a system of matrix equations and a linear matrix equation over arbitrary regular rings with identity abstract in this paper we consider the classical system of matrix equations a 1 xb 1 c 1 a 2 xb 2 c 2 over r an arbitrary regular ring with identity necessary and sufficient conditions for the existence and the expression of the general solution to the system are derived as an application the linear matrix equation axb cyd e over r is considered

modules and matrices abstract noncommutative ring theory was described in terms of matrices in its earliest days we give some examples showing how matrix theory can now in turn profit from the theory of rings and modules

the generalized inverse of a sum with radical element applications abstract an expression for the generalized inverse of a sum with radical element in a ring with unity is generalized to the case of a sum in an additive category c of a morphism with a reflexive von neumann regular inverse 1 2 and a morphism of c which is such that 1x 1 2 is invertible also if c is an additive category with an involution a morphism with moore penrose inverse and such that 1 x 1 x 1 x 1 x 1 1 x 1 x 1 and 1 y 1 y 1 y 1 1 y 1 y 1 are invertible then 1 y 1 x 1 1 x has a moore penrose inverse given by 1 x 1 1 x 1 y 1 relations with results of wynn roth and nashed are discussed

eigenvalue generalized eigenvector assignment by output feedback this note generalizes the previous results of the closed loop eigenstructure assignment via output feedback in linear multivariable systems necessary and sufficient conditions for the closed loop eigenstructure assignment by output feedback are presented some known results on entire eigenstructure assignment are deduced from this result

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

consistency for bi skew symmetric solutions to systems of generalized sylvester equations over a finite central algebra let be a finite dimensional central algebra with an involutorial antiautomorphism and char 2 n n be the set of all n n matrices over a aij n n is called bisymmetric if aij an i 1 n j 1 aji and biskewsymmetric if aij an i 1 n j 1 aji the following systems of generalized sylvester equations over a1x yb1 c1 i asx ybs cs a1xb1 c1xd1 e1 ii asxbs csxds es are considered necessary and sufficient conditions are given for the existence of constant solutions with bi skew symmetric constrains to i and ii as a particular case auxiliary results dealing with the system of sylvester equations are also presented

an inequality between variable wiener index and variable szeged index abstract a well known inequality between the szeged and wiener indices says that sz g e i j e g n e i n e j u v d u v w g for every graph g in the past variable variations of the standard topological indices were defined following this line we study a natural generalisation of the above inequality namely e i j e g n e i n e j u v d u v we show that for all trees the inequality is true if 1 and the opposite inequality holds if 0 n 3 edges but the opposite one does not for general graphs we solve also the case

similarity of matrices over local rings abstract wasow investigated the problem of when for a pair of matrices of analytic functions pointwise similarity implies analytic similarity by using the topology of spec r this is studied for an arbitrary commutative ring results of wasow ohm schneider and friedland are generalized in this setting a canonical form is determined for matrices satisfying the wasow condition

solvability conditions and general solution for mixed sylvester equations in this paper we give some necessary and sufficient solvability conditions for the mixed sylvester matrix equations and parameterize general solution when it is solvable moreover we investigate the maximal and minimal ranks of the general solution and maximal and minimal ranks and inertias of hermitian part of solution respectively

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

roth s theorems for sets of matrices abstract it is shown that roth s theorems on the equivalence and similarity of block diagonal matrices hold for finite sets of matrices over a commutative ring

induced corepresentations of locally compact quantum groups abstract we introduce the construction of induced corepresentations in the setting of locally compact quantum groups and prove that the resulting induced corepresentations are unitary under some mild integrability condition we also establish a quantum analogue of the classical bijective correspondence between quasi invariant measures and certain measures on the larger locally compact group

