Difference between revisions of "MAT5133"

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Introduction to post-quantum Cryptography.
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==Mathematical Biology - MAT4133/5133==
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'''Sample textbook''':
 
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[1] Daniel Bernstein, Johannes Buchmann, Erik Dahmen, ''Post-Quantum Cryptography'', Springer, 2010.
 
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'''Catalog entry'''
 
'''Catalog entry'''
  
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''Prerequisite'': Algebra and Number Systems (MAT 1313), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.
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''Prerequisite'': [[MAT1214]]/[[MAT1213]] Calculus I.
  
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''Contents'':
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''Content'': The Mathematical Biology course is a broad introduction to nonlinear dynamics. Students are assumed to have an operative knowledge of single-variable calculus. Topics are introduced with interrelated biological examples of increasing difficulty. Students are exposed to discrete and continuous models. The mathematical content of the class includes: Flows on the line, linear stability analysis, matrix operations and eigenvalues, flows on the plane, bifurcations, discrete dynamical systems, and higher-dimensional systems. The biological problems studied include: Molecular processes (glycolysis, lactose operon, etc.), physiological processes (single neuron), and ecological processes (predator-prey, competing species, infectious disease modeling).  
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(1) Introduction to post-quantum cryptography.
 
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(2) Quantum computing.
 
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(3) Hash-based Digital Signature Schemes.
 
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(4) Code-based cryptography.
 
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(5) Lattice-based Cryptography
 
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(6) Multivariate Public Key Cryptography.
 
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(7) Homomorphic encryption.
 
  
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Suggested text(s) and/or readings:
 +
* Open educational resources available in the Department of Mathematics for this course, available at https://mathresearch.utsa.edu/wiki
 +
* Britton, Nick. Essential Mathematical Biology. Springer, 2005 (UMS Series).
 +
* Fall, Chistopher P., Marland, Eric S., Wagner, John M., and Tyson, John J. (eds). Computational Cell Biology. Springer, 2002 (IAM Series).
 +
* Keener, J. P., and James Sneyd. Mathematical physiology 1: Cellular physiology. New York, NY: Springer New York, 2009.
 +
* Keener, James, and James Sneyd, eds. Mathematical physiology: II: Systems physiology. New York, NY: Springer New York, 2009.
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* Strogatz, Steven H. Nonlinear Dynamics and Chaos. Westview Press, 2000.
  
