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9,471
Computing in Solvable Matrix Groups
, 1992
"... We announce methods for efficient management of solvable matrix groups over finite fields. We show that solvability and nilpotencecan betestedin polynomialtime. Such efficiency seems unlikely for membershiptesting, which subsumes the discretelog problem. However, assuming that the primes in jGj ..."
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Cited by 29 (1 self)
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We announce methods for efficient management of solvable matrix groups over finite fields. We show that solvability and nilpotencecan betestedin polynomialtime. Such efficiency seems unlikely for membershiptesting, which subsumes the discretelog problem. However, assuming that the primes in j
Notes on thin matrix groups
"... These notes were prepared for the MSRI hot topics workshop on superstrong approximation (2012). We give a brief overview of the developments in the theory, especially the fundamental expansion theorem. Applications to diophantine problems on orbits of integer matrix groups, the affine sieve, group ..."
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Cited by 5 (0 self)
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These notes were prepared for the MSRI hot topics workshop on superstrong approximation (2012). We give a brief overview of the developments in the theory, especially the fundamental expansion theorem. Applications to diophantine problems on orbits of integer matrix groups, the affine sieve, group
An introduction to matrix groups and their applications
"... These notes are intended to provide a first view of Lie theory accessible to mathematics undergraduates. Although the prerequisites are kept as low level as possible, the material is sophisticated and contains many of the key themes of the mature subject. In order to achieve this we confine ourselv ..."
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Cited by 1 (0 self)
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ourselves to matrix groups, i.e., closed subgroups of general linear groups. One of the main results that we prove
NONLINEARITY OF MATRIX GROUPS
, 904
"... The aim of this short note is to answer a question by Guoliang Yu of whether the group EL3(Z〈x,y〉), where Z〈x,y 〉 is the free (noncommutative) ring, has any faithful finite dimensional linear representations over a field. Recall that for every (associative unitary) ring R the group ELn(R) is the su ..."
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Cited by 6 (0 self)
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. There are many results about isomorphisms between various matrix groups over (commutative) rings from the original results of Mal’cev [Ma] to results of O’Meara [OM] to Mostow rigidity results [Mo]. There are also many result about homomorphisms of one general matrix group into another. Churkin [Ch] proved
Finding community structure in networks using the eigenvectors of matrices
, 2006
"... We consider the problem of detecting communities or modules in networks, groups of vertices with a higherthanaverage density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as “modularity ” over possible div ..."
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Cited by 502 (0 self)
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We consider the problem of detecting communities or modules in networks, groups of vertices with a higherthanaverage density of edges connecting them. Previous work indicates that a robust approach to this problem is the maximization of the benefit function known as “modularity ” over possible
Consensus Problems in Networks of Agents with Switching Topology and TimeDelays
, 2003
"... In this paper, we discuss consensus problems for a network of dynamic agents with fixed and switching topologies. We analyze three cases: i) networks with switching topology and no timedelays, ii) networks with fixed topology and communication timedelays, and iii) maxconsensus problems (or leader ..."
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Cited by 1112 (21 self)
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leader determination) for groups of discretetime agents. In each case, we introduce a linear/nonlinear consensus protocol and provide convergence analysis for the proposed distributed algorithm. Moreover, we establish a connection between the Fiedler eigenvalue of the information flow in a network (i
Algebraic Graph Theory
, 2011
"... Algebraic graph theory comprises both the study of algebraic objects arising in connection with graphs, for example, automorphism groups of graphs along with the use of algebraic tools to establish interesting properties of combinatorial objects. One of the oldest themes in the area is the investiga ..."
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Cited by 892 (13 self)
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Algebraic graph theory comprises both the study of algebraic objects arising in connection with graphs, for example, automorphism groups of graphs along with the use of algebraic tools to establish interesting properties of combinatorial objects. One of the oldest themes in the area
Polynomialtime computation in matrix groups
, 1999
"... This dissertation investigates deterministic polynomialtime computation in matrix groups over finite fields. Of particular interest are matrixgroup problems that resemble testing graph isomorphism. The main results are instances where the problems admit polynomialtime solutions and methods that e ..."
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Cited by 5 (3 self)
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This dissertation investigates deterministic polynomialtime computation in matrix groups over finite fields. Of particular interest are matrixgroup problems that resemble testing graph isomorphism. The main results are instances where the problems admit polynomialtime solutions and methods
Practical Algorithms for Polycyclic Matrix Groups
 J. Symb. Comput
, 1997
"... Many fundamental problems are undecidable for infinite matrix groups. Polycyclic matrix groups represent a large class of groups for which these same problems are known to be decidable. In this paper we describe a suite of new algorithms for studying polycyclic matrix groups  algorithms for testi ..."
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Cited by 9 (1 self)
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Many fundamental problems are undecidable for infinite matrix groups. Polycyclic matrix groups represent a large class of groups for which these same problems are known to be decidable. In this paper we describe a suite of new algorithms for studying polycyclic matrix groups  algorithms
Algorithms for matrix groups
 London Math. Soc. Lecture Note Ser
, 2011
"... Existing algorithms have only limited ability to answer structural questions about subgroups G of GL(d, F), where F is a finite field. We discuss new and promising algorithmic approaches, both theoretical and practical, which as a first step construct a chief series for G. 1 ..."
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Cited by 7 (0 self)
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Existing algorithms have only limited ability to answer structural questions about subgroups G of GL(d, F), where F is a finite field. We discuss new and promising algorithmic approaches, both theoretical and practical, which as a first step construct a chief series for G. 1
Results 1  10
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9,471