All Questions
4,828 questions
1
vote
1
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304
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How do maximum norms relatively change in Euclidean translations
Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$
that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible
linear transformation from $\...
- 11
4
votes
1
answer
778
views
Example in dimension theory
Could you give me an example of a complete metric space wiht covering dimension $> n$ all of which compact subsets have covering dimension $\le n$?
- 1,785
5
votes
3
answers
548
views
Quadrics containing many points in special position
Suppose $n$ quadric hypersurfaces cut
out $2^n$ distinct points
$p_1,\ldots,p_{2^n}$ in
$\mathbb{P}^n$. What is the maximal
number of points $p_i$ a quadric can
contain without containing ...
- 28.9k
0
votes
2
answers
251
views
3d width and cross section
Greetings,
We have a horn-shaped 3d body, which is represented as a list of vertices and faces. Each face is a triangle represented by 3 vertices. The body is positioned along the Z-axis (height). We ...
- 4,312
0
votes
2
answers
1k
views
Degenerate case of linear programming duality?
Let's say we have a maximization linear program that looks like this: maximize $\vec{c}\vec{x}$, subject to $\matrix{A}\vec{x} \leq 0$, $\vec{x} \geq 0$. If we take the dual, we have "minimize $0\vec{...
- 837
1
vote
4
answers
978
views
Maximum average value within a rectangular bounding box
The goal is to expedite detection using the sliding window approach. In other words, an object classifier is known and I need to find where the possible locations of this object are in an image. This ...
CommunityBot
- 1
3
votes
1
answer
697
views
Which motion is exclusive in 3D or higher dimensions?
Hi guys,
I have a simple question
Linear movement can be found in 1D, 2D and 3D world objects
Rotation can be found in 2D and 3D world objects.
Now, are there any kind of motion can only be found ...
- 22.3k
8
votes
2
answers
383
views
Do singular values of a point set determine its shape?
Suppose I have $k$ points in $d$ dimensions. Let A be a $k\times d$ matrix with $i$th row giving the coordinates of $i$th point. Do singular values of this matrix have an interpretation as some kind ...
CommunityBot
- 1
1
vote
0
answers
146
views
Are spherical codes algebraic?
Jeffrey Wang in Section 4.2 writes "Since a code is the solution to a number of polynomial equalities between the shortest edges, the coordinates of each rigid point in the code are algebraic and lie ...
- 11
7
votes
1
answer
686
views
Regular simplex in projective space
Is there a reference or a very short argument proving the following statement?
Let $C$ be a set consisting of $r$ points in the real projective space $\mathbb RP ^k$
with its usual round metric. ...
- 85.6k
14
votes
7
answers
3k
views
Cheap, non-constructive, free group generating rotations for Banach-Tarski
Stan Wagon's exposition of Banach-Tarski (for example) includes a beautiful explicit construction of two 2-sphere rotations which generate a free subgroup of the rotation group.
For teaching purposes ...
CommunityBot
- 1
5
votes
2
answers
625
views
Can we alter the axioms of Euclidean space to have $\mathbb{Q}^3$ as a unique model?
I posted this question at math.stackexchange.com but didn't get an answer.
Motivation
Physicists are in search for a model of discrete space(-time) for a long time. So I wondered why not start with a &...
CommunityBot
- 1
3
votes
1
answer
236
views
Non-inherited symmetries of shadows of point sets
Sometimes a point set in Euclidean space may have a shadow with an unexpected symmetry. The purpose here is to ask when this happens or when it doesn't happen (in some generality).
This requires a ...
- 1,277
14
votes
1
answer
886
views
Distance to an apartment of the affine building of GL(N)
Here $F$ is a locally compact non-archimedean non-discrete field.
Let $X$ be the reduced (affine) Bruhat-Tits building of ${\rm GL}(n,F)$. Fix a maximal split torus $T$. Let $B$ be a Borel subgroup ...
- 815
15
votes
3
answers
2k
views
orientations for zero-dimensional manifolds
I am teaching a course on manifolds, and soon I will have to prove the Stokes' theorem which, of course, involves defining oriented manifolds. There are many ways to define an oriented manifold. My ...
- 43.2k
1
vote
1
answer
273
views
Alexandrov's theorem analogue for Galilean kinematics
Let $\mathbb R^4_A$, $\mathbb R^4_B$ be spacetime as seen by two inertial observers $A$, $B$ respectively, and call $f:\mathbb R^4_A \to \mathbb R^4_B$ the change of coordinates.
We assume that $f$ ...
- 32.4k
8
votes
2
answers
577
views
Coiling Rope in a Box: Decidable?
Is the problem Coiling Rope in a Box decidable? To be specific, is this decidable?
Given $L > 0$ and $r \in (0,\frac{1}{2})$,
both rational,
can a rope of length $L$ and radius $r$
fit ...
CommunityBot
- 1
1
vote
1
answer
541
views
What is the definition of product of ideal sheaves?
