王方宇 Fangyu (Fun) WANG

Explored the structural properties of finite sets with small doubling. Provided a detailed proof of Freiman's theorem, demonstrating that any finite subset of integers with a small doubling constant must be contained within a generalized arithmetic progression (GAP). Synthesized interdisciplinary mathematical frameworks, utilizing probability theory (Ruzsa Modeling Lemma), harmonic analysis (Bogolyubov's lemma), and geometry of numbers (Minkowski's theorems).

Project 1: Simple Linear Regression. Investigated the impact of advertising expenditures on product sales, conducting data splitting, model validation, and computing 95% confidence/prediction intervals to optimize budget allocation.
Project 2: Polynomial Regression. Constructed a 4th-order hierarchical polynomial regression model to evaluate the nonlinear relationship between fixed acidity and pH in wine datasets. Successfully identified and mitigated severe multicollinearity (VIF > 5) among higher-order terms via mean-centering transformation. Conducted forward selection via partial F-tests and validated critical model assumptions using R student residuals.

Time Series: Analyzed a 24-year monthly retail sales index using multiplicative decomposition and differencing techniques to achieve time series stationarity. Evaluated AR, MA, ARMA, and ARIMA models via AIC criteria and residual diagnostics (ACF/PACF plots), ultimately selecting ARIMA(1,1,0) to forecast 12-month trend variations.
Statistics: Conducted a comprehensive sales data analysis for a retail chain. Performed hypothesis testing, Single-factor ANOVA, and Tukey's Procedure to analyze average daily quantity sold across product categories and demographics, utilizing Q-Q plots for normality verification.

Projects 1 & 2: Differential Equation Solvers. Developed iterative solvers from scratch using Julia, implementing Newton's Method and Backward/Forward Euler methods to approximate equilibrium solutions for non-linear Initial Value Problems (IVPs). Evaluated numerical stability and error convergence using the 2nd-order Runge-Kutta method coupled with the Adams-Bashforth Three-Step Method against exact trigonometric solutions.
Project 3: Eigenvalue Algorithms. Implemented a pure QR decomposition algorithm to extract matrix eigenvalues. Optimized the computational implementation with a Shifted QR algorithm to successfully resolve complex eigenvalues in ill-conditioned matrices.

Analysis I: Provided rigorous proofs for foundational theorems, including the convergence of bounded monotonic sequences and the sequence defining Euler's number e. Demonstrated applications of the Intermediate Value Theorem for finding real roots of polynomials and Lagrange's Mean Value Theorem for determining function monotonicity. Explored Riemann integrability utilizing Darboux sums and proved the Archimedes-Riemann Theorem. Proved both forms of the Fundamental Theorem of Calculus with practical applications to kinematics.
Analysis II: Investigated the topological properties of Euclidean spaces, proving that the distance between disjoint compact and closed sets is strictly positive. Established the Strong Separating Hyperplane Theorem for convex sets and applied it to derive Farkas' Lemma for linear systems. Included is a comprehensive video presentation detailing the core proofs.

Calculus I: Applied integration and continuous compounding models to analyze retirement annuities under fixed, increasing, and cyclical (sine wave) interest rates. Evaluated perpetuity limits and compared annuity balances with future values.
Calculus II: Calculated 3D solid volumes using double integrals and formally proved Fubini's Theorem in a rectangle utilizing Riemann sums. Analyzed recursive sequence convergence limits mathematically, and demonstrated the Power Iteration method for computing dominant eigenvectors through both rigorous analytic proofs and Python numerical simulations.

Forest Management Modeling: Applied matrix modeling and growth matrices to determine optimal sustainable yield policies, utilizing MATLAB to solve for profit maximization configurations.
Cubic Spline Interpolation: Constructed Natural and Parabolic Runout Cubic Splines from scratch to interpolate Earth's CO2 concentration data. Solved tridiagonal matrix systems for spline coefficients computationally and compared the model estimates directly against empirical NASA datasets.

RLC Circuit Analysis: Modeled electrical closed circuits using second-order linear differential equations based on Kirchhoff's Second Law. Derived homogeneous solutions for varying damping conditions (overdamped, underdamped, critically damped). Analytically solved for the current function over time in a realistic 100 kHz resonating circuit given specific initial value constraints.