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34 votes
2 answers
2k views

Is it always possible to calculate the limit of an elementary function?

I already asked this question on https://math.stackexchange.com/questions/2691331/is-it-always-possible-to-calculate-the-limit-of-an-elementary-function but as I received no answer; maybe it is not as ...
Olivier Esser's user avatar
33 votes
1 answer
2k views

How quickly can the derivative of an everywhere differentiable function change sign?

Let $f : [a,b] \to \Bbb R$ be everywhere differentiable with $f'(a) = 1$ and $f'(b) =-1$. By Darboux theorem, we know that $f'([a,b])$ is an interval containing $[-1,1]$. In particular, the set $\{x \...
Siméon's user avatar
  • 635
19 votes
4 answers
12k views

How did Bernoulli prove L'Hôpital's rule?

To prove L'Hôpital's rule, the standard method is to use use Cauchy's Mean Value Theorem (and note that once you have Cauchy's MVT, you don't need an $\epsilon$-$\delta$ definition of limit to ...
John Palmieri's user avatar
8 votes
1 answer
602 views

Example of a function with a curious property

Denote by $L^1(0,1)$ the space of Lebesgue integrable functions on the interval $(0,1)$. $\textbf{Question:}$ Does there exist a function $F:(0,1)\rightarrow\mathbb{R}$ such that: $\frac{F(x)}{x}\in ...
Tony419's user avatar
  • 421
7 votes
3 answers
905 views

A definition of the fractional derivative

I was investigating the idea of fractional derivatives and devised the following definition. WHich definition is it equivalent to and can I have a reference for it? $$\frac{d^n}{dx^n}f(x) = \lim_{h \...
Halbort's user avatar
  • 1,129
7 votes
1 answer
409 views

A property of $C^2$ functions

Let $f\in C^2(\Bbb R^m), f\geq 0$, Hessian matrix of $f$ is upper bounded by some constant $C$. Do we have $|\nabla f|\leq \alpha \sqrt{f}$ for some $\alpha$, even if the Hessian matrix is degenerate?
zhangwei's user avatar
6 votes
2 answers
409 views

Existence and uniqueness of an Euler-type ODE with varying parameters

Consider this ODE on $[1, \infty)$ $(r^2 - 2ar)f''(r) + 2(r-a) f'(r) - ({4a} + m(m+1))f(r) = -4af(1) $ with initial conditions $\frac{a}{1-2a} f(1) + f'(1) = C, \qquad \lim_{r\to \infty} f(r) = 0$ ...
Laithy's user avatar
  • 969
6 votes
0 answers
136 views

Injectiveness of a monotonic surjective mapping $\mathbb R^n \to \mathbb R^n$ with $\det J \neq 0$

Consider a surjective mapping $F \colon \mathbb R^n \to \mathbb R^n$, $F\in C^1$, $\dfrac{\partial F_i}{\partial x_j} > 0$, and $\det \left(\!\left( \dfrac{\partial F_i}{\partial x_j} \right)\!\...
jokersobak's user avatar
5 votes
0 answers
266 views

Hadamard lemma without integration

Let $I$ be the ideal of smooth germs vanishing at zero. Let $I^{k+1}$ be the ideal generated by $(k+1)$-fold product of such germs. Write $F_k$ for the ideal of $k$-flat germs at zero. By the product ...
Arrow's user avatar
  • 10.5k
5 votes
0 answers
1k views

Boundary of an open, bounded and convex set in $\mathbb{R} ^n$

Let $U$ be an open, bounded and convex set in $\mathbb{R} ^n$. Since $\partial U$ is a rectifiable set it follows that up to a set of $H^{n-1}$-measure zero $\partial U$ is contained in a countable ...
The Convex Man's user avatar
4 votes
1 answer
481 views

Higher-order derivatives of $(e^x + e^{-x})^{-1}$

I am currently trying to build the derivatives of $$f(x) = \frac{1}{e^x+e^{-x}}.$$ It is fairly straightforward to obtain $$ \frac{d^n f}{dx^n} = \frac{P_n(e^x)}{e^{(n-1)\cdot x} (e^x+e^{-x})^{n+1}}, $...
tobias's user avatar
  • 749
4 votes
1 answer
957 views

Derivative is Zero on a dense G_delta set

I have the following question: I have a function $f: \mathbb R \to \mathbb R$ which is differentiable everywhere. I also have a set $G\subset\mathbb R$ which is dense in $\mathbb R$ and a $G_\delta$-...
Neslihan's user avatar
  • 495
4 votes
1 answer
244 views

Does the homeomorphism have a non-negative or non-positive determinant?

