All Questions
4,827 questions
4
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1
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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 ...
- 41
15
votes
3
answers
1k
views
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 \...
- 1,284
7
votes
2
answers
299
views
subsets of products of trees
A subset of a geodesic metric space is called convex if for every two points in the subset one of the geodesics connecting these points lies in the subset. Is it true that every convex subset of a ...
user6976
21
votes
5
answers
1k
views
Is a rhombus rigid on a sphere or torus? And generalizations
If a rectangle is formed from rigid bars for edges and joints
at vertices, then it is flexible in the plane: it can flex
to a parallelogram.
On any smooth surface with a metric, one can define a ...
- 151k
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 ...
10
votes
2
answers
1k
views
Canonical geometric examples
The proofs without words post has some great entries. I'm interested in a similar concept: examples where a problem in math or physics is accompanied by a geometric figure that illuminates some key ...
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 ...
- 1,329
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 ...
- 28.9k
10
votes
3
answers
2k
views
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 ...
- 4,304
7
votes
1
answer
938
views
Which knots' stick numbers are twice their crossing numbers?
Looking at a table of minimum stick numbers for knots (table here),
it seems the known upper bound of $2 c(K)$ in terms of the knot crossing number $c(K)$
is realized by the trefoil $3_1$—it ...
- 151k
0
votes
1
answer
1k
views
What happens to the volume of an ellipsoid as the number of dimensions increases? [closed]
Would the volume of an ellipsoid continuously increase if one keeps adding radii along new dimensions? What is the volume of ellipsoid with infinite dimensions?
- 131
50
votes
3
answers
4k
views
Explicit metrics
Every surface admits metrics of constant curvature, but there is usually a disconnect between
these metrics, the shapes of ordinary objects, and typical mathematical drawings of surfaces.
Can ...
- 25.1k
16
votes
3
answers
2k
views
A random walk on random lines
I am wondering if this random walk remains finite with positive probability.
Start with three lines $A,B,C$ that are extensions of an equilateral triangle.
Let $p_0$ be one corner. Generate a line $...
- 151k
-1
votes
1
answer
502
views
How to formulate such problem mathematically? (line continuation search) [closed]
I have an array of "lines" each defined by 2 points. I am working with only the line segments lying between those points. I need to search lines that could continue one another (relative to ...
- 13
1
vote
1
answer
9k
views
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.
- 111
22
votes
2
answers
3k
views
Missing document request
I received a request for another long-lost document:
I am wondering if there is any way I
might obtain a copy of
The geometry of circles: Voronoi
diagrams, Moebius transformations,
...
- 25.1k
12
votes
3
answers
616
views
Effective contraction of a loop. Reference or a simple proof?
Let $M$ be a compact simply connected R. manifold. Let $x$ be a base point and let $\gamma$ be a smooth loop in $M$ starting and ending at $x$.
Is there a base point preserving retraction of $\...
- 4,188
11
votes
2
answers
2k
views
How many non-equivalent sections of a regular 7-simplex?
Suppose we have a regular 7-simplex in $\mathbb{R}^8$ defined by vertices <1,0,0,...,0>, <0,1,0,..,0>,...,<0,...,0,1>. A section is a 3-dimensional linear subspace of $\mathbb{R}^8$ that ...
7
votes
1
answer
1k
views
Burnside's Lemma and Geometry
I think one of the most interesting results in Elementary Group Theory is the so-called "Burnside's Lemma", counting the numbers of orbits of a (finite) group action.
I wonder if there is any (...
- 1,317
6
votes
1
answer
521
views
The reflex-free hull: Construction?
This is a followup to Bill Thurston's question
about different notions of hulls.
Here I want to raise a question about the
reflex-free hull, which is, intuitively, the smallest
enclosing shape to an ...
- 151k
18
votes
1
answer
2k
views
What can be said about the Shadow hull and the Sight hull?
This is a question implicitly raised by Is the sphere the only surface all of whose projections are circles? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are always ...
- 25.1k
-1
votes
3
answers
464
views
Points in circles that form a given geometric pattern
I am not a specialist in maths, so I thank you very much for any help you can give me.
