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5:57 AM
Q: Why does the velocity gradient tensor have additive decomposition but the deformation gradient does not?

RodriguesIn 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...

The above query did not return any older occurrences.
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century. == Explanation == Modeling an object as a continuum assumes that the substance of the object completely fills the space it occupies. Modeling objects in this way ignores the fact that matter is made of atoms, and so is not continuous; however, on length scales much greater than that of inter-atomic distances, such models are highly...
2 hours later…
7:46 AM
A new tag was created by Rodrigo de Azevedo and added to 14 questions.
Q: Distance between two ellipsoids via Lagrange multipliers

Jack RolphI 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...

Q: How to prove $B-A \succeq 0 \Leftrightarrow$ ellipsoid $x^TBx \leq 1$ contains $x^TAx \leq 1$?

wz0919Assume $A \in \textbf{S}^n_{++}$, 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...

Q: Convexity of the ellipsoid

Yuri SagalaIn 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 ...

Q: Minimum enclosing ellipsoid to maximal enclosed ellipsoid

user3492773Given 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...

Q: Relationship between ellipsoid radii and eigenvalues

chepukhaI 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...

Q: Correspondence between eigenvalues and eigenvectors in ellipsoids

Joe1979think 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...

Q: How can I uniformly draw points from an ellipsoid?

BobSpecifically, 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...

Q: Computing the Minimum Volume Enclosing Ellipsoid using the Ellipsoid Method

user3492773I 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...

Q: Unitary matrices and ellipsoid

David MayarLet $\mathcal{S}$ 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?

Q: Close packing of ellipsoids

KazarkHow 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...

Q: Volume of the intersection of ellipsoids

sdsHow 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...

Q: Show that the ellipsoid is convex

abbasly 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.

Q: Mean width of an ellipsoid

Gilles BonnetLet $E$ 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 $...

Q: Ellipsoid but not quite

peter.petrovI 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...

6 hours later…
1:58 PM
Q: Too many ellipsoid questions being bumped to the front page.

Don ThousandI 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...

5 hours later…
7:19 PM
@MartinSleziak I seem to recall that you posted an SEDE query a while back which ranked users by the number of tags they had created. I can't seem to find it in the transcript here... do you have a link to that query handy?
7:31 PM
@XanderHenderson My "batch" of tag-related queries is here: chat.stackexchange.com/transcript/message/54718583#54718583
@MartinSleziak Ah! Yes! Thank you!
Most frequent tag-creators: main, meta.
Most frequent tag "cleaners": main, meta.
@XanderHenderson The query about tag-creators posted here ^^ is probably what you wanted to see.
I should include these two queries when I post tag-related stuff here from time to time.
I will explicitly mention that I look there at tag names. That means that the count includes the tags which no longer exist. (For example, it includes also the tags which were created by mistake, such as various typos.)
I got what I wanted from the tag creators query, but the "most frequent" is definitely what I wanted. Thanks again.
And I will also point out that the query looks with older occurrence of the tag among the existing posts. So there might be some false positives. (E.g., the question, where the tag was created, was deleted.)
@MartinSleziak But for rough order-of-magnitude questions, it should be about right.
7:36 PM
Yes, it's unlikely that there are many false positives.
You can also check tags created by a specific user: main, meta.
1 hour later…
8:48 PM
Q: Why [tag:derivatives] instead of [tag:differentiation]?

Rodrigo de AzevedoAt 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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