1,787 research outputs found
Permutation groups, simple groups and sieve methods
We show that the number of integers n ≤ x which occur as indices of subgroups of nonabelian finite simple groups, excluding that of An-1 in An, is ∼ hx/log x, for some given constant h. This might be regarded as a noncommutative analogue of the Prime Number Theorem (which counts indices n ≤ x of subgroups of abelian simple groups). We conclude that for most positive integers n, the only quasiprimitive permutation groups of degree n are Sn and An in their natural action. This extends a similar result for primitive permutation groups obtained by Cameron, Neumann and Teague in 1982. Our proof combines group-theoretic and number-theoretic methods. In particular, we use the classification of finite simple groups, and we also apply sieve methods to estimate the size of some interesting sets of primes
A sufficient condition for a number to be the order of a nonsingular derivation of a Lie algebra
A study of the set N_p of positive integers which occur as orders of
nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of
characteristic p>0 was initiated by Shalev and continued by the present author.
The main goal of this paper is to show the abundance of elements of N_p. Our
main result shows that any divisor n of q-1, where q is a power of p, such that
, belongs to N_p. This extends its special
case for p=2 which was proved in a previous paper by a different method.Comment: 10 pages. This version has been revised according to a referee's
suggestions. The additions include a discussion of the (lower) density of the
set N_p, and the results of more extensive machine computations. Note that
the title has also changed. To appear in Israel J. Mat
Adposition and Case Supersenses v2.5: Guidelines for English
This document offers a detailed linguistic description of SNACS (Semantic
Network of Adposition and Case Supersenses; Schneider et al., 2018), an
inventory of 50 semantic labels ("supersenses") that characterize the use of
adpositions and case markers at a somewhat coarse level of granularity, as
demonstrated in the STREUSLE corpus (https://github.com/nert-gu/streusle/;
version 4.3 tracks guidelines version 2.5). Though the SNACS inventory aspires
to be universal, this document is specific to English; documentation for other
languages will be published separately.
Version 2 is a revision of the supersense inventory proposed for English by
Schneider et al. (2015, 2016) (henceforth "v1"), which in turn was based on
previous schemes. The present inventory was developed after extensive review of
the v1 corpus annotations for English, plus previously unanalyzed genitive case
possessives (Blodgett and Schneider, 2018), as well as consideration of
adposition and case phenomena in Hebrew, Hindi, Korean, and German. Hwang et
al. (2017) present the theoretical underpinnings of the v2 scheme. Schneider et
al. (2018) summarize the scheme, its application to English corpus data, and an
automatic disambiguation task
Subgraphs and network motifs in geometric networks
Many real-world networks describe systems in which interactions decay with
the distance between nodes. Examples include systems constrained in real space
such as transportation and communication networks, as well as systems
constrained in abstract spaces such as multivariate biological or economic
datasets and models of social networks. These networks often display network
motifs: subgraphs that recur in the network much more often than in randomized
networks. To understand the origin of the network motifs in these networks, it
is important to study the subgraphs and network motifs that arise solely from
geometric constraints. To address this, we analyze geometric network models, in
which nodes are arranged on a lattice and edges are formed with a probability
that decays with the distance between nodes. We present analytical solutions
for the numbers of all 3 and 4-node subgraphs, in both directed and
non-directed geometric networks. We also analyze geometric networks with
arbitrary degree sequences, and models with a field that biases for directed
edges in one direction. Scaling rules for scaling of subgraph numbers with
system size, lattice dimension and interaction range are given. Several
invariant measures are found, such as the ratio of feedback and feed-forward
loops, which do not depend on system size, dimension or connectivity function.
We find that network motifs in many real-world networks, including social
networks and neuronal networks, are not captured solely by these geometric
models. This is in line with recent evidence that biological network motifs
were selected as basic circuit elements with defined information-processing
functions.Comment: 9 pages, 6 figure
Contextual Object Detection with a Few Relevant Neighbors
A natural way to improve the detection of objects is to consider the
contextual constraints imposed by the detection of additional objects in a
given scene. In this work, we exploit the spatial relations between objects in
order to improve detection capacity, as well as analyze various properties of
the contextual object detection problem. To precisely calculate context-based
probabilities of objects, we developed a model that examines the interactions
between objects in an exact probabilistic setting, in contrast to previous
methods that typically utilize approximations based on pairwise interactions.
