMINANT AND LINEAR ALGEBRA

Matrices, Vectors and their Basic Operations

Determinants

Systems of Linear Equations

Vector Spaces and Linear Algebra

GEOMETRY

BASIC NOTIONS OF GEOMETRY AND EUCLIDEAN GEOMETRY

Introduction

Basic Notions

Euclidean Space

Euclidean Group

Conic Sections

Discrete Groups of Isometries

AFFINE GEOMETRY, PROJECTIVE GEOMETRY, AND NON-EUCLIDEAN GEOMETRY

Affine Geometry

Projective Geometry

Geometries and Groups

Non-Euclidean Geometry

MATHEMATICAL ANALYSIS

MEASURE AND PROBABILITY

Measure

Probability

FUNCTIONAL ANALYSIS AND FUNCTION SPACES

Function Spaces and Some Examples

Basic Concepts in Functional Analysis

Some Advanced Concepts in Functional Analysis

Miscellaneous Function Spaces

FOURIER ANALYSIS AND INTEGRAL TRANSFORMS

Fourier series

Wavelet expansion

Fourier transforms

Fourier analysis on locally compact Abelian groups

Finite Fourier Transform

Integral transforms

OPERATOR THEORY AND OPERATOR ALGEBRA

Hilbert space

Bounded linear operator

Operator theory

Operator algebra

DIFFERENTIAL EQUATIONS OF MATHEMATICALS PHYSICS

A BASIC EXAMPLE OF NONLINEAR EQUATIONS: THE NAVIER-STOKES EQUATIONS

Scaling, hierarchies and formal derivations

Stabilities and instabilities of macroscopic solutions

Turbulence, weak convergence and Wigner measures

Some special properties of the dimension 2

CALCULUS OF VARIATIONS, PARTIAL DIFFERENTIAL EQUATIONS, AND GEOMETRY

An example: minimal surfaces

Phase transitions and interfaces

LINEAR DIFFERENTIAL EQUATIONS

Linearity and Continuity

Examples

Methods

DIFFERENTIAL EQUATIONS AND SYMPLECTIC GEOMETRY

Lagrangian Mechanics

Hamiltonian Systems and Symplectic Geometry

Nonlinear First order Partial Differential Equations

Oscillatory Integrals

Fourier Integral Operators

FROM THE ATOMIC HYPOTHESIS TO MICROLOCAL ANALYSIS

The Schrödinger Equation And Semiclassical Analysis

High Frequency Asymptotics and Microlocal Analysis

COMBINATORICS

Selected Topics in Combinatorics

Introduction

MATHEMATICAL FOUNDATIONS AND INTERPRETATIONS OF PROBABILITY

Finite Probability Spaces

Conditional Probability

Discrete Probability Spaces

Kolmogorov Triplets

ALTERNATIVE PROBABILISTIC SYSTEMS

Early developments

Capacities

The 1970s and 80s

From the 1990s on

CONSTRUCTION OF RANDOM FUNCTIONS AND PATH PROPERTIES

Examples

Definition of the Stochastic Process

Poisson Process

Brownian Motion

STOCHASTIC DIFFERENTIAL EQUATIONS

Existence and Unicity

A Stochastic Chain Rule

A Property of the Solution of a Stochastic Differential Equation

ERGODIC PROPERTIES OF STATIONARY, MARKOV, AND REGENERATIVE PROCESSES

Ergodic Theory for Stationary Processes

Ergodic Properties of Markov Processes

Regenerative Processes

Applications of Ergodic Theorems

STATISTICAL SIMULATION AND NUMERICAL PROCEDURES

Random Number Generation

Non Uniform Random Variate Generation

The Use of Simulation in Statistics

Use of Simulation in Numerical Calculations

INSURANCE MATHEMATICS

Non-life Insurance

Life Insurance

RELIABILITY AND MAINTAINABILITY

Some Reliability Concepts

System Reliability

Availability and Maintainability

Reliability Data Analysis

Towards the 21st Century

INVENTORIES, WATER STORAGE AND QUEUES

Inventory Models

Models for Water Storage

The Queueing System GI /G /S

Queueing Networks

PRELIMINARY DATA ANALYSIS

Univariate Data Sets

Bivariate Data Sets

Multivariate Data Sets

STATISTICAL PARAMETER ESTIMATION

Fundamental Concepts

Optimality Properties

Methods of Parameter Estimation

Classical Confidence Regions

ANALYSIS OF VARIANCE AND ANALYSIS OF COVARIANCE

Analysis of Variance (ANOVA)