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{| class="wikitable"
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! Session !! Lecture !! Session !! Topics !! Book Section !! Competency Required. The student needs to know…  !! Competency Gain. The student will learn to… !! Examples
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| 1 ||  || Introduction to Dynamical Systems. || Historical remarks. Concept of the qualitative analysis of a dynamical system. Linear vs. nonlinear problems. Examples from biology and physics. || Strogatz Chap. 1 || Single-variable calculus. || Define a dynamical system. Understand systems of equations. Provide general examples of dynamical systems in biology and physics. || Poincaré’s three-body problem. Lorentz’s system. Mandelbrot’s fractals.
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==Topics List==
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{| class="wikitable sortable"
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| 2 || L1 || Flows on the Line - Introduction || Simple example dx/dt = sin(x). || Strogatz 2.0, 2.1. || How to solve the separable differential equation dx/dt = sin(x) by integration. || Give a qualitative description of a simple dynamical system. || dx/dt = sin(x)
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! Week !! Topic !! Chapter 1 from the Bernstein-Johannes Buchmann-Dahmen book !! MAT1313, or CS2233/2231, or instructor consent.
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|-
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| 3 || L2 || Fixed Points || Fixed points and stability. || Strogatz  2.2, 2.3. || Single-variable calculus. || Give a qualitative description of a simple dynamical system. || Population dynamics. Logistic equation of population growth
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|-
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| 4 || L3 || Linear Stability Analysis || Linearization about a fixed point. Existence and Uniqueness. || Strogatz 2.4, 2.5, 2.6. || Flows on the line. Taylor series. || Perform a linearization. Describe importance of fixed points. || No examples. Discuss homework
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|-
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| 5 || colspan="7"|  HOMEWORK # 1 
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|-
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| 5 || L4 || Linear Stability Analysis || Linearization about a fixed point.  Hyperbolic vs. non-hyperbolic fixed points. || Strogatz 2.4, 2.5, 2.6. || Flows on the line. Taylor series. || Perform a linearization. Describe importance of fixed points. || dx/dt = r-x^2
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|-
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| 6 || L5 || Bifurcations || Saddle node bifurcation. Taylor expansions and normal forms. || Strogatz 3.0, 3.1. || Single-variable calculus. Taylor series. || Determine changes in a dynamical system with parameter variation. || dx/dt = r+x^2, dx/dt = r-x^2, dx/dt = r+x+e^x
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|-
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| 7 || L6 || Bifurcations || Transcritical bifurcation (supercritical, subcritical), pitchfork bifurcation. || Strogatz 3.2, 3.3, 3.4. || Single-variable calculus. Taylor series. || Determine changes in a dynamical system with parameter variation. Know how to eliminate dimensions of a system. Learn stability analysis of 1-D systems || dx/dt = rx-x^2, dx/dt = rx-x^3
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|-
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| 8 || colspan="7"| COLLECT HOMEWORK # 1 
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|-
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| 8 || L6 || Dimensional analysis and scaling || Dimensionless groups. Buckingham’s Pi theorem. Stability of 1-D discrete systems in the context of the last problem in HW 1. || Murray 1.2, Strogatz 3.7. || Flows on the line. Bifurcations. || Make a dynamical system non-dimensional. || Population dynamics. Insect outbreak model: Spruce budworm.
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|-
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| 9 || L7 || Linear Systems || Introduction. Examples of two-dimensional linear systems. Lexicon. Eigenvalues and eigenvectors.  || Strogatz 5.1.  Class notes. || Single-variable calculus. How to solve the separable differential equation dx/dt = f(x) by integration. || Identify a linear system || Selected set of two examples that demonstrate stiff and non-stiff systems.
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|-
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| 10 || colspan="7"| PRACTICE TEST # 1 
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|-
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| colspan="8"|  REVIEW SESSION 
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|-
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| colspan="8"|  RETURN HOMEWORK # 1 GRADED
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|-
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| 11 || L8 || Linear Systems || Eigenvalues and eigenvectors.  Reduction of second order ODEs to a system of first order ODEs || Strogatz 5.1.  Class notes. || Single-variable calculus. How to solve the separable differential equation dx/dt = f(x) by integration. || Identify a linear system || Selected set of two examples that demonstrate stiff and non-stiff systems.
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|-
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| 12 || L9 || Linear Systems || Classification of linear systems. Types of fixed points. Classification of fixed points. || Strogatz 5.2.  Class notes. || Linear systems. || Classify linear systems. Identify and classify fixed points. || Continue use of selected examples.
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|-
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| 13 || colspan="7"|  COLLECT HOMEWORK # 1 CORRECTIONS 
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|-
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| colspan="8"|  COLLECT PRACTICE TEST # 1
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|-
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| colspan="8"|  TEST  # 1 
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|-
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| 14 || colspan="7"|  SOLUTION TO TEST # 1 
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|-
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| colspan="8"|  GIVE HOMEWORK # 2
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|-
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| 15 || L10 || Two Dimensional Flows || Phase portrait. Consequences of the Hartman-Grossman theorem. Nullclines.  Fixed points and linearization in two and more dimensions. Jacobian matrix. || Strogatz 6.1, 6.2. || Linear systems. || Read a phase portrait. || Selected examples.
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|-
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| 16 || L11 || Two Dimensional Flows || How eigenvalues determine repellers, attractors, saddles, centers, and fixed points.  || Strogatz 6.3 || Linear systems. || Interpret a phase portrait based on eigenvalues. || Simple linear system.
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|-
