In describing the decomposition of the deformation gradient tensor given by $\mathbf{F}=F_{iJ}=\partial x_i/\partial X_J$, Malvern (1969, Introduction to the mechanics of a continuous medium) mentions that due to the finite nature of the displacements implied in the definition of $F_{iJ}$, it doe...
I want to use Lagrange multipliers to find the minimum separating distance between two ellipsoids, both centered at the origin. To illustrate, we start with both ellipsoids sharing the same center: And want to finish with the closest point that the two ellipsoids may be separated: Attempted sol...
Assume $A \in \textbf{S}^n_{++}$, an ellipsoid centered at the origin given by $$\mathcal{E}_A = \{x\mid x^TA^{-1}x \leq 1\}$$ Then we have $\mathcal{E}_A \subseteq \mathcal{E}_B $ if and only if $B-A \succeq 0$. This is the proposition in the Boyd & Vandenberghe's Convex Optimization (pages 45-4...
In Boyd & Vandenberghe, it is mentioned that the ellipsoid is defined by \begin{equation} \mathcal{E} = \left\{ x \in \mathbb R^n \mid (x-x_c)^T P^{-1} (x-x_c) \leq 1 \right\} \end{equation} where $P$ is positive definite. My questions are: How to transform it to the normed (in)equation? How ...
Given a convex body $K$ and an ellipsoid of minimal volume which contains $K$, find the maximal ellipsoid contained in $K$. I have tried to multiply the matrix by 4 (since the eigenvalues are the reciprocals of the squares of the semi-axes, but I don't get the maximal ellipsoid. If you need to...
I should start by saying that I haven't done algebra for very long time. I recently have some work related to algebra, so I need some help to speedup. I went through a theorem in the book stating the relationship between ellipsoid's radii and the eigenvalues. So the ellipsoid is defined as foll...
think of an ellipsoid in the n-dimensional space defined by $$(x-\mu)'A(x-\mu)=1.$$ I was calculating the volumes of n-dimensional ellipsoids like the one from above for a while, which is straightforward once the eigenvalues of matrix $A$ are retrieved. The volume $V$ is then given by (using the...
Specifically, given a positive definite matrix $A \in \mathbb{R}^{n \times n}$, how can I efficiently generate points $x \in \mathbb{R}^n$ that satisfy $x^TAx \leq 1$? I know how to do this when the matrix is just the identity, because then this question reduces to simply generating uniformly fro...
I need your expertise in solving the following the follow problem: Given a convex body P, which is given implicitly by an oracle (either membership or separation), the objective is to find a minimum enclosing ellipsoid using the ellipsoid method. To achieve such task, we need to formulate the p...
Let $\mathcal{S}$ be an ellipsoid in $\mathbb{R}^n$ and let $U\in\mathbb{R}^{n\times n}$ be a unitary matrix. Is it true that $U\mathcal{S}=\{y| \quad \exists x \in\mathcal{S} \quad s.t. \quad y=Ux\}$ is also an ellipsoid?
How can the packing density of a set of congruent ellipsoids be calculated? I'm dealing with prolate spheroids so technically I do not need the general answer for ellipsoids, but my abstract mind loves more general answers. If calculating an exact value is too difficult, an estimate which is accu...
How do I compute the volume of the intersection of two $n$-dimensional ellipsoids? Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid $$E(c,A)=\{x|(A(x-c),x-c)<1\}$$ where $(\cdot,\cdot)$ if the dot product. Then $$\mathrm{vol}(E(c,A))=\frac{u...
Show that an ellipsoid $$\{x\in \mathbb{R}^n \ : \ x^TAx+2b^Tx+c\le 0\},$$ where $A\in \mathbb{S}^n_+$, is a convex set.
Let $E$ be an ellipsoid in $\mathbb{R}^d$ defined by $$\sum_{i=1}^d \frac{x_i^2}{a_i^2}=1$$ Is there a formula to express the mean width (or an approximation of the mean width) of $E$ in term of the lengths $a_i$ of the (semi-principal) axis? The width in the direction of the principal axis are $...
I have an ellipsoid centered at the origin. Assume $a,b,c$ are expressed in millimeters. Say I want to cover it with a uniform coat/layer that is $d$ millimeters thick (uniformly). I just realized that in the general case, the new body/solid is not an ellipsoid. I wonder: How can I calculate th...
I was browsing the front page and noticed that the first 20ish questions were all old questions, edited to include the newly created ellipsoids tag. I was under the impression that there was some limit to editing tags in this way? I don't feel that this is productive to the site's usability. It s...
At the moment, Mathematics SE has tag derivatives (with synonym differentiation) and tag integration. Why the lack of symmetry? Why not have tag differentiation with synonym derivatives instead? Differentiation is about infinitesimal differences — whatever that means. Derivatives? Many things are...
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