Analysis of Covariance

MATHEMATICAL MODELING OF LIFE SUPPORT SYSTEMS: CLASSIFICATION OF MODELS

Mathematical models

Some classes of mathematical models

Linear and nonlinear models

Well-and ill-posed problems

Point models

Distributed models

Discrete models

Imitation modeling

MATHEMATICAL MODELS IN ENERGY SCIENCES AND CHEMICAL PHYSICS

MATHEMATICAL MODELS OF PLASMA PHYSICS

Kinetic models

Transport properties of plasmas

Magnetohydrodynamic models

Mathematical models of thermonuclear plasmas

MATHEMATICAL MODELS IN ENVIRONMENTAL SCIENCES

MATHEMATICAL MODELS FOR PREDICTION OF CLIMATE

Mathematics for climate modeling

Climatic models

Predictability of climate changes

Mathematical model for global transport of persistent organic pollutants in the Northern Hemisphere

Numerical results

MATHEMATICAL MODELS IN FOOD AND AGRICULTURAL SCIENCES

FOOD PRODUCTION AND AGRICULTURAL MODELS: BASIC PRINCIPLES OF DEVELOPMENT

Classification of Agricultural Models

Typical Theoretical Models in Agriculture

Agroecosystem Productivity Models and Simulation Systems

The Use of Models

Experimental Support of Models and Experiment Planning

MATHEMATICAL MODELS OF SOIL IRRIGATION AND SALTING

Balance models of calculation of the irrigation regime and crops productivity.

Simulation of water and salts transport in unsaturated-saturated soils.

The complex simulation models

MATHEMATICAL MODELS IN BIOLOGICAL AND MEDICAL SCIENCES

MATHEMATICAL MODELS IN GLOBAL PROCESSES AND DEVELOPMENT

MATHEMATICAL MODELS AND CONTROL OF CATASTROPHIC PROCESSES

Basic Notions and Examples

Singularity Theory

Singularities in Optimization problems

MODELS AND METHODS OF ACTUARIAL MATHEMATICS

Empirical principles of determination of insurance premiums.