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| 17 || L11 || Limit Cycles || Periodic motion. Poincaré-Bendixon Theorem. Introduction to limit cycles.  Hopf bifurcation || Strogatz 4.0, 4.1, 6.3, 8.2. || Two dimensional flows. || Identify limit cycles. Determine existence of limit cycles. || Simple periodic system with a limit cycle.
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|-
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| 18 || L12 || Two Dimensional Flows || Applications in ecology. Population dynamics.  Rabbit vs. sheep. || Strogatz 6.4. || Linear systems. || Interpret a phase portrait based on eigenvalues. || Species competition. Lokta-Volterra model of  competition.
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|-
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| 19 || colspan="7"|  COLLECT HOMEWORK # 2
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|-
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| 19 || L13 || Limit Cycles || Applications in molecular biology. Glycolytic oscillations. || Strogatz 7.3, 7.5 || Two dimensional flows. || Describe qualitatively and quantitatively shape and period of closed orbits. || Cell biology: Glycolisis. Cell cycle.
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|-
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| 20 || L14  || Numerical Mathematics || Numerical solutions of ODEs || Kreyszig 19.1,  Quarteroni, Sacco, Salieri 11.1, 11.8 || Linear algebra || Compute numerically a system of ODEs || None
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|-
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| 21 || colspan="7"|  RETURN HOMEWORK # 2 GRADED
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| colspan="8"|  ADVISE STUDENTS TO SEARCH FOR PAPERS FOR HW # 5
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|-
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| colspan="8"|  HAVE A BRIEF DISCUSSION ABOUT SCHOLAR DATABASES
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|-
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| 21 || L15 || Numerical Mathematics || Runge-Kutta 2nd order method || Kreyszig 19.3, Quarteroni et al. 11.9 || Linear algebra || Compute numerically a system of ODEs || None
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|-
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| 22 || L16 || Numerical Mathematics || Newton-Raphson method in two dimensions || Kreyszig 17.2 || Linear algebra || Compute numerically the steady states of a system of ODEs || Numerical roots of x^2 – x = 0
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|-
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| 23 || L17 || Numerical Mathematics || Newton-Raphson method in multidimensions || Nash & Sofer 10.3, Quarteroni et al. 7.1 || Linear algebra || Compute numerically the steady states of a system of ODEs || Numerical roots of x^2 – x = 0
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|-
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| 24 || colspan="7"|  COLLECT HOMEWORK # 2 CORRECTIONS
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|-
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|  colspan="8"|    GIVE HOMEWORK # 3 
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|-
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|  colspan="8"|  INTRODUCTION TO COMPUTER SYSTEMS FOR NUMERICAL SOLUTIONS OF ODEs - XPP 
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|-
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| 25 || colspan="7"|  ADVISE STUDENTS TO SEARCH FOR PAPERS FOR HW # 5 
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|-
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| colspan="8"|  PRACTICE TEST # 2 & REVIEW SESSION
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|-
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| 26 || L18 || Applied dynamical systems || Hodgkin-Huxley model.  J. Physiology, 1952 || Fall et al. 2.1, 2.2, 2.5 || Dynamical systems, linear stability, bifurcations || Apply mathematical concepts to biological problems || Hodgkin-Huxley model of neural activity.
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|-
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| 27 || L19 || Applied dynamical systems || Fitzhugh-Nagumo model || Fall et al. 2.6, Strogatz problem 7.5.6 || Dynamical systems, linear stability, bifurcations || Apply mathematical concepts to biological problems || Fitzhugh-Nagumo model of neural activity.
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|-
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| 28 || colspan="7"|  COLLECT PRACTICE TEST # 2 
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|-
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|  colspan="8"|  TEST # 2
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|-
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| 29 || colspan="7"|  SOLUTION TO TEST # 2
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|-
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| 30 || L23 || Limit Cycles || Global bifurcations || Strogatz 8.4 || Dynamical systems, linear stability, bifurcations || Identify global bifurcations || Selected examples from the book
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|-
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| 31 || L20 || Stochastic Systems || Introduce the concept of PDF. Ion channels || Fall et al. 11.1 || Definition of probability || Apply stochastic concepts to dynamical systems || None
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|-
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| 32 || L20 || Stochastic Systems || Ion channels || Fall et al. 11.1 || Definition of probability || Apply stochastic concepts to dynamical systems || None
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|-
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| 33 || L21 || Stochastic Systems || Dwell times || Fall et al. 11.1 || Definition of probability || Apply stochastic concepts to dynamical systems || None
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|-
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| 34 || colspan="7"|  COLLECT HOMEWORK # 3 
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|-
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|  colspan="8"|  GIVE HOMEWORK # 4 
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|-
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| 34 || L22 || Stochastic Systems || Ion channel ensemble || Fall et al. 11.1 || Definition of probability || Apply stochastic concepts to dynamical systems || None
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|-
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| 35 || L24 || Quasi-equilibrium || Michelis-Menten kinetics || Fall et al. 4.7 || Dynamical systems, linear stability, bifurcations || Identify functional responses || Bacterial glucose binding
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|-
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| 36 || colspan="7"| RETURN  HOMEWORK # 3 GRADED
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| 36 || L25 || Chaos || Intro, definitions || Strogatz 9.2, 9.4 || Two dimensional flows. || Understand the concept of chaos. || Lorenz equations.
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|-
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| 37 || L26 || Power-laws || Intro, definitions || PPT || Chaos || Identify when a power law is at play || None
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|-
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| 38 || colspan="7"| COLLECT HOMEWORK # 3 CORRECTIONS
 
