I've started reading up on tensors. Since this lies well outside my usual area, I need some clarifications on some tensor calculus issues.
Let ##A## be a tensor of order ##j > 1##. Suppose that the tensor is cubical, i.e., every mode is of the same size. So for example, if ##A## is of order 3...
Consider the following theorem:
Theorem: Let ##f## be a concave differentiable function and let ##g## be a concave function. Then: ##y \in argmax_{x} {f(x)+g(x)}## if and only if ##y \in argmax_{x} {f(y)+f'(y)(x-y)+g(x)}.##
The intuition is that local maxima and global maxima coincide for...
I'm working on a model which produces a form of concavity which I'm not familiar with. Does anyone know what this form is called and if it has been studied before?
The definition in its differentiable version reads:
Let ##X\subset \mathbb{R}^{n}##. A differentiable function ##f##, defined on...
The Cauchy-Schwartz inequality (\sum_{i=1}^n x_i^2)(\sum_{i=1}^n y_i^2) - (\sum_{i=1}^n x_iy_i)^2 \geq 0 holds with equality (or is as "small" as possible) if there exists an a \gt 0 such that x_i=ay_i for all i=1,...,n .
But when is the inequality as "large" as possible? That is, can we...
Suppose y is a positive vector. Let p and x be two positive matrices with N rows, where ##p_j## and ##x_j## denotes the j:th row in these matrices, so that j = 1,…,N.
Does the following hold:
\inf_{k=1,...,N} [\sup_{l=1,...,N} [p_k(y-x_k)]] = \inf_{k=1,...,N} [p_k(y-x_k)]
where ##p_k(y-x_k)##...