Classification of risk models

Collective risk model

Individual risk model

THE ROLE OF MODELING

Modeling as a Mental Activity

Mathematical Modeling

NONLINEAR PROGRAMMING

Optimality Conditions

Optimization Algorithms

Large Scale Optimization

DISCRETE OPTIMIZATION

Modeling

Solution Methods

THE ROLE OF SOFTWARE IN OPTIMIZATION AND OPERATIONS RESEACH

Historical Perspectives

Obtaining a Solution

Modeling

Computer-Assisted Analysis

Intelligent Mathematical Programming Systems

Beyond the Horizon

Foundations

Seminal Development-Discrete Optimization

COMBINATORIAL OPTIMIZATION AND INTEGER PROGRAMMING

Modeling

Mathematical Foundations

Algorithmic Approaches

Software

GRAPH AND NETWORK OPTIMIZATION

Preliminaries

Shortest Path Problem

The Maximum Flow Problem

The Minimum Cost Flow Problem

The Minimum Spanning Tree Problem

LARGE SCALE OPTIMIZATION

LP Relaxations

Lagrangian Relaxations

Decomposition Methods

Reformulations

Final Remarks

GLOBAL OPTIMIZATION AND META-HEURISTICS

Meta-Heuristic Features

Brief Description of Some Meta-Heuristics

Metaphors of Nature

THE PRINCIPLES OF THE CALCULUS OF VARIATIONS

Classical Theory

Direct Methods

Unstable Critical Points

THE MAXIMUM PRINCIPLE OF PONTRYAGIN

The Maximum Principle

Structure of Optimal Controls

Relation to Dynamic Programming

Numerical Solution Based on the Maximum Principle

DYNAMIC PROGRAMMING AND BELLMAN'S PRINCIPLE

Optimal Control

Value Function and Bellman’s Principle

The Hamilton-Jacobi-Bellman Equation

Optimal Feedback Synthesis

OPTIMIZATION AND CONTROL OF DISTRIBUTED PROCESSES

Optimization Problems Governed by Distributed Processes

Existence and Characterization of Solutions

Discretization of the Problem

Optimization Algorithms

FOUNDATIONS OF NON-COOPERATIVE GAMES

Chess-Like Games

Representations of Non-Cooperative Games

Two-Person Zero-Sum Games

Non-Zero-Sum Games

Games with Incomplete Information

STOCHASTIC AND REPEATED GAMES

Supergames

Repeated Games with Incomplete Information

Stochastic Games

EXPECTED UTILITY THEORY AND ALTERNATIVE APPROACHES

The General Framework

Expected Utility Theory

Non-Expected Utility Theory

DECISION PROBLEMS AND DECISION MODELS

A Classification of Decision Problems

Theories and Models

Decision Trees and Influence Diagrams

MULTIPLE-CRITERIA DECISION MAKING

Value Function Approach

Vector Optimization

Final Remarks

DECISION TREES AND INFLUENCE DIAGRAMS

A Medical Diagnosis Problem

Decision Trees

Influence Diagrams

FRAMING EFFECTS IN THEORY AND IN PRACTICE

Framing Effects in Theory

Framing Effects in Practice

Moderators of Framing Effects

THE DEVELOPMENT OF MATHEMATICS IN A HISTORICAL PERSPECTIVE

Introduction

Measure Theories and Probability

Invariant Measures

Ergodicity and Dynamical Systems

BOURBAKI, AN EPIPHENOMENON IN THE HISTORY OF MATHEMATICS

Introduction

The Origins

The Impact

The Elaboration of the Volumes Constituting the Treatise

BASIC METHODS FOR SOLVING EQUATIONS OF MATHEMATICAL PHYSICS

Analytical methods for problems of mathematical physics

Approximate methods

DISCRETIZATION METHODS FOR PROBLEMS OF MATHEMATICAL PHYSICS

Finite difference methods

Variational methods

Projection methods

VARIATIONAL FORMULATION OF PROBLEMS AND VARIATIONAL METHODS

The variational method

Applications of the Lax-Milgram theorem

Extensions of the variational theory

SOLUTION OF SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS

An Unusable Formula

Direct methods

Iterative methods

Domain decomposition methods

NUMERICAL INTEGRATION

Statements of Problems

Cubature Formulae

NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS AND DYNAMIC SYSTEMS

Dynamic Systems

Analytic Methods

One-step Methods

Stiff Systems

Linear Multistep Methods

Error Estimation

Delay Differential Equations

NUMERICAL METHODS AND ALGORITHMS IN MATHEMATICAL PHYSICS

NUMERICAL METHODS FOR INTEGRAL EQUATIONS

Degenerate Kernels. Projection and Collocation Methods

Iterative methods for linear and nonlinear integral equations

Singular integral equations

NUMERICAL ALGORITHMS FOR INVERSE AND ILL-POSED PROBLEMS

Inverse Problems

Ill-Posed Problems

Numerical Algorithms for Solving Inverse and Ill-Posed Problems

COMPUTATIONAL METHODS AND ALGORITHMS IN CONTINUOUS MEDIUM PROBLEMS

COMPUTATIONAL METHODS FOR COMPRESSIBLE FLOW PROBLEMS

A Brief Description of the Solutions

Numerical Schemes for 1-D Problems

Schemes for Multidimensional Problems

Numerical Examples

METHODS FOR MAGNETOSPHERE AND NEAR-SPACE PROBLEMS

MHD model of solar wind flow around the magnetosphere

Mathematical statement of the flow problem: Basic equations

Thermal anisotropy of the magnetosheath plasma

Reconnection problem

NUMERICAL MODELS AND SIMULATION OF GLOBAL PROBLEMS

NUMERICAL SIMULATION OF CLIMATE PROBLEMS

Climate, Climatic Variability and Climate Changes

Atmosphere & Ocean Circulation Models

Numerical Modeling of Climatic Variability and Climate Changes

RESPONSE ADAPTIVE RANDOMIZATION IN CLINICAL TRIALS

The Design

Likelihood Based Inference

Nonparametric Inference

Regression Models

MULTIVARIATE AND MULTIDIMENSIONAL ANALYSIS

Continuous Outcomes

Non-continuous Outcomes

Graphical Analysis

A Magician at Work?