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| 1-3 
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| 38 || L27 || One-dimensional maps || Introduction. Fixed points and cobwebs. || Strogatz 10.0, 10.1, 10.2, 10.3 || Flows on the line. || Produce and analyze a one-dimensional map. || Population dynamics: Logistic map.
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|| [[Introduction to post-quantum cryptography]]
 
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|| 1
 
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|| MAT1313 or CS2233/2231, or equivalent.
 
 
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|  2-4 
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| 39 || colspan="7"| COLLECT HOMEWORK # 4.
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|| [[Quantum computing]]
 
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|| 2
 
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||  
 
 
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|-
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| 5-6 
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| colspan="8"| GIVE HOMEWORK # 5.
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|| [[Hash-based Digital Signature Schemes]]
 
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|| 3
 
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||
 
 
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| 7-8 
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| 39 || L28 || One-dimensional maps || Age-structured populations in space time. Leslie matrices. || Britton 1.9 || One-dimensional maps || Understand age-structured discrete models. || Physiology:  Red blood cell count.
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|| [[Code-based cryptography]]
 
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|| 4
 
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||  
 
 
|-
 
|-
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| 9-10 
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| 40 || colspan="7"| PRACTICE FINAL & FULL REVIEW SESSION
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|| [[Lattice-based Cryptography]]
 
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|| 5
 
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||  
 
 
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|-
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| 11-12 
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| 41 || colspan="7"| PRACTICE FINAL & FULL REVIEW SESSION
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|| [[Multivariate Public Key Cryptography]]
 
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|| 6
 
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||  
 
 
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|-
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|  13-end 
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| colspan="8" | RETURN HOMEWORK #4
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|| [[Homomorphic encryption]]
 
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||
 
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||
 
 
|}
 
|}

Latest revision as of 20:17, 30 March 2023

Mathematical Biology - MAT4133/5133

Catalog entry

Prerequisite: MAT1214/MAT1213 Calculus I.

Content: The Mathematical Biology course is a broad introduction to nonlinear dynamics. Students are assumed to have an operative knowledge of single-variable calculus. Topics are introduced with interrelated biological examples of increasing difficulty. Students are exposed to discrete and continuous models. The mathematical content of the class includes: Flows on the line, linear stability analysis, matrix operations and eigenvalues, flows on the plane, bifurcations, discrete dynamical systems, and higher-dimensional systems. The biological problems studied include: Molecular processes (glycolysis, lactose operon, etc.), physiological processes (single neuron), and ecological processes (predator-prey, competing species, infectious disease modeling).

Suggested text(s) and/or readings:

