Let $P_i=V_{i}V_{i}^{\top}\in\mathbb{R}^{m\times m}$ where $\forall i\in[T]: V_{i}\in\mathbb{R}^{m\times n}$ is a “tall” matrix (i.e., $m \ge n$) with orthonormal columns. Note that these matrices are symmetric PSD. Is the product of all these matrices, i.e., $P_T P_{T-1}\cdots P_1$, necessarily ...

While working on some properties of partial derivatives and multiply differentiable functions of several variables, I came across the following Hypothesis 1: Let $f: \mathbb{R}^n\to\mathbb{R}$, $\delta>0$, $k\in \mathbb{N}$ and for each partial derivative $D^{k-1}f:=\partial_{i_1}\ldots\partial_{...

In a neighbourhood of $0$ in $\mathbb{R}^n$ a smooth function $h=h(x)$, $h(0)=0$, is given. Take arbitrary real numbers $w,\lambda_1,\dots,\lambda_n\in\mathbb{R}$. The problem is to find a smooth function $u=u(x)$ around $0$ such that $$\sum_i\lambda_i\cdot x^i\,\frac{\partial u}{\partial {x^i}}(...