All Questions
4,827 questions
5
votes
2
answers
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Inequality involving probability measures [closed]
I have been working on a problem(alternate minimization) where I want to establish an inequality in which I am stuck.
An $\alpha$- parameterized version of the divergence(Kullback-Leibler) takes the ...
- 779
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 $\...
- 25.1k
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 (...
- 14.4k
-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: ...
4
votes
1
answer
275
views
Symmetry of the integer gap
Are there results that bound the asymmetry of the duality gap of an integer program? That is to say, if the difference between the LP solution and the IP (primal) solution is $a$, is there a function ...
CommunityBot
- 1
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, ...
- 61.9k
27
votes
6
answers
2k
views
When shorter means smaller?
Assume a convex figure $F\subset \mathbb R^2$ satisfies the following property: if $f:F\to \mathbb R^2$ is a distance-non-increasing map then its image $f(F)$ is congruent to a subset of $F$.
Is it ...
CommunityBot
- 1
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,
...
- 25.1k
6
votes
3
answers
1k
views
How can I embed an N-points metric space to a hypercube with low distortion?
I have a N-point metric space defined by the pairwise distance matrix. I want to encode these N points with binary strings, i.e. each point will be mapped to a vertex in a hypercube. The lengths of ...
- 2,401
1
vote
1
answer
908
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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 (...
- 2,018
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) := \...
- 60.5k
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 ...
- 27.8k
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 ...
- 85.6k
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|\...
- 669
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?
- 68.9k
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 ...
- 68.9k
8
votes
1
answer
700
views
Upper bound for tetrahedron packing?
There have been several recent advances on packing regular tetrahedra in $\mathbb{R}^3$. All the results I've seen have been lower bounds -- first John Conway and Sal Torquato showed that there ...
- 151k
4
votes
4
answers
588
views
Cosine sum problem
Consider any n points on the circumference of a circle. Draw a straight line link between each pair of points with a link weight consisting of the cosine of the angle the link subtends at the centre.
...
- 39.9k
8
votes
1
answer
2k
views
Expected number of steps for a discrete random walk to visit every point on an N-dimensional rectangular lattice
Please imagine a discrete random walk on an N-dimensional rectangular lattice with dimensional lengths $(l_1, ..., l_N) \in L$ and total lattice points $P = \prod{l_i}$, for $i = 1, ..., N$. At each ...
- 4,014
8
votes
1
answer
359
views
Can any rectangle be inscribed in any convex figure?
Can any rectangle be inscribed in any convex figure?
CommunityBot
- 1
6
votes
1
answer
500
views
Can any triangle be inscribed in any convex figure?
Can any triangle be inscribed in any convex figure? i.e. given a convex figure and a triangle can we transpose and scale and rotate that triangle so that its vertices are on the boundary of the ...
- 56.6k
2
votes
2
answers
640
views
Sorting a binary matrix diagonal in polynomial time while preserving rows
Is there a polynomial time solution to sort an arbitrary binary square matrix in polynomial time by rows so that the diagonal contains a 1 if any row contains a 1 in that column?
For example given ...
- 2,210
-2
votes
1
answer
840
views
Generic coordinate system representations [closed]
Please excuse the verboseness which follows, as the question is rather basic, so I would like to state it carefully so that it will not be accidentally neglected as automatically trivial. If, after my ...
- 2,481
0
votes
1
answer
1k
views
For Ax = b, x and b unknown vectors, how do I solve the x that maximizes min(b_i)?
Given a matrix $A$, each element $A_{i,j} \geq 0$, find the vector $\vec x$ that maximizes the minimum element in $\vec b$ ($\vec b = A \vec x$). Note that this is not a linear equation system as I ...
- 9,756
9
votes
2
answers
3k
views
An optimization problem for points on the sphere (master's dissertation)
First, by means of a disclaimer, some background. I am entering the fourth and final year of an undergraduate master's degree in maths, and a quarter of the maximum credit for this year will be for a ...
- 22.3k
8
votes
1
answer
2k
views
Point cloud that maximizes the minimum pairwise distance in Euclidean space
I am interested finding the collection of points in the Euclidean space that has the maximal minimal pairwise distance subject to an average norm constraint, that is, how to maximize
$min_{i \neq j} |...
- 12.6k
5
votes
1
answer
796
views
Minimizing variance of distances between points when mean distance is fixed
In Rd, I have n > d+1 points. The mean distance between pairs of points is 1. How can I minimize the variance of the distances (equivalently, the mean squared distance)? I'm mainly interested in d &...
- 45k
8
votes
2
answers
741
views
Lattice Stick Number vs. Stick Number of Knot
Can the lattice stick number of a knot be bounded
by the stick number of the knot?
The stick number
$S(K)$ of a knot $K$ is the fewest number of segments
needed to realize it by a simple 3D polygon....
- 18.6k
1
vote
1
answer
492
views
Сomplete homogeneous space which is not locally compact
It is well-known theorem that every locally compact, homogeneous, metric space is complete.
Does anybody know example of complete, homogeneous, metric space which is not locally compact?
- 7,442
5
votes
4
answers
954
views
literature on geometrical viewpoint on calculus of variations for physics
What is a good reference for a geometrical viewpoint on the calculus of variations for physics, using differential forms etc. to derive Yang-Mills equations and other topics of the standard model?
...
