MATH 416- Applied Mathematics

MATH 416 — Applied Mathematics

Course Page


Course Information

Item Details
Course Code MATH 416 (Applied Mathematics)
Prerequisite MATH 251 with a minimum grade of “C-” or MATH 261 with a minimum grade of “C-“
Course Description Fourier analysis, solutions of partial differential equations, special functions, and integral transforms.
This course is designated Writing in the Major, emphasizing mathematical exposition, formal proofs,
and algorithmic/computational writing in Mathematica.
Location DMF 260
Days/Time Tuesdays & Thursdays, 2:00–3:15 PM
Office Hours Mon 1:00–2:00 PM (Teams); Tue 3:30–4:30 PM (in person); Thu 3:30–4:30 PM (in person); and by appointment

How to Access Mathematica

Category Details
Platform Mathematica (via Wolfram Cloud)
Step 1 Go to Wolfram Cloud
[1]
Step 2 Use your BSU credentials to sign up (no cost to students)
Step 3 Create and access your personal Mathematica notebooks
Course Computational Policy: All algorithm design, numerical experiments, and code are to be written and executed in Mathematica (Wolfram Language). Pseudocode can be provided, but the executable implementation must be Mathematica. Proofs may be written as text cells in Mathematica; students who prefer LaTeX may attach a short PDF and link it from the notebook.

Textbook

Title Author Publisher Year
A Basis Theory Primer
[2] 		Christopher Heil Birkhäuser 2011

Selected Publications

The following items are recommended references aligned with the course themes (Fourier analysis, PDE, frames, Gabor systems, and wavelets).
They are optional unless explicitly assigned.

Area Publication How it is used in MATH 416
Lie theory & frames V. Oussa, A Bridge Between Lie Theory and Frame Theory: Applications of Lie Theory to Harmonic Analysis, Wiley, 2025.
[11] 		Instructor’s monograph connecting Lie groups/representations to frame constructions; enrichment for the frames/time-frequency portion.
Frame theory R. J. Duffin & A. C. Schaeffer, “A class of nonharmonic Fourier series,” Trans. Amer. Math. Soc. 72 (1952), 341–366.
[12] 		Canonical origin of modern frame theory; motivates frame inequalities and nonharmonic expansions.
Frames (reference text) C. Heil, A Basis Theory Primer, Birkhäuser, 2011.
[2] 		Primary course text for bases/frames and supporting functional-analytic tools.
Frames (reference text) O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, 2003.
[19] 		Supplementary proofs/examples (frames of translates, Gabor frames, wavelet frames).
Time–frequency analysis K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, 2001.
[13] 		Rigorous toolkit for time-frequency shifts and Gabor analysis; excellent for deeper theoretical context.
Gabor analysis (historical) D. Gabor, “Theory of communication. Part 1: The analysis of information,” 1946 (DOI reference).
[14] 		Historical motivation for time-frequency atoms and phase-space thinking.
Wavelets I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992.
[15] 		Foundational wavelet theory; orthonormal wavelet bases and constructions.
Wavelets (applications) S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, 3rd ed., Academic Press, 2009.
[16] 		Computational and signal-processing perspective; supports Mathematica-based labs and numerical experiments.
Fourier analysis E. M. Stein & R. Shakarchi, Fourier Analysis: An Introduction, Princeton University Press, 2003.
[17] 		Supplementary reference for classical Fourier series/transform and connections to PDE.
PDE (reference) L. C. Evans, Partial Differential Equations, 2nd ed., AMS, 2010.
[18] 		Reference for PDE context (heat/wave/Laplace) when used alongside Fourier methods.

Writing in the Major

Each week you will submit a single entry that integrates five components: (1) concise reading notes; (2) one formal proof; (3) one algorithm (pseudocode + Mathematica implementation); (4) one computational lab in Mathematica; and (5) a short reflection. This journal is the primary vehicle for your writing-in-the-major experience.

Component Expectations
Reading Notes ≤ 300 words; key ideas, precise definitions, one question.
Proof Formal, structured, and self-contained.
Algorithm Pseudocode implementation; include inputs/outputs, assumptions, edge cases, and at least one nontrivial test.
Computational Lab Mathematica codes
Reflection ≤ 150 words; what you learned, what remains unclear, verification steps.
Midterm & Final Portfolio: You will perform a midterm audit and submit a curated final paper for this course.