a tree of generalizations of the ordinary singular value decomposition abstract it is shown how to generalize the ordinary singular value decomposition of a matrix into a combined factorization of any number of matrices we propose to call these factorizations generalized singular value decompositions for two matrices this reduces to the product and quotient singular value decompositions one of the factorizations for three matrices is the restricted singular value decomposition these generalizations form a tree of factorizations where at level k for k matrices there are 2 k factorizations not all of which are independent the different levels are related to each other in a recursive fashion any generalized singular value decomposition for k matrices can be constructed from a decomposition for k 1 matrices this results in an inductive proof which uses only the ordinary singular value decomposition several examples are analysed in detail

bounds for the sum balaban index and revised szeged index of regular graphs mathematical properties of many topological indices are investigated knor et al gave an upper bound for the balaban index of r regular graphs on n vertices and a better upper bound for fullerene graphs they also suggested exploring similar bounds for other topological indices in this paper we consider the sum balaban index and the revised szeged index and give upper and lower bounds for these three indices of r regular graphs and also the cubic graphs and fullerene graphs respectively

roth s removal rule revisited abstract the theory of companion matrices is used to give explicit representations for the matrices needed in roth s removal rule these are then used to give simple proofs for the cyclic decomposition theorem as well as for roth s similarity theorem for matrices over a field

the extension of roth s theorem for matrix equations over a ring abstract this paper extends roth s similarity theorem as follows let r be a ring with identity b ki 0 bi i rm q if either r is a division ring and a rn n is algebraic or r is finitely generated as a module over its center then the matrix equation ki 0 aixbi c over r has a solution if and only if

hopf c algebras in this paper we define and study hopf c algebras roughly speaking a hopf c algebra is a c algebra a with a comultiplication a m a a such that the maps a b 7 a 1 b and a b 7 a 1 b have their range in a a and are injective after being extended to a larger natural domain the haagerup tensor product a h a in a purely algebraic setting these conditions onare closely related with the existence of a counit and antipode in this topological context things turn out to be much more subtle but nevertheless one can show the existence of a suitable counit and antipode under these conditions the basic example is the c algebra c0 g of continuous complex functions tending to zero at infinity on a locally compact group where the comultiplication is obtained by dualizing the group multiplication but also the reduced group c algebra c r g of a locally compact group with the well known comultiplication falls in this category in fact all locally compact quantum groups in the sense of 13 and 14 such as the compact and discrete ones as well as most of the known examples are included

complete solution to a problem on the maximal energy of unicyclic bipartite graphs abstract the energy of a simple graph g denoted by e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix denote by c n the cycle and p n 6 the unicyclic graph obtained by connecting a vertex of c 6 with a leaf of p n 6 caporossi et al conjectured that the unicyclic graph with maximal energy is p n 6 for n 8 12 14 and n 16 in hou et al 2002 y hou i gutman c woo unicyclic graphs with maximal energy linear algebra appl 356 2002 27 36 the authors proved that e p n 6 is maximal within the class of the unicyclic bipartite n vertex graphs differing from c n and they also claimed that the energies of c n and p n 6 is quasi order incomparable and left this as an open problem in this paper by utilizing the coulson integral formula and some knowledge of real analysis especially by employing certain combinatorial techniques we show that the energy of p n 6 is greater than that of c n for n 8 12 14 and n 16 which completely solves this open problem and partially solves the above conjecture

on decomposing any matrix as a linear combination of three idempotents abstract in a recent article we gave a full characterization of matrices that can be decomposed as linear combinations of two idempotents with prescribed coefficients in this one we use those results to improve on a recent theorem of rabanovich we establish that every square matrix is a linear combination of three idempotents for an arbitrary coefficient field rather than just one of characteristic 0

the inertia of weighted unicyclic graphs let gw be a weighted graph the inertia of gw is the triple in gw i gw i gw i0 gw where i gw i gw i0 gw are the numbers of the positive negative and zero eigenvalues of the adjacency matrix a gw of gw including their multiplicities respectively i gw i gw are called the positive negative indices of inertia of gw respectively in this paper we present a lower bound for the positive negative indices of weighted unicyclic graphs of order n with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound moreover we characterize the weighted unicyclic graphs of order n with two positive two negative and at least n 6 zero eigenvalues respectively

