27 research outputs found
Small gaps between configurations of prime polynomials
We find arbitrarily large configurations of irreducible polynomials over
finite fields that are separated by low degree polynomials. Our proof adapts an
argument of Pintz from the integers, in which he combines the methods of
Goldston-Pintz-Y\i ld\i r\i m and Green-Tao to find arbitrarily long arithmetic
progressions of generalized twin primes
Simplices over finite fields
We prove that, provided , every sufficiently large subset of
contains an isometric copy of every -simplex that avoids
spanning a nontrivial self-orthogonal subspace. We obtain comparable results
for simplices exhibiting self-orthogonal behavior.Comment: 11 pages; improved and revise
Small unit-distance graphs in the plane
We prove that a graph on up to 9 vertices is a unit-distance graph if and
only if it does not contain one of 74 so-called minimal forbidden graphs. This
extends the work of Chilakamarri and Mahoney (1995), who provide a similar
classification for unit-distance graphs on up to 7 vertices.Comment: (v3) fixed F(9,15,21) and Lemma 1
The optimal packing of eight points in the real projective plane
How can we arrange lines through the origin in three-dimensional
Euclidean space in a way that maximizes the minimum interior angle between
pairs of lines? Conway, Hardin and Sloane (1996) produced line packings for that they conjectured to be within numerical precision of optimal in
this sense, but until now only the cases have been solved. In this
paper, we resolve the case . Drawing inspiration from recent work on the
Tammes problem, we enumerate contact graph candidates for an optimal
configuration and eliminate those that violate various combinatorial and
geometric necessary conditions. The contact graph of the putatively optimal
numerical packing of Conway, Hardin and Sloane is the only graph that survives,
and we recover from this graph an exact expression for the minimum distance of
eight optimally packed points in the real projective plane
Kesten-McKay law for random subensembles of Paley equiangular tight frames
We apply the method of moments to prove a recent conjecture of Haikin, Zamir
and Gavish (2017) concerning the distribution of the singular values of random
subensembles of Paley equiangular tight frames. Our analysis applies more
generally to real equiangular tight frames of redundancy 2, and we suspect
similar ideas will eventually produce more general results for arbitrary
choices of redundancy
Embedding distance graphs in finite field vector spaces
We show that large subsets of vector spaces over finite fields determine
certain point configurations with prescribed distance structure. More
specifically, we consider the complete graph with vertices as the points of and edges assigned the algebraic distance between
pairs of vertices. We prove nontrivial results on locating specified subgraphs
of maximum vertex degree at most in dimensions
Globally optimizing small codes in real projective spaces
For , we classify arrangements of points in
for which the minimum distance is as large as possible. To
do so, we leverage ideas from matrix and convex analysis to determine the best
possible codes that contain equiangular lines, and we introduce a notion of
approximate Positivstellensatz certificates that promotes numerical
approximations of Stengle's Positivstellensatz certificates to honest
certificates.Comment: code included in ancillary file
Exact Line Packings from Numerical Solutions
Recent progress in Zauner's conjecture has leveraged deep conjectures in
algebraic number theory to promote numerical line packings to exact and
verifiable solutions to the line packing problem. We introduce a
numerical-to-exact technique in the real setting that does not require such
conjectures. Our approach is completely reproducible, matching Sloane's
database of putatively optimal numerical line packings with Mathematica's
built-in implementation of cylindrical algebraic decomposition. As a proof of
concept, we promote a putatively optimal numerical packing of eight points in
the real projective plane to an exact packing, whose optimality we establish in
a forthcoming paper.Comment: Mathematica notebook attached as ancillary fil
Lie PCA: Density estimation for symmetric manifolds
We introduce an extension to local principal component analysis for learning
symmetric manifolds. In particular, we use a spectral method to approximate the
Lie algebra corresponding to the symmetry group of the underlying manifold. We
derive the sample complexity of our method for a variety of manifolds before
applying it to various data sets for improved density estimation
Neural collapse with unconstrained features
Neural collapse is an emergent phenomenon in deep learning that was recently
discovered by Papyan, Han and Donoho. We propose a simple "unconstrained
features model" in which neural collapse also emerges empirically. By studying
this model, we provide some explanation for the emergence of neural collapse in
terms of the landscape of empirical risk