Each book on algebraic geometry write I^2 when it deal with nongsingular varieties, here I
is a ideal sheaf. But no one give the definition. I guess it's the sheafification. It's right?
Thanks.
- 39.3k
5
votes
2
answers
565
views
Zorn's Lemma and plane geometry
Given a graph $G$ and a number $n$, Zorn's Lemma immediately implies the existence of a
maximal partial coloring of $G$. Equivalently, one may assign $n+1$ colors to the nodes of $G$ such that nodes ...
- 16.2k
6
votes
2
answers
407
views
Regularity of asymptotic cones
Are there any general conditions guaranteeing that the asymptotic cone of a group/graph is "regular" in some sense? E.g. for $\mathbb{Z}^d$ we get $\mathbb{R}^d$ as the asymptotic cone, which is even ...
- 618
7
votes
1
answer
456
views
Space-discriminating injective curve
Let $f\colon \mathbb R^1\to \mathbb R^3$ be a continuous and injective map. Is $\mathbb R^3\setminus f(\mathbb R^1)$ a path-connected space?
- 32.4k
13
votes
2
answers
876
views
Geodesic metrics that admit dilatation at each point
Consider the class of geodesic metrics $g$ on manifolds, that have the following
property: for each point $x$ there exists a neighbourhood $U_x$ and
a smooth vector field $v_x$ in $U_x$ that ...
- 785
3
votes
2
answers
339
views
A local transitivity property of the automorphism group of a foliated manifold
Let $(M,\mathcal F)$ be a smooth foliated manifold. An automorphism of $(M,\mathcal F)$ is a diffeomorphism of $M$ that takes leaves of $\mathcal F$ onto leaves. Let now $L$ be a leaf of $\mathcal F$. ...
CommunityBot
- 1
9
votes
1
answer
1k
views
Rigidity of triangle comparison in Alexandrov spaces
For $CAT(\kappa)$ spaces $X$ we have following rigidity result: if equality holds in any of the comparison distances between a triangle $\Delta$ in $X$ and the corresponding comparison triangle $\...
- 45k
2
votes
0
answers
160
views
Tubular neighborhood growth of zero set of polynomial of bounded degree in the torus
This question is related to my related post:
Volume growth of tubular neigbhorhood of critical values of an algebraic/differentiable map
The setting here is as follows:
Let $p: \mathbb{R}^{2k} \to \...
CommunityBot
- 1
14
votes
0
answers
4k
views
Minimum tiling of a rectangle by squares
Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes).
Is there an efficient way to calculate this?
- 1,015
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votes
2
answers
1k
views
What is the average center of six points in space
I have three pairs of points in 3D space. These may or may not be coplanar. I want to find a point such that it is equidistant from each pair of points. I know that may or may not be possible ...
- 173
11
votes
2
answers
3k
views
Levy's isoperimetric inequality for sphere
Let me recall subj:
If $s>0$, $A$ and $B$ are two subsets of $\mathbb{S}^{n}$, $|A|=|B|$ ($|\cdot|$ stands for the Lebesgue measure on the sphere) and $B$ is a cup $B=\{ (x_1,x_2,\dots,x_n)\in \...
- 22.3k
5
votes
2
answers
484
views
Better term for a (simplicial) contractible plane continuum
In this joint paper that I should be working on, we make significant use of a certain generalization of a triangulated disk. Many of our important examples are triangulated disks, but we are also ...
- 56.6k
1
vote
1
answer
390
views
Isocontours of depth and magnitude of gradient
We are interested in characterizing a 2D surface $z(x,y)$, where $(x,y)$ is the regular 2D Cartesian grid. Let $\nabla z = (z_x, z_y)$ denote the gradient. The surface is a "general" one, that is, ...
- 39k
2
votes
1
answer
1k
views
Linear Programming Cost Function [closed]
I need to add the following to my LP problem:
If the amount of workers hired in period $t$ ($H_t$) is higher than 25, the hiring cost is only 1 instead of 1.2.
Example: if 30 workers are hired in ...
- 567
8
votes
1
answer
199
views
minimal diameter of full preimage of torus
Given a set $A\subset \mathbb{R}^n$ such that $A\cap (x+\mathbb{Z}^n)\ne \emptyset$ for any $x\in \mathbb{R}^n$ (that is, $p(A)=\mathbb{T}^n$ for the projection $p:\mathbb{R}^n\rightarrow \mathbb{T}^...
- 109k
5
votes
2
answers
459
views
How indepenedent of a chosen metric is the box-counting dimension? Is there a non-integral dimension which is defined for topological spaces?
Question 1. Given a topological space $X$ and two metrics $a$ and $b$ on it, compatible with the topology, what conditions should I impose on them so that box-counting (or other, for example Hausdorff)...
- 41.1k
2
votes
1
answer
743
views
weak metric space
In the definition of a metric space, replace the triangle inequality by the weaker inequality
d (x, z) ≤ C max {d (x, y), d (y, z)},
where C is a positive constant (depending on the "metric", ...