Such a scheme is facilitated by the realistic assumption that the existence of
an object in any given location is influenced by only few informative locations
in space. Based on this assumption, we suggest a method for identifying these
relevant locations and integrating them into a mostly exact calculation of
probability based on their raw detector responses. This scheme is shown to
improve detection results and provides unique insights about the process of
contextual inference for object detection. We show that it is generally
difficult to learn that a particular object reduces the probability of another,
and that in cases when the context and detector strongly disagree this learning
becomes virtually impossible for the purposes of improving the results of an
object detector. Finally, we demonstrate improved detection results through use
of our approach as applied to the PASCAL VOC and COCO datasets
Etching of random solids: hardening dynamics and self-organized fractality
When a finite volume of an etching solution comes in contact with a
disordered solid, a complex dynamics of the solid-solution interface develops.
Since only the weak parts are corroded, the solid surface hardens
progressively. If the etchant is consumed in the chemical reaction, the
corrosion dynamics slows down and stops spontaneously leaving a fractal solid
surface, which reveals the latent percolation criticality hidden in any random
system. Here we introduce and study, both analytically and numerically, a
simple model for this phenomenon. In this way we obtain a detailed description
of the process in terms of percolation theory. In particular we explain the
mechanism of hardening of the surface and connect it to Gradient Percolation.Comment: Latex, aipproc, 6 pages, 3 figures, Proceedings of 6th Granada
Seminar on Computational Physic
Generalization Error in Deep Learning
Deep learning models have lately shown great performance in various fields
such as computer vision, speech recognition, speech translation, and natural
language processing. However, alongside their state-of-the-art performance, it
is still generally unclear what is the source of their generalization ability.
Thus, an important question is what makes deep neural networks able to
generalize well from the training set to new data. In this article, we provide
an overview of the existing theory and bounds for the characterization of the
generalization error of deep neural networks, combining both classical and more
recent theoretical and empirical results
Scalable and Interpretable One-class SVMs with Deep Learning and Random Fourier features
One-class support vector machine (OC-SVM) for a long time has been one of the
most effective anomaly detection methods and extensively adopted in both
research as well as industrial applications. The biggest issue for OC-SVM is
yet the capability to operate with large and high-dimensional datasets due to
optimization complexity. Those problems might be mitigated via dimensionality
reduction techniques such as manifold learning or autoencoder. However,
previous work often treats representation learning and anomaly prediction
separately. In this paper, we propose autoencoder based one-class support
vector machine (AE-1SVM) that brings OC-SVM, with the aid of random Fourier
features to approximate the radial basis kernel, into deep learning context by
combining it with a representation learning architecture and jointly exploit
stochastic gradient descent to obtain end-to-end training. Interestingly, this
also opens up the possible use of gradient-based attribution methods to explain
the decision making for anomaly detection, which has ever been challenging as a
result of the implicit mappings between the input space and the kernel space.
To the best of our knowledge, this is the first work to study the
interpretability of deep learning in anomaly detection. We evaluate our method
on a wide range of unsupervised anomaly detection tasks in which our end-to-end
training architecture achieves a performance significantly better than the
previous work using separate training.Comment: Accepted at European Conference on Machine Learning and Principles
and Practice of Knowledge Discovery in Databases (ECML-PKDD) 201
Primitive Words, Free Factors and Measure Preservation
Let F_k be the free group on k generators. A word w \in F_k is called
primitive if it belongs to some basis of F_k. We investigate two criteria for
primitivity, and consider more generally, subgroups of F_k which are free
factors.
The first criterion is graph-theoretic and uses Stallings core graphs: given
subgroups of finite rank H \le J \le F_k we present a simple procedure to
determine whether H is a free factor of J. This yields, in particular, a
procedure to determine whether a given element in F_k is primitive.
Again let w \in F_k and consider the word map w:G x G x ... x G \to G (from
the direct product of k copies of G to G), where G is an arbitrary finite
group. We call w measure preserving if given uniform measure on G x G x ... x
G, w induces uniform measure on G (for every finite G). This is the second
criterion we investigate: it is not hard to see that primitivity implies
measure preservation and it was conjectured that the two properties are
equivalent. Our combinatorial approach to primitivity allows us to make
progress on this problem and in particular prove the conjecture for k=2.
It was asked whether the primitive elements of F_k form a closed set in the
profinite topology of free groups. Our results provide a positive answer for
F_2.Comment: This is a unified version of two manuscripts: "On Primitive words I:
A New Algorithm", and "On Primitive Words II: Measure Preservation". 42
pages, 14 figures. Some parts of the paper reorganized towards publication in
the Israel J. of Mat
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