The Science of Complexity: From atoms to hurricanes David Quesada, Ph.D. School of Science, Technology, and Engineering Management St. Thomas University, Miami Gardens Mathematics Awareness Month, April 2011.
<ul><li>Mathematics , the language of Science. </li></ul><ul><li>Systems of particles, agents, and bodies. </li></ul><ul><li>Reductionism versus Holism, Certainty versus Uncertainty. </li></ul><ul><li>What do we call simple, complicated, and complex? </li></ul><ul><li>Sand-piles and Self-organized Criticality . </li></ul><ul><li>Graphs and Networks – the environment of complexity. </li></ul><ul><li>Fractals - the geometry of complexity. </li></ul><ul><li>Complex systems in Material Science. </li></ul><ul><li>Complex systems in Earth Sciences. </li></ul><ul><li>Complex systems in Biology and Physiology. </li></ul><ul><li>Complex systems in Economics and Social Sciences. </li></ul><ul><li>Complex systems in Linguistics – Zipf-Mandelbrot law. </li></ul><ul><li>Computational Complexity </li></ul><ul><li>My current research – Applying complexity to study asthma </li></ul>Tentative Outlook Mathematics Awareness Month, April 2011.
Branches of Mathematics Logic Descriptive Statistics Probabilities Geometry Calculus and Differential Equations Mathematics Awareness Month, April 2011.
Some common misconceptions about Mathematics 1. Learning mathematics requires special and rare abilities. 2. Math in modern issues is too complex. 3. Math makes you less sensitive, and is irrelevant to my life 4. Math makes no allowance for creativity. 5. Math provides exact answers. What is Mathematics after all? The word mathematics is derived from the Greek word Mathematikos , which means “ inclined to learn ”. Thus, literally speaking, to be mathematical is to be curious, open-minded, and interested in always learning more !! Do you consider yourself to be either “ math phobic” ( fear of mathematics ) of “ math loathing” ( dislike math )? Many adults harbor fear or loathing of mathematics and, unfortunately, these attitudes are often reinforced by classes that present mathematics as an obscure and sterile subject . Mathematics also may be viewed as a tool for creating models , or representations that allow us to study real phenomena Mathematical Modeling Medicine and Physiology Psychology and Sociology Bioinformatics Engineering Biology and Ecology Computer science and Artificial Intelligence Physics and Chemistry Economics Business Management Atmospheric Physics or Meteorology Mathematics Awareness Month, April 2011.
System of Particles Super Strong Gravitation Strong Electro-Weak Weak Electro-magnetism Isaac Newton, British physicist F = m a Second Law of Newton Mathematics Awareness Month, April 2011. Forces In Nature
Fathers of Statistical Physics – The Physics of many particles Albert Einstein Ludwig Boltzmann Bose – Einstein Condensation Lattice gas of spins Diffusion Limited Aggregation Mathematics Awareness Month, April 2011.
Phase Transitions – From Solid State and Chemistry to Nucleosynthesis Lev Davidovich Landau Russian theoretical physicist <ul><li>Order parameter – emergent property </li></ul><ul><li>Superconducting order parameter </li></ul><ul><li>Spontaneous Magnetization </li></ul><ul><li>New phase of matter </li></ul>Mathematics Awareness Month, April 2011.
Entropy and Thermodynamics out of Equilibrium S - function Mathematics Awareness Month, April 2011.
Statistical Physics out of Equilibrium - Synergetics Hermann Haken, Synergetics Synchronization in the finger motion Norbert Wiener, Control Theory Mathematics Awareness Month, April 2011.
THE LANGUAGE OF COMPLEXITY THE SYSTEM CONCEPT The System concept is a way to break down any large, complex problem into smaller, more easily studied pieces. A System can be defined as any portion of the Universe that can be isolated from the rest of the Universe for the purpose of observing and measuring changes The system can be whatever the observer defines it to be. You choose its limits for the convenience of your study. Mathematics Awareness Month, April 2011.