REPEATED MEASURES AND MULTILEVEL MODELING

General Model

Some Models for Continuous Data

Models for Discrete Data

Generalized Estimating Equations

Discussion

COMPUTER-INTENSIVE STATISTICAL METHODS

Resampling and Monte Carlo methods

Numerical optimization and integration

Density estimation and smoothing

Relaxing least-squares and linearity

SPATIAL STATISTICAL MODELING IN BIOLOGY

Gaussian Random Process Models

Non-Gaussian Random Process Models

Multivariate Spatial Models

Spatiotemporal Models

Computation

Future Directions

EPIDEMIOLOGY METHODS

Types of Investigation

Measures of Association

Common Designs

Discussion

COMMUNICABLE DISEASES AND DATA ANALYSIS

Transmission probability

Basic reproductive number

The dependent happening relation

Population-level effects of intervention

Challenges for the future

STATISTICAL METHODS IN LABORATORY AND BASIC SCIENCE RESEARCH

Theory: Universal Distributions

The Role of Statistics

Statistical Strategies

Case Studies

Closing Remarks

STATISTICAL METHODS FOR TOXICOLOGY

Applications of Biostatistics to Toxicology

General Methods in Dose-Response Modeling

Quantitative Risk Assessment

STATISTICAL METHODOLOGY IN AGRICULTURE AND HORTICULTURE

Current methodology

Future developments

POPULATION GENETICS

Basic Principles

Explanations for Genetic Variation

ENVIRONMETRICS

DESCRIPTIVE MEASURES OF ECOLOGICAL DIVERSITY

Diversity, richness, evenness

General properties of diversity indices

Special indices and families of indices

SAMPLING DESIGNS FOR MONITORING ECOLOGICAL DIVERSITY

Unit sampling

Area sampling

Further developments: two-stage sampling

INFERENCE ON ECOLOGICAL DIVERSITY

Diversity index estimation

Species-abundance curve models

THE INVENTORY AND ESTIMATION OF PLANT SPECIES RICHNESS

Species Inventorying

Estimating and Comparing Species Richness through Samples

GEOSTATISTICS: PAST, PRESENT AND FUTURE

Distribution-Free Methodology

Likelihood-Based Modeling

Model Based Prediction

Discussion and Future Directions

STATISTICAL ANALYSIS OF SPATIAL COUNT DATA

Random Spatial Indices

Non-Random Spatial Indices

Spatial Epidemiology and Disease Mapping

THE ANALYSIS OF PUTATIVE SOURCES OF HEALTH HAZARD

Study Design

Problems of Inference

Modeling the Hazard Exposure Risk

Models for Case Event Data

Models for Count Data

SPATIO-TEMPORAL METHODS IN CLIMATOLOGY

Descriptive Statistical Methods

Future Directions

ENVIRONMENTAL MONITORING

AREA PRECIPITATION MEASUREMENT

The Area Precipitation Measurement Problem

The Kalman Filter Approach

The Cokriging Approach

ECONOMIC ASPECTS OF MONITORING ENVIRONMENTAL FACTORS: A COST-BENEFIT APPROACH

Setting Environmental Standards

Economic Implications of Adopting Environmental Standards

Environmental Valuation

Environmental Policy Regulations

TREND ANALYSIS FOR ENVIRONMENTAL FACTORS: TIME EFFECTS ON NITROUS OXIDE (N2O) LEVELS AT MACE HEAD, IRELAND

The Global Atmospheric Gases Experiment

Nitrous Oxide Levels at Mace Head

Identifying Trends

Trend Analysis for Variance Change

Change-Point Analysis of Nitrous Oxide Levels

RANK TESTS FOR INDEPENDENCE AND RANDOMNESS

Introduction

Rank Tests for Independence

Tests for Randomness against Trend

Contingency Tables

CLASSIFICATION OF MODELS

Discrete time models

Continuous-time Models

BASIC METHODS OF THE DEVELOPMENT AND ANALYSIS OF MATHEMATICAL MODELS

Discrete time models

Continuous time models

MEASUREMENTS IN MATHEMATICAL MODELING AND DATA PROCESSING

Hypothesis Testing

Sufficient Statistics

Signal Detection

Estimation Theory

CONTROLLABILITY, OBSERVABILITY AND STABILITY OF MATHEMATICAL MODELS

Controllability

Stability

Observability

Observers

IDENTIFICATION, ESTIMATION AND RESOLUTION OF MATHEMATICAL MODELS

Problem of identification

Identification procedure

Identification for several other classes of dynamic systems

Research problems

Software for identification

MATHEMATICAL THEORY OF DATA PROCESSING IN MODELS (DATA ASSIMILATION PROBLEMS)

Variational data assimilation

Kalman filtering

CHAOS AND CELLULAR AUTOMATA

Chaos

Cellular automata

MATHEMATICAL MODELS IN WATER SCIENCES

MATHEMATICAL MODELS IN HYDRODYNAMICS

Some fundamentals

Direct Numerical Simulation

Statistical turbulence modeling

Large Eddy simulation

MATHEMATICAL MODELS OF CIRCULATIONS IN OCEANS AND SEAS

Areas of Model Application

Approximate Systems of Equations

Ocean Modeling Concepts

Numerical Aspects

The Quality of Model Results; Validation and Evaluation

Outlook

WAVE MODELING AT THE SERVICE OF SECURITY IN MARINE ENVIRONMENT

Physical principles of free surface waves

Forcing functions for wave modeling

Present applications of wave modeling

Outlook

MATHEMATICAL MODELING OF THE TRANSPORT OF POLLUTION IN WATER

Phemonenology

Experiments

A Short Introduction to Turbulence Theory

Mathematical Modelling of the Transport of Pollution

An Alternative Approach: Lagrangian Tracer Technique (LTT)

