An ordinary differential equation that generalizes the notion of "path of steepest descent." For questions on "gradients" of a function, use (multivariable-calculus) instead.

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### Mean curvature flow: Second time derivative?

Suppose I have a 2D surface in $\mathbb R^3$ undergoing mean curvature flow, i.e., the motion of a point on the surface instantaneously can be described as $$\frac{d\mathbf x}{dt}=-H\mathbf n,$$ where ...
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### Multivariate Chain Rule with Derivatives in Intermediate Functions

I have a function $$G: \mathbb R^d \times \mathbb R \times \mathbb R^d \to \mathbb R$$ where $d$ is a positive integer and the arguments of $G$ are denoted by $({\bf y}, z, {\bf p})$. I'm denoting ...
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### How do we show that narrow convergence is only needed on a spanning uniformly dense subset

At the beginning of Chapter 5 of Ambrosio's Gradient Flows book, he introduces the idea of a narrowly convergent sequence of measures as... "A sequence $(\mu_n) \subset \mathcal{P}(x)$ is ...
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### Existence of projection $P$ equivalent to $P' \circ T \circ P''$ for projections $P', P''$ and smooth translation $T$?

A projection in the linear algebraic sense is a linear map $P$ such that $P^2 = P$. I'm interested in knowing when there is guaranteed to exist a projection $P$ such that $P = P' \circ T \circ P''$, ...
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### Are linear interpolation curves on Wasserstein spaces absolutely continuous?

Let $\mathcal{P}_2$ the space of absolutely continuous probability measures on $\mathbb{R}^d$ with finite second moment equipped with the $2$-Wasserstein metric. Fix $\mu_0, \mu_1 \in \mathcal{P}_2.$ ...
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### (Explaination of) Proof of Global Stable Manifold Theorem in Audin & Damian's book

I am reading Michèle Audin and Mihai Damian's book $\textit{Morse Theory and Floer Homology}$ and I stick at one sentence regarding Global Stable Manifold Theorem which states $W^s(a)$(that is, the ...
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### Help with an article on convex-splitting method

I trying to simulate gradient flow $$\frac{\partial u}{\partial t}=-\nabla_x F(u)$$ where F(u) is (in my case): $$F(u)=2u^4-u^2$$ for u $\in [-0.5,0.5]$. Since the equation is non-linear, I would like ...
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### Solving $\partial_t \gamma_t(x) = - \gamma_t(x) + \frac{\gamma_t''(x)}{\gamma_t'(x)^2}$, a nonlinear PDE on quantile functions

While pondering Wasserstein-2 gradient flows of the Kullback-Leibler divergence functional $\text{KL}(\cdot \mid \nu)$, where $\nu \sim \mathcal N(0, 1)$ is the standard normal distribution (yes, I ...
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### About minimizing dynamic of a convex function

Suppose that we have a strictly convex function $f: \mathbb R \rightarrow \mathbb R$ that admits a unique minimizer $x^*$ and $f(x^*)=0$. Let $x_0< x^*$, clearly $f$ is decreasing on $[x_0,x^*)$ ...
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### Finding the gradient of the squared $L_2$ matrix norm during constrained optimization

I'm reading this paper for a university project, and I'm currently stuck on part B.2 of section 3 where the goal is to minimize $$\frac{1}{N}\sum_ne^{-\rho y_nf(V;x_n)},$$ with respect to the ...
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### When do solution to differential equations belonging to the same parametric family intersect?

General Problem I am interested in studying whether solutions to the Fokker-Planck equation: \tag{1}\label{fp} \frac{\partial p(x, t)}{\partial t} = \textrm{div}(-p(x, t)\nabla\log q(...
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### Gradient flows: Stability of minimizers in infinite dimensions

Suppose I have an $L^2$ gradient system $\dot{x}=-\nabla V(x)$, where $V:H\rightarrow\mathbb{R}$ is an analytic potential on an infinite dimensional Hilbert space $H$. Suppose also that I have ...
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### Proof of equivalence of solutions to initial value problem given same gradient

I have two functions $x(t)$ & $y(t)$ I have $x(0) = y(0)$ and that $\dot{x} = f(x,y)$ & $\dot{y} = g(x,y)$ I have that when $x = y$, $\dot{x} = \dot{y}$. I want to show that $x(t) = y(t)$, I ...
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