Let $ \Omega_1 $ and $ \Omega_2 $ be domains (open and connected) in $ \mathbb{R}^2 $. $ \psi:\Omega_1\to\mathbb{R} $ and $ \phi:\Omega_1\to\mathbb{R} $ are $ C^1 $ functions with two variables. ...
Luis Yanka Annalisc's user avatar
4 votes
1 answer
265 views

Inequality involving sigmoid function

Let $\sigma$ denote the sigmoid function $\sigma(x) = \frac{1}{1+e^{-x}}$, let $x,y \in \mathbb{R}$. Given the following two conditions: $|\sigma(-x) - \sigma(y)| < \epsilon$ and $x - y > c > ...
luw's user avatar
  • 327
4 votes
1 answer
116 views

$AC^p$ curves and pointwise metric speed in abstract metric spaces?

For a fixed "reasonable" metric space $(X,d)$ (say complete, separable, whatever is needed...), a curve $\gamma:[0,1]\to X$ is said to be $AC^p(0,1)$ (absolutely continuous) if $$ d(\gamma(s)...
leo monsaingeon's user avatar
4 votes
2 answers
296 views

Implicit function theorem for subdifferentiable convex functions

I am trying to find a method to apply the implicit function theorem for subdifferential convex functions. The original theorem provides an equation for the partial derivative of the implicit function ...
Programmer1's user avatar
4 votes
0 answers
112 views

Properness of real analytic maps?

Fix a polynomial mapping $\mathbb R^n\overset{f}{\to} \mathbb R$. This answer shows that if the top degree homogeneous component of $f$ is zero only at the origin, then $f$ is proper. Intuitively, ...
Arrow's user avatar
  • 10.5k
3 votes
1 answer
146 views

Behaviour of the solution of a second order ODE

I am currently studying the following second order ODE \begin{cases} \ddot y(x)\left(\ln(x) - 2\ln(y(x))\right) - 2\frac{(\dot y(x))^2}{y(x)} = 0 &\text{in }[0,T]\\ y(0) = 0\\ \dot y(T) = c \end{...
Falcon's user avatar
  • 452
3 votes
1 answer
90 views

Representation of finite differences of order k

We define recursively finite differences $ g_k (x) $ of order $ k $ of function $ f $ as follows: $g_0(x)=f(x)$, $g_n(x)=g_{n-1}(x+h_n)-g_{n-1}(x) (n\in\mathbb{N})$. It is known that all arguments of ...
Paul Ivanov's user avatar
3 votes
1 answer
84 views

Existence and uniqueness of an Euler-type ODE with varying parameters part 2

I am working on some non-local differential equations that appear in geometric analysis. One of which I posted here and was answered by @WillieWong and @losifPinelis. Consider this non-local ...
Laithy's user avatar
  • 969
3 votes
1 answer
2k views

Symmetry of higher order mixed partial derivatives under weaker assumptions

Suppose $U$ is an open subset of $\mathbb{R}^n$, and $f:U\to \mathbb{R}$. When $f$ is $C^2$ we know that the mixed partial derivatives are symmetric, i.e. $\partial_i\partial_jf= \partial_j\...
Mohammad Safdari's user avatar
3 votes
1 answer
163 views

Counting extrema on a simplex

Let $p(x_{1},x_{2},\ldots,x_{n})=\sum_{i,j=1}^{n}{a_{ij}x_{i}x_{j}}$ be a homogenous multivariate polynomial of degree $2$. I would like to know how many extrema $p$ has on the standard simplex ...
Felix Goldberg's user avatar
3 votes
0 answers
100 views

How to compute the partial derivatives of this function?

For any probability measure $\mu$ on $\mathbb R^2$ and $\theta\in [0,2\pi]$, denote by $\mu_\theta$ its projection along $v:=(\cos\theta,\sin\theta)$. Namely, if $X$ is a random variable distributed ...
Fawen90's user avatar
  • 1,389
2 votes
2 answers
328 views

Minimum of an apparently harmless function of two variables

DISCLAIMER: I already posted this question on Mathematics a month ago, here. However, since it has not been solved yet on that platform, I decided to ask it also here on mathoverflow. At a first ...
Paglia's user avatar
  • 837
2 votes
1 answer
264 views

Monotonicity of the integral

Let $R(x)$ be the residual function associated to the normal probability density, i.e. $$R(x)~=~\int_x^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}}dy, \mbox{ for all } x\in R.$$ Define $$\phi(...
CodeGolf's user avatar
  • 1,835
2 votes
2 answers
197 views

About preserving real-rootedness of multivariable polynomials

Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted? If ...
guest's user avatar
  • 31
2 votes
1 answer
165 views

Is Sommerfeld radiation condition invariant under translations?

