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Author Archives: mg50
Infinite Products and Weierstrass Factorization
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Tagged analysis, complex analysis, series
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Discrete Fourier Transform notes
Fast Fourier Transform Assume we have $n$ data points $x_0, x_1, \ldots, x_{n-1}$. For convenience, assume $n$ is a power of two; if it’s different, we can just pad with zeroes. We want to efficiently compute the transformed sequence $y_k … Continue reading
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Notes on linear regression
So suppose we have a set of points $(x_1, y_1), \ldots, (x_n, y_n)$ and we want to find $\beta_0$, $\beta_1$ such that $y_i = \beta_1 x_i + \beta_0$ for each $i$, or in matrix notation $Y = X\bar \beta$, where … Continue reading
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Notes on the t-distribution
Let $X_1, X_2, \ldots, X_n$ be IID normal variables with mean $\mu$ and variance $\sigma^2$. Let $\overline{X}$ be the mean of these variables, and define the deviations $D_i = \overline{X} – X_i$. Let $Y = [X_1 \ X_2 \ \ldots … Continue reading
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Tagged distributions, probability theory, statistics
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Complex analysis notes
Cauchy-Riemann Let $f(z) = f(x + iy) = g(x, y) + ih(x, y)$. If it’s holomorphic, we can differentiate along the real line and the imaginary line: $f’ = \lim\frac{f(x + a, y) – f(x, y)}{a} = \frac{∂g}{∂x} + i\frac{∂h}{∂x}$ $f’ = … Continue reading
Invariant Subspaces
Let M be a matrix, and suppose $v_1, \ldots, v_n$ for a basis for a linearly independent subspace of M. This means $$ Mv_1 = a_{11}v_1 + a_{12}v_2 + \ldots + a_{1n}v_n \\ Mv_2 = a_{21}v_1 + a_{22}v_2 + \ldots … Continue reading