Examples

MATHEMATICAL MODELS IN ENERGY SCIENCES

MATHEMATICAL MODELS IN ELECTRIC POWER SYSTEMS

Basic Concepts

Elements of an Electric Power System

Power System Design, Operation and Control

Equipment Models

Modelling and Simulation of Power System Performance

MATHEMATICAL MODELING AND SIMULATION METHODS IN ENERGY SYSTEMS

Bottom-up versus top-down modeling

Simulation vs. optimization

Technology ranking

Issues in energy modeling

MATHEMATICAL MODELS OF CLIMATE AND GLOBAL CHANGE

MATHEMATICAL MODELS IN METEOROLOGY AND WEATHER FORECASTING

History of Numerical Weather Prediction

Numerical Models

Data Assimilation

Ensemble Forecasting and Predictability

The Future

MATHEMATICAL MODELS OF HUMAN-INDUCED GLOBAL CHANGE

Historical Development

Current Methodology

Strengths and Weaknesses of Climate Models

Future Challenges

MATHEMATICAL SOIL EROSION MODELLING

Surface Hydrology

Soil Erosion Processes

Steady State Solutions of the Rose - Hairsine Model

Dynamic Erosion - Time Dependence

Field Scale

MATHEMATICAL MODELS IN MEDICINE AND PUBLIC HEALTH

MATHEMATICAL MODELS IN EPIDEMIOLOGY

Models for Infectious Diseases

Models for Vector-Born Infections

Models for Parasite Populations

Models with Structure

MATHEMATICAL MODELING AND THE HUMAN GENOME

Modeling DNA

Modeling Genes

MATHEMATICAL MODELS IN DEMOGRAPHY AND ACTUARIAL MATHEMATICS

Life Table Models

Stable Populations

Multistate Population Models

“Two-Sex” Population Models

Dynamic Population Models

ECOLOGICAL AND SOCIO-ECOLOGICAL ECONOMIC MODELS

Ecological-economic interaction models

Dynamic macro and micro simulation models

Optimization and control in simulation models

Game-theoretic models

Equilibrium and optimality in dynamic games

MATHEMATICAL MODELS OF MANAGEMENT OF THE ENVIRONMENT AND ITS NATURAL RESOURCES

Positive and Negative Externalities

Socially Optimum Provision of Environmental Bads

Mechanisms to Achieve the Optimal Level of an Environmental Bad

Socially Optimum Provision of Environmental Public Goods

A Unified Framework for the Optimal Management of Natural Resources

MATHEMATICAL MODELS OF GLOBAL TRENDS AND TECHNOLOGICAL CHANGE

Global Trends and Global Change

Modeling of Global Trends and Global Changes

Models of World Dynamics

Integrated Assessment Global Models

Models of Technological Change

ENTROPY SYSTEMS THEORY

History

Criteria for Entropy Evaluation

Assessing the Past

Future Research

EPISTEMOLOGICAL ASPECTS OF SYSTEMS THEORY RELATED TO BIOLOGICAL EVOLUTION

Integrating Epistemology of Thermodynamics and of Biological Evolutionary Systems

Thermodynamics of Ecosystems and of Biological Evolution

Towards an Evolutionary Physics

