About Past Research Real Calculus Linear Optimization COV

About Me

I received my PhD in Mathematics at Washington State University in April 2013. Prior to finishing my PhD, I took 6 years off from graduate school. During that time I worked at Walla Walla Community College as the Director of Instructional Support and as an adjunct Math Instructor. I also spent one year teaching High School in a small town called Helix, Oregon. I even spent a semester at the University of Miami in Florida (which was so not my favorite place to live). I received my Master's Degree in Mathematics from Kansas State University in 2000 and I received my Bachelor's Degree in Mathematics (with a Physics minor) from Eastern Oregon University in 1998.

I love the outdoors. In fact, each of the background pictures is a place outdoors that I loved so much I had to capture with my camera. One day, I want to go backpacking in the wild for at least a month. That would be my dream vacation. I also love sports: running, soccer, volleyball, snowshoeing, skating (ice and roller).. but I'm not great at any. I also crochet, but just for fun. I love to Batik. I'm sure you've seen some of my work.

My Research

Interests

  • Calculus of Variations and variational techniques for real data
  • Partial Differential Equations
  • Pure and Applied Analysis
  • Nonsmooth Analysis and Optimization

Publications

  • HM and Thomas J. Asaki "A Finite Hyperplane Traversal Algorithm For 1-Dimensional $L^1pTV$ Minimization, For $0< p\leq 1$." COAP Vol. 61(3). PDF
  • Matthew B, Rudd and HVD, "Median Values, 1-Harmonic Functions, and Functions of Least Gradient." CPAA March 2013. PDF
  • HVD, Kevin R. Vixie, and Thomas J. Asaki, "Cone Monotonicity: Structure Theorem, Properties, and Comparisons to Other Notions of Monotonicity." Abstract and Applied Analysis. Vol 2013. Hindawi Pub. Corp., 2013. PDF
  • HVD, "A Study of $p$-Variation and the $p$-Laplacian for $0< p\leq 1$ and Finite Hyperplane Traversal Algorithms for Signal Processing." Dissertation May 2013. PDF

Real Analysis

This course is the first course in a two‐course sequence that provides a theory of the real line, properties of real numbers, and real‐valued functions. Topics include convergence of sequences; open and closed sets; density of sets; Cauchy sequences; monotone convergence theorem; pointwise and uniform convergence of functions; continuity; mean value theorem; intermediate value theorem; compactness; and differentiability.

Below are links for this course (Be sure to refresh the page)

Homework:

NOTE: I will not accept homework submitted without a .tex, .pdf, and all supporting figures (no .aux, .log, nor .syntex.gz are needed).
  • Homework 11pdf tex[Due Tuesday Apr 24 ]

Past Homework

  • Homework 1pdf tex[Due Tuesday Jan 23]
  • Homework 1HW 1: Some Solutions
  • Homework 2pdf tex[Due Thursday Feb 1]
  • Homework: Prepare proofs/disproofs to your assigned problem. Send your work to the class by noon Friday.
  • Homework:Prepare comments for Tuesday about the proofs you received.
  • Homework 3pdf tex[Due Thursday Feb 8]
  • Homework:Proofs of the 4 statements from class on Feb 1 [Due Tuesday Feb 13]
  • Homework 4pdf tex[Due Thursday Feb 15]
  • Homework 5pdf tex[Due Thursday Feb 22]
  • Homework 6+&epsilon: Write the proofs (no LaTeX required) the last two properties (The product of the terms of convergent sequences converges to the product of the limits and The quotient of the terms of convergent sequences converges to the quotient of the limits (with conditions).) And play with inequalities to get our contradiction we were trying to get at the end.
  • Homework 6pdf tex[Due Thursday Mar 8 ]
  • Homework 7pdf tex[Due Thursday Mar 15 ]
  • Exam 1(.tex)[Due Tuesday Mar 20 ]
  • Homework 8pdf tex[Due Wednesday Mar 28 (Yep, during Spring Break)]
  • Homework 9pdf tex[Due Thursday Apr 5 ]
  • Homework 10pdf tex[Due Thursday Apr 12 ]

Linear Algebra

This course provides tools that are useful in physics, engineering, natural sciences, economics and social sciences, and further study in mathematics. Topics include systems of equations, vector and vector spaces, linear operators, basis, change of basis, invertible matrix theorem, determinants, eigenvalues, and eigenvectors.

Below are links for this course (Be sure to refresh the page)

Syllabus

Homework

Past Homework

Calculus 3

This course provides students with the tools to perform calculus on functions of several variables. This is the third course in a three‐course sequence. This class is intended for students in engineering, mathematics, and the sciences. Topics include vector algebra and geometry; functions of several variables; partial and directional derivatives; the gradient; the multivariable chain rule; optimization; multiple and iterated integrals; parametric curves and surfaces in 3‐dimensions; vector fields; divergence and curl; line and surface integrals; Green’s theorem; Stokes’ theorem; and the divergence theorem.

