Determine a solution with period $2$ of the differential-difference equation $$y'(t)+y(t-1) = \cos^2 \pi t$$
Attempted solution
The r.h. side can be rewritten as $\cos^2 \pi t = \dfrac{1}{2} + \dfrac{1}{4}e^{2\pi i t} + \dfrac{1}{4}e^{-2\pi i t}$. With period $2P = 2$ we determine the funda...
In mathematics, a ball is the space bounded by a sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball or hyperball in n dimensions is called an n-ball and is bounded by an (n − 1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional...
how can I make a big summation sign ?
\begin{align}
\cos x = \sum\limits_{n=0}^{\infty} \frac{(ix)^2n}{(2n)!}
\end{align}
this is the code which I'm using for summation
Prove that to each $\epsilon > 0$ there exists a $\delta > 0$ such that $\displaystyle \int_E |f| d \mu < \epsilon$ whenever $\mu (E) < \delta$.
I found this question asked on MSE here, here, and here, but some of the suggestions seem relatively complicated (at least compared to what I did)....
From Theodoridis' Machine Learning, exercise 3.26.
Consider, once more, the same regression as that of Problem 3.8, but with $\boldsymbol\Sigma_{\boldsymbol\eta} = \mathbf{I}_N$.
For context, this is the regression model
$$y_n = \boldsymbol\theta^{T}\mathbf{x}_n + \eta_n\text{, } \qquad n...
