# Tagged Questions

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### Application of a general “Weierstrass theorem”

http://books.google.at/books?id=9OSrV73a40gC&pg=PA45&lpg=PA45 gives a general Weierstrass theorem. Are there notable applications of this theorem, say in the calculus of variations? (I could ...
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### Division of plane into equal area regions

We divide a plane ($\mathbb{R}^2$) into infinite number of regions each of area equal $1$. We can use only (one-dimensional) curves which may meet at points. Fix a point $p$ on a plane and consider ...
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### minimizing a function involving exponential term

Let $w\ge e$ . I want the following $$\min_{r\geq0} r(e^r-w)$$ Is there any way to find it. Thanks.
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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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I am aware of the definition of a Rate of convergence for iterative methods involving a single point $$x_{n+1} = \phi(x_n) ~~~;~~\lim_{n \rightarrow \infty} {{ |x_{n+1} -\alpha|} \over{|x_{n} ... 1answer 39 views ### Approximation in L^2 Let G be a domain assumed smooth enough. I want to show that the mean value m is minimizing  m \rightarrow \| f-m\|_{ L^2(G)}  for  f \in L^2(G). Is it unique? Is it allowed to derive under ... 1answer 342 views ### Arnol'd's trivium problem #68 I came across this blog that says that its French version has answers to most of Arnol'd's trivium problems, and I figured I'd try my hand at some of the ones they don't have. Number 68 raised my ... 1answer 117 views ### how to obtain Euler equation for smoothing spline minimization problem? The question might be trivial, but I don't understand why this minimization problem in Sobolev space$$ \min_{g}\int_{0}^{1}\left\{ f(x)-g(x)\right\}^{2} dx+\lambda\int_{0}^{1}\left\{ ...
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 $h(x) <\epsilon$, ...