# Tagged Questions

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### Existence of a Minimizer $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|}$

given the following functional $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|}$ with $\rho>0$ , $\|\rho\|_1 = 1$ and obviously $\rho\in L^1(\mathbb R^3)$. Can I see ...
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### Finding the shortest path length on a curved surface(hyperboloid)

I wish to find the minimum path length between two points $P_1(\sqrt2,0,-1)$ and $P_2(0,\sqrt2,1)$ on a hyperbolic surface $S =\{(x,y,z)\in R^3\ |\ x^2+y^2-z^2=1\}$ I faintly recall studying ...
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### Calculating the maximum of a function

How can one determine $$\max_{f_0,f_1}\frac{\int_{\mathbb{R}}f_1(y)^xf_0(y)^{1-x}\log\left(\frac{f_1(y)}{f_0(y)}\right)\mbox{d}y}{\int_{\mathbb{R}}f_1(y)^xf_0(y)^{1-x}\mbox{d}y}$$ given ...
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### What methods are available for this optimization problem?

I have an intermediate knowledge of the calculus of variations: I can handle constraints in functional or integral forms and extrapolate to multiple variables and functions. If I dig in my notebooks I ...
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### Calculus of variations for implicitly defined functional

I would like to minimize a functional of the type: $$L[\gamma]=\int_a^b F(T(\gamma(t))dt$$ on the space of paths $\gamma$, where $T=T(\gamma,t)$. Now, usually I would simply apply Euler-Lagrange's ...
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### Is it possible to solve or approximate this second order nonlinear system of differential equations.?

Given initial values $d[0]$ and $k[0]$, I would like to solve for the initial rate of change, $\dot d[0]$, and compare this value against some data. I have the following profit function, which I ...
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### Can this ant find its way back to the nest?

So the puzzle is like this: An ant is out from its nest searching for food. It travels in a straight line from its nest. After this ant gets 40 ft away from the nest, suddenly a rain starts to ...
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### How can Hotelling reduce the Euler-Lagrange equation in his calculus of variations mine problem?

In a 1931 paper Hotelling gives the discounted profit of a mining operation as: $$P=\int_{0}^{\infty} \dot{x} p(x,\dot{x},t) e^{-rt} \:\:dt$$ Note that this is, for the most part, a typical calculus ...
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### Calculus of variations, what is a functional

I'm reading a bit about the calculus of variations, and I've encountered this bit: Suppose the given function $F(.,.,.)$ is twice continuously differentiable with respect to all of its arguments. ...
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### Sufficient conditions for Hessian definiteness for critical points of functionals

Let $C$ be the set of smooth curves from the unit interval into $\mathbb{R}^n$. Let $f : C \rightarrow \mathbb{R}$ be a functional on these curves given by $f(x) := \int_0^1 L(x,\dot{x}) dt$. Define ...
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### How to find $\kappa$ to minimize integral $I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) \mathrm{d}x$

I am trying to find such value $\kappa \in (0,1)$ that would minimize the integral \begin{aligned} I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) ...
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### integer transform

Let be $X$ the following finite set: $X=\{0,1,2,\ldots,63\}$. I want to find two function $f$ and $g$ , where $f,g:X \times X \to Z$. We define $x'=f(x,y)$ and $y'=g(x,y)$. We impose the following ...
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### Gradient/sub-differential of a minimum of a function

Suppose I have a function $F(x,D) = ||y-Dx||_2^2$, such that $x^{*}(D)= \displaystyle arg \min_{x} F(x,D)$ (that is given $y$ and for a fixed $D$) and subject to some constraint $h(x) <\epsilon$, ...
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### Intuition of 'Transversality conditions' needed

I have read different matrials on Calculus of Variations, but I still do not grasp the intuition of transversality condition. From textbooks, I can only roughly get an idea that with a transversality ...
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### Lagrange multipliers — What have I done wrong?

I am trying to find the stationary points of the potential $U(x,y)=x^2+y^2$ with constraint $x^2-2y^2=1$ So I set the Augmented potential $U^*=x^2+y^2+m(x^2-2y^2)$ where $m$ is the Lagrange ...
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### Constraint of a Lagrange multiplier

My question concerns Lagrange multipliers and the possibility to impose constraints on the multipliers themselves. I have a Stokes flow which is solved using the Finite Element Method on a domain ...
It is well known that the problem of minimizing $$J[y] = \int_{0}^{1} \sqrt{y(x)^2 + \dot{y}(x)^2} dx$$ with $y \in C^2[0,1]$ and $y(0) = 1$ and $y(1) = 0$ has no solutions. However, if we remove ...
### How to solve $\max_{f}\int_{0}^{\infty}f\left(x\right)dx$ subject to $\int_{0}^{\infty}xf\left(x\right)dx=x_{0}$?
How to find $\max_{f}\int_{0}^{\infty}f\left(x\right)dx$ subject to $\int_{0}^{\infty}xf\left(x\right)dx=x_{0}$, where $f$ is a function and $x_{0}$ is a constant?