the matrix equation x axb c and an analogue of roth s theorem abstract we give a necessary and sufficient condition for the consistency of the matrix equation x axb c

a new approach to induction and imprimitivity results given a closed quantum subgroup of a locally compact quantum group we study induction of unitary corepresentations of the quantum subgroup to the ambient quantum group more generally we study induction given a coaction of the quantum subgroup on a c algebra we prove imprimitivity theorems that unify the existing theorems for actions and coactions of groups this means that we deflne quantum homogeneous spaces as c algebras and that we prove morita equivalence of crossed products and homogeneous spaces we essentially use von neumann algebraic techniques to prove these morita equivalences between c algebras the theory of induced representations of locally compact l c groups was introduced by mackey 21 who discovered the imprimitivity theorem characterizing induced representations through the presence of a covariant representation of a homogeneous space riefiel 26 provided a modern approach using the language of hilbert c modules after the work of riefiel several induction procedures and imprimitivity results have been obtained both for actions and for coactions of groups we shall brie y review them below the purpose of this paper is to develop such an induction and imprimitivity machinery in the setting of locally compact quantum groups 16 17 in this way we provide a unifled approach to several results on actions and coactions of groups on c algebras at the same time our proofs in the setting of l c quantum groups are simpler than the classical proofs dealing with coactions of groups another motivation comes from quantum group theory meyer and nest have undertaken a reformulation of the baum connes conjecture 24 in which induction and restriction play a crucial role the development of induction and imprimitivity in the current paper should play an equally important role in the formulation of a baum connes conjecture for quantum groups

matrix equivalence and isomorphism of modules abstract it is well known that if a and b are n m matrices over a ring r then coker a coker b does not imply a and b are equivalent an elementary proof is given that the implication does hold if 1 is in the stable range of r furthermore for certain r including commutative rings if a is block diagonal and b is block upper triangular with the same diagonal blocks as a then coker a coker b implies a and b are equivalent under a special equivalence this extends results of roth and gustafson as a corollary a theorem on decomposition of modules is obtained

reiter s properties p1 and p2 for locally compact quantum groups abstract a locally compact group g is amenable if and only if it has reiter s property p p for p 1 or equivalently all p 1 i e there is a net m of non negative norm one functions in l p g such that lim sup x k l x 1 m m p 0 for each compact subset k g l x 1 m stands for the left translate of m by x 1 we extend the definitions of properties p 1 and p 2 from locally compact groups to locally compact quantum groups in the sense of j kustermans and s vaes we show that a locally compact quantum group has p 1 if and only if it is amenable and that it has p 2 if and only if its dual quantum group is co amenable as a consequence p 2 implies p 1

a common solution to a pair of linear matrix equations over a principal ideal domain abstract a necessary and sufficient condition for the existence of a common solution to a pair of linear matrix equations over a principal ideal domain is obtained the equations are of the type a i b i xc i for i 1 2 the solvability condition is that the equations each have a solution and a bilateral linear matrix equation made up of the matrices a i b i and c i has a solution

a lower bound of revised szeged index of bicyclic graphs abstract the revised szeged index of a graph is defined as s z g e u v e n u e n 0 e 2 n v e n 0 e 2 where nu e and nv e are respectively the number of vertices of g lying closer to vertex u than to vertex v and the number of vertices of g lying closer to vertex v than to vertex u and n0 e is the number of vertices equidistant to u and v in the paper we identify the lower bound of revised szeged index among all bicyclic graphs and also characterize the extremal graphs that attain the lower bound

simultaneous solutions of matrix equations and simultaneous equivalence of matrices given matrices ai bi and ci i i of corresponding dimensions over a field f we prove that i if aiciobi are simultaneously similar to aioobi then there exists a simultaneous solution x to the matrix sylvester equations aix xbi ci and ii if aiciobi are simultaneously equivalent to aioobi then there exist simultaneous solutions x y to the matrix equations aix ybi ci we also show that analogous results hold for mixed pairs of matrix sylvester equations a1x1 yb1 c1 a2x2 yb2 c2 and for generalized stein equations x ayb c

