This suggestion came from an edit I am reviewing, by user Tommi Brander. The tag inverse is currently used for a mismatch of different questions and is not helpful at all. Most of these questions involve matrix inverses: they are a basic concept, but do they deserve their own tag as opposed to li...

$\newcommand{\tr}{\operatorname{tr}}$Does submodularity property hold for the trace of a positive-definite hermitian matrix? I.e., does given any real symmetric positive-definite matrices $X,A,B$ $$ \tr X^{-1} + \tr(X+A+B)^{-1} \geq \tr(X+A)^{-1} + \tr(X+B)^{-1} $$ hold? UPD: I have checked it...

Consider the following feasibility problem: Find an $n \times n$ stochastic matrix $L$ such that $L^{-1} M L x$ is a non-negative vector, where $M$ is a known $n \times n$ positive matrix and $x$ is an $n$-dimensional probability vector (i.e., elements of $x$ add to $1$ and are non-negative)....

I have the following problem. If i can get some help, I would greatly appreciate it. I am trying to replicate a particular research paper and came across a problem: Suppose X is a birth death process (represents population size) that evolves by: $X -> X+1 $ if a birth occurs with rate $\mu$ ...

We have a 2D order-2 polynomial, a Gaussian and a 'box' indicator function. Let: $\begin{eqnarray} p(x,y) &=& c_0+c_1x+c_2y+c_3xy+c_4x^2+c_5y^2+c_6xy^2 \\ G(x,y) &=& c_k\cdot\exp\left(\frac{-(x^2+y^2)}{2\sigma^2}\right) \\ \square_a(x,y) &=& \mathbf{1}_{[-a,a]\times[-a,a]}(x,y) ...

I am looking for the associated inverse kernel to the integral transform $T$ defined by $(Tf)(u) = \int_{-\infty}^{+\infty} K(u,t)f(t) \ dt,\ \ u \in \mathbb{R^+}$ whose kernel is $K(u,t) = \frac{2cu}{c^2u^2+4\pi^2(u-t)^2}$ for a given $c > 0$. Edit: this other form of the integral transform ...

Let $C(\mathbb{R})$ be a space of continuous functions, let $f$ be a real valued function $\mathbb{R}\to\mathbb{R}$, and let $g$ be a continuous and differentiable function such that $g$ is invertible and positive. My question here is: $Q_1:$ How do I solve this equation: $e^{f(f(x)}={g^...

As we know there is the expression (f.g) for unary function composition. Is there any reference containing expression for multivariate function composition? I found the only expression searching by 'function composition' in Wikipedia is: $f|_{x_{i}=g}=f[x_{1},x_{2},\cdots,x_{i-1},g(x_{1},\cdots...

Let $f$ be a function such that :$f:\mathbb{R}\to \mathbb{R}$ and $f^{-1}$ is a compositional inverse of $f$. I would'd like to know how do I solve this class of differential equation : $$\displaystyle \ f'= e^{\displaystyle {f}^{-1}}?$$ Note 01: $f' =\displaystyle\frac{df}{dx}$. Edit: $...

Liouville's theorem gives such a proof for antiderivatives of functions like $e^x/x$ or $e^{x^2}$, and differential Galois thory extends that to Bessel functions, say. But what tools exist for implicit functions like Lambert's W?

My question: Is there an equation connecting the two branches $W_0(y)$ and $W_{-1}(y)$ of the Lambert W function for $y \in (-\tfrac 1e,0)$? For example the two square roots $r_1(y)$ and $r_2(y)$ of the equation $x^2=y$ fulfill the equation $r_1(y)=-r_2(y)$. So if one has computed one root, he a...

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map $f^{-1}:Y\to X$ is rational as well? Could you recommend any exact reference to a theorem which g...

I am aware of the inverse function theorem for Lipschitz maps, which uses the notion of generalised derivative $δ_{x_0} f$ of a Lipschitz map $f$, due to F.H.Clarke: Teorem. Let $f : U ⊂ \mathbb{R}^n → \mathbb{R}^n$ be Lipschitz, and suppose that every matrix $A ∈ δ_{x_0}f$ is invertible. Then ...

I have the series $\sum_{n=0}^{\infty}(-1)^{n}a_{n}(\nu)\frac{\sin[\nu\,(m-n)]}{\nu\,(m^2-n^2)}=\frac{1}{m}$, where $m$ is an integer. Is it possible to compute the coeffients $a_{n}(\nu)$? An exact solution is found for $\nu=\pi$: $a_0=1, a_{n}=2\times(-1)^n$. It is related to a wel...

I have been trying to solve the following difference equation for some time now : $$u^3(n+1) = a - b\cdot u^2(n) + u^3(n), \qquad a \ne 0 \ne b$$ I have tried various substitutions, simplifications but nothing did. My last attempt was to transform it into a continuous differential equation by le...