A smooth function $U:\mathbb{R}^3\setminus B_{r_0}(0)\to\mathbb{C}$ (for some $r_0>0$) satisfies the Sommerfeld Radiation Condition with index $k$, denoted $U\in \texttt{SRC}$, whenever $$ \lim_{r\...
GG1's user avatar
  • 146
2 votes
1 answer
168 views

Validity of formula $u(x)=\frac{1}{4\pi}\int_G \nabla_y \frac{1}{\lvert x-y \rvert} \times \omega(y) \, d^3y +A(x)$ for periodic boundary case

I think it is better to provide context in which the previous question Any formula or estimates the Green function for the Laplacian in $3D$ periodic box? has been raised. The motivation is the ...
Isaac's user avatar
  • 3,477
2 votes
1 answer
91 views

Question about optimizing a given function by optimizing an approximation

Let $f$ be a real-valued function. Suppose I want to find a local maximum of $f$, but I decide to work with an ''approximation'' to $f$ --let us call it $g$. What is a suitable notion of ''...
John Doe's user avatar
  • 170
2 votes
0 answers
946 views

On a deceptively tricky calculus problem

Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality Let $A$ be a non-constant operator acting on $C^...
matilda's user avatar
  • 90
2 votes
0 answers
44 views

Derivatives of $G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt$ when $h$ is positive-homogeneous

Let $h:\mathbb R \to \mathbb R$ be a continuous which is positive-homogeneous of order $p \ge 1$, and define $G_h:[-1,1] \to \mathbb R$ by $$ G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt. $...
dohmatob's user avatar
  • 6,853
2 votes
4 answers
584 views

Does the generalised directional derivative satisfy any version of the chain rule?

Is there a chain rule of any kind for the generalised directional derivative (of the Clarke type)? There is certainly a chain rule for the generalised gradient. The generalised directional derivative ...
wlad's user avatar
  • 4,943
2 votes
0 answers
202 views

Universal chord theorem for curves

Let $\mathrm{\gamma} : [0,1] \to \mathbb{R}^2$ be a piecewise smooth, simple plane curve. Assume $\gamma(0) = (0,0)$, $\gamma(1) = (1,0)$ and that the slope of the tangent is not $0$ wherever it's ...
user95393's user avatar
  • 121
1 vote
1 answer
346 views

Where or what is the general formula for the $n$th derivative of the power-exponential function $x^x$?

It is well-known that the power-exponential function $x^x$ and its first few derivatives are often taught in calculus. Does the general formula for the $n$th derivative of the power-exponential ...
qifeng618's user avatar
  • 1,091
1 vote
2 answers
124 views

How to show that this series of rational functions has a maximum at $x=0$ using the “Descartesschen Regel”?

I am reading an old German paper, and at one step they mention that the function \begin{equation*} f(x) := \sum_{k=2}^\infty \frac{(1+x)(k(k-1)^2 + (2+x)(1+x)^2)}{(k+x)^3 (k + 1 + x)^2} \end{equation*}...
user avatar
1 vote
1 answer
135 views

Prove the integral of multi-variable rational fraction is convergent

I have posted this problem in MSE long ago: https://math.stackexchange.com/questions/3782868/multi-variable-rational-fraction-integral. But it hasn't been solved yet so I post it here. Maybe this ...
Houa's user avatar
  • 561
1 vote
1 answer
323 views

A Bessel-like integral

I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $0\le\lambda\le1$, $p\ge0$, $q\ge0$ are ...
valle's user avatar
  • 884
1 vote
0 answers
122 views

Implicit function theorem / Implicit selections when Jacobian not invertible

I saw the attached result in the book by Dontchev and Rockafellar. It requires the Jacobian to be of full rank m. I suspect this condition can be further relaxed. Assume that we know that the columns ...
Ozzy's user avatar
  • 393
1 vote
0 answers
35 views

How to relate this integration with the integral expansion of special functions?

