Questions tagged [differential-calculus]
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169
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Reconstructing an object from its shadow
I'm looking into the section "Reconstructing an object from its shadow" in the book Introduction to the Mathematics of Medical Imaging by Charles L. Epstein.
I have two questions
The ...
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1
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Joint maximizer of a strongly concave function
I have a question that is arising in my research.
Suppose that $f : \mathbb{R}^ 2 \to \mathbb{R}$ is a strongly concave function, satisfying:
For every $x$, the function $y \to f(x, y)$ is maximized ...
CommunityBot
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2
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244
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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 ...
- 21
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Is there a geometric or calculus-based reason why the following system of equations should have only one solution?
Let $x_1,x_2,x_3,x_4>0$. Consider the following cyclic system of equations:
$$ 2 + x_2 + x_3 + x_4 + x_2 x_3 x_4 - 2 \left( \frac{x_2}{ \sqrt{x_1 x_2}} + \frac{x_3}{ \sqrt{x_1 x_3}} + \frac{x_4}{ \...
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Behaviour of the solution of a second order ODE
I am currently studying the following second order ODE
\begin{cases}
\ddot y(x)\left(\ln(x) - 2\ln(y(x))\right) - 2\frac{(\dot y(x))^2}{y(x)} = 0 &\text{in }[0,T]\\
y(0) = 0\\
\dot y(T) = c
\end{...
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On a deceptively tricky calculus problem
Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality
Let $A$ be a non-constant operator acting on $C^...
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Integral of non-Gaussian distributions
In physics, we have an non-Gaussian Distribution which can be simply written as $f(x)=\exp(-ax^2-bx^3)$, and we may need to calculate the integral of this distribution, simply written as $\int_0^\...
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Interpretation of second order term in Fokker-Planck equation
Let $G:\mathbb{R}^d\to\mathbb{R}^{d\times d}$ be a matrix-valued smooth function. Let us define a quantity by
$$
\begin{align*}
\nabla^2\cdot G(x)
&=\sum\limits_{i=1}^{d}\sum\limits_{j=1}^{d}\...
- 178k
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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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How to define the Sobolev quotient space $H^s(Γ)/{\mathbb R}$
Let $\Gamma$ be the boundary of a Lipschitz domain $\Omega\subset \mathbb R^3$. Denote by $H^s(\Gamma)$ the usual scalar Sobolev space for $s\in\mathbb R$. I want to know the definition of the ...
CommunityBot
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Blow-up of solutions to Euler-type ODEs
Let $\ell\in \mathbb{N}$, $a>2$, $C<0$ and $D \in [0,\infty)$. Consider the function $f: [1,\infty) \to \mathbb{R}$ solving
$$[r(r-2/a)f'(r)]' = \frac{f(r) - D}{r(r-2/a)}+ \ell(\ell+1)f(r)$$
$$f(...
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Integration on algebraic curves
Consider the plane algebraic curve
$$f(x, y) = y^4 - (2x - 1)y^2 - (4x - 1) y + x^2 + x + 1 = 0.\tag{1}$$
Its compactification results in a Riemann surface $C_1$ of genus $1$.
Hence, it can be ...
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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 ...
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Does the gradient theorem holds for a continuous function with weak derivatives on a convex set?
Let $\Omega$ be a convex open set in $n$-dimensional Euclidean space whose closure is compact.
Let $f$ be a real-valued continuous function on $\overline{\Omega}$ which also belongs to the Sobolev ...
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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 ...
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Can you help me prove this vector identity?
It could be that the preprint where I found this identity has a typo or that it is simply wrong, but I have been trying to see if this is true:
\begin{equation}
\int \left(\nabla\times F_{\bf B}\...
- 178k
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Solving a limit about sum of series
what's the limit of
$\sqrt{1-t}\sum _{n=0}^{\infty}t^{n^2}$ as $t$ goes to the left of $1$? i.e. $t\to 1^{-}$? I tried several times but failed. Here is my thought:
This is a $0\cdot\infty$ problem, ...
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Does surface integral preserve the curl operation?
Suppose $V\subset \mathbb{R}^3$ be non-empty and at least twice differentiable (Smooth) and let $S$ be the surface that encloses $V$ (for example a sphere). Let $\textbf{F}\in \mathbb{R}^3$ be a ...
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Are all Helmholtz decompositions related?
Suppose $V\subset \mathbb{R}^3$ be non-empty and at least twice differentiable (Smooth) and let $S$ be the surface that encloses $V$ (for example a sphere). Let $\textbf{F}\in \mathbb{R}^3$ be a ...
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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 ...
