on the order of borel subgroups of group amalgams and an application to locally transitive graphs abstract a permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular given nontrivial finite transitive permutation groups l 1 and l 2 with l 1 not semiprimitive we construct an infinite family of rank two amalgams of permutation type l 1 l 2 and borel subgroups of strictly increasing order as an application we show that there is no bound on the order of edge stabilisers in locally l 1 l 2 graphs we also consider the corresponding question for amalgams of rank k 3 we completely resolve this by showing that the order of the borel subgroup is bounded by the permutation type l 1 l k only in the trivial case where each of l 1 l k is regular

finite permutation groups with a transitive minimal normal subgroup a finite permutation group is said to be innately transitive if it contains a transitive minimal normal subgroup in this paper we give a characterisation and structure theorem for the finite innately transitive groups as well as describing those innately transitive groups which preserve a product decomposition the class of innately transitive groups contains all primitive and quasiprimitive groups

a fast algorithm for nonunitary joint diagonalization and its application to blind source separation a fast algorithm named complex valued fast frobenius diagonalization cvffdiag is proposed for seeking the nonunitary approximate joint diagonalizer of a given set of complex valued target matrices it adopts a multiplicative update to minimize the frobenius norm formulation of the approximate joint diagonalization problem at each of multiplicative iterations a strictly diagonally dominant updated matrix is obtained this scheme ensures the invertibility of the diagonalizer the cvffdiag relaxes several constraints on the target matrices and thus has much general applications furthermore the special approximation of the cost function the ingenious utilization of some structures and the adequate notation of concerned variables lead to the high computational efficiency of the proposed algorithm numerical simulations are conducted to illustrate good performances of the cvffdiag

diagonality measures of hermitian positive definite matrices with application to the approximate joint diagonalization problem in this paper we introduce properly invariant diagonality measures of hermitian positive definite matrices these diagonality measures are defined as distances or divergences between a given positive definite matrix and its diagonal part we then give closed form expressions of these diagonality measures and discuss their invariance properties the diagonality measure based on the log determinant divergence is general enough as it includes a diagonality criterion used by the signal processing community as a special case these diagonality measures are then used to formulate minimization problems for finding the approximate joint diagonalizer of a given set of hermitian positive definite matrices numerical computations based on a modified newton method are presented and commented

nonorthogonal joint diagonalization by combining givens and hyperbolic rotations a new algorithm for computing the nonorthogonal joint diagonalization of a set of matrices is proposed for independent component analysis and blind source separation applications this algorithm is an extension of the jacobi like algorithm first proposed in the joint approximate diagonalization of eigenmatrices jade method for orthogonal joint diagonalization the improvement consists mainly in computing a mixing matrix of determinant one and columns of equal norm instead of an orthogonal mixing matrix this target matrix is constructed iteratively by successive multiplications of not only givens rotations but also hyperbolic rotations and diagonal matrices the algorithm performance evaluated on synthetic data compares favorably with existing methods in terms of speed of convergence and complexity

on locally semiprimitive graphs and a theorem of weiss abstract in this paper we investigate graphs that admit a group acting arc transitively such that the local action is semiprimitive with a regular normal nilpotent subgroup this type of semiprimitive group is a generalisation of an affine group we show that if the graph has valency coprime to six then there is a bound on the order of the vertex stabilisers depending on the valency alone we also prove a detailed structure theorem for the vertex stabilisers in the remaining case this is a contribution to an ongoing project to investigate the validity of the potocnik spiga verret conjecture

on groups with every normal subgroup transitive or semiregular we investigate finite solvable permutation groups in which all normal subgroups are transitive or semiregular the motivation comes from universal algebra such groups are examples of collapsing transformation monoids

a direct algorithm for nonorthogonal approximate joint diagonalization while a pair of n n matrices can almost always be exactly and simultaneously diagonalized by a generalized eigendecomposition no exact solution exists in the case of a set with more than two matrices this problem termed approximate joint diagonalization ajd is instrumental in blind signal processing when the set of matrices to be jointly diagonalized includes at least n linearly independent matrices we propose a suboptimal but closed form solution for ajd in the direct least squares sense the corresponding non iterative algorithm is given the acronym diem diagonalization using equivalent matrices extensive numerical simulations show that diem is both fast and accurate compared to the state of the art iterative ajd algorithms