Systems and Complex Systems Key Components of most systems <ul><li>Inputs of things such as matter, energy, or information into the system. </li></ul><ul><li>Flows, or throughputs, of matter, energy, or information within the system at certain rates. </li></ul><ul><li>Stores, or storage areas, within a system where energy, matter, or information can accumulate for various lengths of time before being released. </li></ul><ul><li>Outputs of certain forms of matter, energy, or information that flow out of the system into sinks in the environment. </li></ul>Inputs Outputs Stores Flows Environment Feedback mechanism A feedback loop occurs when an output of matter, energy, or information is fed back into the system as an input that changes The system. Positive feedback loops Negative feedback loops How do time delays affect complex systems? Complex systems often show time delays between the input of a stimulus and the response to it. A long time delay can mean that corrective action come too late. Time delays allow a problem to build up slowly until it reaches a threshold level and causes a fundamental shift in the behavior of a system. What is Synergy, and How can it affect complex systems? A synergistic interaction occurs when two or more processes interact so the combined effect is greater than the sum of their separate effects. The essence of self-organisation is that system structure (at least in part) appears without explicit pressure or constraints from outside the system. In other words, the constraints on form are internal to the system and result from the interactions between the components , whilst being independent of the physical nature of those components. The field of self-organisation seeks to discover the general rules under which such structure appears, the forms which it can take, and methods of predicting the changes to the structure that will result from changes to the underlying systems D. Quesada, La Física de Sistemas Complejos: Un reto a la enseñanza moderna (“ The Physics of Complex Systems: A challenge to modern higher education ”), Proceedings of the Ibero-American workshop on Physics teaching, Univ. of Havana, Jan. 20-24, (1997).
Sand-piles model of Complexity Per Bak legacy Per Bak, Danish theoretical physicist, 1948 – 2002. Co-author with Chao Tang, Kurt Wiesenfeld of the new paradigm of “ Self-organized criticality” via sandpiles. Mathematics Awareness Month, April 2011.
Fractals: The language of Complex Systems Benoit Mandelbrot Mathematics Awareness Month, April 2011.
George Kingsley Zipf, 1902 - 1950 Laws in Quantitative Linguistics – Zipf-Mandelbrot’s Law
Planetary Biology and Inter-species interactions The Web of Life The appearance of plants and other living forms on Earth constituted one of the most important steps in the development of the current chemical make-up of the atmosphere. Mathematics Awareness Month, April 2011.
Logarithmic Spirals – The preferred spirals of Nature
Mathematics and Atmospheric Sciences Mathematics Awareness Month, April 2011.
Slopes, Trigonometric Functions, Average Values, and Global Warming It is worth to notice the periodicity (24 hrs) of these peaks; however it is clear the irregular shape of all these peaks too – Why? Range of variation Cloudiness and Random Fluctuations in the weather are responsible for these irregularities Ongoing research project # 1.1: Climate and Weather Variability
Non-linear dynamics and Chaos Edward Lorenz Mathematics Awareness Month, April 2011.
Slopes, Trigonometric Functions, Average Values, and Global Warming Trigonometric Interpolation Case 1: The free term T o is a constant Case 2: The free term T o is a linear function of time Case 3: The free term To is a quadratic function of time Climate is all about the value of this Integral, known as the average value Weather is all about the values of these Functions at some moments of time, known as the time series Mathematics Awareness Month, April 2011.
Slopes, Trigonometric Functions, Average Values, and Global Warming It is worth to notice how the trigonometric function oscillates around the main value function T o (t). A minimum of 30 years it is needed to make a conclusion about a warming Climate. It is worth to notice also, how short Cold intervals may coexist with a warming trend. Mathematics Awareness Month, April 2011.
A simple grid 16 by 16 illustrates the applicability of these two concepts. Cells shaded in red represent a particular grain, while cells in white are voids. Thus, Porosity equals the amount of white cells over the total of cells, which in this case are 110 / 256 = 0.43 or 43 % of porosity. On the other hand this material is highly permeable because it has several interconnected paths. A paved area usually has a surface layer with very low level of both porosity and permeability, such that the first factor is a very small number. This fact is the reason for too low infiltration rate of water into the ground even if the subsurface soil would contain many porous. Clay (45 – 55 %, less than 0.01 m/day) Fine Sand (30 – 52 %, 0.01 – 10 m/day) Gravel (25 – 40 %, 1000 – 10,000 m/day) Sandstone (5 – 30 %, 0.3 – 3 m/day)
Strong Dynamical Instabilities known as Chaos, restrict the predictability of models… Statistical Analysis
Hurricanes move following a trochoidal curve Mathematics Awareness Month, April 2011.
Project 1.3: Hurricane’s Science Mathematics Awareness Month, April 2011.
How far weather variability influences seasonal asthma episodes: Three years of correlations in Miami Dade, Florida David Quesada School of Science, Technology and Engineering Management, St. Thomas University, Miami Gardens FL 33054 Climatic and environmental changes occurring since the middle of the Twentieth Century as well as the aggravating pollution levels in megacities are exacerbating asthma episodes and the number of hospitalizations due to this disease. Since 1999, in Miami Dade County the hospitalization rates were doubling the Healthy People 2010 objectives in every age group . A comprehensive weather database including outdoor temperature (T), humidity (H), barometric pressure (P), wind direction (θ w ) and speed (v w ) as well as the values of maximum and minimum and the range of all these variables has been created. As a result, a seasonal pattern emerged, with a maximum appearing around the middle of December and a minimum around the middle of March every year for the three years of analysis. Florida Academy of Sciences, 75 th Anniversary Meeting, FIT - Melbourne FL March 2011 Mathematics Awareness Month, April 2011.