Below are links for this course (Be sure to refresh the page)

Syllabus

Homework

Past Homework

Optimization

This course will cover the optimization techniques used to model and solve problems from various disciplines such as business, engineering, sciences, sports, .... In this course students will be introduced optimization methods for linear and integer programming optimization methods. We will emphasize techniques that expand your understanding of Calculus concepts as well as how to formulate a model; interpret problems mathematically and geometrically; solution techniques in cases where Calculus cannot be used. We will also emphasize the theory behind solution techniques; sensitivity analysis; and how to use Octave/Matlab to solve problems.

Below are links for this course (Be sure to refresh the page)

Project Details

You can find information about our expectations for your group projects here:(pdf)(.tex)

Homework

(Check this regularly. I like adding to this list.):

In Class Activities

I've included here, the in-class activities. I've also included the .tex files in case you want to see how to LaTeX some of the things we are doing.

Past Homework

  • Homework 1[DUE: Wednesday Jan 24]
  • Homework 2[DUE: Monday Jan 29]: Finish reading Chapter 1 and complete all in-text exercises not covered in HW 1 (There will be two assignments due this week)
  • Homework 3[Due Monday Feb 5]: Finish 1 and start 2 on this
  • Homework 4[Due Wednesday Feb 7]: Redo 1 on this using this code by adding three more cities in New Mexico. What do you notice?
  • Homework: Determine which constraint in problem 2 still needs discussion. Why does it need discussion? Do number 4 and begin numbers 5 and 6.
  • Find an answer to the last problem in this set. Find a solution to 1 (If you need help putting this into Octave, you can send me a description and I can help) and set up 2 and 3 (you are looking for interesting differences for these) in the Modeling Examples (pdf)(tex)
  • Email around your solutions to number 2. Numbers 4-8 were distributed, prepare a class discussion for your problem. Email me your project idea.Modeling Examples (pdf)(tex) (We won't do Chapter 2 Monday, rather Wednesday)
  • Email around your feedback to make each of the problems solvable. Include me on these emails. Also, write up a good solution to your problem. I will post these solutions. When writing up your solution, I want to see the process...Decision variables clearly defined, Objective in words and then the objective funtion matching, constraints with a word description, the model (in a box) in Octave/Matlab form, the solution and it's meaning. [Due: Wednesday Feb 28]
  • Finish the Sudoku, consider the idea where there are 16 decision variables only taking on the values 1, 2, 3, or 4 and write up a discussion about what happens. What would happen if we tried the same idea for a 9x9 sudoku?
  • Finish the in-class worksheets from March 5.Stationary Points(pdf)(.tex)
  • Finish the second order Newton approximation formula for a multivariable function. Use it to approximate where a stationary point will happen for the function given. (This was Monday's homework). For today, use the code here to see what happens when you try the same examples, but choose a second input (actually at least 3 for each). This means, instead of typing stepit(2), for the second example, you would type stepit(2,5). Don't choose anything bigger than 10, but you can use 10 if you want. Compare this to the results we got in class. What do you see?
  • Complete Gradient Recall(pdf)(.tex)
  • Do 1-3 on Order of Convergence(pdf)(.tex)

Software Code

Below are the links for provided code that can be used on Matlab or Octave (or octave-online.net) that solves any mixed-integer program (linear programs with possibly some integer variable constraints). The relevant files you will need are here: mip.m, ShowSolution.m , myexample.m and classexample.m. Here is the code I wrote for the Scheduling problem.

Calculus of Variations

This course will cover techniques and theory of the Calculus of Variations. In particular, we will look at the theory related to partial differential equations, real analysis, and functional analysis. We will also look at how the Calculus of Variations is used in Image Analysis. This course is an independent study course.

Below are links and info for this course (Be sure to refresh the page)

Syllabus

Homework:

  • Do an exercise on weak limits (the mollifier one would be nice).Finish Exercise 1.3.1 and turn it in. Read Section 2.1 and 2.2 and prepare a chalkboard discussion about it.

Past Homework:

  • For Tuesday Jan 23: Read Chapters 0: Prepare a 20 minute overview of what you read. All questions that arise should be part of this overview.
  • For Tuesday Jan 30: Read Chapter Ch 1 Sections 1 and 2 and do the Exercises. Prepare a presentation of this.
  • For Tuesday Feb 6: Reread Chapter Ch 1 Sections 1 and 2 (keeping in mind what we talked about on Jan 30). Give an example of a function in each of the function spaces: Space of continuous functions, Space of continuous functions that can be continuously extended, and space of continuous functions that vanish at infinity. Look at what you prepared for Jan 30 and adjust as necessary. We will talk more about that.
  • Match up the derivative notation on page 13 with what you saw in Calculus 3. Give an example of a Hölder Continuous function and an example of a function that is not Hölder Continuous.
  • HW for Feb 20
  • Consider the hints for our ongoing proof. Give me your intuition and understanding of: FLCoV (pg 23) and R-L Thm (pg 20). Give the underlying idea for weak derivatives. Finally, do an exercise on weak limits (the mollifier one would be nice).