on the orders of primitive groups abstract almost all primitive permutation groups of degree n have order at most n i 0 log 2 n 1 n 2 i 1 log 2 n or have socle isomorphic to a direct power of some alternating group the mathieu groups m11 m12 m23 and m24 are the four exceptions as a corollary the sharp version of a theorem of praeger and saxl is established where m12 turns out to be the largest primitive group for an application a bound on the orders of permutation groups without large alternating composition factors is given this sharpens a lemma of babai cameron palfy and generalizes a theorem of dixon

distance transitive graphs and finite simple groups introductionthis paper represents the first step in the classification of finite primitive distancetransitive graphs in it we reduce the problem to the case where the automorph ism group is either almost simple or affine let be a simple connected undirected graph with verte oc x se t q e q i fthen d a j8 denotes the distance between 3 a i ann

on the order of uniprimitive permutation groups one of the central problems of 19th century group theory was the estimation of the order of a primitive permutation group g of degree n where g x an we prove i g i exp 4v n log2 n for the case when g is not doubly transitive this result is best possible apart from an o log n factor in the exponent the best result previously known was i g i el h wielandt a similar estimate for the doubly transitive case follows in a subsequent paper for rank 3 groups with subdegree p n 2 we obtain i gi exp 4 n p log2 n in the proof we develop some new combinatorial properties of coherent configurations the results also have relevance to theoretical estimates on the computational complexity of graph isomorphism testing

nonorthogonal joint diagonalization free of degenerate solution the problem of approximate joint diagonalization of a set of matrices is instrumental in numerous statistical signal processing applications for nonorthogonal joint diagonalization based on the weighted least squares wls criterion the trivial zero solution can simply be avoided by adopting some constraint on the diagonalizing matrix or penalty terms however the resultant algorithms may converge to some undesired degenerate solutions nonzero but singular or ill conditioned solutions this paper discusses and analyzes the problem of degenerate solutions in detail to solve this problem a novel nonleast squares criterion for approximate nonorthogonal joint diagonalization is proposed and an efficient algorithm called fast approximate joint diagonalization fajd is developed as compared with the existing nonorthogonal diagonalization algorithms the new algorithm can not only avoid the trivial solution but also any degenerate solutions theoretical analysis shows that the fajd algorithm has some advantages over the existing nonorthogonal diagonalization algorithms simulation results are presented to demonstrate the efficiency of this paper s algorithm

a classification of primitive permutation groups with finite stabilizers abstract we classify all infinite primitive permutation groups possessing a finite point stabilizer thus extending the seminal aschbacher o nan scott theorem to all primitive permutation groups with finite point stabilizers

generalized non orthogonal joint diagonalization with lu decomposition and successive rotations non orthogonal joint diagonalization njd free of prewhitening has been widely studied in the context of blind source separation bss and array signal processing etc however njd is used to retrieve the jointly diagonalizable structure for a single set of target matrices which are mostly formulized with a single dataset and thus is insufficient to handle multiple datasets with inter set dependences a scenario often encountered in joint bss j bss applications as such we present a generalized njd gnjd algorithm to simultaneously perform asymmetric njd upon multiple sets of target matrices with mutually linked loading matrices by using lu decomposition and successive rotations to enable j bss over multiple datasets with indication exploitation of their mutual dependences experiments with synthetic and real world datasets are provided to illustrate the performance of the proposed algorithm

on pyber s base size conjecture let g be a permutation group on a finite set a subset of is a base for g if its pointwise stabilizer in g is trivial the base size of g denoted b g is the smallest size of a base a well known conjecture of pyber from the early 1990s asserts that there exists an absolute constant c such that b g c log g logn for any primitive permutation group g of degree n several special cases have been verified in recent years including the almost simple and diagonal cases in this paper we prove pyber s conjecture for all non affine primitive groups

