All Questions
Tagged with differential-calculus real-analysis
51 questions
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 ...
- 1,389
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-\...
- 23
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{...
- 452
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^...
- 90
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 ...
- 393
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 ...
- 3,477
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 ...
- 1,091
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)\!\...
- 243
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 $n,a,b$ ...
- 11
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. ...
- 1,219
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}^{+\...
- 33
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 ...
- 1,219
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 ...
- 969
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
$\frac{a}{1-2a} f(1) + f'(1) = C, \qquad \lim_{r\to \infty} f(r) = 0$
...
- 969
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?
- 73
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 ...
- 561
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?
$$f(r)=\int_{B_r(0)}\int_{B_r(0)}\ln|x-y|dxdy.$$
- 379
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
$$
G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt.
$...
- 6,853
0
votes
0
answers
81
views
What is the standard terminology for the quantity $\|\nabla f\|_{L^2(\mu)} := \sqrt{\int_{\mathbb R^d}\|\nabla f(x)\|^2d\mu(x)}$?
Let $f:\mathbb R^d \to \mathbb R$ be a continuously differentiable function and let $\mu$ be a probability measure on $\mathbb R^d$.
Question. What is the standard teminology for the quantity $\|\...
- 6,853
4
votes
0
answers
112
views
Properness of real analytic maps?
Fix a polynomial mapping $\mathbb R^n\overset{f}{\to} \mathbb R$. This answer shows that if the top degree homogeneous component of $f$ is zero only at the origin, then $f$ is proper. Intuitively, ...
- 10.5k
-1
votes
1
answer
102
views
Is it true that $\nabla_x \int_0^\infty f(t,0) dt = 0 \implies \nabla_x f(t,0) = 0 \ \forall t>0$? [closed]
Let $f:\mathbb R_+ \times \mathbb R^N \to \mathbb R$ and $$F(x) = \int_0^\infty f(t,x) dt.$$ If $\nabla_x F(0) = 0$ do we have that $\nabla_x f(t,0) = 0$ for all $t \in \mathbb R_+$? If not, which ...
- 131
4
votes
1
answer
481
views
Higher-order derivatives of $(e^x + e^{-x})^{-1}$
I am currently trying to build the derivatives of $$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}}, $...
- 749
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 > ...
- 327
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 ...
- 421
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
$$
\lim_{r\...
- 146
-2
votes
1
answer
968
views
Bounding the product of lipschitz function [closed]
Assume we know that $f(x,y): \mathbb{R}^{a+b} \to \mathbb{R}^d$ is lispchitz w.r.t. $y$, i.e.
$$\|f(x,y) - f(x,y')\| \leq L \|y-y'\|.$$
Is there a way to bound the product $f(x,y)\cdot f(x,y')^T \in\...
- 245
4
votes
1
answer
116
views
$AC^p$ curves and pointwise metric speed in abstract metric spaces?
For a fixed "reasonable" metric space $(X,d)$ (say complete, separable, whatever is needed...), a curve $\gamma:[0,1]\to X$ is said to be $AC^p(0,1)$ (absolutely continuous) if
$$
d(\gamma(s)...
- 5,380
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 ...
- 884
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 ...
- 41
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 ...
- 4,943
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 ...
- 10.5k
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*}...
user145347
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 ...
- 101
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 $...
- 954
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 $g$...
- 671
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 ...
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 ...
- 496
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 ...
- 719
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 \...
- 635
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 ''...
- 170
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\...
- 755
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 $G_\delta$-...
- 495
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 ...
- 121
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)]\}...
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?
$$\frac{d^n}{dx^n}f(x) = \lim_{h \...
- 1,129
2
votes
2
answers
328
views
Minimum of an apparently harmless function of two variables
DISCLAIMER: I already posted this question on Mathematics a month ago, here. However, since it has not been solved yet on that platform, I decided to ask it also here on mathoverflow. At a first ...
- 837
2
votes
1
answer
264
views
Monotonicity of the integral
Let $R(x)$ be the residual function associated to the normal probability density, i.e.
$$R(x)~=~\int_x^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}}dy, \mbox{ for all } x\in R.$$
Define
$$\phi(...
- 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 ...
- 31
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 ...
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
...
- 6,962