leapfrog fullerenes and wiener index fullerene graphs are cubic 3 connected planar graphs with only pentagonal and hexagonal faces a fullerene is called a leapfrog fullerene le f if it can be constructed by a leapfrog transformation from other fullerene graph f here we determine the relation between the wiener index of le f and the wiener index of the original graph f we obtain lower and upper bounds of the wiener index of lei f in terms of the wiener index of the original graph as a consequence starting with any fullerene f and iterating the leapfrog transformation we obtain fullerenes lei f with wiener index of order o n2 64 and n2 36 where n is the number of vertices of lei f these results disprove hua etal 2014 conjecture that the wiener index of fullerene graphs on n vertices is of order n3

some remarks on wiener index of oriented graphs in this paper we study the wiener index i e the total distance or the transmission number of not necessarily strongly connected digraphs in order to do so if there is no directed path from u to v we follow the convention d u v 0 which was independently introduced in several studies of directed networks under this assumption we naturally generalize the wiener theorem as well as a relation between the wiener index and betweenness centrality to directed graphs we formulate and study conjectures about orientations of undirected graphs which achieve the extremal values of wiener index

two sided coupled generalized sylvester matrix equations solving using a simultaneous decomposition for fifteen matrices abstract in this paper we investigate and analyze in detail the structure and properties of a simultaneous decomposition for fifteen matrices a i c p i t i b i c s i q i c i c p i t i 1 d i c s i 1 q i and e i c p i q i i 1 2 3 we show that from this simultaneous decomposition we can derive some necessary and sufficient conditions for the existence of a solution to the system of two sided coupled generalized sylvester matrix equations with four unknowns a i x i b i c i x i 1 d i e i i 1 2 3 apart from proving an expression for the general solutions to this system we derive the range of ranks of these solutions using the ranks of the given matrices a i b i c i d i and e i we provide some numerical examples to illustrate our results moreover we present a similar approach to consider the simultaneous decomposition for 5k matrices and the system of k two sided coupled generalized sylvester matrix equations with k 1 unknowns a i x i b i c i x i 1 d i e i i 1 k k 4 the main results are also valid over the real number field and the real quaternion algebra

roth s solvability criteria for the matrix equations ax xˆb c and x axˆb c over the skew field of quaternions with an involutive automorphism q qˆ abstract the matrix equation a x x b c has a solution if and only if the matrices a c 0 b and a 0 0 b are similar this criterion was proved over a field by w e roth 1952 and over the skew field of quaternions by huang liping 1996 h k wimmer 1988 proved that the matrix equation x a x b c over a field has a solution if and only if the matrices a c 0 i and i 0 0 b are simultaneously equivalent to a 0 0 i and i 0 0 b we extend these criteria to the matrix equations a x x b c and x a x b c over the skew field of quaternions with a fixed involutive automorphism q q

generalized schur methods with condition estimators for solving the generalized sylvester equation stable algorithms are presented for solving the generalized sylvester equation they are based on orthogonal equivalence transformations of the original problem perturbation theory and rounding error analysis are included condition estimators dif sup 1 estimators are developed which when substituted into derived error bounds give accuracy estimates of a computed solution results from numerical experiments on well conditioned and ill conditioned problems are reported

restarted global fom and gmres algorithms for the stein like matrix equation x m x c abstract in this paper we propose the restarted global full orthogonalization method gl fom and global generalized minimum residual gl gmres method to solve the stein like matrix equation x m x c with m x a x b m x a x b m x a x b or m x a x h b respectively where x is an unknown matrix to be solved first by using a real inner product in complex matrix spaces a generalized global arnoldi process is given then we demonstrate how to employ the restarted gl fom and gl gmres algorithms for solving the stein like matrix equation the proposed algorithms take advantage of the shifted structure of the matrix equation and are implemented by the original coefficient matrices finally some numerical examples are given to illustrate the effectiveness with comparison to some existing methods