a theory of semiprimitive groups abstract a transitive permutation group is semiprimitive if each of its normal subgroups is transitive or semiregular interest in this class of groups is motivated by two sources problems arising in universal algebra related to collapsing monoids and the graph restrictive problem for permutation groups here we develop a theory of semiprimitive groups which encompasses their structure their quotient actions and a method by which all finite semiprimitive groups are constructed we also extend some results from the theory of primitive groups to semiprimitive groups and conclude with open problems of a similar nature

a new jacobi like method for joint diagonalization of arbitrary non defective matrices this paper addresses the problem of joint diagonalization of a set of matrices a new jacobi like method that has the advantages of computational efficiency and of generality is presented the proposed algorithm brings the general matrices into normal ones and performs a joint diagonalization by a combination of unitary and shears non unitary transformations it is based on the iterative minimization of an appropriate cost function using generalized jacobi rotation matrices

fast approximate joint diagonalization incorporating weight matrices we propose a new low complexity approximate joint diagonalization ajd algorithm which incorporates nontrivial block diagonal weight matrices into a weighted least squares wls ajd criterion often in blind source separation bss when the sources are nearly separated the optimal weight matrix for wls based ajd takes a nearly block diagonal form based on this observation we show how the new algorithm can be utilized in an iteratively reweighted separation scheme thereby giving rise to fast implementation of asymptotically optimal bss algorithms in various scenarios in particular we consider three specific yet common scenarios involving stationary or block stationary gaussian sources for which the optimal weight matrices can be readily estimated from the sample covariance matrices which are also the target matrices for the ajd comparative simulation results demonstrate the advantages in both speed and accuracy as well as compliance with the theoretically predicted asymptotic optimality of the resulting bss algorithms based on the weighted ajd both on large scale problems with matrices of the size 100times100

nonorthogonal joint diagonalization zero diagonalization for source separation based on time frequency distributions this paper deals with the blind separation of instantaneous mixtures of source signals using time frequency distributions tfds we propose iterative algorithms to perform the nonorthogonal zero diagonalization and or joint diagonalization of given sets of matrices as an application we show that the source separation can be realized by applying one of these algorithms to a set of spatial quadratic tfd matrices corresponding only to the so called cross source terms and or to the so called autosource terms the determination of the above matrices to be jointly decomposed requires first an automatic selection procedure of useful time frequency points regarding this last point we also propose a new selection procedure and a modification of an existing one and provide a comparison with other existing ones the nonorthogonal joint diagonalization and or zero diagonalization algorithm s main advantage is to not require in the blind source separation context a prewhitening stage which allows them to work even with a class of correlated signals and provides generally improved separation performance finally an analytical example and computer simulations are provided in order to illustrate the effectiveness of the proposed approach and to compare it with classical ones

a new algorithm for complex non orthogonal joint diagonalization based on shear and givens rotations this paper introduces a new algorithm to approximate non orthogonal joint diagonalization nojd of a set of complex matrices this algorithm is based on the frobenius norm formulation of the joint diagonalization jd problem and takes advantage from combining givens and shear rotations to attempt the approximate jd it represents a non trivial generalization of the jdi joint diagonalization algorithm souloumiac 2009 to the complex case the jdi is first slightly modified then generalized to the cjdi i e complex jdi using complex to real matrix transformation several methods already exist in the literature consequently we provide herein a brief overview of existing nojd algorithms then we provide an extensive comparative study to illustrate the effectiveness and stability of the cjdi with respect to various system parameters and application contexts

a blind source separation technique using second order statistics separation of sources consists of recovering a set of signals of which only instantaneous linear mixtures are observed in many situations no a priori information on the mixing matrix is available the linear mixture should be blindly processed this typically occurs in narrowband array processing applications when the array manifold is unknown or distorted this paper introduces a new source separation technique exploiting the time coherence of the source signals in contrast with other previously reported techniques the proposed approach relies only on stationary second order statistics that are based on a joint diagonalization of a set of covariance matrices asymptotic performance analysis of this method is carried out some numerical simulations are provided to illustrate the effectiveness of the proposed method

