Questions tagged [differential-calculus]

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Regularity of Volterra convolution integral

I have a question about the regularity of the convolution integral. Let $f: [0,\infty) \to \mathcal{L}(\mathbb{R}^n)$ given by $f(t) = e^{tA}$. Let $g: (0,T) \to \mathbb{R}^n$ such that $g \in L^2(0,T;...
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40 views

Derivative of characteristic polynomial of a graph and derivative of characteristic polynomial of a vertex-deleted subgraph have a common root

Let $G$ be a simple graph and $G-i$ be one of its vertex-deleted subgraphs. Let $\phi(G,x)$ and $\phi(G-i,x)$ be the characteristic polynomials of $G$ and of $G-i$ respectively, with respect to their ...
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31 views

Integration of matrix form of Vasicek variance (Python/Matlab)

$X_t$ is a vector and follows the following Vasicek process. $$ dX_t=(mu-K\cdot X_t)dt+Sigma_x\cdot dZ_t \\ $$ What is the variance of $X_t$? In scalar form the answer is $\frac{Sigma_x^2}{2\cdot K}\...
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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 $n,a,b$ ...
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4 votes
1 answer
219 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. ...
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0 votes
2 answers
63 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}^{+\...
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Finding the maximum of a certain trigonometric series

Let $a_0,_1,a_2,\dots,a_n$ be a finite sequnce of $n $ real numbers and consider the function $$f(t)=\sum_{j,k=0}^{n}a_j a_k \cos\left((j-k)t\right)$$ for $t \in [0,2 \pi]$ .If we want to maximise ...
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1 answer
208 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 ...
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4 votes
1 answer
169 views

Ratio of the first squared and the second moment

Let $G(t)$ be a probability generating function of some integer and non-negative r. v. $X$. Suppose that $$\lim_{t\to1}G'(t)=+\infty.$$ That is $$ \mathbb{E}X=+\infty. $$ Can you show that $$ \lim_{t\...
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3 votes
1 answer
71 views

Inductive proof that $\dot{M}_{n+1}=-M_{n+1}+W^{(n+2)}(0)+vM_{n+2}$

The motivation for the following is to convert the integro-differential equation \begin{equation} \kappa\ddot x+\dot x=-kx+\beta\int_{-\infty}^t W'(x(t)-x(s))e^{s-t}ds, \end{equation} into a ...
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5 votes
3 answers
380 views

Osculating circle

(This question may be too elementary for this site — I'm fine if it needs to be moved to math.stackexchange.) If I approximate a nice planar curve by a straight line, the tangent, then the second ...
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1 answer
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Solution of this differential equation [closed]

I wonder if it is possible to solve analytically the following equation $$ \dot{\alpha}_t = -\frac{2}{m} \alpha^2_t + \frac{1}{2m} (\alpha_t - \alpha_t^*)^2 $$ Where $\alpha_t$ is a complex function, $...
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4 votes
1 answer
499 views

Hegel's disproof of Newton [closed]

I know it's not a very comprehensive question but I've nowhere else to ask. A friend relayed to me a portion of a book from Hegel where he seemingly disproves Newton's way of finding a differential. I ...
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1 vote
0 answers
80 views

In matrix product, differentiate one element with respect to another element

Background Consider a system (roughly) along the lines of those shown in Sims, C. A. (2002). Solving linear rational expectations models, where you have $$ AX_{t+1} = CX_t + M $$ where matrix $M$ is a ...
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1 answer
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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 ...
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6 votes
2 answers
305 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 $\frac{a}{1-2a} f(1) + f'(1) = C, \qquad \lim_{r\to \infty} f(r) = 0$ ...
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3 votes
1 answer
249 views

Laplace-Beltrami of the mean curvature

For a surface $S$ defined in 3D space, denote its mean curvature as $H$, and the Laplace-Beltrami operator as $\Delta_S$. I know that there is a result for the Laplace-Beltrami of coordinate functions:...
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3 votes
0 answers
107 views

Extension of normal vector field to a domain

Let $\Omega \subset \mathbb R^3$ be a bounded regular simply connected domain contained in a ball $S$. Assume also that $\Omega$ is simply connected by surfaces (i.e. every regular closed surface ...
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7 votes
1 answer
175 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?
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1 vote
1 answer
71 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 ...
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1 vote
0 answers
110 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? $$f(r)=\int_{B_r(0)}\int_{B_r(0)}\ln|x-y|dxdy.$$
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2 votes
0 answers
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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. $...
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3 votes
1 answer
159 views

Lower bound of the nth derivative of function

I need to prove that there exists $a> 0$ and $n_0\in\mathbb N$ such that $$\forall n> n_0,\quad \sup\limits_{|x|\leq a} |f^{(n)}(x)| \geq (n!)^{\frac 32}.$$ Where $f$ is defined by $f(x)=\exp(-...
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0 answers
75 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 $\|\...
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2 votes
0 answers
56 views