graphs preserving wiener index upon vertex removal abstract the wiener index w g of a connected graph g is defined as the sum of distances between all pairs of vertices in g in 1991 soltes posed the problem of finding all graphs g such that the equality w g w g v holds for all their vertices v up to now the only known graph with this property is the cycle c11 our main object of study is a relaxed version of this problem find graphs for which wiener index does not change when a particular vertex v is removed in an earlier paper we have shown that there are infinitely many graphs with the vertex v of degree 2 satisfying this property in this paper we focus on removing a higher degree vertex and we show that for any k 3 there are infinitely many graphs with a vertex v of degree k satisfying w g w g v in addition we solve an analogous problem if the degree of v is n 1 or n 2 furthermore we prove that dense graphs cannot be a solutions of soltes s problem we conclude that the relaxed version of soltes s problem is rich with a solutions and we hope that this can provide an insight into the original problem of soltes

digraphs with large maximum wiener index recently the concept of wiener index was extended to digraphs which are not necessarily strongly connected and it was shown that some fundamental results extend naturally within this concept this extension could be applicable in the topics of directed large networks particularly because with this measure one assigns finite values to the average distance and betweenness centrality of the nodes in a directed network it is not hard to show that among digraphs on n vertices the directed cycle c n achieves the maximum wiener index next we investigate digraphs with the second maximum wiener index one can consider this problem in the realm of all digraphs or restricted to those obtained by directing undirected graphs so called antisymmetric digraphs in both situations we obtain that such digraphs are constructed from c n n 6 by adding a single arc we conclude the paper with consideration for possible further works

a qr type reduction for computing the svd of a general matrix product quotient in this paper a qr type reduction technique is developed for the computation of the svd of a general matrix product quotient a a 1 s 1 a 2 s 2 cdots a m s m with a i in bf r n times n hbox and s i 1 or s i 1 first the matrix a is reduced by at most m qr factorizations to the form q 1 11 q 1 21 1 where q 1 11 q 1 21 in bf r n times n and q 1 11 tq 1 11 q 1 21 tq 1 21 i then the svd of a is obtained by computing the csd cosine sine decomposition of q 1 11 and q 1 21 using the matlab command gsvd the performance of the proposed method is verified by some numerical examples

the unitary implementation of a locally compact quantum group action abstract in this paper we study actions of locally compact quantum groups on von neumann algebras and prove that every action has a canonical unitary implementation paralleling haagerup s classical result on the unitary implementation of a locally compact group action this result is an important tool in the study of quantum groups in action we will use it in this paper to study subfactors and inclusions of von neumann algebras when is an action of the locally compact quantum group m on the von neumann algebra n we can give necessary and sufficient conditions under which the inclusion n n m n is a basic construction here n denotes the fixed point algebra and m n is the crossed product when is an outer and integrable action on a factor n we prove that the inclusion n n is irreducible of depth 2 and regular giving a converse to the results of m enock and r nest 1996 j funct anal 137 466 543 1998 j funct anal 154 67 109 finally we prove the equivalence of minimal and outer actions and we generalize the main theorem of yamanouchi 1999 math scand 84 297 319 every integrable outer action with infinite fixed point algebra is a dual action

roth s theorems and decomposition of modules it is shown that there is a connection between roth s theorems on similarity and equivalence of block triangular matrices and decomposition of modules the module property is that if m n mn then n is a summand of m this holds for any commutative ring if m is finitely presented new proofs of roth s theorems are given for commutative rings some results are established in the noncommutative case

on the minimum value of sum balaban index we consider extremal values of sum balaban index among graphs on n vertices we determine that the upper bound for the minimum value of the sum balaban index is at most 4 47934 when n goes to infinity for small values of n we determine the extremal graphs and we observe that they are similar to dumbbell graphs in most cases having one extra edge added to the corresponding extreme for the usual balaban index we show that in the class of balanced dumbbell graphs those with clique sizes 2log 1 2 4n o n have asymptotically the smallest value of sum balaban index we pose several conjectures and problems regarding this topic