SOCIO-TECHNICAL SYSTEMS: HISTORY AND STATE-OF-THE ART

The Role of Automation of Work Processes

The Requirement of Flexible Human Skills: Road to a Socio-Technical View

The Socio-Technical System Approach with Respect to Information- and Communication Technologies

SOFT SYSTEMS METHODOLOGY

Problemology

Soft Systems Methodology - SSM: A General View

SOCIAL PROBLEM DIAGNOSIS: A SOCIOPATHOLOGY IDENTIFICATION MODEL

Anatomy of Sociophysics

Pathology of Socioproblematics

Methodology of Sociodiagnostics

TOTAL SYSTEMS INTERVENTION

Total Systems Intervention (TSI 1)

Local Systemic Intervention (LSI/TSI 2)

Application

Future Challenges

INTEGRATIVE SYSTEMS METHODOLOGY

The State of Systemic Problem-solving

Outline of Integrative Systems Methodology

A Case Study

Reflection

WSR DECISIONS FOR A SUSTAINABLE FUTURE

Philosophy

Methodology

Application

PSYCHOLOGICAL AND CULTURAL DYNAMICS OF SUSTAINABLE HUMAN SYSTEMS

Dimensions of Human Life-support Systems and Sustainability

Can Ecological and Emotional Well-being go together?

THE DYNAMICS OF SOCIAL AND CULTURAL CHANGE

Systems Theory

Sociological Theory

CHAOS: BACK TO "PARADISE LOST": PREDICTABILITY. THE CENTURY OF THE EMERGENCE OF SYSTEMIC THOUGHT AND CHAOS THEORY

The 20th century: the difficult co-existence of Mechanicist Thought and Systemic Thought: emergence of chaos

Structure

A multi-stage modeling process to research on the detection and control of chaos dynamics in the evolution of biological and social systems.

An outstanding example of the chaotic dynamic system: the logistic map

Other important chaotic systems

TRANSDISCIPLINARY UNIFYING THEORY: ITS FORMAL ASPECTS

Rationales to Unifying Transdisciplinarily

External and Internal Constraints

Systemhood Unifying Theories

Unifying the Unifying Theories

Foreseeable Developments

GENERAL SYSTEMS PROBLEM SOLVER

Classification of Systems in GSPS

Systems Problem Solving

Methodological Outcome of the GSPS

HISTORY OF CYBERNETICS

Origins of Cybernetics

Basic Concepts

Future of Cybernetics

EXISTING CYBERNETICS FOUNDATIONS

Organization

Modeling

Information

Control

KNOWLEDGE AND SELF-PRODUCTION PROCESSES IN SOCIAL SYSTEMS

Social Systems

Autopoiesis (Self-Production) of Networks

Knowledge as Coordination of Action

Model of Autopoiesis

Autopoietic Social Systems

Individuals in Networks

SIMULATED ANNEALING: FROM STATISTICAL THERMODYNAMICS TO COMBINATORY PROBLEMS SOLVING

Complexities of Problems and Algorithms

Introduction to Global Search Methods

Contribution of Statistical Physics and Thermodynamics

The Simulated Annealing Algorithm

Examples of Problems Solved Thanks to Simulated Annealing

Comparisons with Other Heuristics and SA Performance Improvements