Consider two circles C1, C2.
Q1: Find the points that are in the intersection of C1 and C2, this is easy !
Q2: ...
88
votes
4
answers
11k
views
Is the sphere the only surface with circular projections? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are circular?
Several ancient arguments suggest a curved Earth, such as
the observation that ships disappear mast-last over the
horizon, and
Eratosthenes'
surprisingly accurate calculation of the size of the
Earth
...
- 236k
3
votes
3
answers
2k
views
Is there a simple criterion to determine if two parallelograms intersect?
Assume we are given two parallelograms in the plane. How can I check if their intersection is nonempty?
Note that I do not need to actually find the intersection.
- 979
65
votes
4
answers
4k
views
Tying knots with reflecting lightrays
Let a lightray bounce around inside a cube whose faces
are (internal) mirrors.
If its slopes are rational, it will eventually form a cycle.
For example, starting with a point $p_0$ in the interior
of ...
1
vote
3
answers
222
views
Generalisation of a multivariable calc problem
I remember the following problem back from my undergraduate days:
Suppose that $f\in C^1(\mathbb{R}^n)$ is a map such that for all p, we have $df(p)\in SO(n)$. Then, $df$ is a constant rotation, ...
- 4,586
2
votes
3
answers
8k
views
How many different rectangles (in terms of area) can fit in a 20-unit-wide square?
How many different rectangles (in terms of area) can fit in a 20-unit-wide square? The rectangles can be squares, and their dimensions are integers.
- 103
21
votes
1
answer
3k
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A circle packing conjecture
Consider $n$ circles with variable radii $r_1,\ldots, r_n$ that pack inside a fixed circle of unit radius. In other words, all $n$ variable-radius circles are contained in the unit radius circle and ...
- 1,085
15
votes
0
answers
753
views
Are all these groups CAT(0) groups?
Given a geodesic metric space $X$ together with a choice of midpoints
$m:X\times X\rightarrow X$ (i.e. $d(m(x,y),x)=d(m(x,y),y)=d(x,y)/2$).
Assume furthermore, that the following nonpositive curvature ...
- 11.1k
11
votes
2
answers
1k
views
Is it a coincidence that the universal parabolic constant shows up in the solution to square point picking?
The expected distance $d$ of randomly selected points within a unit square to the square's center is
$d = \frac{1}{6} P$
where P is the universal parabolic constant
$P = \sqrt{2} + \ln{\left(1+\...
- 1,571
4
votes
2
answers
846
views
Fundamental polygons with infinite pairwise identifications
The topology of a closed surface can be constructed
by identifying edges of a fundamental polygon of an
even number $2n$ of edges.
Labeling the edges and using $\pm 1$ exponents to indicate
direction,
...
- 151k
11
votes
2
answers
2k
views
Wasserstein distance in R^d from one dimensional marginals
This question occurred to me while I was reading Klartag's papers on central limit theorems for convex bodies.
Given probability measures $\mu$, $\nu$ on (the Borel $\sigma$-field of) $R^d$ with ...
73
votes
10
answers
11k
views
Riemannian surfaces with an explicit distance function?
I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...
- 114k
9
votes
2
answers
1k
views
Other norms for lattice reduction techniques (LLL, PSLQ)?
LLL and other lattice reduction techniques (such as PSLQ) try to find a short basis vector relative to the 2-norm, i.e. for a given basis that has $ \varepsilon $ as its shortest vector, $ \varepsilon ...
- 513
8
votes
4
answers
1k
views
Sequences of evenly-distributed points in a product of intervals
Let φ be the golden ratio, (1+√5)/2. Taking the fractional parts of its integer multiples, we obtain a sequence of values in (0,1) which are in some sense "evenly distributed" in a way which ...
- 3,612
20
votes
5
answers
4k
views
Advanced view of the napkin ring problem?
The "napkin-ring problem" sometimes shows up in 2nd-year calculus courses, but it can fit quite neatly into a high-school geometry course via Cavalieri's principle.
However, the conclusion remains ...