Course Outline

Month Topics Important Dates
September Fourier analysis and applications to PDE
[3]

Concepts introduced:

  • Complex numbers: algebraic operations, conjugates, modulus
  • Complex number geometry: polar form, Euler’s formula, De Moivre’s theorem
  • Inequalities for complex numbers: triangle inequality, product/quotient modulus
  • Periodic functions and Fourier series: representation via complex exponentials
  • Fourier coefficients and expansion of periodic signals
  • Parseval’s identity and orthogonality in Fourier series
  • Translation and modulation properties in periodic context
  • Discrete random walks on the integers: recursion, parity, convolution
  • Convolution of measures/sequences and discrete Fourier methods
  • Continuous Fourier transform on the real line
  • Inversion formula for the Fourier transform
  • Riemann-Lebesgue lemma (decay of transform)
  • Operator rules under the Fourier transform (derivative ↔ multiplication, multiplication ↔ differentiation)
  • Applications to differential equations: first-order ODEs, heat equation
 Sep 3 (Tue): Classes begin.
October Bases and frames in Hilbert spaces
[4]

Concepts introduced:

  • Orthonormal systems and bases in Hilbert spaces
  • Frame theory: frame inequalities, bounds, analysis, synthesis, and frame operators
  • Eigenvalues of the frame operator and their relation to bounds
  • Reconstruction via dual frames
 Oct 13 (Mon): No classes.
Oct 22 (Wed): First quarter ends.
Oct 23 (Thu): Second quarter begins.
November
Instructor’s reflection (video)
[7]
Instructor’s reflection (written)
[8] 		Gabor bases and frames
[5]

Concepts introduced:

  • Translation operator
  • Modulation operator
  • Unitarity of translation and modulation
  • Inner product, Non-commutativity of translation and modulation
  • Heisenberg group structure
  • Orthonormal basis of time-frequency atoms
  • Parseval’s identity in time-frequency expansion
  • Delay (time shift) of a signal
  • Frequency shift (modulation) of a signal
  • Coefficients of inner products with atoms
  • Overlap integrals of atoms (time + frequency)
  • Partial reconstruction from time-frequency atoms
  • Polynomial signals and smooth signals in time-frequency context
  • Gibbs-type phenomena from abrupt time/frequency support
  • Time-frequency concentration and decay of coefficients
  • Numerical/experimental implementation of time-frequency atoms
  • Visualization of coefficient magnitudes (heat-map of time vs frequency)
 Nov 11 (Tue): Veterans’ Day — no class.
Nov 12 (Wed): Tuesday schedule runs — no TR class impact.
Nov 26 (Wed): Thanksgiving recess begins after day classes (no TR class that day).
Nov 27 (Thu): Thanksgiving recess.
December Wavelets and wavelet sets
[6]

Concepts introduced:

  • square-integrable functions
  • inner product
  • norm
  • isometry
  • unitary operator
  • translation operator
  • dilation operator
  • affine group
  • unitary representation
  • group relations
  • orbit of a function
  • Fourier transform
  • modulation
  • frequency scaling
  • Plancherel theorem
  • time–frequency localization
  • Haar scaling function
  • Haar wavelet
  • dyadic dilation
  • integer translation
  • Haar system
  • orthonormality
  • orthonormal basis
  • scaling relation
  • Haar expansion
  • Haar coefficients
  • Shannon scaling function
  • Shannon wavelet
  • low-pass filter
  • band-pass filter
  • frequency tiling
  • wavelet set
  • translation tiling
  • dilation tiling
  • minimally supported frequency wavelet (MSF wavelet)
 Dec 1 (Mon): Classes resume.
Dec 9 (Tue): Last TR meeting week.
Dec 11 (Thu): Reading Day (day classes).
Dec 12–18: Finals.
Dec 19 (Fri): Snow Day for finals.
Final grades due Dec 23 (Tue).

Final Assignment: Mathematical Article (≈15 pages)

Here is an example of
an article
written in the customary manner in which mathematicians write professional articles.
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Article Components

Section Purpose Key Requirements
Title Identify the main result Concise; clearly reflects the central theorem.
Abstract High-level overview 150–250 words; gives context, states main theorem, mentions key ideas/methods; no citations or heavy formulas.
Introduction Motivate and situate the work 1–2 pages; motivation/context; precise statement of main theorem (and any corollary); brief roadmap; explain how the two problems are integrated into one narrative.
Preliminaries/Notation Set up language and tools Only include necessary definitions, notation, and background.
Main Results Present and prove the mathematics Clean statement of main theorem; supporting lemmas/propositions with proofs; full proof of main theorem referencing earlier results.
Examples/Applications Illustrate the theory (optional) 1–2 short examples or applications that illuminate the theorem.
Discussion/Future Work Reflect and look ahead ≤ 1 page; discuss limitations, possible extensions, and open questions.
References Credit sources Consistent citation style (e.g., AMS); all referenced works listed.
Appendix Technical overflow (optional) Lengthy computations or auxiliary facts that would interrupt the main flow.

Global Goals & Expectations

Aspect Requirement
Overall Goal Turn two assigned homework problems into one professional-style article proving a single central theorem supported by lemmas/propositions.
Length Approximately 15 pages (excluding references and appendix).
Sources Build the article from the two assigned problems, integrated into a unified narrative.
Originality Exposition must be in your own words; clearly credit any external ideas or sources.
Rigor Define terms; justify every nontrivial step; maintain logical completeness.