CommunityBot
- 1
13
votes
3
answers
2k
views
Metric angles in Riemannian manifolds of low regularity
Given three points $a,b,c$ in a (geodesic) metric space $X$, one defines a comparison angle $\angle(a,b,c)$ by the cosine law:
$$
\angle(a,b,c) = \arccos \frac{|ab|^2 + |ac|^2 - |bc|^2}{2\cdot|ab|\...
8
votes
2
answers
484
views
Preferred embedding of finite metric spaces in riemaniann manifolds of given dimension
In search for a Machian formulation of mechanics I find the following problem. In Machian mechanics absolute space does not exists, and the only real entities are the relative distances between the ...
- 45k
1
vote
2
answers
1k
views
Inequality-constrained linear-regression, what is the covariance of the estimator?
If you do a linear regression: $||Ax - e ||^2$, where e is iid Gaussian, mean 0 and variance 1, then your answer is $x_{hat} = (A' A)^{-1} (A' * e)$ and the covariance of $x_{hat}$ is $(A' A)^{-1}$
...
- 233
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votes
0
answers
2k
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Geometric Proof that Fubini-Study Metric is Round
The Fubini-Study metric d(x,y) on $CP^1$ is defined as follows: for x and y in $CP^1$ let v and w be unit vectors in $C^2$ representing x and y. Then $d(x,y)=2arccos(\langle v,w\rangle)$. The round ...
- 6,247
5
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1
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568
views
Maximum distance to nearest-lattice-point on (hyper-)sphere with unit lat-lon lattice.
Let $U$ be the set of all non-null $n \times 1$ vectors $\mathbf{\mathrm{u}}$, where $u_i \in \lbrace-1, 0, 1\rbrace$. Let $\mathbf{\mathrm{x}}$ be an $n \times 1$ vector in $\mathbf{R}^n$. Let $\...
- 25.7k
10
votes
1
answer
560
views
Are packing-homogeneous spaces homogeneous?
Given a metric space (M,d) define the packing function P(x,R,r) to be the maximum number of non-intersecting balls of radius r with centers in the ball B(x,R). Let’s call M packing-homogeneous if the ...
10
votes
2
answers
4k
views
Morphism between projective varieties
Let $f:X \rightarrow Y$ be a morphism between two smooth projective varieties $X,Y$ which are defined over an algebraically closed field $k$. I am looking for some criteria which guaranties the ...
- 690
1
vote
1
answer
271
views
Synthetic Proof for Ratio of Volumes of Concentric Spheres?
Let $B^n(r)$ be the $n$-ball of radius $r$. A standard (easy) problem for first year calculus students is the following.
$(1)$ Show that $$ \lim_{n\to \infty} \frac{\text{Vol}(B^n(r))}{\text{Vol}...
- 1,302
15
votes
3
answers
1k
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Representation of vectors in $\mathbb{R}^2$ via differences of small vectors.
Is the following fact true?
Let $v_1,\ldots, v_k \in \mathbb{R}^2$, $\|v_i\|\leq 1$, be vectors that add up to zero. Does there exist a permutation $\sigma\in S_k$ and vectors $w_1,\ldots, w_k \...
- 43.2k
3
votes
1
answer
270
views
When is a blow-up a non-trivial product?
Suppose $X$ is an algebraic variety and let $Z \subset X$ be a subvariety. Are there some useful criteria under which the blow-up $Bl_Z X$ becomes a nontrivial product $V \times W$ of the algebraic ...
- 5,137
1
vote
1
answer
9k
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what is the difference between the revised simplex method andthe full tableu?
No to sound naive but they look like they include the same steps to me, one's just the algorithmical representation of the other. Thanks in advance.
- 3,283
4
votes
1
answer
2k
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How to find which subset of bitfields xor to another bitfield?
I have a somewhat coding-oriented problem. I have a bunch of bitfields and would like to calculate what subset of them to xor together to achieve a certain other bitfield, or if there isn't a way to ...
CommunityBot
- 1
9
votes
1
answer
1k
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When completion of locally compact length space is locally compact?
As far as I know the answer to the question:
"Is it true that a completion of a locally compact length space is locally compact?" - Negative.
Does anybody know some metric and/or topological ...
- 618
1
vote
0
answers
139
views
Terminology: metric space with product and unit, and the opposite of a nonexpansive map
Someone I know is trying to figure out if the following concepts already have an established name in the literature, and MO is a great place to ask around.
1) Suppose $X$ is a metric space equipped ...
0
votes
1
answer
253
views
Equidistant points in negatively curved metric spaces
Suppose that $X$ is a simply connected metric space, with a non-positively curved metric (for example, Euclidean or hyperbolic space). Let $A,B,C$ be disjoint, convex sets in $X$, and suppose that the ...
- 2,263
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3
answers
2k
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What spaces have well known horofunctions?
Following Gromov, take a metric space $(X,d)$ and consider $C(X)/\mathbb{R}$ the set of continuous functions to $\mathbb{R}$ with the topology of uniform convergence on compact sets after taking the ...
- 17k