  • Open educational resources available in the Department of Mathematics for this course, available at https://mathresearch.utsa.edu/wiki
  • Britton, Nick. Essential Mathematical Biology. Springer, 2005 (UMS Series).
  • Fall, Chistopher P., Marland, Eric S., Wagner, John M., and Tyson, John J. (eds). Computational Cell Biology. Springer, 2002 (IAM Series).
  • Keener, J. P., and James Sneyd. Mathematical physiology 1: Cellular physiology. New York, NY: Springer New York, 2009.
  • Keener, James, and James Sneyd, eds. Mathematical physiology: II: Systems physiology. New York, NY: Springer New York, 2009.
  • Strogatz, Steven H. Nonlinear Dynamics and Chaos. Westview Press, 2000.
Session Lecture Session Topics Book Section Competency Required. The student needs to know… Competency Gain. The student will learn to… Examples
1 Introduction to Dynamical Systems. Historical remarks. Concept of the qualitative analysis of a dynamical system. Linear vs. nonlinear problems. Examples from biology and physics. Strogatz Chap. 1 Single-variable calculus. Define a dynamical system. Understand systems of equations. Provide general examples of dynamical systems in biology and physics. Poincaré’s three-body problem. Lorentz’s system. Mandelbrot’s fractals.
2 L1 Flows on the Line - Introduction Simple example dx/dt = sin(x). Strogatz 2.0, 2.1. How to solve the separable differential equation dx/dt = sin(x) by integration. Give a qualitative description of a simple dynamical system. dx/dt = sin(x)
3 L2 Fixed Points Fixed points and stability. Strogatz 2.2, 2.3. Single-variable calculus. Give a qualitative description of a simple dynamical system. Population dynamics. Logistic equation of population growth
4 L3 Linear Stability Analysis Linearization about a fixed point. Existence and Uniqueness. Strogatz 2.4, 2.5, 2.6. Flows on the line. Taylor series. Perform a linearization. Describe importance of fixed points. No examples. Discuss homework
5 HOMEWORK # 1
5 L4 Linear Stability Analysis Linearization about a fixed point. Hyperbolic vs. non-hyperbolic fixed points. Strogatz 2.4, 2.5, 2.6. Flows on the line. Taylor series. Perform a linearization. Describe importance of fixed points. dx/dt = r-x^2
6 L5 Bifurcations Saddle node bifurcation. Taylor expansions and normal forms. Strogatz 3.0, 3.1. Single-variable calculus. Taylor series. Determine changes in a dynamical system with parameter variation. dx/dt = r+x^2, dx/dt = r-x^2, dx/dt = r+x+e^x
7 L6 Bifurcations Transcritical bifurcation (supercritical, subcritical), pitchfork bifurcation. Strogatz 3.2, 3.3, 3.4. Single-variable calculus. Taylor series. Determine changes in a dynamical system with parameter variation. Know how to eliminate dimensions of a system. Learn stability analysis of 1-D systems dx/dt = rx-x^2, dx/dt = rx-x^3
8 COLLECT HOMEWORK # 1
8 L6 Dimensional analysis and scaling Dimensionless groups. Buckingham’s Pi theorem. Stability of 1-D discrete systems in the context of the last problem in HW 1. Murray 1.2, Strogatz 3.7. Flows on the line. Bifurcations. Make a dynamical system non-dimensional. Population dynamics. Insect outbreak model: Spruce budworm.
9 L7 Linear Systems Introduction. Examples of two-dimensional linear systems. Lexicon. Eigenvalues and eigenvectors. Strogatz 5.1. Class notes. Single-variable calculus. How to solve the separable differential equation dx/dt = f(x) by integration. Identify a linear system Selected set of two examples that demonstrate stiff and non-stiff systems.
10 PRACTICE TEST # 1
REVIEW SESSION
RETURN HOMEWORK # 1 GRADED
11 L8 Linear Systems Eigenvalues and eigenvectors. Reduction of second order ODEs to a system of first order ODEs Strogatz 5.1. Class notes. Single-variable calculus. How to solve the separable differential equation dx/dt = f(x) by integration. Identify a linear system Selected set of two examples that demonstrate stiff and non-stiff systems.
12 L9 Linear Systems Classification of linear systems. Types of fixed points. Classification of fixed points. Strogatz 5.2. Class notes. Linear systems. Classify linear systems. Identify and classify fixed points. Continue use of selected examples.
13 COLLECT HOMEWORK # 1 CORRECTIONS
COLLECT PRACTICE TEST # 1
TEST # 1
14 SOLUTION TO TEST # 1
GIVE HOMEWORK # 2
15 L10 Two Dimensional Flows Phase portrait. Consequences of the Hartman-Grossman theorem. Nullclines. Fixed points and linearization in two and more dimensions. Jacobian matrix. Strogatz 6.1, 6.2. Linear systems. Read a phase portrait. Selected examples.