I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $p\ge0$, $q\ge0$ are real, and $n,a,b$ ...
Rekha K.'s user avatar
1 vote
0 answers
121 views

Does this integral have a closed form solution? [closed]

Let $x,y\in \mathbb{R}^2$, $B_r(0)=\{x||x|\leq r\}$. Does the following function (denoted as $f(r)$) have a closed form expression? $$f(r)=\int_{B_r(0)}\int_{B_r(0)}\ln|x-y|dxdy.$$
W.J.'s user avatar
  • 379
1 vote
0 answers
136 views

Determining a Closed Formula for the Positive Zeroes of the $n^{th}$ Derivatives of the Function $x↦x^{-x}$

The derivatives of the function $ f(x)=x^{-x}$ have interesting properties, especially when looking at their roots. I am interested in studying the behavior of the roots of the derivatives as the ...
Nathaniel Weidman's user avatar
1 vote
0 answers
104 views

Show this function is strictly concave

Please help me show that $f(w)$ is strictly concave in $w\in[0,\infty)$: $f(w)=\sum_{j=1}^N P_j (w)\cdot u_j $ where $P_j (w)=\sqrt{w}\int _{-\infty}^{\infty}\Pi_{k\neq j}\{\Phi[\sqrt{w}(v-u_k)]\}...
physics_student's user avatar
0 votes
2 answers
121 views

The relation between the convergence of the infinite integral of xf' and f

Question: Let $ f $ be a real-valued function that differentiable on $ [a,+\infty) $. Suppose that $ f $ is monotonically decreasing, $ \lim_{x\to+\infty} f(x) = 0 $ and the integral $ \int_{a}^{+\...
Yu Li's user avatar
  • 33
0 votes
1 answer
139 views

Proving negativeness of function involving $-\log t$

I have been trying to solve the following function is non-increasing with respect $\theta$ \begin{equation} h(t,\beta) = \frac{1-t-\frac{\beta(-\log t)^{\theta}}{\theta(-\log \beta)^{\theta -1}}}{1-\...
MSquared's user avatar
0 votes
1 answer
253 views

$f'(x)>f(f(x))$ implies $f(f(f(x)))\leq0$ for nonnegative $x$

If $f\in C^1(\mathbb R)$ satisfies $f'(x)>f(f(x))$ for all $x\in\mathbb R$, then $f(f(f(x)))\leq0$ for all $x\geq0$. I have some trouble to prove this. I wonder if there's some relations between ...
Luis Yanka Annalisc's user avatar
0 votes
1 answer
91 views

Existence of a certain type of function

Trying to find functions with the given property: Given $M>0, K$ compact in $\mathbf{R^n}$,$f:U\rightarrow\mathbf{R}$ a $C^2$ function, where $U$ open in $\mathbf{R^n}$ and $K\subset U$such that $...
Partha's user avatar
  • 954
0 votes
0 answers
81 views

What is the standard terminology for the quantity $\|\nabla f\|_{L^2(\mu)} := \sqrt{\int_{\mathbb R^d}\|\nabla f(x)\|^2d\mu(x)}$?

Let $f:\mathbb R^d \to \mathbb R$ be a continuously differentiable function and let $\mu$ be a probability measure on $\mathbb R^d$. Question. What is the standard teminology for the quantity $\|\...
dohmatob's user avatar
  • 6,853
0 votes
0 answers
72 views

Looking for example of integral transformations that preserve number of zeros

Let $f:\mathbb{R} \to \mathbb{R} $ have $n<\infty$ zeros. I am looking for non-trivial examples of integral transformation \begin{align} g(x)= \int f(t) h(t,x) dt \end{align} such that $f$ and $g$...
Boby's user avatar
  • 671
0 votes
0 answers
71 views

Existence of local minimizer

For a $f\in C^3$ function, if there is a sufficiently small $\epsilon$ $$\| \nabla F(x) \| < \epsilon$$ and a sufficiently large $\alpha$ where $$\lambda_{\min}[\nabla^2 F(x)] \ge \alpha$$ Can ...
Nikolayevich's user avatar
-1 votes
1 answer
102 views

Is it true that $\nabla_x \int_0^\infty f(t,0) dt = 0 \implies \nabla_x f(t,0) = 0 \ \forall t>0$? [closed]

Let $f:\mathbb R_+ \times \mathbb R^N \to \mathbb R$ and $$F(x) = \int_0^\infty f(t,x) dt.$$ If $\nabla_x F(0) = 0$ do we have that $\nabla_x f(t,0) = 0$ for all $t \in \mathbb R_+$? If not, which ...
Hiro's user avatar
  • 131