Questions tagged [multivariable-calculus]

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2
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0answers
48 views

How to define a function that has these specific properties?

Suppose $x = (x_1,x_2,\dots,x_K) \in \mathbb{Z}^K_{\geq 0}$. For $x,y \in \mathbb{Z}^K_{\geq 0}$, we write $x \succ y$ or $y \prec x$ if $x \neq y$ and \begin{align*} x_{i(x,y)} > y_{i(x,y)...
-1
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0answers
45 views

Class of functions which are defined by their second derivative [closed]

I asked this question on math stackexchange but I didn't get much input. I am looking for classes of differentiable functions that are characterized by their second derivative. For example: convex ...
2
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0answers
26 views

Hessian generating functions

I am looking for a characterization of functions $\Phi: \mathbb{R}^n \to \mathbb{R}^{n \times n}$ such that $\Phi(\mathbf{x}) = \nabla^2 f(\mathbf{x})$ for a function $f$ which is twice continuously ...
1
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0answers
28 views

What relationship exists between samples of a function and samples of its vector gradient field?

A real function $f(x)$ is defined on $N$-dimensional real space where $N \ge 3$. $f(x)$ is differentiable and its gradient with respect to x is $g(x)$. So $g(x)$ is a vector field. Assume we do not ...
1
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0answers
16 views

Creating a Multi-Variable equation to create a “Most Favorable Supplier Index” [closed]

I wanted to create a 'most favorable supplier index', where I want to assign a value to each supplier, ranging from 1-5 or 1-10, whichever, from the most and to the least preferred suppliers. There ...
7
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2answers
378 views

Is a function of several variables convex near a local minimum when the derivatives are non-degenerate?

This is a cross-post. Let $U \subseteq \mathbb R^n$ be an open subset, and let $f:U \to \mathbb R$ be smooth. Suppose that $x \in U$ is a strict local minimum point of $f$. Let $df^k(x):(\mathbb R^n)^...
1
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0answers
63 views

Conditions for a function to vanish almost nowhere on its support?

Let $f:\mathbb{R}^d\rightarrow\mathbb{R}$ be a continuous function and $\mathrm{supp}(f) := \mathrm{cl}\{x\in\mathbb{R}^d\mid f(x)\neq 0\}$ its support. Under which conditions is it true that $f≠0$ (...
1
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0answers
34 views

Differential entropy under the change-of-variable with additive Gaussian noise

I have two Gaussian random variables $$X \sim \mathcal N(0, I), \ \ \ W \sim \mathcal N(0, \sigma\cdot I)$$ and I known a parametric change-of-variable $Y(\theta) = T(X; \theta)$. I would like to ...
3
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1answer
103 views

Gradient condition implies Hörmander condition

We have tempered distribution $K$ in $\mathbb{R}^n$ which coincides with a locally integrable function in $\mathbb{R}^n\setminus \{0\}$. We call the condition $$\int_{|x|>2|y|}|K(x-y)-K(x)|dx\leq ...
0
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1answer
138 views

Is the minimum of a constraint optimization problem differentiable in the constraint parameter?

Let $h:\mathbb R^{>0}\to \mathbb R^{\ge 0}$ be a smooth function, satisfying $h(1)=0$, and suppose that $h(x)$ is strictly increasing on $[1,\infty)$, and strictly decreasing on $(0,1]$. Let $s&...
2
votes
1answer
126 views

Is the optimum of this problem convex in the constraint parameter?

Let $f:\mathbb R^+ \to \mathbb R$ be a smooth function, satisfying $f(1)=0$, and suppose that $|f|$ grows with the distance from $1$: $|f(x)|$ is strictly increasing when $x \ge 1$, and strictly ...
2
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0answers
74 views

The mean along the eccentric anomaly of an ellipse log distance to a point within the ellipse

Conjecture. Let $$ f(r,\alpha,p, \theta) = \ln\left(\left(r\sin\alpha-\sin\theta\right)^{2}\left(1-p\right)^{2}+\left(r\cos\alpha-\cos\theta\right)^{2}\left(1+p\right)^{2}\right). $$ Then for any ...
1
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0answers
41 views

Elasticity tensor in terms of principal stretches

Suppose we are given a frame-indifferent isotropic function $W:GL_+(3) \to [0,\infty)$, where $GL_+(3)$ denotes the set of all real $(3\times 3)$-matrices with positive determinant. We can write $W(F)$...
5
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0answers
412 views

How to compute the volume of a region transformed by a matrix?

This is a rewrite of the OP's question to emphasize what I think are the research level issues here. Let $\mathscr{R}$ be a bounded convex body in $\mathbb{R}^n$ and let $H : \mathbb{R}^n \to \mathbb{...
0
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1answer
200 views

Integral $ g(a)= \int_{0}^{\frac{\pi}{2}} \frac{\arctan(a \tan x)}{\tan x}dx $ [closed]

I am having trouble calculating this integral: $$ g(a) = \int_{0}^{\frac{\pi}{2}} \frac{\arctan(a \tan x)}{\tan x}dx $$ I tried calculating $g'(a)$ but then I get stuck.
2
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3answers
358 views

A Curved/Warped Version of Fubini's Theorem

I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$. Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed ...
0
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0answers
70 views

A question about multivariable calculus and optimization

Consider the function $f(x) :\mathbb{R}\rightarrow \mathbb{R}$, such that $f(x)\geqslant 0\; \forall x\in \mathbb{R}$, and has a set of extremum points at $x_{j}$. Consider the integral : $$\int_{\bar{...