Let $L$ be a finite-dimensional Lie algebra over a field of characteristic zero. It is not difficult to see (and also follows from Theorem 4.4 of [G. Hochschild: Semi-simple algebras and generalized derivations, Amer. J. Math. 64, (1942), 677–694]) that the derivation algebra $D(L)$ of $L$ is sim...

The final result I draw is related to the integral of modified Bessel function of the second kind. But I can not solve it, and I need a explicit solution Are you willing to help me? Thank all $I = \mathop {\lim }\limits_{x \to 0} \frac{\partial }{{\partial x}}\frac{{{K_0}(a\sqrt x )}}{{{K_0}(b\s...

Consider the ring $\mathbb{C}[[X,Y]]$ and its subring $\mathbb{C}[[X+iY]]$, where $i=\sqrt{-1}$. One can show that $f(X,Y):=u(X,Y)+iv(X,Y)\in \mathbb{C}[[X,Y]]$ lies in $\mathbb{C}[[X+iY]]$ iff $u$ and $v$ satisfy the CR-equations formally. The necessary condition follows by straightforward forma...

My goal is to compute this derivative: $$ \frac{d(LL^T)}{dL} $$ Where $L$ is a lower triangular matrix obtained from a Cholesky decomposition. Is there a way to compute it in a compact form?

I would like to prove a real $C^{\infty}$(polynomial) multivariable function $F : (a_1,a_2,...a_n) \rightarrow (b_1,b_2,...b_n) $ is lipchitz of parameter $l$ is it sufficient to prove the norm of dominant eigenvalue of $Jacobian(F)$ is less than $l$. In other words, is the "spectral radius of ...

After this question : Does every real function have this weak continuity property? Natrualy there are an other (more difficult) : Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}$, there exists a real sequence $(x_n)_n$, taking values different from $c$, converging to some...

Let $C^\infty$ denote the collection of functions $f:\mathbb{R}\to\mathbb{R}$ such that for every positive integer $n$, the $n$-th derivative of $f$ exists. For $f\in C^\infty$ we set $f^{(0)} = f$, and $f^{(n+1)} = \big(f^{(n)}\big)'$ for all non-negative integers $n$. Is there $f\in C^\inft...

Let $a> 0, \alpha<0$ and $\beta>0$. How to prove that the function: $$f(x)=\frac{(\Gamma(a)-\Gamma(a,\alpha \ln(\beta x))) (\alpha\ln(x))^a}{(\alpha\ln(\beta x))^a (\Gamma(a)-\Gamma(a,\alpha \ln(x)))},$$ is monotonic. I tried the sign of derivative but is more delicate.

I know practically nothing about fractional calculus so I apologize in advance if the following is a silly question. I already tried on math.stackexchange. I just wanted to ask if there is a notion of fractional derivative that is linear and satisfy the following property $D^u((f)^n) = \alpha D^u...

I want to prove that a differentiable function $f: [0,\infty) \rightarrow \mathbb{R} $, which satisfies the following condition is always non-negative: Assume $f(0)=0$ and whenever $f(a)=0$, then $f'(a) \geq 0$ and $f''(a) \neq 0$. Note that if I change the condition to this: $f(0)=0$ and whenev...

Let $F$ be a continuous convex function on $\mathbb{R}^n$. If the subdifferential $\partial F(x)$ of $F(x)$ admits a continuous selection, for every $x \in \mathbb{R}^n$, does it mean that $F$ is differentiable on $ \mathbb{R}^n$ ? I was trying to use theorem 25.5 and 25.6 (Rockafellar: Convex An...