16 L11 Two Dimensional Flows How eigenvalues determine repellers, attractors, saddles, centers, and fixed points. Strogatz 6.3 Linear systems. Interpret a phase portrait based on eigenvalues. Simple linear system.
17 L11 Limit Cycles Periodic motion. Poincaré-Bendixon Theorem. Introduction to limit cycles. Hopf bifurcation Strogatz 4.0, 4.1, 6.3, 8.2. Two dimensional flows. Identify limit cycles. Determine existence of limit cycles. Simple periodic system with a limit cycle.
18 L12 Two Dimensional Flows Applications in ecology. Population dynamics. Rabbit vs. sheep. Strogatz 6.4. Linear systems. Interpret a phase portrait based on eigenvalues. Species competition. Lokta-Volterra model of competition.
19 COLLECT HOMEWORK # 2
19 L13 Limit Cycles Applications in molecular biology. Glycolytic oscillations. Strogatz 7.3, 7.5 Two dimensional flows. Describe qualitatively and quantitatively shape and period of closed orbits. Cell biology: Glycolisis. Cell cycle.
20 L14 Numerical Mathematics Numerical solutions of ODEs Kreyszig 19.1, Quarteroni, Sacco, Salieri 11.1, 11.8 Linear algebra Compute numerically a system of ODEs None
21 RETURN HOMEWORK # 2 GRADED
ADVISE STUDENTS TO SEARCH FOR PAPERS FOR HW # 5
HAVE A BRIEF DISCUSSION ABOUT SCHOLAR DATABASES
21 L15 Numerical Mathematics Runge-Kutta 2nd order method Kreyszig 19.3, Quarteroni et al. 11.9 Linear algebra Compute numerically a system of ODEs None
22 L16 Numerical Mathematics Newton-Raphson method in two dimensions Kreyszig 17.2 Linear algebra Compute numerically the steady states of a system of ODEs Numerical roots of x^2 – x = 0
23 L17 Numerical Mathematics Newton-Raphson method in multidimensions Nash & Sofer 10.3, Quarteroni et al. 7.1 Linear algebra Compute numerically the steady states of a system of ODEs Numerical roots of x^2 – x = 0
24 COLLECT HOMEWORK # 2 CORRECTIONS
GIVE HOMEWORK # 3
INTRODUCTION TO COMPUTER SYSTEMS FOR NUMERICAL SOLUTIONS OF ODEs - XPP
25 ADVISE STUDENTS TO SEARCH FOR PAPERS FOR HW # 5
PRACTICE TEST # 2 & REVIEW SESSION
26 L18 Applied dynamical systems Hodgkin-Huxley model. J. Physiology, 1952 Fall et al. 2.1, 2.2, 2.5 Dynamical systems, linear stability, bifurcations Apply mathematical concepts to biological problems Hodgkin-Huxley model of neural activity.
27 L19 Applied dynamical systems Fitzhugh-Nagumo model Fall et al. 2.6, Strogatz problem 7.5.6 Dynamical systems, linear stability, bifurcations Apply mathematical concepts to biological problems Fitzhugh-Nagumo model of neural activity.
28 COLLECT PRACTICE TEST # 2
TEST # 2
29 SOLUTION TO TEST # 2
30 L23 Limit Cycles Global bifurcations Strogatz 8.4 Dynamical systems, linear stability, bifurcations Identify global bifurcations Selected examples from the book
31 L20 Stochastic Systems Introduce the concept of PDF. Ion channels Fall et al. 11.1 Definition of probability Apply stochastic concepts to dynamical systems None
32 L20 Stochastic Systems Ion channels Fall et al. 11.1 Definition of probability Apply stochastic concepts to dynamical systems None
33 L21 Stochastic Systems Dwell times Fall et al. 11.1 Definition of probability Apply stochastic concepts to dynamical systems None
34 COLLECT HOMEWORK # 3
GIVE HOMEWORK # 4
34 L22 Stochastic Systems Ion channel ensemble Fall et al. 11.1 Definition of probability Apply stochastic concepts to dynamical systems None
35 L24 Quasi-equilibrium Michelis-Menten kinetics Fall et al. 4.7 Dynamical systems, linear stability, bifurcations Identify functional responses Bacterial glucose binding
36 RETURN HOMEWORK # 3 GRADED
36 L25 Chaos Intro, definitions Strogatz 9.2, 9.4 Two dimensional flows. Understand the concept of chaos. Lorenz equations.
37 L26 Power-laws Intro, definitions PPT Chaos Identify when a power law is at play None
38 COLLECT HOMEWORK # 3 CORRECTIONS
38 L27 One-dimensional maps Introduction. Fixed points and cobwebs. Strogatz 10.0, 10.1, 10.2, 10.3 Flows on the line. Produce and analyze a one-dimensional map. Population dynamics: Logistic map.
39 COLLECT HOMEWORK # 4.
GIVE HOMEWORK # 5.
39 L28 One-dimensional maps Age-structured populations in space time. Leslie matrices. Britton 1.9 One-dimensional maps Understand age-structured discrete models. Physiology: Red blood cell count.
40 PRACTICE FINAL & FULL REVIEW SESSION
41 PRACTICE FINAL & FULL REVIEW SESSION
RETURN HOMEWORK #4
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