Asthma Statistics Worldwide Number of people diagnosed: more than 150 M Europe: the # of cases has doubled USA : the number of cases has increased more than 60% India: between 15 and 20 M Africa: between 11 and 18% population Number of deaths yearly: around 180,000 Miami Dade County , Florida 7.1% Middle and HS children were reported with asthma The number of hospitalizations due to asthma has doubled. The number 1 cause of school absences and 35 % of parents missed work Why to study Asthma? How far Bio-Meteorology may help with? Mathematics Awareness Month, April 2011.
Systems Theory and System Biology Macroscopic Dynamics of breathing Mesoscopic Immune cells population dynamics Microscopic Genes Cooperative effect – Emergent properties <ul><li>Mathematical modeling of the episode of </li></ul><ul><li>Asthma by using dynamical system theory </li></ul>Mathematics Awareness Month, April 2011.
Macroscopic Mechanical Description of Breathing Mathematical expressions appealing to basic laws of aerodynamics describe the basics of breathing and disorders within the lung functioning. Bronchial Hyper-responsiveness : Excessive constriction of smooth muscle that surrounds bronchi and bronchioles, resulting in narrowed airway passage and airflow limitation. Atopy: A genetically determined state of hypersensitivity to environmental allergens that is detected by increased serum immunoglobulin E and/or positive dermal allergen tests Mathematics Awareness Month, April 2011.
Mesoscopic immune description of an asthma episode A system of differential equations describes the population dynamics of each one of the cells involved in an asthma episode. A very complicated Network of cells (IL4, IL3, IL5, IL13- Cytokines, IgE – Immunoglobuline) Interacting and Competing. In asthmatic individuals, antigen presentation is thought to results in the polarization of T-cells towards a T h2 patterns whereas T cells from non atopic, non-asthmatic individuals show the opposing T h1 (interferon- γ and I L2 ) pattern of cytokine secretion Mathematics Awareness Month, April 2011.
Microscopic genetic analysis of asthma incidence Bio-informatics of Asthma The multigenic nature of asthma has greatly hampered efforts to identify the specific genes involved. Genetic heterogeneity across populations, variability in disease expression, phenocopies and uncontrolled environmental influences confound the analysis of asthma and other complex genetic disorders.
BS in Mathematics PREREQUISITE REQUIRED COURSES : 19 credits MAT 205 Applied Statistics (3 credits) MAT 215 Discrete Mathematics (3 credits) MAT 232 Calculus I (4 credits) MAT 233 Calculus II (4 credits) CHE 101/L General Chemistry I + Laboratory (4 credits) CHE 102/L General Chemistry II + Laboratory (4 credits) MAJOR REQUIREMENTS : 35 credits total Core Mathematics Courses : (13 credits) MAT 234 Calculus III (4 credits) MAT 306 Differential Equations (3 credits) MAT 311 Linear Algebra (3 credits) MAT 316 Complex Variables (3 credits) Mathematics Electives : (6 credits) Take two mathematics courses at the 300 or 400 level. Computing Requirement : (6 credits) Take two courses. CIS 230 Introduction to Java Programming (3 credits) CIS 235 Introduction to C++ Programming (3 credits) CIS 302 Advanced C++ Programming (3 credits) CIS 310 Advanced Java Programming (3 credits) CIS 360 Data Structures (3 credits) CIS 351 Systems Analysis and Design (3 credits) CIS 430 Database Management Systems (3 credits) Physical Science Requirements : (10 credits) PHY 207/L University Physics I + Laboratory (5 credits) PHY 208/L University Physics II + Laboratory (5 credits) Sub-Total Credits: 54 GENERAL EDUCATION REQUIREMENTS : 42 credits (Program requirements will satisfy 9 credits of the GER.) GENERAL ELECTIVES : 18 credits Total credits: 120 Mathematics Awareness Month, April 2011.
Science & Mathematics Fellows Program <ul><li>Start Date: August 2008 - 30 freshmen & 30 juniors who transfer with an AA. </li></ul><ul><li>Qualified students may receive financial aid and academic scholarships. </li></ul><ul><li>Research based in state of the art Science & Technology facility. </li></ul>Mathematics Awareness Month, April 2011.