- 4,519
5
votes
2
answers
1k
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Applications of minmax theorem(s)
Intro We suppose $X$ and $Y$ are nonempty sets and f: $X\times Y \rightarrow \mathbb{R}$. A minimax theorem is a theorem that asserts that, under certain conditions,
$$ \inf_Y \sup_X f = \sup_X \...
- 833
2
votes
2
answers
749
views
Projective transformation between polygons.
Extending my earlier question about linear transformations, what's the easiest way to test if there exists a projective linear transformation that takes one polygon to another (in $\mathbb{R}\mathbb{P}...
CommunityBot
- 1
5
votes
2
answers
3k
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Continuous Linear Programming: Estimating a Solution
I have a "continuous" linear programming problem that involves maximizing a linear function over a curved convex space. In typical LP problems, the convex space is a polytope, but in this case the ...
- 8,491
7
votes
1
answer
665
views
What is the Cheeger constant of a cubical subset of the cubic lattice?
The Cheeger constant of a finite graph measures the "bottleneckedness" of the graph, and is defined as:
$$h(G) := \min\Bigg\lbrace\frac{|\partial A|}{|A|} \Bigg| A\subset V, 0<|A|\leq \frac{|V|}{2}...
- 1,916
3
votes
0
answers
192
views
Velcro surface: is it possible to have a surface which is everywhere infinitesimally a cone, but not a normed group?
Is there any example of this velcro-like space? Looking for a LOCALLY COMPACT COMPLETE metric space $(X,d)$ such that:
(A)-it has a metric tangent space $(T_{x}X, d^x)$ at any point $x \in X$,
(...
- 785
3
votes
2
answers
301
views
Optimizing the layout of Infinite Suburbia
Infinite Suburbia is a Euclidean plane, P. All residents live in open unit disks which, like caravans, can travel around but are stationary most of the time. When stationary, these disks lie in a ...
- 785
4
votes
4
answers
589
views
Measures of the complexity of a metric
I am seeking a measure of the "complexity" of a surface $S$,
a quantity that reflects how widely the metric varies from spot to
spot. I am primarily interested in surfaces topologically
equivalent to ...
- 6,825
14
votes
5
answers
3k
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Shortest-path Distances Determining the Metric?
The metric of a Riemannian manifold determines the shortest
distance between any two points.
I assume the reverse holds? That is, if you are given the
shortest distance d(x,y) between every pair of ...
- 151k
1
vote
1
answer
324
views
A question about dissecting spherical triangles
Do there exist spherical triangles which are not isoceles but are the union of a finite collection of
(two or more) congruent triangles with pairwise disjoint (and non-empty) interiors?
- 55.9k
7
votes
1
answer
815
views
Rolling a convex body: Geodesics vs. rolling curves
What are the curves of contact on a convex body $B$ rolling down an inclined plane?
Assume $B$ is smooth, and there is sufficient friction to prevent slippage.
Certainly, one can develop a geodesic ...
- 22.3k
3
votes
1
answer
375
views
Connections between a polytope's symmetry group and the existence of periodic orbits
Given an $n$-dimensional convex polytope $P$, one may set into motion a point-mass, starting on one of the facets of $P$, which travels along a straight trajectory inside $P$ except on collision with ...
- 1,675
5
votes
0
answers
427
views
The Gömböc and monostatic objects
This made the popular news a few years ago. Summary: it's the first homogeneous, convex solid to be found with only one stable and one unstable mechanical equilibrium when resting on a flat surface. I ...
- 3,612
3
votes
1
answer
335
views
When is the neighbourhood of a set a ball?
In euclidean n-space, it's easy to show that given a set $S$ of radius $< r$, the $a$-neighbourhood of $S$ is a ball, for any $a \geq 2r$.
Proof: Let $S$ be contained in $B_r(y)$, $y \in \mathbb{...
- 55.9k
12
votes
1
answer
1k
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Fixed point theorems and equiangular lines
I've been thinking about the equiangular lines (or SIC-POVM) conjecture, and my conclusion is that the best means of attack would be through some kind of fixed point theorem -- I'm thinking ...
- 6,342
4
votes
1
answer
496
views
Is there a standard measure for how close a matrix is to being a distance metric ?
Suppose I have a square n*n, symmetric matrix with positive elements and zero diagonal.
For this to be considered a proper distance metric between n points, the triangle inequality needs to be ...
- 4,462
3
votes
0
answers
559
views
Unprovability of the Steiner-Lehmus theorem
Conway postulated that the Steiner-Lehmus theorem is unprovable using direct methods of proof. Can this be proven directly, that the Steiner-Lehmus theorem cannot be proven directly over Euclidean ...
- 131
15
votes
1
answer
11k
views
Maximum number of mutually equidistant points in an n-dimensional Euclidean space is (n+1). Proof? [closed]
How to prove that the maximum number of mutually equidistant points in an n-dimensional Euclidean space is (n+1)?
- 7,105
1
vote
0
answers
335
views
Irrationality of square root of 2 [closed]
It is possible to explain me the 18th proof of the irrationality of square root of 2 from the following site?
http://www.cut-the-knot.org/proofs/sq_root.shtml
- 27
4
votes
1
answer
1k
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Finding integer points on an N-d convex hull
Suppose we have a convex hull computed as the solution to a linear programming problem (via whatever method you want). Given this convex hull (and the inequalities that formed the convex hull) is ...
- 151k