critical behaviour of the rouse model for gelling polymers it is shown that the traditionally accepted rouse values for the critical exponents at the gelation transition do not arise from the rouse model for gelling polymers the true critical behaviour of the rouse model for gelling polymers is obtained from spectral properties of the connectivity matrix of the fractal clusters that are formed by the molecules the required spectral properties are related to the return probability of a blind ant random walk on the critical percolating cluster the resulting scaling relations express the critical exponents of the shear stress relaxation function and hence those of the shear viscosity and of the first normal stress coefficient in terms of the spectral dimension ds of the critical percolating cluster and the exponents and of the cluster size distribution

quadratic optimization for simultaneous matrix diagonalization simultaneous diagonalization of a set of matrices is a technique that has numerous applications in statistical signal processing and multivariate statistics although objective functions in a least squares sense can be easily formulated their minimization is not trivial because constraints and fourth order terms are usually involved most known optimization algorithms are therefore subject to certain restrictions on the class of problems orthogonal transformations sets of symmetric hermitian or positive definite matrices to name a few in this paper we present a new algorithm called qdiag that splits the overall optimization problem into a sequence of simpler second order subproblems there are no restrictions imposed on the transformation matrix which may be nonorthogonal indefinite or even rectangular and there are no restrictions regarding the symmetry and definiteness of the matrices to be diagonalized except for one of them we apply the new method to second order blind source separation and show that the algorithm converges fast and reliably it allows for an implementation with a complexity independent of the number of matrices and therefore is particularly suitable for problems dealing with large sets of matrices

a fast algorithm for joint diagonalization with non orthogonal transformations and its application to blind source separation a new efficient algorithm is presented for joint diagonalization of several matrices the algorithm is based on the frobenius norm formulation of the joint diagonalization problem and addresses diagonalization with a general non orthogonal transformation the iterative scheme of the algorithm is based on a multiplicative update which ensures the invertibility of the diagonalizer the algorithm s efficiency stems from the special approximation of the cost function resulting in a sparse block diagonal hessian to be used in the computation of the quasi newton update step extensive numerical simulations illustrate the performance of the algorithm and provide a comparison to other leading diagonalization methods the results of such comparison demonstrate that the proposed algorithm is a viable alternative to existing state of the art joint diagonalization algorithms the practical use of our algorithm is shown for blind source separation problems

a comparative study of approximate joint diagonalization algorithms for blind source separation in presence of additive noise a comparative study of approximate joint diagonalization algorithms of a set of matrices is presented using a weighted least squares criterion without the orthogonality constraint an algorithm is compared with an analogous one for blind source separation bss the criterion of the present algorithm is on the separating matrix while the other is on the mixing matrix the convergence of the algorithm is proved under some mild assumptions the performances of the two algorithms are compared with usual standard algorithms using bss simulations results we show that the improvement in estimating the separating matrix resulting from the relaxation of the orthogonality restriction can be achieved in presence of additive noise when the length of observed sequences is sufficiently large and when the mixing matrix is not close to an orthogonal matrix

non orthogonal joint diagonalization in the least squares sense with application in blind source separation approximate joint diagonalization of a set of matrices is an essential tool in many blind source separation bss algorithms a common measure of the attained diagonalization of the set is the weighted least squares wls criterion however most well known algorithms are restricted to finding an orthogonal diagonalizing matrix relying on a whitening phase for the nonorthogonal factor often such an approach implies unbalanced weighting which can result in degraded performance we propose an iterative alternating directions algorithm for minimizing the wls criterion with respect to a general not necessarily orthogonal diagonalizing matrix under some mild assumptions we prove weak convergence in the sense that the norm of parameters update is guaranteed to fall below any arbitrarily small threshold within a finite number of iterations we distinguish between hermitian and symmetrical problems using bss simulations results we demonstrate the improvement in estimating the mixing matrix resulting from the relaxation of the orthogonality restriction