Writing Standards

Aspect Guideline
Notation & Clarity Use clear, consistent notation; explain symbols on first use.
Equations Label important equations; reference them in the text when used.
Structure Use theorem/lemma/proposition environments to organize results.
Focus/Economy Avoid digressions; include only what supports the main theorem and narrative.
Style Precise, mathematically correct language; prefer active voice where possible.
Figures/Tables Provide captions; refer to them explicitly in the text.

Suggested Workflow

Step Task Focus
1 Select & unify problems Choose two homework problems and identify a single central theorem encompassing both.
2 Outline sections Plan Title, Abstract (later), Introduction, Preliminaries, Main Results, etc.
3 List needed lemmas Decide which intermediate results (lemmas/propositions) are required to prove the main theorem.
4 Build proofs Prove lemmas first, then the main theorem; ensure clean logical dependencies.
5 Draft body of paper Write Preliminaries, Main Results, and any Examples/Applications based on the proofs.
6 Write Introduction & Abstract Write these last, once the structure and results are fully clear.
7 Revise Check flow, consistency of notation, cross-references, and formatting; polish style.

Rubric (100 points total)

Criterion Points
Mathematical correctness 30
Synthesis & cohesion 20
Exposition & organization 20
Style & formatting 15
Scholarly practice 10
Polish & professionalism 5

Grading

Item Weight Date
Oral Exam 30% Thu Nov 20 (in class)
Aim: present your plan for the draft of your final paper.
Final Exam 35% Tuesday, Dec. 16
2 – 4 p.m.
Homework 25% Weekly (submission schedule posted with each assignment)
Participation 10% Attendance, engagement, and two rubric-based peer reviews
If you need help computing your grade for the course, use the following 
calculator
(download and open locally in a browser).
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Policies & Standards

  • Reproducibility: Begin labs with a header stating your name, date, Mathematica version. Keep data in a relative data/ folder; avoid absolute paths.
  • Academic Integrity: Idea-level discussion is permitted and must be acknowledged (one line is sufficient). All submitted prose, proofs, and code must be authored by you and reflect your understanding.
  • Accessibility: Proofs may be written in Mathematica text cells or in LaTeX (PDF attached). Figures must be legible and labeled.
  • Late Work: A short grace window may be announced for AMJ entries; beyond that, late submissions require prior arrangement.

References

  1. Wolfram Research, “Wolfram Cloud.” Accessed Dec 23, 2025.
    https://www.wolframcloud.com/
    ↩ back
  2. C. Heil, A Basis Theory Primer (Expanded Edition), Birkhäuser, 2011.
    Amazon listing
    ↩ back
    ↩ back
  3. V. Oussa, “Fourier analysis and applications to PDE.” Accessed Dec 23, 2025.
    vignonoussa.com
    ↩ back
  4. V. Oussa, “Bases and frames in Hilbert spaces.” Accessed Dec 23, 2025.
    vignonoussa.com
    ↩ back
  5. V. Oussa, “Gabor bases and frames.” Accessed Dec 23, 2025.
    vignonoussa.com
    ↩ back
  6. V. Oussa, “Wavelets and wavelet sets.” Accessed Dec 23, 2025.
    vignonoussa.com
    ↩ back
  7. V. Oussa, “Instructor’s reflections (video).” Dropbox. Accessed Dec 23, 2025.
    Dropbox link
    ↩ back
  8. V. Oussa, “MATH 416 — Instructor’s reflections (written).” Accessed Dec 23, 2025.
    vignonoussa.com
    ↩ back
  9. Example article PDF (Dropbox). Accessed Dec 23, 2025.
    Dropbox link
    ↩ back
  10. Course grade calculator (Dropbox). Accessed Dec 23, 2025.
    Dropbox link
    ↩ back

  11. V. Oussa, A Bridge Between Lie Theory and Frame Theory: Applications of Lie Theory to Harmonic Analysis, Wiley, 2025. ISBN 9781119712138.
    Product page
    ↩ back
  12. R. J. Duffin and A. C. Schaeffer, “A class of nonharmonic Fourier series,” Trans. Amer. Math. Soc. 72 (1952), 341–366. DOI: 10.1090/S0002-9947-1952-0047179-6.
    AMS record
    ↩ back
  13. K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, 2001. DOI: 10.1007/978-1-4612-0003-1.
    SpringerLink record
    ↩ back
  14. D. Gabor, “Theory of communication. Part 1: The analysis of information,” 1946. DOI: 10.1049/ji-3-2.1946.0074.
    DOI link
    ↩ back
  15. I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992. ISBN 9780898712742.
    Review record
    ↩ back
  16. S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, 3rd ed., Academic Press, 2009. ISBN 9780123743701.
    Listing
    ↩ back
  17. E. M. Stein and R. Shakarchi, Fourier Analysis: An Introduction, Princeton University Press, 2003. ISBN 9780691113845.
    MAA review
    ↩ back
  18. L. C. Evans, Partial Differential Equations, 2nd ed., American Mathematical Society, 2010. ISBN 9780821849743.
    MAA review
    ↩ back
  19. O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, 2003. DOI: 10.1007/978-0-8176-8224-8.
    SpringerLink record
    ↩ back