1
vote
1
answer
908
views
What are the topological properties of the metric space retained (inherited) for its completion
Let $(X,d)$ be a metric space and $(\bar{X},\bar{d})$ its completion.
There is a list of topological properties
Wikipedia - Topological property
Does anybody know list which of them are retained (...
2
votes
0
answers
281
views
Recovering a piecewise affine function
Lets say I have an piecewise affine convex function $f(x_1,x_2)$, on which the following operations are possible:
Computing $f(x_1,x_2)$.
Computing a subgradient to $f$ at $(x_1,x_2)$
Computing all ...
- 567
2
votes
4
answers
222
views
How to compare finite point sets in normed spaces?
I want to define a "distance" between two subsets $A, B$ of a normed space $(V, \|\cdot\|)$ both with (at most) $n$ elements. A straightforward way for me to do this would be to define
$$ d(A, B) := \...
- 21
6
votes
1
answer
323
views
Stability of midpoints in CAT(0) spaces
Given a CAT(0) space $X$ and a compact, convex subset $A$ of $X$. One can define its midpoint $m(A)$ as the point, at which the following function attains its minimum.
$f:A\rightarrow \mathbb{R}\qquad ...
- 11.1k
11
votes
2
answers
1k
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Which (semi)regular polyhedra are combinations of two others?
The convex combination of convex polytopes is a convex polytope.
An example in $\mathbb{R}^2$ is that a regular octagon
can be obtained as $\frac{1}{2} S + \frac{1}{2} S'$,
where $S$ is a square and $...
- 151k
7
votes
5
answers
5k
views
Distance of a barycentric coordinate from a triangle vertex
I have a triangle $ABC$ with side lengths $a,b,c$ (edges $BC, CA, AB$ respectively).
I have a point $p$ with barycentric coordinates $u:v:w$.
These are normalised: $u+v+w=1$.
$1:0:0$ corresponds to ...
- 173
13
votes
4
answers
1k
views
Can Lipschitz maps increase the Lebesgue dimension ?
Given a map $f:X\rightarrow Y$ of compact metric spaces, such that there is a $C\in \mathbb{R}$ with $d(f(x),f(x'))\le C\cdot d(x,x')$.
Does this already imply, that the Lebesgue dimension of $f(X)$ ...
- 11.1k
3
votes
3
answers
1k
views
An exterior angle theorem for n-dimensional polytopes?
In the plane, the exterior angle of a vertex is $\pi -$ the standard ("interior") angle, which may be negative in some cases. The following is true for non-weird polygons:
The sum of the exterior ...
- 1,843
5
votes
2
answers
438
views
Bounding the product of lengths of basis vectors of a unimodular lattice
Suppose $\Lambda\subset\mathbb{R}^n$ is a unimodular (i.e. volume $1$) lattice in Euclidean space.
Let $v_1,\dots, v_n\in\Lambda$ be a basis of $\Lambda$ such that the product of lengths $A=|v_1|\...
- 7,952
5
votes
2
answers
594
views
Getting rid of exceptional fibers by passing to finite covers?
Consider a Seifert fiber space. Is it always possible to find a finite cover that is a circle bundle and the preimage of any fiber is a finite union of circles?
- 1,414
6
votes
2
answers
1k
views
Light rays bouncing around inside a sphere in d-dimensions
Suppose $S=\mathbb{S}^d$ is a unit sphere in $(d-1)$ dimensional space, with $d=3$ of special interest.
The surface of $S$ is a perfect (internal) mirror.
You stand at point $x$ (not the sphere center ...
- 151k
9
votes
1
answer
1k
views
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 ...
- 141
12
votes
1
answer
595
views
geometry of null homotopies
Given a homotopy class of map $f$ between unit spheres $S^n \to S^m, n>m$, let "stretch" be its "stretch factor" ( = inf over the homotopy class of the sup norm on the ( operator) norm on the first ...
- 1,387
21
votes
2
answers
1k
views
Geometric interpretation of exceptional symmetric spaces
Elie Cartan has classified all compact symmetric spaces admitting a compact simple Lie group as their group of motion.There are 7 infinite series and 12 exceptional cases. The exceptional cases are ...
- 3,022