# Questions tagged [multivariable-calculus]

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• 677
1 vote
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### Practical calculation of Canterbury approximants

I'm looking for references on how to compute Canterbury approximants numerically from a practical point of view. The references on Canterbury approximants that I am aware of all appear rather abstract ...
• 2,768
140 views

### Is the vector field associated with an element of the boundary at infinity on a Hadamard manifold smooth?

A Hadamard manifold $M$ (complete, simply connected, non-positive sectional curvature) has a so-called boundary at infinity $\partial M$ whose elements are equivalence classes of unit-speed geodesic ...
• 23
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• 779
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### Monotonic dependence on an angle of an integral over the $n$-sphere

Let $v,w \in S^{n-1}$ be two $n$ dimensional real vectors on sphere. Consider the following integral: $$\int_{x \in S^{n-1}} \big|\langle x,v \rangle\big|\cdot\big|\langle x,w \rangle\big|\; dx.$$ ...
• 115
212 views

53 views

### Solving nonlinear differential multi-variable equation with block-matrices

Here is the problem: Given a formula $f:\mathbb{R}^{n+k}\rightarrow\mathbb{R}^n$, written as $f(x_1(t),x_2(t),...,x_n(t);a1,...ak)$ with $k$ real unknown parameters $a_1,...,a_k$. For any $(a1,...ak)$,...
182 views

### Generalization(s) of variation diminishing property to multivariate case

Let us first define the variation diminishing property for the Gaussian kernel. Consider a function $f: \mathbb{R} \to \mathbb{R}$ that is sufficiently smooth and define \begin{align} F(x)= \int_{-\...
• 541
561 views

### Any hints on how to prove that the function $\lvert\alpha\;\sin(A)+\sin(A+B)\rvert - \lvert\sin(B)\rvert$ is negative over the half of the total area?

I have this inequality with $0<A,B<\pi$ and a real $\lvert\alpha\rvert<1$: $$f(A,B):=\bigl|\alpha\;\sin(A)+\sin(A+B)\bigr| - \bigl| \sin(B)\bigr| < 0$$ Numerically, I see that ...
• 219
1 vote
184 views

### A non-differentiable function $f(x,y)$ with bounded $f_x$, $f_y$, $f_{xx}$ and $f_{yy}$

Recently I was trying to construct a counterexample to the statement "If there exist $f_{xy}(0,0)$, $f_{yx}(0,0)$ and the functions $f_{xx}$, $f_{yy}$ exist in some neighborhood and are ...
636 views

• 6,843
1 vote
172 views

### Does this non-negative function, with no stationary points, have only descent directions close to a constraint set?

Suppose $P: \mathbb{R}^n \rightarrow \mathbb{R}_{\ge 0}$ is a differentiable map, with $P(x) = 0 \ \forall x \in \mathcal{X}$ and $P(x) > 0 \ \forall x \in \mathcal{X}^c$. Further, suppose $P$ has ...
• 127
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• 383
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### Question about the implicit function theorem. an example of a homogeneous form for which its implicit function satisfies certain conditions

Let $F$ be a homogeneous form with coefficients in $\mathbb{R}$. Suppose it defines a smooth projective variety, in other words at every point other than the origin at least one of the first partial ...
• 3,605
300 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)...
• 195
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### 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 ...
• 121
1 vote
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 ...