on the connective eccentricity index of trees and unicyclic graphs with given diameter abstract the connective eccentricity index cei of a graph g is defined as ce g v v g d v e v where e v d v denote the eccentricity and the degree of v respectively in this paper we investigate the connective eccentricity index of trees and unicyclic graphs on n vertices with diameter d we also characterize the extremal graphs

closed quantum subgroups of locally compact quantum groups abstract we investigate the fundamental concept of a closed quantum subgroup of a locally compact quantum group two definitions one due to s vaes and one due to s l woronowicz are analyzed and relations between them discussed among many reformulations we prove that the former definition can be phrased in terms of quasi equivalence of representations of quantum groups while the latter can be related to an old definition of podle from the theory of compact quantum groups the cases of classical groups duals of classical groups compact and discrete quantum groups are singled out and equivalence of the two definitions is proved in the relevant context a deep relationship with the quantum group generalization of the herz restriction theorem from classical harmonic analysis is also established in particular in the course of our analysis we give a new proof of the herz restriction theorem

on extremal cacti with respect to the szeged index the szeged index of a graph g is defined as sz g e uvenu e nv e where nu e and nv e are respectively the number of vertices of g lying closer to vertex u than to vertex v and the number of vertices of g lying closer to vertex v than to vertex u a cactus is a graph in which any two cycles have at most one common vertex let c n k denote the class of all cacti with order n and k cycles and cnt denote the class of all cacti with order n and t pendant vertices in this paper a lower bound of the szeged index for cacti of order n with k cycles is determined and all the graphs that achieve the lower bound are identified as well the unique graph in cnt with minimum szeged index is characterized

on similarities of class cp and applications to matrix differential equations abstract this paper deals with similarities of class cp in particular reduction of class cp to jordan form and unitary triangularization of class cp of a matrix function the main result is applied to establish a method to lower the order of nonlinear matrix differential equations belonging to a broad class of equations in particular this method permits one to transform nonlinear equations of the first order into an algebraic equation joint to a linear differential equation of the first order the method is applied to the matrix differential equation xx x x x and a particular solution is easily obtained

controllability observability and the solution of ax xb c a closed form finite series representation of the unique solution x of the matrix equation ax xb c is developed using this representation the image kernel and rank of x are related to the controllable and unobservable subspaces of the a c and c b pairs respectively bounds on the rank of x are obtainedin terms of the dimensions of these subspaces in the case that c has unitary rank an exact calculation of rank x is made the generic rank of x with a b fixed and c generic is evaluated

systems of coupled generalized sylvester matrix equations this paper studies some systems of coupled generalized sylvester matrix equations we present some necessary and sufficient conditions for the solvability to these systems we give the expressions of the general solutions to the systems when their solvability conditions are satisfied

the szeged index and the wiener index of partial cubes with applications to chemical graphs in this paper we study the szeged index of partial cubes and hence generalize the result proved by chepoi and klavar who calculated this index for benzenoid systems it is proved that the problem of calculating the szeged index of a partial cube can be reduced to the problem of calculating the szeged indices of weighted quotient graphs with respect to a partition coarser than partition similar result for the wiener index was recently proved by klavar and nadjafi arani furthermore we show that such quotient graphs of partial cubes are again partial cubes since the results can be used to efficiently calculate the wiener index and the szeged index for specific families of chemical graphs we consider c4c8 systems and show that the two indices of these graphs can be computed in linear time

diagonalization of matrices over regular rings abstract square matrices are shown to be diagonalizable over all known classes of von neumann regular rings this diagonalizability is equivalent to a cancellation property for finitely generated projective modules which conceivably holds over all regular rings these results are proved in greater generality namely for matrices and modules over exchange rings where attention is restricted to regular matrices