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Is $\Phi(-a(x+b))+\Phi(-a(x-b))$ log-concave in $x$ over the interval $x \in [0, \infty)$?
Let $f(x)=\Phi(-a(x+b))+\Phi(-a(x-b))$, where $\Phi(\cdot)$ is the c.d.f of the standard normal, and $a>0$.
I would like to know if $\partial^2 \ln(f(x))/\partial x^2<0$ over the interval $x \...
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Example of homeomorphism that lifts to real blow up but not C^1?
Given smooth manifold $M$, let $Bl_\Delta(M\times M)$ be the (say oriented; you can ask this question for the unoriented case too) real blow up of $M\times M$ along the diagonal and let $\pi:Bl_\Delta(...
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Conditions for surface area of surface of revolution to be product of arclengths
Given a circle $C$ in the xz-plane which does not intersect the $z$-axis, we can build a smooth 2-torus with surface area $(2\pi a)(2\pi b)$ where $a$ is the radius of the circle $C$ and $b$ is the ...
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Derivative formula
While trying to understand a paper of Cayley, he left something unexplained, I managed to show that it is equivalent to the following formula, which I got stuck at:
$$k \cdot (f^k)^{(k-1)} = \sum_{j=0}...
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Rotation of the coordinate system for multi-index differentiations
Let $\mathbf{f} = (f_1,\dotsc, f_n)$ be a $C^l$-map from an open subset $U$ of $\mathbb R^d$ to $\mathbb R^n$, and let $x_0 \in U$ be such that $\mathbb R^n$ is spanned by partial derivatives of $f$ ...
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Finding variance-minimizing weights [closed]
I'm trying to solve the following matrix calculus problem:
$\text{argmin}_{v \in R_+^K}(v'\Sigma v) \hspace{0.5pc} \text{subject to} \hspace{0.5pc} 1'v=1$
where $\Sigma$ is a well-behaved (symmetric, ...
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What are the limits of what the theory of time-scale calculus can capture?
Time-scale calculus [0], also called calculus on measure chains (introduced in this widely cited article 1 and also here [2]) unified the concept of derivative of a functions $\mathbb{R}\rightarrow\...
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Time-scale calculus (an similar approaches - measure chains) on more general "time" sets
Time-scale calculus [0], also called calculus on measure chains (introduced in this widely cited article [1] and also here [2]) unified the concept of derivative of a functions $\mathbb{R}\rightarrow\...
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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)\!\...
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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 ...
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How well do Gauss-Legendre quadrature methods fare on "fractal" functions?
The context
I'm making your tipical Mandelbrot set viewer, and I have a function $f: ℂ → ℕ$ that counts how many iterations of
$$
z_0 = 0 \\
z_{i+1} = z_i^2 + c
$$
it takes for a particular point $c$ ...
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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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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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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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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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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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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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$f'(x)>f(f(x))$ 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 ...
- 103k
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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
$$
\lim_{t\...
- 103k
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Density of integers related to the size of its order of appearance in the Fibonacci sequence
Let $z(n)=\min\{k>0 : n\mid F_k\}$. This function is known as the Fibonacci entry point (for example). A result of Sallé gives the sharpest upper bound for $z(n)$, namely, $z(n)\leq 2n$, for all $n$...
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Inductive proof that $\dot{M}_{n+1}=-M_{n+1}+W^{(n+2)}(0)+vM_{n+2}$
The motivation for the following is to convert the integro-differential equation
\begin{equation}
\kappa\ddot x+\dot x=-kx+\beta\int_{-\infty}^t W'(x(t)-x(s))e^{s-t}ds,
\end{equation}
into a ...
- 118k
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Osculating circle
(This question may be too elementary for this site — I'm fine if it needs to be moved to math.stackexchange.)
If I approximate a nice planar curve by a straight line, the tangent, then the second ...
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Solution of this differential equation [closed]
I wonder if it is possible to solve analytically the following equation
$$
\dot{\alpha}_t = -\frac{2}{m} \alpha^2_t + \frac{1}{2m} (\alpha_t - \alpha_t^*)^2
$$
Where $\alpha_t$ is a complex function, $...
- 106k
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Hegel's disproof of Newton [closed]
I know it's not a very comprehensive question but I've nowhere else to ask. A friend relayed to me a portion of a book from Hegel where he seemingly disproves Newton's way of finding a differential. I ...
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In matrix product, differentiate one element with respect to another element
Background
Consider a system (roughly) along the lines of those shown in Sims, C. A. (2002). Solving linear rational expectations models, where you have
$$ AX_{t+1} = CX_t + M $$
where matrix $M$ is a ...
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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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2
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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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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:...
- 106k
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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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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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