1,607 research outputs found
How long does it take to generate a group?
The diameter of a finite group with respect to a generating set is
the smallest non-negative integer such that every element of can be
written as a product of at most elements of . We denote this
invariant by \diam_A(G). It can be interpreted as the diameter of the Cayley
graph induced by on and arises, for instance, in the context of
efficient communication networks.
In this paper we study the diameters of a finite abelian group with
respect to its various generating sets . We determine the maximum possible
value of \diam_A(G) and classify all generating sets for which this maximum
value is attained. Also, we determine the maximum possible cardinality of
subject to the condition that \diam_A(G) is "not too small". Connections with
caps, sum-free sets, and quasi-perfect codes are discussed
A short proof of Kneser's addition theorem for abelian groups
Martin Kneser proved the following addition theorem for every abelian group
. If are finite and nonempty, then where . Here we give a short
proof of this based on a simple intersection union argument.Comment: 3 page
Neural Networks Compression for Language Modeling
In this paper, we consider several compression techniques for the language
modeling problem based on recurrent neural networks (RNNs). It is known that
conventional RNNs, e.g, LSTM-based networks in language modeling, are
characterized with either high space complexity or substantial inference time.
This problem is especially crucial for mobile applications, in which the
constant interaction with the remote server is inappropriate. By using the Penn
Treebank (PTB) dataset we compare pruning, quantization, low-rank
factorization, tensor train decomposition for LSTM networks in terms of model
size and suitability for fast inference.Comment: Keywords: LSTM, RNN, language modeling, low-rank factorization,
pruning, quantization. Published by Springer in the LNCS series, 7th
International Conference on Pattern Recognition and Machine Intelligence,
201
Boundary curves of surfaces with the 4-plane property
Let M be an orientable and irreducible 3-manifold whose boundary is an
incompressible torus. Suppose that M does not contain any closed nonperipheral
embedded incompressible surfaces. We will show in this paper that the immersed
surfaces in M with the 4-plane property can realize only finitely many boundary
slopes. Moreover, we will show that only finitely many Dehn fillings of M can
yield 3-manifolds with nonpositive cubings. This gives the first examples of
hyperbolic 3-manifolds that cannot admit any nonpositive cubings.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol6/paper21.abs.htm
Gauss composition over an arbitrary base
The classical theorems relating integral binary quadratic forms and ideal
classes of quadratic orders have been of tremendous importance in mathematics,
and many authors have given extensions of these theorems to rings other than
the integers. However, such extensions have always included hypotheses on the
rings, and the theorems involve only binary quadratic forms satisfying further
hypotheses. We give a complete statement of the relationship between binary
quadratic forms and modules for quadratic algebras over any base ring, or in
fact base scheme. The result includes all binary quadratic forms, and commutes
with base change. We give global geometric as well as local explicit
descriptions of the relationship between forms and modules.Comment: submitte
Sweepouts of amalgamated 3-manifolds
We show that if two 3-manifolds with toroidal boundary are glued via a
`sufficiently complicated' map then every Heegaard splitting of the resulting
3-manifold is weakly reducible. Additionally, if Z is a manifold obtained by
gluing X and Y, two connected small manifolds with incompressible boundary,
along a closed surface F. Then the genus g(Z) of Z is greater than or equal to
1/2(g(X)+g(Y)-2g(F)). Both results follow from a new technique to simplify the
intersection between an incompressible surface and a strongly irreducible
Heegaard splitting.Comment: This is the version published by Algebraic & Geometric Topology on 24
February 200
The size of triangulations supporting a given link
Let T be a triangulation of S^3 containing a link L in its 1-skeleton. We
give an explicit lower bound for the number of tetrahedra of T in terms of the
bridge number of L. Our proof is based on the theory of almost normal surfaces.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol5/paper13.abs.htm
Schrijver graphs and projective quadrangulations
In a recent paper [J. Combin. Theory Ser. B}, 113 (2015), pp. 1-17], the
authors have extended the concept of quadrangulation of a surface to higher
dimension, and showed that every quadrangulation of the -dimensional
projective space is at least -chromatic, unless it is bipartite.
They conjectured that for any integers and , the
Schrijver graph contains a spanning subgraph which is a
quadrangulation of . The purpose of this paper is to prove the
conjecture
Converting between quadrilateral and standard solution sets in normal surface theory
The enumeration of normal surfaces is a crucial but very slow operation in
algorithmic 3-manifold topology. At the heart of this operation is a polytope
vertex enumeration in a high-dimensional space (standard coordinates).
Tollefson's Q-theory speeds up this operation by using a much smaller space
(quadrilateral coordinates), at the cost of a reduced solution set that might
not always be sufficient for our needs. In this paper we present algorithms for
converting between solution sets in quadrilateral and standard coordinates. As
a consequence we obtain a new algorithm for enumerating all standard vertex
normal surfaces, yielding both the speed of quadrilateral coordinates and the
wider applicability of standard coordinates. Experimentation with the software
package Regina shows this new algorithm to be extremely fast in practice,
improving speed for large cases by factors from thousands up to millions.Comment: 55 pages, 10 figures; v2: minor fixes only, plus a reformat for the
journal styl
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