Second order partial derivatives of Sobolev functions

This has been asked on Mathematics Stack Exchange but apparently received no attention. The question is very basic in nature: Is it true that $W^{2,1}_{\text{loc}}$ functions (after possibly modifying ...
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-1 votes
1 answer
70 views

Solving a fully nonlinear first order PDE

given a symmetric matrix of Holder continuous functions $A(x)$ such that $$ \frac{1}{C} |\xi|^2 \leq \langle A(x)\xi,\xi \rangle \leq C |\xi|^2 $$ find a vector field $\Phi$ such that $$ D \Phi(x)^t D ...
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3 votes
0 answers
96 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, ...
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-2 votes
1 answer
212 views

Gradient Descent for Markov Dynamics [closed]

The closed-loop dynamics of a linear optimal controller are simple but have interesting properties. From a starting state $\mathbf{v}(0)$ the dynamic can be iterated to reach a final state $\mathbf{v}(...
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-1 votes
1 answer
84 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 ...
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5 votes
1 answer
334 views

Signed distance function and level set

For $\phi\in C^1(\mathbb{R}^N)$ with $$\omega_{\phi}=\{x\in\mathbb{R}^N\ |\ \phi(x)>0\}$$ being a bounded set with $\nabla\phi (x)\neq 0,\ \forall\ x\in\phi^{-1}(0)=\partial\omega_{\phi}\neq \...
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0 votes
1 answer
143 views

The derivation of thin plate spline interpolation energy function? [closed]

I am trying to derive the "thin plate energy functional". Given a thin plate $z = z(x,y)$, how does one derive easily the energy functional $$\iint_{\mathbb{R}^2} \,\left[\left(\frac{\...
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4 votes
1 answer
470 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}}, $...
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4 votes
1 answer
149 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 > ...
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1 vote
1 answer
238 views

Fréchet derivative of evaluation-like functional (multivariate)

I'm fairly new to functional calculus but and posting here since the question seems more appropriate than for MSE. When coming across this post I could not help but wonder the following. Let $H$ be ...
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8 votes
1 answer
578 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 ...
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2 votes
1 answer
142 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\...
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-1 votes
1 answer
113 views

An expression for the $k$-th derivative of $f(x)=x^n\exp(-x)$ [closed]

Is there a finite expression for $k$-th derivative of \begin{align} f(x)={x^n}{e^{ - x}} \end{align}
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1 vote
0 answers
42 views

Solving $\frac{dy}{dx}=1+(a_mx^m+a_{m-1}x^{m-1}+...+a_0)y^2$

I have a problem with the following equation, $\frac{dy}{dx}=1+P_m(x)y^2$ Where $P_m(x)$ is a polynomial function. I have solution for $P_m(x)=x$ using Mathematica, and Prof @Claude Leibovici solved ...
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-2 votes
1 answer
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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\...
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3 votes
1 answer
75 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)...
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5 votes
1 answer
297 views

A differential inequality involving gradient and laplacian

Let $V:\mathbb{R}^{n}\to\mathbb{R}$ smooth, such that $\lim_{|x|\to\infty}V(x)=+\infty$. What are conditions on $V$ that guarantee the existence of a function $U:\mathbb{R}^{n}\to\mathbb{R}$ such that ...
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1 vote
1 answer
278 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 ...
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3 votes
0 answers
134 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 ...
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1 vote
3 answers
320 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 ...
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2 votes
1 answer
129 views

Density of integers related to the size of its order of appearance in the Fibonacci sequence

Let $z(n)=\min\{k>0 : n\mid F_k\}$. This function is known as the Fibonacci entry point (for example). A result of Sallé gives the sharpest upper bound for $z(n)$, namely, $z(n)\leq 2n$, for all $n$...
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0 votes
0 answers
108 views

Roots of a family of 4-parameter polynomials

Let $k, \ell, p$ and $q$ be positive integers, with $q>p>1$ and $\gcd(p,q)=1$. Let $f(x)$ the polynomial given by $$ f(x)=x^q-kx^{q-p}-\ell. $$ This polynomial is related to a family of two-...
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5 votes
0 answers
209 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 ...
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  • 9,753
2 votes
1 answer
64 views

Dominant root of a family of odd degree polynomials

Let $q$ be an odd positive integer and denote by $f_q(x)=x^q-x^{q-2}-1$. This family of polynomials appears in some problems related to recurrence sequences. In order to try to use some standard ...
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-2 votes
1 answer
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What are the algebraic steps to transform this expression? (Part of differential calculus) [closed]

sorry if this seems to be too easy for you, but I have struggled a lot with this expression and I don't get it how they got there: How can $\sqrt{2x^2}$ become $4x^2$ ?
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2 votes
2 answers
115 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*}...
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